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abstract: 'We investigate the sparse linear contextual bandit problem where the parameter $\theta$ is sparse. To relieve the sampling inefficiency, we utilize the “perturbed adversary" where the context is generated adversarilly but with small random non-adaptive perturbations. We prove that the simple online Lasso supports sparse linear contextual bandit with regret bound $\mathcal{O}(\sqrt{kT\log d})$ even when $d \gg T$ where $k$ and $d$ are the number of effective and ambient dimension, respectively. Compared to the recent work from [@sivakumar2020structured], our analysis does not rely on the precondition processing, adaptive perturbation (the adaptive perturbation violates the i.i.d perturbation setting) or truncation on the error set. Moreover, the special structures in our results explicitly characterize how the perturbation affects exploration length, guide the design of perturbation together with the fundamental performance limit of perturbation method. Numerical experiments are provided to complement the theoretical analysis.'
author:
- |
Zhiyuan Liu zhiyuan.liu@colorado.edu\
Department of Computer Science, University of Colorado, Boulder Huazheng Wang hw7ww@virginia.edu\
Department of Computer Science, University of Virginia Bo Waggoner bo.waggnor@colorado.edu\
Department of Computer Science, University of Colorado, Boulder Youjian (Eugene) Liu youjian.liu@colorado.edu\
Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder Lijun Chen lijun.chen@colorado.edu\
Department of Computer Science, University of Colorado, Boulder
bibliography:
- 'reference.bib'
title: A Smoothed Analysis of Online Lasso for the Sparse Linear Contextual Bandit Problem
---
Introduction
============
Contextual bandit algorithms have become a referenced solution for sequential decision-making problems such as online recommendations [@li2010contextual], clinical trials [@durand2018contextual], dialogue systems [@upadhyay2019bandit] and anomaly detection [@ding2019interactive]. It adaptively learns the personalized mapping between the observed contextual features and unknown parameters such as user preferences, and addresses the trade-off between exploration and exploitation [@auer2002using; @li2010contextual; @abbasi2011improved; @agrawal2013thompson; @abeille2017linear].
We consider the sparse linear contextual bandit problem where the context is high-dimensional with sparse unknown parameter $\theta$ [@abbasi2012online; @hastie2015statistical; @dash1997feature], i.e., most entries in $\theta$ are zero and thus only a few dimensions of the context feature are relevant to the reward. Due to insufficient data samples, the learning algorithm has to be sampling efficiency to support the sequential decision-making. However, the data from bandit model usually does not satisfy the requirements for sparse recovery such as Null Space condition [@cohen2009compressed], Restricted isometry property (RIP) [@donoho2006compressed], Restricted eigenvalue (RE) condition [@bickel2009simultaneous], Compatibility condition [@van2009conditions] and so on. To achieve the desired performance, current works has to consider the restricted problem settings, e.g., the unit-ball, hypercube or i.i.d. arm set [@carpentier2012bandit; @lattimore2015linear; @kim2019doubly; @bastani2020online], the parameter with Gaussian prior[@gilton2017sparse]. One exception is the Online-to-Confidence-Set Conversions [@abbasi2012online] which considers the general setting but suffers from computation inefficiency.
In this paper, we tackle the sparse linear bandit problem using *smoothed analysis* technique [@spielman2004smoothed; @kannan2018smoothed], which enjoys efficient implementation and mild assumptions. Specifically, we consider the perturbed adversary setting where the context is generated adversarially but perturbed by small random noise. This setting interpolates between an i.i.d. distributional assumption on the input, and the worst-case of fully adversarial contexts. Our results show that with a high probability, the perturbed adversary inherently guarantees the (linearly) strong convex condition for the low dimensional case and the restricted eigenvalue (RE) condition for the high dimensional case, which is a key property required by the standard Lasso regression. We prove that the simple online Lasso supports sparse linear contextual bandits with regret bound $\mathcal{O}(\sqrt{kT\log d})$. We also provide numerical experiments to complement the theoretical analysis.
We also notice the recent work from [@sivakumar2020structured] using smoothed analysis for structured linear contextual bandits. Compared to their work, our proposed method has the following advantages: (1) Our analysis only relies on the simple online Lasso instead of precondition processing and truncation on the error set. Although preconditioning transfers the non-zero singular value to 1, this could amplify the noise, and the preconditioned noises are no longer i.i.d., which makes concentration analysis difficult and the estimation unstable [@jia2015preconditioning]. We also observe this effect in the numeric experiments. (2) Their proof relies on the assumption that perturbations that need to be adaptively generated based on the observed history of the chosen contexts. Instead, our analysis is based on the milder assumption that the perturbation is i.i.d. and non-adaptive. (3) Their regret does not describe the full picture of the effect of variance of the perturbation. Our analysis explicitly show how the perturbation affects the exploration length, guide the design of perturbation together with the fundamental performance limit of perturbation method.
Model and Methodology
=====================
In the bandit problem, at each round $t$, the learner pulls an arm $a_t$ among $m$ arms (we denote the arm sets by $[m]$[^1], that is, $a_t \in [m]$) and receives the corresponding noisy reward $r_{a_t}^t$. The performance of the learner is evaluated by the regret $\mathcal{R}$ which quantifies the total loss because of not choosing the best arm $a_t^*$ during $T$ rounds: $$\begin{aligned}
\mathcal{R}(T) = \sum_{t=1}^{T}(r_{a^*_t}^t - r_{a_t}^t).
\end{aligned}$$ In this paper, we focus on the sparse linear contextual bandit problem. Specially, each arm $i$ at round $t$ is associated with a feature (context) vector $\mu_i^t \in \mathbb{R}^d$. The reward of that arm is assumed to be generated by the noisy linear model which is the inner product of arm feature and an unknown $S$-sparse parameter $\theta^{*}$ where $S$ denotes the set of effective (non-zero) entries and $|S| = k$. That is, $$\begin{aligned}
r_{i}^t = \langle \mu_{i}^t, \theta^{*} \rangle + \eta^t, |\theta^{*}|_{0} = k,
\end{aligned}$$ where $\eta^t$ follows Gaussian distribution $\mathcal{N}(0,\sigma^2)$. To handle the non-convex $L_0$ norm, Lasso regression is the natural way to learn the sparse $\theta^{*}$ with the relaxation from $L_0$ to $L_1$ norm. To achieve the desired performance, the algorithm has to rely on well designed contexts which guarantee sampling efficiency requirements such as Null Space condition [@cohen2009compressed], Restricted isometry property (RIP) [@donoho2006compressed], Restricted eigenvalue (RE) condition [@bickel2009simultaneous], Compatibility condition [@van2009conditions] and so on. However, the data from bandit problems usually does not satisfy these conditions since the contexts could be generated adversarilly. Up to now, deciding on the proper assumptions for sparsity bandit problems is still a challenge [@lattimore2018bandit].
Inspired by the smoothed analysis for greedy algorithm of linear bandit problem [@kannan2018smoothed], we consider the *Perturbed Adversary* defined below for the sparse linear contextual bandit problem.
[**Perturbed Adversary** [@kannan2018smoothed].]{} The perturbed adversary acts as the following at round $t$.
- Given the current context $\mu_1^t, \cdots, \mu_m^t$ which could be chosen adversarially, the perturbation $e_1^t, \cdots, e_m^t$ are drawn independently from certain distribution. Also, each $e_i^t$ is independently (non-adaptively) produced of the context.
- The perturbed adversary outputs the contexts $(x_1^t,\cdots,$ $x_m^t) = (\mu_1^t+e_1^t,\cdots,\mu_m^t+e_m^t)$ as the arm features to the learner.
Let $X \in \mathbb{R}^{d \times t}$ be the context matrix where each column contains one context vector and $Y$ the column vector that contains the corresponding rewards. Based on perturbed adversary setting, we analyze the online Lasso in Algorithm \[algo:1\] for sparse linear contextual bandit.
Initialize $\theta^{0}$, $X$ and $Y$.\
Generally speaking, our analysis considers two cases using different techniques, one for the low dimensional case when $d < T$ and the other for the high dimensional case when $d \gg T$. For the low dimensional case, the analysis utilizes random matrix theory [@tropp2012user] to prove that with a high probability, the minimum eigenvalue of scaled sample covariance matrix is increasing linearly with round $t$; for the high dimensional case, the RE condition is guaranteed with the help of Gaussian perturbation’s property [@raskutti2010restricted] that the nullspace of context matrix under Gaussian perturbation cannot contain any vectors that are “overly” sparse when $t$ is larger than some threshold. The properties of both cases support $\mathcal{O}(\sqrt{\frac{k\log d}{t}})$ parameter recovery of Lasso regression under noisy environment which leads to $\mathcal{O}(\sqrt{kT\log d})$ regret.
Low Dimensional Case
--------------------
We first consider the low dimensional case when $d < T$. Under the perturbed adversary setting, we then define the property named perturbed diversity which is adopted from [@bastani2017mostly].
[**Perturbed Diversity.**]{} Let $e_i^t \sim D$ on $\mathbb{R}^d$. Given any context vector $\mu_i^t$, we call $x_i^t$ perturbed diversity if for $x_i^t = \mu_i^t + e_i^t$, the minimum eigenvalue of sample covariance matrix under perturbations satisfies $$\begin{aligned}
\lambda_{\min}\left( \mathbb{E}_{e_i^t \sim D}\left[x_i^t(x_i^t)^{\top} \right]\right) \geq \lambda_{0},
\end{aligned}$$ where $\lambda_{0}$ is a positive constant.
Intuitively speaking, perturbed diversity guarantees that each context provides at least certain information about all coordinates of $\theta^*$ from the expectation which is helpful to recover the support of the parameter via regularized method. We can find several distributions $D$ that could make the perturbed diversity happen, e.g., the Gaussian distribution. However, without any restriction, $x_i^t$ could be very large and out of the realistic domain. Instead, the value of each dimension (we denote by $x_i^t(j)$ the $j$-th dimension of $x_i^t$) should lie in a bounded interval, in the meanwhile, the total energy of context vector is bounded by certain constant, i.e., $\|x_i^t\|_2^2 \leq R^2$. This motivates us to consider the perturbed diversity under censored perturbed adversary.
\[lemma:1\] Given the context vector $\mu_i^t \in \mathbb{R}^d$ and $|\mu_i^t(j)| \leq q_j$ for each $j \in [d]$, we define the censored perturbed context $x_i^t$ under $e_i^t \sim \mathcal{N}(\bm{0},\sigma_1^2 I)$ as follows: $$\begin{aligned}
x_i^t(j) = \begin{cases} \mu_i^t(j) + e_i^t(j), ~~~\text{if} ~~|\mu_i^t(j)+e_i^t(j)| \leq q_j,\\
~~~q_j, ~~~~~~~~~~~~~\text{if} ~~~\mu_i^t(j)+e_i^t(j) > q_j, \\
~-q_j, ~~~~~~~~~~~\text{if} ~~~\mu_i^t(j)+e_i^t(j) < -q_j. \\
\end{cases}
\end{aligned}$$ Then $x_i^t$ has the perturbed diversity with $\lambda_0 = g(\frac{2q}{\sigma_1},0)\sigma^2$, where $q = \min_j q_j$ and $g(\cdot,\cdot)$ is a composite function of the probability density function $\phi(\cdot)$ and the cumulative distribution function $\Phi(\cdot)$ of the normal distribution. Please refer to equation for more details.
The proof is provided in the appendix and one can easily extend it to the case where $e_i^t \sim \mathcal{N}(\bm{0},\Sigma)$. Based on Lemma \[lemma:1\], we can derive that with a high probability, $\lambda_{\min}(XX^{\top})$ grows at least with a linear rate $t$.
\[lemma:eig\] With the censored perturbed diversity, when $t > \frac{2R^2}{g\left(\frac{2q}{\sigma_1},0\right)\sigma_1^2}\log(dT)$, the following is satisfied with probability $1-\frac{1}{T}$: $\lambda_{\min} (X X^{\top}) \geq g\left(\frac{2q}{\sigma_1},0\right)(1-\tau)\sigma_1^2t,$ where $\tau = \sqrt{\frac{2R^2}{g\left(\frac{2q}{\sigma_1},0\right)\sigma_1^2t}\log(dT)}$.
As one can see from Lemma \[lemma:eig\], after certain number of (implicit) exploration rounds, i.e., $\frac{2R^2}{g\left(2q/\sigma_1,0\right)\sigma_1^2}\log(dT),$ we will have enough information to support the $\mathcal{O}(\sqrt{\frac{k\log d}{t}})$ parameter recovery by Lasso regression. The regret analysis together with the high dimensional case is deferred to the next section.
High Dimensional Case
---------------------
Now we turn to the high dimensional case when $d \gg T$. During the learning process, the scaled sample covariance matrix $XX^{\top}$ is always rank deficiency which means $\lambda_{\min}(XX^{\top}) = 0$ and Lemma \[lemma:eig\] based on random matrix theory can not be applied here any more. We then consider the restricted eigenvalue (RE) condition instead. Here the “restricted” means that the error $\Delta^t := \theta^t - \theta^*$ incurred by Lasso regression is restricted to a set with special structure. That is, $ \Delta^t \in \mathcal{C}(S;\alpha)$ where $$\begin{aligned}
\mathcal{C}(S;\alpha) := \{ \theta \in \mathbb{R}^d | \|\theta_{S^c}\|_1 \leq \alpha \|\theta_S\|_1
\},
\end{aligned}$$ and $\alpha \geq 1$ is determined by the regularized parameter $\lambda^t$. In the following section, we focus on $\mathcal{C}(S;3)$ which could be achieved by setting $\lambda^t = \Theta(2\sigma R \sqrt{2t\log (2d)})$.
The key is to prove that Null space of $X^{\top}$ has no overlapping with $\mathcal{C}(S;3)$. It has been proved that special cases in which contexts are purely sampled from special distributions such as Gaussian and Bernoulli distributions, satisfy this property [@zhou2009restricted; @raskutti2010restricted; @haupt2010toeplitz]. We make a further step to show that nullspace of context matrix under Gaussian perturbation cannot contain any vectors that are “overly” sparse when $t$ is larger than some threshold.
\[theorem:high\] Consider perturbation $e_i^t \sim \mathcal{N}(\bm{0},\Sigma)$ where $\|\Sigma^{1/2}\Delta\|_2 \geq \gamma \|\Delta\|_2$ for $\Delta \in \mathcal{C}(S;3)$. If $ t > \max(\underbrace{\frac{4c'' q(\Sigma)}{\gamma^2} k \log d}_{\textbf{d}},~\underbrace{\frac{8196aR^2 \lambda_{\max}(\Sigma) \log T}{\gamma^4}}_{\textbf{e}})$\[equ:explore\], then with probability $1 - (\frac{c'}{e^{ct}} +\frac{1}{T^a})$, we have $\Delta^{\top} XX^{\top} \Delta \geq h t\|\Delta\|_2^2$, where $c,c',c''$ are universal constants, $q(\Sigma) = \max_i \Sigma_{ii}$ and $h = ( \frac{\gamma^2}{64} - R\|\Delta\|_2^2\sqrt{\frac{2a\lambda_{\max}(\Sigma) \log T}{t}})$.
Moreover, we can design $\gamma^2 = \lambda_{\min}(\Sigma)$. By Rayleigh quotient, one can obtain $\lambda_{\max}(\Sigma) \geq q(\Sigma) =\max_i \Sigma_{ii} \geq \min_i \Sigma_{ii} \geq \lambda_{\min}(\Sigma) = \gamma^2.$
We then discuss how perturbations will affect the exploration length. First, *the larger perturbation does not indicate the less regret.* Results of [@sivakumar2020structured] show that the regret is $\mathcal{O}(\frac{\log T \sqrt{T}}{\sigma_1})$ where $\sigma_1$ is the perturbation’s variance, and suggests choosing larger $\sigma_1$ leads to smaller regret bounds. However this is not the full picture that shows the effect of the perturbation’s variance. Our results show that increasing the variance of the perturbation has limited effect over the necessary exploration and regret, which reveals theoretical limit of the perturbation method. Specifically, in the term **(d)** of Theorem \[theorem:high\], no matter how large the variance is, the term $\frac{q(\Sigma)}{\gamma^2} \geq 1$. So $4c''k\log d$ is the necessary exploration length and cannot be improved. Second, *Condition Number and the SPR (the signal to perturbation ratio)* are important factors. The condition number $\textbf{Cond}(\Sigma)$ controls both term $\textbf{(d)}$ and $\textbf{(e)}$, e.g., $\frac{q(\Sigma)}{\gamma^2} \leq \frac{\lambda_{\max}(\Sigma)}{\lambda_{\min}(\Sigma)} = \textbf{Cond}(\Sigma)$. This also shows that the optimal perturbation design will choose $\Sigma = \sigma_1 I$. In **(e)** of Theorem \[theorem:high\], $\frac{R^2}{\gamma^2}$ can be regarded as the ratio between the energy of the unperturbed context and the perturbation energy. This ratio shows the trade-off between *exploration* and *fidelity*. That is, a large variance not only reduces the exploration ( meanwhile, the lower bound is guaranteed by **(e)**) but also reduces the fidelity of original context.
Regret Analysis
===============
Based on the properties we have proved for the low and high dimensional cases, we can obtain the following recovery guarantee by the techniques from the standard Lasso regression [@hastie2015statistical].
\[lemma:recovery\] If $t > T_e$ and $\lambda^t = 2\sigma R \sqrt{2t\log \frac{2d}{\delta}}$, the Lasso regression under perturbed adversary has the recovery guarantee $\|\theta^t - \theta^{*}\|_2 \leq \frac{3\sigma R}{C}\sqrt{\frac{2k\log 2d/\delta}{t}}$ with probability $1-\delta$, where $T_e = \frac{2R^2}{g\left(\frac{2q}{\sigma_1},0\right)\sigma_1^2}\log(dT)$, $C =g\left(\frac{2q}{\sigma_1},0\right)(1-\tau)\sigma_1^2$ for the low dimensional case and $T_e = \max(\frac{4c'' q(\Sigma)}{\gamma^2} k \log d ,~ \frac{8196aR^2 \lambda_{\max}(\Sigma) \log T}{\gamma^4})$, $C = \frac{\gamma^2}{64} - R\sqrt{\frac{2a\lambda_{\max}(\Sigma) \log T}{t}}$ for the high dimensional case.\
We then get the final result in Theorem \[theorem:regret\] based on all the analysis above.
\[theorem:regret\] The online Lasso for sparse linear contextual bandit under perturbed adversary admits the following regret with probability $1-\delta$. $$\begin{aligned}
Regret \leq 2R\left( T_e +\frac{6\sigma R}{C}\sqrt{2kT\frac{\log 2d}{\delta}} \right) = \mathcal{O}(\sqrt{kT\log d}).
\end{aligned}$$
Conclusion
==========
This paper utilizes the “perturbed adversary” where the context is generated adversarially but with small random non-adaptive perturbations to tackle sparse linear contextual bandit problem. We prove that the simple online Lasso supports sparse linear contextual bandit with regret bound $\mathcal{O}(\sqrt{kT\log d})$ for both low and high dimensional cases and show how the perturbation affects the exploration length and the trade-off between exploration and fidelity. Future work will focus on extending our analysis to more challenge setting, i.e., defending against adversarial attack for contextual bandit model.
\[app:theorem\]
0.2in
Appendix {#appendix .unnumbered}
========
[^1]: In this paper, we denote by $[n]$ the set $[1,\cdots,n]$ for positive integer $n$.
|
---
abstract: 'Real data are constrained to finite sampling rates, which calls for a suitable mathematical description of the corrections to the finite-time estimations of the dynamic equations. Often in the literature, lower order discrete time approximations of the modeling diffusion processes are considered. On the other hand, there is a lack of simple estimating procedures based on higher order approximations. For standard diffusion models, that include additive and multiplicative noise components, we obtain the exact corrections to the empirical finite-time drift and diffusion coefficients, based on Itô-Taylor expansions. These results allow to reconstruct the real hidden coefficients from the empirical estimates. We also derive higher-order finite-time expressions for the third and fourth conditional moments, that furnish extra theoretical checks for that class of diffusive models. The theoretical predictions are compared with the numerical outcomes of some representative artificial time-series.'
author:
- 'C. Anteneodo'
- 'R. Riera'
title: 'Exact corrections for finite-time drift and diffusion coefficients'
---
Introduction
============
Many fluctuating random phenomena, including turbulent diffusion, polymer dynamics or asset price evolution, can be modeled by an univariate Itô-stochastic differential equation (SDE) of the form bellow, characterizing a diffusive model: $$\label{ILE}
dX_t=D_1(X_t)dt+\sqrt{2D_2(X_t)}dW_t \,,$$ where $W_t$ is a Wiener process, $D_1(X_t)$ is the coefficient of the slowly varying component (called drift coefficient) and $D_2(X_t)$ is the coefficient of the rapid one (called diffusion coefficient).
For sufficiently smooth and bounded drift and diffusion coefficients, the associated probability density function (PDF) $P(x,t)\equiv P(X_t=x,t)$ is governed by the corresponding Fokker-Planck equation [@risken] $$\label{FPE}
\partial_t P(x,t)= -\partial_x [D_1(x)P(x,t)]+ \partial_{xx}[D_2(x)P(x,t)] \,.$$
Here, we are concerned with the empirical access to unknown drift and diffusion coefficients of stochastic processes. For an ideal time series $X_t$ generated by Eq. (\[ILE\]) and sampled with a sufficiently high resolution on a long time period, the original coefficients can be perfectly reconstructed. For stationary processes, the coefficients $D_k(x)$, with $k=1,2$ can be directly estimated from the conditional moments [@risken] as: $$\begin{aligned}
\label{ABs}
D_k(x) &=& \lim_{\tau\to 0}\tilde{D}_k(x,\tau)\,,\end{aligned}$$ where $$\begin{aligned}
\label{ABtilde}
\tilde{D}_k(x,\tau) &=& \frac{1}{\tau\,k!} \langle [X_{t+\tau}-X_t]^k \rangle|_{X_t=x} \,,\end{aligned}$$ with $\langle \cdots\rangle$ denoting statistical average and $|_{X_t=x}$ meaning that at time $t$ the stochastic variable assumes the value $x$.
Conversely, for general Markovian stochastic processes, the time evolution of PDFs is governed by a generalization of Eq. (\[FPE\]), namely $$\label{KME}
\partial_t P(x,t)= \sum_{k\ge 0}(-\partial_x)^k [D_k(x)P(x,t)] \,.$$ with coefficients $D_k(x)$ given by Eqs. (\[ABs\])-(\[ABtilde\]), for any integer $k\ge 1$. For diffusive processes, Eq. (\[KME\]) reduces to Eq. (\[FPE\]). Therefore, processes governed by the Itô-Langevin Eq. (\[ILE\]) must furnish null coefficients $\tilde{D}_k$, for $k\ge 3$. Pawula theorem [@risken] simplifies this task by stating that if $D_4$ is null, all other coefficients with $k\ge 3$ are null as well. The coefficient $D_4$ is then a key coefficient to be investigated, in order to establish the validity of the modeling of data series by Eq. (\[ILE\])-(\[FPE\]).
However, due to the finite sampling rate of real data, numerical estimations of $\tilde{D}_k$ can not always be straightforwardly extrapolated to the limit $\tau\to 0$ in Eq. (\[ABs\]). In such cases, one accesses only the finite-$\tau$ estimation of the coefficients given by Eq. (\[ABtilde\]), which may significantly differ from the true coefficients ${D_k}$. This is specially relevant when $\tau$ is large compared to the characteristic timescales of the process.
Some authors [@friedrich4] have introduced finite sampling rate corrections to the coefficients ${D_1}(x,\tau)$ and ${D_2}(x,\tau)$, by deriving expansions for the conditional moments up to some specified low order of $\tau$, directly from the Fokker-Planck equations. Applications of this approach have already been implemented for those coefficients up to second order [@Julia]. The error in the finite-$\tau$ estimated coefficients $\tilde{D}_k$ can also be derived from the stochastic Itô-Taylor expansion [@book] of the integrated form of Eq. (\[ILE\]). Within this line, the first order expansion of drift and diffusion coefficients was recently presented in Ref. [@sura]. However, low order corrections may be inappropriate when the convergence of the limit in Eq. (\[ABs\]) is slow [@friedrich4; @comment]. Moreover, there is no [*a priori*]{} knowledge of whether the sampling rate is fine enough to justify the use of the lowest order approximation.
In the present work, we investigate those issues for diffusion models defined by Eq. (\[ILE\]) with linear drift coefficient, namely, ${D_1}(x)=-a_1x$, representing an harmonic restoring mechanism, and quadratic state-dependent diffusion coefficient, namely, ${D_2}(x)=b_0+b_2x^2$. This class encompasses some of the most common models of the theoretical literature. In fact, this equation is frequently found in a diversity of processes, from turbulence to finance [@turbulence; @turbfinance]. Moreover, the obtained results are also valid for another class of SDEs with additive-multiplicative noises [@multiplicative1; @multiplicative2], given by $$\label{ILE2}
dX_t=-a_1X_tdt+\sqrt{2b_0}dW_t +\sqrt{2b_2}X_tdW^\prime_t \,,$$ where $W_t,\,W^\prime_t$ are uncorrelated Wiener processes.
For discretely sampled data at intervals $\tau$, we will derive, from the stochastic Itô-Taylor expansion, finite-$\tau$ expressions for the parameters $\{{a}_1,{b}_0,{b}_2\}$, up to infinite order. These exact expressions will allow us to reconstruct the true drift and diffusion coefficients from their empirical finite-time estimates. As a corollary, one can determine up to which value of $\tau$ a given order of truncation is reliable (within a fixed tolerance), or reciprocally, which is the sufficient order for a given $\tau$.
Furthermore, as empirical estimates suffer from finite-$\tau$ effects, one always gets non-null $D_4$. Therefore, the evaluation of the corrections for this coefficient is crucial for a suitable probe of the diffusive modeling. In this work, we also derive finite-$\tau$ expressions for coefficients $D_3$ and $D_4$, which furnish extra theoretical tests of consistency for the diffusive models considered.
Our theoretical findings are corroborated by the outcomes of exemplary artificial time-series generated by Eq. (\[ILE\]).
Exact corrections for drift and diffusion coefficients
=======================================================
Let us consider the Itô formula [@book], for a given function $F$ of the stochastic variable $X_t$ $$\begin{aligned}
\nonumber
dF&=&(\partial_t F+D_1\partial_X F +D_2\partial_{XX} F )dt
+ \sqrt{2D_2}\partial_X F dW \\\label{L0L1}
&\equiv& L^0Fdt+L^1FdW,\end{aligned}$$ and its integrated form $$\label{intFform}
F(X_{t+\tau})=F(X_t) + \int_t^{t+\tau}L^0 F(X_s)ds+\int_t^{t+\tau}L^1 F(X_s)dW_s\,.$$
Let $\tau$ be the sampling interval of state space observations. By applying Itô formula (\[intFform\]) to the functions $D_1(X_s)$ and $\sqrt{2D_2(X_s)}$ in the integral form of Eq. (\[ILE\]): $$\label{intXform}
X_{t+\tau}=X_t +\int_t^{t+\tau} D_1(X_s)ds+\int_t^{t+\tau} \sqrt{2D_2(X_s)}dW_s,$$ one finds $$\begin{aligned}
\nonumber
X_{t+\tau}&=& X_t +\int_t^{t+\tau}\biggl(
D_1(X_t)+\int_t^{s}L^0 D_1(X_s')ds' +
\int_t^{s} L^1 D_1(X_{s'})dW_{s'}\biggr)ds \\
&+& \int_t^{t+\tau}\biggl( \sqrt{2D_2(X_t)}+
\int_t^{s}L^0 \sqrt{2D_2(X_s')}ds' +\int_t^{s} L^1
\sqrt{2D_2(X_{s'})}dW_{s'}\biggr)dW_s\,.\label{expansion}\end{aligned}$$ After iterated applications of Itô formula, one gets an expression for the increment of the stochastic variable in terms of multiple stochastic integrals [@book]: $$\label{MSI}
X_{t+\tau}-X_t=\sum_{\alpha_k} c_{\alpha_k}(D_1,D_2)\,I_{\alpha_k},$$ where $\alpha_k=(j_1,j_2,\ldots,j_k)$, with $j_i=0,1$ for all $i$, $c_{\alpha_k}(D_1,D_2)=L^{j_1}L^{j_2}\ldots L^{j_{k-1}}L^{j_k}$ and $I_{\alpha_k}$ are multiple stochastic integrals of the form $I_{\alpha_k}=\int_t^{t+\tau} \int_t^{t+t_{k}} \int_t^{t+t_{k-1}}
\ldots \int_t^{t+t_2} dt_1^{j_1}\ldots dt_{k-1}^{j_{k-1}} dt_{k}^{j_k}$, with $dt_i^0\equiv dt_i$ and $dt_i^1\equiv dW_i$.
By inserting Eq. (\[MSI\]) into Eq. (\[ABtilde\]) and performing the averaging, for $k=1,2$, we achieve analytical expressions for the finite-$\tau$ drift and diffusion coefficients, up to arbitrary order in powers of $\tau$. The resulting expressions preserve the linear and quadratic $x$-dependence, respectively and can be written as: $$\begin{aligned}
\nonumber
\tilde{D}_1(x,\tau) &=& -\tilde{a}_1(\tau)x \\ \label{D1D2til}
\tilde{D}_2(x,\tau) &=& \tilde{b}_0(\tau)+\tilde{b}_2(\tau)x^2 \,.\end{aligned}$$
Hence, we are led to the theoretical relation between the finite-$\tau$ coefficients $\{\tilde{a}_1,\tilde{b}_0,\tilde{b}_2\}$ and the true ones $\{{a}_1,{b}_0,{b}_2\}$, namely, $$\begin{aligned}
\label{a1n}
\tilde{a}_1(\tau)&=& a_1\sum_{j\ge 0} \frac{[-a_1]^j}{(j+1)!}\tau^j \,,\\ \label{b0n}
\tilde{b}_0(\tau) &=& b_0\sum_{j\ge 0} \frac{[(-2(a_1-b_2)]^j}{(j+1)!}\tau^j \,,\\ \label{b2n}
\tilde{b}_2(\tau) &=& \sum_{j\ge0} \frac{ \frac{1}{2}[-2(a_1-b_2)]^{j+1} -[-a_1]^{j+1} }{(j+1)!}
\tau^j \,.\end{aligned}$$ Details of the derivation of Eqs. (\[a1n\])-(\[b2n\]) can be found in the Appendix.
By restricting the expansions (\[a1n\])-(\[b2n\]) to some common finite power $n$, one gets the respective $n$th-order approximation. This result extends previous findings of first [@sura] and second [@Julia] order terms.
Notice that Eq. (\[a1n\]) is uncoupled, meaning that the estimated harmonic stiffness $\tilde{a}_1$ is not affected by the exact noise components. Moreover, from Eq. (\[b2n\]), the estimated multiplicative noise parameter $\tilde{b}_2$ does not depend on the exact additive noise component.
Summing the series in Eqs. (\[a1n\])-(\[b2n\]) up to infinite order, and defining $Z\equiv \exp(-a_1\tau)$ and $W\equiv \exp(-2 b_2 \tau)$, we find the exact finite-$\tau$ expressions: $$\begin{aligned}
\nonumber
\tilde{a}_1 &=& \frac{1-Z}{\tau} \,,\\ \nonumber
\tilde{b}_0 &=& \frac{b_0}{a_1 - b_2} \,\frac{1-Z^2 W}{2\tau} \,,\\ \label{abtils}
\tilde{b}_2 &=& \frac{1-Z}{\tau} \,-\, \frac{1-Z^2 W}{2\tau} \,.\end{aligned}$$ Notice that $\lim_{\tau\to 0} \{\tilde{a}_1,\tilde{b}_0,\tilde{b}_2\} =\{{a}_1,{b}_0,{b}_2\}$ holds.
From Eqs. (\[abtils\]), we obtain an invariant relation among the estimated and exact parameters, namely, $$\label{constraint}
\frac{\tilde{a}_1 - \tilde{b}_2}{\tilde{b}_0} = \frac{a_1 - b_2}{b_0}\,.$$
The meaning of this invariance can be drawn, for instance, from the stationary PDF ${P^*}(x)$ associated to the corresponding Fokker-Planck equation given by Eq. (\[FPE\]). With the present choice of drift and diffusion coefficients, for $a_1 , b_0 >0$, $b_2 \ge 0$, one has: $$\label{pdf}
{P^*}(x)\,=\,P_o/[1+\frac{b_2}{b_0}x^2]^{\frac{a_1}{2b_2}+1} \,,$$ with $P_o$ a normalization constant. This solution is of the $q$-Gaussian form [@multiplicative1], for which, if ${a}_1-{b}_2>0$, the variance is finite with value $\sigma^2=b_0/(a_1-b_2)$. Hence, Eq. (\[constraint\]) represents the uphold of the data variance under changes of sampling intervals. For ${b_2}=0$, one recovers the Gaussian stationary solution and its variance relation. Notice also that, from Eqs. (\[a1n\])-(\[b2n\]), Eq. (\[constraint\]) still holds if one considers partial corrections of the parameters up to any common order $n$ of truncation of the sums.
Let us remark that the results presented in Eqs. (\[D1D2til\])-(\[abtils\]) are valid even when the variance is infinite. However, we will deal only with finite variance cases and consider normalized data (with unitary variance), which only implies a rescaling of $b_0\to b_0/\sigma^2 $. Then, $$\label{constraint2}
{a}_1={b}_0+{b}_2 \,.$$
Taking into account the constraint (\[constraint2\]), from Eqs. (\[abtils\]), the exact finite-$\tau$ expressions are: $$\begin{aligned}
\label{a1tilnor}
\tilde{a}_1 &=& \frac{1-\exp(-a_1\tau)}{\tau} \\ \label{b0tilnor}
\tilde{b}_0 &=& \frac{1-\exp(-2b_0\tau)}{2\tau}
\,.\end{aligned}$$
Eqs. (\[a1tilnor\]) and (\[b0tilnor\]) can be readily inverted to extract the true parameters from their finite-$\tau$ estimates: $$\begin{aligned}
\label{a1}
{a}_1 &=& \frac{\ln( 1 - \tilde{a}_1 \tau))}{-\tau} \,,\\ \label{b0}
{b}_0 &=& \frac{\ln( 1 - 2\tilde{b}_0 \tau))}{-2\tau} \,.\end{aligned}$$ Notice that $\tilde{a}_1\tau$ (and also $2\tilde{b}_0\tau$) can not be greater than unit.
In what follows, we fix the timescale $\tau=1$. A different choice would simply lead to a rescaling of the parameters $(a_1,b_0,b_2)\to (\tau a_1,\tau b_0,\tau b_2)$.
![Drift and diffusion coefficients for the O-U process. Symbols correspond to the numerical computation for artificial series ($10^5$ data), synthetized with the values of $a_1=b_0$ ($b_2=0$, in accord with constraint (\[constraint2\])) indicated on each panel. Lines represent the coefficients given by Eqs. (\[D1D2til\]), using the theoretical $\tau$-expansions (\[a1n\])-(\[b2n\]), at different orders of truncation. The darker the color, the higher the order, from first up to fifth order. The infinite order (exact expression) is represented in thick black lines. The zeroth order, corresponding to the true values, is plotted in dashed lines. []{data-label="fig:b2nullD1D2"}](figure1.eps){width="60.00000%"}
Now we investigate the importance of finite-$\tau$ effects for discretely sampled realizations of representative known diffusive processes. To this end, we generated artificial time-series through numerical integration of Eq. (\[ILE\]), by means of an Euler algorithm with timestep $dt=10^{-3}$, recording the data at each $1/dt$ timesteps, in accord with our choice $\tau=1$. Our theoretical results for ${D_1}$ and ${D_2}$ will be compared to the ones numerically computed from the time-series, through Eq. (\[ABtilde\]).
The particular case $b_2=0$, corresponding to the Orstein-Uhlenbeck (O-U) process and the general case with multiplicative component $b_2>0$ will be investigated separately. Fig. \[fig:b2nullD1D2\] shows the results for the artificial series with known values of the parameters $a_1 = b_0$ ($b_2=0$), together with our theoretical predictions. The exact theoretical expressions reproduce the numerical (finite-time) outcomes. Comparing the panels in Fig. \[fig:b2nullD1D2\], it is clear that, the larger $a_1$, the slower the convergence to the observed coefficients. The results for $a_1 > 1$ also illustrate the entanglement one may find in large-$\tau$ measurements, specifically, an oscillatory convergence of $\tilde {a}_1$ and an alternating signal of $\tilde {b}_2$.
![Drift and diffusion coefficients for the general process with multiplicative noise. Symbols correspond to the numerical computation for artificial series ($10^6$ data), synthetized with the values of $a_1\; (=b_0+b_2$, in accord with constraint (\[constraint2\])) and $b_2$ indicated on each panel. Lines are as in Fig. \[fig:b2nullD1D2\]. []{data-label="fig:b2D1D2"}](figure2.eps){width="60.00000%"}
In Fig. \[fig:b2D1D2\], we plot the numerical computations for artificial time-series together with analytical predictions for $b_2>0$. Again, the theoretical approximations present slower convergence as $a_1$ increases while the exact theoretical expressions agree with finite-time estimates directly obtained from the time-series. Moreover, the actual value of $b_2$ sets the convergence rate of $\tilde {b}_2$.
All these results raise a question about the domain of validity of lower order approximations presented before in the literature. Let us investigate this issue quantitatively. Given $\tilde{a}_1$, obtained from numerical (finite-time) evaluation, the exact value of $a_1$ can be recovered from Eq. (\[a1\]). Approximate values $a_1^{(n)}$ can be obtained by inversion of Eq. (\[a1n\]) truncated at order $n$. Figure (\[fig:orders\]) illustrates $a_1^{(n)}$ as a function of $n$, for different values of $\tilde{a}_1$. Clearly, convergence to the true value $a_1$ is attained (within a given tolerance), at different orders that depend on the value of $\tilde{a}_1$. For instance, for $\tilde{a}_1>0.5$, an order larger than two is required. Convergence is faster for smaller $\tilde{a}_1$, that is, as soon as 1/$\tilde{a}_1$ becomes large compared to the timescale $\tau=1$. For $\tilde{b}_0$ we obtained a very similar convergence scheme (not shown). In Refs. [@sura; @Julia], the fitness of low order expressions for O-U processes results from the particular employment of $\tilde{a}_1\tau<0.5$. However, this may not be the case when dealing with generic empirical data.
![Dependence of ${a}^{(n)}_1$ on the order $n$ of the approximation given by Eq. (\[a1n\]), for different values of $\tilde{a}_1$ (panel (a)). Dotted lines correspond to the respective true values ($a_1$) and missing points denote the absence of real solutions. Panel (b) exhibits the order at which the limiting value is attained (within 5%) as a function of $\tilde{a}_1$. []{data-label="fig:orders"}](figure3.eps){width="50.00000%"}
Higher-order coefficients
=========================
Inserting the Itô-Taylor expansion Eq. (\[MSI\]) into Eq. (\[ABtilde\]) and performing the average for $k=3,4$, we also computed the finite-$\tau$ expansion for $\tilde{D}_3(x,\tau)$ and $\tilde{D}_4(x,\tau)$. The resulting expressions are invariant functions of $x$, namely: $$\begin{aligned}
\label{D3t}
\tilde{D}_3(x,\tau) &=& -\tilde{c}_1(\tau)+ \tilde{c}_3(\tau)x^3 \\ \label{D4t}
\tilde{D}_4(x,\tau) &=& \tilde{d}_0(\tau)+\tilde{d}_2(\tau)x^2 + \tilde{d}_4(\tau)x^4\,.\end{aligned}$$
For the particular case $b_2=0$, we were able to derive the infinite order expansion for the $\tau$-parameters:
$$\begin{aligned}
\nonumber
\tilde{c}_1(\tau) &=& b_0\sum_{j\ge 0} \frac{3^{j+1}-2^{j+1}-1}{2}\frac{[-a_1]^j}{(j+1)!}\tau^j
\,, \nonumber \\
\tilde{c}_3(\tau)&=& \sum_{j\ge 0} \frac{3^{j}-2^{j+1}+1}{2}\frac{[-a_1]^{j+1}}{(j+1)!}\tau^j \,,
\label{ctils} \\
\tilde{d}_0(\tau) &=& b_0^2\sum_{j\ge0} \frac{4^{j}-2^{j}}{2}\frac{ [-a_1]^{j-1}}{(j+1)!} \tau^j \,,
\nonumber \\
\tilde{d}_2(\tau) &=& b_0\sum_{j\ge0} \frac{2\times4^{j}-3^{j+1}+1}{2}\frac{[-a_1]^{j} }{(j+1)!} \tau^j\,,
\nonumber \\
\label{dtils}
\tilde{d}_4(\tau) &=& \sum_{j\ge0} \frac{ 4^{j}-3^{j+1}-1}{6}\frac{[-a_1]^{j+1}}{(j+1)!} \tau^j \,.\end{aligned}$$
Notice that, $\lim_{\tau\to 0} \{\tilde{c}_1,\tilde{c}_3,\tilde{d}_0,\tilde{d}_2,\tilde{d}_4\} =0$, as expected, and that the relevant parameter for the rate of series convergence is $a_1$. Summing the series (\[ctils\])-(\[dtils\]) up to infinite order, and recalling that $Z\equiv \exp(-a_1\tau)$, one obtains $$\begin{aligned}
\nonumber
\tilde{c}_1(\tau) &=& -\frac{b_0}{a_1} \frac{(1-Z)^2(1+Z) }{2\tau} \,,\\ \label{ctilinf}
\tilde{c}_3(\tau) &=& - \frac{(1-Z)^3 }{6\tau}\,,\\ \nonumber
%
\tilde{d}_0(\tau) &=& \frac{b_0^2}{a_1^2} \frac{(1-Z)^2(1+Z)^2}{8\tau} \,,\\ \nonumber
\tilde{d}_2(\tau) &=& \frac{b_0}{a_1} \frac{(1-Z)^3(1+Z)}{4\tau} \,,\\ \label{dtilinf}
\tilde{d}_4(\tau) &=& \frac{(1-Z)^4}{24\tau} \,.\end{aligned}$$
Fig. \[fig:b2nullD3D4\] shows the numerical computation of $\tilde{D}_3(x,\tau)$ and $\tilde{D}_4(x,\tau)$ for the same artificial series as in Fig. \[fig:b2nullD1D2\]. For comparison, the theoretical estimates at different orders of truncation of the series in Eqs. (\[ctils\])-(\[dtils\]) are shown. Notice that, although ${D}_3,{D}_4=0$ for the diffusive processes considered here, their finite-time counterparts have cubic and quadratic forms. Indeed, the exact theoretical expressions given by Eqs. (\[ctilinf\])-(\[dtilinf\]) reproduce the numerical outcomes, validating our approach as furnishing meaningful tests for O-U models. However, for $a_1 \ge 1$, rich pictures for the low order approximations of $\tilde{D}_3$ and $\tilde{D}_4$ arise, which hinder the asymptotic estimation.
![Third and fourth order coefficients for the O-U process. Symbols correspond to the numerical computation for the same artificial series of Fig. \[fig:b2nullD1D2\]. Lines represent the coefficients given by Eqs. (\[D3t\])-(\[D4t\]), using the theoretical $\tau$-expansions (\[ctils\])-(\[dtils\]), at different orders of truncation. Colors as in Fig. \[fig:b2nullD1D2\]. []{data-label="fig:b2nullD3D4"}](figure4.eps){width="60.00000%"}
For the general case with $b_2\ge 0$, we computed the third-order $\tau$-expansions for $\tilde{D}_3(x,\tau)$ and $\tilde{D}_4(x,\tau)$. Each power of $\tau$ of order $j\le3$ has pre-factor denoted by $\tilde{D}_3^{(j)}$ and $\tilde{D}_4^{(j)}$ respectively. We find, for $\tilde{D}_3$: $$\begin{aligned}
\nonumber
\tilde{D}_3^{(0)} &=& 0 \,,\\ \nonumber
\tilde{D}_3^{(1)} &=& -b_0 \alpha x -b_2\alpha x^3 \,,\\ \nonumber
\tilde{D}_3^{(2)} &=& \frac{1}{6}b_0 \alpha (9a_1-16b_2) x
-\frac{1}{6} \alpha(a_1^2-13a_1b_2+16b_2^2) x^3 \,,\\ \label{D3til}
\tilde{D}_3^{(3)} &=& -\frac{1}{6}b_0 \alpha^2 (8a_1-13b_2) x
+\frac{1}{12} \alpha(3a_1^3-32a_1^2b_2+74a_1b_2^2-52b_2^3) x^3 \,,\end{aligned}$$ where $\alpha=a_1-2b_2$. At all orders, $\tilde{D}_3$ vanishes if $a_1=2b_2$.
![Third and fourth order coefficients for the general process with multiplicative noise. Symbols correspond to the numerical computation for the same artificial series of Fig. \[fig:b2D1D2\]. Lines represent the coefficients given by Eqs. (\[D3t\])-(\[D4t\]), using the theoretical $\tau$-expansions (\[D3til\])-(\[D4til\]), at different orders of truncation up to third order. Colors as in previous figures. []{data-label="fig:b2D3D4"}](figure5.eps){width="60.00000%"}
For $\tilde{D}_4$, we have: $$\begin{aligned}
\nonumber
\tilde{D}_4^{(0)} &=& 0 \,,\\ \nonumber
\tilde{D}_4^{(1)} &=& \frac{1}{2}b_0^2 +b_0b_2x^2+\frac{1}{2}b_2^2x^4 \,,\\ \nonumber
\tilde{D}_4^{(2)} &=& -\frac{1}{3}b_0^2(3a_1-7b_2)
+\frac{1}{6}b_0(3a_1^2-30a_1b_2+52b_2^2)x^2
+\frac{1}{6}b_2(3a_1^2-24a_1b_2+38b_2^2)x^4 \,, \\ \nonumber
\tilde{D}_4^{(3)} &=& \frac{1}{6}b_0^2(7a_1^2-34a_1b_2+43b_2^2)
-\frac{1}{6}b_0(6a_1^3-65a_1^2b_2+206a_1b_2^2
-206b_2^3)x^2 \\ \label{D4til}
&&+\frac{1}{24} (a_1^4-36a_1^3b_2+276a_1^2b_2^2-736a_1b_2^3
+652b_2^4)x^4 \,.\end{aligned}$$
In Fig. \[fig:b2D3D4\], we show the numerical computation of $\tilde{D}_3$ and $\tilde{D}_4$ for the same artificial series as in Fig. \[fig:b2D1D2\]. According to our theoretical results, $a_1=2b_2$ is a threshold between positive and negative slopes of $\tilde{D}_3$, as illustrated in Fig. \[fig:b2D3D4\]. For comparison, we also show up to the third order theoretical estimates $\sum_{j\ge 1} \tilde{D}_3^{(j)}$ and $\sum_{j\ge 1} \tilde{D}_4^{(j)}$, according to Eqs. (\[D3til\])-(\[D4til\]).
Coefficientes $\tilde{D}_3$ and $\tilde{D}_4$ provide further tests of validity of the diffusive modeling. From Fig. \[fig:b2D1D2\], third order estimates furnish suitable forecast of the numerical finite-$\tau$ measurements for small enough $a_1$. In such cases, once obtained $a_1,\;b_0$ from Eqs. (\[a1\])-(\[b0\]), these values can be used in theoretical equations for $\tilde{D}_3$ and $\tilde{D}_4$, to check if the corresponding non-null coefficients can be attributed to finite-$\tau$ effects.
Summary and final comments
==========================
For an important class of diffusion models with additive-multiplicative noise, we have derived exact formulas that connect the empirical discrete-time estimates with the actual values of the parameters of drift and diffusion coefficients. Additionally, we also provided theoretical expressions for higher-order coefficients which serve as a further probe for the validity of this class of diffusive models.
Our results allow to access the generating stochastic process. A possible procedure to identify it and its parameters can be summarized as follows. When numerical computation of the coefficients from a real timeseries yields linear and quadratic forms for $D_1$ and $D_2$ (which is a frequent outcome), the values of $\{\tilde{a}_1,\tilde{b}_0,\tilde{b}_2\}$ can be obtained from curve fitting. The present model (with or without multiplicative component) would be adequate when i) $\tilde{a}_1<1$ and ii) ($\tilde{a}_1\simeq\tilde{b}_0+\tilde{b}_2$). For $\tilde{a}_1<0.5$ the second order correction would be enough to recover $a_1$, otherwise larger order corrections should be considered. Once $\tilde{a}_1$ and $\tilde{b}_0$ are known, Eqs. (\[a1\]) and (\[b0\]), allow to obtain, exactly, the original parameters $a_1,b_0$, hence also $b_2=a_1-b_0$. Values such that $b_2<<a_1,b_0$ (hence, $a_1\simeq b_0$) point to a simple O-U process, otherwise a multiplicative term may be also present. In both cases, a further check consists in the analysis of higher order coefficients, e.g., to see whether a non-null $D_4$ can be attributed to finite-time corrections.
By analyzing the O-U process, we also found that a low sampling rate would significantly affect the diffusion coefficient estimate, by adding an extra quadratic term. Thus, the detection of a quadratic $\tilde{D}_2$ does not imply the existence of multiplicative components in the actual process. Moreover, estimations of $D_3$ and $D_4$ from low-order approximations would lead to results inconsistent with the empirical outcomes.
The obtained formulas also allow to quantify the errors induced by a finite sampling rate $\tau$ in the numerically estimated coefficients. The analytical results indicate that, in order to grasp the true values of the parameters from the knowledge of the observed ones, the required correction depends strongly on the (hidden) inverse time $a_1$. Our work shows that one should be careful when applying low-order finite-$\tau$ corrections for diffusion models. Furthermore, as shown in Fig. \[fig:orders\], our results provide a criterion, from the knowledge of $\tilde{a}_1$, to determine the required order $n$, or equivalently, up to which value of $\tau$ the respective approximation is reliable.
Acknoledgements: {#acknoledgements .unnumbered}
================
We acknowledge Brazilian agencies Faperj and CNPq for partial financial support.
Appendix {#appendix .unnumbered}
========
Rewriting Eq. (\[ABtilde\]) according to the notation introduced in Eq. (\[MSI\]), one has $$\nonumber
\tau\tilde{D}_1=\langle \Delta X\rangle = \sum_{\alpha_k} c_{\alpha_k}(D_1,D_2)\langle I_{\alpha_k}\rangle.$$ Only multiple stochastic integrals $I_{\alpha_k}$ such that $\alpha_k=(0,...,0)_k$ have non-null average, being $\langle I_{(0,...,0)_k}\rangle =\tau^k/k!$. From the iterated application of Itô formula to Eq. (\[intXform\]), $c_{(0,...,0)_k}(D_1,D_2)=(L^0)^{k-1} D_1$. These are general results independent of the particular form of $D_1$ and $D_2$. Noticing that, from Eq. (\[L0L1\]), $L^0=\partial_t+D_1\partial_x+D_2\partial_{xx}$, and that $D_1$ is time-independent and linear in $x$, then, $c_{\alpha_k}=D_1(D_1')^{k-1}=(-a_1)^{k} x$. Finally, $$\nonumber
\tilde{D}_1=\langle\Delta X\rangle/\tau =\sum_{k\ge 1} \frac{1}{k!}(-a_1)^k\tau^{k-1}x\,,$$ which is of the same functional form of the true $D_1$ and can be identified with $-\tilde{a}_1x$, so that $$\nonumber
\tilde{a}_1= -\sum_{k\ge 1} \frac{1}{k!}(-a_1)^k\tau^{k-1}\,,$$ which gives Eq. (\[a1n\]).
For the second conditional moment, one has $$\label{Ap_D2til}
2\tau \tilde{D}_2 = \langle (\Delta X)^2 \rangle = \sum_{\alpha_n,\beta_m} c_{\alpha_n}c_{\beta_m}
\langle I_{\alpha_n}I_{\beta_m}\rangle.$$ From the definition of $c_{\alpha_k}$ in Eq. (\[MSI\]), if $b_2=0$ (then $D_2$ is constant), only two classes of terms in Eq. (\[Ap\_D2til\]) are non-null, those with:\
i) $\alpha_n=(0,...,0)_n$ and $\beta_m=(0,...,0)_m$ and\
ii) $\alpha_n=(1,0,...,0)_n$ and $\beta_m=(1,0,...,0)_m$. In those cases, the products $c_{\alpha_n}c_{\beta_m}$ take the values:\
i) $(D_1)^2(D_1')^{k-1}=(-a_1)^{k+1}x^2$ (with $k=m+n-1$),\
ii) $2D_2(D_1')^{k}=2b_0 (-a_1)^{k}$ (with $k=m+n-2$).\
In order to evaluate the averages of products of multiple stochastic integrals, it is useful to recall that $\langle I_{(0,...,0)_n}I_{(0,...,0)_m}\rangle=\frac{\tau^{n+m}}{n!m!}$ and that $\langle I_{(1,...,0)_n}I_{(1,...,0)_m}\rangle=
\frac{\tau^{n+m-1}}{(n+m-1)(n-1)!(m-1)!}$ [@book]. Then, summing over all the pairs $(n,m)$ contributing to the order $\tau^{k+1}$, one obtains:\
i) $\sum \langle I_{\alpha_n}I_{\beta_m}\rangle/\tau^{k+1}=
\frac{1}{(k+1)!} \sum_{n=1}^k \binom{k+1}{n}
=2\frac{2^k-1}{(k+1)!}$,\
ii) $\sum\langle I_{\alpha_n}I_{\beta_m}\rangle/\tau^{k+1}=
\frac{1}{(k+1)!} \sum_{j=0}^k \binom{k}{j}
=\frac{2^k}{(k+1)!}$.\
Finally, from Eq. (\[Ap\_D2til\]), we arrive at $$\tilde{D}_2=\langle(\Delta X)^2\rangle/(2\tau)
=\frac{1}{2}\sum_{k\ge 0}\biggl(
2\frac{2^k-1}{(k+1)!}(-a_1)^{k+1}x^2 +
\frac{2^k}{(k+1)!}2b_0(-a_1)^k
\biggr)\tau^k \,,$$ which can be cast in the form $\tilde{b}_2x^2+\tilde{b}_0$, allowing to identify $\tilde{b}_2$ and $\tilde{b}_0$ with functions of the true parameters, as $$\begin{aligned}
\nonumber
\tilde{b}_0 &=& \frac{1}{2} \sum_{k\ge 0} \frac{2^k}{(k+1)!}2b_0(-a_1)^k \tau^k\,,
\\ \label{bs}
\tilde{b}_2 &=& \sum_{k\ge 0}\frac{2^k-1}{(k+1)!}(-a_1)^{k+1} \tau^k \,. \end{aligned}$$
For the general case $b_2\ge 0$, a similar but tricky derivation leads to Eqs. (\[b0n\])-(\[b2n\]) that generalize the expressions (\[bs\]).
Proceeding with the third order, products of three multiple integrals appear. For $b_2=0$, there are two types of products $I_{\alpha_n}I_{\beta_m}I_{\gamma_l}$ contributing to $\langle(\Delta X)^3\rangle$:\
i) $\alpha_n=(0,...,0)_n$, $\beta_m=(0,...,0)_m$, $\gamma_l=(0,...,0)_l$ and\
ii) $\alpha_n=(0,...,0)_n$, $\beta_m=(1,0,...,0)_m$, $\gamma_l=(1,0,...,0)_l$, with pre-factors proportional to $(D_1)^3(D_1')^{k-2}=(-a_1)^{k+1}x^3$ and to $2D_2D_1(D_1')^{k-1}=2b_0(-a_1)^{k}x$, respectively. Evaluating the averages of products of three multiple stochastic integrals and summing over all the triplets $(n,m,l)$ contributing to the same order in $\tau$, as done for the second order coefficient, one gets $\tilde{D}_3=-\tilde{c}_1(\tau)+ \tilde{c}_3(\tau)x^3$, as in Eq. (\[D3t\]).
Analogously, at fourth order, considering the relevant products of four multiple integrals, for $b_2=0$, three types of contributions appear, yielding $\tilde{D}_4(x,\tau) = \tilde{d}_0(\tau)+\tilde{d}_2(\tau)x^2 + \tilde{d}_4(\tau)x^4$, as in Eq. (\[D4t\]).
Let us recall that the averages of products of $n$ multiple stochastic integrals appearing in the $nth$-order term of the coefficients can be expressed, in general, as multinomial terms, whose summation over all the products has the form $\mu_11^k+\mu_22^k+\mu_3 3^k +....\mu_n n^k$ (with rational $\mu_i$) for the order $k$ in $\tau$. Moreover, products of multiple stochastic integrals can be readily simplified, by means of useful relations between multiple Itô integrals [@book]. Then, although at the cost of increasing the number of indices, the number of factors can be reduced.
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abstract: 'We construct simple standing wave solutions in a 5D space-time with a ghost-like scalar field. The nodes of these standing waves are ‘islands’ of 4D anti-de-Sitter space-time. In the case of increasing (decreasing) warp factor there are a finite (infinite) number of nodes and thus a finite (infinite) number of anti-de-Sitter island-universes having different gravitational and cosmological constants. This is similar to the landscape models, which postulate a large number of universes with different parameters. 0.3cm PACS numbers: 11.25.-w, 11.25.Wx, 11.27.+d'
author:
- |
[**Merab Gogberashvili$^{(1,2)}$ and Douglas Singleton$^{(3,4)}$**]{}\
\
[(1) Andronikashvili Institute of Physics, 6 Tamarashvili Street, Tbilisi 0177, Georgia]{}\
[(2) Javakhishvili State University, 3 Chavchavadze Avenue, Tbilisi 0128, Georgia]{}\
[(3) California State University, Fresno, Physics Department, Fresno, CA 93740-8031, USA]{}\
[(4) VNIIMS, Peoples’ Friendship University of Russia, Moscow 117198, Russia]{}
title: 'Anti-de-Sitter Island-Universes from 5D Standing Waves'
---
0.5cm
Introduction
============
The infinite extra dimension brane theories [@brane1; @brane2; @brane3; @brane4] have been one of the most active areas of study in the past decade. A concise overview of these models can be found in [@Rub]. The original brane models had 5D space-times with 4D $\delta$-function sources which were positive or negative tension thin branes. Later works replaced the $\delta$-function sources with localized energy momentum tensors of finite thickness which were called thick branes. Examples of thick brane models included fluid-like matter sources [@GoSi; @GoSi1], scalar field matter sources [@Bronn; @dzh-sing], and phantom scalar field sources [@Koley; @Koley1; @Koley2; @Phantom; @Phantom1; @Phantom2]. Most brane world models focused on static solutions, i.e. both metric and any fields being time independent. An exception to this are S-brane solutions [@S; @S1; @S2; @S3; @S4] which are used for cosmological studies.
In this paper we present a simple standing gravitational wave solution as a possible 5D brane model. The existence of this solution requires a negative cosmological constant and a phantom/ghost-like scalar field (i.e. a field with a negative sign in front of the field kinetic energy term of the Lagrangian) in the bulk. Such ghost-like scalar fields are generally problematic since they tend to make the system unstable. To counter this criticism we will show that our model can be embedded in a 5-dimensional Weyl model of gravity [@Weyl; @Weyl1; @Weyl2; @Weyl3]. Because of the geometrical origin of the coupling of the scalar to the space-time and since the Weyl scalar does not couple with other matter fields we are able to avoid the usual problems of instability due to ghost fields.
We assume that the ghost-like field vanishes on the brane at the origin. This condition leads to a quantization of the oscillation frequency of the standing wave which in turn makes the scalar field vanish at a finite or infinite number of nodes of the standing wave depending on whether the warp factor of the metric is increasing or decreasing. By taking the matter fields to bind to the standing waves nodes one has a finite or infinite number of anti-de-Sitter island-universes, each with different parameters. In the simple model presented here the different parameters are the effective 4D gravitational constant and the 4D cosmological constant. These two 4D constants are fixed in terms of the 5D constants and the value of the warp factor at the nodes.
For the case of an increasing warp factor, and a finite number of anti-de-Sitter ‘islands’, one could address the family puzzle by taking different fermion generations as bound to different nodes and fixing the number of nodes in the bulk at three. For the case of a decreasing warp factor, and infinite number of anti-de-Sitter island-universes, one has something like the landscape picture in string theory/M theory [@landscape; @landscape1; @landscape2], with its large number of universes with different parameters.
In the model presented in this paper ordinary matter fields are assumed to be bound to one of the anti-de-Sitter ‘islands’. This is an advantage over the original one-brane or two-brane models of [@brane1; @brane2; @brane3; @brane4], where the ordinary matter fields are assumed to be localized to a brane with positive or negative tension. In these models one needs to explain why these brane tensions are not observed.
There are several open questions or problems with the present 5D standing wave solution as brane world model: The first question concerns the stability of the whole 5D solution given the presence of the ghost-like scalar field. This issues of the stability of brane world models in the presence of sources which violate some or all of the usual energy conditions (e.g. negative tension branes [@brane1; @brane2; @brane3; @brane4], or ghost fields [@Koley; @Koley1; @Koley2; @Phantom]) has been known since the beginning of the study of brane world models [@pilo]. In section $3$ we present a possible resolution to this issue by considering Weyl’s generalization of Riemannian geometry. We also offer some discussion in the conclusion section of cases where a ghost field, at least at the classical level, leads to greater stability of brane world models as compared to normal scalar fields. The second question is: “On or near the nodes are the gravitational fluctuations confined so as to give effective 4D gravity near the nodes?". In section $6$ will show that near the nodes one does get effective 4D gravity in much the same way as in [@brane1; @brane2; @brane3; @brane4]. The last question/problem is give a localization mechanism for the matter fields (i.e. fields with spin 0, spin $\frac{1}{2}$, spin 1) on the brane. This localization problem has been an unresolved question for brane world models in general [@Local; @Local1; @Local2; @Local3; @Local4; @Local5; @Local6]. Here we simply postulate that matter fields are bound to the nodes of the standing wave solutions. In section $7$ we give two possible mechanisms for accomplishing this binding of matter fields to the nodes. The first mechanism involves the coupling of the quadrupole moment of the stress-energy tensor of the matter fields to the Riemann tensor. The second mechanism involves coupling of matter fields to the ghost-like field.
Metric and Einstein’s equations
===============================
We study solutions of the 5-dimensional Einstein equations, $$\label{5d-einstein}
R_{ab} - \frac{1}{2} g_{ab} R = \frac{8 \pi G_5}{c^4} T_{ab} -
\Lambda _5 g_{ab}~,$$ where $G_5$ and $\Lambda _5$ are 5D Newton and cosmological constants respectively [@brane1; @brane2; @brane3; @brane4]. Lower case Latin indices $a, b$ run over the full 5D space-time $0,1,2,3,4$. The metric [*ansatz*]{} we use is $$\label{metric}
ds^2 = e^{2a|r|}\left( dt^2 - e^{u}dx^2 - e^{u}dy^2 - e^{-2u}dz^2
\right) -dr^2~,$$ where $a$ is a constant and the [*ansatz*]{} function $u = u(t,r)$ only depends on the time $t$ and the extra coordinate $r$. This [*ansatz*]{} is a combination of the 5D warped brane world model through the $e^{2a|r|}$ term [@brane1; @brane2; @brane3; @brane4] plus an anisotropic ($t,r$)-dependent warping of the brane coordinates, $x,y,z$, through the terms $e^{u(t,r)}, e^{-2u(t,r)}$. The absolute value sign around $r$ gives a $\delta$-function source/brane at $r=0$ exactly like the one brane models of [@brane1; @brane2; @brane3; @brane4]. Since we will focus on standing wave solutions ‘to the right’, i.e. $r>0$, we drop the absolute value sign in . We study both decreasing ($a<0$) and increasing ($a>0$) warp factors. We also find that for both increasing and decreasing warp factors the ansatz function $u(t,r)$ has nodes – $u(t,r) =0$ – at specific values of $r$. At these nodes the space-time in reduces to 4D Minkowski space-time plus a scaled, negative cosmological constant – at the nodes the space-time becomes effectively 4D anti-de-Sitter. As one moves away from these nodes there is an anisotropic stretching/shrinking along the $x,y,z$ directions due to the metric components $e^{u(t,r)}, e^{-2u(t,r)}$. If $u(t,r)$ is positive (negative) as one moves away from the node in $r$ the $x,y$ directions will expand (shrink) as $e^u$ while the $z$ direction will shrink (expand) as $e^{-2u}$. In section $7$ we will use this feature of the metric to propose a possible localization mechanism for matter fields.
For the [*ansatz*]{} we find the non-zero components of the Ricci tensor: $$\begin{aligned}
\label{field-eqns2}
R_{tt} &=& e^{2ar} \left( -\frac 32 e^{-2ar} \dot u ^2 + 4a^2 \right)~, \nonumber \\
R_{xx}&=&R_{yy}=e^{2ar+u}\left( \frac 12 e^{-2ar}\ddot u - 2au'-\frac 12 u'' -4a^2\right)~, \nonumber \\
R_{zz} &=&e^{2ar-2u}\left( - e^{-2ar}\ddot u + 4au' + u'' - 4a^2\right)~, \\
R_{rr} &=& -\frac 32 u'^2 - 4a^2~, \nonumber\\
R_{rt} &=& -\frac 32 \dot uu' ~, \nonumber\end{aligned}$$ where overdots mean time derivatives and primes stand for derivatives with respect to $r$. In there should be terms proportional to $\delta (r)$ coming from the tension of the thin brane at $r=0$. We do not write this explicitly since we focus on $r>0$ and have thus dropped the absolute value in the warp factor of the metric .
Source scalar field
===================
To the metric above we add a massless, non-self interacting scalar phantom/ghost-like field, $\phi (t,z)$ [@phantom; @phantom1], which obeys the Klein-Gordon equation, $$\label{phi}
\frac{1}{\sqrt{g}}~\partial_a (\sqrt{g} g^{a b}\partial_b \phi) =
e^{-2 a r}\ddot \phi - \phi'' - 4 a \phi' = 0 ~.$$ Here $g$ is the determinant of the 5-dimensional background space-time given by . The energy-momentum tensor of the phantom-like field $\phi$ is taken in the form: $$\label{emt-phi}
T_{ab} = - \partial_a \phi\partial_b \phi + \frac 12 g_{a b} \partial^c
\phi~\partial_c \phi~.$$ Strictly speaking $\phi$ is not a phantom field as defined in [@caldwell], where the criterion for a phantom field was $p/\rho
<-1$ ($p$ and $\rho$ are the pressure and energy density of the field respectively). From one can obtain $p$ and $\rho$ for the field $\phi$ and since the field is non-self interacting one does not have $p/\rho <-1$.
To avoid the well-known problems with stability that occur with ghost fields we can associate the ghost-like field $\phi$ with the geometrical scalar field in a five-dimensional, integrable Weyl model. In Weyl’s model a massless scalar appears through the definition of the covariant derivative of the metric tensor, $$\label{D}
D_{c} g_{ab} = g_{ab}\partial_c \phi ~.$$ This is a generalization of the Riemannian case where the covariant derivative of the metric is zero. The result in implies that the length of a vector is altered by parallel transport. Weyl’s scalar field in may imitate a massless scalar field – either an ordinary scalar or ghost-like scalar [@Weyl; @Weyl1; @Weyl2; @Weyl3]. The gravitational action for Weyl’s 5D integrable model can be written as $$\label{grav-weyl}
S_g = \frac{1}{16 \pi G_5} \int d^5x \sqrt{g}\left[R -
(6-5\xi)~\partial^a \phi~\partial_a \phi\right]~,$$ where $\xi$ is an arbitrary constant which can take any value. For example, if one takes $\xi =13/10$ in then the coefficient in front of $\partial^a \phi~\partial_a \phi$ becomes $-1/2$. This would give a ghost field which would lead exactly to the equation of motion and the energy-momentum tensor for our phantom-like scalar field. Thus we can start with a 5D Weyl model and require that we have Riemann geometry on the brane by assuming that the geometrical scalar $\phi$ is independent of brane coordinates $x^i$ and vanishes on the brane. We will show later that our solution has exactly this character – the scalar field $\phi$ vanishes at the nodes of $u (t,r)$ and is independent of $x,y,z$. Associating our scalar field with the geometrical Weyl scalar defined via avoids the usual instability problems of ghost fields since the Weyl model is known to be stable for any value of $\xi$.
The solution
============
For one can rewrite the Einstein equations (\[5d-einstein\]) in the form: $$\label{field-eqns1} R_{ab} = - \partial_a \sigma\partial_b \sigma
+\frac{3}{2} g_{ab} \Lambda _5~,$$ where the gravitational constant has been absorbed via the redefinition of the scalar field, $$\label{sigma}
\sigma =\frac{\sqrt{8 \pi G_5}}{c^2}\phi ~.$$
By combining and one can see that the “constant" terms from $R_{\mu \nu}$ (i.e. those terms that up to some metric factor are $\pm 4 a^2$) can be canceled if the 5D cosmological constant is chosen as $$\label{L=a} \Lambda_ 5 = \frac{8}{3} a^2~.$$ This is the same as the fine tuning used in standard 5D brane models [@brane1; @brane2; @brane3; @brane4].
The terms from $R_{\mu \nu}$ which are quadratic in $u$ can be accounted for if one assumes that $$\label{sigma=u}
\sigma (t,r) = \sqrt{\frac{3}{2}}~u(t,r)~,$$ so that $u(t,r)$ satisfies . A similar equality between the scalar field and metric ansatz function was found for the domain wall plus standing wave solutions of [@domain-wall]. Since our scalar field is proportional to the metric ansatz function $u(t,r)$ the scalar field will vanish wherever $u(t,r)$ has nodes. This requirement, that the scalar field vanish at the zeros of $u(t,r)$, was one of the necessary conditions pointed out in the last section in order to be able to associate our scalar field with a Weyl scalar field.
Because of and , solving Einstein equations has been reduced to finding solutions to the ordinary differential equation, $$\label{field-eqns3}
e^{-2ar}~\ddot u - u'' - 4au' = 0~.$$
We want to have a standing wave solution to , so we use separation of variables writing $$\label{separation}
u(t,r) = C \sin (\omega t) f(r)~,$$ where $C$ and $\omega$ are constants. Because of (\[sigma=u\]) the same separation applies to $\sigma$, but with a different constant, $C \rightarrow \sqrt{\frac{3}{2}} C$. Equation now becomes (\[separation\]) is: $$\label{f}
f'' + 4 a f' + \omega^2 e^{-2 a r} f = 0 ~.$$ The general solution to this equation is: $$\label{solution}
f(r) = A e^{-2ar} J_2 \left( \frac{\omega}{a} e^{-ar} \right) +
B e^{-2ar} N_2 \left( \frac{\omega}{a} e^{-ar} \right) ~,$$ where $A,B$ are constants and $J_2$ and $N_2$ are 2$^{nd}$ order Bessel functions of the first and second kind respectively. Normally the $N_2$ Bessel functions are discarded since they blow up at the origin. But here the functional dependence is $e^{-a r}$, rather than $r$, and neither $J_2$ nor $N_2$ diverges at $r=0$.
Before moving on to the discussion of the physical meaning of the solutions in we add the boundary condition that the ghost-like field should vanish at the brane, $r=0$. To accomplish this we should take either $A=0$ or $B=0$, since the zeros of $J_2$ and $N_2$ do not coincide. Then we set $$\label{omega}
\frac{\omega}{a} = X_{2,n}~,$$ where $X_{2,n}$ is the $n^{th}$ zero of the 2$^{nd}$ order Bessel function $J_2$ or $N_2$, depending on whether one takes $A=0$ or $B=0$ in . The condition quantizes the oscillation frequency, $\omega$.
Standing waves as Anti-de-Sitter ‘islands’
==========================================
We now analyze the physical consequences for metric given the solutions given by , and .
First we note that in terms of the behavior of the $x, y, z$ coordinates gives different asymptotic properties for the increasing ($a>0$) and decreasing ($a<0$) warp factor cases. For the increasing warp factor ($a>0$) the metric function, $u(t,r) \propto
f(r)$, decreases like $e^{-2ar}$ as one moves into the bulk. As one goes to large $r \rightarrow \infty$ this factor of $e^{-2ar}$ drives $u(t,r) \rightarrow 0$ and one has an anti-de-Sitter space-time but scaled up by an overall factor $e^{2 a r}$. For the decreasing factor ($a<0$) the function $u(t,r)$ grows like $e^{2ar}$ into the bulk therefore distances in the $x$ and $y$ directions expand like Exp$(e^{2ar})$, while in the $z$ direction they shrink like Exp$(-2
e^{2ar})$.
The second observation is that $u(t,r)$ is oscillatory and has zeros at various points along the $r$-direction – whenever $f(r)=0$. This happens when $J_2 (r) =0$ or $N_2 (r) =0$, depending on if one is considering $B=0$ or $A=0$. At these zero points, $r_m$, one has $u(t, r_m) =0$ and the metric in reduces to the standard 5D brane metric with an exponential warp factor [@brane1; @brane2; @brane3; @brane4]. This spatial oscillatory behavior of $u (t,r)$ leads to ‘islands’ of anti-de-Sitter space-times in the bulk.
The third observation is that the physical parameters on each node are scaled by the warp factor $e^{ar}$ or some power thereof. For example, taking the 4D reduction of 5D Einstein equations one finds that on the nodes of the anti-de-Sitter ‘islands’ there are effective 4D, negative cosmological constants given by $$\label{cc} \Lambda _4 = e^{2 a r_m} \Lambda _5 ~.$$ Thus the values of the effective 4D cosmological constant on the nodes is set by the scaled 5D cosmological constant. In the usual thin brane models [@brane1; @brane2; @brane3; @brane4] the 5D cosmological constant is fine tuned to exactly cancel the 4D brane tension. Thus on the brane the effective 4D cosmological constant is zero. In our case at the nodes, $r_m$, there is no brane tension and so our effective 4D cosmological constant is non-zero and given by . For a decreasing wrap factor (i.e. $a<0$) one has a cosmological constant which is exponentially suppressed relative to the true 5D value, $\Lambda_5$. Below we argue that these nodes are 4D anti-de-Sitter island universes on which matter fields are bound and on which gravity is effectively 4D.
For the case $a>0$ and $B=0$, there will be $(n+1)$ such ‘islands’ ($n$ is the number of the zero from ). As $r$ runs from $0$ to $\infty$ the argument of $J_2$ in runs from $X_{2, n}$ to $0$ giving $n$ zeros. One additional zero comes from $f(r \rightarrow \infty) =0$. The zeros will occur at $r_m$ where $\omega e^{-a
r_m} / a = X_{2, m}$ with $X_{2, m}$ being a zero of $J_2$ with $m<n$. The case $a>0$ and $A=0$ works out the same with $(n+1)$ ‘islands’ since one also has an additional zero coming from $f(r \rightarrow \infty) \rightarrow 0$ – in this case the divergence coming from $N_2 (0)$ is dominated by the $e^{-2ar}$ pre-factor in . The $\omega e^{-ar}/a $ dependence of $f(r)$ has the effect of stretching out the zeros of the Bessel functions, i.e the spatial oscillation frequency decreases with $r$.
In the case $a<0$ (with either $A=0$ or $B=0$) there are an infinite number of zeros for $u(t,r)$ due to the $\omega e^{ar}/a $ dependence of $f(r)$. In addition the zeros are compressed as $r$ increases, i.e. the spatial oscillation frequency increases with $r$. This compression ($a>0$) and stretching ($a<0$) of the location of the nodes has a connection with the effective thickness of the 4D anti-de-Sitter islands – the compression of the nodes tends to lead to ‘islands’ with a smaller effective thickness while stretching of the nodes leads to ‘islands’ with a large effective thickness. We discuss this further in section $7$ where we examine localization mechanisms for matter fields to the nodes.
Localization of gravity
=======================
Near $r=0$ in metric one has a $node + brane$ as in the 5D single brane models of [@brane1; @brane2; @brane3; @brane4]. The gravitational perturbations around $r=0$ will have one delta-function bound state and continuum states which start from zero mass i.e. which do not have a mass gap. Thus at the $r=0$ node one has effective 4D gravity as in the original models. We now look at the nodes of $u(r,t)$ for $r
> 0$ to see if near these nodes gravity also becomes approximately 4D. This is done by studying the 4D perturbations of the metric near the nodes. If there is a zero mass graviton mode near the node then its exchange between particles localized on the node will lead to a Newtonian potential with corrections coming from the massive modes ($m>0$). Close to the nodes one has $u(r,t) \approx 0$ and the metric takes on the usual 5D warped geometry. Near the nodes small fluctuations around this background can be written as $$\label{node}
ds^2 \approx \left[ e^{2 a r} \eta_{\mu \nu} + h_{\mu \nu} (x_\mu , r) \right] dx^\mu dx^\nu - dr^2 ~,$$ where $x_\mu = (t,x,y,z)$ and Greek indices run over 4D space-time – $\mu , \nu = 0,1,2,3$. The tensor, $h_{\mu \nu}$, gives 4D perturbations around the usual 5D brane world background. Further we fix the gauge so that $h_{\mu \nu}$ is transverse and traceless, $$\partial _\mu h^{\mu} _\nu = h^\mu _\mu =0 ~.$$ Next we separate the tensor perturbation into a 1D and 4D part as $$h_{\mu \nu} (x^\mu, r) = \psi (r) h ^{(4)} _{\mu \nu} (x^\mu )~,$$ with $h ^{(4)} _{\mu \nu} (x^\mu )$ satisfying $$\Box ^{(4)} h ^{(4)} _{\mu \nu} (x^\mu ) = p^2= m^2~.$$ Putting all this together yields the following equation for the 1D part of the perturbations $$\label{pert-1D}
\psi '' - 4 a^2 \psi - m^2 e^{-2ar} \psi =0 ~.$$ In order for gravity to be effectively 4D near the nodes should have a zero mode solution – $m=0$. It is easy to see that it does have a zero mode given by, $$\psi _0 (r) = e^{2ar}~,$$ up to a normalization constant. Equation also has $m
\ne 0$ solutions which are similar to i.e. combinations of second order Bessel functions of the first and second kind, $J_2
, N_2$, [@Rub]. The exchange of the zero mode, $m=0$, between massive particles fixed on the node, will lead to a $1/r$ Newtonian potential while the exchange of the $m \ne 0$ modes will lead to corrections which go as $1/(a^2 r^2)$ [@brane1; @brane2; @brane3; @brane4; @Rub].
The effective 4D Newton’s constant on a particular node can be obtained using the above results. Let us focus on one particular node, $r_m$. The massless gravitational perturbation is $$\label{h-shift}
h_{\mu \nu} = e ^{2 a r} h ^{(4)} _{\mu \nu} (x^\mu ) ~,$$ where $h ^{(4)} _{\mu \nu} (x^\mu )$ is the 4D part. The effective 4D Newton’s constant can be obtained by inserting into the 5D gravitational action and looking at the quadratic part $$\begin{aligned}
\label{grav-action}
S_g &=& \frac{1}{16 \pi G_5} \int _ {r_m - d} ^{r_m +d} \frac{dr}{e^{2 a r}} \int d^4x \left( \partial h \right)^2 \nonumber \\
&=& \frac{1}{16 \pi G_5} \int _{r_m - d} ^{r_m +d} dr ~ e^{2 a r } \int d^4 x \left( \partial h ^{(4)} \right)^2 \\
&=& \frac{e^{2ar_m} \sinh (2 a d)}{16 \pi G_5 a} \int d^4 x \left( \partial h ^{(4)} \right)^2 ~, \nonumber\end{aligned}$$ where $2d$ is the width around the node, $r_m$, within which the matter particles are assumed to be bound. The index $m$ satisfies $m
\le n$ where for $m=n$ one is located at the $r=0$ brane. Using the last expression in the effective 4D Newton’s constant can be written as $$\label{4D-newton}
G_4 = \frac{a G_5}{e^{2ar_m} \sinh (2 a d)} \approx \frac{G_5}{2 ~ d ~e^{2 a r_m}}~.$$ The last expression assumes a small thickness i.e. $d \ll 1$, or $a
d \ll 1$. In the case when $a>0$ one can exponentially suppress the 4D Newton’s constant, $G_4$, relative to the 5D Newton’s constant, $G_5$.
Localization of matter
======================
So far in this paper we have assumed some mechanism for binding matter fields to the nodes of the standing waves. Now we want to give two possible localization mechanisms.
The first mechanism is borrowed from condensed matter physics. It is known that standing electromagnetic waves, so called optical lattices, can provide trapping of various particles by scattering and dipole forces [@Opt; @Opt1]. In [@quadr; @quadr1] localization was demonstrated through quadrupole forces as well. It is also known that the motion of test particles in the field of a gravitational wave is similar to the motion of charged particles in the field of an electromagnetic wave [@Ba-Gr]. Thus standing gravitational waves could also provide confinement of matter via quadrupole forces. As an example let us consider the equations of motion of the system of spinless particles in the quadrupole approximation [@Dix; @Dix1; @Dix2], $$\label{quad}
\frac{Dp^\mu}{ds}= F^\mu_{quad} = -\frac 16
J^{\alpha\beta\gamma\delta}D^\mu R_{\alpha\beta\gamma\delta}~,$$ where $p^\mu$ is the total momentum of the matter field/particle and $J^{\alpha\beta\gamma\delta}$ is the quadrupole moment of the stress-energy tensor for the matter field/particle. The oscillating metric due to gravitational waves should induce a quadrupole moment in the matter fields. If the induced quadrupole moment is out of phase with the gravitational wave the system energy increases in comparison with the resonant case and the fields/particles will feel a quadrupole force, $F^\mu_{quad}$, which ejects them out of the high curvature region i.e. it would localize them at the nodes. This gravitational binding mechanism would be effective for all types of matter fields since gravity couples to all forms of energy-momentum. Since the matter fields are ejected from the high curvature region and toward the nodes the width of the anti-de-Sitter ‘islands’ (i.e. the thickness of the branes) depends on how rapidly the ansatz function $u(t,r) \propto f(r)$ from changes from zero. The width of some anti-de-Sitter ‘island’ will depend in the derivative of $f(r)$ at the node – $f'(r_m)$. The larger $f'(r_m)$ is the smaller the width of the anti-de-Sitter ‘island’. Now $f'(r_m)$ depends on the size of either $A$ or $B$ and as well as $a$, $\omega$ and $r_m$. For some choice of these parameters some nodes might not be phenomenologically acceptable since the brane thickness might be too large. Also in general for the case when $a>0$ there will be fewer phenomenologically acceptable branes since in this situation the nodes are stretched out, as discussed in section $5$, and this tends to decrease $f'(r_m)$ as $r$ increases. On the other hand when $a<0$ there is an increase of $f'(r_m)$ as $r$ increases which leads to more nodes which have a phenomenologically acceptable thickness.
The second mechanism would be to propose some coupling between the ghost-like field, $\sigma$, and matter fields of the form $\sigma A
A$, where $A$ is a scalar, spinor, vector or tensor field. For a normal scalar field such a coupling leads to an attractive force, while for a ghost-like field it leads to repulsion of the matter fields from regions with non-zero $\sigma$. This would force the fields $A$ to congregate at regions were $\sigma$ vanishes, i.e. the nodes of the standing waves. Note that coupling ordinary matter fields to ghost fields is problematic since in general it leads to instabilities. However, as mentioned in the Sec. $3$, our massless ghost-like field can be associated with the geometrical scalar of integrable Weyl models.
Both binding mechanisms have open questions (e.g.“Does the gravitational standing wave induce a quadrupole moment of the matter field stress-energy tensor and if so what is its form?"; “Can one safely couple ordinary matter fields to ghost-like fields?"). A detail analyzes of localization of specific matter fields via these two possible mechanisms will be considered in a forthcoming paper.
Conclusions and discussion
==========================
A simple standing wave solution for 5D spacetime with a ghost-like scalar field plus 5D negative cosmological constant was presented. The requirement that the ghost-like field vanish on the brane quantized the standing wave oscillation frequency in terms of the zeros of the 2$^{nd}$ order Bessel functions. Thus the ghost-like field is not observable on the brane, at $r=0$, nor at any of the nodes, $r_m$, of the standing wave. This is similar to the static brane model [@Koley; @Koley1; @Koley2], where the phantom/ghost field only existed in the bulk.
For the case of increasing (decreasing) warp factors the ghost-like field, $\sigma (t,r)$, and the metric function, $u(t,r)$, vanish at a finite (infinite) number of places in the bulk forming ‘islands’ of 4D anti-de-Sitter space-time. By assuming that the matter fields are bound to these nodes we arrive at a model where each node is an anti-de-Sitter island-universe with different parameters, i.e. an effective 4D Newton’s constant from and an effective 4D cosmological constant from scaled by the value $e^{2 a
r_m}$ at the particular node. One could address the hierarchy problem and the small size of the cosmological constant by taking our universe as one of these Bessel nodes, where the 5D Newton and cosmological constants are appropriately scaled to their effective 4D values. A problem is that the scalings go in oppose directions. For example, with $a>0$ gives a 4D cosmological constant which is exponentially larger than the 5D cosmological constant, while gives a 4D Newton’s constant which is exponentially smaller than the 5D Newton’s constant. For $a<0$ the effective 4D Newton’s constant and the effective 4D cosmological constant have opposite scaling to the $a>0$ case.
There are two distinctly different cases for the standing wave solutions – increasing warp factor ($a>0$) and decreasing warp factor ($a<0$) – corresponding to a finite, or infinite number of anti-de-Sitter ‘islands’ respectively. The finite case could be applied to the generation problem by fixing the model to have three ‘islands’ with different fermion generations bound to different nodes. This is different from brane world models of the generation puzzle like [@gog-sing; @gog-sing1; @gog-sing2], where the generations were associated with different zero modes bound to a single brane. For the infinite branes case one has a simple version of the landscape picture coming from string/M theory, where one has a large number of anti-de-Sitter island-universes each having different parameters e.g. effective 4D gravitational and 4D cosmological constants.
The present models has the same kind of warped geometry as the usual brane models [@brane1; @brane2; @brane3; @brane4]. In distinction from these models the “branes" (i.e. the nodes of $u(r,t)$, excluding the one at $r=0$) of the present standing wave background solution do not have a brane tension but are simply 4D anti-de-Sitter space-times. This is an advantage since for the models of [@brane1; @brane2; @brane3; @brane4] one should explain why the $\delta$-source brane tension is not observed. The disadvantage of the present model is the need for a ghost-like field. The need for the ghost-like field is ameliorated by the fact that it vanishes on the $r=0$ brane and on all the Bessel node anti-de-Sitter ‘islands’.
The issue of instability due to an unusual matter source is also a problem for the original two-brane model which has a negative tension brane. If the negative tension brane is free to oscillate this results in arbitrarily large negative energy modes making the system unstable [@pilo]. A similar problem occurs in the present model if we take our scalar field as an ordinary ghost field. However, if we associate our ghost field with the geometrical scalar of a 5D integrable Weyl model [@Weyl; @Weyl1; @Weyl2; @Weyl3] this alleviates some of the instability problems since the scalar field comes from the metric via . In addition for Weyl models the ghost-like field does not have the standard couplings with the brane energy-momentum and thus does not cause instabilities.
Additionally previous work with standard ghost fields shows – that at least at the classical level – bulk ghost fields may in fact be better at stabilizing brane world models as compared to bulk regular scalar fields. In [@pospelov] bulk ghost fields in 5D are shown to lead to radion stabilization for models with positive tension branes. Essentially the bulk ghost field replaces the negative tension brane. Further it was shown that a bulk scalar field whose sign can vary between regular (positive) and ghost (negative) can be used to model both positive and negative tension branes. Also, there are certain configurations of the bulk ghost fields of reference [@pospelov] which lead to stronger localization of gravity than the original brane models. In references [@maity; @maity1] it is shown that the two brane model of [@brane1; @brane2; @brane3; @brane4] is stabilized by a ghost/tachyon bulk field, while for a regular bulk scalar field the two brane model is unstable. While the stability issue is crucial for all brane world models it has not yet found a complete solution even for the original models of [@brane1; @brane2; @brane3; @brane4]. Here, as in [@Koley; @Koley1; @Koley2; @pospelov] we simply use the ghost scalar to build a brane world model from a 5D standing wave, leaving the generally unresolved question of stability of the solution for a future work. As a final note in [@sen; @sen1; @sen2] it is shown how to construct a consistent field theory with tachyons in the context of D-branes.
Although throughout this paper we have assumed some mechanism for binding matter fields/particles to the anti-de-Sitter ‘islands’ in section $7$ we put forward two possible mechanisms for accomplishing this binding. The first localization mechanism was based on the quadrupole force in which would eject particles from the high curvature regions and thus drive them to the nodes. One might give a heuristic description of this binding mechanism by noting that as one moves away from the nodes where $u(t,r)=0$ that the brane coordinates $x,y,z$ are distorted in an anisotropic way – the $x,y$ coordinates will be stretched/compressed while the $z$ coordinate will be compressed/stretched. This anisotropy causes a force, via , which tends to drive the matter fields toward the nodes. The second mechanism also involved ejecting the matter fields from the anti-nodes and towards the nodes, but in this case the ejection mechanism was accomplished by coupling the matter fields to the ghost-like scalar. This would give rise to a repulsive force which would drive the matter fields toward regions where the ghost-like scalar field was small i.e. toward the nodes. For both mechanisms the thickness of the brane was related to how rapidly the metric and scalar field ansatz functions changed near the nodes i.e. to have a thinner brane one should make $f'(r_m)$ larger. In general the case with $a<0$ would have thinner branes.
[**Acknowledgments:**]{}
M. G. and D. S. are supported by a 2008-2009 Fulbright Scholars Grants. M. G. acknowledges the grant of Georgian National Science Foundation $ST~09.798.4-100$. D. S. would like to thank Vitaly Melnikov for the invitation to work at the Institute of Gravitation and Cosmology at Peoples’ Friendship University of Russia.
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---
abstract: 'In a recent paper \[Phys. Rev. A **82**, 063623 (2010)\] Hallwood *et al.* argued that it is feasible to create large superposition states with strongly interacting bosons in rotating rings. Here we investigate in detail how the superposition states in rotating-ring lattices depend on interaction strength and barrier height. With respect to the latter we find a trade-off between large energy gaps and high cat quality. Most importantly, we go beyond the $\delta$-function approximation for the barrier potential and show that the energy gap decreases exponentially with the number of particles for weak barrier potentials of finite width. These are crucial issues in the design of experiments to realize superposition states.'
author:
- Andreas Nunnenkamp
- Ana Maria Rey
- Keith Burnett
title: Superposition states of ultracold bosons in rotating rings with a realistic potential barrier
---
Ultracold bosons in rotating ring traps are a fascinating subject of research because the particles in this system can form various multi-particle superposition states. Some of these superposition states may play a central role in studies of the quantum-to-classical transition [@Leggett2002] or in applications such as entanglement-enhanced metrology [@Leibfried2004; @Giovannetti2006; @Pezze2009].
Most of the work in this area has focused on the production of so-called NOON states in uniform ring lattices [@Hallwood06a; @Rey07] and superlattices [@Nunnenkamp08]. In these systems there are critical rotation frequencies at which the single-particle spectrum is degenerate so that even weak interactions lead to strong correlations between the particles. The energy gap between ground and excited states for these systems does, as one might expect for multi-particle transitions, decrease exponentially with the number of particles. This does, of course, limit the schemes proposed to generate these quasi-momentum NOON states to operate with a modest number of atoms [@Rey07; @Hallwood06b].
Very recently, the authors of Ref. [@Hallwood2010] showed that for strong interactions the ground state of the Lieb-Liniger model with a $\delta$-function potential barrier is a superposition of states with different total (angular) momenta of the particles. They found that in this regime the energy gap is independent of the number of particles for small barrier heights and proportional to the number of particles for strong barrier heights. They come to the conclusion that the production of superposition states with much larger number of particles will become feasible.
In this paper we study in detail the different superposition states that occur for ultracold bosons in rotating ring lattices with a realistic potential barrier. We show that these superposition states strongly differ in the regimes of weak and strong interaction strength (see also Ref. [@Hallwood2010]) as well as of weak and large potential barriers. Most importantly, we demonstrate that for weak barriers the energy gap decreases exponentially with the number of particles as soon as the finite width of the potential is taken into account. These are important limitations to producing superposition states that must be taken into account in planning experiments.
We focus on systems at a critical rotation frequency where the single-particle spectrum of the uniform ring is doubly degenerate and introduce a potential barrier that breaks translational invariance leading to superposition states of different quasi-momenta. We study systems with much less than one atom per site in order to avoid the superfluid-to-Mott-insulator transition for strong interactions. At low filling factors the particles only feel the quadratic part of the lattice dispersion so that our lattice simulations should be relevant for a continuous loop as studied in Ref. [@Hallwood2010].
We use both exact diagonalization of systems with small numbers of atoms and sites and the Bose-Fermi mapping in the Tonks-Giradeau regime to characterize the many-body ground state. Going beyond Ref. [@Hallwood2010], we calculate the ground-state overlap with three different trail wavefunctions, the spectrum of the single-particle density matrix, and the experimentally accessible correlation functions, such as momentum distribution and momentum noise correlations.
Like Ref. [@Hallwood2010] we find that the superposition states strongly depend on the interaction strength. For weak potential barriers and weak interactions it is a Bose-Einstein condensate, for intermediate strength it is a strongly correlated NOON state, and for strong interactions, where the particles are locked together, it is the superposition of two center-of-mass motional states. In the lattice case we consider that the superpositions also depend on the potential height. We find that they are degraded significantly if the potential barrier is large, which leads to a trade-off between the energy gap and cat quality.
The limit of strong interactions and weak potential barrier is most attractive for generating superpositions with many particles. In this case, the energy gap is independent of the number of particles for a single-site barrier [@Hallwood2010]. However, as soon as we consider a physical potential with finite width, the energy gap decreases exponentially with increasing number of particles. Since any real potential will have a finite width this will give a physical limit to the number of particles in the superposition state.
*Hamiltonian.* We consider a system of $N$ ultracold bosons with mass $M$ confined in a 1D ring lattice of $L$ sites with lattice constant $d$. The ring is rotated in its plane with angular velocity $\Omega$. In the rotating frame the Hamiltonian is [@Bhat2006; @Rey07] $${\hat{H}}
= \sum_{j=1}^L \left[ -J \left({ {\rm e}^{{\rm i} \theta}} {\hat{a}^\dagger}_{j+1} {\hat{a}}_j + \mathrm{H.c.} \right) + \frac{U}{2} \hat{n}_j(\hat{n}_j-1) + V_j \hat{n}_j \right]
\label{Ham}$$ where $\hat{n}_j=\hat{a}_j^{\dagger}\hat{a}_{j}$ and $\hat{a}_{j}$ are the number and bosonic annihilation operators of a particle at site $j$, $\theta = M \Omega L d^2/h$ is the phase twist induced by rotation, $J$ is the hopping energy between nearest-neighbor sites, $U$ the on-site interaction energy, and $V_j$ describes the potential barrier at site $j$.
To understand the effect of rotation on the atoms in the ring lattice, we write the many-body Hamiltonian (\[Ham\]) in terms of the quasi-momentum operators $\hat{b}_q = \frac{1}{\sqrt{L}} \sum_{j=1}^{L} \hat{a}_j e^{-2\pi i q j/L}$, where $2\pi q / d L$ is the quasi-momentum and $q=0,\dots,L-1$ an integer. In this basis the Hamiltonian (\[Ham\]) has the form [@Hallwood06a; @Rey07] $$\begin{aligned}
\hat{H} =
&\sum_{q=0}^{L-1} E_q\hat{b}_q^{\dagger}\hat{b}_{q}+\frac{U}{2L}
\sum_{q,s,l=0}^{L-1}\hat{b}_q^{\dagger} \hat{b}_s^{\dagger}
\hat{b}_l\hat{b}_{[q+s-l] \, \textrm{mod} \, L} \nonumber \\
&+\sum_{\{q,q'\}=0}^{L-1} V_{qq'} \hat{b}_{q'}^{\dagger}\hat{b}_{q}
\label{Ham_mom}\end{aligned}$$ where $E_q=-2J\cos(2\pi q/L-\theta)$ are the single-particle energies, $V_{qq'}$ is the Fourier transform of the single-particle potential $V_{qq'} = \frac{1}{L} \sum_j V_j e^{2\pi i(q-q')j/L}$, and the modulus is taken because in collision processes the quasi-momentum is conserved up to an integer multiple of the reciprocal lattice vector $2\pi/d$, i.e. modulo Umklapp processes. In absence of a potential barrier, i.e. $V_j=0$, the single-particle spectrum is twofold degenerate for $\theta = \pi/L$ which we will refer to as the critical rotation frequency or critical phase twist.
{width="0.66\columnwidth"} {width="0.66\columnwidth"} {width="0.66\columnwidth"} {width="0.66\columnwidth"} {width="0.66\columnwidth"} {width="0.66\columnwidth"}
*Ground state at the critical phase twist.* In the following we will focus on the ground state at the critical phase twist for a small single-site potential barrier ($V_j = 0$ for $j \not= 0$, i.e. $V_{qq'}=V_0 \ll J$). Note that a single-site barrier leads to constant momentum transfer in the first Brillouin zone. Similar results were found in Ref. [@Hallwood2010] which we will extend here.
In Fig. \[fig1\] we show our results for the many-body spectrum and detailed characterization of the ground-state wave function using exact diagonalization of a small system for $N=3$ atoms on $L=5$ lattice sites. We can clearly distinguish three regimes: weak, intermediate, and strong interactions.
In the non-interacting system, $U = 0$, all bosons occupy the lowest-energy single-particle state. In the presence of the potential barrier, $V_0 \not = 0$, translation symmetry is broken and quasi-momentum ceases to be good quantum number. For a weak barrier, $V_0 \ll J$, the ground state of the single-particle Hamiltonian is an equal superposition of the quasi-momentum states ${| q=0 \rangle}$ and ${| q=1 \rangle}$, and the many-body ground state is $$| \psi_0 \rangle = \left( \frac{ {\hat{b}^\dagger}_0 - {\hat{b}^\dagger}_1}{\sqrt{2}} \right)^N {| 0 \rangle}
\label{state1}$$ where ${| 0 \rangle}$ is the many-body vacuum. This is corroborated by our exact-diagonalization results in the weakly-interacting regime: the momentum distribution is $\langle \hat{n}_q \rangle = \langle {\hat{b}^\dagger}_q {\hat{b}}_q \rangle = N/2$ for $q=\{0,1\}$ and $\langle \hat{n}_q \rangle = 0$ otherwise. The distribution of total momentum $P(K)$, i.e. the probability that the many-body system has total quasi-momentum $2\pi K/dL$ is binomial. It can be calculated from $P(K) = \sum_n {| K,n \rangle} {\langle K,n|}$, where ${| K,n \rangle} {\langle K,n|}$ is the projector on the Fock state with $K$ total quasi-momentum quantum number and $n$ is a shorthand for the remaining quantum numbers [@Hallwood2010]. Finally, the spectrum of the single-particle density-matrix (SPDM) $\langle {\hat{a}^\dagger}_i {\hat{a}}_j \rangle$ has one eigenvalue of size $N$ which clearly signals condensation.
As soon as the interactions overcome the energy splitting induced by the potential barrier, the near degenerate many-body states are strongly mixed leading to correlations between the atoms. We discussed this mechanism in detail in the context of NOON-state production in rotating ring lattices and superlattices [@Nunnenkamp2008] as well as quantum vortex nucleation [@Nunnenkamp2010]. Neglecting the effect of the weak potential barrier, the non-interacting many-body spectrum at the critical phase twist is $N+1$-fold degenerate. Most of the degeneracy is lifted by the interactions in first order of perturbation theory, but the states $({\hat{b}^\dagger}_0)^N {| 0 \rangle}$ and $({\hat{b}^\dagger}_1)^N {| 0 \rangle}$ remain degenerate. In the presence of the potential barrier these two states are coupled at some higher order of perturbation theory (even if the system is non-commensurate), the degeneracy is lifted, and the ground state is the quasi-momentum NOON-state [@Hallwood2010] $$| \psi_1 \rangle = \frac{({\hat{b}^\dagger}_0)^N - ({\hat{b}^\dagger}_1)^N}{\sqrt{2}} {| 0 \rangle}.
\label{state2}$$ While the momentum distribution is almost unchanged with respect to the non-interacting case, the momentum noise correlations $\langle \hat{n}_q \hat{n}_{q'} \rangle - \langle \hat{n}_q \rangle \langle\hat{n}_{q'} \rangle$ pick up the large fluctuations in the NOON-state (\[state2\]). The distribution of total quasi-momentum $P(K)$ is bimodal, and the SPDM has two large eigenvalues of size $N/2$ indicating fragmentation of the condensate. Finally, the overlap with the many-body quasi-momentum basis states unambiguously proofs that the ground state is indeed the quasi-momentum NOON-state (\[state2\]).
In passing we note that the presence of NOON-states can be observed in time-of-flight expansion after inducing many-body oscillations with a quench in the rotation frequency [@Nunnenkamp08]. Their frequency is given by twice the energy gap which in the regime of intermediate interactions is exponentially small in the number of particles [@Hallwood06a; @Rey07].
Let us now discuss the strongly-interacting regime. In absence of the potential barrier, i.e. $V_0 = 0$, quasi-momentum is a good quantum number. Strong interactions ($U/J \rightarrow \infty$) lead to fermionization of the bosons within each subspace of total quasi-momentum $K$. In this regime the many-body energy spectrum is equal to the single-particle spectrum (with (anti-)periodic boundary conditions for (even) odd $N$ [@Lieb1961]) which is degenerate at the critical phase twist. The potential barrier then breaks translation symmetry and couples the two degenerate ground states ${| \psi_\infty^{K=0} \rangle}$ and ${| \psi_\infty^{K=N} \rangle}$, which are the ground states of the quasi-momentum subspaces with $K=0$ and $K=N$. We thus call the ground-state wave function a *center-of-mass superposition* $${| \psi_\infty \rangle} = \frac{{| \psi_\infty^{K=0} \rangle} - {| \psi_\infty^{K=N} \rangle}}{\sqrt{2}}.
\label{state3}$$ Compared to the regime of intermediate interactions the two largest eigenvalues of the SPDM are smaller in the strongly-interacting limit pointing to the fact that the center-of-mass superposition (\[state3\]) is only a partially-fragmented state. Since many quasi-momentum basis states contribute to (\[state3\]) the overlap with a quasi-momentum NOON-state (\[state2\]) is also smaller.
Please note that the distribution of total quasi-momentum $P(K)$ is not sensitive to the difference between the states (\[state2\]) and (\[state3\]). This is why going beyond Ref. [@Hallwood2010] with a more detailed analysis in terms of SPDM and overlap is important.
![(Color online) Energy gap $\Delta E$ (solid) and cat quality $Q = 4 P(0) P(N)$ (dashed) versus single-site barrier height $V_0$ for $L = 9$ as well as $N = 2$ (blue), $N = 3$ (green) and $N = 4$ (red) at the critical phase twist in the strongly-interacting limit $U / J \rightarrow \infty$. Insets show distribution of total quasi-momentum $P(K)$ for $V_0/J=10^{-4}$ and $V_0/J=10^{+4}$ for $N = 4$.[]{data-label="fig2"}](fig2.eps){width="0.85\columnwidth"}
*Effect of large barrier heights.* So far we have focused on the case of a small single-site potential barrier $V_0 \ll J$. Let us now study how larger barriers affect the system.
In Fig. \[fig2\] we plot the energy gap $\Delta E$ and the cat quality $Q = 4 P(0) P(N)$ [@Hallwood2010] as a function of the barrier height $V_0$. We find that the gap is small for $V_0 \ll J$, increases sharply around $V_0 = J$, and levels off for $V_0 \gg J$. As Ref. [@Hallwood2010] we see that the energy gap is independent of the number of particles in the small barrier limit and increases with the number of particles in the large barrier limit. However, particularly interesting is the dependence of the cat quality $Q$ and the distribution of total quasi-momentum $P(K)$ as the barrier height increases. Fig. \[fig2\] and its insets clearly show that the properties of the superposition states are degraded if the barrier height is raised to increase the energy gap $\Delta E$. This effect becomes more severe as the number of particles increases.
While we study a lattice system which is different from the continuum model considered in Ref. [@Hallwood2010], we expect that our results qualitatively carry over to the continuum problem as they are obtained at densities well below unit filling.
![(Color online) Energy gap $\Delta E$ versus particle number $N$ for $L = 50$, $V_0/J = 0.01$, $b = \xi/d = 0$ (blue), $b = \xi/d = 2/3$ (green), $b = \xi/d = 4/3$ (red), and $b = \xi/d = 2$ (cyan). Points are the exact single-particle spectrum and lines from perturbative expression (\[gap\]). The inset shows the barrier potential $V_j$ for $L = 50$, $V_0/J = 0.01$, and $b = \xi/d = 2$.[]{data-label="fig3"}](fig3.eps){width="0.8\columnwidth"}
*Number scaling for finite-width potential barrier.* In future experiments the barrier will most likely be provided by a blue-detuned laser beam with Gaussian beam waist. That is why we now generalize the case of a single-site potential barrier and consider a Gaussian potential barrier of finite width $\xi$, i.e. $V_j = {\cal N}^{-1} V_0 e^{-d^2j^2/2\xi^2}$ with the normalization ${\cal N} = \sum_{j=1}^L e^{-d^2j^2/2\xi^2}$. The matrix elements ${V}_{qq'}$ are given by the discrete Fourier transform of the barrier potential $V_j$ which for $d \ll \xi$ becomes the continuous Fourier transform $V_{qq'} = V_0 e^{-2 \pi^2 (\xi/dL)^2 (q-q')^2}/L$ while for $\xi \rightarrow 0$ we recover the single-site barrier limit $V_{qq'} = V_0/L$.
We can get a perturbative expression for the energy gap using the Bose-Fermi mapping [@Lieb1961] and degenerate perturbation theory first order in $V_0$. The energy gap $\Delta E$ between the ground and first excited state in the Tonks limit for $N$ particles is $$\Delta E = 2 \frac{V_0}{L} e^{-2 \pi^2 (\xi/dL)^2 N^2}.
\label{gap}$$ We note that in this weak barrier limit the exponential scaling is independent of the barrier height.
In Fig. \[fig3\] we plot the energy gaps $\Delta E$ which open due to the potential barrier as a function of number of particles $N$ for various barrier widths $\xi/d$. In the small barrier $V_0/J = 0.01$ the accuracy of the perturbative expression (\[gap\]) is excellent. In contrast to the single-site potential barrier, i.e. $\xi/d \ll 1/L$, a potential barrier with finite width $\xi$ leads to exponentially small energy gaps for increasing number of particles $N$. The physical reason is that we need increasingly large momentum transfer to couple the center-of-mass momentum states. Since the matrix elements of a finite-width potential barrier decrease exponentially for increasing momentum transfer, we recover the exponential scaling of the energy gap $\Delta E$ similar of the weakly-interacting regime.
The NIST group [@Ramanathan2011] recently carried out experiments in a toroidal trap with radius of $20 \mu\textrm{m}$ and a $4.3 \mu\textrm{m}$ ($1/e^2$ radius) repulsive barrier. This corresponds to $\xi/dL \approx 1/60$ and a suppression of the gap (\[gap\]) by $e^{-(N/10)^2/2}$. This will severely limit cat state production beyond few tens of particles in the weak barrier regime where the cat quality is good.
*Summary.* In conclusion, we have shown that a weak potential barrier in a rotating ring lattice loaded with ultracold bosons at low filling fraction supports several qualitatively very different superposition states of different total quasi-momentum: a condensate for weak, a NOON-state for intermediate, and a center-of-mass superposition for strong interactions. Raising the potential barrier will increase the energy gap but degrade the superposition states. The energy gap in the strongly-interacting limit is only independent of the number of particles as long as one can neglect the width of the potential barrier. Otherwise one recovers the exponential scaling with increasing number of particles. These are crucial issues in the production of superposition states in rotating atomic rings. They severely limit the size of superposition states in any realizable experimental system.
*Acknowledgements.* AN thanks Steven M. Girvin for a careful reading of the manuscript and acknowledges support from NSF under grant DMR-1004406. AMR acknowledges support from NSF, NIST and a grant from the ARO with funding from the DARPA-OLE.
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abstract: 'We classify nets of conics of rank one in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent problem of classifying the orbits of planes in ${\mathrm{PG}}(5,q)$ which meet the quadric Veronesean in at least one point, under the action of ${\mathrm{PGL}}(3,q) {\leqslant}{\mathrm{PGL}}(6,q)$ (for $q$ odd). Our results complete a partial classification of nets of conics of rank one obtained by A. H. Wilson in the article “The canonical types of nets of modular conics”, [*American Journal of Mathematics*]{} [**36**]{} (1914) 187–210.'
address:
- 'Michel Lavrauw, Sabanc[i]{} University, Istanbul, Turkey; Email: <span style="font-variant:small-caps;">mlavrauw@sabanciuniv.edu</span>'
- 'Tomasz Popiel, Queen Mary University of London, UK; Email: <span style="font-variant:small-caps;">t.popiel@qmul.ac.uk</span>'
- 'John Sheekey, University College Dublin, Ireland; Email: <span style="font-variant:small-caps;">john.sheekey@ucd.ie</span>'
author:
- 'Michel Lavrauw, Tomasz Popiel, John Sheekey'
title: 'Nets of conics of rank one in ${\mathrm{PG}}(2,q)$, $q$ odd\'
---
Introduction
============
The space of forms of degree $d$ on an $n$-dimensional projective space ${\mathrm{PG}}(V)$ comprise a vector space $W$ of dimension ${n+d}\choose{d}$. Subspaces of the projective space ${\mathrm{PG}}(W)$ are called [*linear systems of hypersurfaces of degree $d$*]{}. The problem of classifying linear systems consists of determining the orbits of such subspaces under the induced action of the projectivity group ${\mathrm{PGL}}(V)$ on ${\mathrm{PG}}(W)$. One-dimensional linear systems are called a [*pencils*]{} and two-dimensional linear systems are called [*nets*]{}. In this paper we are concerned with linear systems of conics, namely the case $d=n=2$. Pencils of conics over $\mathbb{C}$ and $\mathbb{R}$ were classified by Jordan [@Jordan1906; @Jordan1907] in 1906–1907, and nets of conics over these fields were treated by Wall [@Wall1977]. For an elementary exposition of the more general case $d=2$ (namely, pencils of quadrics), we refer the reader to chapters 9 and 11 of [@Casas-Alvero2010].
Here we are concerned with linear systems of conics over [*finite*]{} fields. Compared with working over $\mathbb{C}$, complications arise when working over a finite field $\mathbb{F}_q$ (of order $q$) because $\mathbb{F}_q$ is not algebraically closed, resulting in a number of extra orbits which sometimes turn out to be difficult to classify. Pencils of conics over $\mathbb{F}_q$, $q$ odd, were classified by Dickson [@Dickson1908]; an incomplete classification for $q$ even was obtained by Campbell [@Campbell1927]. Our aim is to classify nets of conics over $\mathbb{F}_q$, $q$ odd. This problem was addressed by Wilson [@Wilson1914] using a purely algebraic approach, where the (in)equivalence of nets is studied by means of explicit coordinate transformations. However, as explained in Section \[sec:Wilson\], Wilson’s classification was incomplete. We take a geometric approach, based on the observation that linear systems of conics correspond to subspaces of ${\mathrm{PG}}(5,q)$. We classify the nets of [*rank one*]{}, namely those containing a repeated line, which correspond to planes in ${\mathrm{PG}}(5,q)$ intersecting the quadric Veronesean in at least one point. Our main result is Theorem \[mainthm\]. Here points of ${\mathrm{PG}}(5,q)$ are represented by symmetric $3 \times 3$ matrices, with the quadric Veronesean $\mathcal{V}(\mathbb{F}_q)$ defined by setting all $2 \times 2$ minors equal to zero (see Section \[sec:prelims\]).
\[mainthm\] Let $q$ be a power of an odd prime. There are 15 orbits of planes in $\operatorname{PG}(5,q)$ that meet the quadric Veronesean in at least one point, under the action of ${\mathrm{PGL}}(3,q) {\leqslant}{\mathrm{PGL}}(6,q)$ defined in Section \[ss:PGL3q-action\]. Representatives of these orbits are listed in Table \[table:main\].
As a corollary, we complete Wilson’s classification of nets of rank one, rectifying some of the statements made in his paper (see Section \[sec:Wilson\] for the details).
\[maincor\] There are 15 orbits of nets of conics of rank one in $\operatorname{PG}(2,q)$, $q$ odd.
Our geometric approach provides insight which may be of use for other classification problems and is expected to have further applications in finite geometry. In particular, we are able to deduce further details about the plane orbits, such as the point-orbit distributions (see Definition \[def:dist\] and Table \[table:pt-orbit-dist\]), which serve as important invariants. Data of this kind previously obtained by the first and second authors [@LaPo2017] for [*lines*]{} in $\operatorname{PG}(5,q)$ are used in the proof of Theorem \[mainthm\].
The paper is organised as follows. In Section \[sec:prelims\] we give the necessary preliminaries for the proof of Theorem \[mainthm\], in order to make the paper reasonably self-contained. The problem of classifying nets of conics in the classical projective plane ${\mathrm{PG}}(2,q)$, $q$ odd, is turned into the problem of classifying orbits of planes in 5-dimensional projective space under the action of a copy of the projectivity group ${\mathrm{PGL}}(3,q)$ viewed as a subgroup of ${\mathrm{PGL}}(6,q)$. The classification of points, lines, solids and hyperplanes in ${\mathrm{PG}}(5,q)$ is recalled in Section \[sec:pts\_lines\_sols\_hyps\]. Section \[sec:planes\] introduces some terminology, notation and lemmas used throughout the paper. The classification of planes in ${\mathrm{PG}}(5,q)$ (Theorem \[mainthm\]) is then proved in Sections \[sec:three\_rank\_one\_points\]–\[sec:final\], with the proof divided into several parts. In Section \[sec:three\_rank\_one\_points\] we classify the planes spanned by three points of the quadric Veronesean, giving just two orbits, labelled $\Sigma_1$ and $\Sigma_2$. In Section \[sec:two\_rank\_one\_points\] we classify the planes meeting the quadric Veronesean in exactly two points. There are three such orbits: $\Sigma_3$, $\Sigma_4$ and $\Sigma_5$. The bulk of the classification is done in Section \[sec:one\_rank\_one\_point\_and\_spanned\], which deals with the planes that are spanned by points of the secant variety of the quadric Veronesean and meet the quadric Veronesean in exactly one point. This leads to: eight further orbits $\Sigma_6, \ldots,\Sigma_{13}$ whose existence is independent of the characteristic of the field (as long as it is odd), one orbit $\Sigma_{14}$ which only appears in characteristic $\neq 3$, and one orbit $\Sigma_{14}'$ which only appears in characteristic $3$. In Section \[sec:final\], we show that there is exactly one remaining orbit, $\Sigma_{15}$, consisting of planes that meet the Veronesean but are not spanned by points in the secant variety of the Veronesean. Finally, in Section \[sec:Wilson\], we compare our classification with that of Wilson [@Wilson1914] and deduce Corollary \[maincor\].
[lll]{} Orbit & Representative & Conditions\
$\Sigma_1$ & $\left[ \begin{matrix} \alpha & \gamma & \cdot \\ \gamma & \beta & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]$ &\
$\Sigma_2$ & $\left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & \beta & \cdot \\ \cdot & \cdot & \gamma \end{matrix} \right]$ &\
$\Sigma_3$ & $\left[ \begin{matrix} \alpha & \cdot & \gamma \\ \cdot & \beta & \cdot \\ \gamma & \cdot & \cdot \end{matrix} \right]$&\
$\Sigma_4$ & $\left[ \begin{matrix} \alpha & \cdot & \gamma \\ \cdot & \beta & \gamma \\ \gamma & \gamma & \cdot \end{matrix} \right]$ &\
$\Sigma_5$ & $\left[ \begin{matrix} \alpha & \cdot & \gamma \\ \cdot & \beta & \gamma \\ \gamma & \gamma & \gamma \end{matrix} \right]$ &\
$\Sigma_6$ & $\left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \varepsilon\alpha & \cdot \\ \cdot & \cdot & \gamma \end{matrix} \right]$ & $\varepsilon \in \boxtimes$\
$\Sigma_7$ & $\left[ \begin{matrix} \alpha & \beta & \gamma \\ \beta & \cdot & \cdot \\ \gamma & \cdot & \cdot \end{matrix} \right]$ &\
$\Sigma_8$ & $\left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \cdot & \gamma \\ \cdot & \gamma & \cdot \end{matrix} \right]$ &\
[lll]{} Orbit & Representative & Conditions\
$\Sigma_9$ & $\left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \gamma & \cdot \\ \cdot & \cdot & -\gamma \end{matrix} \right]$ &\
$\Sigma_{10}$ & $\left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \gamma & \cdot \\ \cdot & \cdot & -\varepsilon\gamma \end{matrix} \right]$ & $\varepsilon \in \boxtimes$\
$\Sigma_{11}$ & $\left[ \begin{matrix} \cdot & \beta & \gamma \\ \beta & \alpha & \alpha \\ \gamma & \alpha & \alpha+\gamma \end{matrix} \right]$ &\
$\Sigma_{12}$ & $\left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \gamma & \beta \\ \cdot & \beta & \gamma \end{matrix} \right]$ &\
$\Sigma_{13}$ & $\left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \gamma & \beta \\ \cdot & \beta & \varepsilon\gamma \end{matrix} \right]$ & $\varepsilon \in \boxtimes$\
$\Sigma_{14}$ & $\left[ \begin{matrix} \alpha& \beta & \cdot \\ \beta & c\gamma & \beta-\gamma \\ \cdot & \beta-\gamma & \gamma \end{matrix} \right]$ & $q \not \equiv 0 \pmod 3$, ($\dagger$)\
$\Sigma_{14}'$ & $\left[ \begin{matrix} \alpha+\gamma & \gamma & \gamma \\ \gamma & \beta+\gamma & \gamma \\ \gamma & \gamma & -\beta \end{matrix} \right]$ & $q \equiv 0 \pmod 3$\
$\Sigma_{15}$ & $\left[ \begin{matrix} \alpha & \beta & \gamma \\ \beta & \gamma & \cdot \\ \gamma & \cdot & \cdot \end{matrix} \right]$ &\
Preliminaries {#sec:prelims}
=============
In this section we review some of the theory used in the proof of Theorem \[mainthm\]. Most of it is well known and can be extracted from standard textbooks on projective and algebraic geometry, for example [@Casas-Alvero2010] and [@Harris]. Some parts are from [@LaPo2017]. For a survey of properties of Veronese varieties over fields with non-zero characteristic, we refer the reader to [@Havlicek].
By ${\mathbb{F}}_q$ we denote the finite field of order $q$, and we assume throughout that $q$ is odd. A [*form*]{} on a vector space $V$ (or the projective space ${\mathrm{PG}}(V)$) is a homogeneous polynomial in the polynomial ring over the coefficient field of $V$ in $\dim V$ variables. The zero locus in ${\mathrm{PG}}(V)$ of a form $f$ on ${\mathrm{PG}}(V)$ is denoted by ${\mathcal{Z}}(f)$.
Nets of conics
--------------
A ternary quadratic form $f$ on ${\mathbb{F}}_q^3$ defines a [*conic*]{} ${\mathcal{C}}={\mathcal{Z}}(f)$ in ${\mathrm{PG}}(2,q)$. Each two distinct conics ${\mathcal{Z}}(f_1)$, ${\mathcal{Z}}(f_2)$ define the [*pencil of conics*]{} $$\{{\mathcal{Z}}(af_1+bf_2):a,b\in {\mathbb{F}}_q, (a,b)\neq (0,0)\}.$$ Similarly, a [*net of conics*]{} ${\mathcal{N}}$ is defined by three conics ${\mathcal{C}}_i={\mathcal{Z}}(f_i)$ ($i=1,2,3$) in ${\mathrm{PG}}(2,q)$, not contained in a pencil: $${\mathcal{N}}=\{{\mathcal{Z}}(af_1+bf_2+cf_3):a,b,c\in {\mathbb{F}}_q, (a,b,c)\neq (0,0,0)\}.$$ Given such a net of conics, one can consider $$\label{eqn:qform}
xf_1+yf_2+zf_3=a_{00}(x,y,z)X_0^2+a_{01}(x,y,z)X_0X_1+\dots + a_{22}(x,y,z)X_2^2$$ as a quadratic form whose coefficients are linear forms in $x,y,z$. For each $a,b,c\in {\mathbb{F}}_q$, not all zero, we obtain a conic $${\mathcal{N}}(a,b,c)={\mathcal{Z}}(af_1+bf_2+cf_3).$$ Since $q$ is odd we can consider the matrix $A_{\mathcal{N}}$ of the bilinear form associated to the quadratic form (\[eqn:qform\]). We define the [*discriminant*]{} of the net ${\mathcal{N}}$ as $$\Delta_{\mathcal{N}}=\det (A_{\mathcal{N}}).$$ The discriminant $\Delta_{\mathcal{N}}$ defines a cubic curve ${\mathcal{Z}}(\Delta_{\mathcal{N}})$ in the plane ${\mathrm{PG}}(2,q)$.
The conic ${\mathcal{N}}(a,b,c)$ is singular if and only if $(a,b,c)$ lies on the cubic ${\mathcal{Z}}(\Delta_{\mathcal{N}})$.
Since $q$ is odd, this follows from the fact that the matrix of the linear system obtained by setting the three partial derivatives of ${\mathcal{N}}(a,b,c)$ evaluated at a point belonging to ${\mathcal{N}}(a,b,c)$ equal to zero has determinant equal to $\Delta_{\mathcal{N}}$ evaluated at $(a,b,c)$.
A net has [*rank one*]{} if it contains a repeated line; [*rank two*]{} if it contains no repeated lines but contains a conic which is not absolutely irreducible; and [*rank three*]{} if every conic in the net is absolutely irreducible.
The quadric Veronesean {#ss:V}
----------------------
We represent points $y=(y_0,y_1,y_2,y_3,y_4,y_5,y_6)$ of ${\mathrm{PG}}(5,q)$ by symmetric matrices $$\label{eq:symmetricMatrix}
M_y= \left[
\begin{matrix} y_0 & y_1 & y_2 \\ y_1 & y_3 & y_4\\ y_2 & y_4 & y_5\\\end{matrix}
\right].$$ The Veronese surface ${\mathcal{V}}({\mathbb{F}}_q)$ in ${\mathrm{PG}}(5,q)$ is defined by setting the $2\times 2$ minors of the above matrix equal to zero, and we have the corresponding Veronese map from ${\mathrm{PG}}(2,q)$ to ${\mathcal{V}}({\mathbb{F}}_q) \subset {\mathrm{PG}}(5,q)$: $$\nu:(x_0,x_1,x_2)\mapsto (x_0^2,x_0x_1,x_0x_2,x_1^2,x_1x_2,x_2^2).$$ The [*rank*]{} of a point in ${\mathrm{PG}}(5,q)$ is defined to be the rank of the matrix $M_y$, and we denote by ${\mathcal{P}}_i$ the set of points of rank $i$ for $i=1,2,3$. For convenience, we sometimes denote the rank of a point $x$ by $\operatorname{rank}(x)$. The points contained in ${\mathcal{V}}({\mathbb{F}}_q)$ are points of rank $1$, and the points of rank $2$ are those contained in the secant variety of ${\mathcal{V}}({\mathbb{F}}_q)$.
Let us also partition the set of hyperplanes of ${\mathrm{PG}}(5,q)$ into the following subsets, depending on their intersection with ${\mathcal{V}}({\mathbb{F}}_q)$:
- ${\mathcal{H}}_1$ is the set of hyperplanes intersecting ${\mathcal{V}}({\mathbb{F}}_q)$ in a conic,
- ${\mathcal{H}}_3$ is the set of hyperplanes intersecting ${\mathcal{V}}({\mathbb{F}}_q)$ in a normal rational curve,
- ${\mathcal{H}}_2$ is the set of hyperplanes not contained in ${\mathcal{H}}_1\cup {\mathcal{H}}_3$.
Let $\delta$ denote the map from the set of conics in ${\mathrm{PG}}(2,q)$ to the set of hyperplanes in ${\mathrm{PG}}(5,q)$ which takes the conic ${\mathcal{C}}={\mathcal{Z}}(f)$ with $f(X_0,X_1,X_2)=\sum_{i{\leqslant}j}a_{ij}X_iX_j$ to $$\delta({\mathcal{C}})=H[a_{00},a_{01},a_{02},a_{11},a_{12},a_{22}],$$ where $H[a_0,\ldots,a_5]$ denotes the hyperplane with equation $a_0Y_0+\ldots +a_5Y_5=0$.
\[nu-delta-lemma\] The maps $\nu$ and $\delta$ satisfy the following properties, where ${\mathcal{C}}$ is a conic in ${\mathrm{PG}}(2,q)$.
- A point $x$ in ${\mathrm{PG}}(2,q)$ belongs to ${\mathcal{C}}$ if and only if $\nu(x)\in \delta({\mathcal{C}})$.
- ${\mathcal{C}}$ is a repeated line if and only if $\delta({\mathcal{C}})\in {\mathcal{H}}_1$.
- ${\mathcal{C}}$ is a point if and only if $\delta({\mathcal{C}})\in {\mathcal{H}}_2$ and $\delta({\mathcal{C}})$ intersects ${\mathcal{V}}({\mathbb{F}}_q)$ in a point.
- ${\mathcal{C}}$ is a union of two lines if and only if $\delta({\mathcal{C}})\in {\mathcal{H}}_2$ and $\delta({\mathcal{C}})$ intersects ${\mathcal{V}}({\mathbb{F}}_q)$ in two conics.
- ${\mathcal{C}}$ is non-degenerate if and only if $\delta({\mathcal{C}})\in {\mathcal{H}}_3$.
Furthermore, the image of a line $\ell$ in ${\mathrm{PG}}(2,q)$ under the Veronese map is a conic in ${\mathcal{V}}({\mathbb{F}}_q)$. A plane in ${\mathrm{PG}}(5,q)$ intersecting ${\mathcal{V}}({\mathbb{F}}_q)$ in a conic is called a [*conic plane*]{}. Each conic plane is equal to $\langle \nu(\ell)\rangle$ for some line $\ell$ in ${\mathrm{PG}}(2,q)$. Each two points $x,y \in {\mathcal{V}}({\mathbb{F}}_q)$ lie on a unique conic ${\mathcal{C}}(x,y) \subset {\mathcal{V}}({\mathbb{F}}_q)$ given by $$\label{eqn:C(x,y)}
{\mathcal{C}}(x,y)=\nu(\langle \nu^{-1}(x),\nu^{-1}(y)\rangle).$$ Each rank-$2$ point $z \in \langle {\mathcal{V}}({\mathbb{F}}_q)\rangle$ lies in a unique conic plane $\langle {\mathcal{C}}_z \rangle$. If $z$ is on the secant $\langle x,y\rangle$ then $${\mathcal{C}}_z = {\mathcal{C}}(x,y).$$
We also extend the definition of $\delta$ from the set of conics to the sets of pencils and nets of conics as follows. Given any set $\mathcal S$ of conics in ${\mathrm{PG}}(2,q)$, define $$\delta ({\mathcal{S}})=\cap_{{\mathcal{C}}\in {\mathcal{S}}} \delta({\mathcal{C}}).$$ In this way we obtain the following one-to-one correspondences.
If ${\mathcal{P}}$ is a pencil (respectively, net) of conics in ${\mathrm{PG}}(2,q)$ then $\delta({\mathcal{P}})$ is a solid (respectively, plane) in ${\mathrm{PG}}(5,q)$, and conversely.
The dual map $\delta^*$ of $\delta$ from the set of conics in ${\mathrm{PG}}(2,q)$ to the set of points in ${\mathrm{PG}}(5,q)$ is given by $$\delta^*:{\mathcal{Z}}\Big(\sum_{i{\leqslant}j}a_{ij}X_iX_j\Big)\mapsto
(a_{00},a_{01},a_{02},a_{11},a_{12},a_{22}).$$ If ${\mathcal{P}}$ is a pencil and ${\mathcal{N}}$ is a net of conics in ${\mathrm{PG}}(2,q)$, then $\delta^*({\mathcal{P}})$ is a line and $\delta^*({\mathcal{N}})$ is a plane in ${\mathrm{PG}}(5,q)$.
The action of ${\mathrm{PGL}}(3,q)$ on ${\mathrm{PG}}(5,q)$ {#ss:PGL3q-action}
-----------------------------------------------------------
Recall that we represent points in ${\mathrm{PG}}(5,q)$ by symmetric $3 \times 3$ matrices as per equation (modulo scalars). The action of the group ${\mathrm{PGL}}(3,q)$ on the points of ${\mathrm{PG}}(5,q)$ referred to in Theorem \[mainthm\] is defined as follows.
If $\varphi_A \in {\mathrm{PGL}}(3,q)$ is induced by $A\in {\mathrm{GL}}(3,q)$ then we define $\alpha(\varphi_A) \in {\mathrm{PGL}}(6,q)$ by $$\alpha(\varphi_A):y\mapsto z \quad \text{where} \quad M_z=AM_yA^T.$$ We write $$K:=\alpha({\mathrm{PGL}}(3,q)){\leqslant}{\mathrm{PGL}}(6,q).$$ This also defines an action of ${\mathrm{PGL}}(3,q)$ on subspaces of ${\mathrm{PG}}(5,q)$. The following observation is now readily deduced.
\[prop:nets\_vs\_planes\] The classification of nets of conics in ${\mathrm{PG}}(2,q)$ up to coordinate transformations is equivalent to the classification of the $K$-orbits of planes in ${\mathrm{PG}}(5,q)$.
Properties of the non-degenerate conic in ${\mathrm{PG}}(2,q)$, $q$ odd {#subsec:conic_properties}
-----------------------------------------------------------------------
Let ${\mathcal{C}}$ be a non-degenerate conic in ${\mathrm{PG}}(2,q)$, $q$ odd. Its projective stabiliser $G {\leqslant}{\mathrm{PGL}}(3,q)$ is isomorphic to ${\mathrm{PGL}}(2,q)$. This can be seen using the Veronese map from ${\mathrm{PG}}(1,q)$ to ${\mathrm{PG}}(2,q)$, and it implies that
1. $G$ acts sharply 3-transitively on the points of ${\mathcal{C}}$.
The group $G$ has order $(q+1)q(q-1)$, and ${\mathcal{C}}$ contains $q+1$ points. A point not on ${\mathcal{C}}$ is an [*external point*]{} if it lies on a tangent line to ${\mathcal{C}}$ (in which case it lies on exactly two tangents to ${\mathcal{C}}$), and an [*internal point*]{} otherwise. The set ${\mathcal{E}}$ of external points has size $q(q+1)/2$, and the set ${\mathcal{I}}$ of internal points has size $q(q-1)/2$. If $x,y$ are two external points whose polar lines meet ${\mathcal{C}}$ in points $x_1,x_2$ and $y_1,y_2$ respectively, then by mapping $x_i$ to $y_i$ for both $i=1,2$ we deduce that
1. $G$ acts transitively on ${\mathcal{E}}$, and
2. the stabiliser $G_x {\leqslant}G$ of an external point $x \in {\mathcal{E}}$ acts transitively on the points of ${\mathcal{C}}$ not on the polar line of $x$.
In fact the above argument shows that one only needs 2-transitivity of $G$ on the points on ${\mathcal{C}}$ to deduce transitivity on ${\mathcal{E}}$, so we also have that
1. the stabiliser $G_w {\leqslant}G$ of a point $w\in {\mathcal{C}}$ acts transitively on both (i) the points of ${\mathcal{E}}$ not on the tangent line $t_w({\mathcal{C}})$ of ${\mathcal{C}}$ at $w$, and (ii) the points of $t_w({\mathcal{C}}) \setminus \{w\}$.
If $x\in {\mathcal{E}}$ then the orbit-stabiliser theorem implies that $|G_x|=2(q-1)$. The dual statements of the above properties are:
1. $G$ acts transitively on the secants of ${\mathcal{C}}$, and
2. the stabiliser $G_w {\leqslant}G$ of a point $w\in {\mathcal{C}}$ acts transitively on both (i) the secants not passing through $w$, and (ii) the set of lines through $w$ different from $t_w({\mathcal{C}})$.
We now turn our attention to the action of $G$ on the set of internal points ${\mathcal{I}}$. For this we consider the quadratic extension ${\mathbb{F}}_{q^2}$ of ${\mathbb{F}}_q$, denoting the extension (if well defined) of an object $A$ over ${\mathbb{F}}_q$ to ${\mathbb{F}}_{q^2}$ by $\overline{A}$. An internal point $x$ of ${\mathcal{C}}$ in ${\mathrm{PG}}(2,q)$ becomes an external point $\overline x$ of $\overline {\mathcal{C}}$ in ${\mathrm{PG}}(2,q^2)$. If $\ell$ is the polar line of $x$ then $\overline \ell$ is a secant to $\overline {\mathcal{C}}$. The stabiliser of $\overline \ell$ in $\overline G$ has order $2(q^2-1)$. The points of intersection of $\overline \ell$ and $\overline {\mathcal{C}}$ are conjugate points $p$ and $p^\sigma$, where $\sigma$ denotes the involution in ${\mathrm{P\Gamma L}}(3,q^2)$ induced by the Frobenius map $a\mapsto a^q$. The stabiliser of $\ell\subset \overline \ell$ and the unordered pair $(p,p^\sigma)$ equals the stabiliser of $\ell$ in $G$, which is the group ${\mathrm{PGO}}^-(2,q)$. This group has order $2(q+1)$. In its action on $\overline \ell$ it acts sharply transitively on the points of $\ell$. This implies that the stabiliser $G_x$ of a point $x \in {\mathcal{I}}$ has order $2(q+1)$, and hence that the orbit of $x$ under $G$ has size $q(q-1)/2 = |{\mathcal{I}}|$. This implies that
1. $G$ acts transitively on ${\mathcal{I}}$, and
2. $G$ acts transitively on the set of lines external to ${\mathcal{C}}$ (that is, not intersecting ${\mathcal{C}}$).
Let $G$ be the projective stabiliser of a non-degenerate conic ${\mathcal{C}}$ in ${\mathrm{PG}}(2,q)$, $q$ odd. The stabiliser $G_w {\leqslant}G$ of a point $w\in {\mathcal{C}}$ acts transitively on the set of internal points to ${\mathcal{C}}$.
We continue with the notation introduced above. Consider two distinct internal points $x,y$ with polar lines $\ell,m$. Let $p, p^\sigma$ and $r, r^\sigma$ be the points of intersection of $\overline \ell$ and $\overline m$ with $\overline {\mathcal{C}}$. Let $\alpha$ be the unique element of ${\mathrm{PGL}}(2,q^2)$ mapping the frame $(p',p'^\sigma,w')$ to the frame $(r',r'^\sigma,w')$, where $p', r' , \ldots $ denote the preimages of the points $p, r, \ldots$ under the Veronese map from ${\mathrm{PG}}(1,q^2)$ to $\overline {\mathcal{C}}$. Then $\alpha \sigma \alpha^{-1}\sigma$ fixes the frame $(p',p'^\sigma,w')$ and hence, since $\alpha$ belongs to ${\mathrm{PGL}}(2,q^2)$, which is normal in ${\mathrm{P\Gamma L}}(2,q^2)$, it follows that $\sigma \alpha^{-1}\sigma=\sigma^{-1} \alpha^{-1}\sigma\in {\mathrm{PGL}}(2,q^2)$. Therefore, $\alpha \sigma \alpha^{-1}\sigma$ is the identity, so $\alpha$ commutes with $\sigma$, implying $\alpha \in {\mathrm{PGL}}(2,q)$. It follows that $\alpha$ induces an element in $G_w$ mapping $\ell$ to $m$ and therefore also $x$ to $y$.
We summarise the above results in the following lemma (using again the same notation).
\[lem:Gw\] Let $G$ be the projective stabiliser of a non-degenerate conic ${\mathcal{C}}$ in ${\mathrm{PG}}(2,q)$, $q$ odd, and let $G_w {\leqslant}G$ be the stabiliser of a point $w\in {\mathcal{C}}$. Then the $G_w$-orbits of points in ${\mathrm{PG}}(2,q)$ are precisely $\{w\}$, ${\mathcal{C}}\backslash \{w\}$, ${\mathcal{E}}\backslash t_w({\mathcal{C}})$, $t_w({\mathcal{C}})\backslash \{w\}$, and ${\mathcal{I}}$.
Finally, we note that for $w,u \in {\mathcal{C}}$ the two-point stabiliser $G_{w,u} {\leqslant}G$ acts sharply transitively on points of ${\mathcal{C}}\backslash \{w,u\}$ and has in total $q+6$ orbits on the points of ${\mathrm{PG}}(2,q)$:
- three fixed points $\{w\},\{u\},\{t_u\cap t_w\}$,
- the points of ${\mathcal{C}}\backslash\{u,w\}$,
- the external points on the line ${\langle u,w \rangle}$,
- the internal points on the line ${\langle u,w \rangle}$,
- the points on $t_w({\mathcal{C}}) \backslash \{w,t_u({\mathcal{C}})\cap t_w({\mathcal{C}})\}$,
- the points on $t_u({\mathcal{C}}) \backslash \{u,t_u({\mathcal{C}})\cap t_w({\mathcal{C}})\}$,
- $q-2$ additional orbits of size $q-1$.
Points, lines, solids and hyperplanes of ${\mathrm{PG}}(5,q)$ {#sec:pts_lines_sols_hyps}
=============================================================
As before, let $K$ denote the subgroup $\alpha({\mathrm{PGL}}(3,q))$ of ${\mathrm{PGL}}(6,q)$ defined in Section \[ss:PGL3q-action\]. Before we study the classification of nets of conics, we consider the other linear systems of conics in ${\mathrm{PG}}(2,q)$, or equivalently, the $K$-orbits on subspaces of ${\mathrm{PG}}(5,q)$ of dimension $\neq 2$. A more general version of Proposition \[prop:nets\_vs\_planes\] is the following.
\[lem:duality\] If $q$ is odd then the $K$-orbits of points (respectively, lines) in ${\mathrm{PG}}(5,q)$ are in one-to-one correspondence with the $K$-orbits of hyperplanes (respectively, solids) in ${\mathrm{PG}}(5,q)$.
This is well known; see for example [@LaPo2017 Section 6].
The $K$-orbits on points of ${\mathrm{PG}}(5,q)$ are well understood (see for example [@LaPo2017 Section 2.2]). In the notation established in Section \[ss:V\], the correspondence between the $K$-orbits of points and the $K$-orbits of hyperplanes is as follows.
For $i\in \{1,3\}$, the $K$-orbit ${\mathcal{P}}_i$ corresponds to the $K$-orbit ${\mathcal{H}}_i$. The $K$-orbits on ${\mathcal{P}}_2$ correspond to the $K$-orbits on ${\mathcal{H}}_2$.
Recall from Section \[ss:V\] that each rank-$2$ point $z \in \langle {\mathcal{V}}({\mathbb{F}}_q)\rangle$ lies in a unique conic plane $\langle {\mathcal{C}}_z \rangle$. If $z$ lies on a tangent to the conic ${\mathcal{C}}_z$, it is said to be an [*exterior*]{} point; otherwise, it is said to be an [*interior*]{} point. Since ${\mathbb{F}}_q$ is finite and $q$ is odd, there are two $K$-orbits on ${\mathcal{P}}_2$: the orbit of exterior points ${\mathcal{P}}_{2,e}$ and the orbit of interior points ${\mathcal{P}}_{2,i}$ (see [@LaPo2017 Section 2.2]). There are also two $K$-orbits on hyperplanes in ${\mathcal{H}}_2$: the orbit ${\mathcal{H}}_{2,e}$ of hyperplanes intersecting ${\mathcal{V}}({\mathbb{F}}_q)$ in two conics, and the orbit ${\mathcal{H}}_{2,i}$ of hyperplanes intersecting ${\mathcal{V}}({\mathbb{F}}_q)$ in a point.
The $K$-orbits ${\mathcal{P}}_{2,e}$ and ${\mathcal{P}}_{2,i}$ are in one-to-one correspondence with the $K$-orbits ${\mathcal{H}}_{2,e}$ and ${\mathcal{H}}_{2,i}$, respectively.
[If ${\mathbb{F}}_q$ is replaced by an algebraically closed field then each of ${\mathcal{P}}_2$ and ${\mathcal{H}}_2$ form one $K$-orbit. This is the origin of the complications which arise when working over finite fields.]{.nodecor}
The $K$-orbits on lines in ${\mathrm{PG}}(5,q)$ were determined in [@LaPo2017]. For $q$ odd they are given by Theorem \[linesThm\] (with representatives listed in Table \[table:lines\]). The classification of $K$-orbits of solids in ${\mathrm{PG}}(5,q)$ then follows from Lemma \[lem:duality\].
\[linesThm\] There are 15 $K$-orbits on lines in ${\mathrm{PG}}(5,q)$, $q$ odd. Representatives of these orbits are listed in Table \[table:lines\].
\[remOijNotation\] [ The notation for the line orbits, namely $o_i$ for $i\in \{5,6,9,10,12,16,17\}$ and $o_{i,j}$ for $i\in \{8,13,14,15\}$ and $j\in \{1,2\}$, is used for consistency with [@LaPo2017 Table 2]. To be more specific, in [@LaPo2017] the orbits are denoted by $o_i$ for each $i$ as above, but when $i\in \{8,13,14,15\}$ there are in fact two ‘$o_i$’ orbits. (The reason for the ‘$o_i$’ notation itself is explained in [@LaPo2017].) In the present paper, we instead explicitly label these orbits as $o_{i,j}$ where $j \in \{1,2\}$. In the case $i=15$, the representatives given in Table \[table:lines\] correspond to the representatives in column $j \in \{1,2\}$ of [@LaPo2017 Table 2]. For $i \in \{8,13,14\}$, we have chosen slightly different representatives to those listed in [@LaPo2017 Table 2]. The reason for this change is explained in Remark \[remLineReps\]. ]{}
Orbit Representative Conditions
----------- ----------------------------------------------------------------------------------------------------------------------------------- -----------------------------
$o_5$ $\left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & \beta & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]$
$o_6$ $\left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \cdot & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]$
$o_{8,1}$ $\left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & \beta & \cdot \\ \cdot & \cdot & -\beta \end{matrix} \right]$
$o_{8,2}$ $\left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & \beta & \cdot \\ \cdot & \cdot & -\varepsilon\beta \end{matrix} \right]$ $\varepsilon \in \boxtimes$
$o_9$ $\left[ \begin{matrix} \alpha & \cdot & \beta \\ \cdot & \beta & \cdot \\ \beta & \cdot & \cdot \end{matrix} \right]$
$o_{10}$ $\left[ \begin{matrix} v\alpha & \beta & \cdot \\ \beta & \alpha+u\beta & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]$ ($*$)
$o_{12}$ $\left[ \begin{matrix} \cdot & \alpha & \cdot \\ \alpha & \cdot & \beta \\ \cdot & \beta & \cdot \end{matrix} \right]$
: Matrix representatives of the 15 line orbits in ${\mathrm{PG}}(5,q)$, $q$ odd, under the action of $K={\mathrm{PGL}}(3,q) {\leqslant}{\mathrm{PGL}}(6,q)$ defined in Section \[ss:PGL3q-action\]. Here $\cdot$ denotes $0$, $(\alpha,\beta)$ ranges over all non-zero values in $\mathbb{F}_q^2$, and $\Box$ (respectively, $\boxtimes$) is the set of squares (respectively, non-squares) in $\mathbb{F}_q$. Condition ($*$) is: $v\lambda^2+uv\lambda - 1 \neq 0$ for all $\lambda \in \mathbb{F}_q$. Condition ($**$) is: $\lambda^3+w \lambda^2- u \lambda+ v \neq 0$ for all $\lambda \in \mathbb{F}_q$. []{data-label="table:lines"}
[lll]{} Orbit & Representative & Conditions\
$o_{13,1}$ & $\left[ \begin{matrix} \cdot & \alpha & \cdot \\ \alpha & \beta & \cdot \\ \cdot & \cdot & -\beta \end{matrix} \right]$ &\
$o_{13,2}$ & $\left[ \begin{matrix} \cdot & \alpha & \cdot \\ \alpha & \beta & \cdot \\ \cdot & \cdot & -\varepsilon\beta \end{matrix} \right]$ & $\varepsilon \in \boxtimes$\
$o_{14,1}$ & $\left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & -(\alpha+\beta) & \cdot \\ \cdot & \cdot & \beta \end{matrix} \right]$ &\
$o_{14,2}$ & $\left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & -\varepsilon(\alpha+\beta) & \cdot \\ \cdot & \cdot & \beta \end{matrix} \right]$ & $\varepsilon \in \boxtimes$\
$o_{15,1}$ & $\left[ \begin{matrix} v\beta & \alpha & \cdot \\ \alpha & u\alpha+\beta & \cdot \\ \cdot & \cdot & \alpha \end{matrix} \right]$ & $-v \in \Box$, ($*$)\
$o_{15,2}$ & $\left[ \begin{matrix} v\beta & \alpha & \cdot \\ \alpha & u\alpha+\beta & \cdot \\ \cdot & \cdot & \alpha \end{matrix} \right]$ & $-v \in \boxtimes$, ($*$)\
$o_{16}$ & $\left[ \begin{matrix} \cdot & \cdot & \alpha \\ \cdot & \alpha & \beta \\ \alpha & \beta & \cdot \end{matrix} \right]$ &\
$o_{17}$ & $\left[ \begin{matrix} v^{-1}\alpha & \beta & \cdot \\ \beta & u \beta - w \alpha & \alpha \\ \cdot & \alpha & \beta \end{matrix} \right]$ & ($**$)\
The next lemma gives a useful condition for determining whether a rank-$2$ point in ${\mathrm{PG}}(2,q)$ is an exterior point or an interior point (that is, whether it belongs to the orbit ${\mathcal{P}}_{2,e}$ or the orbit ${\mathcal{P}}_{2,i}$). Here we use the notation $M_{ij}(A)$ to mean the matrix obtained from $A$ by removing the $i$th row and the $j$th column, and $|\cdot|$ denotes the determinant.
\[lem:extcriteria\] A point $y\in {\mathrm{PG}}(5,q)$ belongs to ${\mathcal{P}}_{2,e}$ if and only if $|M_y|=0$ and $-|M_{11}(M_y)|$, $-|M_{22}(M_y)|$, $-|M_{33}(M_y)|$ are all squares with at least one being non-zero.
Note that if the matrix $M_y$ defined by satisfies the given conditions then $y$ is a point of rank $2$. Consider the rank-$2$ point $z_\tau$ with coordinates $(1,0,-\tau,0,0,0)$, where $\tau \in {\mathbb{F}}_q\setminus\{0\}$, and write $$A_\tau=M_{z_\tau}={\left [ \begin{matrix} 1&0&0\\0&-\tau&0\\0&0&0 \end{matrix} \right]}.$$ Note that $z_\tau$ is exterior for $\tau=1$ and interior for $\tau=\epsilon$ a non-square in ${\mathbb{F}}_q$. A rank-$2$ point $y$ is therefore exterior (respectively, interior) if and only if there exists an invertible matrix $X$ such that $M_y=XA_{1}X^T$ (respectively, $M_y=XA_{\epsilon}X^T$). A straightforward calculation shows that for $M_y=XA_{\tau}X^T$ we have $$|M_{11}(M_y)| = -\tau|M_{13}(X)|^2, \quad
|M_{22}(M_y)| = -\tau|M_{23}(X)|^2, \quad
|M_{33}(M_y)| = -\tau|M_{33}(X)|^2,$$ completing the proof.
Planes in ${\mathrm{PG}}(5,q)$, $q$ odd {#sec:planes}
=======================================
We now collect a few final preliminaries before beginning the proof of Theorem \[mainthm\]. Recall that there are four $K$-orbits of points in ${\mathrm{PG}}(5,q)$, namely: ${\mathcal{P}}_1$, the points of rank $1$; ${\mathcal{P}}_{2,e}$, the exterior rank-$2$ points; ${\mathcal{P}}_{2,i}$, the interior rank-$2$ points; and ${\mathcal{P}}_3$, the points of rank $3$.
\[def:dist\]
Let $W$ be a subspace of ${\mathrm{PG}}(5,q)$. The [*point-orbit distribution*]{} of $W$ is the list $[n_1,n_2,n_3,n_4]$, where
- $n_1$ is the number of rank-$1$ points contained in $W$,
- $n_2$ is the number of exterior rank-$2$ points contained in $W$,
- $n_3$ is the number of interior rank-$2$ points contained in $W$,
- $n_4$ is the number of rank-$3$ points contained in $W$.
The [*rank distribution*]{} of $W$ is the list $[m_1,m_2,m_3]$ where $m_i$ is the number of rank-$i$ points of ${\mathrm{PG}}(5,q)$ contained in $W$, for $i \in \{1,2,3\}$. In other words, $[m_1,m_2,m_3] = [n_1,n_2+n_3,n_4]$. Given $i \in \{1,2,3\}$, we say that $W$ has [*constant rank*]{} $i$ if $m_j = 0$ for all $j \neq i$.
The point-orbit distributions of the $K$-orbits of lines in ${\mathrm{PG}}(5,q)$ were previously determined by the first and second authors [@LaPo2017 Tables 1 and 4]. They are summarised for reference in Table \[table:line-rank-dist\]. The point-orbit distributions of the $K$-orbits of planes are determined in a series of lemmas in Sections \[sec:three\_rank\_one\_points\]–\[sec:final\] and are summarised in Table \[table:pt-orbit-dist\].
\[remLineReps\] [ As mentioned in Remark \[remOijNotation\], we have chosen different representatives for the line orbits $o_{i,1}$ and $o_{i,2}$ in the cases $i \in \{8,13,14\}$, compared with [@LaPo2017 Table 2]. The reason for this is that, in the representatives in [@LaPo2017 Table 2], the numbers of interior and exterior points of rank $2$ depend on whether $-1$ is a square in $\mathbb{F}_q$. This is explained in [@LaPo2017 Table 4]. For the representatives given in Table \[table:lines\], the point-orbit distributions are independent of $q$. ]{}
The following notation is used in Table \[table:main\] and throughout the proof of Theorem \[mainthm\].
[ll]{} Orbit & Point-orbit distribution\
$o_5$ & $[2, \frac{q-1}{2}, \frac{q-1}{2}, 0]$\
$o_6$ & $[1, q, 0, 0]$\
$o_{8,1}$ & $[1, 1, 0, q-1]$\
$o_{8,2}$ & $[1, 0, 1, q-1]$\
$o_9$ & $[1, 0, 0, q]$\
$o_{10}$ & $[0, \frac{q+1}{2}, \frac{q+1}{2}, 0]$\
$o_{12}$ & $[0, q+1, 0, 0]$\
$o_{13,1}$ & $[0, 2, 0, q-1]$\
$o_{13,2}$ & $[0, 1, 1, q-1]$\
$o_{14,1}$ & $[0, 3, 0, q-2]$\
$o_{14,2}$ & $[0, 1, 2, q-2]$\
$o_{15,1}$ & $[0, 1, 0, q]$\
$o_{15,2}$ & $[0, 0, 1, q]$\
$o_{16}$ & $[0, 1, 0, q]$\
$o_{17}$ & $[0, 0, 0, q+1]$\
[lll]{} Orbit & Point-orbit distribution & Condition\
$\Sigma_1$ & $[q+1, q(q+1)/2, q(q-1)/2, 0]$ &\
$\Sigma_2$ & $[3, 3(q-1)/2, 3(q-1)/2, q^2-2q+1]$ &\
$\Sigma_3$ & $[2, (3q-1)/2, (q-1)/2, q^2-q]$ &\
$\Sigma_4$ & $[2, (3q-1)/2, (q-1)/2, q^2-q]$ &\
$\Sigma_5$ & $[2, q-1, q-1, q^2-q+1]$ &\
$\Sigma_6$ & $[1, (q+1)/2, (q+1)/2, q^2-1]$ &\
$\Sigma_7$ & $[1, q^2+q, 0, 0]$ &\
$\Sigma_8$ & $[1, 2q, 0, q^2-q]$ &\
$\Sigma_9$ & $[1, 2q, 0, q^2-q]$ &\
$\Sigma_{10}$ & $[1, q, q, q^2-q]$ &\
$\Sigma_{11}$ & $[1, q, 0, q^2]$ &\
$\Sigma_{12}$ & $[1, (q-1)/2, (q-1)/2, q^2+1]$ &\
$\Sigma_{13}$ & $[1, (q+1)/2, (q+1)/2, q^2-1]$ &\
$\Sigma_{14}$ & $[1, (q\mp 1)/2, (q\mp 1)/2, q^2\pm 1]$ & $q \equiv \pm 1 \pmod 3$\
$\Sigma_{14}'$ & $[1, q, 0, q^2]$ & $q \equiv 0 \pmod 3$\
$\Sigma_{15}$ & $[1, q, 0, q^2]$ &\
\[orbitNotation\] [ If a plane in ${\mathrm{PG}}(5,q)$ is spanned by points $x,y,z$ then we represent its $K$-orbit by the matrix $\alpha M_x +\beta M_y + \gamma M_z$, with the convention that zeroes are replaced by dots and the understanding that $(\alpha,\beta,\gamma)$ ranges over all non-zero values in $\mathbb{F}_q^3$. For example, the $K$-orbit $\Sigma_1$ in equation below (and the first row of Table \[table:main\]) is the $K$-orbit of the plane spanned by the points $x=(1,0,0,0,0,0)$, $y=(0,0,0,1,0,0)$ and $z=(0,1,0,0,0,0)$. ]{}
We also note the following preliminary lemmas.
Let $A$ be a $3\times 3$ matrix whose entries are linear forms in variables $x,y,z \in \mathbb{F}_q$. Then the points $(a,b,c)$ for which $A$ evaluated at $(a,b,c)$ has rank $1$ are singular points of the cubic ${\mathcal{Z}}(|A|)$.
A $3 \times 3$ matrix has rank $1$ if and only if its adjugate is zero. The partial derivative of the determinant of a matrix $M$ with polynomial entries with respect to a variable $x$ is given by $$\partial |M|/\partial x=|M| {\mathrm{tr}}(M^{-1}\partial M/\partial x).$$ It follows that if $A$ evaluated at $(a,b,c)$ has rank $1$ then all of $\partial |A|/\partial x$, $\partial |A|/\partial y$, $\partial |A|/\partial z$ evaluated at $(a,b,c)$ are zero. This implies that $(a,b,c)$ is a singular point of ${\mathcal{Z}}(|A|)$.
For the next lemma, recall the notation for the $K$-orbits of lines in ${\mathrm{PG}}(5,q)$ from Theorem \[linesThm\].
\[lem:o6\_plane\] If $\pi$ is a plane in ${\mathrm{PG}}(5,q)$ containing a line of type $o_6$ and a point of rank $3$, then the cubic of points of rank at most $2$ in $\pi$ is either (i) the union of a non-degenerate conic and a tangent line, (ii) the union of a line and a double line, or (iii) a triple line.
By Table \[table:line-rank-dist\], a line of type $o_6$ has rank distribution $[1,q,0]$. Suppose that $\pi={\langle x,y,z \rangle}$ with ${\langle x,y \rangle}$ a line of type $o_6$, $\operatorname{rank}(x)=1$, $\operatorname{rank}(y)=2$ and $\operatorname{rank}(z)=3$. Based on the $o_6$ representative in Table \[table:lines\], we may assume that $x=(1,0,0,0,0,0)$ and $y=(0,1,0,0,0,0)$. The point $z$ then has coordinates $(0,0,a,b,c,d)$ for some $a,b,c,d \in {\mathbb{F}}_q$. The cubic $Q$ of points of rank at most $2$ has equation $X_3 f(X_1,X_2,X_3)=0$, where $$f(X_1,X_2,X_3)=(bd-c^2)X_1X_3-dX_2^2+2acX_2X_3-a^2bX_3^2.$$ If the conic ${\mathcal{C}}={\mathcal{Z}}(f)$ is non-degenerate then $d\neq 0$ and the line ${\langle x,y \rangle}$ is a tangent to ${\mathcal{C}}$. If ${\mathcal{C}}$ is degenerate then $d(bd-c^2)=0$. If $d=0$ then $f(X_1,X_2,X_3)=X_3((bd-c^2)X_1+2acX_2-a^2bX_3)$ and ${\mathcal{C}}$ is the union of two lines, at least one of which is ${\langle x,y \rangle}$. If $d\neq0$ and $bd-c^2=0$ then $f(X_1,X_2,X_3)=-d^{-1}(dX_2-acX_3)^2$, and ${\mathcal{C}}$ is a double line.
Before finally proceeding to the classification of $K$-orbits of planes in ${\mathrm{PG}}(5,q)$, we prove the following lemma, which establishes the non-existence of planes with certain rank distributions.
\[lem:b<q\] There are no planes in ${\mathrm{PG}}(5,q)$ with rank distribution $[2,n_2,n_3]$ where $n_2<q$.
Consider any two distinct points $x,y \in {\mathcal{V}}({\mathbb{F}}_q)$. Since every point on the line $\ell = \langle x,y\rangle$ has rank at most $2$ (as noted in Section \[ss:V\]), a plane $\pi$ through $x$ and $y$ contains at least $q-1$ points of rank $2$. Hence, if $\pi$ has rank distribution $[2,n_2,n_3]$ with $n_2<q$ then we must have $n_2=q-1$. It remains to rule out this case. Since all of the rank-$2$ points in $\pi$ lie on $\ell$, the cubic $Q$ of points of rank at most $2$ is the triple line $\ell$, because $x$ and $y$ are necessarily singular points of that cubic. Hence, if we write $\pi={\langle x,y,z \rangle}$ then the point $z$ has rank $3$ and without loss of generality we may assume that $x=\nu({\langle e_1 \rangle})$, $y=\nu({\langle e_2 \rangle})$ and $z=(0,a,b,0,c,d)$ for some $a,b,c,d\in {\mathbb{F}}_q$, so that $Q$ has equation $X_1X_3(dX_2-c^2X_3)+aX_3^3(ad-bc)+bX_3^2(acX_3-bX_2)=0$. Since this must be equivalent to $X_3^3=0$, it follows that $b=c=d=0$, contradicting $\operatorname{rank}(z)=3$.
Planes spanned by three points of ${\mathcal{V}}({\mathbb{F}}_q)$ {#sec:three_rank_one_points}
=================================================================
[ Throughout the paper we let $e_1,e_2,e_3$ denote the standard basis vectors of ${\mathbb{F}}_q^3$. ]{}
Suppose that a plane $\pi$ in ${\mathrm{PG}}(5,q)$ contains three points $z_1,z_2,z_3 \in {\mathcal{V}}({\mathbb{F}}_q)$, and consider their pre-images $p_1,p_2,p_3$ under the Veronese map $\nu : {\mathrm{PG}}(2,q) \rightarrow {\mathcal{V}}({\mathbb{F}}_q)$. Since lines of ${\mathrm{PG}}(5,q)$ intersect ${\mathcal{V}}({\mathbb{F}}_q)$ in at most two points, the points $z_1,z_2,z_3$ span $\pi$. If $p_1,p_2,p_3$ are collinear then without loss of generality $p_1=\langle e_1\rangle$, $p_2=\langle e_2\rangle$ and $\pi$ is the conic plane $\langle {\mathcal{C}}(z_1,z_2)\rangle$. If $p_1,p_2,p_3$ are not collinear then without loss of generality $p_i=\langle e_i\rangle$ for $i=1,2,3$ and $\pi$ intersects ${\mathcal{V}}({\mathbb{F}}_q)$ in the three points $z_1,z_2,z_3$. These two possibilities give us the following two $K$-orbits (respectively, with notation as established in Section \[sec:planes\]): $$\label{sigma1and2}
\Sigma_1: \left[ \begin{matrix} \alpha & \gamma & \cdot \\ \gamma & \beta & \cdot \\ \cdot & \cdot & \cdot \end{matrix} \right]
\quad \text{and} \quad
\Sigma_2 : \left[ \begin{matrix} \alpha & \cdot & \cdot \\ \cdot & \beta & \cdot \\ \cdot & \cdot & \gamma \end{matrix} \right].$$
\[lem:Sigma\_1 and Sigma\_2\] A plane belonging to the $K$-orbit $\Sigma_1$ has point-orbit distribution $$[q+1,q(q+1)/2,q(q-1)/2,0],$$ and a plane belonging to the $K$-orbit $\Sigma_2$ has point-orbit distribution $$[3,3(q-1)/2,3(q-1)/2,q^2-2q+1].$$
The point-orbit distribution of a plane in the $K$-orbit $\Sigma_1$ follows from Section \[subsec:conic\_properties\]. Let $\pi$ be a plane in the $K$-orbit $\Sigma_2$, and let $z_1,z_2,z_3$ denote the three points of rank $1$ in $\pi$. It is clear from the representative of $\Sigma_2$ that the points of rank $2$ in $\pi$ lie on the secants $\langle z_i,z_j\rangle$, $i\neq j$, which are lines of type $o_5$ by Table \[table:lines\]. It follows from a straightforward calculation and Lemma \[lem:extcriteria\] that each such secant contains $(q-1)/2$ external points and $(q-1)/2$ internal points of the conic ${\mathcal{C}}(z_i,z_j)$ defined as in equation . In total this amounts to $3(q-1)/2$ points of $\pi$ belonging to each of ${\mathcal{P}}_{2,e}$ and ${\mathcal{P}}_{2,i}$.
Planes containing exactly two points of ${\mathcal{V}}({\mathbb{F}}_q)$ {#sec:two_rank_one_points}
=======================================================================
We now classify the planes in ${\mathrm{PG}}(5,q)$ intersecting the quadric Veronesean in exactly two points. By Lemma \[lem:b<q\], we only need to consider planes with rank distribution $[n_1,n_2,n_2]$ where $n_1=2$ and $n_2{\geqslant}q$. Let $\pi=\langle x,y,z\rangle$ be such a plane, with $\operatorname{rank}(x)=\operatorname{rank}(y)=1$. The condition on the rank distribution of $\pi$ implies the existence of a point of rank $2$ which is not on the line $\langle x,y\rangle$. Hence, we may assume that $\operatorname{rank}(z)=2$. Consider the conics ${\mathcal{C}}(x,y)$ and ${\mathcal{C}}_z$ (defined as in Section \[ss:V\]), and let $u = {\mathcal{C}}(x,y)\cap{\mathcal{C}}_z$. There are two possibilities: $u\in \{x,y\}$ and $u\notin \{x,y\}$.
First suppose that $u\in \{x,y\}$, say $u=x$. Then $\langle x,z\rangle$ is the unique tangent $t_x({\mathcal{C}}_z)$ to ${\mathcal{C}}_z$ through $x$ in the plane $\langle {\mathcal{C}}_z\rangle$, because otherwise $\pi$ would contain a third point of ${\mathcal{V}}({\mathbb{F}}_q)$. If $\ell_z$ is the pre-image of ${\mathcal{C}}_z$ under $\nu$, then the stabiliser of $\nu^{-1}(x)$ and $\nu^{-1}(y)$ acts transitively on the points of $\ell_z\setminus \{x\}$, and so the stabiliser of $\{x,y\}$ and ${\mathcal{C}}_z$ acts transitively on the points of $t_x({\mathcal{C}}_z)\setminus\{x\}$. Hence, there is exactly one orbit of such planes, which we call $\Sigma_3$. Taking $x=\nu(e_1)$, $y=\nu(e_2)$, and $\ell_z = \langle e_1,e_3\rangle$ gives the following representative: $$\Sigma_3: \left[ \begin{matrix} \alpha & \cdot & \gamma \\ \cdot & \beta & \cdot \\ \gamma & \cdot & \cdot \end{matrix} \right].$$
\[lem:Sigma\_3\] A plane belonging to the $K$-orbit $\Sigma_3$ has point-orbit distribution $$[ 2 , (3q-1)/2 , (q-1)/2 , q^2-q ].$$
There are $q-1$ points of rank $2$ on the line $\langle x,y \rangle$, half belonging to ${\mathcal{P}}_{2,e}$ and half belonging to ${\mathcal{P}}_{2,i}$. There are also $q$ points of ${\mathcal{P}}_{2,e}$ on the tangent to ${\mathcal{C}}_z$ at $u=x$, giving a total of $q+(q-1)/2 = (3q-1)/2$ points of $\pi$ in ${\mathcal{P}}_{2,e}$. The remaining points of $\pi$ are all of rank $3$.
Now suppose that $u\notin \{x,y\}$. We may fix $x$, $y$, $u$ and ${\mathcal{C}}_z$, and hence ${\mathcal{C}}(x,y)$. The group which pointwise stabilises ${\mathcal{C}}(x,y)$ and also setwise stabilises ${\mathcal{C}}_z$ acts on $\langle{\mathcal{C}}_z\rangle$ as the stabiliser of ${\mathcal{C}}_z$ and $u$ in ${\mathrm{PGL}}(\langle{\mathcal{C}}_z\rangle)$. Since $q$ is odd, this group has three orbits on points of $\langle{\mathcal{C}}_z\rangle \backslash{\mathcal{C}}_z$ (by Lemma \[lem:Gw\]): (i) the points on the tangent to ${\mathcal{C}}_z$ through $u$, (ii) the other external points of ${\mathcal{C}}_z$, and (iii) the internal points of ${\mathcal{C}}_z$. We may choose $x=\nu(e_1)$, $y=\nu(e_2)$, $u=\nu(e_1+e_2)$ , ${\mathcal{C}}_z =\nu({\langle e_1+e_2,e_3 \rangle})$. We now consider separately the cases in which $z$ lies in each of these orbits.
\(i) If $z$ is a point on the tangent to ${\mathcal{C}}_z$ through $u$ then we obtain the orbit $$\Sigma_4 : \left[ \begin{matrix} \alpha & \cdot & \gamma \\ \cdot & \beta & \gamma \\ \gamma & \gamma & \cdot \end{matrix} \right].$$
\[lem:Sigma\_4\] A plane belonging to the $K$-orbit $\Sigma_4$ has point-orbit distribution $$[ 2 , (3q-1)/2 , (q-1)/2 , q^2-q ].$$
The proof of Lemma \[lem:Sigma\_3\] also applies here (except that now $u \neq x$).
[Although planes in $\Sigma_3$ and $\Sigma_4$ have the same point-orbit distribution, these $K$-orbits are distinct. This can be seen by observing that for $\pi \in \Sigma_3$ the $(q-1)/2$ points in ${\mathcal{P}}_{2,i}$ lie on a tangent line to a conic of ${\mathcal{V}}({\mathbb{F}}_q)$ through one of the two points of rank $1$ in $\pi$, while for a plane in $\Sigma_4$ this is not the case.]{.nodecor}
\(ii) If $z$ is an external point of ${\mathcal{C}}_z$ not on the tangent to ${\mathcal{C}}_z$ through $u$ then we may choose $z$ in order to obtain the following representative of a new orbit $\Sigma_5$: $$\Sigma_5 : \left[ \begin{matrix} \alpha & \cdot & \gamma \\ \cdot & \beta & \gamma \\ \gamma & \gamma & \gamma \end{matrix} \right].$$
\[lem:Sigma\_5\] A plane belonging to the $K$-orbit $\Sigma_5$ has point-orbit distribution $$[2,q-1,q-1,q^2-q+1].$$
Let $\pi$ be the plane given above, that is, points of $\pi$ have coordinates $(\alpha,0,\gamma,\beta,\gamma,\gamma)$ with $\alpha, \beta, \gamma \in {\mathbb{F}}_q$ not all zero. The cubic ${\mathcal{Z}}(\Delta_{\mathcal{N}})$ of the net ${\mathcal{N}}$ corresponding to $\pi$ is the union of a non-degenerate conic ${\mathcal{C}}$ and a line $\ell$ secant to ${\mathcal{C}}$. The two points in ${\mathcal{C}}\cap \ell$ are the points $x$ and $y$ from above, and the other points in ${\mathcal{C}}\cup \ell$ are all of rank $2$. This implies that the rank distribution of $\pi$ is $[2,2(q-1),q^2-q+1]$. It remains to show that half of the $2(q-1)$ points of rank $2$ are exterior points and half are interior points. First consider the $q-1$ points of rank $2$ on the line $\ell=\langle x,y\rangle$. Since half of these points are external points of the conic ${\mathcal{C}}(x,y)$ determined by $x$ and $y$, and the other half are internal points of ${\mathcal{C}}(x,y)$, we obtain $(q-1)/2$ points in each of the point orbits ${\mathcal{P}}_{2,e}$ and ${\mathcal{P}}_{2,i}$. Now consider the $q-1$ points of rank $2$ in ${\mathcal{C}}\setminus \{x,y\}$. These points have coordinates $p_{\alpha,\beta} := (\alpha,0,1,\beta,1,1)$ with $\alpha\beta=\alpha+\beta$ and $\alpha,\beta \neq 1$. Lemma \[lem:extcriteria\] implies that $p_{\alpha,\beta}\in {\mathcal{P}}_{2,e}$ if and only if $1-\alpha$, $1-\beta$ and $-\alpha\beta$ are all squares in $\mathbb{F}_q$ (note that they cannot all be zero). The condition $\alpha\beta=\alpha+\beta$ implies that these three elements are either all squares or all non-squares. In particular, $p_{\alpha,\beta}\in {\mathcal{P}}_{2,e}$ if and only if $1-\alpha$ is a square, which occurs for half of the possible $q-1$ values of $\alpha$ (recalling that $\alpha \neq 1$). Hence, we obtain a further $(q-1)/2$ points in each of ${\mathcal{P}}_{2,e}$ and ${\mathcal{P}}_{2,i}$, as claimed.
\(iii) Finally, we show that if $z$ is an internal point of ${\mathcal{C}}_z$ then the plane $\pi$ again belongs to the $K$-orbit $\Sigma_5$. In other words, this case does not yield a new orbit.
Suppose that $\pi={\langle x,y,z \rangle}$ is a plane in ${\mathrm{PG}}(5,q)$ with $\operatorname{rank}(x)=\operatorname{rank}(y)=1$ and $\operatorname{rank}(z)=2$, where $x,y \notin {\mathcal{C}}_z$ and $z$ is an internal point of ${\mathcal{C}}_z$. Then $\pi \in \Sigma_5$.
An argument analogous to the one in the proof of Lemma \[lem:Sigma\_5\] shows that $\pi$ contains a point $z' \in {\mathcal{P}}_{2,e}$ which is not on the line $\langle x,y \rangle$, so $\pi = \langle x,y,z' \rangle \in \Sigma_5$.
Planes meeting ${\mathcal{V}}({\mathbb{F}}_q)$ in one point and spanned by points of rank ${\leqslant}2$ {#sec:one_rank_one_point_and_spanned}
========================================================================================================
Let $\pi={\langle x,y,z \rangle}$ be a plane with $\operatorname{rank}(x)=1$ and $\operatorname{rank}(y)=\operatorname{rank}(z)=2$. Consider the conics ${\mathcal{C}}_y$ and ${\mathcal{C}}_z$ determined by $y$ and $z$ (see Section \[ss:V\]), and let $p_x$ be a point in ${\mathrm{PG}}(2,q)$ and $\ell_y$, $\ell_z$ lines in ${\mathrm{PG}}(2,q)$ such that $x=\nu(p_x)$, ${\mathcal{C}}_y=\nu(\ell_y)$ and ${\mathcal{C}}_z=\nu(\ell_z)$. If $x \in {\mathcal{C}}_y={\mathcal{C}}_z$, then $\pi$ is a conic plane, and so lies in the orbit $\Sigma_1$. The remaining possibilities (up to symmetry) are as follows: (a) $x \notin {\mathcal{C}}_y={\mathcal{C}}_z$, (b) $x = {\mathcal{C}}_y\cap {\mathcal{C}}_z$, (c) $x \in {\mathcal{C}}_y\setminus{\mathcal{C}}_z$, and (d) $x\notin {\mathcal{C}}_y\cup {\mathcal{C}}_z$.
Case [**(a)**]{}
-----------------
If $x \notin {\mathcal{C}}_y={\mathcal{C}}_z$ then $\langle y,z\rangle$ is a line in $\langle{\mathcal{C}}_y\rangle$ external to ${\mathcal{C}}_y$. We may fix $x$ and ${\mathcal{C}}_y$. The group stabilising both $x$ and ${\mathcal{C}}_y$ acts on $\langle{\mathcal{C}}_y\rangle$ as the stabiliser of ${\mathcal{C}}_y$ in ${\mathrm{PGL}}(\langle{\mathcal{C}}_y\rangle)$. This group acts transitively on external lines (by property (C7) of Section \[subsec:conic\_properties\]), and so we have just one orbit arising in this way. Since $q$ is odd we have the following representative: $$\Sigma_6 : \left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \varepsilon\alpha & \cdot \\ \cdot & \cdot & \gamma \end{matrix} \right],
\quad \text{where } \epsilon \in {\mathbb{F}}_q \text{ is a non-square}.$$
\[lem:Sigma\_6\] A plane belonging to the $K$-orbit $\Sigma_6$ has point-orbit distribution $$[ 1 , (q+1)/2 , (q+1)/2 , q^2-1 ].$$ Its points of rank $2$ lie on a line of type $o_{10}$.
The points of rank $2$ are precisely those with $\gamma=0$, namely the points on the line ${\langle y,z \rangle}$ (in the above representative). Since this is a line in $\langle{\mathcal{C}}_y\rangle$ external to ${\mathcal{C}}_y$, half of its points belong to each of ${\mathcal{P}}_{2,e}$ and ${\mathcal{P}}_{2,i}$. By Table \[table:line-rank-dist\], we see that the line has type $o_{10}$.
\[lem:Sigma\_6\_2\] A plane belonging to the $K$-orbit $\Sigma_6$ does not contain a line of constant rank $3$.
This follows immediately from Lemma \[lem:Sigma\_6\], because every line must intersect the line of type $o_{10}$, which is of constant rank $2$.
Case [**(b)**]{}
-----------------
If $x = {\mathcal{C}}_y\cap {\mathcal{C}}_z$ then $\langle x,y\rangle$ is the unique tangent to ${\mathcal{C}}_y$ through $x$ in $\langle {\mathcal{C}}_y \rangle$, and $\langle x,z\rangle$ is the unique tangent to ${\mathcal{C}}_z$ through $x$ in $\langle {\mathcal{C}}_z \rangle$ (because otherwise $\pi$ would contain more than one point of rank $1$). Hence, $\pi=\langle x,y,z\rangle$ is completely determined by $\ell_y$ and $\ell_z$ (because $p_x =\ell_y\cap \ell_z$). Since ${\mathrm{PGL}}(3,q)$ acts transitively on pairs of lines meeting in a point, we obtain just one orbit in this way. A representative is a follows. $$\Sigma_7 : \left[ \begin{matrix} \alpha & \beta & \gamma \\ \beta & \cdot & \cdot \\ \gamma & \cdot & \cdot \end{matrix} \right].$$
\[lem:Sigma\_7\] A plane belonging to the $K$-orbit $\Sigma_7$ is a tangent plane of ${\mathcal{V}}({\mathbb{F}}_q)$ and has point-orbit distribution $ [ 1 , q^2+q,0,0 ]$.
Since such a plane $\pi$ contains two tangents through its unique point $x$ of rank $1$, it follows that $\pi$ is the tangent plane of ${\mathcal{V}}({\mathbb{F}}_q)$ at $x$. The lines through $x$ in $\pi$ are the tangents to the conics of ${\mathcal{V}}({\mathbb{F}}_q)$ through $x$, and so all the rank-$2$ points in $\pi$ are contained in ${\mathcal{P}}_{2,e}$.
Case [**(c)**]{}
----------------
Now suppose that $x \in {\mathcal{C}}_y \setminus {\mathcal{C}}_z$. Then $\langle x,y\rangle$ must be the unique tangent $t_x({\mathcal{C}}_y)$ to ${\mathcal{C}}_y$ through $x$ in $\langle {\mathcal{C}}_y \rangle$. Write $w = {\mathcal{C}}_y\cap {\mathcal{C}}_z$, and $\nu(p_w) = w$. Without loss of generality we may fix $p_x$, $p_w$ and $\ell_z$. The subgroup of $K$ stabilising $x$, $w$ and ${\langle {\mathcal{C}}_z \rangle}$ has three orbits on points of ${\langle {\mathcal{C}}_z \rangle}\backslash {\mathcal{C}}_z$: (i) the points on the unique tangent $t_w({\mathcal{C}}_z)$ to ${\mathcal{C}}_z$ through $w$ in $\langle {\mathcal{C}}_z \rangle$, (ii) the external points of ${\mathcal{C}}_z$ not on $t_w({\mathcal{C}}_z)$, (iii) the internal points of ${\mathcal{C}}_z$. Hence, we obtain at most three $K$-orbits, and we show below that we obtain exactly three orbits $\Sigma_8$, $\Sigma_9$ and $\Sigma_{10}$ (see Remark \[remarkSigma8910\]). The representatives of these orbits given below are obtained by choosing $p_x={\langle e_1 \rangle}$, $p_w={\langle e_2 \rangle}$ and $\ell_z ={\langle e_2,e_3 \rangle}$.
[**(c-i)**]{} If $z$ lies on $t_w({\mathcal{C}}_z)$ then $\pi={\langle t_x,z \rangle}$ and we obtain the orbit $$\Sigma_8 : \left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \cdot & \gamma \\ \cdot & \gamma & \cdot \end{matrix} \right].$$
\[lem:Sigma\_8\] A plane belonging to the $K$-orbit $\Sigma_8$ has point-orbit distribution $ [ 1 , 2q,0,q^2-q ]$. Its points of rank $2$ lie on two lines: a tangent to ${\mathcal{V}}({\mathbb{F}}_q)$, and a line of type $o_{12}$.
Consider the plane $\pi={\langle x,y,z \rangle}$ as above (where $y$ corresponds to $\alpha=\gamma=0$ and $z$ to $\alpha=\beta=0$). The $q$ points other than $x$ on the line ${\langle x,y \rangle}$ belong to ${\mathcal{P}}_{2,e}$. The other points of rank $2$ are the points on the line ${\langle y,z \rangle}$, which belongs to the $K$-orbit $o_{12}$ by Table \[table:lines\].
[**(c-ii) and (c-iii)**]{} If $z$ is an external point of ${\mathcal{C}}_z$ but does not lie on $t_w({\mathcal{C}}_z)$ then we obtain the orbit $$\Sigma_9 : \left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \gamma & \cdot \\ \cdot & \cdot & -\gamma \end{matrix} \right].$$ If $z$ is an internal point of ${\mathcal{C}}_z$ then we obtain the orbit $$\Sigma_{10} : \left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \gamma & \cdot \\ \cdot & \cdot & -\varepsilon\gamma \end{matrix} \right],
\quad \text{where } \epsilon \in {\mathbb{F}}_q \text{ is a non-square}.$$
\[lem:Sigma\_9\_10\] Let $\pi$ be a plane in one of the $K$-orbits $\Sigma_9$ or $\Sigma_{10}$, and let $x$ be the unique point of rank $1$ in $\pi$. The points of rank $2$ in $\pi$ lie on the union of a line $\ell$ through $x$ and a non-degenerate conic meeting $\ell$ in the point $x$. The point-orbit distribution of $\pi$ is $[1,2q,0,q^2-q ]$ or $[1,q,q,q^2-q ]$ according to whether $\pi$ belongs to $\Sigma_9$ or $\Sigma_{10}$.
Suppose that $\pi$ is the representative of $\Sigma_9$ given above, with its unique point $x$ of rank $1$ corresponding to $\beta=\gamma=0$. The points of rank $2$ in $\pi$ lie on the cubic $\gamma(\alpha\gamma-\beta^2)=0$, which is the union of the line $\ell:\gamma=0$ and a non-degenerate conic ${\mathcal{C}}:\alpha\gamma-\beta^2=0$ meeting $\ell$ in the point $x$. The points of rank $2$ on $\ell$ belong to ${\mathcal{P}}_{2,e}$ by Lemma \[lem:extcriteria\]. (Alternatively, observe that they lie on the tangent $t_x({\mathcal{C}}_y)$ to ${\mathcal{C}}_y$ through $x$ in $\langle {\mathcal{C}}_y \rangle$.) Each point of rank $2$ in ${\mathcal{C}}\setminus \ell$ has coordinates $(a^2,a,0,1,0,-1)$ for some $a \in {\mathbb{F}}_q$ and so also belongs to ${\mathcal{P}}_{2,e}$ by Lemma \[lem:extcriteria\]. The proof for $\Sigma_{10}$ is analogous, but now Lemma \[lem:extcriteria\] shows that the points of rank $2$ in ${\mathcal{C}}\setminus \ell$ belong to ${\mathcal{P}}_{2,i}$.
\[remarkSigma8910\] [It follows from Lemmas \[lem:Sigma\_8\] and \[lem:Sigma\_9\_10\] that $\Sigma_8$, $\Sigma_9$ and $\Sigma_{10}$ are three distinct $K$-orbits. Indeed, if $\pi$ is a plane in one of these orbits then its orbit can be determined as follows. If the cubic $Q$ of points of rank at most $2$ in $\pi$ is the union of two lines, then $\pi \in \Sigma_8$. If not, then $Q$ is the union of a line $\ell$ and a non-degenerate conic ${\mathcal{C}}$ meeting $\ell$ in the unique point $x$ of rank $1$ in $\pi$. In this case, consider any line $\ell' \neq \ell$ in $\pi$ through $x$. Then $\ell'$ meets ${\mathcal{C}}$ in a second point, which has rank $2$. If this point belongs to ${\mathcal{P}}_{2,e}$ then $\pi \in \Sigma_9$. Otherwise, $\pi \in \Sigma_{10}$.]{.nodecor}
Case [**(d)**]{} {#ssec:cased}
-----------------
Finally, suppose that $x\notin {\mathcal{C}}_y\cup {\mathcal{C}}_z$, and write $w = {\mathcal{C}}_y\cap {\mathcal{C}}_z$. Without loss of generality, we may fix $p_w$, $p_x$, $\ell_y$ and $\ell_z$. Let us take them as ${\langle e_1 \rangle}$, ${\langle e_2+e_3 \rangle}$, ${\langle e_1,e_2 \rangle}$ and ${\langle e_1,e_3 \rangle}$, respectively. Then the stabiliser of this configuration in ${\mathrm{PG}}(2,q)$ is contained in the group of perspectivities with centre $p_w$ (it fixes three lines through $p_w$, and hence all lines through $p_w$). On the plane $\langle {\mathcal{C}}_y\rangle$ in ${\mathrm{PG}}(5,q)$, the induced group acts as the stabiliser of the conic ${\mathcal{C}}_y$ and the point $w$. We consider separately the cases where (i) $y$ lies on the tangent line $t_w({\mathcal{C}}_y)$ to ${\mathcal{C}}_y$ through $w$ in $\langle {\mathcal{C}}_y\rangle$, and (ii) $y$ does not lie on $t_w({\mathcal{C}}_y)$.
[**(d-i)**]{} Suppose that $y$ is on the tangent line $t_w({\mathcal{C}}_y)$ to ${\mathcal{C}}_y$ through $w$ in $\langle {\mathcal{C}}_y\rangle$. Let $u$ be the unique point on ${\mathcal{C}}_y\backslash\{w\}$ such that $y=t_u({\mathcal{C}}_y)\cap t_w({\mathcal{C}}_y)$. We may take $p_u={\langle e_2 \rangle}$ and $y = (0,1,0,0,0,0)$. Then the stabiliser of $x$, $y$, $w$ and ${\mathcal{C}}_z$ is induced by the group of elations with centre $p_w$ and axis ${\langle p_x,p_u \rangle}$, and acts on $\langle {\mathcal{C}}_z\rangle$ as the stabiliser of ${\mathcal{C}}_z$ and the two points $w$ and $$v:=\nu({\langle p_x,p_u \rangle} \cap \ell_z)$$ of ${\mathcal{C}}_z$. Consider the tangent line $t_{v}({\mathcal{C}}_z)$ to ${\mathcal{C}}_z$ through $v$ in $\langle {\mathcal{C}}_z\rangle$. Note that $v=\nu({\langle e_3 \rangle})$. There are five possibilities for $z$: (A) $z=t_w({\mathcal{C}}_y)\cap t_{v}({\mathcal{C}}_z)$, (B) $z \in t_w({\mathcal{C}}_y) \setminus t_v({\mathcal{C}}_z)$, (C) $z \in t_v({\mathcal{C}}_z) \setminus t_w({\mathcal{C}}_y)$, (D) $z \in {\langle w,v \rangle}$, and (E) $z$ does not lie on ${\langle w,v \rangle}$.
[**(d-i-A)**]{} If $z=t_w({\mathcal{C}}_y)\cap t_v({\mathcal{C}}_z)$ then $z = (0,0,1,0,0,0)$. The plane $\pi={\langle x,y,z \rangle}$ then also contains the point $r = (0,1,1,0,0,0)$, with ${\mathcal{C}}_r=\nu(\langle p_x,p_w \rangle)$. Hence $\pi={\langle x,r,z \rangle}$ with $x\in {\mathcal{C}}_r$, $x\notin {\mathcal{C}}_z$, and $z$ on the tangent $t_w({\mathcal{C}}_z)$ to ${\mathcal{C}}_z$ through $w={\mathcal{C}}_r\cap{\mathcal{C}}_z$ in $\langle {\mathcal{C}}_z \rangle$. This implies that $\pi$ belongs to the $K$-orbit $\Sigma_8$.
[**(d-i-B)**]{} If $z$ lies on $t_w({\mathcal{C}}_y)$ but not on $t_{v}({\mathcal{C}}_z)$ then without loss of generality $z=(1,0,1,0,0,0)$, but then $\pi$ also contains the point $(1,1,1,1,1,1)$, which has rank $1$, a contradiction.
[**(d-i-C)**]{} If $z$ lies on $t_{v}({\mathcal{C}}_z)$ but not on $t_w({\mathcal{C}}_y)$ then without loss of generality $z=(0,0,1,0,0,1)$. This yields a new orbit $\Sigma_{11}$ with the following representative: $$\Sigma_{11} : \left[ \begin{matrix} \cdot & \beta & \gamma \\ \beta & \alpha & \alpha \\ \gamma & \alpha & \alpha+\gamma \end{matrix} \right].$$
\[lem:Sigma\_11\] A plane belonging to the $K$-orbit $\Sigma_{11}$ has point-orbit distribution $ [ 1 ,q,0, q^2]$.
The cubic of points of rank at most $2$ in the plane $\pi$ with the above representative has equation $\alpha(\beta-\gamma)^2-\beta^2\gamma=0$. Consider the lines through the point $x$ of rank $1$, which corresponds to $\beta=\gamma=0$. Each line through $x$ and a point with $\alpha=0$ and $\beta\neq \gamma$ contains exactly one point $s$ of rank $2$, represented by the matrix $$M_s={\left [ \begin{matrix} \cdot&\beta(\beta-\gamma)^2&\gamma(\beta-\gamma)^2\\
\beta(\beta-\gamma)^2& -\beta^2\gamma&-\beta^2\gamma \\
\gamma(\beta-\gamma)^2& -\beta^2\gamma&-\beta^2\gamma+\gamma(\beta-\gamma)^2 \end{matrix} \right]}.$$ In the notation of Lemma \[lem:extcriteria\] we have $-|M_{11}(M_s)|=\beta^2\gamma^2(\beta-\gamma)^2$, $-|M_{22}(M_s)|=\gamma^2(\beta-\gamma)^2$ and $-|M_{33}(M_s)|=\beta^2(\beta-\gamma)^2$, implying that $s \in {\mathcal{P}}_{2,e}$. This amounts to $q$ points of ${\mathcal{P}}_{2,e}$. The line through $x$ and the point with $\alpha=0$ and $\beta=\gamma$ contains no points of rank $2$.
[**(d-i-D)**]{} For convenience, in this section we change our points of reference, instead fixing $p_w$, $p_x$, $p_u$ and $p_v$ as ${\langle e_2 \rangle}$, ${\langle e_1 \rangle}$, ${\langle e_1+e_3 \rangle}$ and ${\langle e_3 \rangle}$, respectively. This implies that $\ell_y= {\langle e_1+e_3,e_2 \rangle}$ and $\ell_z={\langle e_2,e_3 \rangle}$. Note that $y$ is now the point with coordinates $(0,1,0,0,1,0)$. Since we are now assuming that $z$ is on the line ${\langle w,v \rangle}$, we may take $z=(0,0,0,1,0,b)$ for some non-zero $b \in {\mathbb{F}}_q$. If $b$ is a square, say $b=1$, then we obtain the orbit $$\label{Sigma12rep}
\Sigma_{12} : \left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \gamma & \beta \\ \cdot & \beta & \gamma \end{matrix} \right].$$ If $b$ is a non-square, we obtain the orbit $$\label{Sigma13rep}
\Sigma_{13} : \left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & \gamma & \beta \\ \cdot & \beta & \varepsilon\gamma \end{matrix} \right],
\quad \text{where } \varepsilon \in {\mathbb{F}}_q \text{ is a non-square}.$$ Let $\pi_{12}$, $\pi_{13}$ denote these representatives, and let $C_{12}$, $C_{13}$ denote the respective cubic curves of points of rank at most $2$ (defined by setting the determinants of the representatives to zero).
\[lem:Sigma\_12\_13\] A plane belonging to the $K$-orbit $\Sigma_{12}$ has point-orbit distribution $$[1,(q-1)/2,(q-1)/2,q^2+1],$$ and a plane belonging to the $K$-orbit $\Sigma_{13}$ has point-orbit distribution $$[1,(q+1)/2,(q+1)/2,q^2-1 ].$$
The cubic curves $C_{i}$, $i\in \{12,13\}$, are given by $\alpha f_{i}(\beta,\gamma)+g_{i}(\beta,\gamma)=0$, where $f_{12}(\beta,\gamma)=\gamma^2-\beta^2$, $f_{13}(\beta,\gamma)=\epsilon\gamma^2-\beta^2$, $g_{12}=-\beta^2\gamma$ and $g_{13}=-\epsilon\beta^2\gamma$. Consider a line through the point $x$ of rank $1$ (corresponding to $\beta=\gamma=0$) and a point with $\alpha=0$. If $f_i(\beta,\gamma) \neq 0$ then such a line contains a unique point of rank $2$. Since $f_{13}(\beta,\gamma)=0$ has no non-trivial solutions, every line through $x$ in $\pi_{13}$ contains exactly one point of rank $2$ and so $\pi_{13}$ has rank distribution $[1,q+1,q^2-1]$. On the other hand, $f_{12}(\beta,\gamma)=0$ if and only if $\beta = \pm\gamma$, and in these cases $g_{12}(\beta,\gamma) \neq 0$. Hence, in $\pi_{12}$ there are exactly two lines through $x$ which contain no point of rank $2$, and so $\pi_{12}$ has rank distribution $[1,q-1,q^2+1]$. Now, the points of rank $2$ in $\pi_{12}$ are those satisfying $\alpha=\beta^2\gamma/(\gamma^2-\beta^2)$, and Lemma \[lem:extcriteria\] implies that such a point is in ${\mathcal{P}}_{2,e}$ if and only if $\beta^2-\gamma^2$ is a non-zero square. By [@Dickson1901 Theorem 64], the quadratic form $\beta^2-\gamma^2$ evaluates to a non-zero square for precisely $(q-1)^2/2$ inputs $(\beta,\gamma) \in \mathbb{F}_q^2$, so it follows (upon factoring out scalars) that $\pi_{12}$ contains $(q-1)/2$ points in ${\mathcal{P}}_{2,e}$. Similarly a point of rank $2$ in $\pi_{13}$ is in ${\mathcal{P}}_{2,e}$ if and only if $\beta^2-\epsilon \gamma^2$ is a non-zero square. This occurs for $(q+1)(q-1)/2$ inputs $(\beta,\gamma) \in \mathbb{F}_q^2$, so it follows that $\pi_{13}$ contains $(q+1)/2$ points in ${\mathcal{P}}_{2,e}$.
\[lem:Sigma\_12\_13\_cst\_rk\_3\_line\] A plane in either of the $K$-orbits $\Sigma_{12}$ or $\Sigma_{13}$ contains a line of constant rank $3$.
The cubic curve of points of rank at most $2$ in such a plane does not form a blocking set, because it contains no lines and at most $q+2$ points.
\[lem:Sigma\_6\_13\] The $K$-orbits $\Sigma_{6}$, $\Sigma_{12}$ and $\Sigma_{13}$ are distinct.
The $K$-orbits $\Sigma_{12}$ and $\Sigma_{13}$ are distinct by Lemma \[lem:Sigma\_12\_13\]. By Lemma \[lem:Sigma\_6\_2\], a plane in $\Sigma_6$ does not contain a line of constant rank $3$. Hence, $\Sigma_6$ is distinct from $\Sigma_{12}$ and $\Sigma_{13}$ by Lemma \[lem:Sigma\_12\_13\_cst\_rk\_3\_line\].
We also record the following lemma for future reference.
\[lem:Sigma\_13\_o\_13\_line\] A plane in either of the $K$-orbits $\Sigma_{12}$ or $\Sigma_{13}$ contains a line of type $o_{13,1}$ or $o_{13,2}$.
Setting $\alpha=0$ in and yields the lines $$\left[ \begin{matrix} \cdot & \beta & \cdot \\ \beta & \gamma & \beta \\ \cdot & \beta & \gamma \end{matrix} \right]
\quad \text{and} \quad
\left[ \begin{matrix} \cdot & \beta & \cdot \\ \beta & \gamma & \beta \\ \cdot & \beta & \varepsilon\gamma \end{matrix} \right],$$ each of which is $K$-equivalent to either the $o_{13,1}$ representative or the $o_{13,2}$ representative from Table \[table:lines\]. (The exact $K$-orbit of each line depends on whether $-1$ is a square in $\mathbb{F}_q$ or not.)
Before proceeding to the analysis of the case (d-i-E), recall the definitions of the [*Hessian*]{} and the [*points of inflexion*]{} of a cubic curve: the Hessian is the determinant of the $3\times 3$ matrix of second derivatives, and the points of inflexion are the points of intersection of the Hessian and the curve. In the cases considered below, all points of inflexion lie on a common line, which we call the [*line of inflexion*]{}. Cubic curves in characteristic $3$ require separate consideration, so we state some results only for characteristic $\neq 3$. (Lemma \[lem:pi14char3\] confirms that we do not need the any of the analogous results in characteristic $3$.)
Consider again the planes $\pi_{12} \in \Sigma_{12}$ and $\pi_{13} \in \Sigma_{13}$ defined in and , respectively. Recall the definitions of the associated cubic curves $\overline{C}_{12}$ and $\overline{C}_{13}$ in ${\mathrm{PG}}(2,q)$ given immediately before Lemma \[lem:Sigma\_12\_13\]. Given a fixed basis for $\pi_i$, $i \in \{12,13\}$, define a map from $\pi_i$ to ${\mathrm{PG}}(2,q)$ in the natural way, and denote the image of $C_i$ in ${\mathrm{PG}}(2,q)$ by $\overline{C}_i$.
\[lem:Sigma\_13\_o\_13\_inflexion\] Suppose that $q$ is not a power of $3$. Then the cubic curves $\overline{C}_{12}$ and $\overline{C}_{13}$ each have a double point at $P=(1,0,0)$. The tangents at the double point, and the points and lines of inflexion, are as given in the following table:
Plane $\pi_{12}$ $\pi_{13}$
------------------- --------------------------------------- --------------------------------------------------------
Tangents at $P$ $\beta=\pm \gamma$ $\beta= \pm \sqrt{\varepsilon}\gamma$
Inflexion points $\{(0,1,0),(-1,\pm 4\sqrt{-1/3},4)\}$ $\{(0,1,0),(-\varepsilon,\pm 4\sqrt{-\epsilon/3},4)\}$
Line of inflexion $4\alpha+\gamma=0$ $4\alpha+\varepsilon\gamma=0$
In particular, when $-3$ is a square in ${\mathbb{F}}_q$, $\overline{C}_{12}$ has three ${\mathbb{F}}_q$-rational inflexion points, and $\overline{C}_{13}$ has one. When $-3$ is a non-square, $\overline{C}_{12}$ has one ${\mathbb{F}}_q$-rational inflexion point, and $\overline{C}_{13}$ has three.
The cubic $\overline{C}_{12}$ is $\alpha(\gamma^2-\beta^2)-\beta^2\gamma$, so its Hessian is $8\alpha(\gamma^2 - \beta^2) +8 \gamma(2\beta^2 + \gamma^2)$, and the points of inflexion therefore satisfy $\gamma(3\beta^2+\gamma^2)=0$. A similar calculation applies for $\overline{C}_{13}$.
[**(d-i-E)**]{} If the point $z$ is not on the line ${\langle w,v \rangle}$ and not on the tangents $t_w({\mathcal{C}}_y)$ and $t_v({\mathcal{C}}_z)$, then we may assume the line ${\langle z,w \rangle}$ passes through a point $r\in{\mathcal{C}}_z\setminus\{w,v\}$. We retain the same representatives as in case (d-i-D), and without loss of generality we further choose $r = \nu(p_r)$ with $p_r={\langle e_2-e_3 \rangle}$. We may then take $z=z_c:=(0,0,0,c,-1,1)$ where $c\not \in \{0,1\}$ (because $z_0$ lies on $t_v({\mathcal{C}}_z)$ and $z_1 = r$). The plane $\pi_c={\langle x,y,z_c \rangle}$ is then represented by the matrix $$\left[ \begin{matrix} \alpha & \beta & \cdot \\ \beta & c\gamma & \beta-\gamma \\ \cdot & \beta-\gamma & \gamma \end{matrix} \right].$$
\[lem:nice\] Let $\ell$ be a line through the point $x$ in the plane $\pi_c$, $c\in {\mathbb{F}}_q\setminus\{0,1\}$. Then $\ell$ is of type $o_{8,1}$ or $o_{8,2}$ unless $c$ is a square in ${\mathbb{F}}_q$ and $\ell$ contains one of the two points $(0,\beta,0,c\gamma,\beta-\gamma,\gamma)$ with $\beta^2-2\beta\gamma+(1-c)\gamma^2=0$. In this case, $\ell$ is of type $o_9$.
The determinant of $\pi_c$ is $\alpha(\beta^2-2\beta\gamma+(1-c)\gamma^2)+\beta^2\gamma$. Therefore, a line $\ell$ in $\pi_c$ through the point $x=(1,0,0,0,0,0)$ and a point with coordinates $(0,\beta,0,c\gamma,\beta-\gamma,\gamma)$, $(\beta,\gamma)\neq (0,0)$, contains exactly one point of rank $2$ provided that $\beta^2-2\beta\gamma+(1-c)\gamma^2\neq 0$. In this case, $\ell$ has type $o_{8,1}$ or $o_{8,2}$ because by Table \[table:line-rank-dist\] these are the only $K$-orbits of lines with rank distribution $[1,1,q-1]$. If $\beta^2-2\beta\gamma+(1-c)\gamma^2 = 0$ then $c$ must be a square; in this case, $\ell$ contains no points of rank $2$ and hence has type $o_9$.
\[lem:pi14new\] Suppose that $q$ is not a power of $3$ and let $c\in {\mathbb{F}}_q\setminus\{0,1\}$. If $-3c$ is a square in ${\mathbb{F}}_q$ and $\frac{\sqrt{c}+1}{\sqrt{c}-1}$ is not a cube in ${\mathbb{F}}_q(\sqrt{-3})$, then the plane $\pi_c$ is not in any of the $K$-orbits $\Sigma_1,\ldots,\Sigma_{13}$.
Note that we only need to show that $\pi_c \not \in \Sigma_{12} \cup \Sigma_{13}$. Consider the cubic curve $C_{c}$ obtained by setting the determinant $\alpha(\beta^2-2\beta\gamma+(1-c)\gamma^2)+\beta^2\gamma$ of $\pi_{c}$ equal to zero. The points of inflexion are precisely the points of $C_c$ for which $(\beta,\gamma) \neq (0,0)$ and $(1-c)^2\gamma^3 - 3(1-c)\beta^2\gamma + 2\beta^3 = 0$. Since $c \neq 1$, any such point has $\beta \neq 0$, so we may rewrite this equation as $(1-c)^2\theta^3-3(1-c)\theta +2=0$, where $\theta = \gamma/\beta$. By [@Dickson1906], this equation has no solutions in ${\mathbb{F}}_q$ if and only if $-3c$ is a square in ${\mathbb{F}}_q$ and $\frac{\sqrt{c}+1}{\sqrt{c}-1}$ is not a cube in ${\mathbb{F}}_q(\sqrt{-3})$. In this case, $C_c$ has no ${\mathbb{F}}_q$-rational points of inflexion, and so, by Lemma \[lem:Sigma\_13\_o\_13\_inflexion\], $\pi_c$ is not equivalent to a plane in $\Sigma_{12}$ or $\Sigma_{13}$.
\[lem:pi14equiv1213\] Let $c \in {\mathbb{F}}_q \setminus \{0,1\}$, and suppose that either $-3c$ is a non-square in ${\mathbb{F}}_q$ or $\frac{\sqrt{c}+1}{\sqrt{c}-1}$ is a cube in ${\mathbb{F}}_q(\sqrt{-3})$. Then $\pi_c \in \Sigma_{12}$ if $c$ is a square in ${\mathbb{F}}_q$, and $\pi_c \in \Sigma_{13}$ otherwise.
We prove this by explicitly mapping $\pi_{12}$ or $\pi_{13}$ to $\pi_c$ (according to whether $c$ is a square or not). By [@Dickson1906], the equation $4(1-c)^2v^3 + 3c(1 - c)v - c^2=0$ has a solution $v \in {\mathbb{F}}_q$ if and only if $-3c$ is a non-square in ${\mathbb{F}}_q$ or $\frac{\sqrt{c}+1}{\sqrt{c}-1}$ is a cube in ${\mathbb{F}}_q(\sqrt{-3})$. If $c$ is a square in ${\mathbb{F}}_q$, we may therefore define the matrix $$X = \left[ \begin{matrix} 1 & \frac{-zv}{v^2-z^2} & \frac{z^2}{v^2-z^2} \\
0 & v\sqrt{c} & z\sqrt{c} \\
0 & z & v \end{matrix} \right],$$ which satisfies $X\pi_{12}X^T = \pi_c$. Hence, $\pi_c \in \Sigma_{12}$ in this case. If $c$ is a non-square in ${\mathbb{F}}_q$ then $\varepsilon c$ is a square, and so we can instead define $$X = \left[ \begin{matrix} 1 & \frac{-\epsilon zv}{\epsilon v^2-z^2} & \frac{z^2}{\epsilon v^2-z^2} \\
0 & v\sqrt{\varepsilon c} & z\sqrt{c/\varepsilon} \\
0 & z & v \end{matrix} \right].$$ In this case, we have $X\pi_{13}X^T = \pi_c$ and hence $\pi_c \in \Sigma_{13}$.
\[lem:pi14char3\] Suppose that $q$ is a power of $3$, and let $c\in {\mathbb{F}}_q\setminus\{0,1\}$. Then $\pi_c \in \Sigma_{12} \cup \Sigma_{13}$.
Here every element of ${\mathbb{F}}_q(\sqrt{-3})$ is a cube, so the assertion follows from Lemma \[lem:pi14equiv1213\].
If $q$ is not a power of $3$ then there exists $c\in {\mathbb{F}}_q\setminus\{0,1\}$ such that the plane $\pi_c$ is not in any of the $K$-orbits $\Sigma_1,\ldots,\Sigma_{13}$.
By Lemma \[lem:pi14new\], it suffices to show that there exists $c\in {\mathbb{F}}_q$ such that $-3c$ is a square in ${\mathbb{F}}_q$ and $\frac{\sqrt{c}+1}{\sqrt{c}-1}$ is not a cube in ${\mathbb{F}}_q(\sqrt{-3})$. The element $\frac{\sqrt{c}+1}{\sqrt{c}-1}$ is a cube in ${\mathbb{F}}_q(\sqrt{-3})$ if and only if there exists $d\in {\mathbb{F}}_q(\sqrt{-3})$ such that $\sqrt{c}=\frac{d^3+1}{d^3-1}$. The function $f(d)= \big(\frac{d^3+1}{d^3-1}\big)^2$ satisfies $f(d)=f(e)$ if and only if $d^3=e^3$ or $d^3=e^{-3}$, and is therefore six-to-one on ${\mathbb{F}}_q(\sqrt{-3})\backslash\{d:d^7=d\}$. Furthermore, $f(d)$ is a non-zero square in ${\mathbb{F}}_q$ if and only if $d^{3(q-1)}=1$ and $d^6\ne 1$, and a non-square in ${\mathbb{F}}_q$ if and only if $d^{3(q+1)}=1$ and $d^6\ne 1$. If $q\equiv 1\pmod 3$ then ${\mathbb{F}}_q(\sqrt{-3})={\mathbb{F}}_q$, and $f(d)$ is a non-zero square for every $d\in {\mathbb{F}}_q$ such that $d^7\ne d$. Hence, there are $\frac{q-7}{6}+1$ elements of ${\mathbb{F}}_q$ in the image of $f$, so there exists $c\in {\mathbb{F}}_q$ such that $c$ (and hence $-3c$) is a square in ${\mathbb{F}}_q$ and $\frac{\sqrt{c}+1}{\sqrt{c}-1}$ is a non-cube in ${\mathbb{F}}_q(\sqrt{-3})$. If $q\equiv 2\pmod 3$ then ${\mathbb{F}}_q(\sqrt{-3})={\mathbb{F}}_{q^2}$. Since $\gcd(q^2-1,3(q+1))=q-1$, there are $q-5$ solutions to $d^{3(q+1)}=1$, $d^6\ne 1$, so there are $\frac{q-5}{6}$ non-squares of ${\mathbb{F}}_q$ in the image of $f$. Hence, there exists $c\in {\mathbb{F}}_q$ such that $c$ is a non-square (so $-3c$ is a square) in ${\mathbb{F}}_q$ and $\frac{\sqrt{c}+1}{\sqrt{c}-1}$ is a non-cube in ${\mathbb{F}}_q(\sqrt{-3})$.
\[lem:sigma14equiv\] Suppose that $q$ is not a power of $3$ and let $c,d \in {\mathbb{F}}_q\setminus\{0,1\}$ such that $-3c$ and $-3d$ are both squares in ${\mathbb{F}}_q$. Then the planes $\pi_{c}$ and $\pi_{d}$ belong to the same $K$-orbit if one of the following conditions holds:
- $cd=1$,
- $\big(\frac{\sqrt{c}-1}{\sqrt{c}+1}\big)\big(\frac{\sqrt{d}-1}{\sqrt{d}+1}\big)$ is a cube in ${\mathbb{F}}_q(\sqrt{-3})$,
- $\big(\frac{\sqrt{c}+1}{\sqrt{c}-1}\big)\big(\frac{\sqrt{d}-1}{\sqrt{d}+1}\big)$ is a cube in ${\mathbb{F}}_q(\sqrt{-3})$.
If $cd=1$ then $X\pi_{c}X^T=\pi_{d}$ for $$X = \left[ \begin{matrix} 1 & -d & -1 \\ 0 & 0 & -d \\ 0 & -d & 0 \end{matrix} \right],$$ so $\pi_c$ and $\pi_d$ belong to the same $K$-orbit under condition (i). Now consider conditions (ii) and (iii). Since $-3c$ and $-3d$ are both squares in ${\mathbb{F}}_q$, so are $cd$ and $d/c$. Hence, we may instead define $$X = \left[ \begin{matrix} 1 & \pm x_{12} & x_{13} \\ 0 & \pm\theta\sqrt{d/c} & \psi \\ 0 & \pm\psi/\sqrt{cd} & \theta \end{matrix} \right]$$ for some $x_{12}$, $x_{13}$, $\psi$, $\theta$. To satisfy $X\pi_{c}X^T=\pi_{d}$, we require that $$x_{12} = \frac{\psi^3 + 2\psi^2\theta +d\psi\theta^2}{\sqrt{cd}(\psi^2 - d\theta^2)},
\quad
x_{13} = -\theta-1-\frac{2\theta^2(\psi+d\theta)}{\psi^2-d\theta},$$ and that $\psi$, $\theta$ are common solutions to $$\begin{aligned}
0 &= \psi^2+2\psi\theta\pm\sqrt{d/c} \psi +d(\theta^2-\theta), \\
0 &= \psi^2/d+2\psi\theta+\psi +\theta^2-\theta\sqrt{d/c}.\end{aligned}$$ By calculating the resultant of the above polynomials with respect to $\psi$, we find that there is a solution if and only if $$\theta^3-\frac{3d}{4c}\left(\frac{c-1}{d-1}\right)\theta-\frac{d}{4c}\sqrt{\frac{d}{c}}\left(\frac{(\sqrt{cd}\mp1)(c-1)}{(d-1)^2}\right) = 0$$ has a solution $\theta \in {\mathbb{F}}_q$. The discriminant of the above cubic is $$(-3d)\left(\frac{3d(c-1)(1\mp\sqrt{d/c})}{4c(d-1)^2}\right)^2,$$ which is a square in ${\mathbb{F}}_q$ because $-3d$ and $d/c$ are both squares. Hence, by [@Dickson1906], the cubic has a solution in ${\mathbb{F}}_q$ if and only if $$\frac{d\sqrt{d}(c-1)(\sqrt{d}+1)(\sqrt{c}\mp 1)}{8c\sqrt{c}(d-1)^2}$$ is a cube in ${\mathbb{F}}_q(\sqrt{-3})$, which is precisely when $\big(\frac{\sqrt{c}\pm 1}{\sqrt{c}\mp 1}\big)\big(\frac{\sqrt{d}-1}{\sqrt{d}+1}\big)$ is a cube in ${\mathbb{F}}_q(\sqrt{-3})$.
\[Sigma14lemma\] All planes $\pi_c$ such that $\pi_c \notin \Sigma_1 \cup \dots \cup \Sigma_{13}$ belong to the same $K$-orbit.
By Lemma \[lem:pi14char3\], we may assume that $q$ is not a power of $3$. Suppose that $\pi_c$ and $\pi_d$ are two such planes. By Lemma \[lem:pi14equiv1213\], $-3c$ and $-3d$ are both squares in ${\mathbb{F}}_q$, so $cd$ is also a square in ${\mathbb{F}}_q$. Moreover, $\frac{\sqrt{c}+1}{\sqrt{c}-1}$ and $\frac{\sqrt{d}+1}{\sqrt{d}-1}$ are both non-cubes in ${\mathbb{F}}_q(\sqrt{-3})$. Hence, $\frac{\sqrt{c}+1}{\sqrt{c}-1} = \omega^i c_1^3$ and $\frac{\sqrt{d}+1}{\sqrt{d}-1}=\omega^j d_1^3$ for some $c_1,d_1\in {\mathbb{F}}_q(\sqrt{-3})$ and some $i,j\in \{1,2\}$, where $\omega$ is a primitive third root of unity. If $i=j$ then $\big(\frac{\sqrt{c}+1}{\sqrt{c}-1}\big)\big(\frac{\sqrt{d}-1}{\sqrt{d}+1}\big) = \big(\frac{d_1}{c_1}\big)^3$ is a cube in ${\mathbb{F}}_q(\sqrt{-3})$, so $\pi_c$ and $\pi_d$ belong to the same $K$-orbit by Lemma \[lem:sigma14equiv\](iii). If $i\ne j$ then $\big(\frac{\sqrt{c}-1}{\sqrt{c}+1}\big)\big(\frac{\sqrt{d}-1}{\sqrt{d}+1}\big)= \big(\frac{1}{c_1d_1}\big)^3$ is a cube in ${\mathbb{F}}_q(\sqrt{-3})$, so $\pi_c$ and $\pi_d$ belong to the same $K$-orbit by Lemma \[lem:sigma14equiv\](ii).
We denote the $K$-orbit arising from Lemma \[Sigma14lemma\] by $\Sigma_{14}$: $$\Sigma_{14}: \left[ \begin{matrix} \alpha& \beta & \cdot \\ \beta & c\gamma & \beta-\gamma \\ \cdot & \beta-\gamma & \gamma \end{matrix} \right],$$ where $q \not \equiv 0 \pmod 3$, $c \in {\mathbb{F}}_q \setminus \{0,1\}$, $-3c$ is a square in ${\mathbb{F}}_q$, and $\tfrac{\sqrt{c}+1}{\sqrt{c}-1}$ is a not a cube in $\mathbb{F}_q(\sqrt{-3})$.
A plane belonging to the $K$-orbit $\Sigma_{14}$ has point-orbit distribution $$[1, (q \mp 1)/2, (q \mp 1)/2, q^2 \pm 1], \quad \text{where } q \equiv \pm 1 \pmod 3.$$
Denote the above representative of $\Sigma_{14}$ by $\pi_{14}$. Consider the cubic curve defined by setting the determinant of $\pi_{14}$ equal to zero. The tangents through the double point $P:\beta=\gamma=0$ are $\beta=(1+\sqrt{c})\gamma$ and $\beta=(1-\sqrt{c})\gamma$, which are ${\mathbb{F}}_q$-rational if and only if $c$ is a square in ${\mathbb{F}}_q$. If $q\equiv 1 \pmod 3$ then $-3$ is a square, and thus $c$ is a square. Hence, there are $q-1$ points of rank $2$ in $\pi_{14}$. If $q\equiv -1 \pmod 3$ then $-3$ is a non-square, and thus $c$ is a non-square. In this case there are $q+1$ points of rank $2$ in $\pi_{14}$. By Lemma \[lem:extcriteria\], a rank-$2$ point of $\pi_{14}$ with $\beta\ne 0$ is exterior if and only if $(c-1)\gamma^2+2\beta\gamma-\beta^2$ is a non-zero square. This occurs for $(q+1)/2$ choices of $\beta,\gamma$ if $c$ is a non-square, and $(q-1)/2$ choices of $\beta,\gamma$ if $c$ is a square.
[**(d-ii)**]{} The only planes $\pi={\langle x,y,z \rangle}$ with $\operatorname{rank}(x)=1$ and $\operatorname{rank}(y)=\operatorname{rank}(z)=2$ that we have not yet considered are those such that $y$ is not on the tangent $t_w({\mathcal{C}}_y)$ to the conic ${\mathcal{C}}_y$ through the point $w={\mathcal{C}}_y\cap {\mathcal{C}}_z$ in $\langle {\mathcal{C}}_y \rangle$ (and likewise $z$ is not on $t_w({\mathcal{C}}_z)$). We show that there exist such planes not belonging to any of the previously considered $K$-orbits only in the case $q \equiv 0 \pmod 3$, and that all of those planes form a single $K$-orbit.
We begin with a lemma concerning certain lines spanned by points of rank $2$. (Here the notation is as above, but $y$ and $z$ are [*arbitrary*]{} points of rank $2$.)
\[lem:o1314\] Suppose that $\ell$ is a line in ${\mathrm{PG}}(5,q)$ spanned by two points $y$ and $z$ of rank $2$ such that ${\mathcal{C}}_y \ne {\mathcal{C}}_z$. Then $\ell$ is of type $o_{12}$, $o_{13,1}$, $o_{13,2}$, $o_{14,1}$ or $o_{14,2}$. Specifically,
- $\ell$ is of type $o_{12}$ if and only if $y\in t_w({\mathcal{C}}_y)$ and $z\in t_w({\mathcal{C}}_z)$,
- $\ell$ is of type $o_{13,1}$ or $o_{13,2}$ if and only if $y\in t_w({\mathcal{C}}_y)$ and $z\notin t_w({\mathcal{C}}_z)$,
- $\ell$ is of type $o_{14,1}$ or $o_{14,2}$ if and only if $y\notin t_w({\mathcal{C}}_y)$ and $z\notin t_w({\mathcal{C}}_z)$,
where $w= {\mathcal{C}}_y\cap {\mathcal{C}}_z$. Furthermore, if $\ell$ is of type $o_{14,1}$ or $o_{14,2}$ then $\ell$ contains a third point $u$ of rank $2$, and at least one of $y$, $z$, $u$ is an external point to its respective conic ${\mathcal{C}}_y$, ${\mathcal{C}}_z$, ${\mathcal{C}}_u$. Furthermore, $y\in {\langle {\mathcal{C}}_y\cap {\mathcal{C}}_z,{\mathcal{C}}_y\cap {\mathcal{C}}_u \rangle}$, $z\in {\langle {\mathcal{C}}_y\cap {\mathcal{C}}_z,{\mathcal{C}}_z\cap {\mathcal{C}}_u \rangle}$ and $u\in {\langle {\mathcal{C}}_y\cap {\mathcal{C}}_u,{\mathcal{C}}_z\cap {\mathcal{C}}_u \rangle}$.
The hyperplane spanned by the distinct conics ${\mathcal{C}}_y$ and ${\mathcal{C}}_z$ intersects ${\mathcal{V}}({\mathbb{F}}_q)$ in the union of these two conics, so the line $\ell = \langle y,z \rangle$ does not intersect ${\mathcal{V}}({\mathbb{F}}_q)$. Table \[table:line-rank-dist\] then implies that $\ell$ must have type $o_{10}$, $o_{12}$, $o_{13,1}$, $o_{13,2}$, $o_{14,1}$ or $o_{14,2}$. Type $o_{10}$ is ruled out by observing that a line of type $o_{10}$ lies in a conic plane, so any two points $y$ and $z$ of rank $2$ on such a line have ${\mathcal{C}}_y = {\mathcal{C}}_z$, contradicting our assumption. The assertions about the remaining types may be verified by direct calculations using the representatives in Table \[table:lines\].
\[lem:all\_o\_14\] Suppose that $\pi$ is a plane in ${\mathrm{PG}}(5,q)$ containing a point of rank $1$ and spanned by points of rank at most $2$, with $\pi \not \in \Sigma_1 \cup \ldots \cup \Sigma_{14}$. If $\ell$ is a line in $\pi$ spanned by points of rank $2$ and not containing a point of rank $1$, then $\ell$ has type $o_{14,1}$ or $o_{14,2}$.
The result follows from Lemma \[lem:o1314\] (and the remarks preceding it).
Continuing with the above notation, we may now assume that all lines in $\pi$ spanned by points of rank $2$ and not containing a point of rank $1$ are of type $o_{14,1}$ or $o_{14,2}$. Without loss of generality, we may choose $p_x=(1,0,0)$, $p_w=(0,0,1)$, $\ell_y:X_0=0$ and $\ell_z:X_0=X_1$, where $(X_0,X_1,X_2)$ are the homogeneous coordinates in ${\mathrm{PG}}(2,q)$. Since ${\langle y,z \rangle}$ is of type $o_{14,1}$ or $o_{14,2}$, Lemma \[lem:o1314\] implies that $y\in {\langle w,y_1 \rangle}$ and $z\in{\langle w,z_1 \rangle}$ for some $y_1\in {\mathcal{C}}_y$ and $z_1\in{\mathcal{C}}_z$. There are two cases to consider: (A) $p_{z_1} = {\langle p_x,p_{y_1} \rangle}\cap \ell_z$ and (B) $p_{z_1} \neq {\langle p_x,p_{y_1} \rangle}\cap \ell_z$.
[**(d-ii-A)**]{} Here we may take $p_{z_1} = (1,1,0)$, and so $\pi$ is represented by the matrix $$\left[ \begin{matrix} \alpha+\gamma & \gamma & \cdot \\ \gamma & \beta+\gamma & \cdot \\ \cdot & \cdot & b\beta+c\gamma \end{matrix} \right]$$ for some $b,c\in {\mathbb{F}}_q$, where $y = (0,0,0,1,0,b)$ and $z = (1,1,0,1,0,c)$. However, by choosing $\beta,\gamma$ such that $b\beta+c\gamma=0$, we then obtain another point $u\in {\langle y,z \rangle}$ of rank $2$ such that $x\in {\mathcal{C}}_u$. Hence, this case has already been considered (see the discussion at the beginning of Section \[ssec:cased\]).
[**(d-ii-B)**]{} In this case we may take $p_{z_1} = (1,1,1)$. We then have $\pi = \pi_{b,c}$ for some $b,c\in {\mathbb{F}}_q$, where $\pi_{b,c}$ is the plane represented by the matrix $$A_{b,c} = \left[ \begin{matrix} \alpha+\gamma & \gamma & \gamma \\ \gamma & \beta+\gamma & \gamma \\ \gamma & \gamma & b\beta+c\gamma \end{matrix} \right].$$ In other words, $y = (0,0,0,1,0,b)$ and $z = (1,1,1,1,1,c)$. Note that if $c=1$ then $z$ has rank $1$, and if $b=0$ then $y$ has rank $1$, so we assume that $b(c-1)\ne 0$.
\[lem:nice2\] Suppose that $b\neq 0$ and $c\neq 1$, and let $\ell$ be a line in $\pi_{b,c}$ through the point $x = (1,0,0,0,0,0)$. Then $\ell$ is of type $o_{8,1}$ or $o_{8,2}$, unless $\ell$ contains a point with coordinates in $$\{(\gamma,\gamma,\gamma,\beta+\gamma,\gamma, b\beta+c\gamma) : (\beta,\gamma)\in {\mathrm{PG}}(1,q), \; b\beta^2+(b+c)\beta\gamma+(c-1)\gamma^2=0\},$$ in which case $\ell$ is of type $o_9$.
The determinant of the matrix $A_{b,c}$ is $\alpha f_{b,c}(\beta,\gamma)+g_{b,c}(\beta,\gamma)$, where $$f_{b,c}(\beta,\gamma) = b\beta^2+(b+c)\beta\gamma+(c-1)\gamma^2, \quad
g_{b,c}(\beta,\gamma) = \beta\gamma(b\beta+(c-1)\gamma).$$ Since $b\neq 0$ and $c\ne 1$, the zero locus ${\mathcal{Z}}(g_{b,c})$ of $g_{b,c}$ consists of three distinct points in ${\mathrm{PG}}(1,q)$, and none of these points lies on ${\mathcal{Z}}(f_{b,c})$. Hence, if $(\beta_0,\gamma_0)\in {\mathrm{PG}}(1,q)$ with $f_{b,c}(\beta_0,\gamma_0)=0$, then $g_{b,c}(\beta_0,\gamma_0) \neq 0$. For such $(\beta_0,\gamma_0)$, the line through $x$ and the point of $\pi_{b,c}$ parameterised by $(0,\beta_0,\gamma_0)$ contains no points of rank $2$, and is therefore a line of type $o_9$ (by Table \[table:line-rank-dist\]). If $f_{b,c}(\beta_0,\gamma_0)\neq 0$ then this line contains exactly one point of rank $2$, so is of type $o_{8,1}$ or $o_{8,2}$.
\[lem:o13\] If the plane $\pi_{b,c}$ does not belong to any of the $K$-orbits $\Sigma_1,\dots,\Sigma_{14}$ then $\pi_{b,c}$ contains exactly $q$ points of rank $2$.
Consider the quadratic form $f_{b,c}$ defined in the proof of Lemma \[lem:nice2\]. Then ${\mathcal{Z}}(f_{b,c})$ consists of either zero, one or two points in ${\mathrm{PG}}(1,q)$, and in these respective cases, the plane $\pi_{b,c}$ contains $q+1$, $q$, or $q-1$ points of rank $2$. Choose a point $y\in \pi_{b,c}$ of rank $2$. Since any line through $y$ and another point of rank $2$ is of type $o_{14,1}$ or $o_{14,2}$, every such line contains three points of rank $2$ (see Table \[table:line-rank-dist\]). Hence, such lines partition the set of rank-$2$ points other than $y$ into pairs, and so the number of points of rank $2$ in $\pi_{b,c}$ is odd and therefore equal to $q$.
\[lem:o13b\] If $q$ is not a power of $3$ then every plane that intersects ${\mathcal{V}}({\mathbb{F}}_q)$ and is spanned by points of rank at most $2$ belongs to one of the $K$-orbits $\Sigma_1,\dots,\Sigma_{14}$.
We prove the contrapositive. We have already established that any such plane [*not*]{} belonging to any of the $K$-orbits $\Sigma_1,\dots,\Sigma_{14}$ is equivalent to a plane $\pi_{b,c}$ with $b\neq 0$ and $c\neq 1$. Using again the notation from the proof of Lemma \[lem:nice2\], observe that the quadric ${\mathcal{Z}}(f_{b,c})$ in ${\mathrm{PG}}(1,q)$ consists of one point if and only if $(b-c)^2 = -4b$. By Lemma \[lem:o13\] (and its proof), this must be the case. Consider the line $\ell$ of $\pi_{b,c}$ parameterised by $(\alpha,\beta,\gamma)$ with $\alpha+\gamma=0$. Note that $\ell$ does not contain the point $x : \beta=\gamma=0$ of rank $1$, so Lemma \[lem:all\_o\_14\] implies that $\ell$ either contains at most one point of rank $2$, or has type $o_{14,1}$ or $o_{14,2}$, in which case it contains exactly three points of rank $2$ (by Table \[table:line-rank-dist\]). The points of rank $2$ on $\ell$ are determined by the solutions of $\gamma^2((c-1)\gamma+(b+1)\beta)=0$, so $\ell$ contains exactly two points of rank $2$ unless $b=-1$. Therefore, we must have $b=-1$. Putting this into $(b-c)^2 = -4b$ yields $c=-3$ (because $c \neq 1$). However, then the points of rank $2$ on the line of $\pi_{-1,-3}$ parameterised by $(\alpha,\beta,\gamma)$ with $\alpha+\beta=0$ are determined by the solutions of $\beta^2(3\gamma+\beta)=0$. This line, similarly, cannot contain exactly two points of rank $2$, and so $q$ must be a power of $3$.
By Lemma \[lem:o13b\] and its proof, if the plane $\pi_{b,c}$ does [*not*]{} belong to one of the previously considered $K$-orbits $\Sigma_1,\dots,\Sigma_{14}$, then $q$ is a power of $3$ and $(b,c)=(-1,0)$. This yields the following representative of a new orbit which we call $\Sigma_{14}'$: $$\Sigma_{14}': \left[ \begin{matrix} \alpha+\gamma& \gamma & \gamma \\ \gamma & \beta+\gamma & \gamma \\ \gamma & \gamma & -\beta \end{matrix} \right],
\quad \text{for } q \equiv 0 \; (\text{mod } 3).$$
\[Sigma14’-distribution\] If $q$ is a power of $3$ then the $K$-orbit $\Sigma_{14}'$ is distinct from the $K$-orbits $\Sigma_1,\dots,\Sigma_{14}$, and a plane in $\Sigma_{14}'$ has point-orbit distribution $[1,q,0,q^2]$.
Let $\pi$ denote the above representative of $\Sigma_{14}'$. The points of rank $2$ in $\pi$ have the form $$\left[ \begin{matrix} \frac{\gamma^3}{(\beta-\gamma)^2} & \gamma & \gamma \\ \gamma & \beta+\gamma & \gamma \\ \gamma & \gamma & -\beta \end{matrix} \right]
\quad \text{with} \quad \beta \neq \gamma.$$ The three principal minors of this matrix are $$-\left(\frac{\gamma\beta}{\beta-\gamma}\right)^2, \; -(\gamma-\beta)^2, \; -\left(\frac{\gamma(\beta+\gamma)}{\beta-\gamma}\right)^2.$$ Hence, by Lemma \[lem:extcriteria\], all points of rank $2$ in $\pi$ are in ${\mathcal{P}}_{2,e}$, and there are $q$ such points, as claimed. Table \[table:pt-orbit-dist\] now implies that $\pi$ does not belong to any of the previously considered $K$-orbits, with the possible exception of $\Sigma_{11}$. Therefore, it remains to show that $\pi \not \in \Sigma_{11}$. By Lemma \[lem:all\_o\_14\], every line in $\pi$ spanned by points of rank $2$ and not containing the unique point of rank $1$ in $\pi$ has type $o_{14,1}$ or $o_{14,2}$. In particular, every such line contains exactly three points of rank $2$. Consider, on the other hand, the representative of $\Sigma_{11}$ from Table \[table:main\], namely $$\left[ \begin{matrix} \cdot & \beta & \gamma \\ \beta & \alpha & \alpha \\ \gamma & \alpha & \alpha+\gamma \end{matrix} \right].$$ The points in this plane parameterised by $(\alpha,\beta,\gamma) = (0,1,0)$ and $(0,0,1)$ both have rank $2$ and span a line with rank distribution $[0,2,q-1]$. Therefore, $\pi \not \in \Sigma_{11}$, as claimed.
Planes meeting ${\mathcal{V}}({\mathbb{F}}_q)$ in a point and not spanned by points of rank ${\leqslant}2$ {#sec:final}
==========================================================================================================
Finally, we consider the planes in ${\mathrm{PG}}(5,q)$ which contain exactly one point of rank $1$ and are not spanned by points of rank at most $2$. Let $\pi$ be such a plane, and let $x$ denote the unique point of rank $1$ in $\pi$. By assumption, at most one of the lines through $x$ in $\pi$ contains another point of rank at most $2$, so the other lines through $x$ contain no points of rank $2$. By inspecting the possible rank distributions in Table \[table:line-rank-dist\], we see that each such line has type $o_9$. Hence, without loss of generality we can assume that $\pi$ contains the representative of the line orbit $o_9$ given in Table \[table:lines\]. In other words, we can assume that $x=(1,0,0,0,0,0)$ and that $\pi$ contains the rank-$3$ point $z = (0,0,1,1,0,0)$. We may then assume that $\pi = \langle x,y,z \rangle$ where $y$ has coordinates $(0,1,0,a,b,c)$ for some $a,b,c \in \mathbb{F}_q$. Hence, we may represent $\pi$ by the matrix $$\left[ \begin{matrix} \alpha& \beta & \gamma \\ \beta & a\beta+\gamma & b\beta \\ \gamma & b\beta & c\beta \end{matrix} \right].$$ The determinant of this matrix is $\alpha f(\beta, \gamma) + g(\beta, \gamma)$, where $f(\beta,\gamma) = (ac-b^2)\beta^2 + c\beta\gamma$ and $g(\beta,\gamma) = \beta^2(2b\gamma-c\beta) - \gamma^2(a\beta+\gamma)$. There is at most one point ${\langle (\beta_0,\gamma_0) \rangle}$ of ${\mathrm{PG}}(1,q)$ such that $f(\beta_0,\gamma_0)\ne 0$, because any such point corresponds to a point in $\pi$ of rank at most $2$, namely the point parameterised by $(\alpha_0,\beta_0,\gamma_0)$ with $\alpha_0=-{g(\beta_0,\gamma_0)}/{f(\beta_0,\gamma_0)}$. Since $f$ defines a quadric on ${\mathrm{PG}}(1,q)$, this implies that $f(\beta,\gamma)$ must be identically zero. Therefore, $b=c=0$. We claim that $a=0$ also. If not, then the points in $\pi$ of rank $2$ are those parameterised by $(\alpha,\beta,\gamma)$ with $\gamma^2(a\beta+\gamma)=0$, contradicting the fact that the points of rank at most $2$ of lie on a line. Hence, $a=0$. This yields the following representative of the final $K$-orbit, $\Sigma_{15}$: $$\Sigma_{15} : \left[ \begin{matrix} \alpha& \beta & \gamma \\ \beta & \gamma & \cdot \\ \gamma & \cdot & \cdot \end{matrix} \right].$$
\[lem:Sigma\_?\] The $K$-orbit $\Sigma_{15}$ is distinct from all of $\Sigma_1, \dots, \Sigma_{14}$, and from $\Sigma_{14}'$ when $q \equiv 0 \pmod 3$. A plane in $\Sigma_{15}$ has point-orbit distribution $[1,q,0,q^2]$.
The points of rank at most $2$ in the above representative of $\Sigma_{15}$ are all on the line $\langle x,y\rangle$ (namely $\gamma=0$), which has type $o_6$ (see Table \[table:lines\]). Moreover, the points of rank $2$ all lie in ${\mathcal{P}}_{2,e}$ (by Lemma \[lem:extcriteria\]). Hence, this plane has point-orbit distribution $[1,q,0,q^2]$. As explained at the beginning of this section, the plane is not in any of the previously considered $K$-orbits (because it contains a single point of rank $1$ and is not spanned by points of rank at most $2$).
This completes the proof of Theorem \[mainthm\].
Wilson’s classification of nets of rank one {#sec:Wilson}
===========================================
As explained in Section \[sec:prelims\] (see Proposition \[prop:nets\_vs\_planes\]), Theorem \[mainthm\] implies Corollary \[maincor\], namely the classification of nets of conics of rank one in ${\mathrm{PG}}(2,q)$ (for $q$ odd). We now compare the latter classification with that of Wilson [@Wilson1914], and explain why Wilson’s classification was incomplete.
In Part I of his paper [@Wilson1914], Wilson studied the nets of rank one, namely those containing a repeated line. He obtained 11 “canonical types", labelled $\mathrm{I,II,\ldots, XI}$. In Part II, he studied the nets of rank two (those not containing a repeated line, but containing a conic which is not absolutely irreducible), obtaining six canonical types $\mathrm{XII,\ldots, XVII}$. In Part III, he obtained two canonical types $\mathrm{XVIII}$ and $\mathrm{XIX}$ of nets of rank three. Wilson was aware of the fact that he had not completely classified the orbits, pointing out that [*“All questions of inter-relations between these types have been considered and answered, except with respect to the two cases, nets XVI and XVII, and nets XVIII and XIX."*]{} [@Wilson1914 p. 207].
$\Sigma_1$ $\Sigma_2$ $\Sigma_3$ $\Sigma_4$ $\Sigma_5$ $\Sigma_6$ $\Sigma_7$ $\Sigma_8$ $\Sigma_9$ $\Sigma_{10}$ $\Sigma_{11}$ $\Sigma_{12}$ $\Sigma_{13}$ $\Sigma_{14}$ $\Sigma'_{14}$ $\Sigma_{15}$
------------ ---------------- --------------- --------------- --------------- ----------------- -------------- -------------- ---------------- ---------------- --------------- --------------- ----------------- ----------------- ---------------- ---------------
$\mathrm{III}$ $\mathrm{IV}$ $\mathrm{VI}$ $\mathrm{XI}$ $\mathrm{VIII}$ $\mathrm{I}$ $\mathrm{V}$ $\mathrm{VII}$ $\mathrm{VII}$ $\mathrm{IX}$ $\mathrm{XI}$ $\mathrm{X,XI}$ $\mathrm{X,XI}$ $\mathrm{II}$
: Correspondence between $K$-orbits of planes in ${\mathrm{PG}}(5,q)$ and canonical types $\mathrm{I},\ldots,\mathrm{XI}$ of nets of conics of rank one in ${\mathrm{PG}}(2,q)$ obtained by Wilson [@Wilson1914].[]{data-label="Wilson-table"}
Although Wilson’s work was in general very thorough, there are also some other issues with his classification. Besides the aforementioned open cases, there are some inaccuracies, and some missing orbits. Table \[Wilson-table\] shows the correspondence between the $K$-orbits $\Sigma_1,\ldots, \Sigma_{15}$ of planes in ${\mathrm{PG}}(5,q)$ obtained in this paper, and Wilson’s canonical types $\mathrm{I},\ldots,\mathrm{XI}$ of nets of conics of rank one in ${\mathrm{PG}}(2,q)$. As the table illustrates, neither the $K$-orbit $\Sigma_1$ nor the $K$-orbit $\Sigma'_{14}$ (which only arises in characteristic $3$) appear in Wilson’s classification. Moreover, some of Wilson’s canonical types of nets correspond to more than one $K$-orbit. Specifically, type $\mathrm{VII}$ is the union of the $K$-orbits $\Sigma_9$ and $\Sigma_{10}$, and type $\mathrm{XI}$ includes the $K$-orbits $\Sigma_5$, $\Sigma_{12}$, and sometimes $\Sigma_{14}$. The equivalence between the nets of types $\mathrm X$ and $\mathrm{XI}$ is studied on p. 194 of Wilson’s paper, and on p. 196 the author concludes that these types are equivalent except when $q$ has the form $36k+17$. Taking into account that the $K$-orbit $\Sigma_1$ was overlooked and that the nets of type $\mathrm{VII}$ split into two $K$-orbits, this would imply that for such values of $q$ the number of $K$-orbits of planes corresponding to nets of rank one would be $14$, contradicting our classification (in which there are $15$ orbits for every value of $q$).
Acknowledgements {#acknowledgements .unnumbered}
================
The first author acknowledges the support of [*The Scientific and Technological Research Council of Turkey*]{} TÜBİTAK (project no. 118F159).
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---
author:
- |
David R. Wright\
North Carolina State University\
[drwrigh3@ncsu.edu](drwrigh3@ncsu.edu)\
Comments welcome — Please send to the author at the email address above.
title: |
Motivation, Design, and Ubiquity:\
A Discussion of Research Ethics\
and Computer Science
---
This material is based upon work supported by the National Science Foundation under Grant No. 9818359. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).
Copyright 2006, David R. Wright and North Carolina State University. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Abstracting with credit is permitted.
The author acknowledges with great appreciation Gary Comstock, Thomas Honeycutt, and Ed Gehringer for their comments and guidance in preparing this essay.
Introduction {#essay-intro}
============
Computers and the software that makes them work have penetrated into nearly every niche of modern life. Manufacturing, transportation, communications, medical care, financial, and government systems, among others, are highly dependent upon computers and software systems. This places the disciplines of Computer Science and Software Engineering in a particularly critical position, wielding more influence over the day-to-day lives of people all over the world than any other single field of research and application. Writing about engineering in general, Bugliarello[-@b:mach-mod-eth] notes that “...ethics of engineering must go beyond broad generalities and codes of professional good conduct modeled after the Hippocratic oath.” Little of modern engineering could be accomplished without computers and software, and computer scientists and software engineers must be aware of the broad range of ethical responsibilities that come with being direct or indirect enablers of change.
Science and engineering are commonly distinguished as two different sorts of activities. Science, generally speaking, is the pursuit of theoretical results, while engineering seeks to apply those results through the creation and refinement of technology. Computer science, as it is generally organized, taught, and researched, stands in a unique position, intimately involved in both theoretical and applied research. If engineering requires an ethical position beyond professional codes of conduct, as Bugliarello notes, then a discipline that spans both theory and application, and that touches so many facets of life, should be grounded in an equally (or stronger) ethical foundation.
In discussions of ethics and computing, topics such as hacking, computer viruses, software piracy, privacy, and security often come to mind. These subjects are commonly collected under the heading of “Computer Ethics.” This essay, however, is directed towards *researchers* in computer science and software engineering, and is concerned with the ethical conduct of research in these and related fields. Concern for research ethics is growing throughout the research community in general. Given the impact research in computer science has had, and will continue to have, on a global scale, it is important that researchers in this discipline understand potential ethical dilemmas in their research.
In the LANGURE[^1] series of modules, ethics is defined as the study of arguments about right and wrong, good and bad. We have identified four theories that offer guidance in this decision process, and in this essay we assume that the reader has been introduced to these theories and how to apply them. As a basis for analyzing the dilemmas we will discuss, we propose to use a simple and straightforward principle: if everyone engaged in a particular behavior, would we consider that behavior correct and proper? It is important to note that this perspective is concerned both with the act or behavior itself as well as the results. Judging the rightness or wrongness of an act consequentalist and nonconsequentalist considerations is an important task in any area of research. The reader is, of course, free to apply their own ethical analysis of the situations presented.
The intent of this paper is not only to point out concerns and potential dilemmas, but also to build awareness and to provoke discussion of these important issues within the discipline of Computer Science. Computers have become nearly ubiquitous in modern society, and their influence reaches to the most remote parts of the globe. Research in computer science can often find a place in real-world applications far faster than in any other discipline. As Wulf notes, the ripple effects from innovations and technological achievements are often impossible to predict[@w:great-achi], and can affect the lives of millions of people. Furthermore, many sources for research funding require ethical review of research projects, especially when these projects involve human subjects. In many fields the line of demarcation regarding human subjects or participants is clear. This is often not the case in computer science research. How this discipline responds to growing economic and regulatory pressure will directly affect researchers in computer science, and indirectly influence research in other disciplines that depend on innovations in computing for their livelihood.
The immediate goals of this essay are twofold: first, to identify some critical ethical concerns and responsibilities relating to research in computing; and second, to promote awareness and discussion among researchers in this field about ethical concerns and issues. Section \[concerns\] examines practical aspects of computer science research that involve ethical judgement. Section \[negl\] reflects upon the preceding discussion in conjunction with the reader’s own experiences and observations in an attempt to evaluate the current level of ethical awareness in the discipline, and what can (and should) be done to improve it. A hypothetical case is presented in Section \[case\] along with questions to stimulate discussion. Section \[codes\] lists Codes of Conduct for researchers and practitioners of computer science and related fields.
Ethical Concerns in the Conduct of\
Computer Science Research {#concerns}
===================================
In this section we discuss several key ethical issues that impact computer science research. These are practical concerns that most researchers in the field must be aware of and address in the design, implementation, analysis, and reporting of research projects. Most researchers in all fields are aware of the need to take special care and precautions when their research involve human subjects. In computing research, however, there are often individuals at risk of harm that are not at all obvious. Because of the ubiquity and casual use of computing devices today, researchers must carefully assess the possible risks to others using these devices. Researchers must also evaluate the potential for harm to those who generate information sources subsequently used in experimental or case studies. Identifying at-risk individuals and groups is an essential component of ethical research conduct.
The quest for knowledge and understanding is an open-ended journey, requiring researchers to look forward while building upon the work or predecessors. Some areas of computing research are relatively mature and stable while other areas are more dynamic and fluid, and the level of stability is not necessarily dependent upon how long significant research efforts have been underway. In these dynamic areas, researchers are often faced with a difficult choice: corroborating prior work to strengthen the foundations of the research area or “pushing the envelope” while relying on prior work that may be less reliable. The pressure to be original and cutting edge can be enormous, with the potential to make or break a career. These decisions must be made under the influence of the basic scientific principles that define and guide the conduct of research in general.
We begin by looking at the role of software in research, both as the entity under study and as a research tool to examine and measure other entities and/or phenomena. Potential concerns include the objectivity of the software as well as the software developer(s), interactions between different computer programs and/or computer hardware, and human interference. These issues raise significant but often unacknowledged ethical concerns that require extra care in the design, preparation, and execution of software-related experiments.
Software as the subject of research {#soft-subj .unnumbered}
-----------------------------------
Software systems are often placed “under the microscope” of scientific study to evaluate their performance under different situations and stresses. A research problem is identified, and a hypothesis formulated as a solution to the problem. The hypothesis is translated into a computer program, which is then tested against the constraints that characterize the problem. The hypothesis is either corroborated or falsified by the software tests.
To illustrate the potential ethical dilemmas involved in this research, consider the typical software development process. A customer/client identifies a particular problem to be solved or goal to be achieved through the development of a software tool or system, for example, a web-based retail sales portal. The problem or goal is described in detail by a set of requirements stating exactly what the software should and should not do. In this example, the requirements would state privacy and security policies, protocols allowing the application to interact with shipping companies to calculate delivery costs, interfaces with the company’s product and customer databases and accounting systems, user interface design, etc., as well as specific requirements for demonstrating that the expected improvement actually occurs. These requirements are translated into specifications for the system, which form the basis for the design and implementation of the software. Test suites are also developed to evaluate the correct behavior of the delivered product, based on the identified requirements. Software that does not pass a predefined minimal set of tests is judged to be unacceptable because it does not fulfill a minimal set of requirements.
While a research hypothesis usually will not provide a full set of requirements in a formal and explicit manner, those requirements exist implicitly in the description of the problem, the context in which the problem is being investigated, and the solution proposed by the hypothesis. An example might involve the development of a new algorithm for searching databases. The researcher’s hypothesis states that this new algorithm resolves a problem existing algorithms exhibit when searching for particular kinds of information in databases using certain types of structure or organization. The hypothesis specifies an algorithm (with specific requirements as to its implementation), a particular kind of search operation (characterizing the contents of the databases), and the structure and organization of those databases. These requirements define the functional and nonfunctional behaviors and characteristics of the research software just as they do in a commercial development project. This leads to two questions:
1. If the software under study appears to corroborate the hypothesis, is this due to a weak suite of tests, a weakly stated hypothesis, “creative programming,” or accurate corroboration?
2. If the software under study appears to falsify or disprove the hypothesis, is this a result of an error in creating the software (or test suite), an intentional effort to disprove the hypothesis, or an accurate falsification?
Answering these questions is not a trivial task, and will require careful examination of experimental design, software development, environmental constraints, data collection and analysis, and duplicability. Computers are fundamentally deterministic machines: they do what they are told to do. If the results are not what was expected, is it because the expectation was wrong, or does the fault exist in the instructions given? Similarly, if the end results correspond with the expectations, is it because the expectations were valid, or was a particular set of instructions chosen that would lead to those results regardless of the validity of the hypothesis?
Defects in software are the norm rather than the exception, and most of the research in software engineering over the past forty years has sought processes to develop software systems with fewer defects, and techniques to detect defects more reliably. Complicating the defect level is the difficulty in exhaustively testing software systems to guarantee “zero defects” in the delivered product. Based on the historical track record of software development, an unexpected falsification of a research hypothesis is often viewed with a degree of skepticism, spurring more careful examination of the software under study and the environment in which the failure occurred. This may then lead to a correction of defects in the software or test suite, or a re-evaluation of the hypothesis.
Similarly, programming errors can also result in a false positive experimental result. The software under study may not accurately reflect the requirements implied in the hypothesis, or the test suite (experimental setup) may not accurately measure the results, either accidentally or intentionally. Researchers must apply the same level of skepticism and caution to justify positive results as they do to mitigate a negative outcome. Furthermore, detailed records and documentation must be maintained in order to justify the correctness of the experimental approach, software design and implementation, experimental environment (hardware and software), and collected data[@z:rel-res].
In the same paper, Zobel points out several other related ethical problems in empirical studies of software, noting that the actual implementation code itself is not sufficient documentation of an experiment. A researcher’s bias towards a particular algorithm or implementation can alter the results of an experiment. The choice of data, underlying hardware and software, and the quality of the implementations used in an experiment can have significant effects on the results. Programmer skill and experience with the hardware environment can also have critical effects: a programmer with minimal experience using the C programming language on a UNIX or Linux machine may write less reliable code than one with more extensive experience. He also notes that it is not sufficient to note that a particular implementation worked for a particular data set; it must be shown to work consistently on a variety of input data. Additionally, the experimenter must explain *why* the implementation worked, as well as providing detailed information about failures[@z:rel-res]. These are among the standards for research in most other scientific disciplines, and certainly should be in computer science.
Researchers must also exercise care in the selection and application of metrics and analysis methods. The search for reliable metrics continues to be an active research area, motivated, in part, by the need to define a core set of measurements and analyses to measure various aspects of software quality and performance. El Emam, Benlarbi, Goel, and Rai demonstrated that even commonly accepted metrics are not as reliable as expected, identifying a “confounding effect” in certain object-oriented source code metrics[-@ebgr:valid-oo-metric]. These metrics had been shown to correlate with the *fault-proneness* of a class implementation, but using the method they developed to control for the class implementation size, the metrics no longer showed the expected relationship. Repeating prior experiments by other researchers generated the same results (i.e., the lack of correlation between the metrics and the defects), leading them to question the common use of these metrics and to recommend revisiting previous studies to confirm earlier conclusions.
Metrics and analysis methods are primary instruments of computer scientists researching software systems. The development and validation of these metrics and methods is complicated by the dynamic and diverse nature of computing itself: metrics are usually dependent upon the programming language used and the platform where the software is executed. A lack of consensus on the topic of measuring instruments and metrics combined with minimal attention directed towards the quality of measurement results in this discipline, as compared to other science and engineering fields, further complicates the researcher’s work [@as:meas-se]. Jones [-@j:sw-metr] is even more critical: “The software industry is an embarrassment when it comes to measurement and metrics,” noting that the measurements generally reported in the literature lack the precision to duplicate the author’s work, a canonical requirement in other scientific disciplines.
Because software is a human-created entity, and is fundamentally deterministic (although this determinism may be obscured by the system’s complexity), many opportunities for bias, both intentional and accidental, exist in the study of software itself. Furthermore, computer software is an artifact of human creation that exists (as a dynamic phenomenon) within another artificial, human-created environment, and researchers must be careful to ensure the objectivity of experimental designs and implementations. Software written for the expressed purpose of demonstrating a particular performance characteristic should be as neutral as possible, and where implementation biases exist, they *must* be disclosed and their effect on the results explained to allow other researchers and practitioners to relate those results to their own work — this is a hallmark of responsible science. Well-known and understood metrics should be used whenever possible, and any possible biases in measurement, such as those identified in [@ebgr:valid-oo-metric], should also be accounted for in the analysis of the results. The failure to disclose biases, either actual or potential, in the implementation, metrics, or analysis of experimental software can easily be interpreted as laziness or even as an attempt to intentionally deceive others, neither of which are acceptable in scientific research[@nas:resp-sci-v2 p. 168].
Detailed record-keeping is an essential aspect of scientific research, but one that is often neglected in software research. Part of this is attributable to the feeling that “the records are in the software” and associated documentation[@z:rel-res]. As Zobel goes on to point out, this is a very weak assumption on several accounts. The current version of the software, while a result of all the changes that occurred in prior versions, usually does not retain all of the original code. The relationships between the source code, software requirements and design, experimental design, and research hypothesis can easily be obscured, and the source code itself cannot capture and store the entire sequence of design and implementation decisions that led to it: the finished product is a *result* of all of those decisions, but not a *record* of them. Zobel’s recommendation is to adopt the practices of other research disciplines, keeping detailed daily journals of all research-related work, use version control systems to maintain all versions of experimental software, and to record detailed logs of all experimental data, even if there is no expectation of using all of the data generated[@z:rel-res]. The nature of software and the virtual environment in which it exists does not permit short-cutting the essential tasks of researchers. Rather, this nature and environment require as much, if not more care, discipline, and effort than may be common in other fields.
Software as a research tool {#soft-tool .unnumbered}
---------------------------
In addition to being the subject of investigation, software is used in computer science research as a tool for making a variety of measurements in existing or new systems, either hardware or software. The use of software benchmarks (programs that measure the performance of hardware and software systems) became popular in the early 1980s, spurred by the variety of hardware platforms resulting from the introduction of personal computer systems. These results generated by these tools were frequently used in commercial advertising as well as in research environments. Benchmarking software continues to be a widely used method for evaluating hardware and software system performance, particularly in research environments. The problems surrounding the choice of measuring instruments, how the measurements are actually implemented in the system under study, and the interpretation of the results raise several ethics-related issues.
We begin a discussion about the ethical use of software instrumentation with several concerns, based on the work of Levy and Clark[-@lc:bench-perf], regarding the choice of benchmark software. First and foremost, a researcher must understand what it is that the benchmark program actually measures, as well as the specific behavior(s) or attribute(s) to be measured. Pointing out weaknesses in benchmark software, Gustafson identifies several examples of benchmarks that do not necessarily measure what the user might expect. One example checks the results of floating-point calculations to only four decimal places, although the benchmark program itself requires the use of high-precision (64-bit) data[@g:purp-bench]. This approximation saves processing time, but if the user is interested in floating-point mathematical operations and is not aware of this benchmark’s limitations, the interpretation of the results may be in error. Investigating the **`health`** benchmark, Ziles found that this particular tool was not a valid benchmark for the type of measurement it performed, but rather was a “micro-benchmark” and an inefficient one at that[-@z:bench-harm]. The **`health`** benchmark is a health care simulator that models a hierarchy of hospitals and the patients undergoing treatment at each one, and is used to evaluate the performance of linked data structures. Ziles found that the benchmark actually measured traversal time over long linked-lists, and demonstrated an algorithmic modification to the benchmark itself that improved the execution time of the tool by a factor of 200. Furthermore, he also found that the model itself is flawed because the number of patients arriving exceeds the capacity of the hospitals to treat and discharge them; thus as the simulation is run for longer times to collect more data, the reported performance of the system under evaluation will decline as the waiting list grows longer and longer.
A second area of concern that Levy and Clark identify is that of bias in benchmarks as a result of how the programs are implemented, including the particular programming language and/or compiler used to create the program[@lc:bench-perf]. Programming languages have different features and semantics that may offer more efficient ways of coding specific operations. Compilers are platform-specific, and may also perform optimizations tailored to specific hardware architectures. While a researcher might be able to easily recognize optimizations resulting from the choice of programming language or particular features within a language, identifying architecture-specific optimizations performed by a compiler is a much more difficult task. Levy and Clark highlight an example of this kind of bias using several different benchmark programs written in different languages, and compiled and executed on different hardware platforms[@lc:bench-perf]. Their goal was to compare the performance of these tools to identify differences related to the implementation language and execution platform. They concluded that the benchmark performance cannot be attributed to any one factor, and that this type of scalar measurement of execution time is not a reliable way to compare the performance of different hardware and software systems. A direct comparison of two different compilers, using the same hardware architecture and platform, was done recently by Gurumani and Milenkovic, who found that code compiled by the Intel C++ compiler performed better than code generated by Microsoft’s Visual C++ compiler using a recognized standard benchmark suite[@gm:cpp-bench].
The Standard Performance Evaluation Corporation (SPEC) CPU2000 suite of benchmarks is an industry-standardized reference for measuring a computer system’s processor, memory, and compiler support[@spec:cpu2000]. Citron surveyed 173 papers published in three respected computer architecture conferences two years after the suite’s introduction. He found that 115 of these papers cited the use of of a SPEC benchmark, but only 23 of these used then entire suite[@c:misspec]. Of the 92 partial uses, only 27 papers discussed the reason(s) for not using the entire suite. This survey raises two concerns: the limited use of the entire suite of benchmarks that are widely accepted as stable and reliable performance measuring instruments; and the lack of explanation for omitting some of the tests or substituting simulated results in place of actual measurements. Citron notes that this partial use can be misleading to readers, particularly when little or no explanation is given — readers may not have sufficient information to draw realistic conclusions about the research results.
Hennessy, Citron, Patterson, and Sohi[-@hcps:use-abuse-spec] continue this discussion, generally drawing the same conclusions regarding the misuse of the SPEC benchmarks as Citron did in the paper noted above. Sohi, however, argues that there are sometimes extenuating circumstances for not using the entire suite: not all of the tools compile, certain tools evaluate aspects of performance not thought to be related to the problem at hand, etc. He compares this to pharmaceutical researchers who do not try to evaluate a new drug’s performance against all possible diseases. This statement is misleading on two key points. First, as Hennessy points out in the conclusion of this discussion, alternative applications are sometimes found for drugs that are not effective for their original purpose, citing Viagra as an example. Second, Sohi does not recognize that most of the research performed on a new drug prior to \[human\] clinical trials is designed to determine the positive and negative effects of the drug[@fda:beg]. Concern for negative side effects continues throughout the clinical testing phase[@fda:clin], and throughout the life of the drug, as exhibited by the recent removal of an entire group of drugs (including Vioxx, Celebrex, and Bextra) from the market as a result of significant risks to patients[@fda:cox2-talk]. Unfortunately, Sohi seems to ignore the fact that thorough testing can illuminate negative behaviors in computer systems that limited benchmarking may hide.
The third issue that computer scientists who are using benchmarks need to be aware of is how the benchmark program interacts with the system under study. On one hand, a program that consumes excessive resources such as memory, processor time, or disk I/O may significantly affect the measurements made. On the other hand, if the benchmark does not interact sufficiently with the system under study, it may not be able to accurately collect data from the system. The compromise usually involves some kind of data sampling coupled with statistical analysis. Often, this analysis is the computation of a mean value for a data set, although the choice of which mean (arithmetic, harmonic, or geometric) to use is sometimes a matter for dispute. Mashey concludes that the choice of which mean to use for a particular benchmark is less important than understanding the statistical distribution of the data set[@m:war-bench].
Benchmarking tools are a necessary component of computer science research and can effectively and positively influence the maturity of the discipline by providing a common set of experimental evaluations tools and techniques. The availability of standardized, well-defined means of making measurements is an essential element of science and research. This common ground fosters consensus, collaboration, and rigor[@seh:swe-res-bench]. It is critical, however, that the tools themselves are well understood and documented so that researchers can choose the instrument appropriate for the task at hand. Researchers must also take responsibility for the benchmarks they choose to employ, and fully disclose the reasons for making those choices as well as detailed explanations of how the tools are used. It is also essential to accurately and specifically define *what* is measured and why.
Basic principles of scientific inquiry {#basic-prin .unnumbered}
--------------------------------------
Rational discourse, replication, and criticism are cornerstones of modern scientific inquiry, providing the means to advance science and to translate theory into practice and technology. They also help define what is and is not responsible conduct and reporting of research, a critical topic for a relatively young discipline such as computer science. We will consider each of these three subjects as they apply to computer science research and discuss how an awareness of their ethical importance can improve and strengthen research efforts in this discipline.
Rational discourse is the clear and commonly understood communication among scientists of a discipline, and between them and researchers in other disciplines and the “outside” world. On the surface, this communication involves style and standards for publication, and computer science has built largely upon earlier standards established in fields like mathematics and logic. It is important, however, to look beyond the *how* we communicate and examine *what* we discuss and *why*. Computer science has evolved from its origins in mathematics and logic to encompass a broad diversity of research areas, ranging from formal languages and models of computation, stochastic processes, algorithms, and data structures to human-computer interaction, networked and embedded computing, and the many specialties that fall under the umbrella of software engineering. Over the past fifty years, computers have moved from the isolated and controlled “computer room” to a nearly ubiquitous presence in modern life.
This variety of perspectives has led to a corresponding diversity of research in computer science. Without a well-defined body of knowledge, these “branches" have fed primarily on themselves in terms of prior foundational work. While diversity of opinion and research emphasis is not a sign of poor ethical judgement or practice, relying strictly on one path of research introduces additional risk into the research effort itself, particularly when references to work outside that perspective are the exception rather than the rule in new research. Precisely because the body of knowledge is still youthful and growing, it is critically important to relate new research to other investigations into the same or similar problems. It is also important to seek out and relate research in fields outside software engineering to this new work, as results in other fields may have direct and valuable bearing on computer science research.
In his 1993 Turing Award Lecture, Hartmanis argues that computer science is fundamentally different from other sciences because it does not conform to the paradigms of the physical sciences [@h:turing]. The “defining characteristics of computer science” include the unprecedented difference in scale between individual bits of data and programs and the billions of instructions or operations carried out each second by modern computers and the different roles played by theory and experiment. Theories are used to develop methodologies, models, logics, and semantics to be used on program design and implementation, rather than to explain observations and predict new phenomena. Experiments in computer science demonstrate the validity of these methods and models, constrained by their design contexts and actual implementations. Hartmanis concludes that computer science is indeed an “independent new science, but it is intertwined and permeated with engineering concerns and considerations” and a “new form of engineering...the engineering of mathematics or mathematical processes.”[@h:turing] Because of these differences, Hartmanis contends that research in computer science, and the rules of more “traditional” sciences do not equally apply, in particular to empirical research.
Stewart counters Hartmanis’s arguments by first noting that physics deals with multiple scales: phenomena ranging from the tiniest subatomic particles to the entire universe[@s:sci-cs], a span of at least $10^{41}$ in magnitude and eclipsing the “immense differences in scale” that Hartmanis claims are present in computer science[@h:turing]. Stewart also notes that many research areas in computer science are equally capable of developing theories that can be empirically judged rather than just demonstrated, suggesting the common relation between theory and experiment in this discipline is a “fatal flaw to be remedied” rather than a reason for maintaining the *status quo*[@s:sci-cs]. Stewart agrees that computer science is a science in the traditional sense, while sternly criticizing the discipline for its failure to adhere to the basic principles of modern science. In particular, he notes that many subdisciplines rely more on exploratory experimentation rather than stating well-defined problems amicable to theoretical and empirical analysis.
Freeman asserts that an effective computer science research program requires not only a strong core knowledge, but must also interact with other disciplines in a larger, real-world context [@f:eff-cs]. Hartmanis also asserted that computer science must “increase its contact and intellectual interchange with other disciplines,” look for important emerging computer science problems in other fields, avoid focusing on distinctions between basic and applied research and development, and embrace the transfer of knowledge between academic, industrial and social researchers[@h:comp-fut]. Leveson has noted the need to incorporate research in social and cognitive sciences into software engineering research[@l:se-lim-comp]. Glass, Vessy, and Ramesh also concluded that software engineering research is both narrow in research method and approach, and focuses on itself as a reference discipline rather than looking outward for foundational or supportive research[@gvr:rsch-se]. Given the current ubiquity of computing and the prospect that it will become more and more integrated into society, it is imperative that research topics and discourse become more open and active in interdisciplinary research efforts. Computer science must build not only upon itself, but must actively seek out and build upon research and theory from other disciplines. Failure to do so is a breach of the implied moral and social contract between science and society.
The ability to duplicate the work of other researchers is perhaps the most fundamental principle and responsibility of science. Repeating an experiment allows a new result to be corroborated or refuted, as well as providing the means to restate and refine the problem under consideration. Duplicating the prior work of other researchers is often also more than simply recreating the earlier experiment: the later researcher should also be looking for new results that extend or clarify the earlier work or have stated a thesis outlining why the duplication is expected to fail. It is the continual refinement of hypotheses that builds credibility of researchers and results alike.
Rational discourse is a requisite for replicating research. Clear and precise descriptions of experimental setups and protocols, research hypotheses, results anticipated based on theoretical analyses, and complete records of collected data provide the groundwork from which other researchers can attempt to corroborate, refute, and refine existing research. From these attempts to duplicate prior work new, and potentially more interesting, problems become clear. Detailed records can also protect researchers against accusations of fraud or misconduct, just as lapses may indicate their presence.
Given the diversity of research topics within computer science, there cannot be a single standard of judgement for documentation and record keeping. Some experiments, e.g., those studying human reactions or interactions with computing systems, will have requirements closer to those of psychological or cognitive science research, while areas such as experimental algorithmics will not require the same degree of detail[@z:rel-res]. Zobel also points out that, in light of rapidly changing computer technology, it may be nearly impossible to experimentally reproduce results, but that these results should illustrate the same underlying phenomena.
It is the responsibility of computer science researchers to build upon the work of others, while creating new knowledge and understanding to extend and strengthen the discipline. For a young discipline or subarea, it is often critical to look outside the immediate bounds of that research area to similar fields in other areas of investigation. Unfortunately, that outward-looking attitude is the exception rather than the rule in computer science and software engineering: 89% and 98% (respectively) of literature references were within the field of study according to one survey of publications[@grv:rsch-comp-disc]. Not only does this introversion place limits on possible insights and innovations, it reflects a separation from the larger community of science and society.
This leads us to the issue of criticism within the rational discourse of a discipline. Science grows through critical analysis and testing of hypotheses: one need only look at the history of science to see this phenomenon. But critical analysis is only effective when hypotheses are stated clearly and precisely. Generalizations may be easy to corroborate, but ultimately tell us little about the real world, since they can be very difficult to disprove.
Popper identifies four critical lines of testing[@p:logic-sci-disc pp. 32-33] for scientific theories:
- The comparison of conclusions among themselves for internal consistency.
- Investigation of a theory’s logical form to determine if it is empirical/scientific or tautological.
- Comparison with other theories to determine if it represents a scientific advance.
- Testing via empirical application of conclusions derived from the theory.
The identification of these types of testing lead to his definition of the term *falsifiability* as the ability for a scientific system to be refuted by experience[@p:logic-sci-disc pp. 40-41]. Basically, he asserts that scientific theories are not “proven true” by positive results, but only supported temporarily pending a negative result. He also shows that the *empirical content* of a statement increases with the degree of falsifiability — the more specific a theory is (e.g., the more it forbids), the more that theory says about the world of experience[@p:logic-sci-disc p.119]. For example, the statement “there are white ravens” is not falsifiable without examining every raven that exists, past, present, and future, everywhere in the universe. Such statements are considered by Popper to be non-empirical. On the other hand, a statement like “all ravens are black” is falsifiable; one need only find a single raven that is not black to refute the hypothesis. In short, scientific criticism is enabled by testable hypotheses.
Glass, et. al. found that the majority of publications in computer science and software engineering presented abstract and formulative research findings[@grv:rsch-comp-disc; @gvr:rsch-se]. Hartmanis uses this emphasis on the abstract *how* to justify computer science’s distinction as a new kind of science[@h:turing], while Stewart[@s:sci-cs] criticizes the vague and often unstated problems and hypotheses that characterize some areas of research in the discipline. Tichy, Lukowicz, Prechelt, and Heinz are less gentle[-@tlph:exp-eval-cs]: they call the lack of experimental evaluation in computer science “unacceptable, even alarming,” echoing Jones[-@j:sw-metr] characterization of the software industry as “amateurish craft” and an “embarrassment.” They discount explanations regarding the youth of the discipline, the difficulty of experimentation, and the fear of negative career impact, with the unstated implication that the root cause may be simple laziness or not wanting to figuratively rock the boat. The risk of damage to the discipline is great, and because of the intimacy and pervasiveness of computing today, the risk to society as a whole is even greater. To paraphrase Asimov’s Zeroth Law of Robotics[@a:r-and-e]: “A computer scientist may not injure humanity, or, through inaction, allow humanity to come to harm.”
Human participants in computing research {#human .unnumbered}
----------------------------------------
When the topic of human participants in computer science research is discussed, the first area that probably comes to mind is human-computer interaction. Evaluation of educational methods or analytic/modeling techniques for software development would likely follow close behind. There are, however, other research areas that involve human participants in more subtle ways. In this section, we consider some of these areas through discussions of specific cases. We conclude with a comparison of corporate entities and human participants in research, and raise some hitherto unasked questions about potential ethical issues paralleling those concerning human subjects.
The first case we consider is that of research involving *open source software* (OSS). El-Emam points out that OSS is an attractive alternative for researchers desiring to analyze large software systems, as the source code is publicly and freely available[@e:eth-oss]. He also points out that developers of OSS code did not intend for their work to be used as the subject of research, and thus informed consent must be given in order to use the code. However, the task of getting consent from everyone involved in an OSS project is daunting: there may be hundreds or even thousands of contributors, and a researcher cannot know in advance which particular individual’s contributions are going to be the subject of analysis. El-Emam also raises questions regarding minimization of harm and confidentiality. Version control systems used in OSS projects tag contributions with information identifying the developer responsible for each piece of code integrated into the system. If the research resulting in the publication of code segments, particularly in a negative manner, the source code could be traced back to the individual who wrote it, and possibly cause that person professional or personal harm.
Responding to this dilemma, Vinson and Singer note that eliminating personal identifiers from the reported data in order to maintain confidentiality reduces the need for informed consent, but does not necessarily eliminate it[@vs:source-eth]. Furthermore, removing personal identifiers is far from simple. Consider a popular open source e-mail client that is available for a wide variety of hardware platforms and operating systems: Mozilla Thunderbird. Even if the name of this particular program was not stated, the set of possible candidates is small. Including a segment of source code could facilitate the identification of the program with a search of the OSS project’s source code repository as well as the identity of the contributing developer. Vinson and Singer also point out that removing identifiers inhibits replicating the research because subsequent researchers cannot be sure they are evaluating the same software. This also reduces the compatibility of the initial results with later findings for the same reason. They conclude that research with OSS is “fraught with ethical issues,” and encourage researchers to be proactive in the development of ethical guidelines for this type of research.
A second case is described by Storey, Phillips, and Maczewski, where they conducted research, using students as subjects, on web-based learning tools[@spm:web-lt-stu]. Among the ethical issues they raised were the inadvertent coercion of students to participate (by giving course credit to participants), the fairness between students who dropped out and those who did not, and the involvement of the instructor and teaching assistant as experimenters in the research. Participants also faced the stress of learning and switching between new and unfamiliar tools, as well as extraordinary time demands on the teaching staff due to questions about and defects in the systems under evaluation. Davis provides extensive insight into the ethical issues in this case, focusing on the coercion that existed in the experiment[@d:vol]. Davis notes that the students who “volunteered” for the study had registered for the class without knowing about the research. Students registered for the class with the expectation of doing the usual course work, but were then faced with making an unexpected choice between participating in the research or writing a significant review of the tools used in the study. Changing their schedules would have incurred significant financial cost, further coercing students into participating in the study. While this choice is not necessarily wrong, he notes that it does require justification from two perspectives: 1) that the new requirement (participating in the study or writing the review) has positive value for the students, and 2) there is no less-coercive means to achieve the resulting good. Davis concludes by questioning why the institution’s ethics review committee did not identify the coercion in the experiment.
Sieber notes similarities between students and employees as subjects in research, in particular their vulnerability to psychological, social and economic harm[@s:prot-res-sub]. Recruitment by coworkers or supervisors can be coercive; announcements should be made in public forums and volunteers accepted in private to minimize the peer pressure to participate. Singer and Vinson identified other ethical issues involving research in corporate environments with employee subjects[@sv:eth-emp-stud-se]. In the course of their research, managerial approval was required before subjects could be interviewed to ensure that the employees were not bothered during critical work periods, compromising the anonymity of the subjects. Observation of employees at work in most corporate environments (i.e., cubicles) clearly identifies subjects to coworkers and supervisors. Singer and Vinson also emphasized the need to work with large groups of subjects to minimize the chance of identifying individuals from research reports. Finally, they noted that employers sponsoring research, and allowing their employees to participate in it, are often interested in how the subjects are performing. Not only are confidentiality issues at work, but there is potential conflict of interest between the good of the research, the good of the employee/subject, and the good of the company.
Seaman raises new questions about a researcher’s interactions with employees in her case regarding a pair of qualitative studies conducted at two different companies[@s:eth-qual-comm-sw]. To help foster a relaxed relationship with the subjects, the researcher (Seaman) engaged in frequent informal conversation with the employees at both companies. The second company knew of the first study, and in the course of these informal conversations, asked questions about the work practices at the first company. Evading the questions might reinforce the observer-subject relationship and compromise the research, while answering them might cause employees to be concerned about the researcher discussing *their* work habits with a subsequent company, also risking the quality of the research. Gotterbarn responds to this dilemma by asserting that this is not a research ethics problem, but an awkward situation, pointing out that the ethical principles are clear. Only information specifically authorized in the consent agreement can be discussed. Handled tactfully, the researcher can use this situation to build additional trust between herself and the company[@g:eth-qual-comm-sw].
Is Research Ethics a Neglected Topic in the Training of Computer Science Researchers? {#negl}
=====================================================================================
The purpose of this essay has been to point out some ethical issues that can occur in the course of research in computer science. We have purposely avoided topics such as plagiarism, authorship, and more general principles of responsible research conduct, as these are common to all research areas. Issues pertaining to professional ethics have also been omitted, as they relate to the conduct of practitioners in industry rather than researchers, topics that are better addressed in a review directed at professional practice. It is hoped that by focusing attention on these less than obvious issues researchers in computer science and related fields will become more aware of ethical concerns in their research and begin dialogs among their colleagues on the subject.
To highlight the problem, we conclude by briefly discussing a survey conducted to investigate the awareness of research ethics among academic computer science and software engineering researchers in the United Kingdom, with discouraging results. Among the results Hall and Flynn reported, only 16 of the 44 respondents consider monitoring the ethical considerations of software engineering research to be very important[@hf:eth-se-res]. The other respondents stated they did not have feelings either way (39%), they consider such monitoring not important (18%), or they don’t know (7%). Hall and Flynn also reported some striking comments from respondents:
> “I find this questionnaire very worrying because the idea of having to seek ethical approval threatens academic freedom.”
>
> “(Seeking ethical approval) has never arisen and I don’t know why this is an issue.”
>
> “No one is responsible for the ethical approval of CS research.”
This survey was conducted in the UK and the results may not be directly extrapolated to universities in the United States. However, this author’s conversations with graduate students and faculty confirms a more general apathy towards research ethics. Many students, in particular, are not aware of review requirements for human subjects in research. Even more discouraging, however, was the attitude towards research ethics shown by some of the candidates the author had the opportunity to question during a recent faculty search. Several of these candidates (all were currently full professors) did not see the relevance or importance of research ethics training for PhD students in computer science, even when some of the situations discussed in this essay were described to them. Their ethical concerns related to plagiarism and cheating. While these are certainly important topics, there are other very significant ethical issues in computer science research as this essay has tried to illustrate.
Faulty research in computer science may not have the obvious effects that failures in medicine or other disciplines may exhibit. Rather, these effects may be subtle and very difficult to identify when they find their way into real-world applications, but can affect the lives of millions of people. Computers are routinely used to model and forecast weather, crop production, the spread of infectious diseases, economic markets, and many other complex systems. Modern society relies on these models, and subtle flaws or undocumented behaviors resulting from shortcuts in research can have both immediate and long-term effects. Algorithms can be incorporated into commonly used software tools (e.g., spreadsheets, database systems, internet search engines, etc.) based upon the performance reported in research papers, but if the real-world performance does not meet the expectations from the research, time and money are wasted.
It is not hard to imagine scenarios where computers can have significant consequences for people in the remotest parts of the world, people who have never seen a computer themselves. Consider the following example. A deadly disease begins spreading through small, remote villages. Health officials use computers to store and analyze data collected on the disease, tracking its spread to identify infection vectors and characterizing its symptoms to narrow the possible causes. If the software does not perform as expected, based on the research leading to its implementation, critical time is wasted pursuing false leads. Other researchers use computer-aided microscopy and other analysis methods to visualize the virus and map its structure for comparison to known viruses, with more potential for lost time. Drug companies use the collected data to guide data mining software through their repositories of compounds they have developed, searching for potential treatments, cures, and vaccines. A data mining algorithm that was reported to work well on this kind and scale of information, based on partial testing and simulations, does not perform to expectations, but this flaw is unnoticed because the size of the data sources and complexity of the search criteria prevent manual correlation as a backup, and a probable cure goes uninvestigated. These are not “defects” in the usual sense of the term — the software does not crash, visibly erroneous results are not generated — everything seems to be working as designed.
The pervasiveness of computers makes this possible. Computer science is a unique discipline, and needs a unique and comprehensive approach to research ethics. Our modern society relies heavily on computer systems that have the potential to reach into the remotest parts of the world. Researchers must be aware of the immediate consequences of their decisions as well as the long-term and far-reaching effects they can have on the world. Society expects this level of diligence from researchers in medicine, biotechnology, agriculture, energy, and many other disciplines. Because computers are a vital part of virtually every other research area, computer scientists should hold themselves to these high standards and expectations.
Case Study {#case}
==========
Ken and Ann are doctoral students working under Dr. Smith, an internationally respected computer scientist whose specialty is the static analysis of large software systems. Static analysis is used to analyze how a software system will behave without actually executing the program. The results of this analysis can be used to identify programming defects and verify behavior against specifications. Over the past decade, Dr. Smith and his students, along with colleagues from other institutions, have developed and refined a suite of tools for identifying software defects using static analysis. Ken and Ann have each made separate modifications to detect two kinds of defects that are currently not identifiable using this suite of tools. Their experimental setup includes “control” software systems written specifically to include and exclude the particular defects they are interested in, as well as mission-critical systems provided by four vendors who are also providing funding for the research in exchange for access to the resulting data.
Testing their modified tools against the control systems generates the expected results — the intentionally introduced defects are identified, and there are no false positives. Satisfied with these results, Ken and Ann independently run their tools against seven real-world systems, where both notice an interesting anomaly. Their tools do indeed identify the defects, but both also return a statistically significant number of false positives from one vendor’s supplied systems.
Ken attempts to duplicate the false positives by modifying the defect-free control to include the code causing the errors in the vendor systems. All of his trials against these modified controls fail to replicate the false positives generated by the actual systems. Stymied, Ken concludes that there must be some subtleties in other parts of the vendor’s design or implementation that is causing the false positives.
Meanwhile, Ann takes a different track to identify the cause of the anomalous behavior by first running the unmodified analysis tools against the vendor systems. Interestingly, she finds that other analyses also produce false positives with this vendor’s software more consistently than with systems from the other six vendors. Reviewing the results of earlier work with the tool suite by other researchers, Ann finds that this behavior has been noted before, but explained as an artifact of this vendor’s particular programming style. Ann contacts Ken (Dr. Smith is currently out of the country at a conference) and finds that he has noted the same behavior and concluded that the source of the problem is in the software system, not in the analysis tools. Ann is uncomfortable with this position and suspect some kind of bias in the test suite, but a manual analysis of these software systems would be very tedious and time-consuming. As both students are under pressing submission deadlines, they decide to write up their results as is, accepting the previous conclusions for the anomalous behavior and using the published statements as validation for their conclusions.
**Questions for discussion:**
1. The *ACM Code of Ethics and Professional Conduct*[-@acm:ethics] states in section 2.5 that a ACM computing professional will “Give comprehensive and thorough evaluations of computer systems and their impacts, including analysis of possible risks.”
Similarly, the joint ACM/IEEE *Software Engineering Code of Ethics and Professional Practice*[-@acm-ieee:se-ethics] states that a software engineer shall:
> 6.07. Be accurate in stating the characteristics of software on which they work, avoiding not only false claims but also claims that might reasonably be supposed to be speculative, vacuous, deceptive, misleading, or doubtful.
>
> 6.08. Take responsibility for detecting, correcting, and reporting errors in software and associated documents on which they work.
In light of these statements, is it clear that Ann and Ken cannot ignore the anomalous behavior detected in the course of their research?
2. Should anomalous results always be published?
3. If the similar anomalies have been noted by other researchers using the same software tools, data sources, etc., should these prior comments be invoked as the sole explanation for the extraordinary behavior, or does a researcher have an obligation to extend his or her investigation in an attempt to explain the behavior?
4. What other steps could Ken and Ann take to identify the suspected bias in their experiments?
5. What risks do Ann and Ken expose themselves to by attempting to identify any such bias? What are the risks if they do not?
6. What risks exist for their advisor, Dr. Smith, if they take either course of action?
7. What rules of thumb do we have with respect to making these decisions?
Codes of Conduct {#codes}
================
- **ACM Code of Ethics and Professional Conduct**
<http://www.acm.org/constitution/code.html>
- **Software Engineering Code of Ethics and Professional Practice**
<http://www.acm.org/serving/se/code.htm>
- **IEEE Code of Ethics**
<http://www.ieee.org/web/membership/ethics/code_ethics.html>
- **AITP Code of Ethics**
<http://www.aitp.org/organization/about/ethics/ethics.jsp>
- **Australian Computer Society Code of Ethics**
<http://www.acs.org.au/national/pospaper/acs131.htm>
- **Computer Society of India Code of Ethics**
<http://courses.cs.vt.edu/~cs3604/lib/WorldCodes/India.Code.html>
- **DPMA Code of Ethics**
<http://courses.cs.vt.edu/~cs3604/lib/WorldCodes/DPMA.html>
- **An Engineer’s Hippocratic Oath**
<http://courses.cs.vt.edu/~cs3604/lib/WorldCodes/Hippocr.Oath.html>
- **New Zealand Computer Society Code of Ethics & Professional Conduct**
<http://www.nzcs.org.nz/SITE_Default/about_NZCS/Code_of_ethics.asp>
- **NSPE Code of Ethics for Engineers**
<http://www.nspe.org/ethics/eh1-code.asp>
- **American Mathematical Society Ethical Guidelines**
<http://www.ams.org/secretary/ethics.html>
- **Ethical Principles of Psychologists and Code of Conduct**
<http://www.apa.org/ethics/code2002.html>
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[^1]: *LANGURE* is an acronym for Land Grant University Research Ethics, a NSF-funded project to develop a model curriculum in research ethics for doctoral candidates. For more information, please see the program home page at <http://www.chass.ncsu.edu/langure/>.
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abstract: |
We report experimental results for the influence of a tilt angle $\beta$ relative to gravity on turbulent Rayleigh-Bénard convection of cylindrical samples. The measurements were made at Rayleigh numbers $R$ up to $10^{11}$ with two samples of height $L$ equal to the diameter $D$ (aspect ratio $\Gamma \equiv D/L \simeq 1$), one with $L \simeq 0.5$ m (the “large" sample) and the other with $\L \simeq 0.25$ m (the “medium" sample). The fluid was water with a Prandtl number $\sigma = 4.38$.
In contrast to the experiences reported by [@CRCC04] for a similar sample but with $\Gamma \simeq 0.5$ ($D = 0.5$ and $L = 1.0$ m), we found no long relaxation times.
For $R = 9.4\times 10^{10}$ we measured the Nusselt number $ \cal N $ as a function of $\beta$ and obtained a small $\beta$ dependence given by $\ {\cal N}(\beta) = {\cal N}_0 [1 - (3.1\pm 0.1)\times 10^{-2}|\beta|]$ when $\beta$ is in radian. This depression of $\cal N$ is about a factor of 50 smaller than the result found by [@CRCC04] for their $\Gamma \simeq 0.5$ sample.
We measured side-wall temperatures at eight equally spaced azimuthal locations on the horizontal mid-plane of the sample and used them to obtain cross-correlation functions between opposite azimuthal locations. The correlation functions had Gaussian peaks centered about $t_1^{cc} > 0$ that corresponded to half a turn-over time of the large-scale circulation (LSC) and yielded Reynolds numbers $R_e^{cc}$ of the LSC. For the large sample and $R = 9.4\times 10^{10}$ we found $R_e^{cc}(\beta) = R_e^{cc}(0)\times [1 + (1.85\pm 0.21) |\beta| - (5.9\pm 1.7) \beta^2]$. Similar results were obtained also from the auto-correlation functions of individual thermometers. These results are consistent with measurements of the amplitude $\delta$ of the azimuthal side-wall temperature-variation at the mid-plane that gave $\delta(\beta) = \delta(0)\times [1 + (1.84 \pm 0.45) |\beta| - (3.1 \pm 3.9) \beta^2]$ for the same $R$. An important conclusion is that the increase of the speed (i.e. of $R_e$) with $\beta$ of the LSC does not significantly influence the heat transport. Thus the heat transport must be determined primarily by the instability mechanism operative in the boundary layers, rather than by the rate at which “plumes" are carried away by the LSC. This mechanism apparently is independent of $\beta$.
Over the range $10^9 \stackrel {<}{_\sim} R \stackrel {<}{_\sim} 10^{11}$ the enhancement of $R_e^{cc}$ at constant $\beta$ due to the tilt could be described by a power law of $R$ with an exponent of $-1/6$, consistent with a simple model that balances the additional buoyancy due to the tilt angle by the shear stress across the boundary layers.
Even a small tilt angle dramatically suppressed the azimuthal meandering and the sudden reorientations characteristic of the LSC in a sample with $\beta = 0$.
For large $R$ the azimuthal mean of the temperature at the horizontal mid-plane differed significantly form the average of the top- and bottom-plate temperatures due to non-Boussinesq effects, but within our resolution was independent of $\beta$.
author:
- 'Guenter Ahlers, Eric Brown, and Alexei Nikolaenko'
title: 'Absence of slow transients, and the effect of imperfect vertical alignment, in turbulent Rayleigh-Bénard convection'
---
Introduction {#intro}
============
Turbulent convection in a fluid heated from below, known as Rayleigh-Bénard convection (RBC), has been under intense study for some time \[for reviews, see [*e.g.*]{} [@Si94; @Ka01; @AGL02]\]. A central prediction of models for this system \[[@Kr62; @CGHKLTWZZ89; @SS90; @GL01]\] is the heat transported by the fluid. It is usually described in terms of the Nusselt number $${\cal N} = \frac{Q L}{A \lambda \Delta T}
\label{eq:nusselt}$$ where $Q$ is the heat current, $L$ the cell height, $A$ the cross-sectional area, $\lambda$ the thermal conductivity, and $\Delta T$ the applied temperature difference. The Nusselt number depends on the Rayleigh number $$R = \alpha g \Delta T L^3/\kappa \nu
\label{eq:R}$$ and on the Prandtl number $$\sigma = \nu/\kappa\ .$$ Here $\alpha$ is the isobaric thermal expansion coefficient, $g$ the acceleration of gravity, $\kappa$ the thermal diffusivity, and $\nu$ the kinematic viscosity.
An important feature of turbulent RBC is the existence of a large-scale circulation (LSC) of the fluid \[[@KH81]\]. For cylindrical samples of aspect ratio $\Gamma \equiv L/D \simeq 1$ the LSC is known to consist of a single cell, with fluid rising along the wall at some azimuthal location $\theta$ and descending along the wall at a location $\theta + \pi$ \[see, for instance, [@QT01a]\]. As $\Gamma$ decreases, the nature of the LSC is believed to change. For $\Gamma \stackrel {<}{_\sim} 0.5$ it is expected \[[@VC03; @SV05; @SXX05]\] that the LSC consists of two or more convection cells, situated vertically one above the other. Regardless of the LSC structure, the heat transport in turbulent RBC is mediated by the emission of hot (cold) volumes of fluid known as “plumes" from a more or less quiescent boundary layer above (below) the bottom (top) plate. These plumes are swept away laterally by the LSC and rise (fall) primarily near the side wall. Their buoyancy helps to sustain the LSC.
In a recent paper [@CRCC04] reported measurements using a cylindrical sample of water with $\sigma \simeq 2.33$ and with $L = 1$ m and $D = 0.5$ m for $R \simeq 10^{12}$. Their sample thus had an aspect ratio $\Gamma \simeq 0.5$ at the borderline between a single-cell and a multi-cell LSC. They found exceptionally long relaxation times of $\cal N$ that they attributed to a switching of the LSC structure between two states. Multi-stability was observed also in Nusselt-number measurements by [@RCCH04] for a $\Gamma = 0.5$ sample (see also [@NBFA05] for a discussion of these data). Chillá et al. also found that $\cal N$ was reduced by tilting the sample through an angle $\beta$ relative to gravity by an amount given approximately by ${\cal N}(\beta)/ {\cal N}(0) \simeq 1 - 2 \beta$ when $\beta$ is measured in radian. A reduction by two to five percent of $\cal N$ (depending on $R$) due to a tilt by $\beta \simeq 0.035$ of a $\Gamma = 0.5$ sample was reported as well recently by [@SXX05], although in that paper the $\beta$-dependence of this effect was not reported. Chillá et al. developed a simple model that yielded a depression of $\cal N$ for the two-cell structure that was consistent in size with their measurements. Their model also assumes that no depression of $\cal N$ should be found for a sample of aspect ratio near unity where the LSC is believed to consist of a single convection cell; they found some evidence to support this in the work of [@BTL95]. Indeed, recent measurements by [@NBFA05] for $\Gamma = 1$ gave the same $\cal N$ within 0.1 percent for a level sample and a sample tilted by 0.035 rad.
In this paper we report on a long-term study of RBC in a cylindrical sample with $\Gamma \simeq 1$. As expected, we found no long relaxation times because the LSC is uniquely defined. The establishment of a statistically stationary state after a large change of $R$ occurred remarkably quickly, within a couple of hours, and thereafter there were no further long-term drifts over periods of many days.
We also studied the orientation $\theta_0$ of the circulation plane of the LSC by measuring the side-wall temperature at eight azimuthal locations \[[@BNA05b]\]. With the sample carefully leveled ([*i.e.*]{} $\beta = 0$) we found $\theta_0$ to change erratically, with large fluctuations. There were occasional relatively rapid reorientations, as observed before by [@SBN02]. The reorientations usually consisted of relatively rapid rotations, and rarely were reversals involving the cessation of the LSC followed by its re-establishment with a new orientation. This LSC dynamics yielded a broad probability distribution-function $P(\theta_0)$, although a preferred orientation prevailed. When the sample was tilted relative to gravity through an angle $\beta$, a well defined new orientation of the LSC circulation plane was established, $P(\theta_0)$ became much more narrow, and virtually all meandering and reorientation of the LSC was suppressed.
We found that $\cal N$ was reduced very slightly by tilting the sample. We obtained $\ {\cal N}(\beta) = {\cal N}_0 [1 - (3.1\pm 0.1)\times 10^{-2}|\beta|]$. This effect is about a factor of 50 smaller than the one observed by Chillá et al. for their $\Gamma = 0.5$ sample.
From side-wall-temperature measurements at two opposite locations we determined time cross-correlation functions $C_{i,j}$. The $C_{i,j}$ had a peak that could be fitted well by a Gaussian function, centered about a characteristic time $t^{cc}_1$ that we interpreted as corresponding to the transit time needed by long-lived thermal disturbances to travel with the LSC from one side of the sample to the other, i.e. to half a turnover time of the LSC. We found that the $\beta$-dependence of the corresponding Reynolds number $R_e^{cc}$ is given by $R_e^{cc}(\beta) = R_e^{cc}(0)\times [1 + (1.85\pm 0.21) |\beta| - (5.9\pm 1.7) \beta^2]$. A similar result was obtained from the auto-correlation functions of individual thermometers. Thus there is an ${\cal O}(1)$ effect of $\beta$ on $R_e$, and yet the effect of $\beta$ on $\cal N$ was seen to be nearly two orders of magnitude smaller. We also determined the temperature amplitude $\delta$ of the azimuthal temperature variation at the mid-plane. We expect $\delta$ to be a monotonically increasing function of the speed of the LSC passing the mid-plane, i.e. of the Reynolds number. We found $\delta(\beta) = \delta(0)\times [1 + (1.84 \pm 0.45) |\beta| - (3.1 \pm 3.9) \beta^2]$. Thus, for small $\beta$ its $\beta$-dependence is very similar to that of the Reynolds number.
From the large effect of $\beta$ on $R_e$ and the very small effect on $\cal N$ we come to the important conclusion that the heat transport in this system is not influenced significantly by the strength of the LSC. This heat transport thus must be determined primarily by the efficiency of instability mechanisms in the boundary layers. It seems reasonable that these mechanisms should be nearly independent of $\beta$ when $\beta$ is small. This result is consistent with prior measurements by [@CCL96], who studied the LSC and the Nusselt number in a sample with a rectangular cross section. They inserted vertical grids above (below) the bottom (top) plate that suppressed the LSC, and found that within their resolution of a percent or so the heat transport was unaltered. Their shadowgraph visualizations beautifully illustrate that the plumes are swept along laterally by the LSC when there are no grids and rise or fall vertically due to their buoyancy in the presence of the grids. [@CCL96] also studied the effect of tilting their rectangular sample by an angle of 0.17 rad. Consistent with the very small effect of tilting on $\cal N$ found by us, they found that within their resolution the heat transport remained unaltered.
We observed that the sudden reorientations of the LSC that are characteristic of the level sample are strongly suppressed by even a small tilt angle.
Apparatus and Data Analysis {#sec:apparatus}
===========================
For the present work we used the “large" and the “medium" sample and apparatus described in detail by [@BNFA05a]. Copper top and bottom plates each contained five thermistors close to the copper-fluid interface. The bottom plate had imbedded in it a resistive heater capable of delivering up to 1.5 kW uniformly distributed over the plate. The top plate was cooled via temperature-controlled water circulating in a double-spiral channel. For the Nusselt-number measurements a temperature set-point for a digital feedback regulator was specified. The regulator read one of the bottom-plate thermometers at time intervals of a few seconds and provided appropriate power to the heater. The top-plate temperature was determined by the temperature-controlled cooling water from two Neslab RTE740 refrigerated circulators.
Each apparatus was mounted on a base plate that in turn was supported by three legs consisting of long threaded rods passing vertically through the plate. The entire apparatus thus could be tilted by an angle $\beta$ relative to the gravitational acceleration by turning one of the rods. The maximum tilt angle attainable was 0.12 (0.21) rad for the large (medium) sample.
The Nusselt number was calculated using the temperatures recorded in each plate and the power dissipated in the bottom-plate heater. The side wall was plexiglas of thickness 0.64 cm (0.32 cm) for the large (medium) sample. It determined the length $L$ of the sample. Around a circumference the height was uniformly $50.62 \pm 0.01$ cm ( $24.76 \pm 0.01$ cm) for the large (medium) sample. The inside diameter was $D = 49.70 \pm 0.01$ cm ( $D = 24.84 \pm 0.01$ cm) for the large (medium) sample. The end plates had anvils that protruded into the side wall, thus guaranteeing a circular cross section near the ends. For the large sample we made measurements of the outside diameter near the half-height after many months of measurements and found that this diameter varied around the circumference by less than 0.1%.
Imbedded in the side wall and within 0.06 cm of the fluid-plexiglas interface were eight thermistors, equally spaced azimuthally and positioned vertically at half height of the sample. They yielded a relatively high (low) temperature reading at the angular positions where there was up-flow (down-flow) of the LSC. A fit of $$T_i = T_c + \delta ~ cos(i \pi / 4 - \theta_0),\ i = 0, \ldots, 7
\label{eq:T_i}$$ yielded the mean center temperature $T_c$, the angular orientation $\theta_0$ of the LSC (relative to the location of thermistor 0), and a measure $\delta$ of the LSC strength.
We expect the size of $\delta$ to be determined by the heat transport across a viscous boundary layer separating the LSC from the side wall. Thus $\delta$ should be a monotonically increasing function of the LSC Reynolds number $R_e$ because the boundary-layer thickness is expected to decrease with $R_e$ as $1/R_e^{1/2}$, and because the azimuthal temperature variation carried by the LSC near the boundary layer increases with $R$ and thus with $R_e$. However, the precise relationship between $\delta$ and $R_e$ is not obvious. Experimentally we find, over the range $5\times 10^9 < R < 10^{11}$ and for the large sample, that $\delta$ is related to $R$ by an effective power law $\delta \propto R^{0.81}$, whereas $R_e \propto R^{0.50}$ in this range, yielding $\delta \propto R_e^{1.62}$. We would then expect that $\delta$ and $R_e$ will have a similar dependence on $\beta$ (at least for small $\beta$), albeit possibly with somewhat different coefficients.
From time series of the $T_i(t)$ taken at intervals of a few seconds and covering at least one day we determined the cross-correlation functions $C^{i,j}(\tau)$ corresponding to signals at azimuthal positions displaced around the circle by $\pi$ ([*i.e.*]{} $j = i + 4$). These functions are given by $$C^{i,j}(\tau) = \langle [T_i(t) - \langle T_i(t)\rangle_t]\times[T_j(t+\tau) - \langle T_j(t)\rangle_t] \rangle_t\ .
\label{eq:corfunc}$$ We also calculated the auto-correlation functions corresponding to $i = j$ in Eq. \[eq:corfunc\], for all eight thermometers.
Initially each sample was carefully leveled so that the tilt angle relative to gravity was less than $10^{-3}$ radian. Later it was tilted deliberately to study the influence of a non-zero $\beta$ on the heat transport.
The fluid was water at 40$^\circ$C where $\alpha = 3.88\times 10^{-4}$ K$^{-1}$, $\kappa = 1.53\times 10^{-3}$ cm$^2$/s, and $\nu = 6.69\times 10^{-3}$ cm$^2$/s, yielding $\sigma = 4.38$.
The Nusselt number of a vertical sample {#sec:2}
=======================================
Initial transients
------------------
In Fig. \[fig:transients\]a we show the initial evolution of the top and bottom temperatures of the large sample in a typical experiment. Initially the heat current was near zero and $T_b$ and $T_t$ were close to 40$^\circ$C. The sample had been equilibrated under these conditions for over one day. Near $t = 0.6$ h a new temperature set point of 50$^\circ$C was specified for the bottom plate, and the circulator for the top plate was set to provide $T_t \simeq 30^\circ$C. From Fig. \[fig:transients\]a one sees that there were transients that lasted until about 0.9 h (1.2 h) for $T_b$ ($T_t$). These transients are determined by the response time and power capability of the bottom-plate heater and the top-plate cooling water and are unrelated to hydrodynamic phenomena in the liquid. Figures \[fig:transients\]b and c show the evolution of the heat current. After the initial rapid rise until $t \simeq 0.8$ h the current slowly evolved further to a statistically stationary value until $t \simeq 3$ h. A fit of the exponential function $Q(t) = Q_\infty - \Delta Q ~ exp(-t/\tau_Q)$ to the data for $t > 1.2$ h is shown by the solid line in Fig. \[fig:transients\]c and yielded a relaxation time $\tau_Q = 0.48 \pm 0.04$ h. We attribute this transient to the evolution of the fluid flow. It is interesting to compare $\tau_Q$ with intrinsic time scales of the system. The vertical thermal diffusion time $\tau_v \equiv L^2/\kappa$ is 467 hours. Obviously it does not control the establishment of the stationary state. If we consider that it may be reduced by a factor of $1/{\cal N}$ with ${\cal N} = 263$, we still obtain a time sale of 1.78 hours that is longer than $\tau_Q$. We believe that the relatively rapid equilibration is associated with the establishment of the top and bottom boundary layers that involve much shorter lengths $\it l_t$ and $\it l_b$. It also is necessary for the large-scale circulation to establish itself; but, as we shall see, its precise Reynolds number is unimportant for the heat transport. In addition, the LSC can be created relatively fast since this is not a diffusive process.
Results under statistically stationary conditions
-------------------------------------------------
Figure \[fig:transients\] shows the behavior of the system only during the first six hours and does not exclude the slow transients reported by Chillà [*et al.*]{} that occurred over time periods of ${\cal O}(10^2)$ hours. Thus we show in Fig. \[fig:N\_of\_t\] results for ${\cal N}/R^{1/3}$ from a run using the large sample that was continued under constant externally imposed conditions for nine days. Each point corresponds to a value of $\cal N$ based on a time average over two hours of the plate temperatures and the heat current. Note that the vertical range of the entire graph is only 0.8 %. Thus, within a small fraction of 1 %, the results are time independent. Indeed, during nearly a year of data acquisition for a $\Gamma = 1$ sample at various Rayleigh numbers, involving individual runs lasting from one to many days, we have never experienced long-term drifts or changes of $\cal N$ after the first few hours. This differs dramatically from the observations of Chillà [*et al*]{}. who found changes by about 2 % over about 4 days. We conclude that the slow transients observed by them for their $\Gamma = 0.5$ sample do not occur for $\Gamma \simeq 1$.
To document further the stationary nature of the system, we compared results from the large sample for $\cal N$ obtained from many runs, each of one to ten days’ duration, over a period of about five months \[[@NBFA05; @FBNA05]\]. The scatter of the data at a given $R$ is only about 0.1%. This excellent reproducibility would not be expected if there were slow transients due to transitions between different states of the LSC.
Although work in our laboratory with other aspect ratios has been less extensive, we also have not seen any evidence of drifts or transients for the larger $\Gamma = 1.5, 2, 3,$ and 6 \[[@FBNA05; @BNFA05a]\] nor for the smaller $\Gamma = 0.67, 0.43,$ and 0.28 \[[@NBFA05; @BNFA05a]\]. It may be that $\Gamma = 0.5$, being near the borderline between a single-cell LSC and more complicated LSC structures \[[@VC03; @SV05; @SXX05]\], is unique in this respect.
In Fig. \[fig:N\_compare\] we compare results for ${\cal N}/R^{1/3}$ from our large sample \[[@NBFA05]\] with those reported by Chillà [*et al.*]{} (stars). Our results are larger by about 15%. To find a reason for this difference, we first look at the $\Gamma$ and $\sigma$ dependence. The open (solid) circles represent our data for $\sigma = 4.38$ and $\Gamma = 0.67~(0.43)$ and show that the dependence of $\cal N$ on $\Gamma$ is not very strong. The open squares (diamonds) are our results for $\Gamma = 0.67$ and $\sigma = 5.42~(3.62)$ and indicate that $\cal N$ actually [*increases*]{} slightly with $\sigma$. Thus the lower values of $\cal N$ (compared to ours) obtained by Chillá et al. for $\sigma = 2.3$ and $\Gamma = 0.5$ can not be explained in terms of the $\Gamma$ and $\sigma$ dependence of $\cal N$. Some of the difference can be attributed to non-Boussinesq effects that tend to reduce $\cal N$ \[[@FBNA05]\]. However, for the largest $\Delta T$ used by Chillá et al. (31$^\circ$C) we expect this effect to be somewhat less than 1 % \[[@FBNA05]\]. Finally, the effect of the finite conductivity of the top and bottom plates comes to mind. This can reduce $\cal N$ by several % when $\Delta T$ is large \[[@CCC02; @Ve04; @BNFA05a]\], but it is difficult to say precisely by how much. It seems unlikely that this effect can explain the entire difference, particularly at the smaller $R$ (and thus $\Delta T$) where it is relatively small.
Tilt-angle dependence of the Nusselt number
===========================================
In Fig. \[fig:N\_tiltresponse\] we show results for $\cal N$ from the large sample at $R = 9.43\times 10^{10}$. Each data point was obtained from a two-hour average of measurements of the various temperatures and of $Q$. Three data sets, taken in temporal succession, for tilt angles $\beta = 0.000, ~0.087$, and 0.122 are shown. All data were normalized by the mean of the results for $\beta = 0$. Typically, the standard deviation from the mean of the data at a given $\beta$ was 0.13%. The vertical dotted lines and the change in the data symbols show where $\beta$ was changed. One sees that tilting the cell caused a small but measurable reduction on $\cal N$. In Fig. \[fig:N\_of\_beta\] we show the mean value for each tilt angle, obtained from runs of at least a day’s duration a each $\beta$, as a function of $|\beta|$. One sees that $\cal N$ decreases linearly with $\beta$. A fit of a straight line to the data yielded $${\cal N}(\beta) = {\cal N}_0 [1 - (3.1\pm 0.1)\times 10^{-2}|\beta|] \ .
\label{eq:N}$$ with ${\cal N}_0 = 273.5$. Simlar results for the medium cell are compared with the large-cell results in Fig. \[fig:Nred\_of\_beta\]. At the smaller Rayleigh number of the medium sample the effect of $\beta$ on $\cal N$ is somewhat less. Because the effect of $\beta$ on $\cal N$ is so small, we did not make a more detailed investigation of its Rayleigh-number dependence.
Chillà et al. proposed a model that predicts a significant tilt-angle effect on $\cal N$ for $\Gamma = 0.5$ where they assume the existence of two LSC cells, one above the other. They also assumed that there would be no effect for $\Gamma = 1$ where there is only one LSC cell. Although we found an effect for our $\Gamma = 1$ sample, we note that it is a factor of about 50 smaller than the effect observed by Chillà et al. for $\Gamma = 0.5$.
Tilt-angle dependence of the large-scale circulation {#sec:3}
====================================================
The orientation
---------------
In Fig. \[fig:theta+delta\] we show the angular orientation $\theta_0$ (a) and the temperature amplitude $\delta$ (b) of the LSC. For the first 8000 sec shown in the figure, the sample was level ($\beta = 0.000\pm 0.001$). One sees that $\theta_0$ varied irregularly with time. The probability-distribution function $P(\theta_0)$ is shown in Fig. \[fig:P\_of\_theta\] as solid dots. Essentially all angles are sampled by the flow, but there is a preferred direction close to $\theta_0/2\pi = 0.6$. At t = 8000 sec, the sample was tilted through an angle $\beta = 0.087$ radian. The direction of the tilt was chosen deliberately so as to oppose the previously prevailing preferred orientation. As a consequence one sees a sharp transition with a change of $\theta_0$ by approximately $\pi$. The temperature amplitude $\delta$ on average increased slightly, and certainly remained non-zero. From this we conclude that the transition took place via rotation of the LSC, and not by cessation that would have involved a reduction of $\delta$ to zero \[see [@BNA05b]\]. We note that $\theta_0(t)$ fluctuated much less after the tilt. The results for $P(\theta_0)$ after the tilt are shown in Fig. \[fig:P\_of\_theta\] as open circles. They confirm that the maximum was shifted close to $\theta_0 = 0$, and that the distribution was much more narrow.
In Fig. \[fig:Pshift\_of\_theta\] we show $P(\theta_0)$ for $\beta = 0.122,$ (solid circles), 0.044 (open circles), and 0.026 (solid squares). One sees that a reduction of $\beta$ leads to a broadening of $P(\theta_0)$. The square root of the variance of data like those in Fig. \[fig:Pshift\_of\_theta\] is shown in Fig. \[fig:dP\_of\_theta\] on a logarithmic scale as a function of $\beta$ on a linear scale. Even a rather small tilt angle caused severe narrowing of $P(\theta_0)$.
The temperature amplitude
-------------------------
In Fig. \[fig:delta\] we show the temperature-amplitude $\delta(\beta)$ of the LSC as a function of $\beta$. As was the case for $\cal N$, the data are averages over the duration of a run at a given $\beta$ (typically a day or two). The solid (open) circles are for positive (negative) $\beta$. The data can be represented well by either a linear or a quadratic equation. A least-squares fit yielded $$\delta(\beta) = \delta(0)\times [1 + (1.84 \pm 0.45) |\beta| - (3.1 \pm 3.9) \beta^2]
\label{eq:delta}$$ with $ \delta(0) = 0.164$ K.
The Reynolds numbers
--------------------
Using Eq. \[eq:corfunc\], we calculated the auto-correlation functions (AC) $C^{i,i}, i = 0, \ldots, 7$, as well as the cross-correlation functions (CC) $C^{i,j}, j = (i + 4) \% 8, i = 0, \ldots, 7$, of the temperatures measured on opposite sides of the sample. Typical examples are shown in Fig. \[fig:cor\]. The CC has a characteristic peak that we associate with the passage of relatively hot or cold volumes of fluid at the thermometer locations. Such temperature cross-correlations have been shown, [*e.g.*]{} by [@QT01b] and [@QT02], to yield delay times equal to those of velocity-correlation measurements, indicating that warm or cold fluid volumes travel with the LSC. The function $$C^{i,j}(\tau) = - b_0 exp \left ( -\frac{\tau}{\tau^{i,j}_0} \right ) - b_1 exp \left [ -\left ( \frac{\tau - t^{i,j}_1}{\tau^{i,j}_1}\right )^2 \right]\ ,
\label{eq:ccfit}$$ consisting of an exponentially decaying background (that we associate with the random time evolution of $\theta_0$) and a Gaussian peak, was fitted to the data for the CC. The fitted function is shown in Fig. \[fig:cor\] as a solid line over the range of $\tau$ used in the fit. It is an excellent representation of the data and yields the half turnover time ${\cal T}/2 = t^{i,j}_1$ of the LSC. Similarly, we fitted the function $$C^{i,i}(\tau) = b_0 exp\left (- \frac{\tau}{\tau^{i,i}_0}\right ) + b_1 exp \left [- \left (\frac{\tau}{\tau^{i,i}_1} \right )^2 \right ] + b_2 exp \left [-\left ( \frac{\tau - t^{i,i}_2}{\tau^{i,i}_2}\right )^2 \right ]
\label{eq:acfit}$$ to the AC data. It consists of two Gaussian peaks, one centered at $\tau = 0$ and the other at $\tau = t^{i,i}_2$, and the exponential background. We interpret the location $ t^{i,i}_2$ of the second Gaussian peak to correspond to a complete turnover time $\cal T$ of the LSC.
In terms of the averages $<{t^{i,j}_1}>$ and $<{t^{i,i}_2}>$ over all 8 thermometers or thermometer-pair combinations we define \[[@QT02; @GL02]\] the Reynolds numbers $$R^{cc}_{e} = (L/ <t^{i,j}_1>) (L/\nu)
\label{eq:Re_cc}$$ and $$R^{ac}_{e} = (2L/ <t^{i,i}_2>) (L/\nu)\ .
\label{eq:Re_ac}$$ Here the length scale $2L$ was used to convert the turnover time ${\cal T}$ into a LSC speed $2L/{\cal T}$. For $\Gamma = 1$, the length $4L$ might have been used instead, as was done for instance by [@LSZX02]. This would have led to a Reynolds number larger by a factor of two. In Fig. \[fig:Re\]a and b we show $R_e^{cc}(|\beta|)$ and $R_e^{ac}(|\beta|)$ respectively. The solid circles are for positive and the open ones for negative $\beta$. One sees that $R_e^{cc}(|\beta|)$ and $R_e^{ac}(|\beta|)$ initially grow linearly with $\beta$, but the data also reveal some curvature as $|\beta|$ becomes larger. Thus we fitted quadratic equations to the data and obtained $$R_e^{cc}(\beta) = R_e^{cc}(0)\times [1 + (1.85\pm 0.21) |\beta| - (5.9\pm 1.7) \beta^2]
\label{eq:Recc}$$ and $$R_e^{ac}(\beta) = R_e^{ac}(0)\times [1 + (1.72\pm 0.38) |\beta| - (4.1\pm 3.2) \beta^2]
\label{eq:Reac}$$ with $R_e^{cc}(0) = 10467 \pm 43$ and $R_e^{ac}(0) = 10565 \pm 82$ (all parameter errors are 67% confidence limits). The results for $R_e^{cc}(0)$ and $R_e^{ac}(0)$ are about 10% higher than the prediction by [@GL02] for our $\sigma$ and $R$. The excellent agreement between $R_e^{cc}$ and $R_e^{ac}$ is consistent with the idea that the CC yields ${\cal T}/2$ and that the AC gives ${\cal T}$. As expected (see Sect. \[sec:apparatus\]), the $\beta$-dependences of both Reynolds numbers are the same within their uncertainties. It is interesting to see that the coefficients of the linear term also agree with the corresponding coefficient for $\delta$ (Eq. \[eq:delta\]). This suggests that there may be a closer relationship between $\delta $ and $R_e$ than we would have expected [*a priori*]{}. However, the coefficient of the linear term in Eq. \[eq:Recc\] or \[eq:Reac\] is larger by a factor of about 50 than the corresponding coefficient for the Nusselt number in Eq. \[eq:N\].
In Fig. \[fig:Re\_delta\_ratios\] we show measurements of $R_e^{cc}$ and of $\delta$, each normalized by its value at $\beta = 0$, as a function of $\beta$ for the medium sample and $R = 1.13\times 10^{10}$. For this sample we were able to attain larger values of $\beta$ than for the large one. One sees that $\delta$ and $R_e^{cc}$ have about the same $\beta$ dependence for small $\beta$, but that $\delta$ then increases more rapidly than $R_e^{cc}$ as $\beta$ becomes large. Although we do not know the reason for this behavior, it suggests that the larger speed of the LSC enhances the thermal contact between the side wall and the fluid interior.
The Rayleigh-number dependence of $R_e^{cc}$ at constant $\beta$ is shown in Fig. \[fig:Re\_ratios\]. Here the open (solid) circles are from the medium (small) sample. There is consistency between the two samples, and the data can be described by a power law with a small negative exponent. The solid line is drawn to correspond to an exponent of $-1/6$.
A model for the enhancement of the Reynolds number
--------------------------------------------------
As seen in Fig. \[fig:P\_of\_theta\], the LSC assumes an orientation for which gravity enhances the velocity above (below) the bottom (top plate), [*i.e.*]{} the LSC flows “uphill" at the bottom where it is relatively warm and “downhill" at the top where it is relatively cold. This leads to an enhancement of the Reynolds number of the LSC. As suggested by [@CRCC04], one can model this effect by considering the buoyancy force per unit area parallel to the plates. This force can be estimated to be $\rho {\it l}g \beta \alpha \Delta T/2$ where ${\it l}$ is the boundary-layer thickness. It is opposed by the increase of the viscous shear stress across the boundary layer that may be represented by $\rho \nu u'/{\it l}$ where $u'$ is the extra speed gained by the LSC due to the tilt. Equating the two, substituting $${\it l} = L / (2{\cal N})\ ,
\label{eq:BL}$$ solving for $u'$, using Eq. \[eq:R\] for $R$, and defining $R_e' \equiv (L/\nu)u'$ one obtains $$R_e'= \frac{R \beta}{8 \sigma {\cal N}^2}$$ for the enhancement of the Reynolds number of the LSC. From our measurements at large $R$ we found that $R_e$ \[[@ABFN05]\] and $\cal N$ \[[@NBFA05]\] can be represented within experimental uncertainty by $$\begin{aligned}
R_e &=& 0.0345 R^{1/2}\ , \label{eq:Re_exp} \\
{\cal N} &=& 0.0602 R^{1/3}\ , \label{eq:Nu_exp}\end{aligned}$$ giving $$\frac{R_e'}{R_e} = 1.00\times 10^3 R^{-1/6} \sigma^{-1} \beta\ .
\label{eq:Re'}$$ For our $\sigma = 4.38$ and $R = 9.43\times 10^{10}$ one finds $R_e'/R_e = 3.4 \beta$, compared to the experimental value $(1.9\pm 0.2) \beta$ from $R_e^{cc}$ \[Eq. \[eq:Recc\]\] and $(1.7\pm 0.4) \beta$ from $R_e^{ac}$ \[Eq. \[eq:Reac\]\]. We note that the coefficient $1.00\times 10^3$ in Eq. \[eq:Re’\] depends on the definition of $R_e$ given in Eqs. \[eq:Re\_cc\] an \[eq:Re\_ac\] that was used in deriving the result Eq. \[eq:Re\_exp\]. If the length scale $4L$ had been used instead of $2L$ to define the speed of the LSC, as was done for instance by [@LSZX02], this coefficient would have been smaller by a factor of two, yielding near-perfect agreement with the measurements. In Fig. \[fig:Re\_ratios\] one sees that the predicted dependence on $R^{-1/6}$ also is in excellent agreement with the experimental results. However, such good agreement may be somewhat fortuitous, considering the approximations that were made in the model. Particularly the use of Eq. \[eq:BL\] for the boundary-layer thickness is called into question at a quantitative level by measurements of [@LX98] that revealed a significant variation of $\it l$ with lateral position. In addition, it is not obvious that the thermal boundary-layer thickness $\it l$ should be used, as suggested by [@CRCC04], to estimate the shear stress; perhaps the thickness of the viscous BL would be more appropriate.
In discussing their $\Gamma = 0.5$ sample, [@CRCC04] took the additional step of assuming that the relative change due to a finite $\beta$ of $\cal N$ is equal to the relative change of $R_e$. For our sample with $\Gamma = 1$ this assumption does not hold. As we saw above, the relative change of $\cal N$ is a factor of about 50 less than the relative change of $R_e$. The origin of the (small) depression of the Nusselt number is not so obvious. Naively one might replace $g$ in the definition of the Rayleigh number by $g~cos(\beta)$; but this would lead to a correction of order $\beta^2$ whereas the experiment shows that the correction is of order $\beta$, albeit with a coefficient that is smaller than of order one. The linear dependence suggests that the effect of $\beta$ on $\cal N$ may be provoked by the change of $R_e$ with $\beta$, but not in a direct causal relationship.
Tilt-angle dependence of reorientations of the large-scale circulation
======================================================================
It is known from direct numerical simulation \[[@HYK91]\] and from several experiments \[[@CCS97; @NSSD01; @SBN02; @BNA05b]\] that the LSC can undergo relatively sudden reorientations. Not unexpectedly, we find that the tilt angle strongly influences the frequency of such events. For a level sample ($\beta = 0$) we demonstrated elsewhere \[[@BNA05b]\] that reorientations can involve changes of the orientation of the plane of circulation of the LSC through any angular increment $\Delta \theta$, with the probability $P(\Delta \theta)$ increasing with decreasing $\Delta \theta$. Thus, in order to define a “reorientation", we established certain criteria. We required that the magnitude of the net angular change $|\Delta\theta|$ had to be greater than $\Delta\theta_{min}= (2\pi)/8$. In addition we specified that the magnitude of the net average azimuthal rotation rate $|\dot\theta| \equiv |\Delta \theta / \Delta t|$ had to be greater than $\dot\theta_{min} = 0.1/{\cal T}$ where $\cal T$ is the LSC turnover time and $\Delta t$ is the duration of the reorientation (we refer to [@BNA05b] for further details). Using these criteria, we found that the number of reorientation events $n(\beta)$ at constant $R = 9.43\times 10^{10}$ decreased rapidly with increasing $|\beta|$. These results are shown in Fig. \[fig:n\]. It is worth noting that nearly all of these events are rotations of the LSC and very few involved a cessation of the circulation. A least-squares fit of the Gaussian function $$n(\beta) = N_0 exp[-(\beta - \beta_0)^2/w^2]
\label{eq:N_r}$$ to the data yielded $N_0= 1.23 \pm 0.06$ events per hour, $\beta_0 = 0.0093 \pm 0.0010$ rad, and $w = 0.0251 \pm 0.0015$ rad. It is shown by the solid line in the figure.
We note that the distribution function is not centered on $\beta = 0$. The displacement of the center by about 9 mrad is much more than the probable error of $\beta$. We believe that it is caused by the effect of the Coriolis force on the LSC that will be discussed in more detail elsewhere \[[@BNA05c]\].
Tilt-angle dependence of the center temperature
===============================================
We saw from Fig. \[fig:delta\] that the increase of $R_e$ with $\beta$ led to an increase of the amplitude $\delta$ of the azimuthal temperature variation at the horizontal mid-plane. An additional question is whether the tilt-angle effect on this system has an asymmetry between the top and bottom that would lead to a change of the mean center temperature $T_c$ (see Eq. \[eq:T\_i\]). [@CRCC04] report such an effect for their $\Gamma = 0.5$ sample. For a Boussinesq sample with $\beta = 0$ we expect that $T_c = T_m$ with $T_m = (T_t + T_b)/2$ ($T_t$ and $T_b$ are the top and bottom temperatures respectively), or equivalently that $\Delta _t = T_c - T_t$ is equal to $\Delta_b = T_b - T_c$. A difference between $\Delta_b$ and $\Delta_t$ will occur when the fluid properties have a significant temperature dependence \[[@WL91; @ZCL97]\], [*i.e.*]{} when there are significant deviations from the Boussinesq approximation. For the sequence of measurements with the large apparatus and $R = 9.43\times 10^{10}$ as a function of $\beta$ the mean value of $\Delta T = T_b - T_t$ was $19.808 \pm 0.018 {^\circ}$C and $T_c - T_m$ was 0.97$^\circ$C, indicating a significant non-Boussinesq effect. In Fig. \[fig:NOB\]a we show $\Delta_t$ and $\Delta_b$ as a function of $\beta$. One sees that increasing $\beta$ does not have a significant effect for our $\Gamma = 1$ sample. This is shown with greater resolution in Fig. \[fig:NOB\]b where $T_c - T_m = (\Delta_t - \Delta_b)/2$ is shown. We believe that the small variation, over a range of about $5\times 10^{-3} {^\circ}$C, is within possible systematic experimental errors and consistent with the absence of a tilt-angle effect.
Summary
=======
In this paper we reported on an experimental investigation of the influence on turbulent convection of a small tilt angle $\beta$ relative to gravity of the axes of two cylindrical Rayleigh-Bénard samples. The aspect ratios were $\Gamma \simeq 1$.
Where there was overlap, there were significant differences between our results and those obtained by [@CRCC04] for a $\Gamma = 0.5$ sample. We found our system to establish a statistically stationary state quickly, within a couple of hours, after a Rayleigh-number change whereas [@CRCC04] found long transients that they attributed to changes of the LSC structure. We found a very small depression of the Nusselt number $\cal N$ with increasing $\beta$, by about 4% per radian at small $\beta$. [@CRCC04] found a decrease by 200% per radian for their sample.
In contrast to the very small effect of $\beta$ on $\cal N$, we found an increase of the Reynolds number $R_e$ by about 180% per radian for small $\beta$. The small effect on $\cal N$ in the presence of this large change of $R_e$ indicates that the heat transport does not depend strongly on the speed of the LSC sweeping over the boundary layers. Instead, $\cal N$ must be determined by instability mechanisms of the boundary layers, and the associated efficiency of the ejection of hot (cold) volumes (so-called “plumes") of fluid from the bottom (top) boundary layer.
It is interesting to note that the strong dependence of $R_e$ on $\beta$ in the presence of only a very weak dependence of $\cal N$ on $\beta$ can be accommodated quite well within the model of [@GL02]. The Reynolds number can be changed by introducing a $\beta$-dependence of the parameter $a(\beta)$ in their Eqs. (4) and (6). As pointed out by them, a change of $a$ has no influence on the predicted value for $\cal N$.
We also measured the frequency of rapid LSC reorientations that are known to occur for $\beta = 0$. We found that such events are strongly suppressed by a finite $\beta$. Even a mild breaking or the rotational invariance, corresponding to $\beta \simeq 0.04$, suppresses re-orientations almost completely.
Acknowledgment
==============
We are grateful to Siegfried Grossmann and Detlef Lohse for fruitful exchanges. This work was supported by the United States Department of Energy through Grant DE-FG02-03ER46080.
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|
---
author:
- 'E. O. Silva'
- 'F. M. Andrade'
title: 'Remarks on the Aharonov-Casher dynamics in a CPT-odd Lorentz-violating background'
---
Since the construction of the standard model extension (SME), proposed by Colladay and Kostelecký [@PRD.1997.55.6760; @PRD.1998.58.116002; @PRD.2004.69.105009] (see also [@PRD.1999.59.116008; @PRL.1989.63.224; @PRL.1991.66.1811]) quantum field theory systems have been studied in the presence of Lorentz symmetry violation. The SME includes Lorentz-violating (LV) terms in all the sectors of the minimal standard model, becoming a suitable tool to address LV effects in distinct physical systems. Several investigations have been developed in the context of this theoretical framework in the latest years, involving field theories [@PRL.1999.82.3572; @PRL.1999.83.2518; @PRD.1999.60.127901; @PRD.2001.63.105015; @PRD.2001.64.046013; @JPA.2003.36.4937; @PRD.2006.73.65015; @PRD.2009.79.123503; @PD.2010.239.942; @PRD.2012.86.065011; @PRD.2008.78.125013; @PRD.2011.84.076006; @EPL.2011.96.61001; @PRD.2012.85.085023; @PRD.2012.85.105001; @EPL.2012.99.21003; @PRD.2011.84.045008], aspects on the gauge sector of the SME [@NPB.2003.657.214; @NPB.2001.607.247; @PRD.1995.51.5961; @PRD.1998.59.25002; @PLB.1998.435.449; @PRD.2003.67.125011; @PRD.2009.80.125040], quantum electrodynamics [@EPJC.2008.56.571; @PRD.2010.81.105015; @PRD.2011.83.045018; @EPJC.2012.72.2070; @JPG.2012.39.125001; @JPG.2012.39.35002], and astrophysics [@PRD.2002.66.081302; @AR.2009.59.245; @PRD.2011.83.127702]. These many contributions have elucidated the effects induced by Lorentz violation and served to set up stringent upper bounds on the LV coefficients [@RMP.2011.83.11].
Another way to propose and investigate Lorentz violation is considering new interaction terms. In particular, in ref. [@EPJC.2005.41.421], a Lorentz-violating and CPT-odd nonminimal coupling between fermions and the gauge field was firstly proposed in the form $$D_{\mu }=\partial_{\mu }+ieA_{\mu }
+i\frac{g}{2}\epsilon_{\mu \lambda\alpha \nu }
V^{\lambda }F^{\alpha \nu },
\label{eq:cov}$$ in the context of the Dirac equation, $(i\gamma^{\mu}D_{\mu}-m)\Psi =0$. In this case, the fermion spinor is $\Psi$, while $\mathit{V}^{\mu }=(\mathit{V}_{0},\mathbf{V})$ is the Carroll-Field-Jackiw four-vector, and $g$ is the constant that measures the nonminimal coupling magnitude. The analysis of the nonrelativistic limit of Eq. revealed that this nonminmal coupling generates a magnetic dipole moment $(g\mathbf{V})$ even for uncharged particles [@EPJC.2005.41.421], yielding an Aharonov-Casher (AC) phase for its wave function. In these works, after assessing the nonrelativistic regime, one has identified a generalized canonical momentum, $$\boldsymbol{\pi}=\mathbf{p}-e\mathbf{A}+g\mathit{V}_{0}
\mathbf{B}-g\mathbf{V}\times \mathbf{E},
\label{eq:mto}$$ which allows to introduce this nonminimal coupling in an operational way, *i.e.*, just redefining the vector potential and the corresponding magnetic field as indicated below: $$\mathbf{A}\rightarrow
\mathbf{A}+\frac{g}{e}
\left(\mathbf{V}\times\mathbf{E}\right),
\label{eq:a1}$$ $$\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}
\rightarrow \boldsymbol{\nabla}\times \mathbf{A}
- \frac{g}{e}\boldsymbol{\nabla}
\times \left(\mathbf{V}\times \mathbf{E}\right) .
\label{eq:a2}$$ This CPT-odd nonminimal coupling was further analyzed in various contexts in relativistic quantum mechanics [@PRD.2006.74.065009; @PRD.2012.86.045001; @ADP.2011.523.910; @JMP.2011.52.063505; @PRD.2011.83.125025; @PLB.2006.639.675; @EPJC.2009.62.425; @JPG.2012.39.055004; @JPG.2012.39.105004].
The aim of the present work is to study the effect of this CPT-odd LV nonminimal interaction on the AC dynamics. Taking into account Eqs. and we obtain the Schrödinger-Pauli equation $$\hat{H}\Psi =E\Psi,
\label{eq:dedfm}$$ where $$\begin{aligned}
\hat{H} = {} & \frac{1}{2M}
\bigg\{
\left[
\mathbf{p}-
e(\mathbf{A}+\frac{g}{e}\mathbf{V}\times \mathbf{E})
\right]^{2}
+ e U\left( r\right)
% \right.
\nonumber \\
{} &
% \left.
-e\,\boldsymbol{\sigma}\cdot
\left[\boldsymbol{\nabla}\times\mathbf{A}
-\frac{g}{e}\boldsymbol{\nabla}\times
(\mathbf{V}\times\mathbf{E})
\right]
\bigg\},
\label{eq:hdef}\end{aligned}$$ is the Hamiltonian operator. As is well-known, the potential $U(r)$, in cylindrical coordinates, is given by $$U(r) =-\phi\ln \left(\frac{r}{r_{0}}\right) .
\label{eq:ptln}$$ It is very difficult to solve Eq. by including both effects taking into account the logarithmic form of $U(r)$. This potential is relevant if we consider the full Hamiltonian, which is compatible with a charged solenoid. While the AC effect stems from the quantity $\boldsymbol{\sigma}\cdot
[g\boldsymbol{\nabla}\times(\mathbf{V}\times\mathbf{E})]$, one can affirm that the LV background does not contribute to the Aharonov-Bohm (AB) effect. To solve Eq. considering both effects (AB and AC) such potential has to be regarded. Since we are interested only in the background effects, we can examine only the sector generating the AC effect. In this latter case, the field configuration is given by $$\mathbf{E}={\phi}\frac{\boldsymbol{\hat{r}}}{r},~~~
\boldsymbol{\nabla}\cdot\mathbf{E}=\phi\frac{\delta (r)}{r},~~~
\phi=\frac{\lambda}{2\pi \epsilon_{0}},~~~
\mathbf{V}=V{\hat{\mathbf{z}}},
\label{eq:acconf}$$ where $\mathbf{E}$ is the electric field generated by an infinite charge filament and $\lambda$ is the charge density along the $z$-axis. After this identification, the Hamiltonian becomes $$\hat{H}=\frac{1}{2M}
\left[
\left(
\frac{1}{i}\boldsymbol{\nabla}
-\nu\frac{\hat{\mathbf{r}}}{r}
\right)^{2}
+\nu \sigma_{z} \frac{\delta(r)}{r}\right],
\label{eq:hnrf}$$ with $$\nu = g V \phi,
\label{eq:deltac}$$ the coupling constant of the $\delta(r)/r$ potential. Here, we are only interested in the situation in which $\hat{H}$ possesses bound states.
The Hamiltonian in Eq. governs the quantum dynamics of a spin-1/2 neutral particle with a radial electric field, *i.e.*, a spin-1/2 AC problem, with $g\mathbf{V}$ playing the role of a nontrivial magnetic dipole moment. Note the presence of a $\delta$ function singularity at the origin in Eq. which turns it more complicated to be solved. Such kind of point interaction potential can then be addressed by the self-adjoint extension approach [@Book.2004.Albeverio], used here for determining the bound states.
Making use of the underlying rotational symmetry expressed by the fact that $[\hat{H},\hat{J}_{z}]=0$, where $\hat{J}_{z}=-i\partial/\partial_{\phi}+\sigma_{z}/2$ is the total angular momentum operator in the $z$-direction, we decompose the Hilbert space $\mathfrak{H}=L^{2}(\mathbb{R}^{2})$ with respect to the angular momentum $\mathfrak{H}=\mathfrak{H}_{r}\otimes\mathfrak{H}_{\varphi}$, where $\mathfrak{H}_{r}=L^{2}(\mathbb{R}^{+},rdr)$ and $\mathfrak{H}_{\varphi}=L^{2}(\mathcal{S}^{1},d\varphi)$, with $\mathcal{S}^{1}$ denoting the unit sphere in $\mathbb{R}^{2}$. So it is possible to express the eigenfunctions of the two dimensional Hamiltonian in terms of the eigenfunctions of $\hat{J}_{z}$: $$\Psi(r,\varphi)=
\left(
\begin{array}{c}
\psi_{m}(r) e^{i(m_{j}-1/2)\varphi } \\
\chi_{m}(r) e^{i(m_{j}+1/2)\varphi }
\end{array}
\right) ,
\label{eq:wavef}$$ with $m_{j}=m+1/2=\pm 1/2,\pm 3/2,\ldots $, with $m\in
\mathbb{Z}$. By inserting Eq. into Eq. the Schrödinger-Pauli equation for $\psi_{m}(r)$ is found to be $$H\psi_{m}(r)=E\psi_{m}(r), \label{eq:eigen}$$ where $$H=H_{0}+\frac{\nu}{2M} \frac{\delta(r)}{r},
\label{eq:hfull}$$ and $$H_{0}=-\frac{1}{2M}
\left[
\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}
-\frac{(m-\nu)^{2}}{r^{2}}
\right].
\label{eq:hzero}$$
An operator $\mathcal{O}$, with domain $\mathcal{D}(\mathcal{O})$, is said to be self-adjoint if and only if $\mathcal{D}(\mathcal{O}^{\dagger})=\mathcal{D}(\mathcal{O})$ and $\mathcal{O}^{\dagger}=\mathcal{O}$. For smooth functions, $\xi \in C_{0}^{\infty}(\mathbb{R}^2)$ with $\xi(0)=0$, we should have $H \xi = H_{0} \xi$. Hence, it is reasonable to interpret the Hamiltonian as a self-adjoint extension of $H_{0}|_{C_{0}^{\infty}(\mathbb{R}^{2}\setminus \{0\})}$ [@crll.1987.380.87; @JMP.1998.39.47; @LMP.1998.43.43]. It is a well-known fact that the symmetric radial operator $H_{0}$ is essentially self-adjoint for $|m-\nu|\geq 1$ [@Book.1975.Reed.II]. For those values of the $m$ fulfilling $|m-\nu|<1$ it is not essentially self-ajoint, admitting an one-parameter family of self-adjoint extensions, $H_{\theta,0}$, where $\theta\in [0,2\pi)$ is the self-adjoint extension parameter. To characterize this family and determine the bound state energy, we will follow a general approach proposed in ref. [@PRD.2012.85.041701] (cf. also refs. [@AP.2008.323.3150; @AP.2010.325.2529; @JMP.2012.53.122106; @PLB.2013.719.467]), which is based on the boundary conditions that hold at the origin [@CMP.1991.139.103]. The boundary condition is a match of the logarithmic derivatives of the zero-energy ($E=0$) solutions for Eq. and the solutions for the problem defined by $H_{0}$ plus the self-adjoint extension.
Now, the goal is to find the bound states for the Hamiltonian . Then, we temporarily forget the $\delta$ function potential and find the boundary conditions allowed for $H_{0}$. However, the self-adjoint extension provides an infinity of possible boundary conditions, and it can not give us the true physics of the problem. Nevertheless, once the physics at $r=0$ is known [@AP.2008.323.3150; @AP.2010.325.2529], it is possible to determine any arbitrary parameter coming from the self-adjoint extension, and then we have a complete description of the problem. Since we have a singular point we must guarantee that the Hamiltonian is self-adjoint in the region of motion.
One observes that even if the operator is Hermitian $H_{0}^{\dagger}=H_{0}$, its domains could be different. The self-adjoint extension approach consists, essentially, in extending the domain $\mathcal{D}(H_{0})$ to match $\mathcal{D}(H_{0}^{\dagger})$, therefore turning $H_{0}$ self-adjoint. To do this, we must find the the deficiency subspaces, $N_{\pm}$, with dimensions $n_{\pm}$, which are called deficiency indices of $H_{0}$ [@Book.1975.Reed.II]. A necessary and sufficient condition for $H_{0}$ being essentially self-adjoint is that $n_{+}=n_{-}=0$. On the other hand, if $n_{+}=n_{-}\geq 1$, then $H_{0}$ has an infinite number of self-adjoint extensions parametrized by unitary operators $U:N_{+}\to N_{-}$. In order to find the deficiency subspaces of $H_{0}$ in $\mathfrak{H}_{r}$, we must solve the eigenvalue equation $$H_{0}^{\dagger}\psi_{\pm}
=\pm i k_{0} \psi_{\pm},
\label{eq:eigendefs}$$ where $k_{0}\in \mathbb{R}$ is introduced for dimensional reasons. Since $H_{0}^{\dagger }=H_{0}$, the only square-integrable functions which are solutions of Eq. are the modified Bessel functions of second kind, $$\psi_{\pm}=K_{|m-\nu|}(r\sqrt{\mp \varepsilon }),$$ with $\varepsilon=2iM k_{0}$. These functions are square integrable only in the range $|m-\nu|<1$, for which $H_{0}$ is not self-adjoint. The dimension of such deficiency subspace is $(n_{+},n_{-})=(1,1)$. According to the von Neumann-Krein theory, the domain of $H_{\theta,0}$ is given by $$\mathcal{D}(H_{\theta,0})=\mathcal{D}(H_{0}^{\dagger})=
\mathcal{D}(H_{0})\oplus N_{+}\oplus N_{-}.$$ Thus, $\mathcal{D}(H_{\theta,0})$ in $\mathfrak{H}_{r}$ is given by the set of functions [@Book.1975.Reed.II] $$\begin{aligned}
\label{eq:domain}
\psi_{\theta}(r)={}& \psi_{m}(r)\nonumber \\
{} & +c\left[ K_{|m-\nu|}(r\sqrt{-\varepsilon})+
e^{i\theta}K_{|m-\nu|}(r\sqrt{\varepsilon}) \right] ,\end{aligned}$$ where $\psi_{m}(r)$, with $\psi_{m}(0)=\dot{\psi}_{m}(0)=0$ ($\dot{\psi}\equiv d\psi/dr$), is a regular wave function and $\theta \in [0,2\pi)$ represents a choice for the boundary condition.
Now, we are in position to determine a fitting value for $\theta$. To do so, we follow the approach of ref. [@CMP.1991.139.103]. First, one considers the zero-energy solutions $\psi_{0}$ and $\psi_{\theta,0}$ for $H$ and $H_{0}$, respectively, *i.e.*, $$\left[
\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-
\frac{(m-\nu)^{2}}{r^{2}}-\nu \frac{\delta(r)}{r}
\right] \psi_{0}=0,
\label{eq:statictrue}$$ and $$\left[
\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-
\frac{(m-\nu)^{2}}{r^{2}}
\right] \psi_{\theta,0}=0.
\label{eq:thetastatic}$$ The value of $\theta$ is determined by the boundary condition $$\lim_{a\to 0^{+}}
\left(
a\frac{\dot{\psi}_{0}}{\psi_{0}}\Big|_{r=a}-
a\frac{\dot{\psi}_{\theta,0}}{\psi_{\theta,0}}\Big|_{r=a}
\right)=0.
\label{eq:logder}$$ The first term of is obtained by integrating from 0 to $a$. The second term is calculated using the asymptotic representation for the Bessel function $K_{|m-\nu|}$ for small argument. So, from we arrive at $$\frac{\dot{\Upsilon}_{\theta}(a)}{\Upsilon_{\theta}(a)}=\nu,
\label{eq:saepapprox}$$ with $$\Upsilon_{\theta}(r)=D(-\varepsilon)+e^{i\theta} D(+\varepsilon) ,$$ and $$D(\pm \varepsilon)=
\frac
{\left(r\sqrt{\pm \varepsilon}\right)^{-|m-\nu|}}
{2^{-|m-\nu|}\Gamma(1-|m-\nu|)}
-\frac
{\left(r\sqrt{\pm \varepsilon}\right)^{|m-\nu|}}
{2^{|m-\nu|}\Gamma(1+|m-\nu|)}.$$ Eq. gives us the parameter $\theta$ in terms of the physics of the problem, *i.e.*, the correct behavior of the wave functions at the origin.
Next, we will find the bound states of the Hamiltonian $H_{0}$ and, by using , the spectrum of $H$ will be determined without any arbitrary parameter. Then, from $H_{0}\psi_{\theta}=E\psi_{\theta}$ we achieve the modified Bessel equation ($\kappa^{2}=-2ME$) $$\left[
\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-
\frac{(m-\nu)^{2}}{r^{2}}-\kappa ^{2}
\right]\psi_{\theta}(r)=0,
\label{eq:eigenvalue}$$ where $E<0$ (since we are looking for bound states). The general solution for the above equation is $$\psi_{\theta}(r)=K_{|m-\nu|}\left(r\sqrt{-2ME}\right).
\label{eq:sver}$$ Since these solutions belong to $\mathcal{D}(H_{\theta,0})$, they present the form for a $\theta$ selected from the physics of the problem (cf. Eq. ). So, we substitute into and compute $a{\dot{\psi}_{\theta}}/{\psi_{\theta}}|_{r=a}$. After a straightforward calculation, we have the relation $$\frac
{|m-\nu|
\left[
a^{2|m-\nu|}(-ME)^{|m-\nu|}\Theta-1
\right]
}
{a^{2|m-\nu|}(-ME)^{|m-\nu|}\Theta+1}
=\nu,$$ where $\Theta=\Gamma(-|m-\nu|)/(2^{|m-\nu|}\Gamma(|m-\nu|))$. Solving the above equation for $E$, we find the sought energy spectrum $$E=
-\frac{2}{M a^{2}}
\left[
\left(
\frac
{\nu +|m-\nu|}
{\nu -|m-\nu|}
\right)
\frac
{\Gamma (1+|m-\nu|)}
{\Gamma (1-|m-\nu|)}
\right]^{{1}/{|m-\nu|}}.
\label{eq:energy_KS}$$ In the above relation, to ensure that the energy is a real number, we must have $|\nu| \geq |m-\nu|$, and due to $|m-\nu|<1$ it is sufficient to consider $|\nu|\geq 1$. A necessary condition for a $\delta$ function generating an attractive potential, able to support bound states, is that the coupling constant must be negative. Thus, the existence of bound states with real energies requires $$\nu \leq -1.$$ From the above equation and Eq it follows that $g V \lambda < 0$, and there is a minimum value for this product.
In conclusion, we have analyzed the effects of a LV background vector, nonminimally coupled to the gauge and fermion fields, on the AC problem. The self-adjoint extension approach was used to determine the bound states of the particle in terms of the physics of the problem, in a very consistent way and without any arbitrary parameter.
The authors would like to thank M. M. Ferreira Jr. for critical reading the manuscript and helpful discussions. E. O. Silva acknowledges researcher grants by CNPq-(Universal) project No. 484959/2011-5.
|
---
abstract: 'We study a random heteropolymer model with Langevin dynamics, in the supersymmetric formulation. Employing a procedure similar to one that has been used in static calculations, we construct an ensemble in which the affinity of the system for a native state is controlled by a “selection temperature" $T_0$. In the limit of high $T_0$, the model reduces to a random heteropolymer, while for $T_0 \rightarrow 0$ the system is forced into the native state. Within the Gaussian variational approach that we employed previously for the random heteropolymer, we explore the phases of the system for high and low $T_0$. For high $T_0$, the system exhibits a (dynamical) spin glass phase, like that found for the random heteropolymer, below a temperature $T_g$. For low $T_0$, we find an ordered phase, characterized by a nonzero overlap with the native state, below a temperature $T_n \propto 1/T_0 > T_g$. However, the random-globule phase remains locally stable below $T_n$, down to the dynamical glass transition at $T_g$. Thus, in this model, folding is rapid for temperatures between $T_g$ and $T_n$, but below $T_g$ the system can get trapped in conformations uncorrelated with the native state. At a lower temperature, the ordered phase can also undergo a dynamical glass transition, splitting into substates separated by large barriers.'
address: |
$^1$[Department of Applied Physics,\
Chalmers University of Technology and Göteborg University,\
SE 412 96 Göteborg, Sweden]{}\
$^2$[NORDITA, Blegdamsvej 17, DK 2100 København, Denmark]{}
author:
- 'Z. Konkoli$^{1,2}$ and J. Hertz$^{2}$'
title: |
Embedding a Native State into a Random\
Heteropolymer Model: The Dynamic Approach
---
[2]{}
Introduction
============
The protein folding process is relevant for all aspects of life: once read off from the RNA chain, proteins perform a variety of functions, from mechanical work to attacking viruses. [@Boy] The key factor which determines the function of a protein molecule is its 3D structure, which, in turn, is determined by the sequence of amino acids forming the protein chain. [@PGT1; @Wol1; @WE; @Creig] Furthermore, a protein that has been denatured (by stretching it for example) finds its native state relatively quickly. Protein folding has attracted an enormous amount of scientific attention, but still there is no generic understanding of this process. Nevertheless, one thing is clear: a proteins generally has a potential energy surface which results in a stable free energy minimum, corresponding to the native state [@Wol1].
Random heteropolymer models (RHP) have been used extensively as candidate systems which might help us to understand the generic features of the potential energy surfaces of proteins and their connection with thermodynamic [@SG1; @SG2; @GHLO; @GOP; @GLO; @TW; @SGS; @SW] and dynamical [@LT; @Pit; @TPW; @TAB; @PS; @Olem1; @Olem2] properties. The RHP model is characterized by quenched random monomer-monomer interactions, meant to mimic the variety of interactions between amino-acids in random sequences. It turns out that the potential energy surface of the RHP is quite similar to that of a particular class of spin glasses [@SpGl]: Its complex form, with exponentially large numbers of local minima and saddle points, constrains the motion of the system drastically, and it cannot explore its full configuration space and reach Gibbs equilibrium. In a previous paper ([@KHS], henceforth referred to as paper I), we demonstrated, in mean field theory, the existence of a sharp transition to a “dynamical glassy state” in which the equilibration time diverges and the dynamics exhibit aging. (The potential importance of spin glass physics to proteins was first discussed in Ref. [@Wol2]). Obviously, the random heteropolymer model does not describe a protein with a native state, but it alerts us to the need to examine possible glassiness in models for protein dynamics.
Why are real proteins not glassy? Evidently, nature has tuned amino acid sequences to avoid glassy behavior. To understand how such tuning might be done, it is worthwhile to study models which contain competition between glassiness and a tendency to form a native state, by choosing interactions which are not completely random. Several studies along the lines of this suggestion have been made in [ *statics*]{} (using the replica treatment, see, e.g., Ref. [@Wol2]). The tendency toward a particular state can be built in by choosing sequences from a distribution correlated with the native sequence [@PGT1; @RS; @PGT2; @WS]. A dynamical treatment of similar models is highly desirable, not only to help gain insight into results obtained in replica approaches, but also because knowledge of the correct thermodynamics alone may not be sufficient: it is known that in related (mean field models) static and dynamic phase diagrams can be different. Thus (at least on sufficiently short time scales) only a dynamical approach can describe the measurable properties of the system. In this paper we undertake such a study.
We extend the RHP model studied in [@SG1; @SG2] to include the existence of a native state: the original random monomer-monomer interactions are biased so as to favor the native state conformation. The problem is formulated as a Langevin model. To the best of our knowledge, there is so far neither a static nor a dynamic treatment available for a model of this sort: Static studies have been based on random monomer sequences, i.e., using only $N$ random parameters, see Refs. [@PGT1; @RS; @PGT2; @WS], rather than the $N(N-1)/2$ in the RHP model.
Admittedly, the model does not describe a realistic protein (e.g., it does not give rise to secondary structure such as $\alpha$-helices or $\beta$-sheets). However, it does contain important generic features: the polymeric structure and the mixture of attractive and repulsive interactions. Together, these features lead to frustration in the structural dynamics. In our view, ours is the simplest such model that includes competition between glassy and native states. As we will see, it teaches us that one can not get rid of glassiness so easily.
As in paper I, we simplify the model further by omitting three-body interactions in the polymer. (A review describing how to include three-body terms is given in [@GOP].) The price we have to pay for this simplification is that we have to introduce a somewhat arbitrary confining potential, which we take to have a quadratic form. We adjust its strength so that the radius of gyration $R_g$ of a polymer of size $N$ scales like $N^{-1/d}$, where $d$ is the dimensionality of the system. In this way we attempt to describe a globular state. Of course, we can not describe the $\theta$-point transition in such a model, but here we are only interested in transitions between different globular states.
Our formal starting point is the Martin-Siggia-Rose generating functional for the Langevin dynamics of the model [@MSR; @Dom; @Jans1; @Jans2], written, for convenience and compactness, in its supersymmetric form [@Kur]. To derive equations of motion for correlation and response functions we use a variational ansatz with a quadratic action. This approach has been used to study the problem of a manifold in a random potential, in both statics [@MP1; @MP2] and dynamics [@CKD; @CD].
In paper I we showed that the RHP model exhibited broken ergodicity (formally, a spontaneous supersymmetry breaking) in a low-temperature dynamical glassy phase. In the present study, with interactions biased in favor of a native state to a controlled degree, we find, in addition, a well-folded phase, if the bias is strong enough. It can coexist with either the disordered (random-globule) state or the frozen-globule glass phase, depending on the temperature. Furthermore, we find that at low temperature the native phase can itself undergo a dynamical freezing into a different glassy phase. In this phase the conformation of the protein is always highly correlated with the native state, but cooperative kinetic constraints still lead to a divergent equilibration time, as for the frozen-globule state.
The Model
=========
The model is defined as follows. The Langevin dynamics is assumed to be governed by a Hamiltonian $H[x]$, $$\partial x(s,t)/\partial t = - \delta H[x] / \delta x(s,t) + \eta(s,t).
\label{eq:dxdt}$$ Here $x(s,t)$ is the position of monomer $s$ at time $t$. The monomers are numbered continuously from $s=0$ to $s=N$. $\eta(s,t)$ is Gaussian noise $$\langle \eta(s,t)\eta(s',t') \rangle_T = 2T\delta(s-s')\delta(t-t'),
\label{eq:etas}$$ resulting from coupling to a heat bath at temperature $T$.
The Hamiltonian $H[x]$ contains a deterministic part $H_0[x,\mu]$ and a random part $H[x,\{B\}]$. $H_0[x,\mu]$ is defined as $$H_0[x,\mu]= \frac{T}{2} \int_{0}^{N} ds
\{[\partial x(s,t)/\partial s]^2+\mu x(s,t)^2\}.
\label{eq:H0}$$ It describes the elastic properties of the chain and a confinement potential which fixes the density of the protein. The radius of gyration $R_g \sim \mu^{-1/4}$, so, in order that the protein is compact, i.e., $R_g \sim N^{1/d}$, we require $\mu \sim
N^{-4/d}$. Thus, since we are interested in very long proteins (to obtain the thermodynamic limit) we need to solve the model for $\mu$ close to zero.
The random part $H[x,\{B\}]$ describes the quenched random interactions between monomers, $$H[x,\{B\}]= \frac{1}{2} \int_{0}^{N} ds ds'
B_{ss'} V(x(s,t)-x(s',t)).
\label{eq:Hrand}$$ We take $B_{ss'}$ Gaussian, with variance $B^2$. The quenched average over $B_{ss'}$ is performed as $\langle (.) \rangle_B
= \int \prod_{s>s'} dB_{ss'} (.) P(\{B\})$. $V(\Delta x)$ is a short-range potential, and, for simplicity, we take it to have a Gaussian form, as in Ref. [@TPW], $$V(\Delta x)=\left(\frac{1}{2\pi\sigma}\right)^{d/2}
{{\rm e}}^{-(\Delta x)^2/2\sigma}.
\label{eq:V}$$ $d$ is the dimensionality of the system and $\sqrt{\sigma}$ the range of the potential. Large (small) $\sigma$ corresponds to a long (short) range potential. In particular, for $\sigma\rightarrow 0$, $V(\Delta
x)\rightarrow\delta(\Delta x)$, and we recover the potential used in [@SG1; @SG2; @PS]. Here and in the following $\Delta x$ refers to a monomer-monomer distance: $\Delta x=x(s,t)-x(s',t)$ for a pair of monomers $s$, $s'$.
We use reasoning similar to that employed in statics to define $P(\{B\})$ (see Refs. [@PGT1; @RS; @PGT2; @WS]), adapting it to the random-bond model: $$P(\{B\}) \propto {{\rm e}}^{ - \frac{1}{T_0} H[x_0,{B}]
- \frac{1}{2}\int ds ds' B_{ss'}^2 / 2 B^2 }
\label{eq:PB1}$$ $T_0$ is called the selection temperature, and $x_0(s)$ is some arbitrary native state conformation. Thus the symmetric bond distribution of the RHP model is distorted so as to give bigger weight to $B_{ss'}$’s which are attractive between monomers which lie close to each other in the configuration $x_0(s)$. Explicitly, the properly normalized $P(\{B\})$ is given by $$\begin{aligned}
&& P(\{B\}) = (2\pi B^2)^{-N(N-1)/4}
{{\rm e}}^{-\beta_0^2 B^2 /4 \int ds ds' V(x_0(s)-x_0(s'))^2}
\nonumber \\
&& \times {{\rm e}}^{-\beta_0/2\int dsds'B_{ss'} V(x_0(s)-x_0(s'))
-1/2\int dsds'B_{ss'}^2/2B^2},
\label{eq:PB2} \end{aligned}$$ from which we see that the distribution of $B_{ss'}$ is peaked around $B^{max}_{ss'} = - \beta_0 B^2 V(x_0(s)-x_0(s'))$. Thus, if monomers $s$ and $s'$ are close in the native state ($V(x_0(s)-x_0(s'))\ne 0$), their coupling constant $B_{ss'}$ is pulled down, as in a Go model [@Go1; @Go2]. For $T_0\rightarrow\infty$ we recover the RHP model. For $T_0\rightarrow 0$, $P(\{B\})$ picks a specific set of $B_{ss'}$. For this set, by construction, $x_0(s)$ is the deepest minimum of $H[x,\{B\}]$ given in Eq. (\[eq:Hrand\]). This is the mechanism that embeds the native state $x_0(s)$.
This mechanism is somewhat arbitrary. However, the fact that the strength of embedding of the native state is controlled by the single parameter $T_0$ facilitates the study of transitions between random and native-like states (and, as we will show, of possible coexistence of such phases).
So far, the configuration $x_0(s)$ is arbitrary. Thus $x_0(s)$ has to be considered a quenched random function, to be averaged over just like $B_{ss'}$ in order to obtain generic results. We will carry this average out later.
All our results are obtained in the thermodynamic limit, where the length $N$ of the heteropolymer chain goes to infinity. Also, for simplicity, we join the polymer ends to form a ring. This neglect of end effects is valid for a long chain.
Mapping to the Field Theory
===========================
To solve the model we map the Langevin dynamics onto a supersymmetric (SUSY) field theory. Using the standard Martin-Siggia-Rose formalism [@MSR; @Dom; @Jans1; @Jans2] and supersymmetric (SUSY) notation [@Olem1; @Olem2; @Kur; @Olem3], the dynamical average of any observable, for fixed $\{B\}$, can be calculated as (see, e.g., Paper I for details), $$\begin{aligned}
&& \langle {\cal O}[\Phi] \rangle_T =\int D\Phi
{\cal O}[\Phi] e^{-S[\Phi] }, \label{eq:avSUSY} \\
&& S[\Phi] = S_1[\Phi]+S[\Phi, x_0, \{B\}], \label{eq:SSUSY} \end{aligned}$$ where $$\begin{aligned}
& & S_1[\Phi] = 1/2 \int ds d1 ds' d2 \Phi(s,1) K_{12}^{ss'} \Phi(s'2),
\label{eq:S0} \\
& & S[\Phi,x_0,\{B\}] = 1/2 \int d1 ds ds' \times \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \times B_{s,s'} V(\Phi(s,1)-\Phi(s',1)),
\label{eq:Srand} \end{aligned}$$ and $$\begin{aligned}
K_{12}^{ss'} && \equiv \delta_{12} \delta_{ss'} K_1^s \ , \ \
K_1^s = T \left[ \mu-(\partial/\partial s)^2 \right] - D_1^{(2)}, \\
D_1^{(2)} && =2 T \frac{\partial^2}{\partial\theta_1\partial\bar\theta_1}
+ 2 \theta_1 \frac{\partial^2}{\partial\theta_1\partial t_1} -
\frac{\partial}{\partial t_1}. \end{aligned}$$ The $\Phi(s,1)$ denotes a superfield $$\begin{aligned}
& \Phi(s,1) = & x(s,t_1) + \bar\theta_1 \eta(s,t_1) + \nonumber \\
& & + \bar\eta(s,t_1) \theta_1
+ \bar\theta_1\theta_1\tilde x(s,t_1)
\label{Phis1} \end{aligned}$$ containing the physical coordinate $x(s,t)$, the MSR auxiliary field $\tilde x(s,t)$, ghost fields $\eta(s,t)$ and $\bar\eta(s,t)$ that enforce the normalization of the distribution, and Grassmann variables $\theta$ and $\bar\theta$. We use the notation $1\equiv
(\theta_1,\bar\theta_1,t_1)$, likewise $\int d1 \equiv \int d\bar\theta_1
d\theta_1 dt_1$.
Of course, the solution can be obtained without the aid of the supersymmetric formalism, but we find it conveniently compact.
As noticed by De Dominicis [@Dom] the expression in Eq.(\[eq:avSUSY\]) is already normalized, so the average over the quenched disorder $B_{s,s'}$ can be done directly on (\[eq:avSUSY\]): $$\langle\langle {\cal O}[\Phi] \rangle_T\rangle_B = \int D\Phi
{\cal O}[\Phi] {{\rm e}}^{-(S_1[\Phi]+S_2[\Phi, x_0])}
\label{avO},$$ where $\exp(-S_2[\Phi,
x_0])\equiv\langle\exp(-S[\Phi,x_0,\{B\}])\rangle_B$, and $$\begin{aligned}
& S_2[\Phi, x_0] = & -\frac{B^2}{4} \int ds ds'
\left[
\int d1 V(\Phi(s,1)-\Phi(s',1))
\right]^2 \nonumber \\
&& - \frac{\beta_0 B^2}{2} \int ds ds' d1 V(\Phi(s,1)-\Phi(s',1)) \times
\nonumber \\
&& \times V(x_0(s)-x_0(s')). \label{eq:S2} \end{aligned}$$ Thus, the native state $x_0(s)$ enters the action in the second term of Eq. (\[eq:S2\]). Note that there is no term $\beta_0^2V(x_0(s)-x_0(s'))^2$, since it gets cancelled by a similar normalization factor for P({B}) in Eq. (\[eq:PB2\]). It is useful to rewrite Eq. (\[eq:S2\]) as $$\begin{aligned}
& S_2 = & -\frac{B^2}{4} \int d^dx\,d^dy\,d1\,d2\,
A_{12}^{(V)}(x,y) A_{12}^{(\delta)}(x,y) \nonumber \\
& & -\frac{\beta_0B^2}{2} \int d^dx\,d^dy\,d1\,
A_{10}^{(V)}(x,y) A_{10}^{(\delta)}(x,y) \end{aligned}$$ with the notation $A_{12}^{(f)}(x,y) = \int ds f(\Phi(s,1)-x)
f(\Phi(s,2)-y)$, $A_{10}^{(f)}(x,y) = \int ds f(\Phi(s,1)-x)
f(x_0(s)-y)$; $f\in\{V,\delta\}$. In the long-chain limit, as discussed in Paper I (and references therein), one obtains a self-consistent field theoretic formulation, with $S_2$ simplified to, $$\begin{aligned}
&& S_2[\Phi,x_0] = \frac{B^2}{4} \int d^dx d^dy d1 d2
\left[
\langle A_{12}^{(V)}(x,y) \rangle
\langle A_{12}^{(\delta)}(x,y) \rangle -
\right.
\nonumber \\
&& \left.
- A_{12}^{(V)}(x,y) \langle A_{12}^{(\delta)}(x,y) \rangle
- \langle A_{12}^{(V)}(x,y) \rangle A_{12}^{(\delta)}(x,y)
\right] \nonumber \\
&& + \frac{\beta_0B^2}{2} \int d^dx d^dy d1
\left[
\langle A_{10}^{(V)}(x,y) \rangle
\langle A_{10}^{(\delta)}(x,y) \rangle
\right.
\nonumber \\
&& \left.
- A_{10}^{(V)}(x,y) \langle A_{10}^{(\delta)}(x,y) \rangle
- \langle A_{10}^{(V)}(x,y) \rangle A_{10}^{(\delta)}(x,y)
\right].
\label{eq:S'} \end{aligned}$$ All averages of the type $\langle A^{(V,\delta)} \rangle$ have to be calculated self-consistently with $S[\Phi]=S_1[\Phi]+S_2[\Phi]$. (We have abbreviated the double average $\langle \langle.\rangle_T
\rangle_B$ simply by $\langle.\rangle$.) In the limit $N\rightarrow\infty$ Eqs. (\[avO\]) and (\[eq:S’\]) provide an exact description of the dynamics for an arbitrary native state $x_0(s)$.
Average over native state conformations
=======================================
It is impossible to solve the model for a general native state configuration $x_0(s)$. We therefore consider a distribution of native states and perform the average $$\overline{<O[\Phi,x_0]>}=\int Dx_0 <O[\Phi,x_0]> {{\rm e}}^{-S_0[x_0]},$$ where $S_0[x_0]$ weights each native state conformation in the ensemble as $$S_0[x_0]=1/2\int ds x_0(s) K^{ss'}_{00} x_0(s'),$$ with $$K^{ss'}_{00} \equiv \delta_{ss'} ( \mu_0-\partial^2/\partial s'^2).$$ The parameter $\mu_0$ fixes a size of the globule in this ensemble, $$\langle x_0(s)^2 \rangle = \frac{1}{2\sqrt{\mu_0}}
\label{x0^2}$$ Since the polymer ends are joined, there is translational invariance along the coordinate $s$ and $\langle x_0(s)^2\rangle$ does not depend on $s$. Thus, with this procedure, the dynamical generating functional for the problem is calculated as $$e^{-F_{dyn}}=\int Dx_0 D\Phi e^{ -(S_0[x_0]+S_1[\Phi]+S_2[\Phi,x_0])
}.
\label{Fdyn}$$ There is some formal similarity between the dynamical functional $F_{dyn}$ and the static replica partition function. The integration over $Dx_0$ enters in the same way as the extra replica in the static formalism.
Correlation functions
=====================
The SUSY correlation functions $$\begin{aligned}
& & G_{12}^{ss'} \equiv \langle \Phi(s,1) \Phi(s',2) \rangle \label{G12} \\
& & G_{10}^{ss'} \equiv \langle \Phi(s,1) x_0(s') \rangle \label{G10} \\
& & G_{00}^{ss'} \equiv \langle x_0(s)x_0(s') \rangle \label{G00} \end{aligned}$$ contain all the information we are interested in.
$G_{12}^{ss'}$ encodes 16 correlation functions, out of which only two, correlation and response function, are independent and nonzero: $$\begin{aligned}
& G_{12}^{ss'} = & C(s,t_1;s',t_2)
+ (\bar\theta_2-\bar\theta_1) \times \nonumber \\
& & \times [ \theta_2 R(s,t_1;s',t_2) - \theta_1 R(s',t_2;s,t_1) ], \end{aligned}$$ with $$\begin{aligned}
& & C(s,t;s',t') \equiv \langle x(s,t)x(s',t') \rangle, \\
& & R(s,t;s',t') \equiv \langle x(s,t) \tilde x(s',t') \rangle
= \frac{\delta\langle x(s,t)\rangle}{\delta h(s',t')}. \end{aligned}$$ The field $h(s',t')$ entering the description of response function is an arbitrary external field that couples to $x(s',t')$. The fact that only two correlation functions survive is related to Ward identities originating from SUSY invariance of the original action $S$.
The supersymmetry of the theory is associated with equilibrium. One of the Ward identities resulting from SUSY is the fluctuation-dissipation theorem (FDT) which relates correlation and response functions. In the present case, the glassy state manifests itself as a spontaneous breaking of supersymmetry, leading to a modified FDT, as in previous treatments of other models [@Kur; @CKD].
$G_{10}^{ss'}$ describes the overlap with the native state. Due to Ward identities, only a single correlation function survives (see Appendix A for details): $$G_{10}^{ss'} = \langle x(s,t)x_0(s') \rangle \equiv \phi(s,t_1;s').
\label{G10a}$$ Similarly, the native state ensemble is described by $$G_{00}^{ss'} = \langle x_0(s)x_0(s') \rangle \equiv \Gamma(s;s').$$ $G_{12}^{ss'}$ alone is sufficient to describe the RHP model. Here we need the two extra functions $G_{10}^{ss'}$ and $G_{00}^{ss'}$.
Also, in what follows, we exploit the translational invariance along the $s$ coordinate and define Fourier transforms of all correlation functions: $X(s,s')= \int \frac{dk}{2\pi} e^{ik(s-s')}
X_k$ where $X=C,R,\phi,\Gamma$.
Equations of Motion
===================
To solve the model we proceed by making a Gaussian variational ansatz (GVA), assuming that the fields $\Phi$ are described by the approximate action $$\begin{aligned}
& S_{var} = &
\frac{1}{2} \int d1 ds d2 ds'
\Phi(s,1) (G^{-1})_{12}^{ss'} \Phi(s',2) + \nonumber \\
&& + \int d1 ds ds'
\Phi(s,1) (G^{-1})_{10}^{ss'} x_0(s') + \nonumber \\
&& + \frac{1}{2} \int ds ds'
x_0(s) (G^{-1})_{00}^{ss'} x_0(s').
\label{Svar} \end{aligned}$$ Technically, this implies the following approximation for $F_{dyn}$: $$F_{dyn}\approx \langle S \rangle_{var} + F_{var}.
\label{Fdyn1}$$ where $$\begin{aligned}
& & {{\rm e}}^{-F_{var}}\equiv \int Dx_0 D\Phi {{\rm e}}^{-S_{var}} ={{\rm e}}^{(d/2)Tr\ln G}, \\
& & \langle . \rangle_{var}={{\rm e}}^{F_{var}}\int Dx_0 D\Phi( . ){{\rm e}}^{-S_{var}}. \end{aligned}$$ The stationarity condition $$\frac{\delta F_{dyn}}{\delta G_{12}^{ss'}} = 0
\label{dFdG}$$ translates into the equation of motion for Green’s function $G_{12}^{ss'}$ (see Eqs. \[G12k\]-\[G00k\]). We have derived identical equations of motion by using the approach of Ref. [@CD], where standard field theoretic identities (e.g. $\langle \Phi \delta
S/\delta\Phi \rangle=0$ ) are used. It can be shown that for quadratic $S_{var}$ the two procedures give the same result. We omit this analysis here to save space.
In a corresponding equilibrium problem, the stationarity condition is also an extremum condition and provides a bound on the free energy. Here, since $F_{dyn}$ contains integrations over complex fields and Grassmann variables, the GVA does not give a bound on $F_{dyn}$. Nevertheless, it is the first step in a systematic approximation procedure, as outlined in Appendix B.
The GVA has been applied to the problem of a manifold in a random potential, in both statics [@MP1; @MP2] and dynamics [@CKD; @CD]. The method is exact when the dimensionality of the manifold is infinite but is only approximate for finite dimensionality. Nevertheless, even for rather low dimensionality it has been shown to be a very good approximation in the random-manifold problem, where it has been checked numerically [@CD]. We have shown in Paper I that the present model is closely related to the random-manifold problem. Thus, we hope that the GVA will also be reasonable here, although we have not strictly checked its validity.
Using (\[Fdyn1\]), (\[Svar\]) and (\[eq:SSUSY\]) gives the following expression for $F_{dyn}$: $$\begin{aligned}
& & F_{dyn} = \frac{d}{2} \int dsds' K^{ss'}_{00} G^{ss'}_{00}
+ \nonumber \\
& & + \frac{d}{2} \int dsds'd1d2K^{ss'}_{12}G^{ss'}_{12}
- \frac{d}{2} Tr\ln G \nonumber \\
& & - \frac{B^2}{4}
\int d^dxd^dy d1d2 \langle A_{12}^{(V)}(x,y) \rangle
\langle A_{12}^{(\delta)}(x,y) \rangle
\nonumber \\
& & - \frac{\beta_0 B^2}{2}
\int d^dxd^dy d1d2 \langle A_{10}^{(V)}(x,y) \rangle
\langle A_{10}^{(\delta)}(x,y) \rangle ,
\label{Fdyn2} \end{aligned}$$ where all averages are to be calculated using $S_{var}$ (see Eq. \[Svar\]). Performing averages, the fourth and fifth term on the right hand side of (\[Fdyn2\]) become $$\begin{aligned}
& & F_{dyn}^{(4)} =
- \frac{d}{2N}
\int d1d2dsds'{\cal V}\left[ (B_{12}^s+B_{12}^{s'})/2 \right], \\
& & F_{dyn}^{(5)} =
- \frac{\beta_0 d}{N}
\int d1d2dsds'{\cal V}\left[ (B_{10}^s+B_{10}^{s'})/2 \right], \end{aligned}$$ where $$\begin{aligned}
& & B_{12}^s = \langle [ \Phi(s,1)-\Phi(s,2) ]^2 \rangle =
G_{11}^{ss} + G_{22}^{ss} - 2 G_{12}^{ss}, \\
& & B_{10}^s = \langle [ \Phi(s,1)-x_0(s) ]^2 \rangle =
G_{11}^{ss} + G_{00}^{ss} - 2 G_{10}^{ss}, \end{aligned}$$ and $${\cal V}(z) = - \frac{\tilde B^2}{d}(z+\sigma)^{-d/2}\ , \ \
\tilde B^2 = \frac{B^2}{2} \frac{N}{v} (4\pi)^{-d/2}.$$ Performing the variational ansatz (i.e., evaluating Eq. \[dFdG\]) results in the following equations of motion: $$\begin{aligned}
& & \left[ T(\mu+k^2)-D^{(2)}_1 \right] G_{12}^k = \delta_{12}
+ 2 \int d3 {\cal V}'(B_{13}) \times \nonumber \\
& & \times (G_{32}^k-G_{12}^k)
+ 2 \beta_0 {\cal V}'(B_{10}) (G_{02}^k-G_{12}^k),
\label{G12k} \end{aligned}$$ $$\begin{aligned}
& & \left[ T(\mu+k^2)-D^{(2)}_1 \right] G_{10}^k =
2 \int d2 {\cal V}'(B_{12}) \times \nonumber \\
& & (G_{20}^k-G_{10}^k) + 2 \beta_0 {\cal V}'(B_{10})(G_{00}^k-G_{10}^k), \end{aligned}$$ $$\begin{aligned}
& ( \mu_0 + k^2) G_{01}^k = & 2 \beta_0 \int d2 {\cal V}'(B_{20})
(G_{21}^k-G_{01}^k), \end{aligned}$$ $$( \mu_0 + k^2) G_{00}^k = 1 + 2 \beta_0 \int d1 {\cal V}'(B_{10})
(G_{10}^k-G_{00}^k),
\label{G00k}$$ and after disentangling the SUSY notation one gets (see Paper I for related details) $$\begin{aligned}
& [T & (\mu+k^2)+\partial/\partial t] C_k(t,t') = \nonumber \\
& & 2 T R_k(t',t)
+ 2 \int_{0}^{t} dt'' {\cal V}'\left[B(t,t'')\right] R_k(t',t'')
\nonumber \\
& & + 4 \int_{0}^{t} dt'' {\cal V}''\left[B(t,t'')\right]
r(t,t'') \left[ C_k(t,t')-C_k(t'',t') \right] \nonumber \\
& & - 2 \beta_0 {\cal V}'[A(t)] [ C_k(t,t')-\phi_k(t') ] ,
\label{Ck} \end{aligned}$$ $$\begin{aligned}
& [T & (\mu+k^2)+\partial/\partial t] R_k(t,t') = \delta(t-t') + \nonumber \\
& & + 4 \int_{0}^{t} dt'' {\cal V}''\left[B(t,t'')\right] r(t,t'') \left[
R_k(t,t')-R_k(t'',t') \right] \nonumber \\
& & - 2 \beta_0 {\cal V}'[A(t)]R_k(t,t'),
\label{Rk} \end{aligned}$$ $$\begin{aligned}
& [ T & (\mu+ k^2)+\partial/\partial t] \phi_k(t) = \nonumber \\
& & 4 \int_{0}^{t} dt'' {\cal V}''\left[B(t,t'')\right]
r(t,t'') \left[ \phi_k(t)-\phi_k(t'') \right] \nonumber \\
& & + 2 \beta_0 {\cal V}'[A(t)] (\Gamma_k-\phi_k(t)),
\label{Phik1} \end{aligned}$$ $$\begin{aligned}
& & (\mu_0+k^2) \phi_k(t) =
2 \beta_0 \int_{0}^{t}dt''{\cal V}'[A(t'')] R_k(t,t''),
\label{Phik2} \end{aligned}$$ $$\begin{aligned}
& & (\mu_0+k^2) \Gamma_k = 1, \end{aligned}$$ where $B(t,t')$ and $A(t)$ are defined as $B(t,t') = \langle (x(s,t) -
x(s,t'))^2 \rangle = C(s,t;s,t) + C(s,t';s,t') - 2 C(s,t;s,t')$ and $A(t) = \langle (x(s,t) - x_0(s))^2 \rangle = C(s,t;s,t) -
2\phi(s,t;s) + \Gamma(s,s)$. Note that due to translational invariance with respect to $s$ both $B(t,t')$ and $A(t)$ are $s$-independent. The equations of motion for $C_k(t,t')$ and $R_k(t,t')$ are almost identical to those for the pure RHP model. Coupling to the native state enters through the terms proportional to $\beta_0$. Again, for large selection temperature $\beta_0\rightarrow 0$ and one recovers the RHP model.
Extracting order parameters {#sec:order_parameters}
===========================
The equations of motion are coupled integro-differential equations with initial conditions given by $C_k(0,0)$, $\phi(0)$, and (we use Ito’s convention) $R(t+\epsilon,t)\rightarrow 1$ as $\epsilon\rightarrow 0$. To solve the equations analytically we have to consider several assumptions (which can be checked by numerical solution).
First, we make the (rather strong) standard assumptions from aging theory for spin glasses about the asymptotic behavior of the solutions: In the regime of time translational invariance (TTI), $$\begin{aligned}
& & \lim_{t\rightarrow\infty} C_k(t+\tau,t) = C_k(\tau), \\
& & \lim_{t\rightarrow\infty} R_k(t+\tau,t) = R_k(\tau), \end{aligned}$$ and, in the aging regime, $$\begin{aligned}
& & \lim_{t\rightarrow\infty} C_k(t,\lambda t) =
q_k \hat C_k(\lambda), \\
& & \lim_{t\rightarrow\infty} R_k(t,\lambda t) =
\frac{1}{t} \hat R_k(\lambda). \end{aligned}$$ The validity of these assumptions could be checked numerically. Since this has been done for equations of similar type elsewhere [@CD], we omit it in the present analysis.
Second, it is well known that asymptotic solutions of such equations can be characterized by a few order parameters [@CKD; @CD; @CK1; @BCKP; @CK2]. They are defined as $$\begin{aligned}
& & \tilde q_k = \lim_{t\rightarrow\infty} C_k(t,t), \\
& & q_k = \lim_{\tau\rightarrow\infty} C_k(\tau), \\
& & q_{0,k} = \lim_{\lambda\rightarrow 0} q_k \hat C_k(\lambda), \\
& & \varphi_k=\lim_{t\rightarrow\infty}\phi_k(t). \end{aligned}$$ The following $k$-integrated quantities will also be useful: $$\begin{aligned}
& & \tilde q \equiv \int \frac{dk}{2\pi} \tilde q_k =
\lim_{t\rightarrow\infty} \langle x(s,t)x(s,t) \rangle , \\
& & q \equiv \int \frac{dk}{2\pi} q_k =
\lim_{\tau\rightarrow\infty}
\lim_{t\rightarrow\infty} \langle x(s,t)x(s,t+\tau) \rangle ,\\
& & q_0 \equiv \int \frac{dk}{2\pi} q_{0,k} =
\lim_{\lambda\rightarrow 0}
\lim_{t\rightarrow\infty} \langle x(s,t)x(s,\lambda t) \rangle ,\\
& & \varphi \equiv \int \frac{dk}{2\pi} \varphi_k =
\lim_{t\rightarrow\infty} \langle x(s,t) x_0(s) \rangle. \end{aligned}$$ $\tilde q$ measures the size of the globule, $q$ measures the persistent correlation in the TTI regime, $q_0$ the asymptotic correlation in the aging regime, and $\varphi$ the overlap with native state. Also, it is useful to define $$\begin{aligned}
& & b = 2 ( \tilde q - q ) \, , \,\,\, b_0 = 2 ( \tilde q - q_0 ),
\label{bqq0} \\
& & a \equiv \lim_{t\rightarrow\infty}
\langle [x(s,t)-x_0(s)]^2 \rangle
= \tilde q - 2 \varphi + \frac{1}{2\sqrt{\mu_0}}. \end{aligned}$$
Third, we assume that the generalized fluctuation dissipation theorem is valid in the form $$\hat R_k(\lambda) = \frac{x}{T}
\, q_k \frac{ d\hat C_k(\lambda) }{ d\lambda },
\label{GFDT}$$ $x$ could in principle depend on $k$ and $C_k$. However, related models have been studied in detail and they exhibit one step replica symmetry breaking with a $k$-independent $x$. This one step replica symmetry breaking ansatz in our dynamical study translates exactly to Eq. (\[GFDT\]).
Relating order parameters
=========================
For $t=t'$ and $t\rightarrow\infty$ Eq. (\[Ck\]) gives $$\begin{aligned}
& & T(\mu+k^2) \tilde q_k = T
+ \frac{2}{T} {\cal V}'(b) (1-x) ( \tilde q_k - q_k ) \nonumber \\
& & \ \ \ \ \ \
+ \frac{2}{T} {\cal V}'(b_0) x ( \tilde q_k - q_{0,k})
-2\beta_0 {\cal V}'(a)(\tilde q_k - \phi_k).
\label{eq:qtk2} \end{aligned}$$ With $t=t'+\tau$ and $t'\rightarrow\infty$ and then $\tau\rightarrow\infty$ Eq. (\[Ck\]) becomes $$\begin{aligned}
& & T(\mu+k^2) q_k =
\frac{2}{T} ( {\cal V}'(b)
- x {\cal V}'(b_0) ) ( \tilde q_k - q_k ) \nonumber \\
& & \ \ \ \ \ \
+ \frac{2}{T} {\cal V}'(b_0) x ( \tilde q_k - q_{0,k} )
-2\beta_0 {\cal V}'(a)(q_k - \phi_k).
\label{eq:qk2} \end{aligned}$$ Eq. (\[Ck\]) in the aging regime $t'=\lambda t$, first for $t\rightarrow\infty$ and then $\lambda\rightarrow 0$, gives $$\begin{aligned}
& & T(\mu+k^2) q_{0,k} =
\frac{2}{T} {\cal V}'(b_0) (1-x) ( \tilde q_k - q_k ) \nonumber \\
& & \ \ \ \ \ \
+ \frac{2}{T} {\cal V}'(b_0) x ( \tilde q_k - q_{0,k} )
-2\beta_0 {\cal V}'(a)(q_{0,k} - \phi_k).
\label{eq:q0k2} \end{aligned}$$ Eqs. (\[Phik1\]) and (\[Phik2\]) result in two equations for $\varphi_k$, $$\begin{aligned}
& T ( \mu + k^2 ) \varphi_k = &
\frac{2}{T_0} {\cal V}'(a) ( \Gamma_k - \varphi_k )
\label{varphik1} \\
& (\mu_0 + k^2 ) \varphi_k = &
\frac{2}{TT_0} {\cal V}'(a) x ( q_k - q_{o,k} ) \nonumber \\
& & + \frac{2}{TT_0} {\cal V}'(a) ( \tilde q_k - q_k )
\label{varphik2} \end{aligned}$$ They are equivalent; one can chose to solve for the order parameters working with either (\[varphik1\]) or (\[varphik2\]). This seems a rather remarkable coincidence. We believe that it originates from the SUSY invariance of the original action $S$. For example, a similar comment holds for equations (\[Ck\]) and (\[Rk\]); they are equivalent in the TTI regime and one can derive one from the other. The ‘conspiracy’ of (\[Phik1\]) and (\[Phik2\]) not contradicting each other is very likely a similar phenomenon. Eq.(\[Rk\]) for $\lambda=1$ reduces to $$\hat R_k(1) (\tilde\mu+k^2+\Sigma) = - (\tilde q_k - q_k )
\frac{4{\cal V}''(b)}{T^2} \hat r(1),
\label{MSCk}$$ where $$\hat r(\lambda) \equiv \int \frac{dk}{2\pi} \hat R_k(\lambda)$$ and $\Sigma$ is defined by $$\Sigma = x \frac{2}{T^2} \left( {\cal V}'(b) - {\cal V}'(b_0) \right).
\label{Sigma}$$ Solving these equations for the order parameters gives $$\begin{aligned}
& & b = \frac{1}{\sqrt{\tilde\mu+\Sigma}}, \label{qtq} \end{aligned}$$ $$\begin{aligned}
& & b_0 =
\frac{1}{x} \frac{1}{\sqrt{\tilde\mu}}
+ \frac{x-1}{x} \frac{1}{\sqrt{\tilde\mu+\Sigma}}
\label{qq0}, \end{aligned}$$ $$\begin{aligned}
& \tilde q = & \frac{b_0}{2}
+ \frac{{\cal V}'(b_0)}{4T^2\tilde\mu^{3/2}}
+ \frac{1}{4\sqrt{\mu_0}}
\left(
\frac{1-\frac{\mu}{\tilde\mu}}{1-\frac{\mu_0}{\tilde\mu}}
\right)^2 \times \nonumber \\
& & \times \left(2+\sqrt{\frac{\mu_0}{\tilde\mu}} \right)
\left( 1 -\sqrt{\frac{\mu_0}{\tilde\mu}} \right)^2,
\label{qt} \end{aligned}$$ $$\begin{aligned}
& a = & \frac{b_0}{2}
+ \frac{{\cal V}'(b_0)}{4T^2\tilde\mu^{3/2}}
+ \frac{1}{4\sqrt{\mu_0}}
\frac{1}{
\left( 1+\sqrt{\frac{\mu_0}{\tilde\mu}} \right)^2
} \times \nonumber \\
& & \times
\left[
\sqrt{\frac{\mu_0}{\tilde\mu}}
\left( 1+2\sqrt{\frac{\mu_0}{\tilde\mu}} \right)
+ 2 \frac{\mu}{\tilde\mu}\sqrt{\frac{\mu_0}{\tilde\mu}}
\right.
\nonumber \\
& & \left.
\,\,\,\,\,\,\,\,\,\, + \left( \frac{\mu}{\tilde\mu} \right)^2
\left( 2+\sqrt{\frac{\mu_0}{\tilde\mu}} \right)
\right],
\label{a} \end{aligned}$$ $$\begin{aligned}
& & \tilde\mu = \mu + \frac{2}{TT_0}{\cal V}'(a), \label{mut} \end{aligned}$$ and the combination of Eq. (\[qtq\]) and (\[MSCk\]) gives $$\begin{aligned}
&& 0 = \hat r(1) \left[ T^2 + b^3 {\cal V}''(b) \right]. \label{MSC} \end{aligned}$$ Furthermore, the overlap $\varphi$ with the native state is given by $$\varphi = \frac{1}{2\sqrt{\mu_0}}
\frac{
1-\frac{\mu}{\tilde\mu}
}{
1+\sqrt{
\frac{\mu_0}{\tilde\mu}
}
}.
\label{varphi}$$ All overlap order parameters are positive. However, this result is not obvious and has to be obtained after some algebra.
These equations have two kinds of solutions. In one kind, $b=b_0$, so there is no glassiness (aging). For this kind of solution, the parameter $x$ is irrelevant. We call such solutions “ergodic”. (While it will turn out that some of them are not truly ergodic, in the sense of describing states where the entire configuration space is visited with Boltzmann probabilities, they violate ergodicity in a rather trivial way, like a ferromagnet below the Curie temperature. We could call them “non-glassy”, but we prefer not to use a negative term.)
For an ergodic solution, with $b=b_0$, $\Sigma=0$. Furthermore, $\hat
r(\lambda)=0$, so Eqn. (\[MSC\]) is trivially satisfied. One then has to solve the four equations (\[qtq\]) and (\[qt\]-\[mut\]) for $b$, $\tilde q$, $a$ and $\tilde \mu$.
The stability of such a phase against glassiness can be determined using the analysis we presented in Paper I (see Fig. 1). There, we studied a model with no native-state bias in its interactions ($T_0 =
\infty$) for finite $\mu$. The boundary of the glassy state as a function of $\mu$ has a form qualitatively like that in the p-spin glass as a function of field [@CS; @CHS]. In the present model, the presence of the native state enters the calculation solely through the replacement of $\mu$ by ${{\tilde \mu}}$. Therefore, if a particular $T$ and ${{\tilde \mu}}$ fall in the glassy regime (the region below the full and dashed lines) in Fig. 1, the ergodic ansatz has to be given up.
The instability can occur in two ways, according to whether ${{\tilde \mu}}$ is bigger or smaller than the critical value ${{\tilde \mu}}_c$. Above ${{\tilde \mu}}_c$, the line separating glassy from ergodic regions is an Almeida-Thouless (AT) line; below it the stability condition $$T^2 + b^3 {\cal V}''(b) >0, \label{eq:ATinequality}$$ is violated. For ${{\tilde \mu}}< {{\tilde \mu}}_c$, there is no AT instability. The transition is like that for the completely random heteropolymer. To find such a transition, we have to solve for a glassy phase, characterized in part by a value of the FDT-violation parameter $x <
1$ and then find where in the parameter space $x \rightarrow 1$. In the region where the $x<1$ solution exists, the associated ergodic phase is unstable and is replaced by the glassy one.
In a glassy phase, aging is present: $\hat r(1)\ne 0$, so the quantity in brackets in Eq. (\[MSC\]) has to vanish, i.e., the AT condition has to be satisfied as an equality, rather than an inequality. This so-called marginal stability condition determines $b$ as a function of temperature. In this case we have three more unknowns, $\Sigma$, $b_0$ and $x$, making a total of seven, and seven equations, (\[qtq\]-\[MSC\]), to solve for them.
We look for ergodic solutions first in the next section, and we examine their stability. Then, in the following section, we study glassy solutions (within the 1-step aging ansatz of section VII) and identify the regions in the parameter space where they hold.
Ergodic phases {#sec:erg_phase}
==============
For ergodic phases, Eqns. (\[qtq\]-\[mut\]) reduce to $$\begin{aligned}
&& b=b_0=\frac{1}{\sqrt{\tilde\mu}} \label{berg} \\
& \tilde q = & \frac{1}{2\sqrt{\tilde\mu}}
+ \frac{{\cal V}'(1/\sqrt{\tilde\mu})}{4T^2\tilde\mu^{3/2}}
+ \frac{1}{4\sqrt{\mu_0}}
\left(
\frac{1-\frac{\mu}{\tilde\mu}}{1-\frac{\mu_0}{\tilde\mu}}
\right)^2 \times \nonumber \\
& & \times \left(2+\sqrt{\frac{\mu_0}{\tilde\mu}} \right)
\left( 1- \sqrt{\frac{\mu_0}{\tilde\mu}} \right)^2
\label{qterg} \\
& a = & \frac{1}{2\sqrt{\tilde\mu}}
+ \frac{{\cal V}'(\frac{1}{\sqrt{\tilde\mu}})}{4T^2\tilde\mu^{3/2}}
+ \frac{1}{4\sqrt{\mu_0}}
\frac{1}{
\left( 1+\sqrt{\frac{\mu_0}{\tilde\mu}} \right)^2
} \times \nonumber \\
& & \times
\left[
\sqrt{\frac{\mu_0}{\tilde\mu}}
\left( 1+2\sqrt{\frac{\mu_0}{\tilde\mu}} \right)
+ 2 \frac{\mu}{\tilde\mu}\sqrt{\frac{\mu_0}{\tilde\mu}}
\right.
\nonumber \\
& & \left.
\,\,\,\,\,\,\,\,\,\, + \left( \frac{\mu}{\tilde\mu} \right)^2
\left( 2+\sqrt{\frac{\mu_0}{\tilde\mu}} \right)
\right]
\label{aerg} \\
& & \tilde\mu = \mu + \frac{2}{TT_0} {\cal V}'(a) \label{muterg} \end{aligned}$$ They can be solved numerically: given $\mu$, $\mu_0$, $T$ and $T_0$ one can find $\tilde\mu$, which in turn determines $\tilde q$, $b=b_0$ (equivalently $q=q_0$), and $\varphi$. However, it is possible to gain some analytic understanding in a few soluble limits.
In this discussion we will concentrate on the limit of small $\mu$. As we noted in paper I, if we want to confine $N$ monomers within a gyration radius $\sqrt{\tilde q} \propto \mu^{-1/4}$, we need $\mu
\propto N^{-4/d}$. Thus, for a long polymer $\mu \rightarrow 0$. We will also take $\mu=\mu_0$ to simplify the algebra a bit.
The pair of equations (\[aerg\]) and (\[muterg\]) fully determine ${{\tilde \mu}}$ as function of $T$ and $T_0$. For $\mu_0=\mu$ they take the form a() &=& +\
&+& ( 1 + ) [eq:a]{}\
(a) &=& + [eq:tmu]{} Given ${{\tilde \mu}}$, $T$, $T_0$ one can find the overlap with the native state $\varphi$ and the size of the polymer from (\[varphi\]).
Random-globule state
--------------------
It is immediately evident that when both the temperature $T$ and the selection temperature $T_0$ are large, $\tilde \mu \approx \mu$ in (\[muterg\]), leading to a random-globule solution $a = b =
\mu^{-1/2}$, $\tilde q = \mu^{-1/2}/2$, $\varphi = 0$. What is not so obvious is that in the $\mu \rightarrow 0$ limit a solution very close to this exists all the way down to very low temperatures, even for small $T_0$. In this subsection we examine this state in detail.
We look first for solutions of Eqs. (\[eq:a\]) and (\[eq:tmu\]) with the ansatz $\alpha \equiv {{\tilde \mu}}/\mu$ fixed and $\mu\rightarrow
0$. We call this the random globule ansatz, since, as will be shown, the polymer does not have any fixed conformation (it is melted), and on the average the conformations it adopts have zero overlap with the native state. (Strictly speaking, this is the only truly ergodic phase we find.) For $a$ we get, $$a = \frac{1}{\sqrt{\mu}}
\left[
\frac{
3+\frac{1}{\alpha}
}{
4\sqrt{\alpha}
}
+
{\cal O}(\mu^{(d-2)/2})
\right],$$ which, after inserting into (\[eq:tmu\]), gives $$\alpha \approx 1 + \frac{{{\tilde B}}^2}{TT_0} \mu^{(d-2)/4}
\left(
\frac{
4 \sqrt{\alpha}
}{
3 + \frac{1}{\alpha}
}
\right)^{d/2+1}.
\label{alpha}$$ Eqn. (\[alpha\]) can be used to calculate $\alpha$ as a function of $\mu$. One can see easily that $\alpha\rightarrow 1$ when $\mu\rightarrow 0$. This shows that our ansatz is self-consistent in the limit of small $\mu$. Also, (\[qterg\]) and (\[varphi\]) become $$\begin{aligned}
& & \varphi=\frac{1}{2\sqrt{\mu}}
\left[\frac{\alpha-1}{2}+{\cal O}(\alpha-1)^2\right]
\label{varphi2} \\
& & \tilde q=\frac{1}{2\sqrt{\mu}}\left[1+{\cal O}(\alpha-1)\right].
\label{qterg2} \end{aligned}$$
The normalized overlap between the polymer conformation and the native state is: $$\cos \theta =
\frac{
\lim_{t\rightarrow\infty}\langle x(s,t)x_0(s)\rangle
}{
\sqrt{
\lim_{t\rightarrow\infty} \langle x(s,t)^2 \rangle
\langle x_0(s)^2 \rangle
}
} =
\frac{\varphi
}{
\sqrt{\tilde q (\frac{1}{2\sqrt{\mu}}})
}.
\label{costh}$$ >From (\[varphi2\]) and (\[qterg2\]) we get $\cos(\theta)\sim(\alpha-1)\sim\mu^{(d-2)/4}$. Thus there is no overlap with native state as $\mu\rightarrow 0$.
Furthermore, to check that polymer does not freeze into some other conformation, we calculate the normalized overlap between two configurations taken at very different times, $$\cos \theta' =
\frac{
\lim_{\tau,t\rightarrow\infty} \langle x(s,t)x(s,t+\tau)\rangle
}{
\sqrt{ [\lim_{t\rightarrow\infty} \langle x(s,t)^2 \rangle ]^2 }
} =
\frac{q}{\tilde q}.
\label{costh'}$$ After rewriting $$q/\tilde q=1-\frac{b}{2\tilde q}=
1-\frac{1}{2\sqrt{{{\tilde \mu}}\tilde q}},
\label{q/qt}$$ and, using (\[qterg2\]), we get $\cos \theta'={\cal
O}(\alpha-1)$. Again, as $\mu\rightarrow 0$, $\cos \theta' \rightarrow
0$. This confirms that the ansatz $\alpha ={\cal O}(1)$ and $\mu
\rightarrow 0$ leads to a melted random-globule-like phase. This phase is identical to that found at high temperatures for the completely random heteropolymer in paper I.
The validity of the present ansatz rests upon the fact that we can solve Eq. (\[alpha\]). Clearly, for $\mu\rightarrow 0$ a solution can always be found, namely $\alpha=1$. Since the physically relevant $\mu$ is $\propto N^{-4/d}$, we can always satisfy this equation, for any $T_0$, in the limit $N \rightarrow\infty$.
We now address briefly the question of what happens for finite $N$ (and $\mu$). One can easily see that Eq. (\[alpha\]) has two solutions when $\mu^{(d-2)/4}/(TT_0)$ is not too large (e.g. by plotting the left- and right-hand side as functions of $\alpha$). The solution close to 1 is lost when the slopes of the left- and right-hand sides become roughly equal. Evaluating these slopes leads to the condition (+1)\^ <1 \[eq:coilcond\] for the existence of a random-globule-like state.
Some caution is in order. Working this out for finite $N$, $d=3$, and an average density of $1$, we find that the inequality (\[eq:coilcond\]) is violated below a temperature T\_x = ( )\^[1/2]{} N\^[-1/3]{}. With the small power of $N^{-1}$, one has to go to quite large $N$ to make this temperature very low. Thus our statement that the random-globule-like state exists for all temperatures in the $\mu
\rightarrow 0$ limit may be of limited relevance for real 3-dimensional heteropolymers of the length of typical proteins. Nevertheless, here we are just considering this simple limit.
We now discuss the stability of this solution. In the large-$N$ limit, it is locally stable against spontaneous formation of a native-like state at any $T$ and $T_0$. However, it is unstable against glass formation at low temperatures: Since it is identical with the random-globule solution of the completely random heteropolymer problem, we can take over the result from paper I that it is unstable below a temperature $T_g \propto {{\tilde B}}$, with the constant of proportionality of order 1. This glass temperature is independent of $T_0$. (In Fig. 1 this is the transition at ${{\tilde \mu}}\rightarrow 0$.) Thus, wherever the system is in a random-globule-like state at $T >
T_g$, it will no longer equilibrate if the temperature is lowered below $T_g$. Instead, it will become glassy and its dynamics will show aging.
Ergodic native state
--------------------
At low $T_0$ and $T$, one expects that the polymer should be very close to its native state, i.e., small $a$. Therefore we also look for such solutions of Eqs. (\[eq:a\]) and (\[eq:tmu\]). We will try to solve equations (\[eq:a\]) and (\[eq:tmu\]) in the limit where $\mu \rightarrow 0$ and ${{\tilde \mu}}$ stays finite. The limit $\mu\rightarrow 0$ turns out not to involve any subtleties when ${{\tilde \mu}}$ is kept constant, so we will just set $\mu = 0$ from the outset. Eqs. (\[eq:a\]) and (\[eq:tmu\]) become $$\begin{aligned}
a({{\tilde \mu}}) &=& \frac{3}{4\sqrt{{{\tilde \mu}}}} +\frac{{{\tilde B}}^2}{8T^2 {{\tilde \mu}}^{3/2}
(\sigma + {{\tilde \mu}}^{-1/2})^{\frac{d}{2}+1}} {\label}{eq:a1} \\
{{\tilde \mu}}(a) &=& \frac{{{\tilde B}}^2}{TT_0 (\sigma + a)^{\frac{d}{2}+1}}
{\label}{eq:tmu1} \end{aligned}$$ These equations can be solved for ${{\tilde \mu}}$ as function of $T$ and $T_0$. However, one has to keep in mind that $\mu\rightarrow 0$ has been taken. This implies that (\[varphi\]) and (\[qterg\]) become $$\varphi \approx \frac{1}{2\sqrt{\mu}} , \ \ \
\tilde q \approx \frac{1}{2\sqrt{\mu}}
\label{varphiqt},$$ and, inserting (\[varphiqt\]) into (\[costh\]), the normalized overlap between native state and polymer conformations, becomes $\cos
\theta \approx 1$. Furthermore, because of its large overlap with the native state, the polymer is essentially frozen. This can be seen by calculating the normalized overlap between two polymer conformations after a very long time interval, as in previous section. Inserting (\[varphiqt\]) into (\[costh’\]) and (\[q/qt\]) gives $\cos
\theta' \approx 1-\sqrt{\mu/{{\tilde \mu}}}\rightarrow 1$.
There is interesting behavior associated with the limit $\mu\rightarrow 0$ for very long polymers. When the polymer gets longer and longer ($N\rightarrow\infty$) a finite part of the chain is not in the native state conformation, since $a$ stays constant. The rest of the chain is in the native state, which can be seen from the fact that overlap with native state approaches 1. Thus, in the limit of a very long polymer, the fraction of chain not in the native state conformation becomes negligible: the recipe for biasing the coupling constants $B_{ss'}$ described in chapter II works best for long polymers.
In the following we will proceed with the solution of equations (\[eq:a1\]) and (\[eq:tmu1\]). Before continuing, it will be useful to compactify notation a bit. Making the change of variables $\hat X = X / \sigma$ for $X=b,\,b_0,\,a,\,q,\,\tilde q$; $\hat Y = Y
\sigma^2$ for $Y=\mu,\,{{\tilde \mu}}$; and $\hat Z=Z\sigma^{(d-2)/4}/{{\tilde B}}$ for $Z=T,\,T_0$, we get equations of the same form, with $X \rightarrow
\hat X$, $Y \rightarrow \hat Y$ and $Z \rightarrow \hat Z$, but with $\sigma =1$ and ${{\tilde B}}=1$. Thus, without loss of generality, we can choose units with $\sigma=1$ and ${{\tilde B}}=1$ (and remove the hats). From now on we do this.
The working strategy for solving the equations is as follows. For fixed $T$, one can consider $T_0$ as a function of ${{\tilde \mu}}$. This can be easily done by inserting the expression for $a$ from Eq. (\[eq:a1\]) into (\[eq:tmu1\]), thus writing $\beta_0 = 1/T_0$ as $$\beta_0({{\tilde \mu}},T) = T{{\tilde \mu}}[1+a({{\tilde \mu}},T)]^{d/2+1}
\label{invt0tmu}$$ The four panels of Fig. 2 shows the shape of $\beta_0({{\tilde \mu}},T)$ as a function of ${{\tilde \mu}}$ for four different temperatures. We want ultimately to construct a phase diagram in the $(\beta_0,T)$ plane. Therefore we have to specify $T$ (one panel of the figure) and $\beta_0$ and ask whether one or more solutions, i.e., particular values of ${{\tilde \mu}}$ which solve Eq. (\[invt0tmu\]), exist. For example, in panel (a) in Fig. 2, a horizontal line at $\beta_0>\beta_0^{\rm min}$ intersects $\beta_0({{\tilde \mu}},T)$ curve at two places, indicating two solutions ${{\tilde \mu}}={{\tilde \mu}}_{1}, {{\tilde \mu}}_{2}$. To make the figures more readable we have shown such a horizontal line, at a particular values of $\beta_0$, only in panel (a). If this horizontal line is moved below $\beta_0^{\rm min}$, it will never intersect the $\beta_0({{\tilde \mu}},T)$ curve. Thus, we can see that for every $T$, there is a value $\beta_0^{\rm min}(T)$ below which no solutions exist.
We proceed with the analysis of Fig. 2. For sufficiently high temperatures (panels (a)-(c)) there are exactly two solutions for all $\beta_0 > \beta_0^{\rm min}$. Of these, the one with the larger value of ${{\tilde \mu}}$ is a stable solution (local free energy minimum) describing the ergodic native phase. For example, the solution labeled ${{\tilde \mu}}_2$ in panel (a) is of this sort. The one with the smaller value of ${{\tilde \mu}}$ (e.g., the one labeled by ${{\tilde \mu}}_1$ in panel (a)) is unstable. It describes a free energy maximum between the minima at the random-globule and ergodic native states. We will call such states “unstable stationary” (abbreviated US). (We have not done a static calculation to show this, but the situation here is analogous to that in an ordinary ferromagnet below the Curie temperature. There, one has three solutions of the mean field equations, one with positive, one with negative, and one with zero magnetization. The middle one, with zero magnetization, is unstable). The US state has a lower overlap with the native conformation than the ergodic-native solution does, because it has a smaller value of ${{\tilde \mu}}$. As $\beta_0$ is increased from below through $\beta_0^{\rm min}$, the native-state and US-state solutions appear together and separate. For the temperatures of panels (a)-(c), they both exist for all $\beta_0 >\beta_0^{\rm min}$.
Panel (d) (at the lowest of the temperatures) shows a more complex behavior where double-minimum structure appears. We have found numerically that this happens below $T\approx 0.20$. Here the behavior around $\beta_0^{\rm min}$ is just as in the other cases, but we note that at this temperature $\beta_0({{\tilde \mu}},T)$ has a second local minima at a smaller value of ${{\tilde \mu}}$. Thus there is a range $\beta_1^{\rm min}<\beta_0<\beta_1^{\rm max}$ for which there are four solutions. The rightmost one is stable and describes the ergodic-native phase, as before. Moving from right to left, the solutions alternate between stability and instability. Thus the second solution from the left represents a locally stable conformation. It is also correlated with the native state, since ${{\tilde \mu}}$ is finite (though we always find ${{\tilde \mu}}\ll 1$ in 3 dimensions). The remaining two solutions (with $\partial \beta_0/\partial {{\tilde \mu}}<0$) represent US states (local free energy maxima) between it and the random-globule phase in one direction and the ergodic-native phase in the other.
Plotting $\beta_0^{\rm min}$ against $T$, we obtain the stability boundary indicated by the thick solid curve in Fig. 3. Within our present assumption of ergodicity, everywhere to the right of this line the ergodic-native phase is dynamically stable. One can invert the relation $\beta_0^{\rm min}(T)$, obtaining a transition temperature $T_n(\beta_0)$, the maximum temperature for which the ergodic native phase is dynamically stable. It is separated from the (also stable) random-globule phase by a barrier, the top of which is described by the unstable solution.
In Fig. 3 we also indicate the region in the $(\beta_0,T)$ plane where the second locally-stable solution is found. This region has the form of a kind of sliver extending out toward large $\beta_0$ at low temperatures.
So far we have not examined the stability of these solutions against glassiness. As indicated above, we do this with the help of Fig. 1: Stable solutions can not lie in the range ${{\tilde \mu}}_{min} < {{\tilde \mu}}<
{{\tilde \mu}}_{AT}$. In Fig. 2, these limits are marked on the ${{\tilde \mu}}$ axes. We thus see, for example, in Panel (c), that the native-state solutions found for the range of $\beta_0$ corresponding to values of ${{\tilde \mu}}$ between ${{\tilde \mu}}_*$ and ${{\tilde \mu}}_{AT}$ are not acceptable: they violate the AT stability condition (\[eq:ATinequality\]).
Similarly, in panel (b) the US solutions found for a range of $\beta_0$ values can also be seen to lie in the forbidden region. And the intermediate locally-stable states that we identified in panel (d) always lie in a glassy region.
In Fig. 3 we also plot the AT line (\[MSC\]) in the $(\beta_0,T)$ plane, indicating the regions where the various kinds of ergodic solutions are forbidden. For the native-phase solutions, the forbidden region is a strip mostly at low values of $T$ (diagonally cross-hatched region between thick and AT line). However, it “wraps around” at the leftmost part of the region where those solutions are found.
The forbidden region for the US solutions occupies most of the region where these solutions occur below $T_{max}$, the maximum temperature for a glass transition shown in Fig. 1, including the entire portion of it below $T_g$, the glass instability temperature of the random-globule state.
The structure in a tiny region near the minimum value of $\beta_0$ for which ergodic-native solution are found is a bit complicated and cannot be seen in the top panel of Fig. 3. Therefore, the lower panel shows an enlargement of this region.
In summary, we have found four kinds of ergodic solutions. One essentially describes a random globule state. It is locally stable (in the limit of a large globule) at all $T$ and $\beta_0$ against condensation into a native-like state, but unstable against glass formation everywhere below a transition temperature $T_g$. The second kind of solution describes a phase which is highly correlated with the native state conformation, and it is stable in most of the region where the solution exists. The third kind of solution describes a locally stable state, correlated with the native state but more weakly so than the ergodic native phase just described. It is never stable against glass formation. Finally, there are unstable solutions, found whenever the ergodic-native solutions exist. They describe US states, free energy maxima between pairs of the previously-described solutions. However, in a large part of the region where these solutions are found (roughly, everywhere below $T_{max} \approx T_g$) they violate the AT stability condition and so are not physically relevant.
Outside the regions where these ergodic solutions are allowed, we have to look for glassy solutions. We do this in the next section.
Glassy phases
=============
In a glassy phase, $\hat r(1)\ne 0$ and Eq. (\[MSC\]) has to be kept, which gives, $$T^2 = - b^3 {\cal V}''(b)
\label{b(T)}$$ Also, equations (\[Sigma\]), (\[qtq\]) and (\[qq0\]) can be rewritten in the form $$\begin{aligned}
&& \frac{ {\cal V}'(b) - {\cal V}'(b_0) }{ b_0 - b } = \frac{T^2}{2}
\frac{\sqrt{\tilde\mu}}{b}
\left( \frac{1}{b} + \sqrt{\tilde\mu} \right),
\label{b0b} \\
&& b_0 - b = \frac{1}{x} \left( \frac{1}{\sqrt{\tilde\mu}} - b \right).
\label{xbb0} \end{aligned}$$ and, with $\mu_0=\mu$, (\[a\]) and (\[mut\]) become $$\begin{aligned}
a &=& \frac{b_0}{2} +\frac{1}{8T^2 {{\tilde \mu}}^{3/2}
(1 + b_0)^{\frac{d}{2}+1}} + \frac{1}{4\sqrt{{{\tilde \mu}}}}
\left( 1 + \frac{\mu}{{{\tilde \mu}}}\right)
{\label}{eq:a2} \\
{{\tilde \mu}}&=& \mu + \frac{1}{TT_0 (1 + a)^{\frac{d}{2}+1}}
{\label}{eq:tmu2} \end{aligned}$$ The above equations can be solved as follows. Eq. (\[b(T)\]) gives $b$ as a function of $T$, and then (\[b0b\]), (\[eq:a2\]) and (\[eq:tmu2\]) can be used to find $b_0$ and ${{\tilde \mu}}$ as functions of $T$ and $T_0$. Once $b_0$ and ${{\tilde \mu}}$ are found one can calculate $\tilde q$ as $$\begin{aligned}
& \tilde q = & \frac{b_0}{2} +
\frac{1}{8T^2 \tilde\mu^{3/2}
(1 + b_0)^{\frac{d}{2}+1}} + \nonumber \\
& & +\frac{1}{4\sqrt{\mu}} ( 2 + \sqrt{\frac{\mu}{\tilde\mu}})
(1-\sqrt{\frac{\mu}{\tilde\mu}})
\label{qtglas1} \end{aligned}$$
As in our analysis of ergodic solutions in the preceding section, we will try two types of ansatz: one with ${{\tilde \mu}}/\mu=const$ as $\mu\rightarrow 0$ and one with ${{\tilde \mu}}=const$ as $\mu\rightarrow 0$ leading to what we call frozen-globule and glassy native phases, respectively.
Frozen-globule phase
--------------------
The limit where $\alpha={{\tilde \mu}}/\mu$ is kept constant and $\mu\rightarrow
0$ is easily treated. Eq. (\[b(T)\]) stay the same, while Eq. (\[b0b\]) gives $$\frac{{\cal V}(b)-{\cal V}(b_0)}{b_0-b}
\approx\frac{T^2\sqrt{\alpha}}{2b^2}\sqrt{\mu}.$$ Since $b$ is kept fixed the only solution of equation above is $b_0\rightarrow\infty$ as $$b_0 \approx \frac{\psi(b)}{\sqrt{\alpha\mu}},
\label{b0glas}$$ where $\psi(b)$ is a function which depends only on $b$, as $\mu$ is sent to $0$. Inserting (\[b0glas\]) into (\[eq:tmu2\]) gives $$\alpha = 1 + {\cal O}(\mu^{(d-2)/4})$$ and $\alpha$ stays very close to $1$, as in the ergodic random globule case. Also, $\varphi$ is given by (\[varphi2\]), while (\[qtglas1\]) gives $$\tilde q = \frac{1}{2\sqrt{\mu}} \left[ \psi(b)
+ {\cal O}(\alpha-1) \right]
\label{qtglas2}$$ which can be compared with ergodic globule result, Eq. (\[qterg2\]). Eq. (\[costh\]) stays the same, and one gets $\cos \theta\sim
\alpha-1$ which goes to zero as $\mu\rightarrow 0$. There is no overlap with native state. Does the system freeze into some other configuration? To find out, we calculate overlap angles between configurations at time $t$ and a much later time $t'$. As discussed in section \[sec:order\_parameters\] there are two ways in which the limit $t,t\rightarrow\infty$ can be taken, leading to $q_0\ne q$.
In the first limit, the equivalent of Eq. (\[costh’\]) for the ansatz used here reads $$\cos \theta'_g =
\frac{
\lim_{\lambda,t\rightarrow\infty} \langle x(s,t)x(s,\lambda t)\rangle
}{
\sqrt{ [\lim_{t\rightarrow\infty} \langle x(s,t)^2 \rangle ]^2 }
} =
\frac{q_0}{\tilde q}
\label{costh'g}$$ Using Eq. (\[bqq0\]), we can write $\cos \theta'_g =1-b_0/2\tilde
q$, and Eqs (\[qtglas2\]) and (\[b0glas\]) give $\cos \theta'_g
\sim \alpha-1$ which goes to $0$ as $\mu\rightarrow 0$. (This behavior is analogous to that found in p-spin glasses.) However, at not-too-long time scales (shorter than the waiting time), as in equation (\[costh”g\]), the polymer is frozen: $$\cos \theta''_g =
\frac{
\lim_{\tau,t\rightarrow\infty} \langle x(s,t)x(s,t+\tau)\rangle
}{
\sqrt{ [\lim_{t\rightarrow\infty} \langle x(s,t)^2 \rangle ]^2 }
} =
\frac{q}{\tilde q}
\label{costh''g}$$ Using Eq. (\[bqq0\]), we can write $\cos \theta''_g=1-b/2\tilde q$, and Eq. (\[qtglas2\]) gives $\cos \theta''_g\sim
1-\sqrt{\mu}b/\psi(b)$, which goes to $1$ as $\mu\rightarrow 0$. Thus, this glassy phase has no overlap with the native state.
As discussed above, there is an upper temperature limit $T_g$ (independent of $\beta_0$) above which this phase melts, leaving the system in the random-globule state. $T_g$ can be found from Eqs. (\[b(T)\]) and (\[b0b\]), using $b_0 \rightarrow \infty$ and (\[xbb0\]) with $x \rightarrow1$. This leads to a value $b =
2/({\mbox{$\frac{1}{2}$}}d - 1) = {\cal O}(1)$ at the transition and $T_g = 2 ({\mbox{$\frac{1}{2}$}}d
-1)^{{\mbox{$\frac{1}{2}$}}({\mbox{$\frac{1}{2}$}}d - 1)}/({\mbox{$\frac{1}{2}$}}d +1)^{{\mbox{$\frac{1}{2}$}}( {\mbox{$\frac{1}{2}$}}d +1)}$. For $d=3$, $T_g \approx 0.535$.
Glassy native states
--------------------
We also have to study the possible glassy phases with overlap with the native state, i.e., with finite ${{\tilde \mu}}$ (and, accordingly, finite $a$) when $\mu\rightarrow 0$. In such a phase, as in the ergodic native-like states described above, the system moves only in the neighborhood of the native state configuration. However, in a “glassy native” state even this restricted motion is strongly suppressed by the complexity of the local potential energy surface, and a glassy phase results.
As in the ergodic ansatz, the limit $\mu\rightarrow 0$ introduces no problems. Eqns. (\[b(T)\]) and (\[b0b\]) remain the same as in the frozen-globule case, while the equations for $a$ and ${{\tilde \mu}}$, (\[eq:a2\]) and (\[eq:tmu2\]) become $$\begin{aligned}
a({{\tilde \mu}}) &=& \frac{b_0}{2} +\frac{1}{8T^2 {{\tilde \mu}}^{3/2}
(1 + b_0)^{\frac{d}{2}+1}} + \frac{1}{4\sqrt{{{\tilde \mu}}}}
{\label}{eq:a3} \\
{{\tilde \mu}}(a) &=& \frac{1}{TT_0 (1 + a)^{\frac{d}{2}+1}}
{\label}{eq:tmu3} \end{aligned}$$ Again, Eq. (\[b(T)\]) specifies $b$ as a function of $T$, and (\[b0b\]), (\[eq:a3\]) and (\[eq:tmu3\]) determine $b_0$ and ${{\tilde \mu}}$ as functions of $T$ and $T_0$. $\tilde q$ and $\varphi$ are given by $\varphi,\tilde q\approx 1/(2\sqrt{\mu})$.
The overlap with the native state is the largest possible: $\cos
\theta_g=1$, as can be easily seen from Eq. (\[costh\]) and the values for $\varphi$ and $\tilde q$ we have just given. The overlap between two conformations at very different times also takes its largest possible value. From Eqs. (\[costh’g\]) and (\[costh”g\]), knowing that $b_0$ and $b$ do not depend on $\mu$ we have $\cos \theta'_g =1-b_0/2\tilde q\approx 1 - b_0
\sqrt{\mu}\rightarrow 1$ and $\cos \theta''_g =1-b/2\tilde q\approx
1-b\sqrt{\mu}\rightarrow 1$. Thus, the polymer is frozen almost everywhere into the native conformation. However, the freezing is not total, since $a$ in (\[eq:a3\]) is not zero. Furthermore, there is aging in the system, since $x$ in Eq. (\[xbb0\]) is not equal to 1.
We turn now to the solution of the equations (\[b(T)\]), (\[b0b\]), (\[eq:a3\]) and (\[eq:tmu3\]). As for the corresponding ergodic phases we have to resort to numerical solution; here we describe the analysis. The working strategy is similar to the one presented in subsection IX.B; the goal is to find $\beta_0$ as function of ${{\tilde \mu}}$ for fixed $T$ since, as in the ergodic native case, extrema of the function $\beta_0({{\tilde \mu}},T)$ govern the phase boundaries.
The procedure for finding value of the function $T_0({{\tilde \mu}},T)$ is as follows. Eq (\[b(T)\]) determines $b$ as a function of $T$, to be referred to as $b(T)$. Once $b(T)$ is found from (\[b(T)\]) it is inserted into Eq. (\[b0b\]), which determines $b_0(T,{{\tilde \mu}})$. The value found for $b_0$ is inserted into Eq. (\[eq:a3\]) to find $a$, and finally $\beta_0 =1/T_0$ is calculated from Eq. (\[eq:tmu3\]). Thus, at each temperature for which glassy solutions are possible, we can construct a graph of $\beta_0({{\tilde \mu}})$, as we did for ergodic solutions in Fig. 2. We have used Mathematica to do these calculations. We can use these curves, together with the ergodic ones previously analyzed, to identify the possible states of the system at a given temperature and $\beta_0$ (Fig. 4). The procedure is fairly simple. At any given ${{\tilde \mu}}$, only one of the solutions is physical: In the region ${{\tilde \mu}}_{min}<{{\tilde \mu}}<{{\tilde \mu}}_{AT}$ one has to follow the glassy $\beta_0({{\tilde \mu}})$ curve, while outside it one follows the ergodic one. In Fig. 4 the physical solution is indicated as the thick dashed curve. One then looks for solutions as intersections of this curve with a horizontal line at a given value of $\beta_0$, as done previously (e.g., as in Fig. 2, panel (a)) within the ergodic ansatz.
In Fig. 4 this procedure is shown for several different values of $T$. In the first panel, $T$ lies just a little below $T_{max}$ (as in Panel (b) of Fig. 2). Suppose we start in the ergodic native phase at large $\beta_0$ and then lower $\beta_0$. (In Fig. 5, this would correspond to moving along a horizontal line (constant $T$) slightly above point B in panel (b) or (c)). We can lower $\beta_0$ all the way down to $\beta_0^{\rm min}$ without encountering an AT instability. So, just as in the ergodic analysis of section X.B, beyond $\beta_0^{\rm min}$ the ergodic native phase melts into the random-globule phase.
In the same panel we can also analyze the what happens to the unstable stationary state in the same range of $\beta_0$ for this temperature. At very large $\beta_0$ we have an ergodic solution, but as we lower $\beta_0$ we pass through a range of ${{\tilde \mu}}$, between ${{\tilde \mu}}_{min}$ and ${{\tilde \mu}}_{AT}$, where the ergodic solution is unstable against glassiness. In this region we must follow the glassy curve instead of the ergodic one. We interpret this glassy solution in the following way: The free energy landscape near the US maximum becomes rough in this range of values of $\beta_0$ (at this temperature), the same way the free energy landscape near the minimum corresponding to a thermodynamic phase becomes rough in a glassy state. We call it a “glassy US state”.
The next panel is for a slightly lower temperature (but still above $T_g$). Here, as we lower $\beta_0$ in the ergodic-native phase, we reach an AT instability before we get all the way down to the minimum on the ergodic curve. (In Fig. 5, this would correspond to moving on a line of constant $T$, meeting the AT line somewhere between points A and B in panel (b) or (c)). Furthermore, the only available glassy solution for ${{\tilde \mu}}< {{\tilde \mu}}_{AT}$ is one with negative $\partial \beta_0/\partial {{\tilde \mu}}$, that is, it corresponds to the kind of glassy US state discussed above. As this is not a stable phase, we conclude that for this $T$, the minimum value of $\beta_0$ lies at this AT line, and beyond it there is no stable native-like state. We can follow the glassy US state back up to larger $\beta_0$, seeing that we eventually cross over to a normal (non-glassy) transition state.
In the last panel, the temperature is lowered a bit more (below $T_g$). Again, starting in the ergodic native phase at large $\beta_0$ and lowering $\beta_0$, we encounter an AT instability and a glassy solution appears. (Equivalently, in Fig. 5 one moves on a horizontal line somewhere below $T_g$ until meeting the AT line for the first time.) For smaller $\beta_0$, we switch to the glassy curve, which has positive $\partial \beta_0/\partial {{\tilde \mu}}$, describing a glassy native phase. We can follow this curve down to its minimum $\beta_0$, beyond which no phases correlated with the native phase exist. But, of course, following it back up toward large $\beta_0$ on the unstable branch, we can identify the glassy US state between the phase correlated with the native state and the one uncorrelated with it. (Above $T_g$, the latter is the random-globule phase; below it, it is the frozen-globule phase.)
Features of the phase diagram
=============================
The phase structure implied by this simple model is not so simple. Fig. 5 shows the phase diagram constructed from the above analysis. For clarity, we show in the top panel only the solutions that correspond to stable phases. The second panel shows the details in the region where the ergodic-native, glassy-native, random-globule and frozen globule states come together (or nearly so). The third panel shows the regions where ergodic and glassy US states are found.
There are six distinct regions in the phase diagram. In region I (high $T$, small $\beta_0$), the only stable phase is the random globule. In region II (small $\beta_0$, $T <T_g$) it undergoes a glass transition to the frozen-globule phase. The properties of the system in this part of the phase diagram are the same as in the completely random heteropolymer model of paper I; the bias of the interactions toward a native state does not have any effect until a (temperature-dependent) threshold $\beta_0^c(T)$ is reached. This threshold is marked on the diagram by the lines separating region I from regions III and V and region II from region VI.
To help thinking about these phases, we offer the schematic free energy-surface pictures of Fig. 6. They show how we imagine the free energy varies as a function of the native-state overlap coordinate $\varphi$. Fig. 6A depicts this cross-section through the free-energy surface in region I, where there is a single smooth minimum around $\varphi = 0$, representing the random-globule phase. Fig. 6B shows what happens in region II, where this phase is replaced by the frozen-globule phase. We represent this by drawing the free energy surface with many local minima. Fig. 6C shows what happens in the middle of region III, where there are two (smooth) minima, the new one corresponding to the ergodic native phase. (It will lie above or below that at $\varphi = 0$ according to whether we are above or below a first-order transition that we expect to occur at a temperature $T_1
<T_n$, see below.) In Fig. 6D we depict the situation in region IV, where both the $\varphi \approx 0$ region and the region around the maximum become rough. Fig. 6E represents region V, with the native valley and the maximum rough, but the region near $\varphi = 0$ still smooth. In Fig. 6F (region VI), that, too, becomes rough.
An important feature is the fact that the random-globule and frozen-globule phases remain (dynamically) stable in their respective temperature ranges for all $\beta_0$. Thus the horizontal line separating region I from region II continues across the diagram, separating region V from region VI and region II from region IV.
In each of regions II, IV, V and VI (Fig. 6, panels (b), (d), (e), (f)) there are two stable states. Thus, regions I and III are separated by the line $T_n(\beta_0)$, below or to the right of which, in addition to the random-globule state, an ergodic native state exists and is stable. In region IV, the frozen-globule and this ergodic native state are both stable. In region V, the random globule state and a glassy native state are stable, while in region VI the stable states are the frozen globule and the glassy native phase.
The diagram shows some interesting fine structure in the neighborhood of the region where regions I, III, and V meet (second panel, point A). In particular, below point A, the boundary between regions III and V (i.e., between the ergodic-native and glassy-native state) is an AT line. It comes about as can be seen in the last panel of Fig. 4, where, following the ergodic phase down from high $\beta_0$ and ${{\tilde \mu}}$, we reach ${{\tilde \mu}}_{AT}$ and thereafter have to switch to the glassy solution.
On the other hand, the boundary $T_n(\beta_0)$ above point B is reached as in the first panel of Fig. 4: one can come all the way down to the minimum value of $\beta_0$ for the ergodic solution before reaching ${{\tilde \mu}}_{AT}$. The full line continuing upwards and to the right of point B is an AT line which goes over into an $x=1$ line at its maximum, $T_{max}$. Below this line US solutions become glassy.
Between points A and B, the boundary is marked by reaching ${{\tilde \mu}}_{AT}$ in the way shown in the second panel of Fig. 4: There is no stable ergodic native solution to the left of this line, since, upon lowering $\beta_0$ below $\beta_0^{\rm min}$ in Panel (b), the solution with $\partial \beta_0/\partial {{\tilde \mu}}>0$ is lost. Thus in this region the line AB is an AT line for both the native phase and the US solutions. The dashed line marks the boundary found if one ignores the AT line (i.e., it is a portion of the boundary found using the ergodic ansatz and shown in Fig. 3).
Everywhere below the AT line (both the portion AB and its extension upward to the right) is the region where the US solutions are glassy, as shown in the last panel of Fig. 5. At a given $\beta_0$, these features all have an onset at a temperature between $T_g$ and $T_{max}$. As can be seen in Fig. 1, this is a very small temperature range. This is also the reason why there is fine structure, as shown in both the second and the third panels of Fig. 5, in such a small temperature range in the phase diagrams.
As remarked above, we have not done an equilibrium calculation, but we expect that the first-order transition temperature $T_1(\beta_0)$ where the free energies of the random-globule and ergodic-native phases are equal will also rise with $\beta_0$. For large $\beta_0$, we expect $T_1$, like $T_n$, to be proportional to $\beta_0$ (but $T_1
< T_n$, of course).
Thus, at fairly large $\beta_0$, we expect the following sequence of stable states as we lower $T$ from a high value. Initially, only the random-globule state is stable. Then, below $T_n$, the ergodic native state is also stable, and below $T_1$ it becomes the lowest-free energy state. Going further down in $T$, we cross the boundary (last panel in Fig. 5) where the US state between the ergodic native and random-globule states becomes glassy (i.e., acquires a rough local free energy landscape). Very soon thereafter, we cross $T_g$, where the random-globule state undergoes glassy freezing. Continuing, we reach a temperature where the ergodic native state undergoes glassy freezing. Finally, we reach the stability limit of this glassy-native phase, leaving the system with nowhere to go but the frozen-globule state.
What lessons are there in these findings for protein folding? We start from the assumption that the initial state in the folding process is uncorrelated with the native state (i.e., in Fig. 6 we start in a local minimum at $\varphi\approx 0$). Folding requires the system to find its way to the (ergodic) native state.
One feature that is evident is that such a path in configuration space always requires an uphill free-energy step. This is because either the random-globule or frozen-globule phases is always locally stable.
If we stick to our mean-field dynamical picture, where barriers are infinite, folding is, strictly speaking, impossible. In dynamical terms the “infinite” barriers translate into the fact that the equations which govern the motion of order parameters, $\varphi$ included, have basins of attraction corresponding to the plots shown in Fig. 6. For example, starting from $\varphi$ somewhere close to $0$ and given a free energy profile like that in Fig. 6C, dynamical equations will never carry $\varphi$ to the large value describing the native state. On the contrary, $\varphi$ will approach $0$ as time goes on.
But, if we relax this assumption and imagine finite barriers (associated with local nucleation of a native phase[@BW90; @TakWol97; @PTW01]), we may ask (informally) when activated motion over the barrier to nucleation is least hindered. We argue that the free energy landscape features present globally in our mean-field picture will also be relevant locally: when our calculation here finds a glassy US state, we expect that free energy surface near the true transition state will also be rough. Thus, from the preceding description of the phases and the transition states between them, we can see that folding should be easiest for large $\beta_0$ in a window between $T_1(\beta_0)$ and the upper boundary of the region where the US states become glassy (and passage across the transition region is kinetically impeded by the tortuous nature of the local free energy landscape). The latter boundary lies, in turn, just barely above $T_g$, where the landscape in the (large) portion of the configuration space uncorrelated with the native state also becomes rough, further impeding escape from it. At still lower temperatures, things become even worse, first with the onset of glassiness in the native-like region of configuration space itself and finally with the disappearance of native-like solutions. But these features probably have minor consequences, since folding will already have been so strongly impeded by the effects (with onset near $T_g$) that tend to confine it in a region of configuration space uncorrelated with the native state.
Discussion
==========
We have introduced what we might call a generic model for a protein, based on what seems to us to be the simplest way to incorporate a tendency to form a native state in an otherwise random heteropolymer model. To make it possible to calculate typical properties, we follow previous authors [@PGT1; @RS; @PGT2; @WS] and do not specify a particular native state, but rather an ensemble of them, constrained only by chain-entropic constraints and confinement to the appropriate volume. This ensemble is characterized by the selection temperature $T_0$. Our model differs from previous ones in that they are based on random-sequence heteropolymers, while we start from a model [@SG1; @SG2] in which each monomer-monomer interaction is an independent random variable.
While it might be argued that random-sequence models are more relevant to proteins, they approach the model we consider here in the limit where the number of monomer types becomes large. Thus, what we find out about our model may be relevant to proteins (with 20 different amino acids). Of course, it is also important to study what happens away from the large-monomer-type limit; our analysis here can help in solving that more difficult problem.
Furthermore, naively, one might assume that by adjusting $N(N-1)/2$ parameters one could imprint a native state more strongly than for models with only $N$ parameters. Our model shows that this is not necessarily true. Parts of the phase diagram are glassy, even for very low selection temperature $T_0$, when the native state should be strongly imprinted into the model.
Instead of the quadratic confinement term $\mu x(s,t)^2$ one could add three-body terms, which are commonly used to fix the globule density. It would be interesting to extend the analysis presented here to such models. Also, in our treatment, translational invariance within the globule is put in by hand. Keeping three-body terms would lead to automatic translational invariance. We have seen that if translational averaging is omitted (see paper I) then the equations become coupled in the $k$ variable and are thus a lot harder to solve.
Within out model, we have made just two approximations: the Gaussian variational ansatz of section VI, and the assumption of 1-step ergodicity breaking (analogous to 1-step replica symmetry breaking in the replica approach). Otherwise the solution is complete and exact to the accuracy we were able to achieve numerically.
Our most important result is the existence of the various different phases at large ${{\tilde B}}/T_0$, where the interactions that favor a native state are strong. While it is natural to anticipate that the native-like configurations will be thermally disrupted above a temperature of order ${{\tilde B}}^2/T_0$, it is not so obvious that at low temperatures there will be other impediments to efficient folding. We identify two of these:
\(1) The frozen-globule state, which is uncorrelated with the native state, always exists below $T_g$, no matter how big $\beta_0$ is. This means that in a large part of configuration space, the system may be trapped in a rough energy landscape and never (in MFT) get to the native-state region where it can fold rapidly. Furthermore, in almost the same temperature, range, we expect that the energy landscape is also rough around the transition region on the way to the correctly-folded state, further impeding the folding process. Thus, while lower temperature favors well-folded over random-globule-like configurations energetically, the rough energy landscape of the glassy phase will hinder correct folding. Our conclusion here is consistent with that of Goldstein [*et al.*]{}[@GLW92], who found (albeit in a different kind of model) that a large $T_n/T_g$ (or $T_1/T_g$) ratio favors folding.
\(2) At even lower temperatures, the native state itself is unstable against a glass transition where it splits into a large number of substates. Transitions between these substates are blocked by high barriers (infinite, in MFT). A phase of this kind was found earlier by Bryngelson and Wolynes in a phenomenological model[@BW87]. It is tempting to associate the substates with the glassy conformations observed at temperatures below 200 K in myoglobin [@Ibenetal].
Of course, MFT is an approximation. The escape from the tortuous part of the energy landscape to the smooth region will not take forever, nor will transitions between low-$T$ substates. Nevertheless, MFT does indicate when we can expect relaxational dynamics, including folding, to be slow or fast, as well as give us some insight into the physics behind these differences.
Our analysis here is a purely dynamical one. We do not compute equilibrium partition functions. A complete analysis would include such calculations, but we defer them to future work. Nevertheless, the purely dynamical analysis can reveal important properties of the system that cannot be seen in an equilibrium analysis. For example, it has been know for a long time that for a large class of models — namely, those which have a glass transition where $x \rightarrow 1$ — the dynamic and static glass transition temperatures are different [@CK1; @CS; @CHS]. This is expected to be the case for the transition at $T_g$ in our model: The equilibrium glass transition temperature is lower than the dynamical one. Thus, in a temperature range just below the dynamical $T_g$, the equilibrium analysis does not reveal the slow dynamics (accompanied by aging) that we are able to identify and analyze here.
In other glassy models for which it has been possible to do a more complete analysis [@CKD; @CD; @CK1; @CS; @CHS], the static and dynamic transitions coincide when they occur as a result of an Almeida-Thouless instability (the marginal stability condition, Eqn \[MSC\]) [@AT]. This is the case here at the phase boundary where the native-like state becomes glassy.
Gillin and Sherrington [@GS] and Gillin [*et al.*]{} [@GNS] have been able to analyze both the statics and the dynamics of several classes of mean-field spin glass models with a competition between glassy and ferromagnetic states (see also ref. [@HSN] for a special case). Some features of the phase diagram of our model that we have been able to discover so far are also seen in their models.
Gillin [*et al.*]{} also studied full (as well as 1-step) replica symmetry breaking solutions, which we have not. In some of their models, the counterpart of our glassy native phase undergoes full RSB at low temperatures, and the counterpart of our II-VI boundary becomes vertical. It is possible in our model as well that, in particular regions of the phase diagram, our 1-step solutions are not stable and full RSB is necessary. More generally, it will be an interesting problem to try to explore what kinds and degrees of universality there are in the phase structures of various systems where the glassiness induced by disorder competes with some kind of order analogous to the native state in our problem or the ferromagnetic state in theirs.
It is a pleasure to thank Silvio Franz for discussions leading to our formulation of this problem.
correlation function $G_{10}^{ss'}$
====================================
Here we derive Eq. (\[G10a\]). Inserting Eq. (\[Phis1\]) for the superfield into (\[G10\]) gives $$\begin{aligned}
G_{10}^{ss'}
& = & \langle x(s,t_1) x_0(s') \rangle
+ \langle \bar\eta(s,t_1) x_0(s') \rangle \theta_1 + \nonumber \\
& & + \bar\theta_1 \langle \eta(s,t_1) x_0(s') \rangle
+ \bar\theta_1\theta_1 \langle \tilde x(s,t_1) x_0(s')
\rangle .
\label{G10b} \end{aligned}$$ One can show that the action of the dynamical generating functional $F_{dyn}$ (see Eq. \[Fdyn\]) is invariant under the infinitesimal transformation (BRS symmetry) $$\delta \Phi(s,t,\theta,\bar\theta) = \epsilon
\frac{\partial}{\partial\bar\theta} \Phi(s,t,\theta,\bar\theta)
\label{deltaPhi}.$$ This follows in two steps. First one notices that for any function $f$ $$\delta\int d\bar\theta f(\Phi(\bar\theta)) =
\epsilon \int d\bar\theta \frac{\partial}{\partial\bar\theta}
f(\Phi(\bar\theta)) = 0,$$ due to the identity $\int d\bar\theta
\frac{\partial}{\partial\bar\theta} = 0$. This means that any term involving a local function of $\Phi$ (i.e., not containing derivatives over $\theta$ and $\bar\theta$), e.g., $S_2[\Phi,x_0]$ (see Eq. \[eq:S2\]), is invariant under the transformation (\[deltaPhi\]). $S_0[x_0]$ is trivially invariant since it does not contain the superfield $\Phi$. (The same reasoning holds for a transformation like (\[deltaPhi\]) with a derivative with respect to $\theta$ instead of with respect to $\bar\theta$.) The only term left is the $S_1[\Phi]$ (i.e., the part of the action quadratic in the superfield) and it is straightforward to see that this term is also invariant under (\[deltaPhi\]) (though not under the transformation involving the derivative with respect to $\theta$).
The fact that the action is invariant under (\[deltaPhi\]) implies the Ward identity $$\frac{\partial}{\partial\bar\theta_1} G_{12}^{ss'} = 0
\label{ward1}$$ which gives $0=\langle \eta x_0 \rangle + \theta_1 \langle \tilde
x x_0 \rangle$ (we have suppressed the arguments of the fields to simplify the notation). This implies that separately one has $$\langle \eta x_0 \rangle=0, \ \ \langle \tilde x x_0 \rangle=0
\label{ward2}$$ Inserting (\[ward2\]) into (\[G10b\]) gives the desired result, Eq. (\[G10a\]).
Details on the GVA and how to improve it
========================================
Here we give more background on the use of Eq. (\[dFdG\]). In the dynamical calculation, the fields are complex and contain Grassmann variables; thus, $F_{dyn}$ is not a real number. This means that any interpretation of Eq. (\[dFdG\]) as an extremum condition for $F_{dyn}$ has to be given up. Nevertheless, we can still make some sense of the GVA as the first step in a systematic approximation scheme.
Formally, one starts from Eq. (\[Fdyn\]), which we rewrite in the shorter form $$e^{-F_{dyn}}=\int D\Psi e^{ -S[\Psi] }.
\label{Fdyn3}$$ where $\Psi$ stands for the pair $(x_0,\Phi)$ and, likewise, $D\Psi$ for $Dx_0D\Phi$. One can express $F_{dyn}$ in a slightly different form $$e^{-F_{dyn}} = \langle e^{-(S-S_{var})} \rangle_{var} e^{-F_{var}}
\label{Fdyn4}$$ where $$e^{-F_{var}}=\int D\Psi e^{-S_{var}}$$ and $$\langle (...) \rangle_{var}=
\frac{
\int D\Psi ( ... ) e^{-S_{var}}
}{
\int D\Psi e^{-S_{var}}
}$$ In a static calculation one would proceed with the inequality $$e^{-F} \ge e^{ - \langle (S-S_{var}) \rangle_{var} }
e^{ -F_{var} }
\label{Fdyn5}$$ to conclude that $$F \le \langle S-S_{var} \rangle_{var} + F_{var}
\label{Fdyn6}.$$ Thus, in a static calculation, the variationally-obtained $F$ gives an upper bound on the true $F$. What is allowed to vary is the form of $S_{var}$, most often, the parameters describing it. (In the GVA, $S_{var}$ is specified by $G_{12}^{ss'}$ and $G_{10}^{ss'}$.)
In the dynamical problem we follow another route, starting exactly at the problematic step, Eq. (\[Fdyn5\]), along the lines of ref. [@Kleinert]. Instead of the inequality (\[Fdyn6\]) we use Eq. (\[Fdyn4\]) in a slightly modified form $$F_{dyn} = F_{var}
- \ln \langle e^{-\Delta S} \rangle_{var}
\label{Fdyn7},$$ where $\Delta S=S-S_{var}$. Applying a cumulant expansion $$\begin{aligned}
\langle {\rm exp}(-\Delta S)\rangle_{var} & & =
{\rm exp}\left[ -\langle \Delta S \rangle_{var} + \right. \nonumber \\
& & \left. + \frac{1}{2} \left(
\langle \Delta S^2 \rangle_{var}
- \langle \Delta S \rangle_{var}^2
\right)
+ \cdots \right],
\label{Fdyn8} \end{aligned}$$ one gets $$F_{dyn} = F_{var} + \langle S-S_{var} \rangle_{var} + \Delta F$$ where $\Delta F$ contains second and higher order corrections in $\Delta S$. In any approximation made by keeping a finite number of terms in (\[Fdyn8\]) (the simplest being to set $\Delta
F=0$), $F_{dyn}$ depends on $G_{12}^{ss'}$. To minimize this dependence, we chose $G_{12}^{ss'}$ so that the derivative of the approximate form for $F_{dyn}$ with respect to $G_{12}^{ss'}$ vanishes. This gives Eq. (\[dFdG\]). Furthermore, if all terms in $\Delta F$ are kept, this procedure, by construction, formally gives back the exact $F_{dyn}$.
The meaning of minimizing the dependence with respect to quantities involving Grassmann numbers may seem obscure, but we note that we are using the SUSY representation only for compactness. The entire GVA calculation could have been presented equivalently without any Grassmann variables, with no change in meaning or result. Thus, we are really only minimizing the dependence on parameters of physically well-defined correlation and response functions.
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---
abstract: 'We propose to correlate transmittance maps and spectral-density maps of planar junctions, in order to analyze quantitatively and in detail spin-dependent transport calculations. Since spectral-density maps can be resolved with respect to atom, angular momentum, and spin, the resulting correlation coefficients reveal unequivocally, e.g., which layers or which orbitals determine the tunnel conductances. Our method can be used for transport calculations within the Landauer-Büttiker formalism. Its properties and features will be discussed by means of a pure bcc Fe(001) lead as well as an extensively studied Fe(001)/MgO/Fe(001) planar tunnel junction.'
author:
- 'P. Bose'
- 'P. Zahn'
- 'I. Mertig'
- 'J. Henk'
bibliography:
- 'papers.bib'
title: 'Correlating transmission and local electronic structure in planar junctions: A tool for analyzing spin-dependent transport calculations'
---
Motivation
==========
Spin electronics—or spintronics for short—is one of the major topics in contemporary physics (see for example Refs. and ). With respect to both device applications and fundamental physics, planar junctions have been and are being investigated experimentally and theoretically with great effort. The spin-dependent transport properties of magnetic tunnel junctions show up as , that is the change of the conductance upon reversal of the magnetization direction in one of the two electrodes[@Maekawa02; @Zutic04]. Replacing the insulating barrier by a ferroelectric, a effect can be observed in addition. In this case, the conductance depends as well on the orientation of the electrical polarization in the ferroelectric[@Tsymbal06].
In experiments, the current that is flowing through a tunnel device is detected in dependence on external fields and device parameters (e.g. bias voltage and individual layer thicknesses). The current-voltage characteristics of tunnel devices are often interpreted within the Jullière model[@Julliere75], perhaps due to its simplicity. The validity of this model has been severely questioned[@MacLaren97] because it relates the ratio exclusively with the spin polarization of the electrodes and, thus, neglects the interface and barrier regions completely. If the Jullière model would be valid, the tunnel conductance would not depend on the interface material at all, in contrast to observations. In other words, the interface region is essential and has to be described in theory as good as possible.
From the preceding it is evident that a reliable description of transport in planar junctions must capture the essential properties on an atomistic level. Hence, present theoretical investigations of transport rely on sophisticated first-principles approaches to the electronic and magnetic structures.
Advanced first-principles approaches to transport allow a very detailed analysis of the electronic structure and the conductance. In particular, the probably most important question—which orbitals in which layer determine the transport properties—can be answered. However, the amount of output data that is produced by modern computer codes becomes often unmanageable for human beings. Therefore, one restricts oneself to representative subsets or comprises the numerical data into manageable representations. For example, one puts the focus of the analysis on a single wavevector in the , typically to the center[@Butler01]. By doing so, one should be aware that such a restriction could fail because considerable parts of the may contribute to the conductance. Examples for data representations are transmittance and spectral-density maps in the (introduced in Section \[sec:theoretical\_analysis\]) which have become established analysis tools. They allow in principle to answer the above question. Because they are compared visually, they leave space for speculation and interpretation; or phrased differently, they introduce ambiguity. Apparently, there is need for an improved analysis tool which allows to analyze transport properties unequivocally and quantitatively, rather than ambiguously and qualitatively.
In this paper, we propose a quantitative analysis of transport properties which goes beyond the approaches sketched above. We propose to correlate transmittance and spectral-density maps. The resulting correlation coefficients reveal unequivocally, for example, which layers or which orbitals determine the tunnel conductances. Our method can be used for any computer code which relies on a Landauer-Büttiker-type approach, thus being applicable in most of the present-day transport calculations. Its properties will be discussed for two junctions exhibiting a planar geometry.
The paper is organized as follows. In Section \[sec:theoretical\_analysis\] we introduce our approach for analyzing transport calculations. Section \[sec:conductance\_calculations\] gives a brief overview of our numerical approach. The correlation analysis is applied in Section \[sec:applications\] to a pure Fe(001) lead (\[sec:Fe001\]) and to an Fe(001)/MgO/Fe(001) (\[sec:FeMgOFe\]). Concluding remarks are given in Section \[sec:conclusions\].
Theoretical analysis of the tunnel conductance {#sec:theoretical_analysis}
==============================================
We consider a planar junction which consists of a left electrode $\mathcal{L}$, an interface region $\mathcal{I}$, and a right electrode $\mathcal{R}$. Due to the translational invariance parallel to the interface region, the Bloch states in the electrodes are indexed by the (in-plane) wavevector ${\boldsymbol{k}}_{\parallel}$ in the ; ${\boldsymbol{k}}_{\parallel}$ is conserved in the scattering process.
The bias voltage $V$ opens an ‘energy window of transport’; the chemical potentials $\mu_{\mathcal{L}}$ and $\mu_{\mathcal{R}}$ of $\mathcal{L}$ and $\mathcal{R}$, respectively, differ by $e V = \mu_{\mathcal{L}} - \mu_{\mathcal{R}}$. Without loss of generality we consider the case $V > 0$, for which incoming occupied Bloch states in $\mathcal{L}$ can be transmitted into outgoing unoccupied Bloch states in $\mathcal{R}$.
According to Landauer and Büttiker[@Buettiker85; @Imry99], the conductance $C$ is given by $$\begin{aligned}
C(V) & = \frac{e^{2}}{h}
\int_{\mu_{\mathcal{R}}}^{\mu_{\mathcal{L}}}
\int_{\mathrm{2BZ}}
T^{\mathcal{L} \to \mathcal{R}}(V; E, {\boldsymbol{k}}_{\parallel})\,\mathrm{d}^{2}{\boldsymbol{k}}\,\mathrm{d}E.\end{aligned}$$ The wavevector integral is over the . The transmittance $T^{\mathcal{L} \to \mathcal{R}}(V; E, {\boldsymbol{k}}_{\parallel})$ is the sum over the transmission probabilities of all incoming occupied states $\lambda$ in $\mathcal{L}$ and outgoing unoccupied states $\rho$ in $\mathcal{R}$. It is related to the scattering matrix $S$ of the interface region by $$\begin{aligned}
T^{\mathcal{L} \to \mathcal{R}}(V; E, {\boldsymbol{k}}_{\parallel})
& = \sum_{\lambda \rho} \left| S_{\lambda \rho}^{\mathcal{L} \to \mathcal{R}}(V; E, {\boldsymbol{k}}_{\parallel}) \right|^{2}.
\label{eq:transmission}\end{aligned}$$
The above transmittance is a key quantity in the proposed analysis; it is conveniently displayed versus ${\boldsymbol{k}}_{\parallel}$ at fixed $E$ and $V$, in so-called transmittance maps ($T$ maps). Note that $T$ can be regarded as a global quantity since it depends on the electronic structure of the entire junction, that is both electrodes (incoming and outgoing Bloch states) and the interface region (scattering matrix).
The local electronic structure of the device is described in terms of the spectral density $$\begin{aligned}
N_{\alpha}(V; E, {\boldsymbol{k}}_{\parallel})
& = -\frac{1}{\pi} {\mathrm{Im}}\, {\mathrm{Tr}}_{\alpha} G^{+}(V; E, {\boldsymbol{k}}_{\parallel}).
\label{eq:SD}\end{aligned}$$ $\alpha$ is a compound index which can comprise for example layer, atom, orbital, angular momentum, spin indices or point-group representation. For a given division of the complete index space, the subsets of $\alpha$ are disjoint. The trace ${\mathrm{Tr}}_{\alpha}$ of the Green function $G^{+}$ is restricted to the given $\alpha$ and, thus, allows a detailed analysis of the electronic structure. As for the transmittance, the spectral density is displayed in maps versus ${\boldsymbol{k}}_{\parallel}$ ($N$ maps).
In previous publications we have analyzed the conductance by relating pronounced features in a transmittance map with features in the associated spectral-density maps. If these features show up in both the $T$ map and in an $N_{\alpha}$ map we concluded that the chosen set $\alpha$ determines the transmittance in this $(V,E)$ region. This way we could show that for Fe/Mn/vacuum/Fe junctions the topmost Mn layer governs their ratio[@Bose07]. However, for more complicated systems it turned out that the visual inspection of the maps becomes ambiguous, tedious, and not very reliable. As a consequence, we propose to correlate properly normalized $T$ and $N$ maps by means of a projection (inner product). This procedure results in a set of a few unambiguous numbers which allow to determine rapidly the set of significant $\alpha$ indices.
For given energy $E$ and bias voltage $V$ we define the average value of a function $X(V; E, {\boldsymbol{k}}_{\parallel})$ of the transmittance $T(V; E, {\boldsymbol{k}}_{\parallel})$ and the spectral density $N_{\alpha}(V; E, {\boldsymbol{k}}_{\parallel})$ as the average over the , $$\begin{aligned}
A_{X}
& \equiv
\frac{1}{\Omega_{\mathrm{2BZ}}}
\int_{\mathrm{2BZ}}
X(V; E, {\boldsymbol{k}}_{\parallel})\
\,\mathrm{d}^{2}{\boldsymbol{k}},\end{aligned}$$ where $\Omega_{\mathrm{2BZ}}$ is the area of the [^1]. The correlation coefficient $c_{\alpha}(V; E)$ is then defined by $$\begin{aligned}
c_{\alpha}(V; E)
& \equiv
\frac{A_{T N_{\alpha}}}{\sqrt{A_{T^{2}} A_{N_{\alpha}^{2}}}}.\end{aligned}$$ This quantity can be interpreted as an inner product of the normalized $T(V; E, {\boldsymbol{k}}_{\parallel})$ and $N_{\alpha}(V; E, {\boldsymbol{k}}_{\parallel})$. Since the latter are semi-positive for all ${\boldsymbol{k}}_{\parallel}$, $0 \leq c_{\alpha}(V; E) \leq 1$.
Note that $c_{\alpha}(V; E)$ is invariant with respect to scaling $T(V; E, {\boldsymbol{k}}_{\parallel})$ and $N_{\alpha}(V; E, {\boldsymbol{k}}_{\parallel})$ (explicitly $T(V; E, {\boldsymbol{k}}_{\parallel}) \to \tau T(V; E, {\boldsymbol{k}}_{\parallel})$ and $N_{\alpha}(V; E, {\boldsymbol{k}}_{\parallel}) \to \nu N_{\alpha}(V; E, {\boldsymbol{k}}_{\parallel})$). Further, for a visual inspection and comparison of the $X(V; E, {\boldsymbol{k}}_{\parallel})$ on the same scale it is convenient to normalize them by $$\begin{aligned}
\widetilde{X}(V; E, {\boldsymbol{k}}_{\parallel})
& \equiv \frac{X(V; E, {\boldsymbol{k}}_{\parallel})}{\sqrt{A_{X^2}}}.
\label{eqn:normX}\end{aligned}$$
The correlation coefficient, alternatively expressable as $c_{\alpha}(V; E) = A_{\widetilde{T} \widetilde{N}_{\alpha}}$, is a measure for the ‘overlap’ of $T$ and $N_{\alpha}$ or of $\widetilde{T}$ and $\widetilde{N}_{\alpha}$. (i) Consider a constant transmittance and a constant spectral density, $T(V; E, {\boldsymbol{k}}_{\parallel}) = t$ and $N_{\alpha}(V; E, {\boldsymbol{k}}_{\parallel}) = n_{\alpha}$, which can be viewed as ‘completely overlapping’. Then $A_{T^{2}} = t^{2}$, $A_{N_{\alpha}^{2}} = n_{\alpha}^{2}$, and $A_{T N_{\alpha}} = t n_{\alpha}$. Consequently, $c_{\alpha} = 1$ which we will consider as perfect correlation. (ii) Consider a $T(V; E, {\boldsymbol{k}}_{\parallel})$ which is nonzero only in a region $\Omega_{T}$ of the : $$\begin{aligned}
T(V; E, {\boldsymbol{k}}_{\parallel})
& = T(V; E, {\boldsymbol{k}}_{\parallel}) 1_{\Omega_{T}}({\boldsymbol{k}}_{\parallel}),\end{aligned}$$ with the indicator $$\begin{aligned}
1_{\Omega_{T}}({\boldsymbol{k}}_{\parallel})
& =
\begin{cases}
1 & {\boldsymbol{k}}_{\parallel} \in \Omega_{T}
\\
0 & {\boldsymbol{k}}_{\parallel} \not \in \Omega_{T}
\end{cases}.\end{aligned}$$ Likewise, $N_{\alpha}(V; E, {\boldsymbol{k}}_{\parallel})$ is assumed nonzero in a region $\Omega_{N_{\alpha}}$ which is disjoint with $\Omega_{T}$ ($\Omega_{T} \cap \Omega_{N_{\alpha}} = \varnothing$; ‘zero overlap’). Consequently, $A_{T^{2}} \not= 0$, $A_{N_{\alpha}^{2}} \not= 0$, and $A_{T N_{\alpha}} = 0$, giving $c_{\alpha} = 0$. We consider this case as perfectly uncorrelated. We note in passing that there may be other definitions of correlation coefficients, as it is the case for the correlation of random variables[@Bickel77].
Finally, the $_{\alpha} = A_{N_{\alpha}}$ and the $= \sum_{\alpha}A_{N_{\alpha}}$ ($\equiv~A_{N_{\Sigma}}$) are computed based on $N_{\alpha}$, respectively.
Conductance calculations {#sec:conductance_calculations}
========================
Because our theoretical approach to spin-dependent tunneling has been described in detail elsewhere[@Tusche05; @Bose07; @Bose08; @Khan08; @Bose2010] we restrict ourselves to a brief survey here. The electronic structure of a tunnel junction is computed within the local spin-density approximation to density-functional theory, as is formulated in multiple-scattering theory[@Zabloudil05]. Our spin-polarized relativistic layer method[@omni; @Henk02] provides on the one hand the Green function of the entire system, from which the spectral densities $N_{\alpha}(V; E, {\boldsymbol{k}}_{\parallel})$ can be computed, eq. (\[eq:SD\]). On the other hand, the Green function can be used to calculate the transmission $T(V; E, {\boldsymbol{k}}_{\parallel})$, eq. (\[eq:transmission\]), within the framework of the Landauer-Büttiker theory[@Henk06; @Saha08]. We can also follow the approach introduced by MacLaren and Butler[@MacLaren99] in which the Bloch states in the electrodes and the scattering matrix of the interface region are computed by means of layer- algorithms[@MacLaren89a]. Here, the Green function is not computed explicitly.
The $T$ and $N$ maps have been computed on identical ${\boldsymbol{k}}_{\parallel}$ meshes in the entire , with at least $40\,000$ points. In the following applications we restrict ourselves to the case of zero bias ($V = 0$ and $\mu_{\mathcal{L}} = \mu_{\mathcal{R}}$).
Applications {#sec:applications}
============
Fe(001) {#sec:Fe001}
-------
The properties of the proposed correlation analysis are best introduced by an—admittedly—trivial case: the spin-resolved Sharvin conductance[@Sharvin65] of Fe(001). Because the three regions of the junction—$\mathcal{L}$, $\mathcal{I}$, and $\mathcal{R}$—are identical, the scattering matrix in eq. (\[eq:transmission\]) is $S_{\lambda\rho} = \delta_{\lambda\rho}$ and $T({\boldsymbol{k}}_{\parallel})$ is an integer. Accordingly, the $N$ map is a projection of the Fermi surface onto the (001) plane. Without spin-orbit coupling, spin is a good quantum number; hence, we treat the majority and minority channels separately.
{width="95.00000%"}
Figure \[fig:Fe001\] displays the normalized ${\boldsymbol{k}}_{\parallel}$-resolved transmittance and spectral density maps. According to Table \[tab\_decomposition\], the later are ordered based on their {s, p, d} or $\{\Delta_{1}, \Delta_{5}, \Delta_{2}+\Delta_{2'}\}$-like orbital contributions.
$\alpha$ $\ell$ $m$ orbitals
-------------------- --------- ------- ------------------------------------
$\Delta_1$-like 0, 1, 2 0 s, p$_z$, d$_{3z^2-r^2}$
$\Delta_5$-like 1, 2 -1, 1 p$_x$, p$_y$, d$_{3xz}$, d$_{3yz}$
$\Delta_2$-like 2 -2 d$_{x^2-y^2}$
$\Delta_{2'}$-like 2 2 d$_{3xy}$
: Decomposition $\alpha$ of the spectral density with respect to angular momentum $(\ell)$ and orbital $(m)$ quantum numbers according to Eq. . Incrementing $\ell$ provides a classification by means of s, p, and d orbitals. An alternative decomposition can be obtained with respect to the irreducible representations of the point group $C_{4\mathrm{v}}$ \[\]. Because such a decomposition holds strictly speaking only for the center ($\overline{\Gamma}$, ${\boldsymbol{k}}_{\parallel} = 0$), we refer to ‘$\Delta_{1}$-like’ maps etc.
\[tab\_decomposition\]
The assignment of the angular momentum orbitals to the different subsets $\alpha$ is motivated by the irreducible representations of the $C_{4\mathrm{v}}$ symmetry group. This decomposition is given in Table \[tab\_decomposition\].
\(i) The map of the total spectral density agrees nicely with the associated transmittance map in the majority channel (top row). In particular, all features are present and the correlation is consequently sizable ($c_{\Sigma} = 0.802$).
\(ii) Other maps which essentially capture all features within the majority channel are the $\widetilde{N}_{\mathrm{d}}$ and $\widetilde{N}_{\mathrm{\Delta_{5}\mathrm{-like}}}$ maps. A visual comparison of both with the transmittance could lead to the conclusion that the former matches slightly better than the latter. The correlation analysis, however, shows that this may be a misinterpretation; both coefficients indicate sizeable correlations ($c_{\mathrm{d}} = 0.798$ and $c_{\Delta_5\mathrm{-like}} = 0.828$) but that of the $\Delta_{5}\mathrm{-like}$ subset is slightly larger.
\(iii) As a result of the relatively small correlation coefficients of s ($c_{\mathrm{s}} = 0.365$) and p ($c_{\mathrm{p}} = 0.432$) we identify the d majority states as the dominating conducting channels. Further, with the help of Table \[tab\_decomposition\] and the $c_{\alpha}$ of the $\Delta$-like maps, a hierarchy of conducting d states can be specified. Since $c_{\Delta_1\mathrm{-like}} = 0.484$ is quite small, the d$_{3z^2-r^2}$ states seem to play no significant role. Due to the large $c_{\Delta_5\mathrm{-like}}$ the d$_{3xz}$ and d$_{3yz}$ orbitals appear to form the leading transport channels, followed by the d$_{x^2-y^2}$ and d$_{xy}$ states ($c_{(\Delta_2+\Delta_2')\mathrm{-like}} = 0.656$).
\(iv) Similar observations can be made for the minority channel (bottom row). Again, the d states represent the main conducting channels. But in comparison to the majority channel, the role of (d$_{3xz}$, d$_{3yz}$) and (d$_{x^2-y^2}$, d$_{3xy}$) orbitals is interchanged.
This example shows that the correlation analysis provides a powerful analysis tool which on one hand fits nicely to the visual interpretation of $T$ and $N$ maps. On the other hand, it clearly reveals possible misinterpretations, as has become evident in point (ii).
With 94% (majority) and 66% (minority) the d states constitute by far the main contributions to the . In this example the hierarchy of correlation coefficients often reflects the hierarchy of partial contributions to the (see Fig. \[fig:Fe001\]). This ordering becomes incorrect for instance in the case of $\Delta_{5}$-like and $(\Delta_{2}+\Delta_{2'})$-like states in the minority channel. Here, the $\Delta_{5}$-like states represent with 64% most of the , but exhibit with a $c_{\Delta_5\mathrm{-like}}$ of 0.788 a smaller correlation than the $(\Delta_{2}+\Delta_{2'})$-like contributions (26%, $c_{\Delta_2+\Delta_{2'}} = 0.847$).
This last finding indicates already that a one-to-one mapping of hierarchies and correlation coefficients is not viable. In general, transmittances depend not only on the number of available states but rely also on other conditions like e.g. the wavefunction matching at interfaces. Hence, as a prototype which exhibits more complicated correlations between transport and electronic structures properties, Fe/MgO/Fe will be discussed now.
Fe(001)/MgO/Fe(001) {#sec:FeMgOFe}
-------------------
In the following an Fe(001)/MgO/Fe(001) comprising six monolayers MgO is analysed. The magnetic directions within both Fe leads are collinearly aligned to each other and considered for the case of a parallel magnetic configuration. Below we discuss the correlation analysis of the majority channel in more detail, because the current is dominated by the spin-up carriers[@Butler01].
The corresponding transmittance map is displayed in Fig. \[fig:maj-transmittance\].
As typical for $\widetilde{T}({\boldsymbol{k}}_{\parallel})$ in the majority channel \[\] a Gaussian-like, radially symmetric distribution is found around the center.
Let us suppose briefly that the Jullière[@Julliere75] model is valid and that the transport properties can be interpreted exclusively based on the available $_{\alpha}$. Then the spin-polarized conductances are estimated with the product of the densities of states at the Fermi energy, strictly speaking with the $_{\alpha}$ inside the left $(\mathcal{L})$ and right lead $(\mathcal{R})$. In this picture, the conducting channels would be, according to the previous discussion of iron, mainly determined by d-like or $\{\Delta_{5}, \Delta_{2}+\Delta_{2'}\}$-like Bloch states of the Fe electrodes. But a visual inspection and comparison with the majority maps in Fig. \[fig:Fe001\] reveals at first glance no similarity in the structures with neither the $\widetilde{T}({\boldsymbol{k}}_{\parallel})$ nor the $\widetilde{N}_{\mathrm{d}}({\boldsymbol{k}}_{\parallel})$ maps.
Consequently, one could ask which states are essential for the transport processes if the predominant Fe bulk states do not play a decisive role. A closer look at the $\widetilde{N}_{\mathrm{s}}$ and $\widetilde{N}_{\mathrm{\Delta_{1}\mathrm{-like}}}$ maps in Fig. \[fig:Fe001\] leads to the identification of centrosymmetric blobs like that in Fig. \[fig:maj-transmittance\]. But the associated Bloch states exhibit small $c_{\alpha}$ and marginal contributions to the in Fe bulk. Do these states play a decisive role for the transport within the ?
To answer this question we consider in a next step the layer-wise contributions of the $_{\alpha}$ to the respective at the Fermi energy (see Fig. \[fig:PDOS-contribution\]). In particular, the $\nicefrac{A_{\widetilde{N}_{\alpha}}}{A_{\widetilde{N}_{\Sigma}}}$ percentages are shown for one half of the symmetric and are classified again by means of {s, p, d} or $\{\Delta_{1}, \Delta_{5}, \Delta_{2}+\Delta_{2'}\}$-like orbital decompositions.
The values for the outermost Fe layers in Fig. \[fig:PDOS-contribution\] (a) and (b) are identical to those for the pure Fe lead in Fig. \[fig:Fe001\], indicating the bulk-like character of those layers far from the interfaces. The corresponding d states represent with 80%-95% the most numerous parts of the up to the second Fe layers adjacent to the MgO interfaces. At these interfaces the number of d states is drastically reduced. Within the MgO film the decrease is continued down to about 10% inside the middle region of the tunnel barrier.
On the other hand, the s and p contributions which are apparently vanishingly small within the Fe electrodes, obtain substantial weight inside the MgO spacer. In particular, the p states exhibit a maximal percentage of 60% within the 1. MgO monolayer. Deeper inside the MgO, this fraction reduces to about 40%. In contrast, the s fractions reach a level of 20% at the Fe/MgO interfaces and increase monotonously to roughly about 40%. Thus, in the middle of the MgO the number of p and s states are comparably large, with a slight advantage of the former.
Further, a decomposition in $\{\Delta_{1}, \Delta_{5}, \Delta_{2}+\Delta_{2'}\}$-like contributions reveals completely different characteristics in Fig. \[fig:PDOS-contribution\]b. Here, the $\Delta_{1}$-like contributions, which are of minor importance within the Fe leads, experience a massive increase at the Fe/MgO interfaces and reach levels of about 80% within the tunnel barrier. The remaining 20% are predominantly occupied with $\Delta_{5}$-like and few $(\Delta_{2}+\Delta_{2'})$-like states. This hierarchical order of $_\alpha$ within the MgO reflects the well known fact of symmetry-selective decay lengths of the evanescent Bloch states within the tunnel barrier[@Butler01; @Heiliger06b].
Due to their sizable presence within the bottle neck of the junction, i. e. the MgO barrier, s and $\Delta_{1}$-like states might indeed characterize the transport. But whether this predominance also results in a dominance of the conducting channels can only be answered by an analysis of the transmittance map in Fig. \[fig:maj-transmittance\] with the respective $\widetilde{N}_{\alpha}$ maps. In order to specify exactly which layer-resolved spectral densities fits best the structures of the transmittance map, the $\widetilde{N}_{\alpha}$ maps of the whole Fe/MgO/Fe tunnel junction have to be inspected. These maps are shown in Fig. \[fig:FEMGOcorrelation-maps\]a.
One can see immediately that $\widetilde{N}_{\Sigma}$, $\widetilde{N}_{\mathrm{s}}$, $\widetilde{N}_{\mathrm{p}}$, and $\widetilde{N}_{\Delta_1\mathrm{-like}}$ within the MgO layers exhibit a great similarity with the structure of $\widetilde{T}({\boldsymbol{k}}_{\parallel})$ in Fig. \[fig:maj-transmittance\]. On the other hand, the similarities of $\widetilde{N}_{\mathrm{d}}$, $\widetilde{N}_{\Delta_5\mathrm{-like}}$ and $\widetilde{N}_{(\Delta_2+\Delta_{2'})\mathrm{-like}}$ within the same layers appear rather small. Due to the great dissimilarities of the spectral density maps within the Fe layers it is reasonable to expect low correlations there. But, based on visual comparisons it is hard to make definitive statements about similarities of the structures and hence which states provide the largest contributions to the electronic transport of the whole MTJ.
At this point, the tool of the correlation analysis provides the potential to gain clearer statements. The computed correlation coefficients which represent a measure for the similarity of two maps are summarized in Fig. \[fig:FEMGOcorrelation-maps\]b and \[fig:FEMGOcorrelation-maps\]c.
\(i) Considering Figure \[fig:FEMGOcorrelation-maps\]b, it turns out that s states exhibit the most prominent correlations in all layers. For each Fe layer the $c_{\mathrm{s}}$ can be quantified with about 0.1-0.2 as nearly double as large as those of p and d states. Together with the significantly high correlations of 0.7 (1. MgO layer) up to 0.95 (3. MgO layer) inside the tunnel barrier, the previously assumed dominant role of the s states can be regarded as proven for the entire MTJ.
\(ii) However, inside the MgO layers the p and d states exhibit sizeable increases of their correlation coefficients, too. In particular, the $c_{\mathrm{p}}$ are with 0.2 (1. MgO layer), 0.4 (2. MgO layer) and 0.7 (3. MgO layer) approximately twice as high as the $c_{\mathrm{d}}$. In the discussion of the $c_{\Delta_{1}\mathrm{-like}}$ coefficients below it will become evident that the increasing $c_{\mathrm{p}}$ and $c_{\mathrm{d}}$ are mainly related to states exhibiting p$_{\mathrm{z}}$ and d$_{3z^2-r^2}$ orbital character. (iii) Since the maps comprise structures of s states closley correlated to the transmission map and less correlated p and d states, the correlation values of $c_{\Sigma}$ are always lower than those for the s states.
\(iv) Along the atomic layers of the MTJ, the characteristics of the $c_{\Delta_{1}\mathrm{-like}}$ coefficients in Fig. \[fig:FEMGOcorrelation-maps\]c show a qualitatively similar dominance as it was found for the $c_{\mathrm{s}}$ coefficients in Fig. \[fig:FEMGOcorrelation-maps\]b. Within the Fe electrodes the correlations of the $\Delta_{1}$-like states are just slightly but noticeable larger than those of the s states in these layers, indicating an additional relevance of p$_{\mathrm{z}}$ and d$_{3z^2-r^2}$ orbitals in the electronic transport. This finding is substantiated by relatively small correlations of both other $\Delta$-like representations inside the tunnel barrier. Since the latter exhibit only {p$_x$, p$_y$, d$_{3xz}$, d$_{3yz}$} and {d$_{x^2-y^2}$, d$_{xy}$} orbital character (see Table \[tab\_decomposition\]), it is reasonable to assume that the rising importance of p and d states inside the MgO is identical to an increasing significance of p$_{\mathrm{z}}$ and d$_{3z^2-r^2}$ orbitals.
The comparison of the results of the correlation analysis with the discussion of available PDOS$_{\alpha}$ in Fig. \[fig:PDOS-contribution\] shows common features but reveals also significant differences. A common outcome of both approaches is the principal importance of the tunnel barrier for the electronic transport of the MTJ. In particular, both discussions end up with a conclusion that $\Delta_{1}$-like states— i. e. s, p$_z$ and d$_{3x^2-r^2}$ orbitals which are preferentially aligned along the transport direction— tunnel most effectively and consequently carry the dominant part of the tunnel current.
The fact, that $\Delta_{1}$-like states, especially those with s orbital character, show their decisive role also within the Fe leads, represents a qualitative different outcome of the correlation analysis. In contrast, these states exhibit the lowest PDOS$_{\alpha}$ contributions within the Fe layers in Fig. \[fig:PDOS-contribution\].
In principle, the symmetry-selective filtering of the MgO tunnel barrier shows up by means of the hierarchical order of the {$\Delta_{1}, \Delta_{5},
\Delta_{2}+\Delta_{2'}$}-like PDOS$_{\alpha}$ fractions in Fig. \[fig:PDOS-contribution\]. But deeper within the MgO these percentages stay rather constant. The layer-wise increase of the effective tunneling processes of $\Delta_{1}$-like states inside the MgO can only be seen by the increasing characteristics of the $c_{\Delta_{1}\mathrm{-like}}$ coefficients in Fig. \[fig:FEMGOcorrelation-maps\]c.
Conclusions and remarks {#sec:conclusions}
=======================
The analysis of transport properties within contacts that exhibit planar geometries is often accompanied with a visual comparison of large ensembles of ${\boldsymbol{k}}_{\parallel}$-resolved local spectral density and transmission maps. In this article we presented an analysis tool which helps to avoid laborious inspections and potential misinterpretations by providing exact measures. Applying the proposed correlation analysis the contributions of atoms or orbitals can be quantified unambigously. The dominance of the s states in the tunnel current of Fe/MgO/Fe could be proven even for the states in the Fe leads, where the contribution of states to the is only marginal.
Further, the method extends the popular discussion of transport proporties at the $\bar{\Gamma}$ point in Fe/MgO/Fe to a more comprehensive anlysis which comprises the entire . In principle, the technique is not restricted to planar junctions. Beside more complex layered structures it can be applied to other nano structures like point contacts or nano wires. For these atomic-sized contacts, we expect that the corresponding correlation coefficients provide the same physical insights as one would obtain from an analysis of the conduction eigenchannels \[\].
This work is supported by the DFG’s *Sonderforschungsbereich* 762 ‘Functionality of Oxide Interfaces’.
[^1]: The considered area $\Omega$ has not necessarily to be extended over the whole 2BZ. Principally, $\Omega$ can be restricted to arbitrary parts of it.
|
---
abstract: 'The ATLAS experiment at the Large Hadron Collider will incorporate discrete, high-resolution tracking sub-systems in the form of segmented silicon detectors with 40MHz radiation-hard readout electronics. In the region closest to the $pp$ interaction point, the thin silicon tiles will be segmented into a pixel geometry providing two-dimensional space-point information. The current status of the ATLAS pixel project will be presented with an emphasis on the performance of the front-end electronics and prototype sensors.'
address: Lawrence Berkeley Laboratory
author:
- John Richardson
- representing the ATLAS Pixel Collaboration
title: The ATLAS Pixel Project
---
Introduction
============
The ATLAS experiment [@TP] at the CERN Large Hadron Collider will function as an exploratory tool for the Higgs boson along with possible new physics phenomena beyond the scope of the Standard Model such as supersymmetry. An average of 23 minimum-bias events will be produced every 25ns at the design luminosity ($\cal{L}$ = 10$^{34}$cm$^{-2}$s$^{-1}$). Hence interesting signatures will be buried amongst extraordinary degrees of background. In order to realise fully the physics potential of 14TeV $pp$ collisions at this luminosity, ATLAS must be capable of high-efficiency track reconstruction with excellent impact-parameter resolution for 3D-vertexing and b-tagging. Even at lower luminosities of 10$^{33}$cm$^{-2}$s$^{-1}$, ATLAS will encounter millions of $t\bar{t}$ pairs per year for top-quark analyses along with a plethora of $b$-quarks enabling the possibility of CP-violation studies in the $b$-sector. For these areas of study, the tracking system must again deliver high spatial precision and reconstruction efficiency along with excellent momentum resolution.
In addition to the performance requirements, the extremely harsh radiation environment defines perhaps the greatest challenge of all to the detector elements in ATLAS. At a radius of 30cm in the barrel region we expect to encounter a hadronic fluence[^1] of $\approx$ 1.3$\times$10$^{14}n_{\mbox{eq}}$cm$^{-2}$ after 10 years of operation. At 10cm the figure is closer to 1.0$\times$10$^{15}n_{\mbox{eq}}$cm$^{-2}$.
The Inner Detector of ATLAS [@TDR1] will incorporate high-resolution, low mass discrete tracking components in the form of silicon microstrip and silicon pixel [@TDR2] detectors. The pixel elements will form the innermost layers by virtue of their higher granularity and because they are more suited to providing pattern recognition capability at the greatest track densities. Also, having a greater degree of segmentation than their strip-geometry counterparts naturally endows them with lower noise-per-channel and greater immunity to radiation damage.
Layout of the Pixel System
==========================
Figure \[fi:tiles\] shows the relative positioning of the individual modules within the pixel sub-detector. There are 3 barrel layers at radii of 4.3, 10.1 and 13.2cm along with 5 disc structures which extend to $\pm$78cm in the forward directions; providing pseudorapidity coverage to 2.5. The innermost barrel layer, (known as the $B$-layer), is not expected to survive the 10-year duration of the experiment and will therefore be replaced periodically. A single module comprises a silicon sensor [*tile*]{} with 16 front-end readout chips bump-bonded to it along with one [*Module Controller Chip*]{} (MCC). The pixel detector will incorporate 2228 such modules with a total active area of 2.3m$^2$.
13 cm
-.2 cm
Module structure and hybridization
==================================
There are currently two technology options being pursued for module hybridization known as the [*flex-hybrid*]{} approach and the [*MCM-D*]{} technique[^2]. The former incorporates a thin kapton hybrid which is glued to the sensor backplane to provide the necessary bussing between the FE-chips, MCC and optical transmission components. The front-end chips, which are bump-bonded to the opposite face of the sensor, protrude along the long-edges of the module in order that their 48 power,data and control bond-pads may be wire-bonded up to the hybrid. The MCC sits on top of the hybrid along with the other electrical components such as decoupling capacitors, optical-link termination devices and temperature sensor.
The MCM-D method eliminates the need for wire bonds as the bussing is processed on to the sensor surface (active-side) in four layers of copper metallisation. The long sensor dimension is extended to provide a region of ‘dead’ silicon upon which the MCC and peripheral components are mounted. The sensor is also wider in order that the connections from the front-end chips to the power and control bus’ may be formed using bump-bonds in the same way that the individual pixel implants are connected to the preamp inputs.
Bump-bonding at 50$\mu$m pitch was until recent years considered to be a serious issue. However, there are several vendors with whom very promising experience has been gained recently in both the Indium and $PbSn$-solder technologies with pre-flip-chip yields of order 10$^{-5}$. For the Indium process, the flip-chip stage involves pressing together the electronics chip and sensor, (both of which have the bumps grown on them), to form a cold weld. Solder bumps are grown only on the electronics surface using a plating and re-flow technique. The recipient sensor surface is plated with a wettable metallisation. Flip-chipping in this case involves bringing the two surfaces into accurately aligned contact and re-flowing the solder at 250$^{\mbox{o}}$C.
Prototype Sensors
=================
The pixel sensors will be formed from 250$\mu$m-thick high-$\rho$ Silicon except within the $B$-layer where it is hoped to utilise material as thin as 200$\mu$m. High-$\rho$ silicon which begins as $n^-$-type inverts to $p$-type after a hadronic dose of $\approx$ 2.5$\times$10$^{12}n_{\mbox{eq}}$cm$^{-2}$. Beyond this dose the depletion voltage rises as the effective carrier concentration continues to increase. After 10-years of radiation exposure, the required reverse-bias voltage for full depletion of the pixel sensors will be in excess of 1000V. Since the maximum operating voltage is specified to be 600V, we will be forced to operate them in a partially-depleted regime. For this reason all of the prototype designs are based on $n^+$-type pixel implants so that post-inversion, the junction resides on the active side.
The first-phase of sensor prototyping was conducted with two vendors; Seiko in Japan and C.I.S. in Germany. Identical wafer designs were submitted to both and featured two full-size tile designs. One of these had [*p-stop*]{} isolation structures, (known as ‘Tile-1’), whilst the other, (Tile-2), used the [*p-spray*]{} [@Tilman] technique to over-compensate the electron accumulation-layer which otherwise has the effect of shorting $n^+$-implants together. Along with these devices there were a variety of smaller detector designs with a pixel-array corresponding to that of a single readout chip. Some of these matched the Tile-1 and Tile-2 designs (known as ST1 and ST2 respectively). Figure \[fi:st1st2\] shows the layout of these two designs in detail. The ST1 design is on the left and the $p^+$-type isolation rings may be seen surrounding the $n^+$-pixel implants. In the ST2 design there are also ring-like structures around each pixel but these are now $n^+$-type and have the purpose of reducing the inter-pixel capacitance. The ST2 design also incorporates reach-through biasing which is not possible to implement with p-stop isolation.
6.2cm 3.6cm 6.2cm 3.6cm 6.2cm 3.6cm
0.5cm
The third design shown in figure \[fi:st1st2\] is known as SSG[^3]. This also utilises p-spray isolation but does not include any intermediate $n^+$ structures. Neighbouring pixels rather have just small gaps between them.
Front-End Readout Electronics
=============================
There are many demands on the front-end readout electronics. Insensitivity to the changes in transistor thresholds and transconductance brought on by a total ionising dose of 30MRad, (in layer-1 after 10-years of operation), must be built into the design. In addition, the front-ends must be able to deliver high-efficiency performance when the operation of the sensors has been compromised by hadronic damage. This will involve leakage-current tolerence to the level of 100nA/channel whilst maintaining sensitivity to charge spectra much reduced by partial-depletion operation and charge-trapping effects. The power consumption must not exceed 40$\mu$W per channel, the timewalk must be contained to the 25ns bunch-crossing period of the LHC and the cross-coupling between neighbouring pixels should account for a charge loss of no more than 5%.
Over the last year and a half, the [*demonstrator*]{} development programme has followed two concurrent design directions. The two designs, known as FE-A and FE-B, were aimed at the DMILL and Honeywell radiation-hard process’ respectively. To date both have been realised in radiation-soft technologies, (namely AMS and HP), and evaluated extensively in laboratory and testbeam environments. The FE-A design, (which was initially a BiCMOS design), was also laid out and fabricated as a pure CMOS chip which is referred to as the FE-C. The two design efforts have joined forces and are now on the threshold of the first radiation-hard submission (to DMILL), of a common demonstrator design, (FE-D), which is largely based on combined features of the FE-A and FE-B approaches. Later in 1999 it is planned that a common design submission also be made for the Honeywell SOI process (FE-H). All of the demonstrator chips were designed to be fully pin-compatible enabling a common hardware test system to be used for all cases. Only minor software and firmware options are required to switch between different chips.
There were several philosophical operational differences between FE-A and FE-B not least of which was the use of a dual-threshold discriminator for the FE-B front-ends as opposed to the faster single discriminator approach used in FE-A. Time information is derived from a lower time-threshold whilst the upper threshold is resposible for hit-adjudication. Both approaches incorporate negative polarity preamps for the $n^+$ in $n^-$ sensors, LVDS inputs and outputs for clock and control lines, a serial command protocol and serial readout scheme for line reduction and local data storage in the form of end-of-column-pair buffer sets and a FIFO for hits which have matched incoming triggers. They also both incorporate eight 8-bit local DACs for front-end biases, a local chopper circuit for calibration charge injection, 7-bits of digital charge measurement based on time-over-threshold, channel-by-channel masking capability for separate strobe and readout selection and a 3-bit local threshold-tuning DAC, (TDAC), in every pixel cell for threshold-dispersion reduction.
Results from demonstrator-assembly testing
==========================================
Throughout 1998 the collaboration followed a rich program of testing of ‘single-chip assemblies’ which comprised the radiation-soft demonstrator electronics bump-bonded to the first-generation prototype sensors, (both irradiated and non-irradiated samples). Testing has also progressed on a smaller number of full-scale modules with 16 FE readout-chips connected to a single tile.
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The top-left plot in figure \[fi:tdactun\] shows the distribution of thresholds for a bare chip. This is formed from scanning the input calibration-pulse amplitude in fine steps and deriving an ‘s-curve’ of efficiency versus charge (knowing the magnitude of the calibration capacitance) pixel-by-pixel. The width of this distribution is 265e- with all of the 3-bit TDACs set to the same value. On the right of this plot the TDAC values for all channels have been assigned in order to minimise the dispersion. This process results in a post-tune distribution width of 94e-.
The two lower plots on the left of figure \[fi:tdactun\] show the threshold values plotted according to ‘channel-number’ which is defined to be 160$\times$column number + row number. The four rightmost plots show the tuned-threshold distribution and noise distribution for a typical FE-B single-chip assembly incorporating an ST2 sensor. The equivalent noise charge (ENC) is measured to be 105e-. Similar results are obtained for ST1 assemblies whereas the noise for an SSG device was measured to be 170e- due to the higher load capacitance.
A range of assemblies were evaluated in the H8 testbeam of CERN’s SPS accelerator during four periods of 1998 running. The H8 facility offers a four cross-plane silicon microstrip hodoscope, providing an extrapolation resolution of 3$\mu$m for charged pions with momenta in excess of 100GeV/c. This enabled detailed studies of hit-efficiency, charge-collection efficiency, resolution and noise occupancy to form part of the comparative process for the different prototype sensor designs both before and after irradiation. The left-most plots in figure \[fi:h81\] show the charge spectra obtained from the three principal design types, ST2, SSG and ST1 for tracks of normal incidence. Each plot has two histograms, one corresponding to the case where only a single channel was seen to fire and the other for double-channel clusters. In all cases except for ST2, the spectra show the usual Landau/Gaussian-convolution shape, peaking at around the 21,000e- value expected for the 280$\mu$m thickness of Silicon. For the ST2 case the double-channel distribution has a distinctly lower peak value indicating some degree of charge-loss. In the central plots, a mean-charge surface is plotted as a function of the local extrapolation point within a 2-pixel cell.
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For the ST1 and SSG cases it is clear that there is a slight degree of charge-collection inefficiency in the region between neighbouring columns. For the ST2 design, the loss is extreme in this region and further loss is apparent all around the pixel-cell outline. This effect is directly attributable to the atoll $n^+$-rings which surround each pixel in the ST2 design. The leftmost plots in figure \[fi:h82\] show residual distributions for an SSG assembly for tracks of normal incidence. The upper right distribution corresponds to the overall binary case in the $r\phi$ dimension which yields an r.m.s. of 12.9$\mu$m. This is slightly better than the expected 50$\mu$m/$\sqrt{12}$ due to a small admixture of double-channel clusters. Similar results were obtained for all of the designs pre-irradiation.
The table overleaf summarises the overall efficiency figures obtained for the ST1, ST2 and SSG assemblies for thresholds of approximately 2ke- and 3ke-. The values are at least equal to 99% in almost all cases upon application of a range of quality cuts in time and space, the single exception being ST2 at 3ke- threshold which registers 98.8%. This is due to the worse charge-collection efficiency mentioned earlier.
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Irradiated ST1 and ST2 samples were also tested in H8 at -10$^{\mbox{o}}$C. ST2 samples which had received 5.0$\times$10$^{14}n_{\mbox{eq}}$cm$^{-2}$ and 1.0$\times$10$^{15}n_{\mbox{eq}}$cm$^{-2}$ both performed extremely well for various reverse-bias voltages up to 600V, as tabulated in the right-hand table below. An efficiency of 98.2% was recorded for the maximum dose sample when the analysis was restricted to a region within each pixel cell which eliminates the charge-loss effects due to the atoll $n^+$ structures. The fake-occupancy probability for this device was recorded to be 0.9$\times$10$^{-7}$ and half of those ‘hits’ were identified as originating from physics. On the right of figure \[fi:h82\] are the distributions of residuals for the high-dose ST2 sample. The overall binary resolution is measured to be 16.5$\mu$m. The corresponding ST1 irradiated samples in contrast showed poorer performance. Extremely large numbers of noise hits were recorded for even the 5$\times$10$^{14}n_{\mbox{eq}}$cm$^{-2}$ sample for applied detector biases in excess of 10V.
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Noise analyses of the irradiated ST1 samples in the laboratory indicate severe micro-discharge occurance once the backplane bias is raised much above 100V. At 250V a broad spectrum of noise values are recorded up to 15,000e-. Even the highly dosed ST2 assembly on the other hand yields a very tight noise distribution at 600V in the laboratory with a mean of just 260e-, (see figure \[fi:irrnoise\]).
Figure \[fi:bigiv\] shows the results of I-V characterisations of the high-dose irradiated samples. The ST1 assembly breaks down at an applied bias of around 630V whereas the ST2 device shows no signs of breakdown even up to 1kV. All of these results strongly reinforce the view that p-spray isolation leads to lower surface fields than the p-stop approach and thus greater micro-discharge and breakdown immunity when operated at high-voltage. The next generation sensor prototypes reflect these observations with the principal design based on p-spray isolation without any intermediate $n^+$ structures. This design looks like the SSG approach except that the gaps are made slightly wider in order to optimise the noise performance.
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Conclusions
===========
The ATLAS Pixel demonstrator program is progressing well with very encouraging results obtained from radiation-soft versions of the front-end readout electronics. The success of the electronics program has enabled evaluation of a series of prototype sensor designs which have provided the collaboration with a clear concept for a new prototype layout. This new design is optimised for high-voltage operation post-irradiation whilst maintaining good charge-collection efficiency. The experience obtained with the two front-end design concepts (FE-A and FE-B) has been of great benefit in converging on combined designs for the upcoming radiation hard submissions.
ATLAS Technical Proposal [**CERN/LHCC/94-43**]{} (1994). ATLAS Inner Detector Technical Design Report [**CERN/LHCC/97-16**]{} (1997). ATLAS Pixel Detector Technical Design Report [**CERN/LHCC/98-11**]{} (1998). T.Rohe et al., Nucl. Inst. Meth. in Phys. Res. [**A,**]{} 224-228 (1998).
[^1]: expressed in terms of the 1MeV-neutron NIEL equivalent damage
[^2]: [**M**]{}ulti [**C**]{}hip [**M**]{}odule - [**D**]{}irect.
[^3]: [**S**]{}ingle [**S**]{}mall [**G**]{}ap
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---
abstract: 'The Bell-wave (B-wave) supposition has been introduced in an attempt to investigate Bell’s conjecture (according to which behind the scenes something is going faster than light). Here it is shown, for the case of two entangled photons, that if it is further assumed that the B-waves propagate with superluminal but finite velocity then it is possible, at least in principle, to have faster-than-light (FTL) communication.'
author:
- |
Luiz Carlos Ryff\
*Instituto de Física, Universidade Federal do Rio de Janeiro,*\
*Caixa Postal 68528, 21941-972 RJ, Brazil*
title: 'Bell’s Conjecture and Faster-Than-Light Communication'
---
To try to explain the Einstein-Podolsky-Rosen (EPR) correlations, John Bell conjectured that something should be propagating with superluminal velocity, and suggested the reintroduction of the idea of an aether, a preferred frame of reference . However, as far as I know, he never elaborated on this idea. B-waves have been assumed in an attempt to investigate Bell’s conjecture . Considering a two-photon entangled state, a B-wave is created when the first photon of the pair is detected in the preferred frame. It then propagates and reaches the second photon, forcing it into a well-defined state. But the state in which the first photon is found is not necessarily the state into which the second photon will be forced. It will depend not only on the initial entangled state, but also on the optical devices the photons will find on their way to the detectors. How is the correct information conveyed? Assuming that there cannot be any sort of conspiracy of nature, or, in other words, that nature is blind, this can only take place in a purely mechanical or automatic way, so to speak. A possibility is to have the B-wave following the path of the first photon backwards to the source and then following the path of the second photon. Each time it passes through an optical device its state is changed, eventually reaching the second photon in the correct state. This simple mechanism can, in principle, reproduce the results of all Bell inequalities tests with pairs of photons, and is consistent with the following aspect of the quantum mechanical formalism. For instance, let us consider the two-photon polarization-entangled state $\mid \psi \rangle $ = (1/$\sqrt{2}$)($\mid $V$\rangle _{1}\mid $H$\rangle _{2}-\mid $H$\rangle _{1}\mid $V$\rangle _{2}$) $\equiv $ (1/$\sqrt{2}$)($\mid $V$\rangle \mid $H$\rangle -\mid $H$\rangle
\mid $V$\rangle $). If photon 1 passes through a $\lambda /2$-plate, such that $\mid $V$\rangle $($\mid $H$\rangle $) $\rightarrow $ $\mid $H$\rangle $($\mid $V$\rangle $), then $\mid \psi \rangle $ $\rightarrow $ (1/$\sqrt{2}$)($\mid $H$\rangle \mid $H$\rangle -\mid $V$\rangle \mid $V$\rangle $). That is, only the potential states of photon 1 are changed (this is valid for any optical device, not only for wave plates). It seems that the communication between the photons – if it exists – must only occur at the moment of detection.
It has already been shown that the assumption of finite-speed ($v$) superluminal communication leads, under specific circumstances and for more than two entangled particles, to FTL signalling . A simple example is the following. Let us consider the state of three qubits $\mid $GHZ$\rangle $ = (1/$\sqrt{2}$)($\mid $0$\rangle _{1}\mid $0$\rangle _{2}\mid
$0$\rangle _{3}$ +$\mid $1$\rangle _{1}\mid $1$\rangle _{2}\mid $1$\rangle
_{3}$). Particle 1 is sent to Alice (**A**) and particles 2 and 3 are sent to Bob (**B**) and Charlie (**C**), respectively, who work not far away from each other in the same lab.. At instant $t_{A}$ (in the preferred frame), **A** may decide to measure the state, $\mid $0$\rangle $ or $\mid $1$\rangle $, of particle 1, or not; and, at instant $t_{L}>t_{A}$ (also in the preferred frame), **B** and **C** will measure the state, $\mid $0$\rangle $ or $\mid $1$\rangle $, of their particles. The condition $v>l/(t_{L}-t_{A})>c$ has to be fulfilled, where $l$ is the distance from **A** to** B** and from **A** to **C**. Supposing that the correlations are purely nonlocal, whenever **B** and **C** perform their measurements, but **A** does not perform hers, the probability of **B** and **C** observing the same outcome is $1/2$, since there can be no communication between them ($v<\infty $). On the other hand, whenever **B** and **C** perform their measurements, and **A** performs hers, this probability is equal to $1$, since the first measurement forces the other two particles into the same state. Therefore, if we have in the left lab. many **A**s, and in the right distant lab. the corresponding **B**s and **C**s, and the **A**s combine to take the same decision together, that is, to perform a measurement or not, the **B**s and **C**s will know (comparing their results and disregarding improbable statistical fluctuations) what has been decided in the left lab. before this information could reach them transmitted by a light signal.
For two particles, and assuming the existence of B-waves with the properties mentioned above, the demonstration is as follows. Let us imagine the following experiment performed in the preferred frame. A source $S$ emits entangled photons, $\nu _{1}$ and $\nu _{2}$, in state $$\frac{1}{\sqrt{2}}\left( \mid V\rangle \mid H\rangle -\mid H\rangle \mid
V\rangle \right) . \tag{1}$$$\nu _{1}$ and $\nu _{2}$ are emitted in opposite directions, reaching two-channel polarizers with orientations $\mathbf{a}$** **and $\mathbf{b}$**,** respectively. The condition $$x_{b}>x_{a}+2y \tag{2}$$is fulfilled, where $x_{b}$ is the distance followed by $\nu _{2}$ from $S$ to detector $D_{2}$($D_{2^{\prime }}$), placed on the transmission (reflection) channel, and $x_{a}+2y$ is the distance followed by $\nu _{1}$ from $S$ to detector$\ D_{1}$($D_{1^{\prime }}$), placed on the transmission (reflection) channel, and where $y$ is the height of a detour introduced in $\nu _{1}$’s path. Therefore, $\nu _{1}$ is always detected before $\nu _{2}$. Between $S$ and the detour there is a Pockels cell. Then, introducing $t_{l}$ and $t_{B}$ as $$t_{l}=\frac{x}{c} \tag{3}$$and $$t_{B}=\frac{x+2y}{v_{B}}, \tag{4}$$where $x$ is the distance from the Pockels cell ($PC$) to $D_{1}$, and $v_{B} $ is the velocity of the Bell-wave, the condition $$t_{l}<t_{B} \tag{5}$$has to be fulfilled, which leads, using $(3)$ and $(4)$, to $$y>\left( \frac{v_{B}-c}{c}\right) \frac{x}{2}. \tag{6}$$Therefore, it is possible to have the detection of $\nu _{1}$ triggering a light signal that activates the $PC$ just before the passage of the B-wave, which, as a result, has its state modified. Since $v_{B}>c$, the B-wave reaches $\nu _{2}$ before a light wave sent from $D_{1}$ or $D_{1^{\prime }}$ at the moment of detection. Now, let us assume that the activating signal is only triggered when $\nu _{1}$ is registered at $D_{1^{\prime }}$. The detection probabilities are then given by $$p_{12}=\frac{1}{2}\sin ^{2}(a,b),\text{ \ \ \ \ \ \ \ \ \ \ \ }p_{12^{\prime
}}=\frac{1}{2}\cos ^{2}(a,b), \tag{7}$$$$\text{ \ \ \ \ \ \ \ }p_{1^{\prime }2}=\frac{1}{2}\cos ^{2}(a,b^{\prime }),\text{ and \ \ \ }p_{1^{\prime }2^{\prime }}=\frac{1}{2}\sin
^{2}(a,b^{\prime }),\text{\ \ \ \ \ \ \ \ } \tag{8}$$where $\mathbf{b}^{\prime }\neq \mathbf{b}$, since the state of the B-wave has been modified in accordance with our purposes. Hence, we obtain $$p_{12}+p_{12^{\prime }}=p_{1}=\frac{1}{2}, \tag{9}$$$$p_{1^{\prime }2}+p_{1^{\prime }2^{\prime }}=p_{1^{\prime }}=\frac{1}{2},
\tag{10}$$$$p_{12}+p_{1^{\prime }2}=p_{2}=\frac{1}{2}\left[ \sin ^{2}\left( a,b\right)
+\cos ^{2}\left( a,b^{\prime }\right) \right] , \tag{11}$$and $$p_{12^{\prime }}+p_{1^{\prime }2^{\prime }}=p_{2^{\prime }}=\frac{1}{2}\left[
\cos ^{2}\left( a,b\right) +\sin ^{2}\left( a,b^{\prime }\right) \right] .
\tag{12}$$If the activating signal is not triggered, $\mathbf{b}^{\prime }\rightarrow
\mathbf{b}$, which leads to $p_{2}=p_{2^{\prime }}=1/2$; on the other hand, if it is, we have $p_{2}\neq p_{2^{\prime }}$. Therefore, comparing the detections on the left side of the experimental apparatus it is possible to know what decision was taken on the right side (to trigger the activating signal or not); and this information can be transmitted with superluminal velocity .
The above discussion might be seen as an argument against the Bell conjecture. However, if a preferred frame is assumed, the possibility of FTL signaling can not be discarded; in particular, no causal paradoxes will necessarily arise from this . But some obstacles would make the realization of the experiment difficult. In particular, we don’t know how to determine the preferred frame and the superluminal speed. Using recent data , $(6)$ leads to $y>10^{4}x/2$, which suggests the use of optical fibers, to keep the detour within the dimensions of the laboratory!
Actually, it can always be conjectured that the communication between the entangled photons does not occur through ordinary three-dimensional space; however, this should not be an impedimentto the investigation of simple – perhaps far too naive– and experimentally testable alternatives , as the one discussed here.
[9]{} Interview with J. S. Bell, in *The Ghost in the Atom,*** **eds P. C. W. Davies and J. R. Brown (Cambridge University Press, 1989). Quoting John Bell: The reason I want to go back to the idea of an aether here is because in these EPR experiments there is the suggestion that behind the scenes something is going faster than light. Now, if all Lorentz frames are equivalent, that also means that things can go backward in time. \[This\] introduces great problems, paradoxes of causality and so on. And so, it’s precisely to avoid these that I want to say there is a real causal sequence which is defined in the aether. In P. Caban and J. Rembieliński, *Phys. Rev. A*** 59**, 4187 (1999), and in J. Rembieliński and K. A. Smoliński, *Phys. Rev. A*** 66**, 052114 (2002) the possibility of introducing and identifying a preferred frame is examined. A. J. Leggett, *Found. Phys.* **33**, 1469 (2003), has suggested some reasonable criteria that a nonlocal theory should satisfy and has proposed a nonlocal model which has not passed the experimental test: S. Gröblacher, T. Paterek, R. Kaltenbaek, Č. Brukner, M. Żukowski, M. Aspelmeyer, and A. Zeilinger, *Nature* **446**, 871 (2007); C. Branciard, A. Ling, N. Gisin, C. Kurtsiefer, A. Lamas-Linares, and V. Scarani, *Phys. Rev. Lett.* **99**, 210407 (2007).
L. C. Ryff, arXiv: quant-ph/0809.0530v1. As opposed to the approaches mentioned in ref. 1, here the idea is to investigate a possible *mechanism* behind the nonlocal correlations.
V. Scarani and N. Gisin, *Braz. J. Phys.* **35**, 328 (2005) and *Phys. Lett. A* **295**, 167 (2002).
P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, *Phys. Rev. Lett.* **75**, 4337 (1995).
A similar situation, in the case of a null measurement, has been discussed in: L. C. Ryff and C. H. Monken,*Quantum Semiclass. Opt.* **1**, 345 (1999).
On this point, the interview with D. Bohm in ref. 1 is worth reading, and also sec. 12.8 in D. Bohm and B. J. Hiley, *The Undivided Universe* (Routledge,* *1993).
A possibility is to consider the reference frame in which the microwave background radiation is most isotropic, as in V. Scarani, W. Tittel, H. Zbinden, and N. Gisin, *Phys. Lett. A* **276**, 1 (2000).
D. Salart, A. Baas, C. Branciard, N. Gisin, and H. Zbinden, *Nature* **454**, 861 (2008); also J. Kofler, R. Ursin, C. Brukner, and A. Zeilinger, arXiv: quant-ph/ 0810.4452, for a comment, and D. Salart, A. Bass, C. Branciard, N. Gisin, and H. Zbinden, arXiv: quant-ph/ 0810.4607, for the reply. One of the criticisms presented in the comment is based on the argument that in a Franson-type experiment the Clauser-Horne-Shimony-Holt Bell inequality is not applicable even with perfect detectors because of the inherent postselection(S. Aerts, P. Kwiat, J.-Å. Larsson, and M. Żukowski, *Phys. Rev. Lett.*** 83**, 2872 (1999)); however, see L. C. Ryff, *Phys. Rev. Lett.*** 86**, 1908 (2001), and arXiv: quant-ph/ 0810.4825.
|
---
abstract: 'We consider the low energy description of five dimensional models of supergravity with boundaries comprising a vector multiplet and the universal hypermultiplet in the bulk. We analyse the spontaneous breaking of supersymmetry induced by the vacuum expectation value of superpotentials on the boundary branes. When supersymmetry is broken, the moduli corresponding to the radion, the zero mode of the vector multiplet scalar field and the dilaton develop a potential in the effective action. We compute the resulting soft breaking terms and give some indications on the features of the corresponding particle spectrum. We consider some of the possible phenomenological implications when supersymmetry is broken on the hidden brane.'
author:
- 'Ph. Brax'
- 'N. Chatillon'
title: '**Soft Supersymmetry Breaking on the Brane**'
---
1 cm
1 cm
Introduction
============
Supersymmetry breaking is one of the unsolved challenges of particle physics. A proper understanding of the origin of supersymmetry breaking would certainly increase the prospects of an experimental discovery of supersymmetry and shed new light on thorny issues such as the cosmological constant problem. Many models of supersymmetry breaking have been proposed so far. Amongst the most popular are the gravity mediated and gauge mediated scenarios (see [@gravity_gauge_mediated] for reviews). Each have interesting features although none provide a completely satisfactory framework. Recently brane models have been introduced and address both the hierarchy problem [@ADD; @RS1] and the cosmological constant problem [@selftuning]. Brane models have been originally built up in a non-supersymmetric setting. The supersymmetrisation of the Randall-Sundrum model [@Altendorfer:2000rr; @Falkowski:2000er] and models with a bulk scalar field [@kallosh; @offshell; @flp2] provide a justification for certain fine–tunings used in brane models in order to find flat brane solutions. Later it has been noticed that supersymmetry is in fact compatible with branes of lower tension than the Randall-Sundrum case [@Brax:2001xf; @susyRS_arbitrary_tensions; @Brax:2002vs; @Lalak:2002kx; @twisted_sugra; @twist_warp_sugra]. When the tensions are detuned, the 5d action of the theory is supersymmetric, but can be built so that the warped background solutions of the equations of motion either break supersymmetry or not. Hence this may realize a spontaneous breaking of supersymmetry in five dimensions. Other types of supersymmetry breaking solutions with tension detuning, which do not correspond to static warped backgrounds with straight branes, can lead to strongly 4d Lorentz violating effects [@kinetic_susy_breaking; @detuned].
At sufficiently low energy, well below the brane tensions, supersymmetric brane models can be described by a 4d supersymmetric effective action [@luty1; @susy_radion; @flp3; @effective_detuned; @effective_sugra]. This 4d effective action is determined by a Kähler potential for the moduli fields. When the bulk contains a vector multiplet and the universal hypermultiplet, the moduli describe the radion, the zero mode of the vector multiplet scalar field and the dilaton, i.e. the zero mode of the hypermultiplet. The moduli coming from the vector multiplet bulk scalars are associated to the axion–like fields originating from the fifth components of the bulk vector fields. The existence of these massless moduli may imply strong deviations from general relativity [@testsGR]. The cosmology of these tensor–scalar theories is also interesting [@rev1; @rev2] and may be probed using the CMB anisotropies [@cosmoduli; @CMB].
The supersymmetric low energy action in the Randall-Sundrum case has been nicely spelt out in [@effective_detuned]. Detuning the brane tensions by including constant superpotentials leads to an effective potential for the radion. It corresponds to the fact that the detuned boundary conditions are no longer compatible with 4d Poincaré supersymmetry . The brane system is then subject to a back-reaction effect which can be analysed using the equations of motion of the low-energy action. When the radion effective potential admits a minimum, this value of the radion is equal to the one obtained by solving the 5d equations with detuned tensions. At the minimum, the potential is negative and supersymmetry is preserved for specific values of the radion imaginary part, as the F–terms of the radion vanishes, corresponding to $AdS_4$ supergravity. This is the same result as obtained analysing the 5d equations of motion and Killing spinor equations. Notice that the resulting configuration breaks Poincaré invariance.
In the present article, we study the low energy action of brane models with bulk scalar fields belonging to a vector multiplet and the universal hypermultiplet of 5d $N=2$ supergravity. We include the effects of a detuning of the brane tensions in the form of superpotentials on the branes. Supersymmetry is then broken by the $F$–terms associated to the moduli. We analyse both the cosmological and the particle physics consequences of such a breaking.
Our analysis of the breaking of supersymmetry extends to the case with a vector multiplet the results of [@flp3] where the Randall–Sundrum model with a hypermultiplet in the bulk was considered. The phenomenology of this model has been spelt out in [@casas] with particular emphasis on the electro–weak symmetry breaking, for matter on the negative tension brane and specific assumptions about moduli stabilisation. On the contrary, we consider the case where matter lives on the positive tension brane with no assumption about moduli stabilisation. We envisage the case when moduli may not be stabilised and take into account the corresponding solar system constraints.
A particular stumbling block of supersymmetric models is the origin of the hierarchy between a large scale such as the Planck mass and the $\mu$ term. In the following we will show that when a $\mu$ term is included on the positive tension brane, a large hierarchy can be induced thanks to the presence of the vector multiplet scalar field in the bulk. Similarly, when supersymmetry is broken on the hidden brane of negative tension, and no $\mu$ term is included in the superpotential of the positive tension brane, an effective $\mu$ term results from anomaly mediation.
Breaking supersymmetry using boundary superpotentials has been investigated in [@luty2; @zwirner] in the flat case where the brane tensions vanish. In the flat case, a $N=1$ superspace formulation of $D=5$ supergravity coupled to boundary branes has been elaborated in [@phil1] and used to compute quantum corrections to the soft breaking terms [@phil2]. In the case that we consider, the bulk background is warped. We analyse the non–trivial effects induced by such a warping on the breaking of supersymmetry. In particular all our results apply to the Randall–Sundrum setting where matter lives on the positive tension brane.
When supersymmetry is broken on the hidden brane of negative tension and matter lives on the positive tension brane, we find that soft supersymmetry breaking terms are of two sorts. First of all the soft trilinear $A$ terms and the gaugino masses receive a non–vanishing contribution at one–loop level from anomaly mediation. Secondly and contrarily to anomaly mediation scenarios, the soft masses are non-vanishing at tree level. Therefore they do not suffer from the tachyons of anomaly mediated models.
In section 2 we present the models including a bulk vector multiplet and study the zero–modes, i.e. we give different parametrisations of the low energy moduli. In section 3 we analyse the low energy action computing the Kähler potential and the superpotential when coupling 5d supergravity with a vector multiplet in the bulk to matter on the branes. We also discuss the dilaton arising from the bulk hypermultiplet. In section 4 we focus on the case with no hypermultiplet and compute both the moduli potential and the soft breaking terms at the classical level. We then introduce the dilaton field in section 5 and discuss race–track models. In section 6, we discuss the one–loop anomaly mediated soft terms. In section 7, we focus on supersymmetry breaking on the hidden brane, making explicit the soft breaking terms and the associated phenomenology. In section 8, we introduce an explicitly supersymmetry breaking step by taking into account charges on the branes in order to bypass the cosmological constant problem. This leads to an extra contribution to the moduli potential. Finally in section 9, we discuss the cosmological consequences and the gravitational constraints on the models. We also include two appendices. In a first appendix, we discuss the Randall-Sundrum case and radion stabilisation. In the second one, we give the Randall-Sundrum soft terms.
Supergravity with Boundary Branes
=================================
Supergravity Construction
-------------------------
For the bulk theory with no brane coupling, ${\mathcal N}=2$ $D=5$ pure supergravity was first constructed in [@5d; @pure; @sugra] ; vector multiplets were added in [@vector; @coupling], and finally vector multiplets, hypermultiplets and tensor multiplets were treated together in [@vector-tensor-hyper]. Gauged supergravity with boundary branes in five dimensions has been elegantly constructed when vector multiplets live in the bulk [@kallosh; @offshell]. The supergravity multiplet comprises the metric tensor $g_{ab}$, $a,b=1\dots 5$, the gravitini $\psi_a^A$ where $A=1,2$ is an $SU(2)_R$ index and the graviphoton field $A_a$. The ${\mathcal N}$=2 vector multiplets in the bulk possess one vector field, a $SU(2)_R$ doublet of symplectic Majorana spinors and one real scalar. When considering $n$ vectors multiplets, it is convenient to denote by $A^I_a$, $ I=1\dots
n+1$, the $(n+1)$ vector fields.
The two boundaries are fixed points of a $Z_2$ orbifold like in the Randall-Sundrum model and its supersymmetrisation. The action on each brane depends on two ingredients. First the branes couple to the bulk, i.e. to gravity and the real scalar fields. Then ordinary matter is confined to either of the branes. In the following, we will first describe the bulk and brane theory without matter.
The supergravity theory with boundaries differs from usual five-dimensional non supersymmetric theories with boundaries as new superpartner fields are introduced in order to close the supersymmetry algebra and ensure the invariance of the Lagrangian. The vector multiplets comprise scalar fields $\phi^i$ parameterizing the manifold $$C_{IJK}h^I(\phi) h^J(\phi) h^K(\phi)=1$$ with the functions $h^I(\phi), I=1\dots n+1$ playing the role of auxiliary variables. In heterotic M–theory [@ovrut] the symmetric tensor $C_{IJK}$ has the meaning of an intersection tensor. Defining the metric $$G_{IJ}=-2C_{IJK}h^{K}+3h_Ih_J$$ where $h_I=C_{IJK}h^Jh^K$, the bosonic part of the Lagrangian (vector fields not included) reads $$S_{bulk}=\frac{1}{2\kappa_5^2}\int d^5x \sqrt {-g_5}\Big({\mathcal
R} -\frac{3}{4}(g_{ij}\partial_{\mu}
\phi^i\partial^{\mu}\phi^j+V)\Big) \label{lag}$$ where the sigma-model metric $g_{ij}$ is $$g_{ij}=2 G_{IJ}\frac{\partial h^I}{\partial \phi^i}\frac{\partial h^J}{\partial \phi^j}$$ and the potential is given by $$V=U_iU^i-U^2$$ using the sigma-model metric $g_{ij}$. The superpotential $U$ defines the dynamics of the theory. It is given by $$U=4\sqrt\frac{2}{3}g h^IV_I$$ where $g$ is a gauge coupling constant and the $V_I$’s are real numbers such that the $U(1)$ gauge field is $A^I_a V_I$.
The boundary action depends on two new fields. There is a supersymmetry singlet $G$ and a four form $A_{\mu\nu\rho\sigma}$ [@kallosh]. One also modifies the bulk action by replacing $g\to G$ and adding a direct coupling $$S_A=\frac{2}{4! \kappa_5^2}\int d^5x \epsilon^{abcde}A_{abcd}\partial_{e}G.$$ The boundary action is taken as
$$S_{bound}=-\frac{1}{\kappa_5^2}\int d^5x
(\delta_{x_5}-\delta_{x_5-R}) (\sqrt {-g_4}\frac{3}{2}U
+\frac{2g}{4!}\epsilon^{\mu\nu\rho\sigma}A_{\mu\nu\rho\sigma}).
\label{boundary action}$$
where $\mu,\nu,\rho,\sigma$ are four-dimensional indices on the branes. Notice that the four-form $A_{abcd}$ is not dynamical. According to this action the branes can be seen as charged under this bulk four-form with a charge $\pm g$. In section \[charged branes\] we will consider branes charged under a new bulk four-form with kinetic terms and arbitrary charge, in order to cancel the vacuum energy.
The supersymmetry algebra closes on shell where $$G(x)=g\epsilon (x_5),$$ and $\epsilon (x_5)$ jumps from -1 to 1 at the origin of the fifth dimension. On shell the bosonic Lagrangian reduces to the bulk Lagrangian coupled to the boundaries as, $$S_{bound}=-\frac{3}{2\kappa_5^2}\int d^5x
(\delta_{x_5}-\delta_{x_5-R})\sqrt {-g_4}U,$$ Crucially, the boundary branes couple directly to the bulk superpotential. Notice that the two branes have opposite (field-dependent) tensions $$\lambda_{\pm}=\pm \frac{3}{2\kappa_5^2}U$$ where the first brane has positive tension.
Let us focus on the case of a single vector multiplet $n=1$. The equations of motion can be written in a first order BPS form $$\frac{a'}{a}=-\frac{U}{4},\ \phi'=\frac{\partial U}{\partial \phi},$$ where $'=d/dz$ for a metric of the form $$\label{background}
ds^2 = dz^2 + a^2(z)\eta_{\mu\nu}dx^\mu dx^\nu.$$ The boundary conditions are automatically satisfied implying that the positions of the two boundary branes are not specified. Moreover the BPS background preserves $N=1$ Poincaré supersymmetry from the four dimensional point of view. Indeed denote by $\epsilon^A$ the supersymmetry parameter of $N=2$ 5d supergravity. The Killing spinor solutions of $\delta_\epsilon \psi_a^A=0$ satisfy [@kallosh] $$\epsilon ^A (z,x^\mu) =a^{1/2}(z) \epsilon^A$$ where $\epsilon^A$ is a constant spinor such that $\gamma_5
\epsilon^A =(\sigma_3)^A_B\epsilon^B$ implying that only one chirality of the original supersymmetries is preserved. Having obtained Killing spinors corresponding to $N=1$ 4d supersymmetry, we will explicitly find that the low energy Lagrangian can be written in a 4d supersymmetric way.
Let us give the simplest example of models of supergravity with a single scalar field [@BD]. We choose only one vector multiplet and the only component for the symmetric tensor $C_{IJK}$ is $C_{122}=1$. The moduli space of vector multiplets is then defined by the algebraic relation $$3 h^1(h^2)^2=1.$$ This allows to parameterize this manifold using the coordinate $\phi$ such that $h^1$ is proportional to $e^{\sqrt{\frac{1}{3}}\phi}$ and $h^2$ to $e^{-\phi/2\sqrt 3}$. The induced metric $g_{\phi\phi}$ can be seen to be one. The most general superpotential is a linear combination of the two exponentials $U=ae^{\sqrt{\frac{1}{3}}\phi}+be^{-\phi/2\sqrt 3}$. In the following we will focus on models where the superpotential $U$ can be expressed as an exponential of the normalised scalar field $\phi$ $$\label{potential}
U=4k e^{\alpha \phi}.$$ The values $\alpha =1/\sqrt 3,-1/\sqrt {12}$ correspond to the previous example. The metric in the bulk depends on the scale factor $$\label{scale}
a(z)=(1-4k\alpha^2z)^{\frac{1}{4\alpha^2}},$$ while the scalar field solution is $$\label{psi}
\phi = -\frac{1}{\alpha}\ln\left(1-4k\alpha^2z\right).$$ In the $\alpha\to 0$ we retrieve the AdS profile $$a(z)=e^{-kz}.$$ corresponding to the supersymmetric Randall-Sundrum model with no vector multiplet in the bulk.
The zero–modes
--------------
The BPS configurations have zero modes solving the linearised equations of motion together with the boundary conditions at the branes. The linearised Einstein and Klein–Gordon equations have been thoroughly studied in [@chris1] following an earlier work [@christos] in the Randall–Sundrum case. The end result is that there are two scalar modes and one spin two mode. They correspond to either the radion and the zero mode of the bulk scalar field or the two brane positions. The spin two zero mode, i.e. the graviton, is associated to 4d gravity at low energy.
The number of moduli can be inferred by a counting argument based on supersymmetry. At low energy, the only imprint of the two bulk vector fields $A_a^I,\ I=1,2,$ are two pseudo–scalar fields $A_5^I$ (the vector fields are projected out by the $Z_2$ symmetry). These two axion–like fields are combined with the radion and the bulk scalar field to form the scalar fields comprising two chiral superfields. Already in the Randall–Sundrum case, the radion can be seen as the fluctuation of the $g_{55}$ component of the bulk metric and is associated to the zero mode of the gravi–photon $A_5$. This remains true and is complemented by the association of the bulk scalar field zero mode of the vector multiplet to the corresponding axion–like field in the vector multiplet.
The two scalar zero modes can be viewed as the two brane positions $$t_1=\xi_1(x),\ t_2= r+\xi_2(x)$$ representing the massless fluctuations with respect to fixed branes at $0$ and $r$ $$\Box^{(4)}\xi_{1,2}=0$$ where the bulk metric is unperturbed.
Equivalently the two branes can be considered as fixed and the metric is perturbed $$ds^2= a^2(G(x,z))g_{\mu\nu}dx^\mu dx^\nu + (\partial_z G)^2 dz^2$$ where $g_{\mu\nu}= \eta_{\mu\nu} +h_{\mu\nu}$ is the perturbed 4d metric and $$G(x,z)= z +\frac{\xi(x)}{a^2(z)} +\xi_0(x)$$ where $$\Box^{(4)} \xi= 0, \ \Box^{(4)} \xi_0= 0$$ and $$\xi_1=\frac{\xi}{a_1^2}+\xi_0,\ \xi_2=\frac{\xi}{a_2^2}+\xi_0$$ The radion is related to $\xi$ as $$t(x)=(\frac{1}{a_2^2}-\frac{1}{a_1^2})\xi(x)$$ where $a_{1,2}$ are the scale factors of the first and second branes. The mode $\xi_0$ is associated to a global translation of the bulk–brane system. Similarly the scalar field is perturbed as $$\phi(x,z)=\phi_{BPS}(G(x,z))$$ No intrinsic zero–mode is associated to the scalar field.
To linear order, this parametrisation is equivalent to $$\delta g_{55}= -2 \frac{a'}{a^3} \xi$$ confirming the link between $\delta g_{55}$ and the radion. Similarly the perturbed scalar field is $$\frac{\delta \phi}{\phi'_{BPS}}= 2\frac{\delta g_{55}}{U}
+\frac{\partial U}{\partial \phi}\xi_0$$ picking contributions from $\delta g_{55}$ and $\xi_0$.
Finally, one can also use a parametrisation generalising the one of [@effective_detuned] $$ds^2= A^2(x,z) \Omega^2 (x,z) g_{\mu\nu} dx^\mu dx^\nu + \frac{(
a^2+ 2aa' \xi_0)^2}{A^4}dz^2$$ where $$A^2= a^2 + 2\frac{a'}{a}\xi +2 aa' \xi_0$$ and $$\Omega^2= (1- \frac{a_1^2-a_2^2}{\int_0^r a^2 dz} \xi_0)^{-1}$$ The dimensional reduction using this ansatz leads to an action in the Einstein frame for the two moduli $\xi_0$ and $\xi$.
In the following, we will work exclusively with the two moduli $t_{1,2}$ as the parametrisation is simpler.
The Low Energy Action
=====================
The vector multiplet sector
---------------------------
At low energy the brane and bulk system is amenable to a four–dimensional treatment where the dynamics are captured by the slow motion of moduli fields. Two of the moduli of the system are the brane positions as they are not specified by the equations of motions. At low energy, one considers small deformations of the static configuration allowing the moduli to be space–time dependent. We denote the position of brane 1 by $t_1(x^\mu)$ and the position of brane 2 by $t_2(x^\mu)$. We consider the case where the evolution of the brane is slow. This means that in constructing the effective four–dimensional theory we neglect terms of order higher than two in a derivative expansion. Moreover the non-linearity of the Einstein equations on the branes in the matter energy–momentum tensors [@project] are neglected. Such a regime is only valid at low energy well below the brane tensions of both branes. For instance, putting the standard model matter on the positive tension brane leads to an effective action valid only up to the positive brane tension. In addition to the brane positions, we need to include the graviton zero mode, which can be done by replacing $\eta_{\mu\nu}$ with a space–time dependent tensor $g_{\mu\nu}(x^\mu)$.
After integrating over the fifth dimension, one obtains a 4d effective action. The Einstein-Hilbert term in 4d follows from the 5d term and reads $$S_{\rm bulk} = \int d^4 x \sqrt{-g_4} f(t_1,t_2) {\cal R}^{(4)},$$ with $$f(t_1,t_2) = \frac{1}{\kappa_5^2} \int^{t_2}_{t_1} dz a^2 (z)$$ For the exponential superpotential $U$, this is $$f(t_1,t_2)=
\frac{a(t_1)^{2+4\alpha^2}-a(t_2)^{2+4\alpha^2}}{(2+4\alpha^2)k\kappa_5^2}.$$ Notice that the action is in the brane frame different from the Einstein frame. Including the boundary terms leads to the following effective action [@cosmoduli]
$$\begin{aligned}
S_{} = \int d^4 x \sqrt{-g_4}\left[ f(t_1,t_2) {\cal R}^{(4)} +
\frac{3}{4}a^2(t_1)\frac{U(t_1)}{\kappa_5^2}(\partial t_1)^2 -
\frac{3}{4} a^2(t_2)\frac{U(t_2)}{\kappa_5^2}(\partial t_2)^2
\right].\end{aligned}$$
As expected the moduli are free scalar fields.
The effective action for the two moduli $t_1$ and $t_2$ is written in an explicit supergravity form. This follows from the fact that the two-brane system satisfies BPS conditions. At low energy the bulk and brane system preserves one of the original supersymmetries. Indeed one can write the Einstein-Hilbert term and the kinetic terms of the moduli as
$$\int d^4 x \sqrt{-g_4}\left[ f(t_1,t_2) {\cal R}^{(4)}
+ 6 \partial_{T_1}\partial_{\bar T_1}f\partial_\mu T_1\partial^\mu
\bar T_1 + 6\partial_{T_2}\partial_{\bar T_2}f\partial_\mu T_2
\partial^\mu \bar T_2\right ],$$
where $$t_1=\frac{1}{2}(T_1+\bar T_1),\ t_2=\frac{1}{2}(T_2+\bar T_2).$$ This allows us to identify the fields $T_1$ and $T_2$ as the scalar parts of two chiral multiplets whose dynamics are captured by the function $f(t_1,t_2)$ $$\int d^4 x d^4 \theta E^{-1} f(\frac{T_1+\bar T_1}{2}\
,\frac{T_2+\bar T_2}{2})$$ where $E$ is the vielbein determinant superfield. From this action one can read off the Kähler potential for the two moduli fields in the Einstein frame $$K=-3\ln (\kappa_4^2 f).$$ which depends on the two moduli. Notice that the Kähler potential possesses two global symmetries $$T_i\to T_i +ib_i .$$ coming from the independence of the background geometry on the axion fields. A detailed analysis of the Kähler geometry with an arbitrary number of vector multiplets is under study [@kahler].
Let us now concentrate on the Randall–Sundrum case $\alpha=0$, the Kähler potential reads $$K=-3\ln ( e^{-k(T_1+\bar T_1)}-e^{-k(T_2+\bar T_2)}).$$ The Kähler potential can be written as $$K=3k(T_1+\bar T_1)-3\ln(1-e^{-k(T+\bar T)}),$$ where $$T=T_2- T_1$$ is the radion superfield. In the Randall–Sundrum case, the field $T_1$ can be eliminated by a Kähler transformation; this shows that one of the two moduli decouples, leaving only the radion as the relevant physical field.
The hypermultiplet sector
-------------------------
We can now introduce another ingredient, i.e. a hypermultiplet living in the bulk. We will focus on the universal hypermultiplet comprising four scalar fields, two being odd under the orbifold parity [@flp2]. We also assume that the hypermultiplet is not charged under the gauged $U(1)_R$ symmetry in such a way that no contribution from the hypermultiplet appears in the bulk potential. At low energy, the two even scalar fields become the scalar part of a chiral multiplet $S$ which will be called the dilaton in the following. The low energy dynamics of the dilaton is determined by the Kähler potential $$K(S,\bar S)=- \ln (S+\bar S)$$ Notice that the full moduli Kähler potential is then just the sum of the vector multiplet and hypermultiplet contributions.
The coupling to matter \[section coupling\]
-------------------------------------------
Let us now introduce matter on the boundary branes. We couple the matter fields to the induced metric on the ith–brane leading to an action for the matter scalar field $s$ coupled to the brane position $t_i$ $$\int d^4 x \sqrt{-g} \Big( a^2(t) (\partial s\partial \bar s)+ a^4(t)
\vert\frac{\partial w(s)}{\partial s}\vert ^2 \Big)$$ up to derivative terms in $t_i$. We have denoted by $w$ the superpotential of the supersymmetric theory on the brane. As we are supersymmetrising the matter action only at zeroth order in $\kappa_4$, we have suppressed the non-renormalizable terms in the matter fields for fixed moduli, hence the globally supersymmetric form of the potential. Such an action can be supersymmetrised (we follow the conventions of [@weinberg] for the definitions of the Kähler potentials) $$-3\int d^4 x d^4 \theta E^{-1} \Big( f- a^2(\frac{T_i +\bar
T_i}{2})\Sigma \bar \Sigma\Big)$$ modifying the Kähler potential of the moduli $$K=-3\ln (\kappa_4^2 f - \frac{1}{3}\kappa_4^2 a^2(\frac{T_i+\bar
T_i}{2})\Sigma \bar \Sigma) \label{kha}$$ where $\Sigma =s+\dots $ is the chiral superfield of matter on the brane (not to be confused with the hypermultiplet superfield S). Similarly the potential on the brane follows from $$\int d^4x d^2\theta \Phi^3 W(T_i,S,\Sigma)$$ where $$W(T_i,S)= a^3(T_i)w( S,\Sigma)$$ and $\Phi$ is the chiral compensator whose $F$–term is the gravitational auxiliary field. At low energy this leads to a direct coupling between matter fields and the moduli. This is crucial when discussing supersymmetry breaking. Moreover the coupling to the brane breaks the global symmetries $T_i\to T_i +
ib_i$. This allows to break supersymmetry using the imaginary parts (axions) of the moduli fields. In the Randall–Sundrum case, the symmetry is extended to $SL(2,\mathbb{R})$ which is not broken by the boundary superpotential [@effective_sugra]. This leads to the absence of any axionic dependence in the scalar potential [@effective_detuned].
When including matter fields on several branes, the superpotential is simply a sum of all the contributions coming from the different branes. The Kähler potential is obtained by summing the different contributions from the branes inside the logarithmic term.\
Let us now turn to the gauge sectors. We assume that each brane carries gauge fields $A_\mu^{1,2}$ associated to the gauge groups $G_{1,2}$. As the gauge kinetic terms are conformally invariant we find that the gauge coupling constant does not acquire a moduli dependence. However, we need a non constant gauge coupling to generate gaugino masses. For example, one can make it explicitly dependent on the brane value of the bulk scalar field $\phi(T_i)=
-4\alpha \ln(a(T_i))$ in the 5d action. Contrarily to the brane localized potential of the bulk scalar field (the tension), which is related to its bulk potential, this dependence is not constrained by local brane-bulk supersymmetry and is thus arbitrary. In that case however, by 5d locality the gauge coupling function on brane $i$ can only depend (analytically) on the $T_i$ modulus (and the dilaton S). In the absence of more information on the coupling of the bulk scalar field to the brane gauge kinetic terms, we will allow for a general coupling to the moduli fields $$\int d^4 x d^2 \theta (f_1(T_1,S) {\mathcal W}_{1\alpha} {\mathcal
W}^\alpha_1 +f_2(T_2,S) {\mathcal W}_{2\alpha} {\mathcal
W}^\alpha_2)$$ Another possibility is to consider the anomalous breaking of the (super-)conformal invariance of the gauge kinetic term [@anomaly]. First, conformal invariance of the action before gauge fixing the conformal compensator superfield $\Phi$ to $1+\theta^2 F_{\Phi}$ implies that $\Phi$ must multiply any cut-off dependence appearing during the renormalisation process. Furthermore, after the gauge-fixing of $\Phi$, a R-symmetry transformation of the action results in an anomalous shift of the gauge $\theta$-angle given by the imaginary part of the gauge coupling. Compensation of this shift by a R-symmetry phase rotation of $\Phi$ before gauge-fixing dictates the $\Phi$ dependence for the gauge coupling function $f_a$ evaluated at the scale $E$ on either of the branes $$f_a(E)=f_a^0 + \frac{b}{2\pi} \ln(\frac{E}{\Lambda_{UV}(T_1,T_2)
\Phi})$$ where the effective cut-off $\Lambda_{UV}$ depends on the moduli $T_i$ in the Einstein frame. Anomaly mediation contributes to the gaugino masses both through $F_{\Phi}$ and $F^{T_i}$. We present this possibility in section \[AMSB\].
Supersymmetry Breaking at Tree Level
=====================================
F–type supersymmetry breaking
-----------------------------
We will consider the case of a single vector multiplet in the bulk and its coupling to the two boundary branes carrying matter superfields $\Sigma_{1,2}^i$. The case including the universal hypermultiplet will be dealt with later. The low energy theory depends on the Kähler potential and the superpotential. The Kähler potential includes both the matter fields and moduli
$$\begin{aligned}
K &=& -3\ln \Big[\kappa_4^2 f(\frac{T_1+ \bar
T_1}{2},\frac{T_2+\bar T_2}{2}) - \frac{1}{3}\kappa_4^2 a^2(\frac{T_1+\bar
T_1}{2}) (\Sigma_1^i \bar \Sigma_1^{\bar i}+
\frac{\lambda^1_{ij}\Sigma_1^i\Sigma_1^j+ cc}{2})
\nonumber \\
&& - \frac{1}{3}\kappa_4^2 a^2(\frac{T_2+\bar T_2}{2})(\Sigma_2^i \bar
\Sigma_2^{\bar i}+ \frac{\lambda^2_{ij}\Sigma_2^i\Sigma_2^j+
cc}{2})\Big] \label{kha_with_GM}\end{aligned}$$
We consider diagonal kinetic terms and Giudice-Masiero mixing terms [@GM] $\lambda^{1,2}_{ij}$ for phenomenological purpose, i.e in order to address the so–called $\mu$ problem. We expand the superpotential $$W=a(T_1)^3 w(\Sigma_1^i)+a(T_2)^3w(\Sigma_2^i)$$ Notice that the two sectors on the branes are decoupled in the superpotential. The matter superpotentials are $$w(\Sigma_{1,2}^i)= w_{1,2} + \frac{1}{2}\mu_{ij}^{1,2} \Sigma_{1,2}^i \Sigma_{1,2}^j
+\frac{1}{6}\lambda_{ijk}^{1,2} \Sigma_{1,2}^i \Sigma_{1,2}^j\Sigma_{1,2}^k$$ where the constant pieces $w_{1,2}$ give a negative contribution to the brane cosmological constants, i.e. the brane tensions. Incorporating constant terms in the superpotentials of each brane, one obtains a brane configuration with detuned tensions. The detuning of the brane tensions is responsible for supersymmetry breaking when the F–terms of $T_1$ or $T_2$ are non-vanishing.
Using the previous ingredients one can work out the soft supersymmetry breaking terms in the Einstein frame. In the following we will focus on the exponential coupling $U=4k
e^{\alpha \phi}$. We identify the inverse squared Planck mass as $$\kappa_4^2= 2k (1+2\alpha^2) \kappa_5^2$$ The gravitino mass is given by $m_{3/2}=\kappa_4^2 <e^{K/2}\vert
W\vert >$ where the non-vanishing vev is provided by the constant terms in the superpotentials. Moreover, the gravitino mass becomes a function of the moduli $$m_{3/2}= \kappa_4^2 \Delta^{-3/2} \vert a(T_1)^3 w_1 +a(T_2)^3
w_2\vert$$ where $\Delta=a(t_1)^{2+4\alpha^2}-a(t_2)^{2+4\alpha^2}$. In the Randall-Sundrum case with $\alpha=0$, this reduces to $$m_{3/2}= \kappa_4^2 \frac{\vert w_1 +e^{-3kT}
w_2\vert}{(1-e^{-2kt})^{3/2}}$$ It is interesting to compute the F–terms associated to the breaking of supersymmetry. The non–vanishing F–terms associated to the two moduli are
$$\begin{aligned}
F^{T_1}&=& <\frac{W}{|W|}>\Big[2 \kappa_5^2\Delta^{-1/2}a(\bar
T_1)^{3-4\alpha^2} a(t_1)^{-2+4\alpha^2}\bar w_1
\nonumber \\
&&+ 4 (1+2\alpha^2)\Delta^{-3/2}
k \kappa_5^2 \ i\ {\cal I}m T_1\ a(\bar T_1)^{3-4\alpha^2}
a(t_1)^{4\alpha^2}\bar w_1
\nonumber \\
&&+ 4 (1+2\alpha^2)\Delta^{-3/2} k \kappa_5^2 \ i\ {\cal I}m T_2\
a(\bar T_2)^{3-4\alpha^2} a(t_1)^{4\alpha^2}\bar w_2\Big]
\nonumber \\
F^{T_2}&=& <\frac{W}{|W|}>\Big[-2 \kappa_5^2\Delta^{-1/2}a(\bar
T_2)^{3-4\alpha^2} a(t_2)^{-2+4\alpha^2}\bar w_2
\nonumber \\
&&+ 4 (1+2\alpha^2)\Delta^{-3/2}
k \kappa_5^2 \ i\ {\cal I}m T_2\ a(\bar T_2)^{3-4\alpha^2}
a(t_2)^{4\alpha^2}\bar w_2
\nonumber \\
&&+ 4 (1+2\alpha^2)\Delta^{-3/2} k \kappa_5^2 \ i\ {\cal I}m T_1\
a(\bar T_1)^{3-4\alpha^2} a(t_2)^{4\alpha^2}\bar w_1\Big]
\nonumber \\\end{aligned}$$
where we have defined F-terms using the following phase convention : $$F^i \equiv e^{K/2}|W| G^{i \bar j}G_{\bar j} =
\frac{W}{|W|}e^{K/2}K^{i \bar j}(\overline{W}_{\bar j} +
\kappa_4^2 \overline{W} K_{\bar j}).$$ As the moduli are of length dimension one, the F–terms are dimension–less. In the case of vanishing imaginary parts of the moduli $T_i$, they simplify to $$\begin{aligned}
F^{T_1}&=& 2 \kappa_5^2\Delta^{-1/2}a(t_1) \bar w_1 <\frac{W}{|W|}> \nonumber \\
F^{T_2}&=& -2 \kappa_5^2\Delta^{-1/2} a(t_2) \bar w_2 <\frac{W}{|W|}> \nonumber \\\end{aligned}$$ Notice that in this case, each modulus breaks supersymmetry when the constant superpotential on the corresponding brane is non–zero. The soft supersymmetry breaking terms are a direct consequence of the detuning of the brane tensions. Note that the dependence of supersymmetry breaking on the vev of the radion imaginary part has also been studied in [@wilson] in the case of the detuned Randall-Sundrum model.
The moduli potential \[potential section\]
------------------------------------------
The breaking of supersymmetry leads to a potential for the moduli fields given by $V=\kappa_4^{-4}e^{G}(G_iG^i -3)$ where $G= K +\ln
\kappa_4^6 \vert W\vert^2$. This gives the potential
$$\begin{aligned}
V&=&\frac{3\kappa_4^2}{1+2\alpha^2}\Delta^{-2}\Big[ \vert
a(T_2)^{3-4\alpha^2} w_2\vert^2 a(t_2)^{-2+4\alpha^2}-\vert
a(T_1)^{3-4\alpha^2} w_1\vert^2 a(t_1)^{-2+4\alpha^2}\Big]
\nonumber \\
&&+24\alpha^2 k^2 \kappa_4^2 \Delta^{-3}\vert a(T_1)^{3-4\alpha^2}
w_1 {\cal I}m T_1 +a(T_2)^{3-4\alpha^2} w_2 {\cal I}m T_2\vert^2 .
\label{pot with im non zero}\end{aligned}$$
We have distinguished the real terms $a(t_{1,2})$ from the complex terms $a(T_{1,2})$ which depend on the axion–like fields. Notice that as soon as one of the $w_i$ is vanishing, the corresponding ${\cal I}m T_i$ becomes a flat direction in agreement with the restoration of the global symmetry $T_i \to T_i
+ib_i$.
Let us first consider the Randall-Sundrum case $\alpha=0$. The potential does not depend on the axion–like fields at all $$V= {3\kappa_4^2}\frac{|w_2|^2 \rho^{4}-|w_1|^2}{(1-\rho^{2})^2} .$$ We have defined $\rho =a(t_2)/a(t_1)$. Hence the axion–like field ${\cal I}m T$ is a flat direction while the radion flat direction is lifted.
Notice that for $\alpha \ne 0$ the flat directions for the axion–like fields ${\cal I} m T_i$ are lifted. One can show that $$({\cal I} m T_1,{\cal I} m T_2)=(0,0)$$ is an extremum of the potential. In the scenario of hidden brane supersymmetry breaking ($w_1=0$, $w_2\neq 0$) that we will consider later, this extremum is stable in ${\cal I} m T_2$, while ${\cal I} m T_1$ becomes a flat direction of the potential as already mentioned.
The potential becomes then $$V=
\frac{3\kappa_4^2}{1+2\alpha^2}a(t_1)^{-12\alpha^2}\frac{|w_2|^2
\rho^{4-4\alpha^2}-|w_1|^2}{(1-\rho^{2+4\alpha^2})^2} .
\label{pot}$$ This is equivalent to the potential obtained by detuning the brane tensions $$\lambda_{\pm}=\pm \delta _{1,2} \frac{3}{2\kappa_5^2} U .$$ where the detuning parameter $\delta_{1,2}$ is less than one $$\delta_{1,2}= 1-\kappa_5^2 |w_{1,2}|^2 .$$ Notice that the brane tension in supergravity is always less than the tuned tension with no supersymmetry breaking [@Brax:2001xf; @susyRS_arbitrary_tensions].
Several cases may be distinguished, which are summarized in figures \[figure potentiels 1\], \[figure potentiels 3\] and \[figure potentiels 6\] for $Q=0$. Note that only one half of the different profiles is actually possible for a vanishing $Q$ parameter (whose significance as a brane charge will be explained in section \[charged branes\]). When $w_1\ne 0$ and $|w_2|\leq |w_1|$ , the theory is unstable, with an unbounded from below potential in the limit $\rho \to 1$, i.e. for nearly colliding branes. This corresponds to figure \[figure potentiels 3\]. For $w_1\ne 0$ and $|w_2|>|w_1|$, the potential has a global minimum but at a negative energy ; this is figure \[figure potentiels 1\]. Finally, the case $w_1=0$ is phenomenologically more interesting as the potential is positive and bounded from below ; this is figure \[figure potentiels 6\]. All the other figures correspond to $Q\ne 0$ and will be considered later in section \[charged branes\].
![The moduli potential as a function of $\rho=a(t_2)/a(t_1)$ for ${\mathcal I}m T_{1,2}=0$ and $ \vert w_2\vert > \vert w_1 \vert > \vert Q\vert$. The radion may be stabilised with a negative vacuum energy.[]{data-label="figure potentiels 1"}](potradion11.eps){height="8cm"}
![The moduli potential as a function of $\rho=a(t_2)/a(t_1)$ for ${\mathcal I}m T_{1,2}=0$ and $ \vert w_2\vert < \vert w_1 \vert < \vert Q\vert$. The radion has a metastable minimum for an infinite interbrane distance, and an unbounded from below branch leading to colliding branes.[]{data-label="figure potentiels 2"}](potradion12.eps){height="8cm"}
![The moduli potential as a function of $\rho=a(t_2)/a(t_1)$ for ${\mathcal I}m T_{1,2}=0$, $ \vert w_2\vert \le \vert w_1 \vert $ and $\vert w_1\vert >
\vert Q\vert$. The radion rolls down leading to colliding branes with an infinite negative energy. []{data-label="figure potentiels 3"}](potradion21.eps){height="8cm"}
![The moduli potential as a function of $\rho=a(t_2)/a(t_1)$ for ${\mathcal I}m T_{1,2}=0$, $ \vert
w_2\vert \ge \vert w_1 \vert$ and $\vert w_1 \vert < \vert
Q\vert$. The radion is attracted towards a stable minimum with an infinite interbrane distance, whose positive energy can be fine-tuned to match the observed vacuum energy.[]{data-label="figure potentiels 4"}](potradion22.eps){height="8cm"}
![The moduli potential as function of $\rho=a(t_2)/a(t_1)$ for ${\mathcal I}m T_{1,2}=0$ and $ \vert w_2\vert <
\vert w_1 \vert = \vert Q\vert$. The radion rolls down towards colliding branes with an infinite negative energy.[]{data-label="figure potentiels 5"}](potradion31.eps){height="8cm"}
![The moduli potential as function of $\rho=a(t_2)/a(t_1)$ for ${\mathcal I}m T_{1,2}=0$ and $ \vert w_2\vert > \vert w_1 \vert = \vert Q\vert$. There is a minimum with zero energy for an infinite interbrane distance.[]{data-label="figure potentiels 6"}](potradion32.eps){height="8cm"}
Soft terms \[soft terms section\]
---------------------------------
Let us now discuss the soft terms for general $w_i$. We restrict ourselves to the situation where matter is confined on the *positive tension brane*. This will allow us to evade some of the gravitational constraints due to the presence of very light moduli. We write the soft Lagrangian for the *canonically normalised* complex matter scalars $\tilde{s}_i$ as
$${\cal L}_{soft}= -m^2_{i\bar j} \tilde{s}_i \overline{\tilde{s}_j} - (\frac{1}{2}B_{ij}\tilde{s}_i \tilde{s}_j + hc)
- (\frac{1}{6} A_{ijk}\tilde{s}_i \tilde{s}_j \tilde{s}_k + hc)$$
Note that non vanishing soft terms do not necessarily imply supersymmetry breaking (defined by non zero F–terms) when the vacuum energy cannot be neglected, as can be seen from the general expression of the soft masses (\[m2\]) and B–terms (\[b\]) ; we will however keep the denomination of soft breaking terms for simplicity.
Using the usual formulae for the soft breaking terms [@soft], the un-normalised A–terms read
$$\begin{aligned}
(A_{ijk})^{un.}&=&\frac{\bar W}{|W|}e^{K/2}F^{T_M}\Big[ \partial_{T_M} \lambda_{ijk}(T_p)
+K_{T_M} \lambda_{ijk}(T_p) -\Big((\partial_{T_M}K_{i\bar m}) K^{\bar mn}
\lambda_{njk}(T_p)
\nonumber \\
&&+ (i \leftrightarrow j) + (i \leftrightarrow k)\Big)\Big]
\nonumber \\
\Rightarrow A_{ijk} &=&12 \kappa_4^2 \lambda_{ijk} \alpha^2 (k {\mathcal I}m T_1)\Big( 2 \bar
w_2 a(\bar T_2)^{3-4\alpha^2} a(T_1)^{3-4\alpha^2} a(t_1)^{-6}k
{\mathcal I}m T_2
\nonumber \\
&&-\bar w_1 a(\bar T_1)^{6-4\alpha^2}a(T_1)^{-4\alpha^2} a(t_1)^{-6}
(\frac{\Delta a(t_1)^{-2}}{1+2\alpha^2}+2 i k {\mathcal I}m T_1) \Big) .\end{aligned}$$
In this formula the Yukawa couplings $\lambda_{ijk}(T_p)=a^3(T_p)
\lambda_{ijk}$ are the moduli-dependent ones including the cubed analytic warp factors, while $\lambda_{ijk}$ has no modulus dependence. The notation will be the same for the next terms. Notice that $$A_{ijk}\equiv 0$$ when the imaginary parts ${\cal I}m T_i$ vanish. This result is akin to the vanishing of the $A$ terms in no–scale supergravity [@noscale], defined by the Kähler potential $K=-3 ln
\Big(\frac{\kappa_4^2}{\kappa_5^2}(T+\bar
T)-\frac{1}{3}\kappa_4^2\Sigma \bar \Sigma \Big)$. This can be understood as in the limit of vanishing bulk curvature $k
\rightarrow 0$, the Kähler potential (\[kha\]) takes the no-scale form.
The un-normalised soft masses for the scalars
$$(m^2_{i\bar j})^{un.}= (m_{3/2}^2+\kappa_4^2<V>) K_{i\bar j}
-\overline{F^{T_M}}\Big(\partial_{\bar T_M}\partial_{T_N} K_{i\bar
j} - (\partial_{T_N} K_{i \bar k})K^{\bar k l}(\partial_{\bar T_M}
K_{l \bar j}) \Big)F^{T_N} \label{m2}$$
are diagonal and read
$$\begin{aligned}
m^2_{i \bar j} &=& \frac{2}{3}\kappa_4^2 <V> \delta_{i\bar j}+ 2
\alpha^2\kappa_4^4 \delta_{i\bar j}\Big|\frac{1}{1+2\alpha^2}\Delta^{-1/2}a(\bar
T_1 )^{3-4\alpha^2}a(t_1)^{-2}\bar w_1
\nonumber \\
&&+2\Delta^{-3/2}(k i{\mathcal I}m T_1)a(\bar
T_1)^{3-4\alpha^2}\bar w_1+2\Delta^{-3/2}(k i{\mathcal I}m
T_2)a(\bar T_2)^{3-4\alpha^2}\bar w_2\Big|^2
\nonumber \\\end{aligned}$$
reducing to
$$m^2_{i\bar j}= \delta_{i \bar j}
\frac{2\kappa_4^4}{1+2\alpha^2}a(t_1)^{-12\alpha^2}\frac{|w_2|^2
\rho^{4-4\alpha^2}-|w_1|^2(1-\frac{\alpha^2}{1+2\alpha^2}(1-\rho^{2+4\alpha^2}))}{(1-\rho^{2+4\alpha^2})^2}$$
for the canonically normalised matter fields living on the first brane, and for $Im T_i=0$. The supergravity vacuum energy contribution $<V>$ to the soft masses is given in equation (\[pot with im non zero\]), or (\[pot\]) for vanishing $Im
T_i$.
The general un-normalised effective $\mu$ term on the first brane is given by $$(\mu^{eff}_{ij})^{un.}=\frac{\bar W}{|W|}e^{K/2}\mu_{ij}(T_p) + m_{3/2}\lambda_{ij}(T_p) - \overline{F^{T_M}}
\partial_{\bar T_M}\lambda_{ij}(T_p)$$ where $\lambda_{ij}(T_p)=a^2(t_p) \lambda_{ij}$ are the (moduli-dependent) Giudice-Masiero couplings given in (\[kha\_with\_GM\]), not to be confused with the Yukawa couplings $\lambda_{ijk}$. The normalised effective $\mu$ term is $$\mu^{eff}_{ij}=
<\frac{\overline{W}}{|W|}>\frac{a(T_1)^3}{a(t_1)^2\Delta^{1/2}}\Big(
\mu_{ij} +\lambda_{ij}\frac{\kappa_4^2
w_1a(T_1)^{-4\alpha^2}}{1+2\alpha^2}\Big)$$ reducing to $$\mu_{ij}^{eff}=
<\frac{\overline{W}}{|W|}>\frac{a(t_1)}{\Delta^{1/2}}\Big(
\mu_{ij} +\lambda_{ij}\frac{\kappa_4^2
w_1a(t_1)^{-4\alpha^2}}{1+2\alpha^2}\Big)$$ for vanishing imaginary parts. Note that it does not depend on the hidden brane detuning $w_2$, except non-relevantly through its phase pre–factor.
The general un-normalised $B$ terms are
$$\begin{aligned}
(B_{ij})^{un.}&=&\frac{\bar W}{|W|}e^{K/2}\Big[F^{T_M}\Big(\partial_{T_M}
\mu_{ij}(T_p) + K_{T_M}\mu_{ij}(T_p)-( \mu_{ik}(T_p)K^{k\bar l}\partial_{T_M}K_{j\bar l}
+ i \leftrightarrow j)\Big)
\nonumber \\
&& -m_{3/2}\mu_{ij}(T_p)\Big]+ (2m_{3/2}^2+\kappa_4^2 <V>)\lambda_{ij}(T_p) - m_{3/2}\overline{F^{T_M}}
\partial_{\bar T_M} \lambda_{ij}(T_p)
\nonumber \\
&& + m_{3/2}F^{T_M}\Big[\partial_{T_M}
\lambda_{ij}(T_p)-( \lambda_{ik}(T_p)K^{k\bar l}\partial_{T_M}K_{j\bar l}
+ i \leftrightarrow j)\Big]
\nonumber \\
&& - \overline{F^{T_M}}F^{T_N}\Big[\partial_{\bar T_M}
\partial_{T_N} \lambda_{ij}(T_p) -(K^{\bar k
l}\partial_{T_N}K_{i\bar k} \partial_{\bar T_M} \lambda_{lj}(T_p)
+ i \leftrightarrow j)\Big] \label{b}\end{aligned}$$
In our case the normalised $B$ terms become
$$\begin{aligned}
B_{ij}&=&
-\kappa_4^2(\mu^{eff}_{ij})_{\lambda=0}<\frac{W}{|W|}>\Big[\bar
w_1 \frac{a(\bar T_1)^{3-4\alpha^2}}{(1+2\alpha^2)a(t_1)^2
\Delta^{1/2}}
+\bar w_1 \frac{a(\bar T_1)^{3-4\alpha^2}}{a(t_1)^{4\alpha^2}
\Delta^{3/2}}(4 k \alpha^2 i {\mathcal I}m
T_1)\Big(\frac{3}{2\alpha^2}(4k\alpha^2 i{\mathcal I}m T_1)
\nonumber \\
&&+\frac{3}{1+2\alpha^2}(a(t_1)^{4\alpha^2}+a(T_1)^{4\alpha^2}+\Delta
a(t_1)^{-2} \Big)
+\bar w_2 \frac{a(\bar T_2)^{3-4\alpha^2}}{a(t_1)^{4\alpha^2}
\Delta^{3/2}}\frac{3}{2\alpha^2}(4k\alpha^2 i{\mathcal I}m
T_1)(4k\alpha^2 i{\mathcal I}m T_2)\Big]
\nonumber \\
&&+2\lambda_{ij}
\Delta^{-3}|w_2|^2 |a(T_2)|^{6-8\alpha^2}\Big[12\alpha^2 k^2({\cal I}m T_2)^2
+\frac{1}{1+2\alpha^2}a(t_2)^{-2+4\alpha^2}\Delta\Big]
\nonumber \\
&&+\lambda_{ij}[w_1\neq 0 \rm\ terms] .\end{aligned}$$
For ${\mathcal I}m(T_i)=0$ we obtain
$$\begin{aligned}
B_{ij} &=& -\Delta^{-2} a(t_1)\mu_{ij} \Big[ \kappa_4^2\bar w_1
a(t_1)^3\frac{1-\rho^{2+4\alpha^2}}{1+2\alpha^2}\Big]+ \lambda_{ij}
\kappa_4^2<V>
\nonumber \\
&&+ 2\lambda_{ij} \kappa_4^4 \Delta^{-3}{\cal R}e\Big(w_1
<\overline W>\Big)a(t_1)^3\frac{1-\rho^{2+4\alpha^2}}{1+2\alpha^2}
\nonumber \\
&& - \frac{\lambda_{ij}}{1+2\alpha^2} \kappa_4^4 \Delta^{-2}
a(t_2)^{2+4\alpha^2} |w_1 a(t_1)^{1-4\alpha^2}+w_2
a(t_2)^{1-4\alpha^2}|^2 .\end{aligned}$$
We will give a simplified expression when discussing the phenomenology of these models. The effective Yukawa couplings are given by $$\lambda_{ijk}^{eff}=<\frac{\overline{W}}{|W|}>\frac{a(T_1)^3}{a(t_1)^3}\lambda_{ijk}$$ reducing to $$\lambda_{ijk}^{eff} =<\frac{\overline{W}}{|W|}>\lambda_{ijk} .$$ when the imaginary parts vanish.
Finally the classical gaugino masses for gauge fields living on the first brane are $$m_a=-\frac{k}{2} \sum_{i=1}^2 \beta_a^i a(t_i)^{-4\alpha^2}
F^{T_i} \label{gaugino masses}$$ where we have defined $\beta_a^i=\frac{\partial \ln f_a}{\partial
\ln a(T_i) }$ depending on the gauge coupling function $f_a$ for each gauge group $G_a$. As discussed in subsection \[soft terms section\], the case $w_1=0$ (positive potential) will be the one relevant phenomenologically. Considering that classically, five dimensional locality implies $\beta^2_a=0$ (as discussed in subsection \[section coupling\]), we find that $$m_a=0$$ as long as ${\cal I}m T_i=0$. There are different possibilities to generate non zero gaugino masses. For example one may add a new source of supersymmetry breaking due to a new field, like in the next section ; later in section \[AMSB\] we will also consider the contribution of anomaly mediated supersymmetry breaking to the gaugino masses. In the following section we will explore the possibilities opened up by the presence of a dilaton in order to generalise our construction.
Supersymmetry Breaking and the Dilaton
======================================
The moduli potential
--------------------
So far we have concentrated on the case where only one vector multiplet is present in the bulk. In this section we will focus on the case where both a vector multiplet and the universal hypermultiplet are present. The nature of the scalar potential in that case changes drastically. It becomes very intricate to study the stability of the possible extrema.
We will consider the moduli Kähler potential $$K=-\ln(S+\bar S) - 3 \ln\Big(a(\frac{T_1+\bar
T_1}{2})^{2+4\alpha^2}-a(\frac{T_2+\bar
T_2}{2})^{2+4\alpha^2}\Big)$$ and an arbitrary superpotential first $$W=W(T_1,T_2,S)$$ where we have suppressed the dependence on the matter fields. The scalar potential reads now
$$\begin{aligned}
V&=&\frac{|(S+\bar S)W_S - W|^2}{(S+\bar
S)\Delta^3}+\frac{|W_{T_2}|^2 a(t_2)^{-2+4\alpha^2}-|W_{T_1}|^2
a(t_1)^{-2+4\alpha^2}}{3(1+2\alpha^2)k^2(S+\bar S)\Delta^2}
\nonumber \\
&&+
\frac{|W_{T_1}a(t_1)^{4\alpha^2}+W_{T_2}a(t_2)^{4\alpha^2}+3kW|^2}{6\alpha^2k^2(S+\bar
S)\Delta^3} .\end{aligned}$$
Notice the similarity with case when no dilaton is present. Choosing the superpotential to be of the form $$W=w_1(S)a(T_1)^3+w_2(S)a(T_2)^3$$ where the two functions $w_i(S)$ replace the constant superpotentials $w_i$ of the previous section, we find that
$$\begin{aligned}
V&=&\frac{|(S+\bar S)W_S - W|^2}{(S+\bar S)\Delta^3}
\nonumber \\
&&+\frac{3\kappa_4^2}{(1+2\alpha^2)(S+\bar S)\Delta^2}\Big[ \vert
a(T_2)^{3-4\alpha^2} w_2(S)\vert^2 a(t_2)^{-2+4\alpha^2}-\vert
a(T_1)^{3-4\alpha^2} w_1(S)\vert^2 a(t_1)^{-2+4\alpha^2}\Big]
\nonumber \\
&&+\frac{24\alpha^2 k^2 \kappa_4^2}{(S+\bar S)\Delta^3}\vert
a(T_1)^{3-4\alpha^2} w_1(S) {\cal I}m T_1 +a(T_2)^{3-4\alpha^2}
w_2(S) {\cal I}m T_2\vert^2.\end{aligned}$$
The structure of the potential when $w_i(S)$ are not constant is very difficult to analyse. In the following we will study the case where only one of the terms $w_2(S)$ is present.
Race–track models
------------------
Let us assume that strong coupling effects lead to a potential for the dilaton on the second brane. Including matter fields the superpotential reads now $$W=a(T_1)^3(\frac{1}{2}\mu \Sigma^2 + \frac{1}{6}y\Sigma^3)
+a(T_2)^3 w_2 h(S) \label{racetrack W}$$ where $h(S)$ results from strong gauge coupling effects. In that case the scalar potential for the moduli reads $$V = \kappa_4^{-4} e^{\hat G}\frac{|(S+\bar S)h'(S)-h(S)|^2}{S+\bar
S} +\frac{|h(S)|^2}{S+\bar S}\hat V .$$ where hatted quantities refer to the case with no dilaton. This potential admits extrema in $S$ satisfying
$$\Big((S+\bar S)\overline{h'(S)}-\overline{h(S)}\Big) h''(S)(S+\bar
S)^2 =-\Big((S+\bar S)h'(S)-h(S)\Big)\overline{h(S)}(1+\kappa_4^4
e^{-\hat G} \hat V) .$$
There are two types of extrema. First of all when $$(S+\bar S)h'(S)=h(S)$$ we find that $$F^S=0$$ Notice that $S$ is determined independently of the other moduli. In that case the potential reduces to $$V=\frac{|h(S)|^2}{(S+\bar S)}\hat V$$ at the extremum. The other extrema satisfy the necessary condition $$|\frac{h''(S)}{h(S)}|(S+\bar S)^2 = |1+\kappa_4^4 e^{-\hat G}\hat
V| .$$ Let us concentrate on the gaugino condensation case (see [@gaugino; @condensate] for a recent review) where $$h(S)=\exp(-\frac{3}{2b_0}S) \label{racetrack W 2}$$ The potential has then either no extremum or a maximum, i.e. the dilaton has a run–away potential, see figures 7 and 8.
{height="8cm"}
{height="8cm"}
Another relevant case is the race–track superpotential on the second brane where $$h(S)=\lambda_1 \exp(-b_1 S) + \lambda_2 \exp(-b_2 S)$$ with $b_1,b_2 \geq 0$. This factor originates from the gaugino condensation in two different gauge groups, and is especially motivated in our case as being known to allow for stabilization [@racetrack]. The extrema with $F^S=0$ satisfy $$\begin{aligned}
\Big|\frac{\lambda_2}{\lambda_1}\Big|\exp\Big((b_1-b_2)Re(S)\Big)=\frac{1+2b_1
Re(S)}{1+2b_2 Re(S)}
\nonumber \\
(b_1-b_2)Im(S)= \pi +arg(\frac{\lambda_1}{\lambda_2})\ (mod\ 2\pi)\end{aligned}$$ with a unique solution in Re(S), at least when $|\frac{\lambda_2}{\lambda_1}|=1$. The stability of this configuration depends on the moduli $T_i$ and deserves further study.
The soft terms can be related to the soft terms when no dilaton is present. One can of course evaluate them at the various extrema.
$$\begin{aligned}
y_{eff} &=& (S+\bar S)^{-1/2}\hat y_{eff}
\nonumber \\
\mu_{eff} &=& (S+\bar S)^{-1/2}\hat \mu_{eff}^{no\ GM} +
|h(S)|(S+\bar S)^{-1/2}\hat \mu_{eff}^{GM\ only}
\nonumber \\
m_{3/2} &=& |h(S)|(S+\bar S)^{-1/2} \hat m_{3/2}
\nonumber \\
m^2 &=&|h(S)|^2(S+\bar S)^{-1} \hat m^2 +\kappa_4^{-2}
\frac{|F^S|^2}{(S+\bar S)^2}
\nonumber \\
A &=& |h(S)|(S+\bar S)^{-1} \hat A
\nonumber \\
B &=& |h(S)|(S+\bar S)^{-1} \hat B^{no\ GM} +
\lambda_{GM}\kappa_4^{-2} \frac{|F^S|^2}{(S+\bar
S)^2}+|h(S)|^2(S+\bar S)^{-1} \hat B^{GM\ only}
\nonumber \\
m_a &=& |h(S)|(S+\bar S)^{-1/2}\hat m_a \label{dilaton rescaling}\end{aligned}$$
For most terms, this results in a simple rescaling with S-dependent factors. Note that accordingly the Giudice-Masiero parts of the effective $\mu$-term and the B-term are not rescaled like their ordinary parts. The soft masses and B-terms also have new additive contributions in $|F^S|^2$ from their vacuum energy term.
To conclude the racetrack case, we find that as long as the imaginary parts of the $T_i$ moduli vanish and $S$ sits at the extremum where $F^S=0$, the gauginos are still massless $m_a=0$. In section \[AMSB\] we will see however that the gauginos can pick up a mass via anomaly mediation.
Anomaly Mediated Supersymmetry Breaking \[AMSB\]
================================================
We have seen that the tree level action does not mix the moduli dependence of the gauge coupling functions coming from each brane. This is due to the locality of the coupling to the bulk scalar field in five dimensions. Now there is a quantum conformal anomaly at one loop which leads to a coupling of both moduli to the gauge sectors on each brane. We will follow closely the superfield method of [@anomaly] in order to derive its consequences on the soft breaking terms, especially the gaugino masses and the $A$ terms.
Let us consider the supergravity action written with the chiral compensator formalism. To simplify the discussion we only consider a single matter $\Sigma$ field on the first brane
$$\begin{aligned}
&\int d^4x d^2 \theta d^2 \bar\theta\ \Big[-3 f(t_1,t_2)+a^2(t_1)
\Big(|\Sigma|^2+\frac{1}{2}(\lambda\Sigma^2+\bar \lambda\bar
\Sigma^2)\Big)\Big]|\Phi|^2 &
\nonumber \\
&+\int d^4x d^2\theta\ \Phi^3 a^3(T_1)[\frac{1}{2}\mu \Sigma^2 +
\frac{1}{6}y\Sigma^3] + h.c.& .\end{aligned}$$
This is the action in the brane frame as can be seen from the non-canonical term $-3 f(t_1,t_2)|\Phi|^2$. The coupling to the moduli has been determined in section 2. We have included a Giudice-Masiero term involving the coupling $\lambda$. The $\theta=0$ component of the conformal compensator has not yet been gauge-fixed.
In the following we will factorise real superfields $R$ $$R(\theta,\bar \theta)= R_0 +R_1 \theta^2 +\bar R_1 \bar \theta^2
+R_2 \theta^2 \bar\theta^2$$ as $$R(\theta,\bar\theta)= R_0(1+\frac{R_1}{R_0}\theta^2) (1+\frac{\bar
R_1}{\bar R_0}\bar\theta^2)(1 +D_R\theta^2\bar\theta^2)$$ where $D_R=\frac{R_2}{R_0}-\frac{|R_1|^2}{R_0^2}$. We will denote by $${\cal {R}}=R_0^{1/2}(1+\frac{R_1}{R_0}\theta^2)$$ the chiral part of the factorisation of $R(\theta,\bar\theta)$.
The change to the superspace Einstein frame is realized by the chiral superfield redefinition $$\tilde{\Phi}={\mathcal F} \Phi$$ where ${\cal F}=f^{1/2}|_{\theta=0}(1+\theta^2
\frac{1}{2}\frac{f_{t_i}}{f}F^{T_i})$. The action becomes
$$\begin{aligned}
&\int d^4x d^2 \theta d^2 \bar \theta\
\Big[-3 (1+D_f\theta^2\bar\theta^2)|\tilde\Phi|^2+\frac{|\tilde\Phi|^2}{|{\mathcal
F}|^2}a^2(t_1) \Big(|\Sigma|^2+\frac{1}{2}(\lambda\Sigma^2+\bar
\lambda\bar \Sigma^2)\Big)\Big]&
\nonumber \\
&+\int d^4x d^2\theta\ \frac{\tilde \Phi^3}{{\mathcal F}^3}
a^3(T_1)[\frac{1}{2}\mu \Sigma^2 + \frac{1}{6}y\Sigma^3] + h.c.&\end{aligned}$$
The $D_f$ term contributes to the classical action only. One can now gauge-fix $\tilde{\Phi}|_{\theta=0}=1$ corresponding to the Einstein frame. Let us also factorise the real superfield $a^2(t_1)$.
$$\begin{aligned}
&\int d^4x d^2 \theta d^2 \bar \theta\ \Big[-3|\tilde
\Phi|^2(1+D_f\frac{1}{4}\theta^2\bar\theta^2)] +\frac{|\tilde
\Phi|^2 |{\cal A}|^2}{|{\cal F}|^2}(1+ D_{a^2}\theta^2\bar
\theta^2) \Big(|\Sigma|^2+\frac{1}{2}(\lambda\Sigma^2+\bar
\lambda\bar \Sigma^2)\Big)\Big]&
\nonumber \\
&+\int d^4x d^2\theta\ \frac{\tilde{\Phi}^3 a^3(T_1)}{{\mathcal
F}^3}[\frac{1}{2}\mu \Sigma^2 + \frac{1}{6}y\Sigma^3] + h.c.&
.\end{aligned}$$
where ${\cal A}=a(t_1)|_{\theta=0}(1+\frac{\partial \ln
a(t_1)}{\partial t_i} F^{T_i})$. Now we can redefine the matter fields $$\tilde \Sigma= \frac{\tilde \Phi{\cal A}}{\cal F} \Sigma\equiv
{\cal G}\Sigma$$ Explicitly this reads $$\tilde \Sigma =\frac{a(t_1)}{f^{1/2}}\Big|_{\theta=0} (1+F_{\cal
G}\theta^2)\Sigma$$ where $$F_{\cal G}= (F_{\tilde\Phi}+\frac{\partial \ln\
a(t_1)f^{-1/2}}{\partial t_i}F^{T_i}).$$ Notice that according to this definition of $F_{\cal G}$, the $F$-term of $\cal{G}$ is actually $\frac{a(t_1)}{f^{1/2}}\Big|_{\theta=0}F_{\cal G}$. The matter action becomes
$$\begin{aligned}
&\int d^4x d^2 \theta d^2 \bar\theta
\Big(1+\theta^2\bar
\theta^2|F^{T_1}|^2\frac{1}{a(t_1)}\frac{\partial^2
a(t_1)}{(\partial
t_1)^2}\Big)\Big(|\tilde\Sigma|^2+\frac{1}{2}\lambda
\frac{\bar{\mathcal G}}{\mathcal G}\tilde\Sigma^2+h.c.\Big)&
\nonumber \\
&+\int d^4x d^2\theta\ \frac{\tilde{\Phi}^3 a^3(T_1)}{{\mathcal
F}^3{\mathcal G}^3} [\frac{1}{2}\mu{\mathcal G} \tilde \Sigma^2 +
\frac{1}{6}y\tilde\Sigma^3] + h.c.&
\label{classical superspace soft br action}\end{aligned}$$
This is the classical action in terms of the normalised matter fields in the Einstein frame. The superpotential action can be rewritten as $$\int d^4x d^2\theta\ \frac{a^3(T_1)}{{\mathcal A}^3} [\frac{1}{2}\mu{\mathcal G} \tilde \Sigma^2 +
\frac{1}{6}y\tilde\Sigma^3] + h.c.
\label{classical superspace superpot rewritten}$$ The field redefinitions that we have performed are all anomalous. Let us now write the action with renormalized couplings, evaluated at a given energy scale $E$. The superpotential couplings are not renormalised. The effective ultra-violet field-dependent cut-off is $\Lambda_{UV}{\mathcal
G}$ and is superfield dependent. This superfield ${\mathcal G}$ breaking supersymmetry when $F_{\mathcal G}\ne 0$, the rescaling generates anomalous soft terms. We thus have the kinetic part of the action
$$S_{qu}\supset \int
d^4x d^2 \theta d^2 \bar \theta
\Big(Z(\frac{E}{\Lambda_{UV}|{\mathcal
G}|})|\tilde\Sigma|^2+\frac{1}{2}\lambda(\frac{E}{\Lambda_{UV}|{\mathcal
G}|})\tilde\Sigma^2+h.c.\Big)$$
where we have discarded the $\theta^2\bar \theta^2 |F^{T_1}|^2\frac{1}{a(t_1)}\frac{\partial^2
a(t_1)}{(\partial
t_1)^2}$ and $\frac{\bar{\mathcal G}}{{\mathcal G}}|_{\theta\ne 0}$ terms of (\[classical superspace soft br action\]) as contributing only to the tree level soft breaking action, already computed in the previous sections, and not to the anomalous terms. Similarly, in the classical superpotential action (\[classical superspace superpot rewritten\]), the $\frac{a^3(T_1)}{{\mathcal A}^3}|_{\theta \ne 0}$ and ${\mathcal G}|_{\theta \ne 0}$ terms will be eliminated from now on.
Now the wave function normalization Z contains $\theta^2$ and $\bar \theta^2$ terms
$$Z(\frac{E}{\Lambda_{UV}|{\mathcal G}|})=Z(\frac{E
f^{1/2}(t_1,t_2)}{\Lambda_{UV}a(t_1)})\Big|_{\theta=0}\Big|1-\frac{\gamma}{2}\theta^2F_{\mathcal
G}\Big|^2 \Big(1+\frac{1}{8}\theta^2\bar \theta^2(\frac{\partial
\gamma}{\partial g}\beta_g+\frac{\partial \gamma}{\partial \ln
\lambda}\xi)|F_{\mathcal G}|^2\Big)$$
We can redefine the matter fields using $$\hat \Sigma=
Z^{1/2}|_{\theta=0}\Big(1-\frac{\gamma}{2}\theta^2F_{\mathcal
G}\Big)\tilde\Sigma .$$
where we have defined $\gamma\equiv \frac{\partial ln Z}{\partial
ln E}$ the anomalous dimension of $\tilde \Sigma$, similarly $\xi
\equiv \frac{\partial ln \lambda}{\partial ln E}$, and $\beta_g\equiv \frac{\partial g}{\partial ln E}$ where $g$ is a gauge coupling. The resulting action involves several blocks each leading to soft breaking terms. The soft masses come from $$\int d^4x d^2 \theta d^2 \bar \theta\ (1-\theta^2\bar \theta^2
m^2_{an.})|\hat\Sigma|^2$$ and read $$m^2_{an.,\bar i j}=-\frac{1}{8}\delta_{\bar i j}(\frac{\partial
\gamma_{i}}{\partial g}\beta_g+\frac{\partial \gamma_{i}}{\partial
\ln \lambda_{kl}}\xi_{kl})|F_{\mathcal G}|^2$$ We have restored the index structure in the final formulae to take into account the case with several chiral multiplets, with the simplification assumption of diagonal wave function renormalisation $Z_{\bar i j}=Z_i \delta_{\bar i j}$. Notice that this is the generalisation of the result of Randall and Sundrum where the contribution from the moduli fields has been taken into account. The $B$ and the $\mu$ terms are also generated at one loop $$\begin{aligned}
&\int d^4x d^2 \theta d^2\bar \theta \frac{1}{2}\Big((\lambda_R+
\bar\theta^2\mu_{an.}-\theta^2\bar\theta^2
B_{an.}^{G.M.})\hat\Sigma^2+h.c.\Big)&
\nonumber \\
&+\int d^4x d^2\theta \frac{1}{2}(\mu_R +\theta^2 B^\mu_{an.})\hat
\Sigma^2 + h.c.&\end{aligned}$$ where $ \lambda_{R, ij}= \frac{\lambda_{ij}}{\sqrt{Z_iZ_j}}$ and are given by $$B_{an, ij}= B^{\mu}_{an, ij}+B_{an, ij}^{G.M.}$$ with $$B_{an., ij}^\mu=\frac{\gamma_i+\gamma_j}{2} F_{\mathcal G}\mu_{R,
ij}$$ where $\mu_{R, ij}=
\frac{a^3(T_1)}{a^2(t_1)f^{1/2}\sqrt{Z_iZ_j}}\mu_{ij}$ and $$B_{an., ij}^{G.M.}= \frac{1}{4}\Big(\frac{1}{2}\frac{\partial
\xi_{ij}}{\partial g}\beta_g+\frac{1}{2}\frac{\partial
\xi_{ij}}{\partial \ln
\lambda_{kl}}\xi_{kl}-(\gamma_i+\gamma_j)\xi_{ij}\Big)|F_{\mathcal
G}|^2\lambda_{R, ij}$$ The anomalous effective $\mu$ term is given by $$\mu_{an., ij}=- \frac{\xi_{ij}}{2} F_{\mathcal G} \lambda_{R, ij}.$$ Notice that when the tree level $\mu_{eff}=0$, one can generate a loop contribution to the $\mu$ term via the Giudice–Masiero term. Finally the $A$ terms are also generated at one loop from $$\int d^4x d^2\theta \frac{1}{6}(y_R+\theta^2 A_{an.})\hat\Sigma^3
+ h.c.$$ where $y_{R,ijk}=
\frac{a^3(T_1)}{a^3(t_1)\sqrt{Z_iZ_jZ_k}}|_{\theta=0}y_{ijk}$ leading to $$A_{an., ijk}=\frac{\gamma_i+\gamma_j +\gamma_k}{2}F_{\mathcal
G}y_{R,ijk}$$ The order parameter for anomaly mediation is $F_{\mathcal G}$. The contribution in $F^{T_i}$ arising in $F_{\mathcal G}$ springs from the conformal anomaly in going from the brane frame to the Einstein frame. We have already computed $F^{T_i}$, we find now
$$F_{\tilde \Phi}=\frac{2\kappa_4^2}{3\Delta^{3/2}}\Big(ik{\cal I}m
T_1 a(\bar T_1)^{3-4\alpha^2} \bar w_1 +ik{\cal I}m T_2 a(\bar
T_2)^{3-4\alpha^2} \bar w_2 \Big)
+\frac{1}{2}\frac{f_{t_i}}{f}F^{T_i}$$
This can be obtained by noticing that the superspace volume element $E^{-1}\supset \bar \Phi \Phi$.
Note that in principle the running couplings are now evaluated for a cut-off $$\Lambda_{UV}\Big|1-\frac{\gamma}{2}\theta^2 F_{\mathcal G}\Big|$$ which is again $\theta$-dependent. This dependence should be expanded, yielding extra contributions to the soft terms, but these would be of higher order and were consequently neglected.
Let us conclude with the anomalous gaugino masses, deduced from the running gauge coupling function
$$f(E)=f^0 + \frac{b}{2\pi} \ln(\frac{E}{\Lambda_{UV}{\mathcal G}})
= f_i^0 + \frac{b}{2\pi} \ln(\frac{E
f^{1/2}(t_1,t_2)}{\Lambda_{UV}a(t_1)}|_{\theta=0}) -
\frac{b}{2\pi}\theta^2 F_{\mathcal G} .$$
Thus the anomalous contribution to the gaugino masses is $$m^{an.}_{a}=\frac{b_a}{2\pi}F_{\mathcal G} .$$ Notice that all the soft terms emanating from anomaly mediation are of order $F_{\mathcal G}$.
Soft Supersymmetry Breaking from the Hidden Brane
=================================================
We come back to the case where no hypermultiplet is in the bulk. Having analysed the breaking of supersymmetry in the brane models with a vector multiplet scalar field, we will extract ingredients with phenomenological relevance. We will focus on the case where the potential admits a global minimum for $\rho=0$, $w_1=0$, i.e. supersymmetry breaking is only due to the negative tension brane. The axions will be taken to be at the extremum ${\mathcal I}m
T_i=0$, which in this case is stable for ${\mathcal I}m T_2$ and flat for ${\mathcal I}m T_1$. Notice that $F^{T_2}\ne 0$ implying that supersymmetry is broken.
The supersymmetry breaking can be better understood using the normalised moduli fields. We redefine the non-axionic fields in the following way: $$\begin{aligned}
a(t_1)^{1+2\alpha^2} &=& e^\sigma \cosh r
\nonumber \\
a(t_2)^{1+2\alpha^2} &=& e^\sigma \sinh r\end{aligned}$$ Notice that $r=0$ when the second brane hits the singularity while $r=\infty$ when the two branes collide. The sliding field $\sigma$ describes the position of the centre of mass of the two branes. In the Einstein frame the kinetic terms are expressed as $$S=\frac{1}{2\kappa_4^2}\int d^4 x \sqrt {-g}( {\cal R}^{(4)}
-\frac{12\alpha^2}{1+2\alpha^2}(\partial \sigma)^2
-\frac{6}{1+2\alpha^2} (\partial r)^2)$$ This is a sigma–model Lagrangian with normalisation matrix $\gamma_{rr}=\frac{6}{1+2\alpha^2}$ and $\gamma_{\sigma\sigma}=\frac{12\alpha^2}{1+2\alpha^2}$. Notice that $\sigma$ decouples in the RS model as the centre of mass of the brane system is irrelevant. Moreover in the RS model, $$\tanh r=e^{-kt}$$ where $t$ is the radion.
Before analysing the structure of the soft terms, let us concentrate on the $\mu$ problem. Indeed the $\mu$ term must be of the order of the weak scale. There are two sources for the $\mu$ term here, the supersymmetric $\mu$ term in the superpotential and the Giudice-Masiero term in the Kähler potential. It turns out that one cannot use the Giudice-Masiero mechanism to obtain a small $\mu$ term indeed $$\mu^{eff}\approx \mu e^{-\frac{2\alpha^2}{1+2\alpha^2}\sigma}$$ is independent of $\lambda_{GM}$. However it is possible to use the field $\sigma$ to generate the electroweak hierarchy between a Planck scale fundamental $\mu$-term and the weak scale effective $\mu$-term. This new mechanism using the sliding field $\sigma$ is not possible in the pure Randall-Sundrum model with $\alpha=0$.
Another interesting scenario can be obtained when the classical fundamental $\mu$ term vanishes. At one loop the $\mu$ term depends on $F_{\mathcal G}$ $$F_{\mathcal G} \approx 2 \kappa_4^2 \bar w_2
e^{-\frac{6\alpha^2\sigma}{1+2\alpha^2}}r^{\frac{3}{1+2\alpha^2}}$$ We thus obtain that $$\mu=-\frac{\xi}{2} \lambda_{GM} F_{\mathcal G}$$ with $\xi\equiv \frac{\partial ln \lambda_{GM}}{\partial ln E}$. The $\mu$ term can be of the order of the electro–weak scale provided the $F_{\mathcal G}$ breaking term is in TeV range.
The soft masses are given by $$m^2 \approx \frac{1}{\lambda_{GM}}B$$ where the tree level contribution dominates. Hence the $B$ terms are naturally of order of the soft masses. Now the soft masses are related to $F_{\mathcal G}$ breaking term via $$m^2
\approx
\frac{1}{2(1+2\alpha^2)} \frac{|F_{\mathcal G}|^2}{ r^{2}} .$$ Notice that for small $r$ the soft masses can be larger than the TeV range. Of course this can be modified provided $\xi \lambda$ is large enough. In that case $\mu\approx \xi \lambda F_{\mathcal
G}\approx F_{\mathcal G}/r\approx m$ may both be at the electro–weak scale even for small $r$, at the cost of a unnatural relation between the parameters $r_{now}$ and $\xi\lambda$.
Supersymmetry is broken and the gravitino mass becomes $$m_{3/2} \approx \frac{F_{\mathcal G}}{2}$$ implying that the gravitino mass can also be within the $TeV$ range.
Now the tree level gaugino masses vanish as ${\cal I}m T_i=0$. A non-vanishing contribution results from anomaly mediation. Therefore the anomalous contribution becomes $$m^{an.}_a \approx \sqrt{\frac{1+2\alpha^2}{2}}\frac{b_a}{\pi} m
r$$ This is suppressed by a $r$ factor with respect to the soft masses.
Finally the classical A term vanishes. The $A$ term picks a quantum contribution $$A_{an}\approx 3\gamma y \sqrt{\frac{1+2\alpha^2}{2}}m r$$ which is suppressed by $\gamma r$ with respect to the soft masses.
We have thus characterized the mass spectrum of the superpartners. The soft terms are all determined by the supersymmetry breaking scale $F_{\mathcal G}$ and the value of the radion field $r$ where $r$ is driven to zero according to the scalar potential. The conclusions are unmodified for the Randall-Sundrum case $\alpha=0$.
Notice that the potential for the radion $r$ is now $$V=\frac{3\kappa_4^2}{1+2\alpha^2}e^{-\frac{12\alpha^2}{1+2\alpha^2}\sigma}
|w_2|^2\sinh^{\frac{4-4\alpha^2}{1+2\alpha^2}}(r)$$ driving the $r$ field to zero. As the field rolls down to its minimum, the vacuum energy becomes smaller and smaller. Notice that for small $r$ $$V\approx \frac{3}{4(1+2\alpha^2)}\frac{\vert F_{\mathcal
G}\vert^2}{\kappa_4^2} r^{-2}$$ Fine–tuning the cosmological constant for $\vert F_{\mathcal
G}\vert\approx 1$ TeV would require $r_{now}\approx 10^{45}$. To remedy this problem, we now introduce an explicit supersymmetry breaking step. This will allow us to obtain a very small vacuum energy, so that the soft terms have the usual physical interpretation as masses and coupling constants in an almost flat space-time.
Charged Branes \[charged branes\]
=================================
The branes that we have considered so far are neutral branes. One can also consider the case where a four–form lives in the bulk and couples to the bulk scalar field. This induces a new contribution to the potential for the moduli which turns out to be of the same form as the potential obtained with $w_i\ne 0$ [@detuned]. The main difference here springs from the fact the we introduce the brane charges in an explicitly supersymmetry breaking form. It would be nice to embedd the charged case in a fully supersymmetric framework. A possibility would consist in using the Lagrange multiplier four–form of (\[boundary action\]), but then the added kinetic terms would have to be supersymmetrised too.
Let us consider a four–form $C_{abcd}$ living in the bulk. We define its field strength $F=dC$ which is a five–form dual to a scalar field $* F$. The branes have charges $\pm Q$. The global charge is zero as the fifth dimension is compact. We choose the Lagrangian to be
$$S_C=-\frac{1}{2} \int \frac{1}{U^2}F\wedge *F
-\frac{\sqrt{6(1-\alpha^2)}}{1+2\alpha^2}\frac{\kappa_5 Q}{\sqrt
k}\int_+ C +\frac{\sqrt{6(1-\alpha^2)}}{1+2\alpha^2}\frac{\kappa_5
Q}{\sqrt k}\int_-C \label{action}$$
The prefactors are chosen for convenience. Notice that the coupling constant for the four–form is proportional to $U$. One can easily find the solutions of the equations of motion for $C$ $$*F=\frac{\sqrt{6(1-\alpha^2)}}{1+2\alpha^2}\frac{\kappa_5 Q}{\sqrt
k}\epsilon (z) U^2$$ where $\epsilon (z)$ is the odd function jumping from -1 to 1 at the first brane and 1 to -1 at the second brane. Notice that the boundary term depends on $\int F$ and vanishes altogether. The action (\[action\]) when evaluated yields a potential in the Einstein frame which turns out to be $$V= -\frac{3\kappa_4^2}{1+2\alpha^2}a(t_1)^{-12\alpha^2}\frac{Q^2
\rho^{4-4\alpha^2}-Q^2}{(1-\rho^{2+4\alpha^2})^2} \label{pot1}$$ This is nothing but the potential for the moduli when substituting $|w_{1,2}|^2 \to -Q^2$. In particular the total potential taking into account both the tension detuning and the explicit supersymmetry breaking by the charge $Q$ is now $$V=
\frac{3\kappa_4^2}{1+2\alpha^2}a(t_1)^{-12\alpha^2}\frac{(|w_2|^2-Q^2)
\rho^{4-4\alpha^2}-(|w_1|^2-Q^2)}{(1-\rho^{2+4\alpha^2})^2}
\label{pot2}$$ This potential has a much richer structure. We can distinguish five cases. We will discuss the nature of the potential as a function of $\rho=a(t_2)/a(t_1)$ (see figures \[figure potentiels 1\], \[figure potentiels 2\], \[figure potentiels 3\], 4, 5 and 6). .5 cm
[*i) $\Big( |w_2| \leq |w_1|$ and $|Q| < |w_1|\Big)$ or $|w_2|<|w_1|=|Q|$*]{} .5 cm
The potential is negative and unbounded from below. This case is not favoured phenomenologically. This corresponds to figures \[figure potentiels 3\] and \[figure potentiels 5\].
.5 cm
[*ii) $\vert Q\vert< \vert w_1\vert <
\vert w_2 \vert $*]{}
.5 cm
The potential admits a negative minimum leading to anti de Sitter space. This is not a viable phenomenological case. Note that this is the only case where radion stabilisation occurs. This corresponds to figure \[figure potentiels 1\] on the left hand side. .5 cm
[*iii) $\vert Q\vert> \vert w_1\vert >
\vert w_2 \vert $*]{}
.5 cm
The potential is unbounded from below as $\rho \to
1$. Nevertheless it admits a local minimum at the origin corresponding to a positive cosmological constant. The de Sitter phase of the theory with $\rho=0$ is a metastable state protected from the unstable branch of the potential by a local maximum. Eventually the Universe would tunnel through this finite energy barrier. The resulting phase would result in a big crunch singularity due to the brane collision. This case corresponds to figure \[figure potentiels 2\]. .5 cm
[*iv) $\Big(|w_1|\leq |w_2|$ and $|w_1|<Q \Big)$ or $|w_2|>|w_1|=|Q|$*]{}
.5 cm
In that case the potential is bounded from below. Its minimum is at the origin $\rho=0$. The corresponding cosmological constant is positive and can be fine–tuned to match the present value of the vacuum energy. This corresponds to figures \[figure potentiels 4\] and \[figure potentiels 6\] on the right hand sides. .5 cm
[*v) $|Q|=|w_1|=|w_2|$*]{}
.5 cm The potential vanishes altogether, a situation reminiscent of a BPS bound.
Cosmological and Gravitational Consequences \[grav and cosmo consequences\]
===========================================================================
We will now investigate the cosmological consequences of the previous model where supersymmetry is broken on the hidden brane. The potential reads
$$V=\frac{3\kappa_4^2}{1+2\alpha^2}e^{-\frac{12\alpha^2}{1+2\alpha^2}\sigma}
[Q^2 \cosh^{\frac{4-4\alpha^2}{1+2\alpha^2}}(r)
-(Q^2-|w_2|^2)\sinh^{\frac{4-4\alpha^2}{1+2\alpha^2}}(r)]$$
Note that $\rho=\frac{a(t_2)}{a(t_1)}=\tanh^{\frac{1}{1+2\alpha^2}}(r)$. The field $r$ is driven towards its minimum at $r=0$. For small $r$ the potential reduces to an exponential potential for the sliding field $\sigma$ $$V=\frac{3\kappa_4^2
\tilde Q^2(r)}{1+2\alpha^2}e^{-\frac{12\alpha^2}{1+2\alpha^2}\sigma}$$ where we have defined $\tilde Q^2(r) \equiv Q^2
\cosh^{\frac{4-4\alpha^2}{1+2\alpha^2}}(r)
-(Q^2-|w_2|^2)\sinh^{\frac{4-4\alpha^2}{1+2\alpha^2}}(r) > 0$, which reduces to the constant $Q^2$ for small enough $r$. This is a typical example of a quintessence potential for $\sigma$. The phenomenology of this type of potential is well–known. Let us summarize some of its salient features.
The exponential potential admits an attractor with scale factor $$a=a_0 t^{\frac{1+2\alpha^2}{3\alpha^2}}$$ which is accelerating as soon as $\alpha <1$. Even in the presence of cosmological matter such as cold dark matter, the above attractor still exists and attracts the energy fraction carried by the sliding field towards $$\Omega_\sigma =1$$ i.e. the Universe becomes scalar field dominated eventually. On the attractor the equation of state of the Universe is constant $$w=-1+ \frac{4\alpha^2}{1+2\alpha^2}$$ We will see that $\alpha$ is constrained by solar system experiments leading to a tight bound on the equation of state.
These results are equivalent to solving the full 5d equations of motion with a positive detuning of the tension on the positive tension brane. In 5d the motion of the positive tension brane is at constant speed in conformal coordinates with speed $v$ $$\sqrt{1-v^2}= \frac{1}{1+ \kappa_5^2 \tilde Q^2}$$ For small $\tilde Q$, the brane is non–relativistic and $v=\pm \sqrt 2
\kappa_5 \tilde Q$.
Of course, the Universe cannot be on the attractor now as one expects $\Omega_\sigma \approx .7$. Hence one must fine–tune the value of the sliding field now in such a way that $$\tilde Q e^{-\frac{6\alpha^2}{1+2\alpha^2}\sigma_{now}}\approx\sqrt{\frac{1+2\alpha^2}{3\kappa_4^2}}\sqrt
{\Omega_\Lambda \rho_c}$$ where $\Omega_\Lambda \approx 0.7$ and $\rho_c\approx 10^{-48}$ GeV$^4$. Numerically this leads to $$\tilde Q e^{-\frac{6\alpha^2}{1+2\alpha^2}\sigma_{now}}\approx 10^{-6}
\hbox{GeV}^3$$ This is nothing but the usual fine–tuning of the cosmological constant. In particular, due to the smallness of the vacuum energy we can interpret the soft terms in the conventional way where the cosmological constant is set to zero. Notice that this is achieved thanks to the charge Q.
The fields $\sigma$ and $r$ are extremely light fields of masses of the order of the Hubble rate $H_0\approx 10^{-42}$ GeV. These extremely light fields may lead to large deviations from gravity in solar system experiments. The solar system experiments are mainly sensitive to the variation of the nucleon masses as a function of the moduli. The bulk of the nucleon masses is given by the QCD condensate $\Lambda_{QCD}$ expressed in the Einstein frame. This condensate can be estimated from the pole of the strong coupling constant at one loop and reads $$\Lambda_{QCD}=A(\sigma,r) e^{-\frac{2\pi}{b_3}{\mathcal R}e\ f_3(\sigma,r,{\cal I}m
T_1)} \Lambda_0$$ where $f_3$ is the strong coupling function in the Lagrangian and $b_3$ is the QCD renormalisation group coefficient (the 3 index signaling the SU(3) gauge group). The overall factor $A(\sigma,
r)$ accounts for the change of frame between the brane frame and the Einstein frame. The scale $e^{-\frac{2\pi}{b_3}{\mathcal R}e\
f_3(\Lambda_0)}\Lambda_0$ is the renormalisation group invariant QCD scale in the brane frame where ordinary matter is minimally coupled to the brane metric. Explicitly we find that $$A(\sigma ,r)= e^{-\frac{2\alpha^2}{1+2\alpha^2}\sigma}
\cosh^{\frac{1}{1+2\alpha^2}}(r)$$ The solar system experiments give tight bounds on the Eddington parameter $\vert \gamma_{ED}-1\vert \le 10^{-5}$ [@testsGR]. The Eddington parameter is related to $\gamma_{ED}-1\approx -2
\theta$ where $\theta$ depends the coupling constants $$\alpha_{r}=\frac{\partial \ln \Lambda_{QCD}}{\partial
r},\alpha_{\sigma}=\frac{\partial \ln \Lambda_{QCD}}{\partial
\sigma},\alpha_{{\cal I}m T_1}=\frac{\partial \ln \Lambda_{QCD}}{\partial
{\cal I}m T_1}$$ and the normalisation of the fields $\gamma_{ij}$, with inverse matrix $\gamma^{ij}$. It is defined by
$$\theta= \gamma^{ij}\alpha_i\alpha_j=\gamma^{rr}\alpha_r^2
+\gamma^{\sigma \sigma}\alpha_\sigma^2 +\frac{1}{2}K^{T_1\bar
T_1}\alpha_{{\cal I}m T_1}^2 .$$
This constraint is valid for massless scalar fields, in the sense as their inverse mass is larger than solar system scales. This is the case for the moduli we consider whose inverse mass is typically of the size of the Hubble radius. Explicitly this leads to
$$\begin{aligned}
\theta&=&\frac{1}{3}\frac{\alpha^2}{1+2\alpha^2}\Big(1+\frac{\pi}{b_3}{\mathcal R}ef_3\frac{\beta_3}{\alpha^2}(\frac{a(t_1)}{a(T_1)})^{4\alpha^2}\Big)^2+
\frac{\tanh^2
r}{6(1+2\alpha^2)}\Big(1-\frac{2\pi}{b_3}{\mathcal R}ef_3\ \beta_3(\frac{a(t_1)}{a(T_1)})^{4\alpha^2}\Big)^2
\nonumber \\
&&+\frac{1}{12(1+2\alpha^2)}\Big(1+2\alpha^2 \tanh^2(r)\Big)
\frac{4\pi^2}{b_3^2} \Big( {\mathcal R}ef_3\frac{\beta_3}{\alpha} (\frac{a(t_1)}{a(T_1)})^{4\alpha^2}\Big)^2 . \label{Eddington}\end{aligned}$$
Note that $\beta^i_a \equiv \frac{\partial \ln
{\mathcal R}e f^i_a}{\partial \ln a(T_i)}$ is now defined from the *real* gauge coupling ${\mathcal R}e f^i_a$ and not any longer from its complex version.
Imposing the experimental bounds leads to $$\alpha <10^{-2},\ r_{now}\le 10^{-2}$$ together with the fact that $\frac{{\mathcal R}e
f_3}{\alpha}\vert\beta_3\vert=\frac{1}{\alpha}|\frac{\partial
{\mathcal R}e f_3}{\partial ln a(T_1)}|$ cannot be too large compared to one. Considering the part of moduli-dependence due to the coupling of the gauge kinetic terms with the brane value of the bulk scalar field $\phi(T_1)=-4\alpha\ \ln(a(T_1))$, one has $\frac{{\mathcal R}e f_3}{\alpha} |\beta_3|=4|\frac{\partial
{\mathcal R}e f_3}{\partial \phi(T_1)}|$ which has to be small compared to one.
The QCD gauge coupling $f_3$ is of order one and the factor $|\frac{a(t_1)}{a(T_1)}|$ is always smaller than one. The bound on $r_{now}$ can be fulfilled as the value $r=0$ is a minimum of the potential. Choosing $\alpha$ small enough, we find that the scalar–tensor theory is attracted towards general relativity. Notice that this leads to a very tight constraint on the equation of state of matter on the attractor $w_\sigma\le -0.96$ hardly distinguishable from a pure cosmological constant.
Now let us come back to the supergravity case with a vector multiplet where $\alpha^2=1/12,1/3$. As can be seen the sliding field $\sigma$ leads to large deviations from ordinary gravity which are not compatible with experiments. The $\gamma^{\sigma\sigma}\alpha_{\sigma}^2$ contribution to the $\theta$ parameter is indeed the one proportional to $\alpha^2$ and thus too large in the vector multiplet case. Of course this is the usual stabilisation problem as already appearing for the dilaton. One possibility indeed would be to stabilise the $\sigma$ field. This would require to find a potential with an appropriate minimum. This is a notoriously difficult problem. Let us conclude with the pure Randall-Sundrum case $\alpha=0$. The quintessence field $\sigma$ disappears so that the gravitational constraints are now satisfied ; the vacuum energy is adjusted to its present value through the $\tilde Q$ parameter. This is the fine–tuning of the cosmological constant.
Hence in the Randall-Sundrum case with supersymmetric matter on the positive tension brane and detuning of the negative tension brane, the gravitational constraints are satisfied for far–away branes. In that case, the soft terms are determined by the anomaly mediation breaking term $F_{\cal G}$ and the value of the radion now. The radion is driven to zero implying that gravity is retrieved and the soft masses are very large compared to the gaugino masses and the $\mu$ term. For small values of $r$ this would lead to a naturalness problem for the electro–weak breaking. The detailed analysis of the phenomenology of these models is left for future work.
Conclusion
==========
We have analysed the supersymmetry breaking of 5d supergravity models with boundary branes in the presence of one vector multiplet and the universal hypermultiplet. As a particular case, this includes the supersymmetric Randall–Sundrum model. We have discussed the soft breaking terms and the moduli potential when supersymmetry is broken on both branes. We have focused on the case where matter is present on the positive tension brane and supersymmetry is broken on the negative tension brane. We have seen that the spectrum is determined both by the anomaly mediation $F_{\cal G}$ term and the normalised radion field $r$. In particular the radion $r$ is driven to zero where general relativity is retrieved. In that case the soft masses are much larger than the gaugino masses leading to a possible naturalness problem.
We have also discussed the case with the universal hypermultiplet and in particular brane race–track models. These models are very intricate and would deserve further study. In particular one may hope that the moduli may become stabilised at appropriate values.
We have seen that the supergravity models with a vector multiplet are disfavoured as leading to strong deviations from general relativity. One way of modifying this result would be for the moduli to become chameleon fields whereby their masses depend on the environment [@chameleon]. In the solar system, this could be enough to evade the gravitational constraints and enlarge the class of phenomenologically relevant models. This is left for future work.\
We would like to thank U. Ellwanger and Z. Lalak for useful comments and discussions.
$AdS_4$ Supersymmetry and Radion stabilisation
==============================================
Let us consider the Randall-Sundrum brane system with detuned tensions $\lambda_{1,2}=\pm \delta_{1,2}6k$ where $$\delta_{1,2}=1-\kappa_5^2|w_{1,2}|^2$$ where we assume that the detuning is small $$\kappa_5^2|w_{1,2}|^2 \ll 1 .$$ The solution of the 5d equations of motion including the brane boundary condition leads to a fixed value of the interbrane distance $$kt=\frac{1}{2}\ln\Big(\frac{(\lambda+\lambda_1)(\lambda-\lambda_2)}{(\lambda+\lambda_2)(\lambda-\lambda_1)}\Big)$$ where $\lambda=6k$ is the tuned RS tension. For small detuning this gives $$kt=\ln|\frac{w_2}{w_1}| .$$ Notice that stabilisation of the radion can only be achieved when $|w_2|>|w_1|$. In that case, the configuration is known to respect $AdS_4$ supersymmetry. We will compare this 5d analysis to the effective action approach.
In 4d at low energy, the Kähler potential $$K=-3 \ln (e^{-k(T_1+\bar T_1)}-e^{-k(T_2+\bar T_2)})$$ complemented with the superpotential $$W=w_1 e^{-3kT_1} + w_2 e^{-3k T_2}$$ is equivalent to $$\begin{aligned}
K&=&-3 \ln (1-e^{-k(T+\bar T)}) \nonumber \\
W&=&w_1+w_2e^{-3kT}\end{aligned}$$ up to a Kähler transformation, with $T=T_2-T_1$. The scalar potential admits a minimum when $|\frac{w_2}{w_1}|>1$ where $$kt=\ln|\frac{w_2}{w_1}|$$ coinciding with the 5d result. At the minimum the $F^T$ term reads $$F^T=-\frac{W}{|W|}\frac{\kappa_4^2}{k} \bar
w_1(1-|\frac{w_1}{w_2}|^2)^{-1/2}(1+e^{3 i k {\cal I}m T-i\eta})$$ where $\eta$ is the phase of $\frac{w_2}{w_1}$. This vanishes for $$3k{\cal I}m T=\eta + \pi (mod\ 2\pi)$$ leading to a configuration with a negative potential energy $$V=-3\frac{m_{3/2}^2}{\kappa_4^2}$$ and a non zero gravitino mass. The solution of Einstein’s equations is then $AdS_4$ which breaks Poincaré invariance.\
Soft terms in the Randall-Sundrum case $\alpha=0$
=================================================
In the Randall-Sundrum limit, only one chiral superfield out of the $T_{1,2}$ remains physical. We start from $$\begin{aligned}
K&=&-3 ln(1-e^{-k(T+\bar T)}) \nonumber \\
W&=&w_1+w_2 e^{-3kT}\end{aligned}$$ where $T=T_2-T_1$ is the radion superfield, and we will write $T=t+i{\cal I}m T$. As can be checked the results will be the same as those obtained from our general calculation by taking the limit $\alpha=0$. We write the soft Lagrangian for the canonically normalised complex matter scalars $\tilde{s}_i$ on the *positive* tension brane as $${\cal L}_{soft}= -m^2_{i\bar j} \tilde{s}_i \overline{\tilde{s}_j}
- (\frac{1}{2}B_{ij}\tilde{s}_i \tilde{s}_j + hc) - (\frac{1}{6}
A_{ijk}\tilde{s}_i \tilde{s}_j \tilde{s}_k + hc) .$$
The radion potential $$V=-3\kappa_4^2\frac{|w_1|^2-e^{-4kt}|w_2|^2}{(1-e^{2kt})^2}$$ does not depend on ${\cal I}m T$. The gravitino mass is $$m_{3/2}=\kappa_4^2\frac{|w_1+w_2 e^{-3kT}|}{(1-e^{-2kt})^{3/2}} .$$ The radion F–term is $$F^T=-\frac{\bar W}{|W|}\frac{\kappa_4^2}{k}\frac{\bar w_1+\bar w_2
e^{-kt+3ik {\cal I}m T}}{(1-e^{-2kt})^{1/2}} .$$ The effective normalised superpotential is given by $$\begin{aligned}
\mu_{ij}^{eff}&=&\frac{\bar W}{|W|}\frac{1}{(1-e^{-2kt})^{1/2}}(\mu_{ij}+\kappa_4^2 \lambda_{ij} w_1)
\nonumber \\
\lambda^{eff}_{ijk}&=&\frac{\bar W}{|W|}\lambda_{ijk} .\end{aligned}$$ The normalised soft mass is $$m^2_{i\bar j}=\delta_{i \bar j}\frac{2}{3}\kappa_4^2
V=-2\delta_{i\bar
j}\kappa_4^4\frac{|w_1|^2-e^{-4kt}|w_2|^2}{(1-e^{2kt})^2}$$ and the normalised soft bilinear term is $$\begin{aligned}
B_{ij}&=& - \frac{\kappa_4^2 \bar w_1 }{1-e^{-2kt}}\mu_{ij}
+\kappa_4^4 \lambda_{ij}\frac{1}{(1-e^{-2kt})^2}
\nonumber \\
&&\Big(-|w_1|^2(1+e^{-2kt})
+2|w_2|^2e^{-4kt}\Big)
.\end{aligned}$$ The normalised soft trilinear term is vanishing : $$A_{ijk}= 0 .$$
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|
=23.7cm =16.5cm =-2.0cm =-1.4cm
[*Conformal symmetry, anomaly and effective action*]{}
[*for metric-scalar gravity with torsion* ]{}
[**J.A. Helayel-Neto**]{}[^1]\
Centro Brasileiro de Pesquisas Físicas-CBPF-CNPq\
Universidade Católica de Petrópolis (UCP)\
[^2]\
Centro Brasileiro de Pesquisas Físicas-CBPF-CNPq\
Faculdade de Educacao da Universidade Federal do Rio de Janeiro, (UFRJ)\
[^3]\
Departamento de Fisica – ICE, Universidade Federal de Juiz de Fora\
Juiz de Fora, 36036-330, MG, Brazil\
Tomsk State Pedagogical University, Tomsk, Russia
*[*ABSTRACT*]{}*
We consider some aspects of conformal symmetry in a metric-scalar-torsion system. It is shown that, for some special choice of the action, torsion acts as a compensating field and the full theory is conformally equivalent to General Relativity on classical level. Due to the introduction of torsion, this equivalence can be provided for the positively-defined gravitational and scalar actions. One-loop divergences arising from the scalar loop are calculated and both the consequent anomaly and the anomaly-induced effective action are derived.
Introduction
============
The studies in the framework of gravity with torsion have a long history and many interesting achievements. In particular, the relation between torsion and local conformal transformation (see, for example, [@histo1; @hehl-PR] for a recent reviews with broad lists of references) is a topic of special relevance. Torsion appears naturally in the effective action of string and, below the Planck scale, should be treated, along with the metric, as part of the gravitational background for the quantized matter fields. In this paper, we are going to investigate some details of conformal symmetry that come up in some metric-scalar-torsion systems. First, we briefly consider, following earlier papers [@deser; @conf; @duco], a similar torsionless system and then investigate the theory with torsion; proceeding further, we derive the one-loop divergences, conformal anomaly and the anomaly-induced effective action.
Since we are going to consider the issues related to conformal transformation and conformal symmetry in gravity, it is worthy to make some general comments. The conformal transformations in gravity may be used for two different purposes. First one can, by means of the conformal transformation, change the so-called conformal frame in the scalar-metric theory. Indeed, this choice is not arbitrary since the masses of the fields must be introduced. There may be various reasons to choose one or another frame as physical; one may find a detailed discussion on this issue in [@histo1]. All theories for which this consideration usually applies do not have conformal symmetry, even before the masses are introduced. Second, one can consider the theory with conformal symmetry.
In the present paper, we are going to discuss only the second case, that is, massless theories with unbroken conformal symmetry. Then, the change of the conformal frame is nothing but the invariant (at least, at the classical level) choice of the dynamical variables. It is worth mentioning that the theories with conformal symmetry have prominent significance in many areas of modern theoretical physics. The most important, at least for the sake of applications, is that at the quantum level this symmetry is violated by anomaly. As examples of the application of the trace anomaly, one may recall the derivation of the effective equations for strings in background fields (which are nothing but conditions for the anomaly cancellation) [@CFMP], the study of the renormalization group flows (see, for example, [@anselmi] for the recent developments and list of references), the derivation of the black hole evaporation in the semiclassical approach [@black], first inflationary model [@star] and some others (we recommend [@histo2] for a general review on conformal anomaly). Recently, the application of anomalies for $n=4$ has been improved by the use of the anomaly-induced effective action obtained long ago [@rei; @frts]. In particular, this led to a better understanding of the anomaly-generated inflation [@anju] and allowed to perform a systematic classification of the vacuum states in the semiclassical approach to the black hole evaporation [@balsan]. Anomaly-induced effective action has been used to develop the quantum theory for the conformal factor [@antmot] and for the consequent study of the back reaction of gravity to the matter fields [@brv].
The generalization of the anomaly and the anomaly-induced action for the case of completely antisymmetric torsion has been given in [@buodsh]. Here we investigate a non-trivial conformal properties of the metric-scalar-torsion system, which modify the Noether identity corresponding to the conformal symmetry. This modification is due to the fact that, in the model under discussion, torsion [*does transform*]{}, while its antisymmetric part considered before [@buodsh] does not. As a result of this new feature, the conformal anomaly is not the anomaly of the Energy-Momentum Tensor trace, but rather the trace of some modified quantity. However, using a suitable parametrization for the fields one can, as it will be shown below, reduce the calculation of this new anomaly, and obtain the expressions for both the anomaly and the anomaly-induced effective action using corresponding results from [@rei; @frts; @buodsh].
For the sake of generality, we present all classical formulae in an $n$-dimensional space-time, for $n\neq 2$ (see the Appendix of [@duco] for the discussion of the special $n=2$-case in the torsion-free theory). The divergences and anomaly are all evaluated around $n=4$.
Brief review of the torsionless theory.
=======================================
Our purpose here is to show that conformally invariant second-derivative $n$-dimensional metric-dilaton model is conformally equivalent to General Relativity. This equivalence has been originally demonstrated and discussed in [@deser] for the four-dimensional space and free scalar field, and later generalized in [@conf; @duco] for the interacting scalar field in an arbitrary $n\neq 2$ dimensions. In this section, we review the result of [@deser; @conf] presenting calculations in a slightly different manner. Our starting point will be the Einstein-Hilbert action with cosmological constant. For further convenience we take this action with negative sign. S\_[EH]{}\[g\_\] = d\^nx { [R]{} + } \[0.1\] The above action depends on the metric $\,{\hat g}_{\mu\nu}\,$ and we now set $\,{\hat g}_{\mu\nu} = g_{\mu\nu}\cdot e^{2\si(x)}$. In order to describe the local conformal transformation in the theory, one needs some relations between geometric quantities of the original and transformed metrics: = e\^[n]{}, = e\^[-2]{} \[n1\] Substituting (\[n1\]) into (\[0.1\]), after integration by parts, we arrive at: S\_[EH]{}\[g\_\] = d\^nx { e\^[(n-2)]{}()\^2 +R + e\^[n]{}}, \[n2\] where $(\na \si)^2
= g^{\mu\nu}\partial_\mu\si \partial_\nu\si $. If one denotes = e\^ , \[n3\] the action (\[0.1\]) becomes S=d\^nx { ()\^2+R\^2+( )\^ \^ } \[n5\] And so, the General Relativity with cosmological constant is equivalent to the metric-dilaton theory described by the action of eq. (\[n5\]). One has to notice that the latter exhibits an extra local conformal symmetry, which compensates an extra (with respect to (\[0.1\])) scalar degree of freedom. Moreover, (\[n5\]) is a particular case of a family of similar actions, linked to each other by the reparametrization of the scalar or (and) the conformal transformation of the metric [@duco]. The symmetry transformation which leaves the action (\[n5\]) stable, g\^\_=g\_e\^[2(x)]{} , \^ = e\^[(1-)(x)]{}, \[nnn\] degenerates at $\,\,n \to 2\,\,$ and that is why this limit cannot be trivially achieved [@duco]. If we start from the positively defined gravitational action (\[0.1\]), the sign of the scalar action (\[n5\]) should be negative, indicating to the well known instability of the conformal mode of General Relativity (see [@wet; @mod] for the recent account of this problem and further references).
Conformal invariance in metric-scalar-torsion theory
====================================================
In this section, we are going to build up the conformally symmetric action with additional torsion field. Our notations are similar to the ones accepted in [@book] for the $n = 4$ case. Torsion is defined as $${\wt \Gamma}^{\al}_{\be\ga} - {\wt \Gamma}^{\al}_{\ga\be} =
T^{\al}_{\be\ga}\,,$$ where ${\wt \Gamma}^{\al}_{\be\ga}$ is the non-symmetric affine connection in the space provided by independent metric and torsion fields. The covariant derivative, $\,{\wt\na}_\al\,$, satisfies metricity condition, $\,{\wt\na}_\al g_{\mu\nu}=0$. Torsion tensor may be decomposed into three irreducible parts: \_[, ]{} - \_[, ]{} = T\_[, ]{} = ( T\_g\_ - T\_g\_ ) - \_[\_1 ... \_[n-3]{}]{} S\^[\_1 ... \_[n-3]{}]{} + q\_. \[irr\] Here $\,T_\al = {T^\be}_{\al\be}\,$ is the trace of the torsion tensor $\,{T^\be}_{\al\ga}\,$. The tensor $\,S^{{\mu}_1 ... {\mu}_{n-3}}\,$ is completely antisymmetric and, in the case of a purely antisymmetric torsion tensor, $\,T_{\al\be\ga}$ is its dual. $\,\varepsilon_{\alpha\beta\ga{\mu}_1 ... {\mu}_{n-3}}$ is the maximal antisymmetric tensor density in the $\,n$-dimensional space-time. The sign of the $S$-dependent term corresponds to an even $\,n$. We notice that, in four dimensions, the axial tensor $\,S_{\mu_1 ... \mu_{n-3}}\,$ reduces to the usual $S_\mu$ – axial vector [@book]. In $n$ dimensions, the number of distinct components of the $\,S_{\mu_1 ... \mu_{n-3}}\,$ tensor is $\,\frac{n^3-3n^2+2n}{6}$. The tensor $\,{q^\be}_{\al\ga}\,$ satisfies, as in the $n=4$ case, the two constraints: $$\,{q^\be}_{\al\be}=0\,\,\,\,\,\,\,\,\,\,\,\,\, {\rm and }
\,\,\,\,\,\,\,\,\,\,\,\,\,\,
q_{\al\be\ga}\cdot \ep^{\al\be\ga\mu_1 ... \mu_{n-3}}=0$$ and has $\,\frac{n^3-4n}{3}\,$ distinct components. We shall denote, as previously, Riemannian covariant derivative and scalar curvature by $\na_\al$ and $R$ respectively, and keep the notation with tilde for the geometric quantities with torsion.
The purpose of the present work is to describe and discuss conformal symmetry in the metric-scalar-torsion theory. It is well-known that torsion does not interact minimally with scalar fields, but one can formulate such an interaction in a non-minimal way (see [@book] for the introduction). Moreover, this interaction between scalar field and torsion is necessary element of the renormalizable quantum field theory in curved space-time with torsion [@bush]. One may construct a general non-minimal action for the scalar field coupled to metric and torsion as below: S =d\^nx { g\^\_ \_+ \_i P\_i \^2 - \^ }. \[n7\] Here, the non-minimal sector is described by five structures: P\_1 = R , P\_2=\_T\^, P\_3=T\_T\^, P\_4=S\_[\_1 ... \_[n-3]{}]{}S\^[\_1 ... \_[n-3]{}]{} , P\_5=q\_q\^; \[nn8\] in the torsionless case the only $\,\xi_1 R\,$ term is present. $\xi_i$ are the non-minimal parameters which are typical for the theory in external field. Renormalization of these parameters is necessary to remove corresponding divergences which really take place in the interacting theory [@bush]. Furthermore, the renormalizable theory always includes some vacuum action. This action must incorporate all structures that may show up in the counterterms. The general expression for the vacuum action in the case of a metric-torsion background has been obtained in [@chris]; it contains 168 terms. In fact, one can always reduce this number, because not all the terms with the allowed dimension really appear as counterterms. The discussion of the vacuum renormalization for the external gravitational field with torsion has been previously given in [@buodsh] for a purely antisymmetric torsion. In the present article, we will be interested in the special case of the (\[n7\]) action, which possesses an interesting conformal symmetry. The appropriate expression for the corresponding vacuum action will be given in Section 4, after we derive vacuum counterterms.
Using the decomposition (\[irr\]) given above, the equations of motion for the torsion tensor can be split into three independent equations written for the components $T_\al,\,S_\al,\,q_{\al\be\ga}$; they yield: T\_= , S\_[\_1 ... \_[n-3]{}]{} = q\_ = 0. \[n9\] Replacing these expressions back into the action (\[n7\]), we obtain the on-shell action S = d\^nx { (1-) g\^\_ \_+ \_1 \^2R -\^ }, \[onshell\] that can be immediately reduced to (\[n5\]), by an obvious change of variables, whenever \_1=(1-). \[n8\] Therefore, we notice that the version of the Brans-Dicke theory with torsion (\[n7\]) is conformally equivalent to General Relativity (\[0.1\]) provided that the new condition (\[n8\]) is satisfied and no sources for the components $T_\al,\,S_{\mu_1 ... \mu_{n-3}}\,$ and $\,q_{\al\be\ga}$ of the torsion tensor are included. In fact, the introduction of external conformally covariant sources for $\,S_{\mu_1 ... \mu_{n-3}}$ , $\,{q^{\alpha}}_{\beta\gamma}\,$ or to the transverse component of $\,T_{\al}\,$ does not spoil the conformal symmetry.
One has to remark, that the theory with torsion provides, for $\,\frac{\xi^2_2}{\xi_3}-1 > 0$, the equivalence of the positively defined scalar action (\[n7\]) to the action (\[0.1\]) with the negative sign. The negative sign in (\[0.1\]) signifies, in turn, the positively defined gravitational action. Without torsion one can achieve positivity in the gravitational action only by the expense of the negative kinetic energy for the scalar action in (\[n5\]). Thus, the introduction of torsion may lead to some theoretical advantage.
The equation of motion for $T_\al$ may be regarded as a constraint that fixes the conformal transformation for this vector to be consistent with the one for the metric and scalar. Then, instead of (\[nnn\]), one has g\^\_=g\_e\^[2(x)]{}, \^ = e\^[(1-)(x)]{} , T\^\_= T\_+ (1-)\_(x) \[mmm\] Now, in order to be sure about the number of degrees of freedom in this theory, let us compute the remaining field equations for the whole set of fields. From this instant we consider free theory and put the coupling constant $\,\la=0$, because scalar self-interaction does not lead to the change of the qualitative results.
The dynamical equations for the theory encompass eqs. (\[n9\]) along with &&\_1\^2(R\_-g\_R ) -g\_()\^2 + \_\_\
&+& \_2\^2 (T\_T\_- g\_T\_ T\^ ) + \_3\^2( \_T\_- g\_\_T\^)=0 \[teq\] and (-\_1R-\_2\_T\^-\_3T\_T\^) =0. In case of the on-shell torsion (\[n9\]), equations (\[teq\]) reduce to: & &\^2 (R\_-g\_R )-g\_()\^2+ \_\_=0,\
& & -()R=0. \[equa\] Taking the trace of the first equation, it can be readily noticed that this equation is exactly the same as the one for scalar field. This indeed justifies our procedure for replacing (\[n9\]) into the action. It is easy to check, by direct inspection, that even off-shell, the theory with torsion (\[n7\]), satisfying the relation (\[n8\]), may be conformally invariant whenever we define the transformation law for the torsion trace according to (\[mmm\]): and also postulate that the other pieces of the torsion, $S_\mu$ and ${q^\al}_{\be\ga}$, do not transform. The quantities $\,\sqrt{-g}\,$ and $\,R\,$ transform as in (\[n1\]). One may introduce into the action other conformal invariant terms depending on the torsion. For instance: S = - d\^nx\^ T\_ T\^, where $T_{\al\be}=\partial_\al T_\be-\partial_\be T_\al$. This term reduces to the usual vector action when $n\to 4$. It is not difficult to propose other conformally invariant terms containing other components of the torsion tensor in (\[irr\]).
One can better understand the equivalence between GR and conformal metric-scalar-torsion theory (\[n7\]), (\[n8\]) after presenting an alternative form for the symmetric action. It proves a useful way to divide the torsion trace $T_{\mu}$ into longitudinal and transverse parts: T\_ = T\^\_ + \_T , \[dividi\] where $\,\na^\mu T^{\bot}_{\mu} = 0\,$ and $\,T\,$ is the scalar component of the torsion trace, $T_\mu$. The coefficient $\,\frac{\xi_2}{\xi_3}\,$ has been introduced for the sake of convenience. Under the conformal transformation (\[mmm\]), the transverse part is inert and the scalar component transforms as $\,T^\prime = T - \si$. $\,$ Now, we can see that the conformal invariant components, $\,S,T^{\bot}\,$ and $\,q$, appear in the action (\[n7\]) in the combination \^2 = \_3(T\^\_)\^2 + \_4 S\_[\_1\_2 ... \_[n-3]{}]{}S\^[\_1\_2 ... \_[n-3]{}]{} + \_5 q\_\^2, \[combi\] which has the conformal transformation, $\,{{\cal M}^2}^\prime
\rightarrow e^{(2-n)\rho (x)}\cdot {\cal M}^2\,$ similar to that of the square of the scalar field. In fact, the active compensating role of the torsion trace, $T_{\mu}$, is accommodated in the scalar mode $T$. Other components of the torsion tensor, including $T^{\bot}_{\mu}$, appear only in the form (\[combi\]) and allow to create some kind of [*conformally covariant*]{} “mass”. On the mass-shell, this “mass” disappears because all its constituents, $\,T^{\bot}_{\mu}, S_{\mu_1 ...\mu_{n-3}}\,$ and $\,q_{\mu\nu\la}$, vanish.
The next observation is that all torsion-dependent terms may be unified in the expression $$P = -\frac{n-2}{4(n-1)}\,\frac{\xi^2_2}{\xi_3}\,R +
\xi_2\,(\na_\mu T^\mu) +
\xi_3\,T_\mu T^\mu +
\xi_4\, S_{\mu_1\mu_2 ... \mu_{n-3}}^2
+ \xi_5\,q_{\mu\nu\la}^2$$ = - + [M]{}\^2 . \[P\] It is easy to check that the transformation law for this $\,P\,$ is the same as the one for $\,{\cal M}^2$. Using new definitions, the invariant action becomes S\_[inv]{} = d\^nx { g\^\_ \_+ R\^2 + 12P\^2 }. \[inv\] Let us make a change of variables in the last action: $$T = \ln \psi\,,$$ with obvious transformation law, $\psi' = \psi\cdot e^{-\sigma}$, for the new scalar $\,\psi$. After some small algebra, one can cast the action (\[inv\]) in the form of a 2-component conformally invariant sigma-model: S\_[inv]{} = 12 d\^nx { - \_2 +[M]{}\^2\^2 + (\^2\^[-1]{})\_2 }, \[inv-new\] where $$\De_2 = {\Box} - \frac{n-2}{4(n-1)}\,R$$ is a second-derivative conformally covariant operator acting on scalars. It is easy to check that if one takes, for example, the conformally flat metric, $g_{\mu\nu}=\eta_{\mu\nu}\cdot \chi^2(x)$, one arrives at three-scalar non-linear sigma-model, but two of these three scalars are dependent. The last expression for the action (\[inv-new\]) shows explicitly the conformal invariance of the action and also confirms the conformal covariance of the quantity $\,P$.
One may construct a trivial 2-scalar sigma-model conformally equivalent to GR, by making a substitution $\,\si \rightarrow \si_1 + \si_2\,$ in the action (\[n2\]), and after regarding $\,\si_1(x)\,$ and $\,\si_2(x)\,$ as distinct fields. A detailed analysis shows that the theory (\[inv-new\]) can not be reduced to such a 2-scalar sigma-model by a change of variables. In order to understand why this is so, we need one more representation for the metric-scalar-torsion action with local conformal symmetry.
Let us start, once again, from the action (\[n7\]), (\[n8\]) and perform only part of the transformations (\[mmm\]): \^ = e\^[(1-)(x)]{} , T\_T\^\_= T\_+ (1-)\_(x). \[partofmmm\] Of course, if we supplement (\[partofmmm\]) by the transformation of the metric, we arrive at (\[mmm\]) and the action does not change. On the other hand, the results of Section 2 suggest that (\[partofmmm\]) may lead to an alternative conformally equivalent description of the theory. Taking $\,\ph \cdot e^{(1-\frac{n}{2})\rho(x)} =
\frac{8(n-1)}{G\,(n-2)}\,\left(1-\frac{\xi^2_2}{\xi_3}\right)
= const$, we obtain, after some algebra, the following action: $$S = \frac{1}{G}\,\int d^nx \sqrt{-g}\;R +$$ + d\^nx { \_4 S\_[\_1 ... \_[n-3]{}]{}\^2 + \_5 q\_\^2 + \_3 (T\_-\_)\^2}. \[oneq\] This form of the action does not contain interaction between the curvature and the scalar field. At the same time, the latter field is present until we use the equations of motion (\[n9\]) for torsion. Torsion trace looks here like a Lagrange multiplier, and only using its corresponding equation of motion (and also for other components of torsion), we can obtain the action of GR. It is clear that one can arrive at the same action (\[oneq\]) making the transformation of the metric as in (\[mmm\]) instead of (\[partofmmm\]).
To complete this part of our consideration, we mention that the direct generalization of the Einstein-Cartan theory including an extra scalar may be conformally equivalent to General Relativity, provided that the non-minimal parameter takes an appropriate value. To see this, one uses the relation = R - 2\_ T\^- T\_T\^+ q\_ q\^ + S\^[\_1 ... \_[n-3]{}]{} S\_[\_1 ... \_[n-3]{}]{} \[curvator\] and replace it into the “minimal” action S\_[ECBD]{} = d\^nx { g\^\_ \_+[R]{}\^2 } . \[mini\] It is easy to see that the condition (\[n8\]) is satisfied for the special value = . \[special\] In particular, in the four-dimensional case, the symmetric version of the theory corresponds to $\xi = \frac13$, contrary to the famous $\xi = \frac16$ in the torsionless case. The effect of changing conformal value of $\xi$ due to the non-trivial transformation of torsion has been discussed earlier in [@payo] (see also further references there).
In the next section we shall see how the non-trivial conformal transformation for torsion changes the Noether identity and the quantum conformal anomaly.
Divergences, anomaly and induced effective action
==================================================
In four dimensions, by integrating over the free scalar field (even without self-interaction), one meets vacuum divergences and the resulting trace anomaly breaks conformal invariance. As it was already mentioned in the Introduction, the anomaly and its application is one of the most important aspects of the conformal theories. The anomaly is a consequence of the quantization procedure, and it appears due to the lack of a completely invariant regularization. In particular, trace anomaly is usually related to the one-loop divergences [@duff]. At the same time, one has to be careful, because the non-critical use of this relation may, in principle, lead to mistakes.
Consider the renormalization and anomaly for the conformal metric-scalar theory with torsion formulated above. The renormalizability of the theory requires the vacuum action to be introduced, which has to be (as it was already noticed in Section 3) of the form of the possible counterterms. The total four-dimensional action including the vacuum term can be presented as: S\_t = S\_[inv]{} + S\_[vac]{}, \[tot\] where $S_{inv}$ has been given in (\[inv\]) and the form of the vacuum action will be established later.
Before going on to calculate the divergences and anomaly, one has to write a functional form for the conformal symmetry. It is easy to see that the Noether identity corresponding to (\[mmm\]) looks like 2 g\_ +\_ - =0 . \[neuth\] Now, if we are considering both metric and torsion as external fields, and only the scalar as a quantum field, in the vacuum sector we meet the first two terms of (\[neuth\]). This means (due to the invariance of the vacuum divergences) that the vacuum action may be chosen in such a way that -[T]{} = 2 g\_ +\_ =0 . \[vacu\] The new form (\[vacu\]) of the conformal Noether identity indicates to a serious modification in the conformal anomaly. In the theory under discussion, the anomaly would mean $\,<{\cal T}>\,\neq\,0$ instead of usual $\,<{T}^\mu_\mu>\,\neq\,0$. Therefore, at first sight, we meet here some special case and one can not directly use the relation between the one-loop counterterms and the conformal anomaly derived in [@duff], because this relation does not take into account the non-trivial transformation law for the torsion field. It is reasonable to remind that, for the case of completely antisymmetric torsion discussed in [@buodsh], this problem did not show up just because $S_\mu$ is inert under conformal transformation.
Let us formulate a more general statement about conformal transformation and anomaly. When the classical metric background is enriched by the fields which do not transform, the Noether identity corresponding to the conformal symmetry remains the same as for the pure metric background. In this case one can safely use the standard relations [@duff] between divergences and conformal anomaly. However, if the new background field has a non-trivial transformation law, one has to care about possible modifications of the Noether identity and consequent change of anomaly.
In our case, the anomaly is modified, because it has a new functional form, $\,<{\cal T}>\,\neq\,0$. One can derive this new anomaly using, for instance, the methods described in [@duff] or [@book]. However, it is possible to find $\,<{\cal T}>\,$ in a more economic way, using some special decomposition of the background fields. As we shall see, the practical calculation of a new anomaly and even the anomaly-induced action may be reduced to the results known from [@duff; @rei; @frts] and especially [@buodsh], where the theory of the antisymmetric torsion was investigated.
Our purpose is to change the background variables in such a way that the transformation of torsion is absorbed by that of the metric. The crucial observation is that $P$, from (\[P\]), transforms[^4] under (\[mmm\]) as $P^\prime = P \cdot
e^{-2\rho(x)}$. Therefore, the non-trivial transformation of torsion is completely absorbed by $P$. Since $P$ only depends on the background fields, we can present it in any useful form. One can imagine, for instance, $P$ to be of the form $\,P=g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu\,$ where vector $\,\Pi_\mu\,$ doesn’t transform, and then the calculation readily reduces to the case of an antisymmetric torsion [@buodsh]. In particular, we can now use standard results for the relation between divergences and anomaly [@duff].
In the framework of the Schwinger-DeWitt technique, we find the 1-loop counterterms in the form \^[(1)]{}\_[div]{} = - d\^nx{C\^2-E + R + P + P\^2 }, \[div\] Here $$C^2 = C_{\mu\nu\al\be}C^{\mu\nu\al\be}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
{\rm and} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
E = R_{\mu\nu\al\be}R^{\mu\nu\al\be} - 4R_{\mu\nu\al\be}R^{\mu\nu}
+R^2$$ are the square of the Weyl tensor, and the scalar integrand of the Gauss-Bonnet term. The eq. (\[div\]) gives, as a by-product, the list of necessary terms in the vacuum action. Taking into account the arguments presented above, we can immediately cast the anomaly under the form <[T]{}> = - . \[tracean\] One may proceed and, following [@rei; @frts; @buodsh], derive the conformal non-invariant part of the effective action of the vacuum, which is responsible for the anomaly (\[tracean\]). Taking into account our previous treatment of the conformal transformation of torsion, we consider it is hidden inside the quantity $P$ of eq. (\[P\]), and again imagine $P$ to be of the form $\,P=g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu\,$. Then the equation for the effective action $\Ga [g_{\mu\nu},\Pi_\al]\,$ is [^5] - g\_ = <[T]{}>. \[mainequation\] In order to find the solution for $\,\Ga$, we can factor out the conformal piece of the metric ${g}_{\mu\nu} = {\bar g}_{\mu\nu}\cdot
e^{2\si}$, where ${\bar g}_{\mu\nu}$ has fixed determinant and put $\,P = {\bar P}\cdot e^{-2\si(x)}$, that corresponds to $\,{\bar \Pi}_\al = \Pi_\al$. This transformation for artificial variable $\Pi_\al$ is identical to the one for the $S_\al$ pseudovector, that is the case considered in [@buodsh]. Then the result can be obtained directly from the effective action derived in [@buodsh], and we get $${\Ga}\, = \,S_c[{\bar g}_{\mu\nu}; {\bar P}]
\,- \,\frac{1}{12}\cdot\frac{1}{270 (4\pi)^2}\,\int d^4 x
\sqrt{-g (x)}\,R^2(x) + \frac{1}{(4\pi)^2}\,
\int d^4 x\sqrt{-{\bar g}}\,\left\{
\,\si\cdot [ \frac{1}{120}\,{\bar C}^2 -
\right.$$ . - ([|E]{} - 23[|]{}\^2 [|R]{}) + 12[|P]{}\^2 \] + -16([|]{}\_)[|]{}\^ +16[|P]{}([|]{}\_)\^2}, \[efac\] where $S_c[{\bar g}_{\mu\nu}; {\bar P}]\,$ is an unknown functional of the metric ${\bar g}_{\mu\nu}(x)$ and $\,{\bar P}\,$, which acts as an integration constant for any solution of (\[mainequation\]).
Now, one has to rewrite (\[efac\]) in terms of the original field variables, $g_{\mu\nu}, {T^\al}_{\be\ga}$. Here, we meet a small problem, because we only have, for the moment, the definition $\Pi_\al = {\bar \Pi}_\al$ for the artificial variable $\Pi_\al$, but not for the torsion. Using the previous result (\[mmm\]), we can define \_ = [[|T]{}\^]{}\_ -, \[mmm-new\] so that $\,{{\bar T}^\al}_{\,\,\be\ga}\,$ is an arbitrary tensor. Also, we call ${\bar T}^\al = {\bar g}^{\al\be}\,{\bar T}_\be$ etc. Now, we can rewrite (\[efac\]) in terms of metric and torsion components $${\Ga}\, = \,S_c[{\bar g}_{\mu\nu}; {{\bar T}^\al}_{\,\,\be\ga}]
\,- \,\frac{1}{12}\cdot\frac{1}{270 (4\pi)^2}\,\int d^4 x \sqrt{-g
(x)}\,R^2(x) +$$$$+ \frac{1}{(4\pi)^2}\, \int d^4 x\sqrt{-{\bar
g}}\,\left\{ \,+ \frac{1}{180}\,\si {\bar \De}\si +
\frac{1}{120}\,{\bar C}^2\,\si - \frac{1}{360}\,({\bar E} -
\frac23\,{\bar \na}^2 {\bar R})\,\si \right.$$$$\left. +
\frac{1}{72}\,\si\,\left[ \,-\frac{\xi^2_2}{\xi_3}\,{\bar R} +
6\xi_2\,({\bar \na}_\mu {\bar T}^\mu) + 6\xi_3\,{\bar T}_\mu {\bar
T}^\mu + 6\xi_4\, {\bar S}_{\mu}{\bar S}^{\mu} + 6\xi_5\,{\bar
q}_{\mu\nu\la}{\bar q}^{\mu\nu\la} \right]^2 + \right.$$ . 16}, \[efac-final\] This effective action is nothing but the generalization of the similar expressions of [@rei; @frts; @buodsh] for the case of general metric-torsion background and conformal symmetry described in Section 3. The curvature dependence in the last two terms appears due to the non-trivial transformation law for torsion.
Conclusions.
============
We have considered the conformal properties of the second-derivative metric-torsion-dilaton gravity, on both classical and quantum level. The main results, achieved above, can be summarized as follows:
i\) If the parameters $\,\xi_i\,$ of the general metric-torsion-dilaton theory (\[n7\]) satisfy the relation (\[n8\]), there is conformal invariance of the new type, more general than the one considered in [@obukhov; @payo] and different from the ones considered in [@bush; @book].
ii\) In this case the vector trace of torsion plays the role of a compensating field, providing the classical conformal on-shell equivalence between this theory and General Relativity. The conformal invariant scalar-metric-torsion action can be presented in alternative forms (\[inv\]), (\[inv-new\]), (\[oneq\]), while (\[inv-new\]) is some new nonlinear $n$-dimensional conformal invariant sigma-model. If the nonminmal parameters satisfy an additional relation $\,\,{\xi^2_2} > {\xi_3}$, the equivalence holds between the positively defined General Relativity and scalar field theory. This indicates to the stabilization of the conformal mode in gravity with torsion and may indicate that the introduction of torsion can be some useful alternative to other approaches to this problem (see [@wet; @mod]).
iii\) The conformal Noether identity (\[neuth\]) indicates that the conformal anomaly with torsion is not the anomaly of the trace of the Energy-Momentum Tensor, but rather the anomaly of some modified quantity $\,{\cal T}$. But, using the appropriate decomposition of the background variables, the calculation of this new anomaly can be readily reduced to the use of the known results of [@duff] and [@buodsh], and the derivation of the anomaly-induced effective action can be done through the similar decomposition of variables and the use of the known results of [@rei; @frts; @buodsh]. As in other known cases, the induced action (\[efac-final\]) breaks the conformal equivalence between metric-torsion-scalar action and (\[0.1\]) and therefore may lead to the nontrivial applications to inflationary cosmology.
[**Acknowledgments.**]{} I.Sh. is grateful to the CNPq for permanent support. His work was partially supported by RFFI (project 99-02-16617). A.P.F. and J.A.H.-N. are grateful to G. de Berredo Peixoto for discussions and careful reading of the work.
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[^1]: e-mail: helayel@cat.cbpf.br
[^2]: e-mail: apenna@rio.nutecnet.com.br
[^3]: e-mail: shapiro@fisica.ufjf.br
[^4]: As a consequence, the action $\,\int\sqrt{-g}P\phi^2\,$ is conformal invariant. This fact has been originally discovered in [@obukhov].
[^5]: We remark that this equation is valid only for the “artificial” effective action $\Ga [g_{\mu\nu},\Pi_\al]\,$, while the effective action in original variables $g_{\mu\nu}, T^\al_{\be\ga}$ would satisfy the modified equation (\[vacu\]). The standard form of the equation for the effective action is achieved through the special decomposition of the external fields.
|
---
abstract: 'We prove that any smooth complex projective variety with generic vanishing index bigger or equal than $2$ has birational bicanonical map. Therefore, if $X$ is a smooth complex projective variety $X$ with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of $X$ is fibred by subvarieties of codimension at most $1$ of an abelian subvariety of $\operatorname{Alb}X$.'
address: 'Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona. Gran Via, 585, 08007 Barcelona. Spain'
author:
- Martí Lahoz
title: Generic Vanishing Index and the Birationality of the Bicanonical Map of Irregular Varieties
---
[^1]
Introduction
============
In the study of smooth complex algebraic varieties, the natural maps provided by the holomorphic forms defined in the variety, have a special importance. For example, the invertible sheaf $\omega_X$ of differential $n$-forms (where $n$ is the dimension of $X$) produces a map to a projective space, known as the canonical map. The multiples of this canonical sheaf $\omega_X^{\otimes m}$ produce in this way the pluricanonical maps. $$\varphi_m:X\dashrightarrow {\mathbb{P}}^N= {\mathbb{P}}(H^0(X,\omega_X^{\otimes m})^\vee).$$ When $\varphi_m$ gives a birational equivalence between $X$ and its image, we will simply say that $\varphi_m$ is birational. We say that $X$ is of general type if for some $m>0$ the rational map $\varphi_m$ is birational.
For example, the curves of general type are those of genus $g\geq 2$. The tricanonical map $\varphi_3$ is always birational for such curves and the bicanonical $\varphi_2$ is also birational once that $g\geq 3$. Moreover, the canonical map is birational as soon as the curve is non-hyperelliptic.
For surfaces, Bombieri [[@Bom]]{} have given sharp numerical conditions for the birationality of $\varphi_m$ for $m\geq 3$. The bicanonical map has revealed to be more complicated and has studied by many algebraic geometers. In fact, the surfaces with irregularity $q(S)\leq 1$ and $\chi(S,\omega_S)=1$ are not completely understood and there is no classification about which ones have birational $\varphi_2$. For a modern review of the state of the art in the surface case, we refer to [@BCP Thm. 8].
For higher dimensions not many results are known in general. Nevertheless, the example of the bicanonical map on surfaces shows that for small irregularity $q(X)=h^0(X,\Omega_X^1)$, the classification becomes more difficult. For complex varieties, recall that the differential $1$-forms give rise to the Albanese map $$\operatorname{alb}: X\to \operatorname{Alb}X= H^0(X,\Omega_X^1)^\vee/H_1(X,\mathbb{Z}).$$ from $X$ to an abelian variety of dimension $q(X)=h^0(X,\Omega_X^1)$. We say that $X$ is *irregular* if, and only if, $\operatorname{Alb}X$ is not trivial, i.e. $q(X)>0$. And we say that $X$ is of *maximal Albanese dimension (m.A.d)* if, and only if, the Albanese map $\operatorname{alb}: X\to \operatorname{Alb}X$ is generically finite onto its image.
It turns out that some properties of m.A.d varieties seem to behave independently of the dimension and, indeed, Chen-Hacon showed that this is the case for their pluricanonical maps.
$\;$
1. $X$ m.A.d and $\chi(\omega_X)>0$ $\Rightarrow$ $X$ is of general type, furthermore, $\varphi_3$ is birational.
2. $X$ m.A.d $\Rightarrow$ $\varphi_6$ is the stable pluricanonical map.
For $\varphi_2$, we cannot expect to use $\chi(\omega_X)$ to control directly its birationality. For example, if $C$ is a curve of genus $2$, then the bicanonical map of the product $C\times Y$ is never birational. In fact, it is clear that any variety that admits a fibration whose general fibre has non-birational $\varphi_2$ will have a non-birational bicanonical map. This should be considered, at least at first glance, as the standard case for higher dimensional varieties.
The following theorem provides geometric constraints for the non-birationality of the bicanonical map (see Theorem \[thm:corfibr\]).
\[thm:geomfibrA\] Let $X$ be a smooth projective complex variety of maximal Albanese dimension such that the bicanonical map is not birational. Then, the Albanese image of $X$ is fibred by subvarieties of codimension at most $1$ of an abelian subvariety of $\operatorname{Alb}X$. The base of the fibration is also of maximal Albanese dimension.
That is, $X$ admits a fibration onto a normal projective variety $Y$ with $0 \leq \dim Y <
\dim X$, such that any smooth model $\tilde Y$ of $Y$ is of maximal Albanese dimension and $$q(X) - \dim X \leq q(\tilde Y) - \dim Y +1.$$ Hence, when $q(X)>\dim X+1$ implies the existence of an actual fibration, i.e. $\dim Y>0$, whose general fibre is mapped generically finite through the Albanese map of $X$ either onto a fixed abelian subvariety of $\operatorname{Alb}X$, or onto a divisor of this fixed abelian subvariety. When $\dim Y=0$ the theorem simply says that the image of $X$ in $\operatorname{Alb}X$ has codimension at most $1$.
In particular, when $X$ does not admit any fibration and $q(X)>\dim X$, there is only one possible case, i.e. $X$ is birationally equivalent to a theta-divisor of an indecomposable principally polarized abelian variety (see [[@BLNP Thm. A]]{}). When $X$ does not admit any fibration and $q(X)=\dim X$, there is only one known case of variety of general type and non-birational bicanonical map: a double cover of a principally polarized abelian variety $(A,\Theta)$ branched along a reduced divisor $B\in {\left\vert2\Theta\right\vert}$. Is this the only case? The answer is affirmative in the case of surfaces due to Ciliberto-Mendes Lopes [@CM Thm 1.1].
To deduce Theorem \[thm:geomfibrA\] it is useful to consider the generic vanishing index introduced by Pareschi–Popa in [@PPCdF Def. 3.1] $$\operatorname{gv}(\omega_X)=\min_{i>0}{\left\{\operatorname{codim}_{\operatorname{Pic}^0X} V^i(\omega_X)-i\right\}},$$ where $V^i(\omega_X)={\left\{\alpha\in\operatorname{Pic}^0X{\;\vline\;}h^i(X,\omega_X\otimes\alpha)>0\right\}}$. As a consequence of Generic Vanishing Theorem of Green–Lazarsfeld [@GL1 Thm. 1], we have that for any irregular variety $1-\dim X\leq \operatorname{gv}(\omega_X)\leq q(X)-\dim X$.
Moreover, the negative values of $\operatorname{gv}(\omega_X)$ can be interpreted in terms of the dimension of the generic fibre of the Albanese map (see Theorem \[thm:GL1\]) and $X$ is a m.A.d variety if, and only if, $\operatorname{gv}(\omega_X)\geq 0$. Due to the work of Pareschi–Popa [@PPCdF] we can interpret the positive values of $\operatorname{gv}(\omega_X)$ in terms of the local properties of the Fourier-Mukai transform of the structural sheaf (see Theorem \[thm:posit\_gv\]). They have also proved that the positive values of $\operatorname{gv}(\omega_X)$ give a lower bound for the Euler characteristic $\chi(\omega_X)$ (see Theorem \[thm:CdF\]).
Using the generic vanishing index we have the following more synthetic result.
Let $X$ be a smooth projective complex variety such that $\operatorname{gv}(\omega_X)\geq 2$. Then, the rational map associated to $\omega_X^{2}\otimes \alpha$ is birational onto its image for every $\alpha\in\operatorname{Pic}^0 X$.
Theorem \[thm:geomfibrA\] is deduced from this result by an argument of Pareschi-Popa. On the other hand, this result (see Theorem \[thm:fibr\]) is proved using a birationality criterion (see Lemma \[lem:birprov\]) that is a slight modification of [@BLNP Thm. 4.13].
For curves, $\operatorname{gv}(\omega_C)\geq 2$ is equivalent to $g(C) \geq 3$. For surfaces, $\operatorname{gv}(\omega_S)\geq 2$ is equivalent to suppose that $q(S)\geq 4$ and does not admit an irregular fibration to a curve of genus $\leq q(S)-3$ (see Example \[ex:exemples\]).\
[**Acknowledgments.**]{} This is part of my Ph.D. thesis. I would like to thank my advisors, M.A. Barja and J.C. Naranjo, for their valuable suggestions. I am also grateful to G. Pareschi for helpful conversations during my Ph.D studies. I would like to thank also R. Pardini for her useful comments.
Generalized Fourier-Mukai transform
===================================
$X$ will be a smooth projective variety over an algebraically closed field $k$ (from section \[sub:char0\] on, we will restrict to $k={\mathbb{C}}$). It will be equipped with a morphism $a:X\to A$ to a non-trivial abelian variety $A$, in particular, $X$ will be irregular. Let ${\mathcal{P}}$ be a Poincaré line bundle on $A\times \operatorname{Pic}^0A$. We will denote $$\label{eq:PoicPa}
P_a=(a\times \operatorname{id}_{\operatorname{Pic}^0X})^*{\mathcal{P}},$$ the *induced Poincaré line bundle* in $X\times \operatorname{Pic}^0A$. When $a=\operatorname{alb}$, the Albanese map of $X$, then the map $\operatorname{alb}^*$ identifies $\operatorname{Pic}^0(\operatorname{Alb}X)$ to $\operatorname{Pic}^0X$ and the line bundle $P_{\operatorname{alb}}$ will be simply denoted by $P$.
Letting $p$ and $q$ the two projections of $X\times \operatorname{Pic}^0 A$, we consider the left exact functor $\Phi_{P_a} {\mathcal{F}}=q_*(p^*{\mathcal{F}}\otimes P_a)$, and its right derived functors $$\label{eq:FMPa} R^i\Phi_{P_a} {\mathcal{F}}= R^iq_*(p^*{\mathcal{F}}\otimes P_a).$$ Sometimes we will have to consider the analogous derived functor $R^i\Phi_{P_a^{-1}}{\mathcal{F}}$ as well. By the Seesaw Theorem [@MAV Cor. 6, pg. 54], ${\mathcal{P}}^{-1}=
(1_A\times
(-1)_{\operatorname{Pic}^0 A})^*{\mathcal{P}}$, so $$\label{eq:FMPdual} R^i\Phi_{P_a^{-1}}{\mathcal{F}}= (-1_{\operatorname{Pic}^0 A})^*R^i\Phi_{P_a}{\mathcal{F}}\qquad\text{for any }i.$$ Given a coherent sheaf ${\mathcal{F}}$ on $X$, its *i-th cohomological support locus with respect to $a$* is $$V^i_a({\mathcal{F}})={\left\{\alpha \in \operatorname{Pic}^0A {\;\vline\;}h^i({\mathcal{F}}\otimes a^*\alpha)>0\right\}}$$ Again, when $a$ is the Albanese map of $X$, we will omit the subscript, simply writing $V^i({\mathcal{F}})$. By base change, these loci contain the set-theoretical support of $R^i\Phi_{P_a}{\mathcal{F}}$, i.e. $\operatorname{supp}R^i\Phi_{P_a}{\mathcal{F}}\subseteq V^i_a({\mathcal{F}})$.
A way to measure the size of all the $V^i_a({\mathcal{F}})$’s is provided by the following invariant introduced by Pareschi–Popa.
\[def:gvindex\] Given a coherent sheaf ${\mathcal{F}}$ on $X$, the *generic vanishing index* of ${\mathcal{F}}$ (with respect to $a$) is $$\operatorname{gv}_a({\mathcal{F}}):=\min_{i>0}{\left\{\operatorname{codim}_{\operatorname{Pic}^0 A}V^i_a({\mathcal{F}})-i\right\}}.$$ By convention we define $\operatorname{gv}_a({\mathcal{F}})=\infty$, when $V^i_a({\mathcal{F}})=\emptyset$ for every $i>0$. When $a$ is the Albanese map of $X$, we will omit the subscript, simply writing $\operatorname{gv}({\mathcal{F}})$.
By base change (see [@PPCdF Lem. 2.1]) it is easy to see that $\operatorname{gv}_a({\mathcal{F}})$ can be also defined as the $\min_{i>0}{\left\{\operatorname{codim}_{\operatorname{Pic}^0 A}\operatorname{supp}R^i\Phi_{P_a}{\mathcal{F}}- i\right\}}$.
Generic vanishing index of the canonical sheaf
==============================================
Relations between $\operatorname{gv}(\omega_X)$ and the Fourier-Mukai transform of ${\mathcal{O}}_X$
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Here we specialize some general results of Pareschi–Popa [@PPCdF; @PPGVsheaves] to the canonical sheaf of a smooth projective variety of dimension $d$. Some of these results were previously obtained by Hacon (see [@Happroach]).
The negative values of the $\operatorname{gv}$-index are related with the vanishing of the lowest cohomologies of the Fourier-Mukai transform of its Grothendieck dual. In the case of $\omega_X$ this can be stressed simply as:
\[thm:GVvsWIT\] The following are equivalent,
$\operatorname{gv}_a(\omega_X)\geq -e$ for $e\geq 0$;
$R^i\Phi_{P_a}{\mathcal{O}}_X =0$ for all $i\neq d-e,\ldots,d$.
Hence, when $\operatorname{gv}_a(\omega_X)\geq 0$, $R^i\Phi_{P_a}{\mathcal{O}}_X=0$ for all $i\neq d$, and we usually denote $$\widehat{{\mathcal{O}}_X}=R^d\Phi_{P_a} {\mathcal{O}}_X.$$ Note that, in this case, $H^i(X,\omega_X\otimes a^*\alpha)=0$ for all $i>0$ and general $\alpha\in
\operatorname{Pic}^0 A$. Therefore, by deformation-invariance of $\chi$, the generic value of $h^0(X,\omega_X\otimes a^*\alpha)$ equals $\chi(\omega_X)$, in particular $\chi(\omega_X)\geq 0$. Since, by base-change, the fibre of $\widehat{{\mathcal{O}}_X}$ at a general point $\alpha\in \operatorname{Pic}^0 A$ is isomorphic to $H^{d}(X,a^*\alpha)\cong H^0(X,\omega_X \otimes a^*\alpha^{-1})^*$, the (generic) rank of $\widehat{{\mathcal{O}}_X}$ is $\operatorname{rk}\widehat{{\mathcal{O}}_X}=\chi(\omega_X)$.
From Grothendieck-Verdier duality [@conrad Thm. 4.3.1] and Theorem \[thm:GVvsWIT\] it follows that,
\[cor:duality\] If $\operatorname{gv}_a(\omega_X)\geq 0$ then $\operatorname{\mathcal{E}\mathit{xt}}^i_{{\mathcal{O}}_{\operatorname{Pic}^0 A}}((-1_{\operatorname{Pic}^0 A})^*\widehat{{\mathcal{O}}_X},{\mathcal{O}}_{\operatorname{Pic}^0 A})\cong
R^i\Phi_{P_a}\omega_X$.
The following result of Pareschi–Popa gives a dictionary between the positive values of $\operatorname{gv}_a(\omega_X)$ and the local properties of the Fourier-Mukai transform of $\widehat{{\mathcal{O}}_X}$.
\[thm:posit\_gv\] Assume that $\operatorname{gv}_a(\omega_X)\geq 0$. Then, $$\operatorname{gv}_a(\omega_X)\geq m\;\text{ if, and only if, }\;\widehat{{\mathcal{O}}_X}\text{ is a }m\text{-syzygy sheaf.}$$ In particular, $\operatorname{gv}_a(\omega_X)\geq 1$ is equivalent to $\widehat{{\mathcal{O}}_X}$ being torsion-free and $\operatorname{gv}_a(\omega_X)\geq 2$ to $\widehat{{\mathcal{O}}_X}$ being reflexive.
Using the Evans–Griffith Syzygy Theorem and the previous theorem, Pareschi–Popa obtain the following bound on the Euler holomorphic characteristic that generalizes to higher dimensions the Castelnuovo-de Franchis inequality.
\[thm:CdF\] Assume that $\operatorname{gv}_a(\omega_X)\geq 0$. Then, $\chi(\omega_X) \geq \operatorname{gv}_a(\omega_X)$.
\[rem:CdF\] In fact, the theorem of Pareschi–Popa is more general, namely that for any coherent sheaf ${\mathcal{F}}$ if $\infty>\operatorname{gv}_a({\mathcal{F}})\geq 0$, then $\chi({\mathcal{F}})\geq \operatorname{gv}_a({\mathcal{F}})$. As a consequence, we easily obtain that for any non-zero coherent sheaf ${\mathcal{F}}$, $\operatorname{gv}_a({\mathcal{F}})\geq 1\Rightarrow
\chi({\mathcal{F}})\geq 1$. Observe also that if $a$ is non-trivial, we always have $\operatorname{gv}_a(\omega_X)<\infty$.
Top Fourier-Mukai transform of the canonical sheaf
--------------------------------------------------
In the case of abelian varieties (or complex torus) the following result is well-known and crucial in the proof of the Mukai Equivalence Theorem [@Mdual Thm 2.2]. We will need it in the proof of Theorem \[thm:fibr\].
\[prop:IrrMukai\] If $a^*:\operatorname{Pic}^0A\to \operatorname{Pic}^0X$ is an embedding, then $$R^d\Phi_{P_a} \omega_X\cong k(\hat 0).$$
Generic vanishing theorem of Green–Lazarsfeld {#sub:char0}
---------------------------------------------
The name of the $\operatorname{gv}$-index comes from the well-known Generic Vanishing Theorem of Green–Lazarsfeld. As other general vanishing theorems, it requires $\operatorname{char}k=0$ so from now on we will restrict ourselves to the case $k={\mathbb{C}}$. Basically, the following theorem is [@GL1 Thm. 1]. The converse implication was proven independently in [@LP Thm. B] and [@BLNP Prop. 2.7].
\[thm:GL1\] For any $e>0$, the following are equivalent:
the generic fibre of $a\colon X\to A$ has dimension $e$,
$\operatorname{gv}_a(\omega_X)= -e$.
Moreover $\operatorname{gv}_a(\omega_X)\geq 0$ if, and only if, $a\colon X\to A$ is generically finite onto its image.
In particular, observe that for any irregular variety $1-\dim X\leq \operatorname{gv}(\omega_X)\leq q(X)-\dim X$.
\[rem:gt\] If $\operatorname{gv}_a(\omega_X)\geq 0$ and $\chi(\omega_X)>0$, then $X$ is a variety of general type. Indeed, by the previous result $a:X\to A$ is generically finite and since $\chi(\omega_X)>0$, we have that $V_a^0(\omega_X)=\operatorname{Pic}^0A$, so by [@CHpm Cor.2.4], $\kappa(X)=\dim X$. In particular, if $\operatorname{gv}_a(\omega_X)\geq 1$, then $X$ is of general type.
Subtorus theorem of Green–Lazarsfeld and Simpson
------------------------------------------------
The following theorem is due to Green and Lazarsfeld [@GL2 Thm 0.1] with an important addition due to Simpson [@simpson §4,6,7].
\[thm:GL2\] Let $W$ an irreducible component of $V^i(\omega_X)$ for some $i$. Then,
There exists a torsion point $\beta\in \operatorname{Pic}^0X$ and a subtorus $B$ of $\operatorname{Pic}^0X$ such that $W=\beta +B$.
There exists a normal variety $Y$ of dimension $\leq d-i$, such that any smooth model of $Y$ has maximal Albanese dimension and a morphism with connected fibres $f\colon X\to Y$ such that $B$ is contained in $f^*\operatorname{Pic}^0Y$.
\[rem:Steinfact\] It is useful to recall that the morphism $f\colon X\to Y$ in the second part of the previous theorem, arises as the Stein factorization of the morphism $\pi\circ \operatorname{alb}\colon X\to \operatorname{Pic}^0W$, where $\pi\colon\operatorname{Alb}X\to \operatorname{Pic}^0W$ is the dual map of the inclusion $W\subseteq \operatorname{Pic}^0X$. Hence, the key point of the second part of the theorem is the dimensional bound for $Y$.
Birationality criterion for maximal Albanese dimension varieties {#sec:bir2}
================================================================
In this section, we will assume that $a:X\to A$ is a generically finite morphism onto its image, where $A$ is an abelian variety. We introduce another piece of notation.
\[term:last\] Let ${\mathcal{F}}$ be a subsheaf of a line bundle and suppose that $\operatorname{gv}_a({\mathcal{F}})\geq 1$.
We denote $U_{\mathcal{F}}$, the open subset where $h^0({\mathcal{F}}\otimes a^*\alpha)$ has the minimal value, i.e. $\chi({\mathcal{F}})$.
Let $Z$ be the exceptional locus of $a:X\to A$, that is $Z = a^{-1} (T)$, where $T$ is the locus of points in $A$ over which the fibre of $a$ has positive dimension.
We define $${\mathcal{B}}^{\mathcal{F}}_a(x)={\left\{\alpha\in U_{\mathcal{F}}{\;\vline\;}x\text{ is a base point of } {\left\vert{\mathcal{F}}\otimes a^*\alpha\right\vert}\right\}}.$$ By Remark \[rem:CdF\], $\chi({\mathcal{F}})\geq 1$. So, by semicontinuity, it makes sense to speak of the base locus of ${\mathcal{F}}\otimes a^*\alpha$ for all $\alpha\in\operatorname{Pic}^0A$.
The following lemma is a slight modification of [@BLNP Thm. 4.13] and it is based on [@PPreg1 Prop. 2.12 and 2.13].
\[lem:birprov\] Suppose that $a:X\to A$ is a generically finite morphism onto its image and let ${\mathcal{F}}$ be a subsheaf of a line bundle such that $\operatorname{gv}_a({\mathcal{F}})\geq 1$ and $R^ia_*{\mathcal{F}}=0$ for all $i>0$. Suppose that for a general $x\in X$, $$\operatorname{codim}_{U_{\mathcal{F}}} {\mathcal{B}}^{{\mathcal{F}}}_a(x)\geq 2.$$ Then, the rational map associated to the linear system ${\left\vert{\mathcal{F}}\otimes L\right\vert}$ is birational for every line bundle $L$ such that $\operatorname{gv}_a(L)\geq 1$.
We first compare the Fourier-Mukai transform of ${\mathcal{F}}\otimes {\mathcal{I}}_x$ and ${\mathcal{F}}$.
**Claim.** Let $x\in X$ be a closed point out of $Z$. Then $R^ia_*({\mathcal{F}}\otimes {\mathcal{I}}_x \otimes a^*\alpha)=0$ for $i>0$. This follows immediately from the exact sequence $$\label{eq:standard}
0\to {\mathcal{F}}\otimes {\mathcal{I}}_x \to {\mathcal{F}}\to k(x)\to 0$$ and the hypothesis that $R^ia_*{\mathcal{F}}=0$, $a$ is generically finite and $x\not\in Z$. Hence, the degeneration of the Leray spectral sequence yields to $$\label{eq:GRp}V^i_a({\mathcal{F}}\otimes {\mathcal{I}}_x)=V^i(a_*({\mathcal{F}}\otimes {\mathcal{I}}_x)).$$ By sequence , tensored by $a^*\alpha$, it follows that $$\label{eq:equality}V^i_a({\mathcal{F}}\otimes {\mathcal{I}}_x)=V^i_a({\mathcal{F}})\qquad\text{for all } i\geq 2.$$ For $i=1$ we have the surjection $H^1({\mathcal{F}}\otimes {\mathcal{I}}_x\otimes a^*\alpha)\twoheadrightarrow
H^1({\mathcal{F}}\otimes a^*\alpha)$, that is an isomorphism if, and only if, $x$ is not a base point of ${\left\vert{\mathcal{F}}\otimes a^*\alpha\right\vert}$. In other words $V^1_a({\mathcal{F}}\otimes {\mathcal{I}}_x)\subseteq {\mathcal{B}}^{\mathcal{F}}_a(x)\cup V^1_a({\mathcal{F}})$. Since $\operatorname{gv}_a({\mathcal{F}})\geq 1$, the hypothesis on ${\mathcal{B}}^{{\mathcal{F}}}_a(x)$ guarantees that $$\label{eq:codim} \operatorname{codim}V^1_a({\mathcal{F}}\otimes {\mathcal{I}}_x)\geq 2,$$ for a general $x\in X\setminus Z$. Hence by , and , $\operatorname{gv}(a_*({\mathcal{F}}\otimes {\mathcal{I}}_x))\geq 1$. By [@PPreg1 Prop. 2.13], $a_*({\mathcal{F}}\otimes {\mathcal{I}}_x)$ is continuously globally generated (CGG, see [@PPreg1]). Therefore ${\mathcal{F}}\otimes {\mathcal{I}}_x$ itself is CGG outside $Z$ (with respect to $a$). Since the same is true for $L$, it follows from [@PPreg1 Prop 2.12] that for all $\alpha\in\operatorname{Pic}^0 A$, ${\mathcal{F}}\otimes
L\otimes{\mathcal{I}}_x$ is globally generated outside $Z$. So the rational map associated to ${\left\vert{\mathcal{F}}\otimes L\right\vert}$ is birational.
\[rem:birvsva\] From the proof we see that if $\operatorname{codim}_{U_{\mathcal{F}}} {\mathcal{B}}^{{\mathcal{F}}}_a(x)\geq 2$ for every $x\in X\setminus Z$, then ${\mathcal{F}}\otimes L$ is very ample out of $Z$, the exceptional locus of $a$.
Adjoint line bundles {#subsect:adj}
--------------------
When ${\mathcal{F}}=\omega_X$ we will call $U_{\mathcal{F}}$ simply $U_0$ and ${\mathcal{B}}^{\omega_X}_a(x)$ simply by $${\mathcal{B}}_a(x)={\left\{\alpha\in U_0{\;\vline\;}x\text{ is a base point of }\omega_X \otimes a^*\alpha \right\}}.$$ Throughout subsections §\[subsect:adj\] and §\[subsect:decomp\], we will assume that $\operatorname{gv}_a(\omega_X)\geq 1$.
\[pr-df:birprov\] Let $X$ be a variety such that $\operatorname{gv}_a(\omega_X)\geq 1$ and let $L$ be any line bundle on $X$ such that $\operatorname{gv}_a(L)\geq 1$. Suppose that there exists $\alpha\in \operatorname{Pic}^0A$ such that $\omega_X\otimes
L\otimes a^*\alpha$ is not birational. Then, $$\operatorname{codim}_{X\times U_0} {\left\{(x,\alpha)\in X\times U_0{\;\vline\;}x\text{ is a base point of }\omega_X \otimes
a^*\alpha \right\}}=1,$$ and its divisorial part is dominant on $X$ and surjects on $U_0$ via the projections $p$ and $q$. We endow this set with the natural subscheme structure given by the image of the relative evaluation map $q^*(q_*{\mathcal{L}})\otimes{\mathcal{L}}^{-1}\to {\mathcal{O}}_{X\times U_0}$, where ${\mathcal{L}}= {\left.\left(p^*\omega_X)\otimes
P_a\right)\right|_{X\times U_0}}$ and we call ${\mathcal{Y}}$ the union of its divisorial components that dominate $U_0$. Let $\overline{{\mathcal{Y}}}$ be its closure in $X\times \operatorname{Pic}^0 X$. Then
$X$ is covered by the scheme-theoretic fibres of the projection $\overline{{\mathcal{Y}}}\to U_0$, that we will call $F_\alpha$, for $\alpha$ varying in $U_0$. By definition, at a *general* point $\alpha\in U_0$, $F_\alpha$ is the fixed divisor of $\omega_X\otimes a^*\alpha$.
For a general $x\in X$, the fibre of the projection $\overline{\mathcal{Y}}\to X$ is a divisor, that we will call ${\mathcal{D}}_x$. By definition, ${\mathcal{D}}_x$ is the closure of the union of the divisorial components of the locus of $\alpha\in U_0$ such that $x\in \operatorname{Bs}(\omega_X\otimes a^*\alpha)$.
Everything follows from taking ${\mathcal{F}}=\omega_X$ in Lemma \[lem:birprov\]. The surjectivity of the projection to $U_0$ is consequence of the Castelnuovo-de Franchis inequality \[thm:CdF\], i.e. $\chi(\omega_X)\geq \operatorname{gv}_a(\omega_X)\geq 1$.
Decomposition {#subsect:decomp}
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In the sequel we will need $a^*:\operatorname{Pic}^0A\to
\operatorname{Pic}^0X$ to be an embedding. However, for simplicity we will go one step further and we will simply suppose that $A=\operatorname{Alb}X$. Suppose that we are under the hypotheses of the previous Proposition-Definition and consider a fixed point $\alpha_0\in U_0$, and the map $$\label{eq:f}f_{\alpha_0}\colon U_0 \to \operatorname{Pic}^0 X\qquad \alpha\mapsto
{\mathcal{O}}_X(F_\alpha-F_{\alpha_0}),$$ where $F_\alpha$ is the divisor defined in Proposition-Definition \[pr-df:birprov\][*(a)*]{}. For $\alpha \in U_0$, all the $F_\alpha$ are algebraically equivalent since they are the fibres of $\overline{{\mathcal{Y}}}\to U_0$, so the map is well-defined.
The following lemma shows that this map induces a decomposition of $\operatorname{Pic}^0X$ and that the divisors $F_\alpha$ move algebraically along a non-trivial factor of $\operatorname{Pic}^0X$. Although the proof is basically the same as [@BLNP Lem. 5.1], we do not require $V^1(\omega_X)$ to be a finite set, but only a proper subvariety.
\[lem:propKerf\] The map defined in , induces an homomorphism $f:\operatorname{Pic}^0
X\to\operatorname{Pic}^0X$ such that,
$f^2=f$ and $\operatorname{Pic}^0 X$ decomposes as $\operatorname{Pic}^0 X\cong \ker f\times \ker(\operatorname{id}- f).$ Moreover $\dim \ker(\operatorname{id}-f)>0$.
Fix $\bar\beta\in \ker f$ such that $ U_0\cap\left({\left\{\bar\beta\right\}}\times\ker(\operatorname{id}-f)\right)$ is non-empty. Then, for $\gamma\in U_0\cap \ker({\rm id}-f)$ the line bundle ${\mathcal{O}}_X(F_{\bar\beta\otimes \gamma})\otimes \gamma^{-1}$ does not depend on $\gamma$. Since it is effective by semicontinuity, we call it ${\mathcal{O}}_X(F)$.
For all $(\beta,\gamma)\in \ker f\times\ker({\rm id}-f)\cong\operatorname{Pic}^0 X$ such that $\beta\otimes\gamma\in U_0$, ${\left\vert{\mathcal{O}}_X(F)\otimes\gamma\right\vert}$ is contained in the fixed divisor of $\omega_X\otimes\beta\otimes\gamma$.
Let ${\mathcal{O}}_X(M_\alpha)=\omega_X\otimes a^*\alpha \otimes {\mathcal{O}}_X(-F_\alpha)$. Then, the proof of [*(a)*]{} is the same as [@BLNP Lem. 5.1][*(a)*]{}. Item [*(b)*]{} follows directly from the definition of $f$. To prove [*(c)*]{}, let $(\beta,\gamma)\in \ker f\times\ker(\operatorname{id}-f) $ such that $\beta\otimes\gamma\in U_0$ and $E\in {\left\vert{\mathcal{O}}_X(F)\otimes\gamma\right\vert}$. Then ${\mathcal{O}}_X(F_{\beta\otimes\gamma}-E)\cong{\mathcal{O}}_X(F_{\beta\otimes\gamma}-F_{\bar\beta\otimes\gamma}
)=f(\beta\otimes\bar\beta^{-1})={\mathcal{O}}_X$. Since $F_{\beta\otimes\gamma}$ is a fixed divisor of ${\left\vert\omega_X\otimes\beta\otimes\gamma\right\vert}$, also $E=F_{\bar\beta\otimes\gamma}$ is a fixed divisor in ${\left\vert\omega_X\otimes\beta\otimes\gamma\right\vert}$.
Using the decomposition given by the previous Lemma we give an explicit description of the “half” Poincaré line bundle.
\[lem:halfpoinc\]\[lem:decomp\] We call $B=\operatorname{Pic}^0(\ker f)$ and $C=\operatorname{Pic}^0(\ker (\operatorname{id}- f))$ so that $$\operatorname{Alb}X\cong B\times C\qquad \text{ and }\qquad\operatorname{Pic}^0 X\cong \operatorname{Pic}^0 B\times \operatorname{Pic}^0C,$$ with $\dim C>0$. Then we have the following description of “half” Poincaré line bundle. $$(\operatorname{alb}\times \operatorname{id}_{\operatorname{Pic}^0 X})^*({\mathcal{O}}_{B\times\operatorname{Pic}^0 B}\boxtimes{\mathcal{P}}_C)\cong
{\mathcal{O}}_{X\times\operatorname{Pic}^0 X}(\overline{\mathcal{Y}})\otimes p^*{\mathcal{O}}_X(-F)\otimes q^*{\mathcal{O}}_{\operatorname{Pic}^0 X}(-{\mathcal{D}}_{\bar x}),$$ where $\bar x$ is such that $\operatorname{alb}(\bar x)=0$ in $\operatorname{Alb}X$ and ${\mathcal{P}}_C$ is the Poincaré line bundle in $C\times \operatorname{Pic}^0C$.
The decomposition of $\operatorname{Pic}^0X$ comes directly from Lemma \[lem:propKerf\][*(a)*]{}. By the definition of $\overline{\mathcal{Y}}$ (see Proposition-Definition \[pr-df:birprov\]) and the definition of $F$ (see Lemma \[lem:propKerf\][*(b)*]{}) we have that the line bundle $${\mathcal{O}}_{X\times\operatorname{Pic}^0 X}(\overline{\mathcal{Y}})\otimes p^*{\mathcal{O}}_X(-F)\otimes q^*{\mathcal{O}}_{\operatorname{Pic}^0
X}(-{\mathcal{D}}_{\bar p}),$$
restricted to $X\times{\left\{\beta\otimes\gamma\right\}}$ is isomorphic to ${\mathcal{O}}_X(F_{\beta\otimes\gamma}-F)=\gamma$, for all $(\beta,\gamma)\in U_0\subseteq \ker f\times\ker(\operatorname{id}-f)$;
restricted to ${\left\{\bar x\right\}}\times\operatorname{Pic}^0 X$ is isomorphic to ${\mathcal{O}}_{\operatorname{Pic}^0X}({\mathcal{D}}_{\bar
x})\otimes{\mathcal{O}}_{\operatorname{Pic}^0X}(-{\mathcal{D}}_{\bar x})$, i.e. trivial.
On the other hand, $(\operatorname{alb}\times \operatorname{id}_{\operatorname{Pic}^0 X})^*({\mathcal{O}}_{B\times\operatorname{Pic}^0 B}\boxtimes{\mathcal{P}}_C)$,
restricted to $X\times{\left\{\beta\otimes\gamma\right\}}$ is isomorphic to $\gamma$, for all $(\beta,\gamma)\in\ker f\times\ker(\operatorname{id}-f)$;
restricted to ${\left\{\bar x\right\}}\times\operatorname{Pic}^0 X$ is isomorphic to ${\mathcal{O}}_{\operatorname{Pic}^0X}$, i.e. trivial.
Then, the Lemma follows from the see-saw principle.
The bicanonical map of irregular varieties {#sec:bic}
==========================================
The next theorem gives a sufficient numerical condition for the birationality of the bicanonical map, analogous to Pareschi–Popa Theorem [[@PPreg3 Thm. 6.1]]{} for the tricanonical map.
\[thm:fibr\] Let $X$ be a smooth projective complex variety such that $\operatorname{gv}(\omega_X)\geq 2$. Then, the rational map associated to $\omega_X^{2}\otimes \alpha$ is birational onto its image for every $\alpha\in\operatorname{Pic}^0 X$.
As a first corollary we have the following geometric interpretation.
\[thm:corfibr\] Let $X$ be a smooth projective complex variety of maximal Albanese dimension such that the bicanonical map is not birational. Then $0 \leq \operatorname{gv}(\omega_X)\leq 1$. Moreover, it admits a fibration onto a normal projective variety $Y$ with $0 \leq \dim Y < \dim X$, any smooth model $\tilde Y$ of $Y$ is of maximal Albanese dimension and $$q(X) - \dim X \leq q(\tilde Y) - \dim Y +\operatorname{gv}(\omega_X).$$
By Theorems \[thm:GL1\] and \[thm:fibr\], it is clear that $0 \leq
\operatorname{gv}(\omega_X)\leq 1$. Now, the proof is the same as the proof of [@PPCdF Thm. B].
\[ex:exemples\] We would like to show examples of varieties with $\operatorname{gv}(\omega_X)\geq 2$. For curves $C$, this is equivalent to $g(C) \geq 3$. For surfaces $S$, is equivalent to suppose that $q(S)\geq 4$ and $S$ does not admit an irregular fibration to a curve of genus $\leq q(S)-3$ (see [@Be Cor. 2.3]).
On the other hand, if $A$ is a simple abelian variety, then every subvariety $X$ of codimension $\geq 2$ has $\operatorname{gv}(\omega_X)\geq 2$. Moreover, the property of having $\operatorname{gv}(\omega_X)\geq 2$ is closed under taking products and cyclic coverings induced by a torsion point $\alpha\in \operatorname{Pic}^0X-V^1(\omega_X)$.
The rest of the paper is devoted to the proof of Theorem \[thm:fibr\].
Assume that $\operatorname{gv}(\omega_X)\geq 1$ and there exists $\alpha\in \operatorname{Pic}^0X$ such that $\omega_X^{\otimes 2}\otimes \alpha$ is non-birational. Then, we want to see that $\operatorname{gv}(\omega_X)=1$. Under these hypotheses we can apply Proposition-Definition \[pr-df:birprov\] and Lemma \[lem:halfpoinc\], so $\operatorname{Alb}X\cong B\times C$, where $B=\operatorname{Pic}^0(\ker (\operatorname{id}-f))$ and $C=\operatorname{Pic}^0(\ker
f)$. We have the following commutative diagram $$\label{eq:2diagrames}
\xymatrix{
\operatorname{Pic}^0X\ar[d]_{p_{\hat{b}}}&X\times \operatorname{Pic}^0X\ar[l]_q \ar[r]^{\operatorname{alb}\times\operatorname{id}}\ar[d]^{\operatorname{id}\times p_{\hat
b}} &\operatorname{Alb}X\times \operatorname{Pic}^0X\ar[d]^{p_b\times p_{\hat{b}}}\\
\operatorname{Pic}^0B& X\times\operatorname{Pic}^0B\ar[l]^q \ar[r]_{b\times \operatorname{id}} & B\times\operatorname{Pic}^0B}$$ where
$p_b:\operatorname{Alb}X\to B$ and $p_{\hat b}:\operatorname{Pic}^0X\to\operatorname{Pic}^0B$ are the corresponding projections,
$b$ is the composition by $b:X\stackrel{\operatorname{alb}}{\to}\operatorname{Alb}X \stackrel{p_b}{\to} B$, and
abusing notation we also call $q$ either the projection $X\times\operatorname{Pic}^0X\to \operatorname{Pic}^0X$ or $X\times \operatorname{Pic}^0B\to\operatorname{Pic}^0B$ and $p$ the projections $X\times\operatorname{Pic}^0X\to X$ or $X\times \operatorname{Pic}^0B\to
X$.
The effectiveness of $\overline{\mathcal{Y}}$ give us the following short exact sequence on $X\times \operatorname{Pic}^0 X$ $$0\to (\operatorname{alb}\times \operatorname{id})^*({\mathcal{O}}_{B\times\operatorname{Pic}^0 B}\boxtimes{\mathcal{P}}_C)^{-1} \stackrel{\cdot \overline{\mathcal{Y}}}\to
p^*{\mathcal{O}}_X(F)\otimes q^*{\mathcal{O}}({\mathcal D}_{\bar x})\to {\left.\left(p^*{\mathcal{O}}_X(F)\otimes q^*{\mathcal{O}}({\mathcal
D}_{\bar x})\right)\right|_{\overline{\mathcal{Y}}}}\to 0.$$ Recall that $P=(\operatorname{alb}\times \operatorname{id}_{\operatorname{Pic}^0 X})^*({\mathcal{P}}_B\boxtimes{\mathcal{P}}_C)$ since the Poincaré line bundle ${\mathcal{P}}$ in $\operatorname{Alb}X\times\operatorname{Pic}^0X$ is isomorphic to ${\mathcal{P}}_B\boxtimes{\mathcal{P}}_C$. We apply the functor $R^d
q_*(\>\cdot\>\otimes(\operatorname{alb}\times\operatorname{id}_{\operatorname{Pic}^0 X})^*({\mathcal{P}}_B^{-1}\boxtimes{\mathcal{O}}_{C\times\operatorname{Pic}^0 C}))$, that is, we tensor by the other “half” Poincaré line bundle and we consider the top direct image. We get $$\begin{aligned}
\cdots\to &R^d\Phi_{P^{-1}}({\mathcal{O}}_X)\to R^d q_*\left(p^*{\mathcal{O}}_X(F)\otimes (\operatorname{alb}\times\operatorname{id}_{\operatorname{Pic}^0
X})^*({\mathcal{P}}_B^{-1}\boxtimes{\mathcal{O}}_{C\times\operatorname{Pic}^0 C})\right)\otimes {\mathcal{O}}_{\operatorname{Pic}^0X}(\mathcal{D}_{\bar x})\to \\
&\to R^d q_*\left({\left.(p^*{\mathcal{O}}_X(F)\otimes q^*{\mathcal{O}}_{\operatorname{Pic}^0X}({\mathcal{D}}_{\bar x}))\right|_{\overline{\mathcal{Y}}}}\otimes
(\operatorname{alb}\times\operatorname{id}_{\operatorname{Pic}^0 X})^*({\mathcal{P}}_B^{-1}\boxtimes{\mathcal{O}}_{C\times\operatorname{Pic}^0 C})\right)\to 0 \end{aligned}$$ Using that $R^i\Phi_{P^{-1}}\cong (-1)_{\operatorname{Pic}^0 X}^*R^i\Phi_{P}$ (see ), we have the following short exact sequence, $$\label{eq:fibrshort} 0\to (-1)^*_{\operatorname{Pic}^0 X}\widehat{{\mathcal{O}}_X}\buildrel\mu\over\to
{\mathcal{E}}({\mathcal D}_{\bar x})\to {\mathcal{T}}\to 0$$ where:
By base change, ${\mathcal{E}}= R^d q_*(p^*{\mathcal{O}}_X(F)\otimes(\operatorname{alb}\times\operatorname{id}_{\operatorname{Pic}^0
X})^*({\mathcal{P}}_B^{-1}\boxtimes{\mathcal{O}}_{C\times\operatorname{Pic}^0 C}))$ is a coherent sheaf of rank $h^d({\mathcal{O}}_X(F)\otimes\beta^{-1})$ by a general $\beta\in \ker f$, i.e $h^0(\omega_X\otimes{\mathcal{O}}_X(-F)\otimes\beta)=\chi(\omega_X)$ by Lemma \[lem:propKerf\][*(c)*]{}. Then, $$\begin{aligned}
{\mathcal{E}}&= R^d q_*(p^*{\mathcal{O}}_X(F)\otimes(\operatorname{alb}\times\operatorname{id})^*({\mathcal{P}}_B^{-1}\boxtimes{\mathcal{O}}_{C\times\operatorname{Pic}^0 C}))\\
&= R^d q_*(p^*{\mathcal{O}}_X(F)\otimes (\operatorname{id}\times p_{\hat b})^*(b\times\operatorname{id})^*{\mathcal{P}}_B^{-1}) &\text{right square of
\eqref{eq:2diagrames}}\\
&= R^d q_*(\operatorname{id}\times p_{\hat b})^* (p^*{\mathcal{O}}_X(F)\otimes (b\times\operatorname{id})^*{\mathcal{P}}_B^{-1}) &\text{abuse of
notation on }p\\
&= p_{\hat b}^*R^d q_*(p^*{\mathcal{O}}_X(F)\otimes (b\times\operatorname{id})^*{\mathcal{P}}_B^{-1}) &\text{flat base change}\\
&= p_{\hat b}^*R^d\Phi_{P_b^{-1}}({\mathcal{O}}_X(F)),\end{aligned}$$ following the notation of and .
${\mathcal{T}}=R^d q_*\left({\left.(p^*{\mathcal{O}}_X(F)\otimes q^*{\mathcal{O}}_{\operatorname{Pic}^0X}({\mathcal{D}}_{\bar x}))\right|_{\overline{\mathcal{Y}}}}\otimes
(\operatorname{alb}\times\operatorname{id}_{\operatorname{Pic}^0 X})^*({\mathcal{P}}_B^{-1}\boxtimes{\mathcal{O}}_{C\times\operatorname{Pic}^0 C})\right)$ is supported at the locus of the $\alpha\in\operatorname{Pic}^0 X$ such that the fibre of the projection $q\colon\overline{\mathcal{Y}}\to \operatorname{Pic}^0
X$ has dimension $d$, i.e. it coincides with $X$. Such locus is contained in $V^1(\omega_X)$, therefore, since $\operatorname{gv}(\omega_X)\geq 1$, $\operatorname{codim}\operatorname{supp}{\mathcal{T}}\geq 2$.
The map $\mu$ is injective since it is a generically surjective map of sheaves of the same rank (recall that $\operatorname{rk}\widehat{{\mathcal{O}}_X}=\chi(\omega_X)$), and, as $\operatorname{gv}(\omega_X)\geq 1$, the source $\widehat{{\mathcal{O}}_X}$ is torsion-free (Thm. \[thm:posit\_gv\]).
$\mu$ is $R^d q_*(m_s)$, where $m_s$ is the multiplication by the section defining $\overline{\mathcal{Y}}$. By base change [@MAV Cor. 3, pg. 53], $R^d q_*(m_s)\otimes {\mathbb{C}}(\alpha)= H^d({\left.m_s\right|_{q^{-1}{\left\{\alpha\right\}}}})$ where $q$ is the projection $q\colon\overline{\mathcal{Y}}\to \operatorname{Pic}^0 X$. When $q^{-1}{\left\{\alpha\right\}}=X$, ${\left.m_s\right|_{q^{-1}{\left\{\alpha\right\}}}}=0$, so in these points $R^d
q_*(m_s)\otimes {\mathbb{C}}(\alpha)=0$.
${\mathcal{T}}\neq 0$.
Suppose that ${\mathcal{T}}=0$, so $\mu$ is an isomorphism. Taking $\operatorname{\mathcal{E}\mathit{xt}}^d(\>\cdot\>,{\mathcal{O}}_{\operatorname{Pic}^0X})$ we get $$\begin{aligned}
k(\hat 0) &=R^d\Phi_{P}\omega_X&\text{Prop. \ref{prop:IrrMukai}}\\
&=\operatorname{\mathcal{E}\mathit{xt}}^d({\mathcal{E}},{\mathcal{O}}_{\operatorname{Pic}^0X})\otimes{\mathcal{O}}(-{\mathcal{D}}_{\bar x}) &\operatorname{\mathcal{E}\mathit{xt}}^d(\mu,{\mathcal{O}}_{\operatorname{Pic}^0X})\text{ and Cor.
\ref{cor:duality}}\\
&=p_{\hat b}^*\operatorname{\mathcal{E}\mathit{xt}}^d(R^d\Phi_{P_b}({\mathcal{O}}_X(F)),{\mathcal{O}}_{\operatorname{Pic}^0B})\otimes{\mathcal{O}}(-{\mathcal{D}}_{\bar x}) &\text{see item
{\it
(a)},}\end{aligned}$$ which implies that $\operatorname{codim}_{\operatorname{Alb}X} B=\dim \ker(\operatorname{id}-f)=0$ contradicting Lemma \[lem:decomp\].
Let $\tau({\mathcal{E}}({\mathcal{D}}_{\bar x}))$ be the torsion part of ${\mathcal{E}}({\mathcal{D}}_{\bar x})$ and $\widetilde{{\mathcal{E}}({\mathcal{D}}_{\bar
x})}$ the quotient of ${\mathcal{E}}({\mathcal{D}}_{\bar x})$ by its torsion part. Hence $\widetilde{{\mathcal{E}}({\mathcal{D}}_{\bar x})}$ is torsion-free. Now consider the following composition $$\xymatrix@C=1.8pc@R=1.2pc{
(-1)^*_{\operatorname{Pic}^0 X}\widehat{{\mathcal{O}}_X}\ar[r]^(.6)\mu\ar[rd]_{\tilde\mu} & {\mathcal{E}}({\mathcal D}_{\bar
x})\ar@{>>}[d]\\
&\widetilde{{\mathcal{E}}({\mathcal{D}}_{\bar x})}.}$$ Since $\tilde\mu$ is generically surjective and $(-1)^*_{\operatorname{Pic}^0 X}\widehat{{\mathcal{O}}_X}$ is torsion-free (recall that $\operatorname{gv}(\omega_X)\geq 1$), we have that $\tilde\mu$ is injective. Completing the diagram we get, $$\label{eq:tautilde}
\xymatrix@C=1.3pc@R=1.pc{&& 0\ar[d]&0\ar[d]\\
&& \tau({\mathcal{E}}({\mathcal{D}}_{\bar x}))\ar[d]\ar@{=}[r]&\tau({\mathcal{E}}({\mathcal{D}}_{\bar x}))\ar[d]\\
0\ar[r] &(-1)^*_{\operatorname{Pic}^0 X}\widehat{{\mathcal{O}}_X}\ar[r]^(.6)\mu\ar@{=}[d] & {\mathcal{E}}({\mathcal D}_{\bar
x})\ar@{>>}[d]\ar[r]& {\mathcal{T}}\ar[r]\ar[d]&0\\
0\ar[r] &(-1)^*_{\operatorname{Pic}^0 X}\widehat{{\mathcal{O}}_X}\ar[r]^(.6){\tilde\mu} &\widetilde{{\mathcal{E}}({\mathcal{D}}_{\bar
x})}\ar[d]\ar[r]& \widetilde{\mathcal{T}}\ar[r]\ar[d]&0\\
&&0&0
}$$ If $\widetilde{\mathcal{T}}= 0$, then the middle horizontal short exact sequence splits. But, for $\alpha$ a closed point in the support of ${\mathcal{T}}$ (by the previous claim we know that ${\mathcal{T}}\neq 0$), $\mu\otimes{\mathbb{C}}(\alpha)=0$ by item [*(d)*]{}, so $\mu$ cannot split. Therefore $\widetilde{\mathcal{T}}\neq 0$.
Let $e=\operatorname{codim}_{\operatorname{Pic}^0X} \operatorname{supp}\widetilde {\mathcal{T}}\geq 2$ (see item [*(c)*]{}). Then $\operatorname{codim}_{\operatorname{Pic}^0X}\operatorname{supp}\operatorname{\mathcal{E}\mathit{xt}}^e(\widetilde {\mathcal{T}},{\mathcal{O}}_{\operatorname{Pic}^0X})=e$. Now, we apply the functor $\operatorname{\mathcal{E}\mathit{xt}}^i(\>\cdot\>,{\mathcal{O}}_{\operatorname{Pic}^0X})$ to the bottom row of using Corollary \[cor:duality\] $$\ldots\to R^{e-1}\Phi_{P}\omega_X\to \operatorname{\mathcal{E}\mathit{xt}}^e(\widetilde
{\mathcal{T}},{\mathcal{O}}_{\operatorname{Pic}^0X})\to\operatorname{\mathcal{E}\mathit{xt}}^e(\widetilde{{\mathcal{E}}({\mathcal{D}}_{\bar x})},{\mathcal{O}}_{\operatorname{Pic}^0X})\to\ldots$$ Since $\widetilde{{\mathcal{E}}({\mathcal{D}}_{\bar x})}$ is torsion-free, $\operatorname{codim}_{\operatorname{Pic}^0X}\operatorname{supp}\operatorname{\mathcal{E}\mathit{xt}}^e(\widetilde{{\mathcal{E}}({\mathcal{D}}_{\bar x})},{\mathcal{O}}_{\operatorname{Pic}^0X})>e$. Therefore, we must have $\operatorname{codim}_{\operatorname{Pic}^0X}\operatorname{supp}R^{e-1}\Phi_{P}\omega_X=e$ and $\operatorname{gv}(\omega_X)\leq 1$.
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[^1]: The author has been partially supported by the Proyecto de Investigación MTM2009-14163-C02-01. This paper was revised while the author was supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’.
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[The gauge theory on a set of D3-branes at a toric Calabi-Yau singularity can be encoded in a tiling of the 2-torus denoted dimer diagram (or brane tiling). We use these techniques to describe the effect on the gauge theory of geometric operations partially smoothing the singularity at which D3-branes sit, namely partial resolutions and complex deformations. More specifically, we describe the effect of arbitrary partial resolutions, including those which split the original singularity into two separated. The gauge theory correspondingly splits into two sectors (associated to branes in either singularity) decoupled at the level of massless states. We also describe the effect of complex deformations, associated to geometric transitions triggered by the presence of fractional branes with confinement in their infrared. We provide tools to easily obtain the remaining gauge theory after such partial confinement. ]{}
Introduction
============
The study of the $N=1$ supersymmetric gauge theory on a stack of D3-branes probing a Calabi-Yau threefold conical singularity is a fruitful source of new insights into brane dynamics [@Douglas:1996sw; @Douglas:1997de], the AdS/CFT correspondence [@Klebanov:1998hh], and its extensions to non-conformal situations by the addition of fractional branes [@Klebanov:2000hb].
Although present techniques do not allow this analysis for a general singular Calabi-Yau [^1], several techniques have been successfully applied to the understanding of D3-branes at toric singularities: Partial resolutions (e.g. [@Morrison:1998cs; @Beasley:1999uz; @Feng:2000mi]), mirror symmetry (e.g. [@Hanany:2001py; @Feng:2002kk]), un-Higgsing [@Feng:2002fv; @Hanany:2005hq], etc. These tools have provided precise checks of AdS/CFT for quiver conformal field theories [@Bertolini:2004xf; @Benvenuti:2004dy; @Benvenuti:2005ja; @Franco:2005sm; @Butti:2005sw; @Butti:2005vn], and interesting information on related non-conformal systems [@Franco:2004jz; @Herzog:2004tr; @Franco:2005fd; @Berenstein:2005xa; @Franco:2005zu; @Bertolini:2005di].
A recent great improvement in the study of the system of D3-branes at a toric singularity has been the introduction of the so-called brane tilings or dimer diagrams [@Hanany:2005ve; @Franco:2005rj; @Hanany:2005ss; @Feng:2005gw; @Franco:2006gc]. These techniques have provided new viewpoints on the D3-brane gauge theory (e.g. its moduli space, its Seiberg dualities [^2], etc). In addition they lead to interesting mathematical implications, like the description of the system in the large volume regime via exceptional collections [@Hanany:2006nm], or a generalization of the McKay correspondence [@Franco:2005rj].
One expects that, although not completely general, toric singularities are representative enough for the physics of D3-branes at general singularities.
One of the basic features of string theory at Calabi-Yau singularities is the existence of localized modes which can smooth out the singularity. When D3-branes are located at such singularities, a natural question is what is the gauge theory interpretation of these smoothings (in other words, how do the D-branes experience this smoothing). A first kind of smoothings corresponds to partial resolutions and are associated to Kahler parameters. It has been known for some time that these modes couple as Fayet-Iliopoulos terms to the D-brane gauge theory [@Douglas:1997de]. In concrete examples it has been shown that they hence trigger a partial Higgsing of the gauge theory reducing it to the gauge theory in the left-over singularity [@Morrison:1998cs]. The observation that minimal partial resolutions (those removing a triangle from the toric diagram) admit a simple description in terms of brane tilings [@Franco:2005rj], suggests the existence of a simple description of general partial resolutions in this language. In this paper we provide such a description, in terms of dimer diagram concepts and directly on the gauge theory side (by providing the relevant Higgsing vevs associated to a partial resolution). Our results significantly improve the understanding of partial resolutions in the literature, and can be used to easily analyze complicated resolutions which e.g. split the original singularity into two singularities.
A second kind of smoothing corresponds to complex deformations. These are extremely interesting from the gauge theory viewpoint, since they are related to geometric transitions triggered by fractional branes which experience confinement at the infrared. The prototypical situation is the conifold singularity with fractional branes [@Klebanov:2000hb; @Vafa:2000wi], but similar behaviour has been discussed in more generality (see e.g. [@Cachazo:2001sg], and [@Aganagic:2001ug; @Franco:2005fd] for general toric singularities).
In [@Franco:2005fd] the gauge theory process of confinement of a subset of gauge factors was translated in an ad hoc manner to the dimer diagram language. In this paper we provide a detailed description of the effect of these geometric transitions in the gauge theory, allowing us to derive simple dimer diagram rules to obtain the remaining theory after infrared confinement of the fractional brane gauge groups. Moreover, this description provides a new insight into the nature of other kinds of fractional branes, which are known not to confine and trigger a complex deformation, but rather remove the supersymmetric groundstate [@Berenstein:2005xa; @Franco:2005zu; @Bertolini:2005di; @Intriligator:2005aw].
The two kinds of smoothings are most easily described in terms of the web diagrams of the toric singularity [@Aharony:1997ju; @Aharony:1997bh; @Leung:1997tw]. Our main tool in finding the gauge theory counterpart of these geometric operations is the close relation between the dimer diagrams and the web diagram of the associated singularity [@Feng:2005gw; @Hanany:2005ss].
The paper is organized as follows. In Section \[review\] we review some aspects of the dimer diagram techniques to study the gauge theory on D3-branes at toric singularities. This Section contains all the background material on dimers we need, so Sections \[splitting\] and \[complexdeformations\] can be read independently. In Section \[splitting\] we describe the effect of a general partial resolution of the singularity on the gauge theory. In Section \[dconi\] and \[moreexamples\] we work out several examples in detail, describing the effects of the partial resolution on the dimer diagram via simple operations in its zig-zag paths. In Section \[fieldtheory\] we translate the dimer diagram rules for partial resolution to explicit vevs for the gauge theory bi-fundamentals, triggering the corresponding Higgs mechanism. The proof of their flatness is postponed to Appendix \[proof\]. In Section \[matchings\] we provide a different view on the dimer description of partial resolutions in terms of perfect matchings. Finally, in Section \[fractional\] we discuss partial resolution in the presence of fractional branes, and possible obstructions. In Section \[complexdeformations\] we describe complex deformations as geometric transitions triggered by fractional branes with infrared confining behaviour. In Section \[dconicomp\] and \[moreexamplescomplex\] we work out several examples in detail, describing the effect of such geometric transitions on the dimer diagram via simple operations on its zig-zag paths. In Section \[fieldcomp\] we describe the field theory interpretation of the complex deformation, and describe in terms of the dimer diagrams the gauge theory analysis of branes probing the confining theory. In Section \[pmcomp\] we provide a different view of the complex deformation in the dimer [^3] in terms of perfect matchings. Finally in Section \[conclu\] we offer some final remarks.
Review of dimer diagrams {#review}
========================
In this Section we review some background material on dimer diagrams and their relevant to quiver gauge theories. Reviews of the mathematical aspects of dimers can be found in [@kenyon1; @kenyon2].
Quiver gauge theories and dimer diagrams {#quiver}
----------------------------------------
The gauge theory of D3-branes probing toric threefold singularities is determined by a set of unitary gauge factors (of equal rank in the absence of fractional branes, which we do not consider for the moment), chiral multiplets in bi-fundamental representations, and a superpotential given by a sum of traces of products of such bi-fundamental fields. The gauge group and matter content of such gauge theories can be encoded in a quiver diagram, such as that shown in Figure \[dconidimer\_quiver\]a, with nodes corresponding to gauge factors, and arrows to bi-fundamentals. The superpotential terms correspond to closed loops of arrows, but the quiver does not fully encode the superpotential.
Recently it has been shown that all the gauge theory information, including the gauge group, the matter content and the superpotential, can be encoded in a so-called brane tiling or dimer graph [@Hanany:2005ve; @Franco:2005rj] [^4]. This is a tiling of $\IT^2$ defined by a bipartite graph, namely one whose nodes can be colored black and white, with no edges connecting nodes of the same color . The dictionary associates faces in the dimer diagram to gauge factors in the field theory, edges with bi-fundamental fields (fields in the adjoint in the case that the same face is at both sides of the edge), and nodes with superpotential terms. The bipartite character of the diagram is important in that it defines an orientation for edges (e.g. from black to white nodes), which determines the chirality of the bi-fundamental fields. Also, the color of a node determines the sign of the corresponding superpotential term.
The explicit mapping between this bipartite graph and the gauge theory, is illustrated in one example in Figure \[dconidimer\_quiver\]. Many interesting features of the gauge theory have been described in terms of dimers by now.
![Quiver and dimer for a $\IZ_2$ orbifold of the conifold. Faces in the dimer correspond to gauge groups, edges correspond to bifundamentals and each vertex corresponds to a superpotential term. Edges have an orientation determined by the coloring of the adjacent nodes.[]{data-label="dconidimer_quiver"}](dconidimer_quiver2.eps)
Dimer diagrams and the mirror Riemann surface {#mirror}
---------------------------------------------
There are two interesting ways to relate physically the dimer diagram with the gauge theory. As described in [@Franco:2005rj] the diagram can be considered to specify a configuration of NS5- and D5-branes, generalizing the brane box [@Hanany:1997tb; @Hanany:1998ru; @Hanany:1998it] and brane diamond models [@Aganagic:1999fe]. The NS5-branes extend in the $0123$ directions and wrap a holomorphic curve in the $4567$ directions. The D5-branes span the $012346$ directions and are bounded by the NS branes in the 46 directions. These directions are compact and parametrize a torus. The NS branes thus generate a tiling of this $\IT^2$, represented by a bipartite graph of the kind described above, hence the name brane tiling.
A second useful and more explicit viewpoint on correspondence between the gauge theory on D3-branes at toric singularities and dimer diagrams was provided in [@Feng:2005gw] by using mirror symmetry, as we now describe [^5]. The mirror geometry to a Calabi-Yau singularity $\mathcal{M}$ is specified by a double fibration over the complex plane W given by W &=& P(z,w) \
W &=& uv with $w,z \ \in \IC^*$ and $u,v \ \in \IC$.Here $P(z,w)$ is the Newton polynomial of the toric diagram of $\mathcal{M}$. The surface $W = P(z,w)$ describes a genus $g$ Riemann surface $\Sigma_W$ with punctures, fibered over $W$. The genus $g$ equals the number of internal points of the toric diagram. The fiber over $W=0$, denoted simply $\Sigma$, will be important for our purposes. It corresponds to a smooth Riemann surface which can be thought of as a thickening of the web diagram [@Aharony:1997ju; @Aharony:1997bh; @Leung:1997tw] dual to the toric diagram, see Figure \[thickweb\].
![a) An example of a web diagram (for the theory in Figure \[dconidimer\_quiver\]); b) the corresponding Riemann surface $\Sigma$ in the mirror geometry.[]{data-label="thickweb"}](thickweb.eps)
At critical points $W=W^*$, a cycle in $\Sigma_W$ degenerates and pinches off. Also, at $W=0$ the $S^1$ in $W=uv$ degenerates. One can use these degenerations to construct non-trivial 3-cycles in the mirror geometry as follows. Consider the segment in the $W$-plane which joins $W=0$ with one of the critical points $W=W^*$, and fiber over it the $\IS^1$ in $W=uv$ times the 1-cycle in $\Sigma_W$ degenerating at $W=W^*$, see Figure \[s3fiber2\]. The result is a 3-cycle with an $\IS^3$ topology. The number of such degenerations of $\Sigma_W$, and hence the number of such 3-cycles, is given by twice the area of the toric diagram.
Mirror symmetry specifies that the different gauge factors on the D3-branes in the original singularity arise from D6-branes wrapping the different 3-cycles. The 3-cycles on which the D6-branes wrap intersect over $W=0$, precisely at the intersection points of the 1-cycles in $\Sigma_{W=0}$. Open strings at such intersections lead to the chiral bi-fundamental fields. Moreover, disks in $\Sigma$ bounded by pieces of different 1-cycles lead to superpotential terms generated by world-sheet instantons.
Hence, the structure of the 3-cycles, and hence of the gauge theory, is determined by the 1-cycles in the fiber $\Sigma$ over $W=0$. This structure, which is naturally embedded in a $\IT^3$ (from the $\IT^3$ fibration structure of the mirror geometry), admits a natural projection to a $\IT^2$, upon which the 1-cycles end up providing a tiling of $\IT^2$ by a bipartite graph, which is precisely the dimer diagram of the gauge theory.
This last process is perhaps better understood (and of more practical use) by recovering the Riemann surface $\Sigma$ from the dimer diagram of the gauge theory, as follows. Given a dimer diagram, one can define zig-zag paths (these, along with the related rhombi paths, were introduced in the mathematical literature on dimers in [@kenyon1; @kenyon2], and applied to the quiver gauge theory context in [@Hanany:2005ss]), as paths composed of edges, and which turn maximally to the right at e.g. black nodes and maximally to the left at white nodes. They can be conveniently shown as oriented lines that cross once at each edge and turn at each vertex, as shown in Figure \[conidimer\_zigzag\].
Notice that at each edge two zig-zag paths must have opposite orientations. For dimer models describing toric gauge theories, these zig-zag paths never intersect themselves and form closed loops wrapping $(p,q)$ cycles on the $\IT^2$. This is shown for the conifold in Figure \[conidimer\_zigzag\] where the zig-zag paths A, B, C and D have charges (0,1), (-1,1), (1,-1), (0,-1) respectively.
As shown in [@Feng:2005gw], the zig-zag paths of the dimer diagram associated to D3-branes at a singularity lead to a tiling of the Riemann surface $\Sigma$ in the mirror geometry. Specifically, each zig-zag path encloses a face of the tiling of $\Sigma$ which includes a puncture, and the $(p,q)$ charge of the associated leg in the web diagram is the $(p,q)$ homology charge of the zig-zag path in the $\IT^2$. The touching of two of these faces in the tiling of $\Sigma$ corresponds to the coincidence of the corresponding zig-zag paths along an edge of the dimer diagram. The tiling of $\Sigma$ for the conifold is shown in Figure \[coniriem\]a, while the corresponding web diagram is shown in Figure \[coniriem\]b.
The dimer diagram moreover encodes the 1-cycles in the mirror Riemann surface, associated to the different gauge factors in the gauge theory. Consider a gauge factor associated to a face in the dimer diagram. One can consider the ordered sequence of zig-zag path pieces that appear on the interior side of the edges enclosing this face. By following these pieces in the tiling of $\Sigma$ one obtains a non-trivial 1-cycle in $\Sigma$ which corresponds precisely to that used to define the 3-cycle wrapped by the mirror D6-branes carrying that gauge factor. Using this map, it is possible to verify all dimer diagram rules (edges are bi-fundamentals, nodes are superpotential terms) mentioned at the beginning. An amusing feature is that these non-trivial 1-cycles in $\Sigma$ are given by zig-zag paths of the tiling of $\Sigma$. The non-trivial 1-cycles in the mirror Riemann surface for the case of the conifold are shown in Figure \[conisurf\].
Perfect matchings {#pmatchings}
-----------------
A last concept we would like to discuss is that of perfect matchings for a dimer diagram. A perfect matching is a subset of edges of the dimer diagram, such that every vertex of the graph is the endpoint of exactly one such edge. In Figure \[conipmatch\] we show the four perfect matchings for the conifold. For future convenience, we consider the edges in each perfect matching to carry an orientation, e.g. from black to white nodes.
As discussed in [@Franco:2005rj] and proved in [@Franco:2006gc], there is a one to one correspondence between the perfect matchings for a dimer diagram and the linear sigma model fields that arise in the construction of the moduli space of the quiver gauge theory [@Douglas:1997de; @Morrison:1998cs; @Feng:2000mi]. This implies that each perfect matching has an associated location in the toric diagram of the corresponding singularity. This can be obtained as follows. Fix a given perfect matching as reference matching, denoted $p_0$. Then for any perfect matching $p_i$ we can consider the path $p_i-p_0$, obtained by superimposing the edges of $p_i$ and those of $p_0$, with flipped orientation for the latter. With the convention that repeated edges with opposite orientation annihilate, we obtain a (possibly trivial, or even empty) path in the dimer diagram, carrying a (possibly trivial) $\IT^2$ homology charge $(n_i,m_i)$. Then the location of the matching $p_i$ in the toric diagram is given by $(-m_i,n_i)$. This definition is equivalent to that using the height functions, so we denote $(h_{x,i},h_{y,i})=(-m_i,n_i)$ the slope of $p_i$. Clearly the choice of reference matching simply amounts to a choice of origin in the toric diagram.
For completeness, let us mention the direct definition of the slope for a perfect matching, see e.g. [@Franco:2005rj]. The dimer path $p_i-p_0$ divides the infinite tiling of $\IR^2$ into regions that can be labeled by an integer, with the rule that each crossing of the path changes the label by one unit (up- or downwards depending on orientation of the crossing). This label assignment, when regarded in the torus, is multivalued. The holonomies around the two fundamental 1-cycles are denoted $(h_x,h_y)$ and are called the slope of $p_i$.
In Figure \[coniheight\]a we have shown the paths $p_i-p_1$ in the dimer diagram for the conifold (along with the labeling by integers to obtain the slopes). The location of the perfect matchings in the toric diagram in shown in Figure \[coniheight\]b.
Although not emphasized in the literature, there is a beautiful interpretation of pairs of perfect matchings. From a construction similar to the above, to any pair of perfect matchings $p_i$, $p_j$ one can associate a path (which we call ‘difference path’) $p_{ij}=p_j-p_i$ in the dimer diagram, with $\IT^2$ homology charge $(\Delta n, \Delta m)$. In the toric diagram this is associated to the segment joining the location of $p_i$ to that of $p_j$, which as a vector is given by the slope difference $(\Delta h_x,\Delta h_y)=(-\Delta m, \Delta n)$. Now clearly, the homology charge $(\Delta n, \Delta m)$ is precisely the $(p,q)$ charge of the segment in the web diagram dual to that segment in the toric diagram. This suggests a natural interpretation of $p_j-p_i$ in the mirror Riemann surface. Indeed, by lifting the dimer path $p_j-p_i$ to the mirror Riemann surface (using the tiling of the latter) one obtains a non-trivial 1-cycle which winds around the tube corresponding to the thickening of the leg in the web diagram. This is illustrated in Figure \[conipmatch2\] for the case of the conifold.
Clearly, the dimer paths associated to adjacent external matchings (i.e. matchings which are at adjacent locations on the boundary of the toric diagram) carry the same charges as zig-zag paths (although in general may not coincide edge by edge with them). This hence shows the equivalence of the two ways we have described to obtain the toric diagram associated to a dimer diagram [^6] (namely, construction of the web diagram by using charges of zig-zag paths, and construction of the toric diagram using height functions).
Partial resolution {#splitting}
==================
In this Section we provide a description of the effect of general partial resolutions on the gauge theory using dimer diagram techniques. The simplest class of partial resolutions corresponds to the removal of a triangle in the toric diagram. There are however more involved possibilities, like the splitting of a singularity into two singularities (examples will come later). Clearly, the former can be regarded as a particular case of the latter, with the second ‘singularity’ being a smooth patch of the final geometry.
Minimal partial resolutions (those removing one triangle of the toric diagram) and their relation to the D3-brane gauge theory (appearance of a Fayet-Iliopoulos term triggering a Higgs mechanism) were discussed in [@Morrison:1998cs]. This process was described as removal of edges in the dimer diagram in [@Franco:2005rj]. However, in both cases the mapping between a particular operation on the gauge theory/dimer diagram and the resulting geometry, was manifest only upon computation of the moduli space of the gauge theory. This makes it difficult to obtain the gauge theory corresponding to a given partial resolution, and requires some trial and error. Also, the description of more involved partial resolutions (splitting the singularities) and their gauge theory/dimer diagram counterpart was not provided. Moreover, as pointed out and explained in [@Hanany:2005ss], arbitrary addition/removal of edges in a dimer diagram can lead to inconsistent theories.
In this Section we consider an arbitrary partial resolution of a toric singularity, typically splitting it into several. We consider the original set of D3-branes to split accordingly into subsets located at the daughter singular points. Hence one expects that the original gauge theory splits (via a Higgs mechanism) into several gauge sectors, decoupled at the level of massless modes, and correspondingly that the original dimer diagram splits into several sub-dimers associated to the subsets of D3-branes at the daughter singularities. We provide a simple construction of the splitting of dimer diagrams that corresponds to a given partial resolution. In addition, we provide a simple recipe for the bi-fundamental vevs that trigger the corresponding Higgsing in the gauge theory.
As a prototypical example we consider partial resolutions splitting a singularity into two. Other cases, like minimal partial resolutions, can be recovered as a particular case as mentioned above. Splitting into more than two daughter singularities can be easily obtained by iteration of our procedure.
An example in detail {#dconi}
--------------------
Let us start with a simple example of the splitting via partial resolution of a singularity into two singularities, using concepts and techniques from dimers.
Consider the singularity whose toric diagram and web diagram are shown in Figure \[dconitoric\]. This singularity, and the gauge theory on D3-branes located on it, have been studied in [@Uranga:1998vf; @Aganagic:1999fe]. We refer to it as the double conifold. The dimer diagram shown in Figure \[dconidimer\] provides (a toric phase of) the gauge theory on D3-branes at this double conifold singularity.
The above singularity admits partial resolutions to geometries with two separated singularities. One such partial resolution is illustrated in Figure \[dconisplit\]a, and corresponds to a large blow up of an $\IS^2$, smoothing the initial geometry to two isolated conifold singularities. A different splitting, into two $\IC^2/\IZ_2$ (times $\IC)$ singularities, is shown in Figure \[dconisplit\]b.
The partial resolution corresponds to turning on Fayet-Iliopoulos terms in the D3-brane gauge theory [^7]. These FI terms force some of the bi-fundamental scalars to acquire a vev, breaking the gauge symmetry. For the case of a partial resolution splitting a singularity, the left over field theory must correspond to two gauge sectors, corresponding to the gauge theories on stacks of D3-branes at the two singularities. These two sectors are decoupled at the level of massless states. Namely, the only states charged under both sectors are massive, with mass controlled by the bi-fundamental vevs, and hence to the size of the 2-sphere responsible for the splitting. This agrees with the picture of open strings stretching between the two stacks of D3-branes. In section \[fieldtheory\] we will be more explicit about the precise set of vevs corresponding to splitting singularities.
In this section, our aim is to provide a simple recipe that implements the effect of the resolution on the gauge field theory. This will be expressed in terms of a simple operation that, starting from the dimer of the initial singularity, leads to two sub-dimers corresponding to the gauge theories in the two daughter singularities.
The geometrical effect of partial resolutions is most manifest in the web diagram. Let us for concreteness consider the partial resolution of the double conifold to two conifolds, Figure \[dconisplit\]a. As described in Section \[mirror\] the web diagram is encoded in the dimer diagram via its structure of zig-zag paths [@Hanany:2005ss; @Feng:2005gw]. The zig-zag paths corresponding to the dimer in Figure \[dconidimer\] are shown in Figure \[dconipath\]. The corresponding asymptotic legs in the web diagram, and the tiling of the mirror Riemann surface $\Sigma$, are shown in Figure \[dconiriem\].
In this language, it is easy to realize that the partial resolution corresponds to factorizing the Riemann surface $\Sigma$ by an elongated tube, as in Figure \[dconifactor\]a. The structure of the two left over singularities can be determined by analyzing the local structure of the two daughter Riemann surfaces. Due to the factorization along the infinite tube, each daughter Riemann surface has a new puncture, denoted G, which must correspond to a new zig-zag path in the corresponding daughter dimer diagram. In particular, the decomposition of the tiling of $\Sigma$ upon this factorization, shown in Figure \[dconifactor\]b, leads to two sets of zig-zag paths, namely C, D, E, G and A, B, F, G, respectively, with specific adjacency relations. This information can be used to construct two dimer diagrams, which encode the gauge theories on D3-branes at the two singular points in the geometry after partial resolution.
In Figure \[dconidaug1\] we show the two sets of zig-zag paths. For convenience, the inherited paths are drawn in the locations corresponding to the original dimer. The information from the zig-zag paths allows to construct the dimer diagram corresponding to D3-brane at each of the left-over singularities after partial resolution. The dimer diagrams are also shown in the picture.
It is easy to convince oneself that the two theories are isomorphic (as expected from the symmetric factorization of the Riemann surface, or of the web diagram). Hence, it is enough to focus in one of them, say that shown in Figure \[dconimass1\]a. Since this theory has a bi-valent node, one should integrate out the corresponding massive matter, with the result shown in Figure \[dconimass1\]b. This can be redrawn as in Figure \[dconimass1\]c, and one recognizes the dimer diagram for the conifold singularity, as expected. Hence the above technique of zig-zag paths provides a simple tool to determine the effect of a splitting by partial resolution on the dimer diagram of the D3-brane gauge theory, as a specific splitting of the initial dimer into two sub-dimers. Moreover, in section \[fieldtheory\] we will show that the operation in the dimer diagram encodes in a simple manner the set of bi-fundamental vevs that corresponds in the gauge field theory to the partial resolution of the singularity.
The whole procedure can be subsumed in a simple operation in the dimer diagram, without the need to go through the Riemann surface. In terms of the dimer diagram the previous discussion amounts to drawing the old zig-zag paths that define the remaining singularity we are interested in (C, D and E in the example above), and then completing with new zig-zag paths (this will be G) until all edges have two zig-zag paths going through them. The number of required new paths is given by the decrease in the number of holes in the factorization plus one. In the remaining examples we will obtain our results by using this simple shortcut.
Further examples and comments {#moreexamples}
-----------------------------
### Double conifold to two $\IC^2/\IZ_2$ singularities
The technique we have described in the above example is fully general, and can be applied to any partial resolution. To provide an additional example, consider for instance the splitting of the above singularity into two $\IC^2/\IZ_2$ singularities, Figure \[dconisplit\]b. Starting with the zig-zag paths in Figure \[dconipath\], the partial resolution corresponds to a factorization of the mirror Riemann surface splitting the set of paths into two subsets, namely A, C, F, and B, D, E. Each set, along with a new path H from the new puncture in the daughter Riemann surface, allow to read off the dimer diagrams (and hence the quiver gauge theories) for D3-branes in the two left-over singularities. This is shown in Figure \[dconidaug2\], where one indeed recognizes the dimer diagrams of two $\IC^2/\IZ_2$ theories.
### From $dP_3$ to two SPP’s
Before concluding this section, we present a further example, where the factorization lowers the genus of the mirror Riemann surfaces. Namely, the factorization implies elongating several segments in the web diagram. Consider for instance the splitting of the complex cone over $dP_3$ to two suspended pinch point (SPP) singularities, shown in Figure \[dp3toric\].
The dimer diagram for (a toric phase of) the gauge theory on D3-branes at the cone over $dP_3$ is shown in Figure \[dp3dimer\]. The unit cell of the corresponding dimer diagram is shown in Figure \[dp3path\], where we also show the zig-zag paths. The partial resolution in Figure \[dp3toric\] has the effect of splitting this dimer diagram into the two dimer diagrams in Figure \[dp3daug\]. After integrating out massive fields, they can be shown to correspond to the gauge theories of D3-branes at SPP singularities, in agreement with the underlying geometric picture. Although this example follows from exactly the same rules as previous ones, we encounter the new feature that the splitting of the dimer involves two new zig-zag paths (denoted G and H) rather than one. This simply reflects the fact that the factorization of the Riemann surface involves two elongated tubes, hence two new punctures for each daughter Riemann surface.
### Minimal partial resolution {#minimalresolution}
To conclude this section, we would like to mention that this technique can be applied to asymmetric splittings, where the two daughter geometries are not the same. One particular extremal case is a minimal partial resolution (removing only one triangle from the toric diagram). Hence, only one singularity is left over after the partial resolution (namely the second singularity turns out to be a smooth patch). Let us describe this more explicitly.
In terms of the web diagram, this simply corresponds to elongating a tube that separates two external legs from the rest of the web. Using the zig-zag paths, it is easy to show that the left-over singularity corresponds to a dimer diagram obtained from the initial one by the removal of some edges. These edges are precisely those over which the two zig-zag paths associated to the removed legs overlap [^8]
To provide one particular example, we describe the partial resolution of the double conifold to an SPP singularity via the removal of one triangle in the corresponding toric diagram. Concretely consider separating the legs A and F from the rest of the web diagram, by stretching the intermediate segment. Since the corresponding zig-zag paths overlap over the lower left edge in the dimer diagram, this is the edge to be removed. In field theoretic terms this means that the corresponding bifundamental gets a vev, and the two faces (gauge groups) sharing the edge join (gauge factors break to the diagonal combination). The resulting dimer diagram is that of the SPP theory, as can be checked by computing the gauge theory data.
We hope these examples suffice to illustrate the general validity of the above prescription.
Field theory interpretation {#fieldtheory}
---------------------------
As discussed in [@Morrison:1998cs], partial resolutions of singularities correspond to turning on Fayet-Iliopoulos terms on the gauge theory on D3-branes sitting at them. These FI terms force some of the bi-fundamental scalars to acquire vevs, preserving supersymmetry but partially breaking gauge symmetry, in precise agreement with the quiver gauge theory on D3-branes at the final left-over singularity.
In this section we show that the operation of splitting a dimer, as described in previous section, encodes in a very precise fashion the field theory data corresponding to the Higgs mechanism and gauge symmetry breaking. Moreover we show that dimer techniques can be efficiently used to show the F- and D-flatness of such vevs.
For simplicity, we center on a gauge theory with all gauge factors having equal rank $N$. Discussion of other situations (fractional branes) is postponed until section \[fractional\]. We also consider that after the splitting, $N_1$ D3-branes remain at the first singularity and $N_2$ remain at the second.
In order to describe the bi-fundamental vevs in the field theory, we notice that in the dimer splitting there are three different kinds of bi-fundamental fields, according to the behaviour of the corresponding edge: a) those appearing in the two daughter dimers; b) those not appearing in the first sub-dimer, but present in the second; c) those not appearing in the second, but present in the first. This suggests the following ansatz for their vevs, which we denote $V_0$, $V_1$, $V_2$, respectively: V\_0 = ; V\_1 = ; V\_2 = \[vevansatz\] where bi-fundamental fields are regarded as $N\times N$ matrices, and the entries are blocks of dimension appropriate to the partition $N=N_1+N_2$. Here we take $v$ to be adimensional, and we consider that a dimensionful constant enters into the vev of each bi-fundamental, exponentiated to the appropriate power to match its conformal dimension. This factor does not change the discussion of flatness, hence we ignore it in what follows.
The interpretation of this ansatz is very clear. The $N_1$, $N_2$ entries in the diagonal determine the pattern of gauge symmetry breaking triggered by that bi-fundamental for the set of the $N_1$, $N_2$ D3-branes in the first, resp. second dimer. An edge absent in a sub-dimer implies a local recombination of the corresponding set of D3-branes across the associated bi-fundamental. Namely, there is a non-vanishing vev in the corresponding set of entries. Similarly, for edges present in a sub-dimer there is no vev in the corresponding entries of the associated bi-fundamentals.
The proof that the above assignment of vevs satisfies the flatness conditions in the field theory is provided in appendix \[proof\]. However it is useful to work out an explicit example, so consider for instance the splitting of the double conifold to two conifold singularities. Using the information in Figure \[dconidimer\] for the initial dimer, and Figure \[dconidaug1\] for the sub-dimers, we obtain the following set of vevs \_[12]{} = V\_2 , \_[23]{} = V\_2 , \_[34]{} = V\_0 , \_[41]{} = V\_0\
\_[21]{} = V\_0 , \_[32]{} = V\_0 , \_[43]{} = V\_1 , \_[14]{} = V\_1 \[thevevs\] where we have introduced the notation $\Phi_{ij}$ for a bi-fundamental $({\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}_i,{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}}_j)$, and take $\Phi_{12}$ to correspond to the vertical edge in the left part of the depicted unit cell.
It is now straightforward to analyze the flatness conditions on the set of vevs for this example. Concerning the F-term conditions, all nodes are 4-valent, hence the superpotential is a sum of quartic terms. Moreover, any such term contains at least two fields without vev. Hence, the F-terms conditions are automatically satisfied. Concerning the non-abelian D-term conditions, we write the generators of $SU(N)$ as T= and obtain that the D-term contributions for the $SU(N)_i$ factors are & SU(N)\_1 (\_[12]{}\^T \_[12]{}) + (\_[14]{}\^T \_[14]{}) = |v|\^2 ( T\_[11]{} + T\_[22]{}) = 0\
& SU(N)\_2 -(\_[12]{}\^T \_[12]{}) + (\_[23]{}\^T \_[23]{}) = |v|\^2 ( T\_[22]{}- T\_[22]{}) = 0\
& SU(N)\_3 -(\_[23]{}\^T \_[23]{}) - (\_[43]{}\^T \_[43]{}) = -|v|\^2 ( T\_[11]{} + T\_[22]{}) = 0\
& SU(N)\_4 (\_[43]{}\^T \_[43]{}) - (\_[14]{}\^T \_[14]{}) = |v|\^2 ( T\_[11]{}- T\_[11]{}) = 0 \[dflat\] where we have used tracelessness of $SU(N)$ generators. Finally, concerning the abelian D-term conditions, the above vevs lead to non-zero contributions which are suitably canceled by the non-zero FI terms. This effective absence of $U(1)$ D-term constraints can be equivalently regarded as the statement that there are $B\wedge F$ couplings (related to the FI terms by supersymmetry) which render the $U(1)$’s massive, so that they are not present at low energies and hence no D-term constraints have to be imposed, see footnote \[uunos\].
Notice that the description in this section generalizes in a straightforward fashion to the splitting of a dimer into more than two sub-dimers.
Effect on perfect matchings {#matchings}
---------------------------
It is interesting to consider the effect of partial resolution on perfect matchings. This can be easily analyzed at the level of the dimer diagrams, as we do in what follows in a particular example. Consider the double conifold, whose dimer diagram is shown in Figure \[dconidimer\]. The eight perfect matchings for this diagram are shown in Figure \[dconipmatch\]. The location of these matchings in the toric diagram, obtained as described in Section \[pmatchings\], using $p_1$ as reference matching, are shown in Figure \[dconidescend\]a.
Consider the partial resolution of the double conifold to two conifolds, studied in Section \[dconi\], whose two resulting sub-dimers are shown in Figure \[dconidaug1\]. The splitting of the dimer into sub-dimers implies that the perfect matchings of the original dimer fall into different classes:
- The perfect matchings $p_4,p_5$ descend to perfect matchings of the first sub-dimer.
- The perfect matchings $p_1,p_8$ descend to perfect matchings of the second sub-dimer.
- The perfect matchings $p_2,p_7$ correspond to perfect matchings of both the first and second sub-dimer.
- The perfect matchings $p_3,p_6$ do not correspond to perfect matchings of either sub-dimer.
This correspondence becomes nicely meaningful when one considers the location of the different perfect matchings in the toric diagram. The partial resolution splits the toric diagram in two pieces, separated by a common internal segment. Perfect matchings of the original dimer which descend to perfect matchings of a given sub-dimer are located at points on the piece of the toric diagram describing the corresponding daughter singularity. Perfect matchings descending to matchings of both singularities are located along the common segment in the toric diagram. This is described for the double conifold in Figure \[dconidescend\].
It is possible to show that this pattern is completely general, and that for a general partial resolution perfect matchings fall into one of these four classes. Namely, we label the edges of the dimer diagram with labels 1, 2 and 3, according to whether it is present in sub-dimer one, or in sub-dimer 2, or in both. Perfect matchings involving edges of type 1 and 3 end up in the interior of the toric sub-diagram 1; perfect matchings involving edges of type 2 and 3 end up in the interior of the toric sub-diagram 2; perfect matchings only involving edges of type 3 appear on both toric sub-diagrams, along their common boundary; perfect matchings with edges of type 1 and 2 (and possibly 3) disappear.
One can also obtain the effect of the partial resolution on the perfect matchings from the viewpoint of the Riemann surface. For that, one can use the relation described in Section \[pmatchings\] between pairs of perfect matchings and 1-cycles on the mirror Riemann surface. The first observation is that a partial resolution corresponds to the introduction of a segment joining two external non-adjacent perfect matchings $p$, $p'$ in the toric diagram. This is just dual to separating the web diagram by elongating the leg dual to that segment. Notice that cases where there are multiple matchings at the corresponding points in the toric diagram simply correspond to cases where there are several parallel legs in the web diagram, and correspondingly several possibilities to perform the partial resolution. For instance, in our above example, the partial resolution corresponds to choosing the perfect matchings $p_2$ and $p_7$.
To such a pair of perfect matchings one can associate a path $p'-p$ in the dimer diagram and a 1-cycle in the mirror Riemann surface. In fact, this 1-cycle wraps around the tube which becomes infinitely elongated in the partial resolution process. In terms of the dimer diagram, it means that the path in the dimer diagram becomes the new zig-zag path (denoted G in our example in Section \[dconi\]) introduced to construct the new sub-dimers.
Given that this 1-cycle separates the Riemann surfaces in two pieces, which are naturally associated to the two daughter singularities, it is possible to interpret the four classes of perfect matchings in terms of their behaviour on the Riemann surface $\Sigma$. Consider one of the external perfect matchings e.g. $p$. For any other matching $p_i$ one can consider the 1-cycles associated to $p_{i}-p$ obtained using the tiling of $\Sigma$. If the whole of such 1-cycle lies on one component of $\Sigma$, the matching $p_i$ will correspond to a perfect matching of the corresponding sub-dimer, and to a point in the corresponding toric sub-diagram. If all pieces of the 1-cycle are included in the 1-cycle $p'-p$, then $p_i$ will correspond to a perfect matching of both sub-dimers, and will appear in both toric diagrams (concretely, along the common boundary). Finally if the 1-cycle contains pieces lying in both components of $\Sigma$, the corresponding perfect matching disappears in the process of partial resolution.
These properties are easily explicitly checked in our above example, and can be generalized to any partial resolution. We leave the discussion as an exercise for the interested reader.
Partial resolutions with fractional branes {#fractional}
------------------------------------------
In this section we would like to study partial resolutions for singularities in the presence of fractional branes, and their description using dimers. For concreteness we center on a particular example, although our conclusions are of general validity.
Let us consider the splitting of the double conifold to two conifold singularities. The dimer diagram for the double conifold, with the most general set of fractional branes, is shown in Figure \[dconifract\]a. Since the field theory is non-chiral, there are no restrictions on the gauge factor ranks, and hence there are three kinds of fractional branes.
When the singularity is split into two conifolds, the latter may contain fractional branes as well. The most general possibility is shown in Figure \[dconifract\]b, c. Since each conifold allows for one kind of fractional brane, there are two possible fractional branes in the final system.
![a) The dimer for the double conifold with the most general set of fractional branes. b,c) The sub-dimers for the daughter conifold singularities, with their fractional branes. []{data-label="dconifract"}](dconifract.eps)
It is thus a natural question to ask what happens with the third kind of fractional brane. The answer, that we can recover from different viewpoints, is that it obstructs the partial resolution. A pictorial way to derive this result is to compare the original dimer and the daughter sub-dimers in Figures \[dconifract\]a and b,c, respectively. In order to have a proper splitting, the number of branes in a given face of the original dimer must agree with the sum of the numbers of branes in the corresponding location in the sub-dimers. In our particular case, this implies N=N\_1+N\_2 , M=M\_1 , P=0 , Q=M\_2 Hence we see that the splitting necessarily forces the fractional brane changing the rank of the gauge group $3$ to be absent, in the sense that only in the absence of such brane the splitting is possible. More precisely, what obstructs the splitting is the fractional brane which controls the difference between the ranks of the gauge factors 1 and 3.
In what follows we present several interpretations for this fact. From the viewpoint of the field theory of the initial singularity, it means that the theory with different ranks for the factors 1 and 3 does not have the corresponding flat direction. This can be argued in general, but is suffices to discuss one particular example, for instance $M=Q=0$, $P\neq 0$. It is simple to show that the D-term conditions for gauge factor 3 cannot be satisfied. Indeed, the natural ansatz is similar to (\[thevevs\]), with the only difference that for non-square matrices, the entries in the $M\times P$ additional submatrix are taken to be zero. In computing the D-term, as in (\[dflat\]), for the gauge factor 3, one notices that the non-zero vevs do not suffice to complete the full $SU(M+P)$ trace, and hence the D-term does not vanish.
One can regard the dimer as realizing a physical construction of the gauge theory in terms of NS-branes and D5-branes [@Franco:2005rj], similar to brane box [@Hanany:1997tb; @Hanany:1998ru; @Hanany:1998it] or brane diamond models [@Aganagic:1999fe]. In that context, the relation between obstructions to splitting the brane configuration and D-terms in the gauge theory is familiar and well-known. Intuitively, the motion of the NS-branes drags a subset of the D5-branes, increasing their tension and breaking supersymmetry (equivalently, misaligning the phase of their BPS charge with respect to the others). In the absence of fractional branes, D5-branes on opposite sides of the NS-brane can recombine and snap back, restoring supersymmetry at the price of breaking gauge symmetry (equivalently, forming a bound state with appropriately aligned phase). For certain fractional branes the possibility of recombination is not possible, leading to an obstruction to the brane motion.
To conclude this section, we provide an interpretation of the obstruction in terms of the mirror geometry, where a very explicit version of the above picture can be derived. In order to do that, consider the 1-cycles on the mirror Riemann surface $\Sigma$ which correspond to the different fractional branes, in our case, to the different faces in the dimer. These are sketched in Figure \[dconi3cycles\].
The structure of these 1-cycles, and in particular their winding around the punctures of $\Sigma$, leads to a natural explanation of the obstruction. Consider introducing only fractional branes changing the rank of the gauge factor 3. In the mirror this corresponds to introducing D-branes along the cycle that surrounds the punctures B, C. These punctures end up in different daughter Riemann surfaces in the splitting, see figure \[dconifactor\], hence in trying to perform the partial resolution, the D-brane stretches along the elongated tube, hence increases its tension and breaks supersymmetry. Moreover, it is not possible to express this 1-cycle in terms of a combination of brane cycles not stretching along the tube, hence no process restoring supersymmetry can take place. The same argument goes through if one considers only fractional branes of type 1, since they surround the punctures A, D. Notice that there is no problem if one considers instead fractional branes of type 2 or of type 4, since they do not correspond to cycles stretching along the tube.
Finally, consider introducing the same number of fractional branes of type 1 and 3. This case leads to equal rank for gauge factors 1 and 3, and hence we expect no obstruction. Indeed, although the brane correspond to cycles stretching along the tube, it is possible to deform them topologically to a sum of cycles of type 2 and 4, which do not stretch.
As mentioned above, this picture generalizes to more involved situations. The general lesson is that sets of fractional branes associated to cycles stretching along the tubes which elongate in the factorization of the Riemann surface lead to obstructions to the partial resolution.
Complex deformations {#complexdeformations}
====================
In this section we discuss the smoothing of singular geometries via a different process, namely complex deformations. Again, our analysis is general and valid for complex deformations which partially smooth out a singularity, or which split it into two daughter singularities.
Complex deformations of toric singularities have been discussed in diverse contexts. They are easily characterized in terms of the web diagram, as a splitting of the web into two sub-webs in equilibrium. Pictorially, the segment suspended between the two sub-webs after splitting represents the 3-cycle in the deformed geometry. This description, phrased in terms of a physical realization as a fivebrane web in [@Aharony:1997ju; @Aharony:1997bh] and in toric language in [@Leung:1997tw; @Aganagic:2001ug], is actually based on the mathematical theory of complex deformations [@altmann1; @altmann2; @altmann3].
In what concerns the relation between complex deformations and the gauge theory on D3-branes, the situation is very different from partial resolutions, and has been studied in [@Franco:2005fd] (generalizing the conifold case in [@Klebanov:2000hb]). The D-branes always live in the resolution phase of the singular geometry. Complex deformation phases are realized after geometric transitions triggered by the back-reaction of a large number of fractional branes [^9]. Namely, the homology class wrapped by the fractional brane disappears, and it is replaced by a 3-cycle in a complex deformation phase, which supports 3-form fluxes. This is similar to the Klebanov-Strassler proposal for the conifold with fractional branes [@Klebanov:2000hb].
The description of ‘deformation’ fractional branes in terms of the dimer diagrams was provided in [@Franco:2005zu]. Moreover a simple procedure was suggested to transform the original dimer diagram into the dimer diagram of the singular geometry after a complex deformation. Deformation branes correspond to a clusters of faces, touching at their corners, and complex deformation has the effect of smoothing certain touching edges and collapsing the painted regions to zero size. Although efficient, this procedure was rather [*[ad hoc]{}*]{}, with no derivation from first principles, and no clear rules on which intersections to smooth out, etc. In this section we fill this gap, and find clear rules based on physical principles of geometric transitions.
An example in the double conifold {#dconicomp}
---------------------------------
Let us start by considering a simple example, namely the complex deformation of the double conifold singularity into a conifold. In order to emphasize the physical ideas, we start with the gauge theory description and derive the effect on the dimer and the description in the web diagram.
Consider the gauge theory for D3-branes at the double conifold singularity (whose dimer diagram is in Figure \[dconidimer\]) in the presence of additional fractional branes, of the kind that increase the rank of the gauge factor 2. The geometric transition triggered by fractional branes is most easily seen in the mirror picture, in terms of an operation on the mirror Riemann surface. In figure \[dconibreak2\]a we show the mirror Riemann surface, along with the 1-cycle on which the corresponding fractional brane wraps. It corresponds to a 1-cycle winding around the punctures labeled A and B. The geometric transition corresponds to cutting the Riemann surface along this 1-cycle, as shown in Figure \[dconibreak2\]b, and gluing the boundary to get a new Riemann surface $\Sigma_1$, as shown in Figure \[dconibreak2\]c.
Notice that in fact this process leads to to two disjoint Riemann surfaces. The second (denoted $\Sigma_2$) corresponds to the removed pieces A, B, with a suitable closing of its boundary. In cases where the 1-cycle splits off two punctures, like in our example, this second Riemann surface is somewhat degenerate and we do not shown it in the pictures.
The distance between the two Riemann surfaces in the ambient space corresponds to the size of the new cycle, which is a 3-cycle in the original D3-brane configuration. The 1-cycle on which the fractional D-branes were wrapping has disappeared as required. The process can be regarded as a splitting of the original web diagram into two sub-webs in equilibrium, given by the sets C, D, E, F and A, B respectively. Hence, from the physics of geometric transitions we recover the usual description in terms of the web diagram as shown in Figure \[dconicomplex\]. A bonus of our argument is that it allows a direct identification of which web splitting corresponds to the geometric transition triggered by a given fractional brane. This piece of the dictionary was missing from previous analysis [@Franco:2005fd], which supplemented it with suitable guesswork.
The new Riemann surfaces moreover allow us to construct the dimer theories of the theories in the left over geometry after complex deformation. Namely, for each of the Riemann surfaces, the left over zig-zag paths, along with the adjacency relations (including those required by the gluing of boundaries in figure \[dconibreak2\]), can be used to define the left-over dimer diagram [^10]. This is shown in Figure \[dconidimdef\] for $\Sigma_1$. In figure \[dconidimdef\]a we depict the zig-zag paths of the original dimer diagram which are associated to punctures in the daughter Riemann surface $\Sigma_1$. In Figure \[dconidimdef\]b we implement the new adjacency relations implied by the gluing of the boundary in Figure \[dconibreak2\]c. We also show the dimer diagram that corresponds to this final configuration. After integrating out the matter massive due to the bi-valent node, the diagram is easily shown to correspond to the conifold theory, hence describes the gauge theory on D3-branes sitting at the left over conifold singularity after complex deformation.
Notice that this operation on the dimer is equivalent to the proposal in [@Franco:2005fd] of collapsing some faces to zero, while splitting some nodes open, shown in Figure \[dconiadhoc\]. The key difference is that in our analysis the rules are derived from first principles thanks to a proper understanding of the geometric transition and its implications on the dimer.
An example in $dP_3$ {#moreexamplescomplex}
--------------------
Let us consider another example, in fact one of the complex deformations discussed in [@Franco:2005fd]. Consider the gauge theory on D3-branes at the complex cone over $dP_3$, whose dimer diagram is shown in Figure \[dp3dimer\], with a number of fractional branes increasing the rank of the gauge factors 1, 3 and 5 by the same amount. The gauge theory analysis in [@Franco:2005fd] suggests that these fractional branes trigger a complex deformation smoothing the singularity completely.
We can easily recover this result following the procedure described above in the dimer diagrams. Thus one constructs the 1-cycle in the mirror Riemann surface that is the homological sum of the 1-cycles corresponding to the faces 1, 3 and 5. Working this out as in the above example, this total 1-cycle winds around the punctures A, C and E, namely separates the Riemann surface in two regions, which include the punctures A, C, E and B, D, F, respectively. This is shown in figure \[dp3break\]. The geometric transition triggered by the fractional branes should make this cycle disappear, hence we cut the Riemann surface along this cycle, and glue the boundary of each to yield two daughter Riemann surfaces. Each has the topology of a 2-sphere with three punctures. Clearly this process corresponds to a splitting of the web diagram in two sub-webs, in particular to the splitting shown in Figure \[dp3complex\]. This had already been obtained in [@Franco:2005fd] by guesswork.
The above operations in the 1-cycles in the mirror Riemann surface, have a direct effect on the set of zig-zag paths of the original theory. In fact, the resulting dimers after the complex deformation can be recovered by directly operating on these. Consider the zig-zag paths of the original $dP_3$ theory, shown in Figure \[dp3path\]. The cutting of the Riemann surface separates them into two daughter sets, that will correspond to the zig-zag paths of the daughter theories. They are shown in the Figure \[dp3dimdef\]a. The gluing of the boundaries of the pieces of the original Riemann surface to form the daughter ones imply new adjacency relations between the unpaired pieces of the zig-zag paths. The result is two sets of consistent zig-zag paths, which can be used to construct the daughter dimer diagrams. This is shown in Figure \[dp3dimdef\]b, where the two daughter dimer diagrams are seen to describe $N=4$ supersymmetric theories. This corresponds to two sets of D3-branes sitting at different smooth points in the deformed geometry, in agreement with the geometrical picture discussed above.
This example makes manifest one subtle feature. In cutting the Riemann surface, it is not only the homology class of the 1-cycle that is important, rather it is crucial to use the particular representative associated with the tiling of $\Sigma$. In the above example the particular representative cuts the Riemann surface into two disks, while a more generic representative in the same homology class (e.g. that obtained by smoothing the intersections in figure \[dp3break\]) would not yield that result.
This observation is important in a last respect. It is known that there exist several kinds of fractional branes at singularities, dubbed ‘$N=2$’, ‘deformation’ and ‘DSB’ fractional branes in [@Franco:2005zu]. Different fractional branes lead to different behaviour of the infrared gauge theory. Interestingly, their differences become manifest in the brane tiling and in the associated 1-cycle in the mirror Riemann surface. Namely, only deformation branes lead to 1-cycles which separate the Riemann mirror surface into two pieces. Hence only deformation branes can lead to a splitting of the Riemann surface making the 1-cycle trivial, as required by the physical interpretation of the geometric transition. For this connection to hold it is crucial [*not*]{} to allow to replace the 1-cycle determined by the tiling by another representative in the same homology class.
A last remark is in order. Notice that it is straightforward to derive the rule that complex deformations correspond to splitting off a sub-web in equilibrium. This follows from the fact that the cluster of dimer diagram faces associated to a deformation fractional brane have the topology of a disk. Hence, the total 1-cycle associated to the fractional brane has zero total $(p,q)$ charge. Thus the set of punctures it surrounds, and which correspond to the removed legs in the web diagram, have zero total $(p,q)$ charge, leading to a sub-web in equilibrium.
Field theory interpretation {#fieldcomp}
---------------------------
The gauge theory interpretation of complex deformations has been discussed for relatively general toric singularities in [@Franco:2005fd], generalizing the discussion of the conifold in [@Klebanov:2000hb]. Complex deformations of the geometry are related to confinement of the gauge factors associated to certain fractional branes. In [@Franco:2005fd], the appearance of complex deformations in such situations was tested by introducing additional D3-brane probes. Confinement of the fractional brane gauge factors leads to a quantum deformed moduli space for these probes, hence reproducing the complex deformation of the underlying geometry. In fact, if the complex deformed space contains a left over singularity, the gauge theory on the D3-brane probe reduces, after confinement, to the quiver gauge theory associated to that singularity.
In this section we provide a description of the gauge theory analysis of the dynamics of the additional D3-brane probes, along the lines of [@Franco:2005fd], in terms of dimer diagrams. We show that one can recover the dimer diagram of the left over singularities by simple operations in the dimer, with a clear gauge theory interpretation. Moreover, this recipe agrees with the procedure based on zig-zag paths described in Sections \[dconicomp\], \[moreexamplescomplex\].
Let us consider the example in Section \[dconicomp\] of the double conifold deforming to a conifold. As discussed above, the complex deformation is triggered by introducing fractional branes changing the rank of the gauge factor 2. The complex deformation can be tested by introducing additional D3-branes probes. Hence we consider, as in [@Franco:2005fd], the gauge theory with rank vector $(M,2M,M,M)$, and eventually we will focus on $M=1$.
For convenience we sketch the different steps in the gauge theory analysis (referring the reader to [@Franco:2005fd] for details), and their implementation in dimer diagrams, as shown in Figure \[dconifhu\]. The starting dimer diagram with fractional branes is shown for convenience in Figure \[dconifhu\]a.
The gauge factor 2 confines, hence one must introduce the corresponding mesons and baryons. The dynamics of the D3-brane probes is manifest on the mesonic branch, so the baryons will play no role and are set to zero. The mesons are however crucial so it is convenient to introduce them in an early stage. Figure \[dconifhu\]b shows the gauge theory dimer diagram with the mesons of gauge factor 2 already introduced, even before taking into account the full effects of confinement. The mesons correspond to new edges appearing from the corners of face 2. This intermediate operation is formally similar to the implementation of Seiberg duality in dimer diagrams [@Franco:2005rj].
Since the gauge factor 2 has the same number of colors and flavours, the classical constraint on the meson matrix has a quantum modification to ${\rm det}\, {\cal M}=1$. This constraint can be implemented in the superpotential via a Lagrange multiplier. Actually, the D3-brane probe theory is manifest when the Lagrange multiplier has actually a non-zero vev. At this stage, all the non-abelian dynamics has been introduced and we can simply consider the abelian situation $M=1$. In this case, the non-perturbative contribution to the superpotential simply amounts to two quadratic terms in the mesons. The dimer diagram resulting from the above operations is shown in Figure \[dconifhu\]c. The gauge factor 2 has confined and disappeared. It leaves behind the mesons with new interaction terms between mesons from opposite corners. The latter implies that the diagram cannot be drawn on a torus, since two edges must cross without intersection. This simply reflects the fact that the moduli space of the gauge theory is not a toric variety, in agreement with the fact that a complex deformation of a toric space is not itself a toric space. The dimer diagram clearly knows that after confinement a quantum deformation of the moduli space is taking place!
Although the full deformed geometry is not a toric space, one can zoom into a neighbourhood of the D3-brane probe and find a toric description for it. If one chooses mesonic vevs corresponding to locating the D3-brane probe at the left over singularity in the deformed space, one should then recover the gauge theory of D3-branes at this singularity. At the level of the dimer diagram, a choice of mesonic vevs saturating the constraint ${\rm det}\, {\cal M}=1$ should lead to the dimer diagram of the left over singularity. In Figure \[dconifhu\]d we show the result of this, where the removal of a line of edges corresponds to giving vevs to the associated mesons. After integrating out the bi-valent nodes, one obtains Figure \[dconifhu\]e, which corresponds to the dimer diagram of the conifold theory.
Hence, we have provided a set of simple dimer diagram rules which reproduce the gauge theory analysis of the D3-brane probe theory in [@Franco:2005fd]. Both techniques show that the geometry probed by the D3-brane is the complex deformation of the double conifold to the conifold.
Notice that the gauge theory operations we have just described nicely dovetail the procedure in terms of zig-zag paths described in previous sections. Namely the removal of the sub-web legs corresponds to confinement of the fractional brane. The edges which lose one of their zig-zag paths correspond to fields charged under the fractional brane group, and the process of closing them by new adjacency relations corresponds to constructing the mesons of the confining theory. More precisely, to the mesons that survive after satisfying the quantum constraint via introduction of vevs. Hence, the zig-zag path method provides in one step the final dimer diagram of D3-branes at the left over singularity. Notice also the nice relation between our gauge theory process as described above, and the [*ad hoc*]{} dimer diagram rules in [@Franco:2005fd], see Figure \[dconiadhoc\].
We conclude this section by showing the dimer diagram representation of the gauge theory analysis of D3-branes probes in the complex deformation of the cone over $dP_3$ to a smooth space. The relevant steps are illustrated in Figure \[dp3fhu\] and correspond to the gauge theory discussion in Section 4.3 in [@Franco:2005fd]. A sketch of this gauge theory analysis is provided along with the picture. The fact that the dimer diagram rules reproduce in a few easy steps an involved gauge theory analysis, such as this one, illustrates the power of these representations. Moreover, it is easy to show that the complete process is reproduced in a one-step fashion by the operations using zig-zag diagrams described in Section \[moreexamplescomplex\].
Effect on perfect matchings {#pmcomp}
---------------------------
In this section we describe complex deformations from another useful viewpoint, in terms of perfect matchings. As we show below, this has a very direct connection with the decomposition of the toric diagram as a Minkowski sum, used in the mathematical literature on complex deformations [@altmann1; @altmann2; @altmann3] (see also Appendix in [@Franco:2005zu] for a short description).
For concreteness let us consider an example which illustrates the general idea. Consider the deformation of the complex cone over $dP_3$ to a smooth space, as described in Section \[moreexamplescomplex\]. The dimer diagram for the $dP_3$ theory is shown in Figure \[dp3dimer\], and its twelve perfect matchings are shown in Figure \[dp3pmatch\].
The location of these perfect matchings in the toric diagram, obtained as described in Section \[pmatchings\] is shown in Figure \[dp3heights\]a.
The effect of the complex deformation on the perfect matchings can be determined as follows. Recall that to each pair of perfect matchings we can associated a 1-cycle $p_j-p_i$ in $\Sigma$ obtained by superimposing them. The complex deformation amounts to cutting $\Sigma$ in two pieces and gluing the boundary of each piece separately to obtain $\Sigma_1$ and $\Sigma_2$. In this process, some of the homology classes of $\Sigma$ become trivial in e.g. $\Sigma_1$. This implies non-trivial equivalence relations between difference paths of perfect matchings. For instance, matchings whose difference path is a combination of the 1-cycles becoming trivial become equivalent in $\Sigma_1$. In addition to these identifications, the equivalence relations also imply changes in the slopes of the matchings, and thus an induced change in the toric diagram.
To illustrate the procedure let us consider the deformation of the complex cone over $dP_3$ to a smooth space. In this example, studied above, $\Sigma_1$ is obtained by removing the zig-zag paths A, C, E, enclosed by the fractional brane associated to the gauge factors 2, 4, 6. It is easy to obtain the fate of the different 1-cycles of $\Sigma$ in $\Sigma_1$, by drawing them on the Riemann surface. For instance, the zig-zag paths A, C, E become trivial in $\Sigma_1$, whereas B, D, F remain non-trivial zig-zag paths in the daughter surface. Another interesting set of 1-cycles is given by those associated to the $i^{th}$ gauge factor, which we denote by $f_i$. One can check that $f_2$, $f_4$, $f_6$ become trivial in $\Sigma_1$, while $f_1$, $f_3$, $f_5$ become identical to $-$B, $-$D, $-$F in $\Sigma_1$. Then one can easily check relations like $p_1-p_3=-A-E+f_2\simeq 0$, etc. Following this, the matchings fall into three equivalence classes. We have p\_1=p\_3=p\_[10]{}=p\_[12]{} ; p\_2=p\_4=p\_6=p\_8 ; p\_5=p\_7=p\_9=p\_[11]{} In addition one can obtain relations like $p_1-p_2=F+f_2\simeq F$, etc, which can be used to obtain the new relative slopes, and the positions of the (equivalence classes of) perfect matchings in the new toric diagram. The result of this operation is shown in Figure \[dp3mink\]a. The result is indeed the diagram corresponds to a smooth space, in fact the dual to the subweb corresponding to the legs B, D, F. The resulting diagram can be regarded as a contraction of the original toric diagram along a triangle, as illustrated in Figure \[dp3mink\]b.
Clearly, one can operate in a similar manner with the other daughter Riemann surface $\Sigma_2$, and define another set of equivalences of perfect matchings. The end result is that the equivalence classes describe a toric diagram dual to the sub-web diagram corresponding to the legs A, C, E, as expected for $\Sigma_2$.
The splitting of the toric diagram into several sub-diagrams is dual to the splitting of the web diagram into sub-webs in equilibrium. Now the splitting of the toric diagram into sub-diagrams, each of which can be regarded as a contraction of the initial diagram (along the other sub-diagrams) has a nice connection with the mathematical operation known as decomposition of a toric polygon into Minkowski summands [@altmann1; @altmann2; @altmann3]. In fact, this operation lies at the heart of the mathematical characterization of complex deformations of toric singularities. It is a pleasant surprise to find a direct realization of the Minkowski decomposition of the toric diagram in the dimer language. We hope this relation brings the mathematics literature somewhat closer to the physical realization of deformations via geometric transitions (see [@Pinansky:2005ex] for an example of this).
As a final remark, we would like to point out that the use of perfect matchings also provides an interesting new way to recover the dimer diagrams of the daughter theories. Namely, after determining the equivalence classes of perfect matchings for a given Riemann surface, one can pick one representative of each class, and consider the set of edges involved in this set of matchings. The diagram obtained by superimposing all of them is exactly the dimer diagram that corresponds to the daughter theory.
Conclusions {#conclu}
===========
In this paper we have provided tools to describe in full generality the processes that smooth out toric singularities, from the viewpoint of the gauge theory on D3-branes probing such geometries. The results are nicely cast in the language of dimer diagrams, in particular making use of the connection between the dimer diagram and the web diagram (via zig-zag paths and/or perfect matchings).
For partial resolutions our tools should allow a quick construction of the gauge theory for any toric singularity. It would suffice to implement out optimized tools using zig-zag paths to the partial resolution algorithm introduced in [@Morrison:1998cs], with the advantage of being able to carry out a complicated partial resolution in one step.
We have also provided a detailed gauge theory interpretation of the splitting of a dimer diagram into sub-dimers in the partial resolution process. It corresponds to a specific Higgs mechanism which splits the gauge theory into two gauge sectors decoupled at the level of massless states. We envision interesting model building applications of such systems.
An interesting open question is to understand the inverse process of combining different toric singularities into a single one, by inverting the partial resolution. Namely, by adjoining the corresponding web diagrams along one or several legs, and to shrink the resulting segments. Progress in this direction should deal with ambiguities in the precise dimer diagrams to be combined, since the latter are defined modulo integration of bi-valent nodes.
For complex deformations we have provided a dictionary between the fractional branes and the precise splitting of the web diagram into sub-webs. Moreover, we have provided a simple set of dimer rules that reproduce the involved non-perturbative gauge theory analysis which describes D3-branes probes at such geometries. In addition the net result of these gauge theory operations is subsumed in extremely simple operations on the dimer in terms of zig-zag paths. It would be interesting to develop similar tools to analyze other infrared behaviours, like the removal of the supersymmetric vacuum by DSB branes.
We hope these tools are useful for these and other interesting purposes.
**Acknowledgements**
We thank S. Franco and A. Hanany for illuminating conversations. A.M.U. thanks M. González for kind encouragement and support. F.S. and I.G.-E. thank CERN TH for hospitality during completion of this project. This work has been partially supported by CICYT (Spain) under project FPA-2003-02877, and the RTN networks MRTN-CT-2004-503369 ‘The Quest for Unification: Theory confronts Experiment’, and MRTN-CT-2004-005104 ‘Constituents, Fundamental Forces and Symmetries of the Universe’. The research of F.S. is supported by the Ministerio de Educación y Ciencia through an F.P.U grant. The research of I.G.-E. is supported by the Gobierno Vasco PhD fellowship program and the Marie Curie EST program.
Proof of flatness {#proof}
=================
The flatness conditions can be checked in the general case, by a slight generalization of the analysis in the example in Section \[fieldtheory\]. We recall that in this section we are considering original dimer diagrams not containing bi-valent nodes (hence they have been integrated out if originally present).
[**F-flatness conditions:**]{} As described in Section \[matchings\], in partial resolutions each subdimer contains at least one perfect matching of the original dimer diagram. This implies that every sub-dimer contains all the nodes of the original dimer diagram. From this follows that in any sub-dimer, for any node there are at least two edges ending on it in every sub-dimer. At the level of the gauge theory, this implies that for each superpotential term of the original theory there are a sufficient number of bi-fundamentals with zero vevs to automatically satisfy the F-term conditions. Hence the assignment of vevs dictated by the dimer rules is F-flat.
[**Non-abelian D-flatness conditions:**]{} As described in section \[splitting\], we divide the set of zig-zag paths into two disjoint sets, where each set admits a dual interpretation as the set of external legs in the web diagram that we take to infinity. Let us denote collectively the elements belonging to the first set as 1 and those belonging to the other set as 2.
Consider a given face in the dimer diagram, and orient its edges by running through them e.g. counterclockwise. Each edge can then be classified into 4 types depending on which kind of zig-zag paths intersect over it. We will denote the four kinds as type 1, 2, 3 and $3'$, see figure \[zzedge\], where 3 and $3'$ are distinguished by the orientation [^11].
![The four possible types of edges, classified according to the zig-zag paths meeting at the edge.[]{data-label="zzedge"}](zzedge.eps "fig:") ![The four possible types of edges, classified according to the zig-zag paths meeting at the edge.[]{data-label="zzedge"}](zzedge.eps "fig:") ![The four possible types of edges, classified according to the zig-zag paths meeting at the edge.[]{data-label="zzedge"}](zzedge.eps "fig:") ![The four possible types of edges, classified according to the zig-zag paths meeting at the edge.[]{data-label="zzedge"}](zzedge.eps "fig:")
In this fashion, we assign to each face a (periodic) string of symbols given by the kind of edges we encounter when traversing the face counterclockwise. A typical string will then look like:$$\ldots 3'1323'3 \ldots$$ where we have written just the period. It is easy to realize that any valid string should satisfy a few rules which we can read from the dimer diagram. Namely there are some sequences of symbols that are not allowed, for example $3'2$. To see this, focus on the zig-zag paths “interior” to the edge. The given sequence would tell us that a type 1 zig-zag path exits the $3'$ vertex from the right, and then joins a type 2 zig-zag path in the next edge, see figure \[zzbadedge\]. This is obviously not allowed. The other disallowed sequences are $13'$, $23$, $31$, $33$, $3'3'$, $12$ and $21$.
![Inconsistent pasting of edges ($3'2$). Note that the only constraints come from the joining of the interior zig-zag paths. The exterior ones can be arbitrary as they do not need to be joined (in the absence of bi-valent nodes) since they “run off” along some extra edge, denoted by the dashed line in the drawing.[]{data-label="zzbadedge"}](zzbadedge.eps)
We can then associate to the most general face in a dimer a sequence of symbols not containing these forbidden words. It is easy to convince oneself that in any such string, at least one of the following substitutions applies and gives rise to another consistent sequence with two symbols removed in the period (“$\cdot$” denotes the empty word): $$\begin{aligned}
11 & \longrightarrow \cdot \quad\quad ; \quad\quad & 22 \longrightarrow
\cdot
\quad\quad ; \quad\quad 33' \longrightarrow \cdot \quad\quad ;
\quad\quad 3'3 \longrightarrow \cdot \\
132 & \longrightarrow 3 \quad\quad ; \quad\quad
& 23'1 \longrightarrow 3'\end{aligned}$$ As an example, applying the rules one would get the following sequence of strings:$$3'133'1132 \lra 3'11132 \lra 3'132 \lra 3'3 \lra \cdot$$
Since we can always apply one of these rules, and all of them reduce the length of the string by two, we have found that it is always possible to reduce an arbitrary string to nothing [^12]. The interesting fact about these operations is that on the field theory side they do not change the value of the D-term. Essentially, the first four rules preserve the D-term value because the disappeared edges correspond to a fundamental and an antifundamental with the same vev, hence with canceling contributions to the D-term. For the last two rules, the disappeared edges have vevs whose contributions add up to the trace of an $SU(N)$ generator, which is zero. One can in this way easily translate between the language used in equation \[dflat\] and this language of sequences. What this means is that the value of the D-term for all possible faces in a dimer is given by the D-term of the empty sequence, which is equal to zero.
As an example, let us study the configuration depicted in figure \[dtermface\]. The periodic string we associate with the face is given by $\ldots 223'132 \ldots$. Applying the rules we have described a possible reduction to nothing would be:$$223'132 \lra 23'32 \lra 22 \lra \cdot$$ This proves that the D-term for the relevant gauge group vanishes.
![A possible face in a dimer, where we have indicated the relevant classification for the zig-zag paths. External edges are denoted by the dashed lines, and the arrow indicates the traversal direction used in the text when enumerating the edges.[]{data-label="dtermface"}](dtermface.eps)
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[^1]: For certain simple singularities (like complex cones over del Pezzo surfaces) which include some non-toric cases, general methods based on exceptional collections have been applied, see e.g. [@Wijnholt:2002qz; @Herzog:2003zc]
[^2]: See [@Beasley:2001zp; @Feng:2001bn; @Berenstein:2002fi; @Herzog:2004qw] for earlier descriptions of Seiberg dualities for D3-branes at singularities.
[^3]: We will frequently abuse language and simply call *dimer* the complete dimer graph. Strictly speaking a dimer is what we call an edge.
[^4]: The brane tiling / dimer diagram can be dualized to an improved quiver diagram, the periodic quiver, which also encodes all this information.
[^5]: Earlier applications of mirror symmetry to D3-branes at singularities can be found in [@Hanany:2001py; @Feng:2002kk].
[^6]: Yet another equivalent way, not used in this paper, is the computation of the determinant of the Kasteleyn matrix, see e.g. [@Hanany:2005ve; @Franco:2005rj].
[^7]: \[uunos\] This is if the $U(1)$ factors are included in the gauge theory. In fact, these $U(1)$ factors are generically massive, hence disappear from the low energy effective theory. In this viewpoint, the partial resolution corresponds to vevs for suitable baryonic operators. We however find it more convenient to include the $U(1)$’s explicitly, and consider the couplings rendering them massive at a subsequent stage. See [@Ibanez:1998qp; @Morrison:1998cs] for a more detailed discussion.
[^8]: This description explains as in [@Hanany:2005ss] the possibility of the appearance of inconsistent dimer diagrams by arbitrary addition/removal of edges. For instance, consider a minimal partial resolution involving two zig-zag paths overlapping over more than one edge. The removal on only one of these edges does not correspond to a consistent separation of zig-zag paths and leads to an inconsistent diagram.
[^9]: Here by fractional brane we mean any anomaly-free assignment of ranks to the quiver gauge theory, which does not correspond to all factors having equal rank. Also it is important to point out that we center on the case of ‘deformation branes’ in the sense of [@Franco:2005zu], since there exist other kinds of fractional branes with different dynamical behaviour, like the removal of supersymmetric vacuum by strong infrared effects [@Berenstein:2005xa; @Franco:2005zu; @Bertolini:2005di], see also [@Intriligator:2005aw].
[^10]: For Riemann surfaces with two punctures the dimer diagram is somewhat peculiar, with one face, one edge and no nodes. The corresponding gauge theory can however be properly obtained by recalling the physical realization of the dimer in terms of NS- and D5-branes. The special rules for this diagram are due to the extended supersymmetry of the configuration.
[^11]: The similar notation for edges and zig-zag paths is introduced to (hopefully) improve the readability. In the rest of this section we mostly deal with edges, so this should not cause too much confusion.
[^12]: The sequences always have even length, consistently with anomaly cancellation.
|
---
abstract: 'We investigate the relation between weighted quasi-metric Spaces and Finsler Spaces. We show that the induced metric of a Randers space with reversible geodesics is a weighted quasi-metric space.'
author:
- 'Sorin V. SABAU, Kazuhiro SHIBUYA and Hideo SHIMADA'
title: 'Metric structures associated to Finsler metrics [^1] [^2] '
---
Introduction and Motivation
===========================
Riemannian spaces can be represented as metric spaces. Indeed, for a Riemannian space $(M,a)$ we can define the induced metric space $(M,d_\alpha)$, with the metric $$d_\alpha:M\times M\to [0,\infty),\quad
d_\alpha(x,y):=\inf_{\gamma\in \Gamma_{xy}}\int_a^b\alpha(\gamma(t),\dot\gamma(t))dt,$$ where $\Gamma_{xy}:=\{ \gamma:[a,b]\to M\ |\ \gamma \textrm{ (piecewise) C}^\infty \textrm{-curve},\gamma(a)=x,\gamma(b)=y\}$ is the set of curves joining points $x$ and $y$, $\dot\gamma(t):=\frac{d\gamma(t)}{dt}$ the tangent vector to $\gamma$ at $\gamma(t)$, and $\alpha(x,X)$ the Riemannian norm of the vector $X\in T_xM$. It is easy to see that $d_\alpha$ is a metric on $M$, i.e. it satisfies the axioms:
1. Positiveness: $d_\alpha(x,y)>0$ if $x\neq y$, $d_\alpha(x,x)=0$,
2. Symmetry: $d_\alpha(x,y)=d_\alpha(y,x)$,
3. Triangle inequality: $d_\alpha(x,y)\leq d_\alpha(x,z)+d_\alpha(z,y)$,
for any $x,y,z\in M$.
More general structures than Riemannian ones are Finsler structures (see [@BCS00], [@S01], [@MHSS01] for definitions).
Similarly with the Riemannian case, one can define the induced metric of a Finsler space $(M,F)$ by $$\label{Finslerian distance}
d_F:M\times M\to [0,\infty),\quad
d_F(x,y):=\inf_{\gamma\in \Gamma_{xy}}\int_a^bF(\gamma(t),\dot\gamma(t))dt,$$ but in this case, unlike the Riemannian counterpart, $d_F$ lacks the Symmetry condition 3 above. In fact $d_F$ is a special case of [*quasi-metric space*]{}.
We recall here that a [*quasi-metric*]{} $d$ on a set $X$ is a function $d:X\times X\to [0,\infty)$ that satisfies the axioms:
1. Positiveness: $d(x,y)>0$ if $x\neq y$, $d(x,x)=0$,
2. Triangle inequality: $d(x,y)\leq d(x,z)+d(z,y)$,
3. Separation axiom: $d(x,y)=d(y,x)=0\quad \Rightarrow \quad x=y,$
for any $x,y,z\in X$.
Remark that in the definition of quasi-metric spaces, it is commonly used $d_F(x,y)=0 \Rightarrow x=y$, without assuming that both $d_F(x,y)$ and $d_F(y,x)$ are zero (Def 2.1 in [@JLP]). This guarantees that the distance is zero only in the diagonal. Our definition here is stronger than the usual one. In general, the distance associated to a Finsler metric is a generalized metric, namely, a quasi-metric such that the forward and backward topology coincide, see Remark 2.2 in [@JLP].
A quasi-metric that satisfies the symmetry axiom $d(x,y)=d(y,x)$ for all $x,y\in M$ is a metric on $M$. One can easily see that this happens in the case of Riemannian and absolute homogeneous Finsler metrics, i.e. Finsler norms $F$ for which $F(x,y)=F(x,-y)$, for all $(x,y)\in TM\setminus\{0\}$.
Finsler manifolds have a richer geometrical structure than Riemannian ones. Indeed, for a given Finsler manifold $(M,F)$ we can define the [*reverse*]{} Finsler structure $(M,\bar F)$, where $\bar F(x,y):=F(x,-y)$. This means that on $M$ we obtain the reverse (or the dual) quasi-metric $d_{\bar F}(x,y):=d_F(y,x)$. Moreover, from the metric point of view we can define
- the symmetrization of $d_F$, namely $$\label{symmetrized dist}
\rho(x,y):=\dfrac{d_F(x,y)+d_F(y,x)}{2},$$
- the max-metric $$\label{max metric}
d^*(x,y):=\max\{d_F(x,y),d_F(y,x)\},$$
for any $x,y\in M$. One can easily see that these are metrics on $M$. We also recall that because of the lack of symmetry of distance function $d_F$ it is customary to make distinction between balls, neighborhoods, etc. by calling them [*forward*]{} or [*backward*]{}, respectively.
One special class of quasi-metric spaces are the so called [*weighted quasi-metric spaces*]{} $(M,d,w)$, namely $d$ is a quasi-metric on $M$ for each there exists a function $w:M\to [0,\infty)$, called the [*weight*]{} of $d$, that satisfies
1. Weightability: $d(x,y)+w(x)=d(y,x)+w(y),\quad \forall x,y\in M.$
In the case the weight function $w$ is ${\ensuremath{\mathbb{R}}}$-valued, $w$ is called [*generalized weight*]{}.
If $(M,d)$ is a metric space, then it can be regarded as a weighted metric space with a weight function $w=constant$.
The weighted quasi-metric spaces were initially introduced in the context of theoretical computer science [@M94] and their topological properties are extensively studied (see [@KV94] and references herein). It is worth mentioning that the study of denotational semantics of programming languages imposes a topological model defined on a weighted quasi-metric space. From topological point of view, a topological $T_0$-space $X$ admits a weighted quasi-metric provided it has a base $\bigcup_{n\in {\ensuremath{\mathbb{N}}}}\mathcal B_n$ for its topology with the property that for each $n\in {\ensuremath{\mathbb{N}}}$ there is an $m_n\in {\ensuremath{\mathbb{N}}}$ such that each point of $X$ belong to at most $m_n$ elements on $\mathcal B_n$ ([@KV94]). We recall that a topological space is a $T_0$-space or Kolmogorov space if for every pair of distinct points of $X$, at least one of them has an open neighborhood not containing the other. This condition is one of the separation axioms in topology and its intuitive meaning is that the points of $X$ are topologically distinguishable.
More recently, it has been shown ([@SY09]) that weighted quasi-metric spaces are essential for sequence comparison in molecular biology and bioinformatics. The comparison of biological sequences (especially proteins) is the fundamental method for the investigations of the origin and function of peptide fragments with evolutionary conserved sequence. The primary method used here is the similarity search: find similar fragments to a given query amino acids sequence and infer the function refereeing to the known functions of search results. The representative tool for similarity search is BLAST that can be accessed from NCBI, DDJ or ENSEMBL web sites.
It is also known that similarity search of biological sequences can be geometrically formalized by defining a sort of “distance ” on the free monoid $\Sigma^{*}$ over a non-empty finite set $\Sigma$. Concretely, in the case of peptide fragments comparison, $\Sigma$ is the set of proteinogenic amino acids (arginine, histidine, lysine; aspartic acid, glutamic acid; serine, threonine, asparagine, glutamine; cysteine, glycine, proline; alanine, isoleucine, leucine, methionine, phenylalanine, tryptophan, tyrosine, valine), where each amino acid is denoted by a letter and $\Sigma^{*}$ contains all finite sequences of zero or more elements from $\Sigma$.
Remarkably, the evolutionary distance induced by local or global alignments of peptide fragments is actually a weighted quasi-metric on $\Sigma^{*}$. Therefore, the study of sequence comparison reduces to the geometry of weighted quasi-metrics, where minimizing similarities between peptide fragments is equivalent to minimizing weighted quasi-distances between elements of the free monoid $\Sigma^{*}$ ([@SY09]).
However, despite of extensive investigations of weighted quasi-metric spaces from topological point of view and different applications, a study of these spaces from differential geometry point of view cannot be found in literature.
In the present paper we will show that the metric structure induced by a Finsler metric with reversible geodesics is actually a weighted quasi-metric. This clarifies the geometrical meaning of weighted quasi-structures.
Moreover, we obtain several interesting geometrical properties of Finsler metrics with reversible geodesics and weighted quasi-metric spaces.
[**Acknowledgements.**]{} The authors are extremely grateful to the referee for several useful suggestions that have considerable improved the paper. We also thanks to Miguel Angel Javaloyes for pointing out some inaccuracies in a preliminary version.
Finsler metrics and weighted quasi-metrics
==========================================
Recall that a Finsler metric $F$ on a $n$-dimensional differential manifold $M$ is called [*with reversible geodesics*]{} if and only if for any geodesic $\gamma:[0,1]\to M$ of $F$, the reverse curve $\bar\gamma(t):=\gamma(1-t)$ is also a geodesic of $F$.
We point out that even a Finsler space is with reversible geodesics, the Finslerian distance function $d_{F}$ is not symmetric, except for the absolute homogeneous case.
We have (see [@MSS10], [@MSS13], [@SS12] and references herein)
\[rev\_geod\] Let $(M,F=F_{0}+\beta)$ be a Finsler space whose fundamental function is obtained by a Randers change of an absolute homogeneous Finsler metric $F_{0}$ by a one-form $\beta$. Then $(M,F)$ is with reversible geodesics if and only $\beta$ is closed.
The intuitive meaning of the Randers change $$\label{Randers change}
F=F_{0}+\beta,\quad d\beta=0$$ is that the $F$-geodesics coincide with the $F_{0}$-geodesics as set of points, i.e. $F$ and $F_{0}$ are projectively equivalent. Remark that this Randers change is a special Randers change with $\beta$ closed. Hereafter, when referring to this formula, we always mean [*Randers change with $\beta$ closed*]{}.
1. A special case is the case of Randers metrics $F=\alpha+\beta$, where $\alpha=(a_{ij}(x))$ is a Riemannian metric and $\beta$ closed one-form. It is known that a Randers metric is positive definite if and only if the Riemannian length of the vector $b_{i}(x)$ is less than one, i.e. $b(x):=\sqrt{a_{ij}(x)b^i(x)b^j(x)}< 1$, for $\forall x\in M$. This property also holds in the more general case of an arbitrary Randers change.
\[theorem 1\] Let $M$ be an $n$-dimensional simply connected smooth manifold.
A Finsler metric $F$ induces a generalized weighted quasi-distance $d_F$ on $M$ if and only if it is the Randers change of an absolute homogeneous Finsler space $F_0$ by an exact one-form $\beta$.
We assume that $F=F_0+\beta$, where $F_0$ is an absolute homogeneous Finsler metric on $M$ and $\beta$ an exact one-form.
Let $\gamma_{xy} \in \Gamma_{xy}$ be an $F$-geodesic, which is in the same time an $F_{0}$-geodesic, then from we have $$\label{distances formula}
d_{F}(x,y)=\int_{a}^{b}F_{0}(\gamma_{xy}(t),\dot\gamma_{xy}(t))dt+
\int_{a}^{b}b_{i}(\gamma_{xy}(t))\dot\gamma_{xy}^{i}(t)dt
=d_{F_{0}}(x,y)+\int_{\gamma_{xy}}\beta.$$
Let us consider a fixed point $a\in M$ and define the function $w_{a}:M\to {\ensuremath{\mathbb{R}}},\quad w_{a}(x):=d_{F}(a,x)-d_{F}(x,a).$ From it follows $$\label{w_a}
w_{a}(x)=\int_{\gamma_{ax}}\beta-\int_{\gamma_{xa}}\beta=2\int_{\gamma_{ax}}\beta=
-2\int_{\gamma_{xa}}\beta,$$ where we have used Stokes’ theorem for the one-form $\beta$ on the closed domain $D$ with boundary $\partial D:=\gamma_{ax}\cup \gamma_{xa}$.
One can see that $w_a$ is an anti-derivative of $\beta$. This is well defined if and only if the path integral in right hand side of is path independent, that is $\beta$ must be exact.
Then $d_{F}$ is a weighted quasi-metric with generalized weight $w_{a}$. Indeed, we have $$\label{rel 1}
d_{F}(x,y)+w_{a}(x)=d_{F_{0}}(x,y)+\int_{\gamma_{xy}}\beta+\int_{\gamma_{ax}}\beta-\int_{\gamma_{xa}}\beta=d_{F_{0}}(x,y)-\int_{\gamma_{xa}}\beta-\int_{\gamma_{ya}}\beta,$$ where we have used again Stokes’ theorem for the one-form $\beta$ on the closed domain with boundary $\gamma_{ax}\cup \gamma_{xy}\cup \gamma_{ya}$.
Similarly, $$\label{rel 2}
d_{F}(y,x)+w_{a}(y)=d_{F_{0}}(y,x)-\int_{\gamma_{ya}}\beta-\int_{\gamma_{xa}}\beta,$$ and hence $d_{F}$ is weighted quasi-metric with generalized weight $w_{a}$.
Conversely, we assume that $(M,F)$ is a Finsler metric whose induced distance function $d_F$ is a weighted quasi-metric on $M$ with weight $w:M\to [0,\infty)$. For simplicity we assume that $w$ is a smooth function.
Let $\gamma:[0,\varepsilon)\to M$ be a [*short*]{} $C^{1}$ curve that emanates from the point $p:=\gamma(0)\in M$ with initial velocity $v:=v^{i}\frac{\partial}{\partial x^{i}}\in T_{p}M$. Then by Busemann-Mayer Theorem (see for example [@BCS00]) we have $$\label{F(p,v)}
F(p,v)=\lim_{t\to 0^{+}}\frac{d_F(p,\gamma(t))}{t}.$$
In a local chart $U$ around the point $p\in M$ the manifold $M$ looks locally as the Euclidean space, therefore we can write $$\label{F(p,-v)}
F(p,-v)=\lim_{t\to 0^{+}}\frac{d_F(\gamma(t),p)}{t},$$ and hence, from , and condition of weightability it results $$\begin{split}
F(p,v)-F(p,-v)&=\lim_{t\to 0^{+}}\frac{d_F(p,\gamma(t))-d_F(\gamma(t),p)}{t}
=\lim_{t\to 0^{+}}\frac{w(\gamma(t))-w(p)}{t}\\
&=\frac{1}{2}\frac{\partial w}{\partial x^{i}}(p)v^{i}=dw_{p}(v).
\end{split}$$
Then, we have $$F(p,v)=\frac{1}{2}[F(p,v)-F(p,-v)]+\frac{1}{2}[F(p,v)+F(p,-v)]
=F_{0}(p,v)+\beta(p,v),$$ where $F_0=\frac{1}{2}[F(p,v)+F(p,-v)]$, and $\beta(p,v)=\frac{1}{2}dw_{p}(v)$.
The geodesic reversibility is now obvious from Proposition \[rev\_geod\]. ${\hfill \Box}$
Moreover, if for the arbitrary chosen point $a\in M$, there exists a constant $l_{a}$ such that $l_{a}\leq d_{F}(a,x)-d_{F}(x,a),\quad \forall x\in M,$ then by putting $\widetilde w_{a}(x):=w_{a}(x)-l_{a}$ it follows that $(M,d_{F},\widetilde w_{a})$ is a weighted quasi-metric space. Obviously, when for example $M$ is compact, such an $l_{a}$ always exists (compare with the reversibility function used in [@R2004]).
For later use we recall ([@V95]) the following lemma.
\[quasi-metric prop\] Let $(M,d)$ be any quasi-metric space. Then $d$ is weightable if and only if there exists $w:M\to [0,\infty)$ such that $$\label{formula d}
d(x,y)=\rho(x,y)+\frac{1}{2}[w(y)-w(x)], \quad \forall x,y\in M,$$ where $\rho$ is the symmetrized distance of $d$. Moreover, we have $$\label{w bound}
\frac{1}{2}|w(x)-w(y)|\leq \rho(x,y), \quad \forall x,y\in M.$$
The proof is trivial from the definition of a weighted quasi-metric.
If $(M,F)$ is a Finsler space given by the Randers change , then the induced quasi-metric $d_F$ and the symmetrized metric $\rho$ induce the same topology on $M$. This follows immediately from [@KV94], Lemma 4.
From Lemma \[quasi-metric prop\] it can be seen that the assumption of $w$ to be smooth is not essential. Indeed, from Lemma \[quasi-metric prop\] it can be seen that if $d_F$ is a weighted quasi-metric, the function $w$ is 1-locally Lipschitz, that is differentiable almost everywhere on $M$. Therefore, the one-form $\beta$ exists almost everywhere on $M$.
See [@M12] for a very interesting discussion on the completeness of a Randers change by means of an exact one-form $\beta$.
We discuss an interesting geometric property concerning the geodesic triangles.
\[Prop 1.6\] Let $(M,F)$ be a Finsler metric given by the Randers change . Then the perimeter length of any geodesic triangle on $M$ does not depend on the orientation, that is $$\label{perimeter}
d_F(x,y)+d_F(y,z)+d_F(z,x)=d_F(x,z)+d_F(z,y)+d_F(y,x),\quad \forall x,y,z\in M.$$
(5, 3) (3,1) (2.8,0.7)[$x$]{} (3,1)(3,3)(4,4) (4,4) (3,1)(4,3)(4.94,3.4) (4,4)(4.9,4)(4.94,3.4) (4.94,3.4) (3.7,4)[$y$]{} (3.35,3)[(-1,-2)[0.04]{}]{} (4.35,3)[(1,1)[0.04]{}]{} (4.5,3.94)[(-3,1)[0.04]{}]{} (5,3.2)[$z$]{} (0,1) (-0.2,0.7)[$x$]{} (0,1)(0,3)(1,4) (1,4) (0,1)(1,3)(1.94,3.4) (1,4)(1.9,4)(1.94,3.4) (1.94,3.4) (0.7,4)[$y$]{} (0.35,3)[(1,2)[0.04]{}]{} (1.35,3)[(-1,-1)[0.04]{}]{} (1.5,3.94)[(3,-1)[0.04]{}]{} (2,3.2)[$z$]{}
[**Figure 2.**]{} The perimeter of the triangle $\Delta xyz$ is independent of the orientation.
In other words, even though the distance between two points $x$ and $y$ depends on the orientation of a minimizing geodesic joining points $x$ and $y$, i.e. $d_F(x,y)\neq d_F(y,x)$, the sum of distances between three points $x$, $y$, $z$ on $M$ do not depend on the direction we trace out the perimeter of the geodesic triangle $\Delta xyz$. We point out that weighted quasi-metric spaces can be characterized by this property without the explicit use of the weight function. Indeed, a quasi-metric $d$ is weightable if and only if relation holds.
The proof is almost trivial. Since $F$ it is the Randers change , then from Theorem \[theorem 1\] it follows that the quasi-metric is weightable and therefore holds good. By using this formula an elementary computation proves . ${\hfill \Box}$
Moreover, we have
Let $(M,F)$ be a Finsler space that satisfies . Then $F$ can be written as the Randers change of an absolute homogeneous Finsler metric $F_0$ by an exact one-form $\beta$.
It is easy to see that if a quasi-metric satisfies then it is weightable (see [@V99]). Then conclusion follows from Theorem \[theorem 1\]. ${\hfill \Box}$
It should be clear that not any quasi-metric space is weightable. In fact, it can be shown that the class of weightable quasi-metric spaces are exactly those quasi-metric spaces that satisfy relation (see [@V99]).
Isometric embeddings of Finsler spaces
======================================
If $(X,q,w)$ and $(Y,p,u)$ are two weighted quasi-metric spaces, the mapping $\varphi:X\to Y$ with the properties $$\begin{aligned}
\label{metric ineq}
& & p(\varphi(x),\varphi(y))\leq q(x,y),\quad \forall x,y\in X\\
& & u(\varphi(x))\leq w(x), \quad \forall x\in X \label{weight ineq}\end{aligned}$$ is called a [*morphism*]{} of weighted quasi-metric spaces.
In the case we have equality in relation , then the morphism $\varphi$ is called an [*isometric morphism*]{}. In this case $w$ and $u \circ \varphi$ differ by a constant only.
Moreover, an [*isomorphism*]{} of the weighted quasi-metric spaces $(X,q,w)$ and $(Y,p,u)$ is a bijective function $\varphi:X\to Y$ that preserves both the quasi-metric and the weight function. Finally, an [*embedding*]{} of $(X,q,w)$ into $(G,Q,W)$ is an isomorphism of $(X,q,w)$ onto a subspace of $(G,Q,W)$. Here, a [*subspace $(Y,p,u)$ of a weighted quasi-metric space*]{} $(G,Q,W)$ is a subset $Y\subset G$, the function $p$ and $u$ are the restriction of $Q$ and $W$ to $Y\times Y$ and $Y$, respectively.
\[space times ray\] Consider a metric space $(S,d)$ and the half ray $I:=[0,\infty)$. Then the product space $
G:=S\times I
$ inherits a natural structure of (generalized) weighted quasi-metric space $(G,Q,W)$, where $$\label{Q,W for space times ray}
\begin{split}
& Q:G\times G\to [0,\infty), \quad Q(u,v):=d(x,y)+\eta-\xi,\\
& W:G\to [0,\infty),\quad W(u):=2\xi,\quad \forall u=(x,\xi),v=(y,\eta)\in S\times I.
\end{split}$$
The generalized weighted quasi-metric space $(S\times I,Q,W)$ constructed in Example \[space times ray\] is sometimes called [*the bundle over $(S,d)$*]{} (see [@V99]).
\[Graph of a function\] We consider the case of the graph of a non-negative valued function $f:S\to [0,\infty)$ defined on a metric space $(S,d)$.
Indeed, if we denote the graph of $f$ by $G_f:=\{(x,f(x)):x\in S\}$ then $(G_f,Q,W)$ is a naturally induced weighted quasi-metric space structure defined by $$\begin{split}
& Q:G_f\times G_f\to [0,\infty), \quad Q(u,v):=d(x,y)+f(y)-f(x),\\
& W:G_f\to [0,\infty),\quad W(u):=2f(x), \quad \forall u=(x,f(x)),v=(y,f(y))\in G_f.
\end{split}$$
Based on these, one has
\[embedding theorem\] Every weighted quasi-metric space $(X,q,w)$ is embeddable in a bundle over a suitable metric space $(S,d)$.
The idea of the proof is simple. Following Example \[space times ray\], given the weighted quasi-metric space $(X,q,w)$ one constructs a naturally associated product of a metric space $(S,d)$ and a half line.
The obvious choice for $(S,d)$ is the symmetrization of the quasi-metric space $(X,q)$. Therefore one has the natural weighted quasi-metric space $(G,Q,W)$, where $G:=X\times [0,\infty)$, $Q$ and $W$ are defined in .
One defines now the function $\varphi:X\to G, \ \varphi(x):=(x,\frac{1}{2}w(x)),$ and show that this is indeed an embedding.
A fundamental result is that any weighted quasi-metric space can be constructed starting from a metric space $(S,d)$ and a 1-Lipschitz function $f:S\to [0,\infty)$ defined on it, i.e. $$\label{Lipschitz condition}
|f(x)-f(y)|\leq d(x,y), \quad \forall x,y\in S.$$
\[induced constr\]
1. Let $(S,d)$ be a metric space and $f:S\to [0,\infty)$ a 1-Lipschitz function. Then the graph of $f$ is a weighted quasi-metric space $(G_f,Q,W)$.
2. Conversely, every weighted quasi-metric space $(X,q,w)$ can be constructed in this way.
The proof is also quite obvious. Statement 1 is straightforward from Example \[Graph of a function\]. We point out that Lipschitz condition guaranties that $Q(u,v)\geq 0$, i.e. $(G_f,Q,W)$ is actually a weighted quasi-metric space.
Statement 2 follows from proof of Theorem \[embedding theorem\]. Indeed, given a weighted quasi-metric space $(X,q,w)$ one can construct
- a metric space $(S,d):=(X,\rho)$, where $\rho$ is the symmetrization of $q$,
- a Lipschitz function $f:S\to [0,\infty)$, $f(x):=\frac{1}{2}w(x)$.
One can see that this $f$ always satisfies the Lipschitz condition on $(S,d)$ because of . Moreover, due to Theorem \[embedding theorem\] there is an embedding of $(X,q,w)$ onto $(G_f,Q,W)$. That is recover the original weighted quasi-metric space $(X,q,w)$ from $(G_f,Q,W)$ by identifying $X$ with a subspace of $G$ obtained by the obvious projection and restricting $Q$ and $W$ to this $X$.
Next, we recall the differential manifold structure of the graph of a smooth function.
Let us consider a $C^{\infty}$ function $
f:M\to [0,\infty),\ x\mapsto f(x)
$ and the graph of $f$ denoted by $G_{f}=\{(x,f(x)):x\in M\}\subset M\times {\ensuremath{\mathbb{R}}}$. Then it is known that $G_{f}$ is a $C^{\infty}$ submanifold of the product manifold $M\times {\ensuremath{\mathbb{R}}}$ that is actually diffeomorphic to $M$. Indeed, the mapping $$\varphi:M\to G_f,\quad x\mapsto \varphi(x)=(x,f(x))$$ with the inverse $$\label{inverse psi}
\psi:G_f\to M,\quad u=(x,f(x))\mapsto \psi(x,f(x))=x$$ is a diffeomophism. Remark that $\psi$ is nothing else than the projection onto the first factor.
Any given weighted quasi-metric space $(M,q,w)$ that satisfies some supplementary metrizability condition induced a Finsler structure $(M,F=F_{0}+df)$ on $M$ and conversely, every given weighted quasi-metric space $(M,q,w)$ can be constructed in this way.
We recall the smooth approximation of Lipschitz functions on a Finsler manifold:
\[smooth approx\] Let $(M,F)$ be a Finsler manifold and $f:M\to {\ensuremath{\mathbb{R}}}$ a 1-Lipshitz function, i.e. $$|f(x)-f(y)|\leq d_{F}(x,y), \quad \forall x,y\in M,$$ where $d_{F}$ is the Finslerian induced quasi-distance on $M$. Then, for any small positive $\varepsilon_{1}$, $\varepsilon_{2}$ there exists a smooth function $\widetilde{f}:M\to {\ensuremath{\mathbb{R}}}$ such that
1. $|\widetilde{f}(x)-f(x)|<\varepsilon_{1}$, $\forall x\in M$,
2. $\widetilde{f}$ is $(1+\varepsilon_{2})$-Lipschitz, i.e. $|\widetilde{f}(x)-\widetilde{f}(y)|\leq (1+\varepsilon_{2})d_{F}(x,y), \quad \forall x,y\in M.$
The Riemannian version of this lemma can be found in [@A07].
Then, we have
1. Let $(M,F=F_{0}+df)$ be a Finsler space, where $f$ is a $C^{\infty}$ non-negative function $f$ on $M$. Then the graph manifold $G_{f}$ inherits a natural structure of weighted quasi-metric space $(G_{f},Q,W)$ that coincides with the weighted quasi-metric space $(M,d_{F}, 2f)$ induced by $F$, up to an isomorphism.
2. For any given weighted quasi-metric space $(M,q,w)$, whose symmetrized metric is $C^{\infty}$-Riemannian (or absolutely homogeneous Finsler) metrizable, there exist
1. a smooth approximation function $\widetilde{f}:M\to [0,\infty)$ of $w$,
2. a weighted quasi-metric space $(G_{\widetilde f},\widetilde{Q}, \widetilde{W})$, called the smooth approximation of $(G_{f},Q,W)$, that coincides with the weighted quasi-metric space $(M,d_{F}, 2\widetilde{f})$ induced by a (not necessarily positive definite) Randers metric $F=\widetilde\alpha+d\widetilde{f}$ (or Randers change $F=F_{0}+d\widetilde{f}$), up to an isomorphism.
1\. If we start with the Finsler structure $(M,F=F_{0}+df)$, then this is with reversible geodesics, and therefore from Theorem \[theorem 1\] it follows that $M$ becomes a weighted quasi-metric space $(M,d_{F}, 2f)$, where $d_{F}(x,y)=d_{F_{0}}(x,y)+f(y)-f(x),\quad \forall x,y\in M.$ The symmetrized metric $\rho$ of $d_{F}$ coincides with $d_{F_{0}}$ and therefore the graph manifold $G_{f}$ becomes a weighted quasi-metric space $(G_{f},Q,W)$, where $$Q(u,v)=\rho(x,y)+f(y)-f(x),\ W(u)=2f(x),\quad \forall u=(x,f(x)),v=(y,f(y))\in G_{f}.$$
It can be easily seen that this $(G_{f},Q,W)$ is indeed a weighted quasi-metric space and that $\psi:(G_{f},Q,W)\to (M,d_{F}, 2f)$ defined in is an isometric embedding.
2\. Starting with an arbitrary weighted quasi-metric space $(M,q,w)$ remark that the weight $w:M\to [0,\infty)$ is a 1-Lipschitz function (see ) with respect to the symmetrized metric $\rho$. This is not good enough to define a $C^{\infty}$-Finsler metric because $w$ has a measure zero set of points where it fails to be differentiable.
We are going to use the smooth approximation of Lipschitz functions on Riemannian manifolds ([@A07]) or Finsler manifolds (see Lemma \[smooth approx\]). For the sake of simplicity we present only the Riemannian case here. We put $f:=\frac{1}{2}w$ and denote the smooth approximation of $f$ by $\widetilde f$. It follows that $G_{\widetilde f}$ is a smooth manifold that inherits a natural structure of weighted quasi-metric space from $(G_{f},Q,W)$ constructed in Theorem \[induced constr\], 1. Namely, we put $$\widetilde Q(u,v):=(1+\varepsilon_{2})\rho(x,y)+\widetilde f(y)-\widetilde f(x),\
\widetilde W(u):=2\widetilde f(x),\ \forall u=(x,\widetilde f(x), v=(y,\widetilde f(y))\in G_{\widetilde f}.$$ Since $\widetilde f$ is $(1+\varepsilon_{2})$-Lipschitz with respect to $\rho$ it follows that $\widetilde Q(u,v)\geq 0$ for any $u,v\in G_{\widetilde f}$. Elementary computations shows that indeed $(G_{\widetilde f},\widetilde{Q}, \widetilde{W})$ is a weighted quasi-metric space that smoothly approximates $(G_{f},Q,W)$ in the sense that $$|\widetilde Q(u,v)-Q(u,v)|\leq 2\varepsilon_{1},\quad
|\widetilde W(u)-W(u)|\leq 2\varepsilon_{1}, \quad \forall u,v\in G_{\widetilde f}.$$
We define now $\widetilde a_{ij}(x):=(1+\varepsilon_{2})^{2}a_{ij}(x)$, where $(M,a)$ is the Riemannian metric corresponding to the metric space $(M,\rho)$, for $\forall i,j\in\{1,2,\dots,n\}$, and $\forall x\in M$.
The Randers space $(M,F=\widetilde\alpha+d\widetilde f)$ induces a structure of weighted quasi-metric space $(M,d_{F}, 2\widetilde{f})$ on $M$ which is isometrically embeddable into $(G_{\widetilde f},\widetilde{Q}, \widetilde{W})$ as shown above. Obviously, this Randers space is positive definite if and only if the Riemannian length of the gradient vector $grad\ \widetilde f$ is less than one.
The proof is identical if $(M,\rho)$ is absolutely homogeneous Finsler metrizable. ${\hfill \Box}$
For any given weighted quasi-metric space $(M,q,w)$, whose symmetrized metric is $C^{\infty}$-Riemannian (or absolutely homogeneous Finsler) metrizable, and whose weight $w$ is a smooth function, the weighted quasi-metric space $(M,d_{F}, 2{f})$ induced by the Randers metric $F=\alpha+d{f}$ (or Randers change $F=F_{0}+d{f}$) coincides with $(M,q,w)$.
Obviously not any metric space $(M,d)$ is Riemannian metrizable. Here Riemannian metrizable means that $M$ is a differentiable manifold and that there exists a $C^\infty$-Riemannian metric $a=a_{ij}(x)$ on $M$ whose associated distance function $d_{\alpha}$ coincides with $d$.
General conditions for a metric space to be Riemannian metrizable can be found in [@N99]. Similar conditions can be easily established for metric spaces to be absolute homogeneous Finsler metrizable.
More generally one can study conditions for quasi-metric space to be Finsler metrizable. Metrizability of metric or quasi-metric spaces is a complex subject that we intend to discuss in a forthcoming paper.
We discuss now another representation of Finsler spaces.
We recall that for a metric space the [*Hausdorff distance*]{} is a distance function between subsets of $M$. Indeed, if $(M,d)$ is a metric space, then the mapping $$d_H:2^M\times 2^M\to [0,\infty),\quad d_H(A,B)=\max\{\sup_{a\in A}d(a,B),
\sup_{b\in B}d(b,A)\}$$ is called the [*Hausdorff metric*]{}, where $2^M$ is the set of all subsets of $M$.
In the case of $2^M$ the function $d_H$ is only a semi-metric. However, the pair $(\mathcal{P}_0(M,d),d_H)$ is a metric space, where $\mathcal{P}_0(M,d)$ is the set of non-empty closed subsets of $M$ ([@BBI01]).
This notion can be easily extended to the more general case of quasi-metric spaces as follows. Let $(M,q)$ be a quasi-metric space. We define the mappings: $$\begin{split}
& d_H^f(A,B):=\sup_{a\in A}q(a,B),\quad
d_H^b(A,B):=\sup_{b\in B}q(A,b),\\
& d_H(A,B):=\max\{d_H^f(A,B),d_H^b(A,B)\},\quad \forall A,B\in \mathcal{P}_0(M,d).
\end{split}$$
It can be seen that $d_H^f$, $d_H^b$, $d_H$ are extended quasi-metrics on $\mathcal{P}_0(M,q)$ and that they become quasi-metrics when restricted to $\mathcal K_0(M,q)$, that is the set of all non-empty compact subsets of $(M,q)$.
The pair $( \mathcal K_0(M,q),d_H)$ is called the associated [*quasi-Hausdorff metric*]{} of $(M,q)$ (compare with [@S01]).
Many of the geometrical properties of Hausdorff distance extend to the case of quasi-Hausdorff distances (a detailed study of these together with the Gromov-Hausdorff distance will be given elsewhere).
Let ($M,q)$ be a quasi-metric space, and construct the metric space $(X,\delta)$, where $X:=M\times [0,\infty)$, and $\delta((x,\xi),(y,\eta))=d^*(x,y)+|\xi-\eta|,$ for $\forall (x,\xi),(y,\eta)\in X$. Here $d^*$ is the max-metric .
It can be easily seen that indeed $(X,\delta)$ is a metric space and that for $\forall z\in M$, the set $E(z):=\{(y,\eta)\in X:d(y,z)\leq \eta\}$ is a non-empty closed subset in $(X,\delta)$, i.e. $E(z)\in \mathcal{P}_0(X,\delta)$. We write here $\delta$ in order to make explicit the topology where the set are closed.
We recall ([@V95]) that if $(M,d)$ is a quasi-metric space, then the mapping $E:(M,d)\to (\mathcal{P}_0(X,\delta),d_H^f), \ z\mapsto E(z)$ is an isometry of $M$ onto a subspace $E(M)$ of $\mathcal{P}_0(X,\delta)$. Indeed, it can be seen that $E$ is injective, and $d(x,y)=d_H^f(E(x),E(y))$, for $\forall x,y\in M$.
We obtain
Let $(M,F)$ be a Finsler space with associated quasi-metric $d_F$. Then the quasi-metric space $(M,d_F)$ is isometric to a subspace $E(M)$ of $(\mathcal{P}_0(M),d_H^f)$.
Let us assume now that $d_F$ is a weighted quasi-metric. In this case we have $d_H^f(A,B)=d_H^\rho(A,B)+\frac{1}{2}\Bigl(\inf_{b\in B}w-\sup_{a\in A}w
\Bigr),$ where we have used $d(a,B)=\inf_{b\in B}d(a,b)=\rho(a,B)+\frac{1}{2}\Bigl(\inf_{b\in B}w-w(a)
\Bigr).$ Here $d_H^\rho(A,b)$ is the usual Hausdorff distance of the symmetrized metric $\rho$.
It can be seen that the forward Hausdorff distance can not exceed the $\rho$-Hausdorff distance.
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[^1]: Mathematics Subject Classification (2010):53C60, 53C22.
[^2]: Keywords: Randers metrics, geodesics, distance.
|
---
abstract: 'We investigate the feasibility of sequence-level knowledge distillation of Sequence-to-Sequence (Seq2Seq) models for Large Vocabulary Continuous Speech Recognition (LVSCR). We first use a pre-trained larger teacher model to generate multiple hypotheses per utterance with beam search. With the same input, we then train the student model using these hypotheses generated from the teacher as pseudo labels in place of the original ground truth labels. We evaluate our proposed method using Wall Street Journal (WSJ) corpus. It achieved up to $ 9.8 \times$ parameter reduction with accuracy loss of up to 7.0% word-error rate (WER) increase.'
address: |
Tokyo Institute of Technology\
raden@ks.c.titech.ac.jp, inoue@ks.c.titech.ac.jp, shinoda@c.titech.ac.jp
bibliography:
- 'refs.bib'
title: 'Sequence-Level Knowledge Distillation for Model Compression of Attention-based Sequence-to-Sequence Speech Recognition'
---
speech recognition, large vocabulary continuous speech recognition, sequence-to-sequence, attention model, knowledge distillation, sequence-level knowledge distillation
Introduction {#sec:intro}
============
In recent years, end-to-end deep neural networks for Large Vocabulary Continuous Speech Recognition (LVSCR) have been steadily improving their accuracy, rivalling the traditional Gaussian Mixture Model-Hidden Markov Models (GMM-HMM) and hybrid models of deep neural networks and HMMs (DNN-HMM). While these models have an acoustic model and a language model which are trained separately, for end-to-end training, the whole model is trained using backpropagation with audio-transcription pairs [@sutskever2014sequence].
Several end-to-end architectures for speech recognition have been proposed. Their examples include Connectionist Temporal Classification (CTC) [@graves2013speech], Recurrent Neural Network-Transducer (RNN-T) [@graves2014towards], and Sequence-to-Sequence (Seq2Seq) with attention [@bahdanau2016end]. Unlike CTC and RNN-T, Seq2Seq with attention does not make any prior assumptions on the output sequence alignment given an input; it jointly learns how to align while learning to encode the input and decode its result into the output.
In machine learning, model compression [@han2015deep] is a way to significantly compress a model by reducing the number of its parameters while having negligible accuracy loss. This is important for deploying trained models on memory and compute-constrained devices such as mobile phones and embedded systems, and when energy efficiency is needed on large-scale deployment. Several model compression methods exist, such as pruning, quantization [@han2015deep], and knowledge distillation (KD) [@hinton2015distilling]. KD is the focus of this work, where a smaller student model is trained by using the output distribution of a larger teacher model. Recently KD for Seq2Seq models with attention was proposed for neural machine translation (NMT) task [@kim2016sequence] and proved to be effective. Also, Seq2Seq models for CTC-based speech recognition was recently proposed [@takashima2018ctc].
In this paper, we propose a sequence-level KD method for Seq2Seq speech recognition models with attention. Different from the previous work for NMT [@kim2016sequence], we extract the hypotheses from a pre-trained teacher Seq2Seq model using beam search, and train student Seq2Seq models using the hypotheses as pseudo labels on the sequence-level cross-entropy criterion. To the best of our knowledge, there is no prior work on applying KD in Seq2Seq-based speech recognition models.
Knowledge Distillation for Seq2Seq Models
=========================================
Sequence-to-Sequence Learning
-----------------------------
The Sequence-to-Sequence (Seq2Seq) [@sutskever2014sequence] is neural network architecture which directly models conditional probability $p(\mathbf{y}|\mathbf{x})$ where $\mathbf{x} = [x_1, ..., x_S]$ is the source sequence with length $S$ and $\mathbf{y} = [y_1, ..., y_T]$ is the target sequence with length $T$.
Figure \[fig:seq2seq\] illustrates the Seq2Seq model. It consists of an encoder, a decoder and an attention module. The encoder processes an input sequence $\mathbf{x}$ and outputs encoded hidden representation $\mathbf{h^e} = [h^e_1, ...,h^e_S]$ for the decoder [@tjandra2017attention]. The attention module is a network that assists the decoder to find relevant information on the encoder side based on the current decoder hidden states [@bahdanau2016end]. The attention module does this by producing a context vector $c_t$ at time $t$ based on the encoder and decoder hidden states: $$\begin{aligned}
c_t &= \sum_{s=1}^{S} a_t(s) * h^e_s , \\
a_t(s) & = \frac{\exp(\text{Score}(h^e_s, h^d_t))}{\sum_{s=1}^{S}\exp(\text{Score}(h^e_s, h^d_t))} ,\end{aligned}$$ Several variations exist for $\text{Score}(h^e_s, h^d_t)$: $$\begin{aligned}
\text{Score}(h_s^e, h_t^d) =
\begin{cases}
\langle h_s^e, h_t^d\rangle ,& \text{: dot product} \\
h_s^{e\intercal} W_{s} h_t^d ,& \text{: bilinear} \\
V_s^{\intercal} \tanh(W_{s} [h_s^e, h_t^d]) ,& \text{: MLP} \label{eq:mlpscore} \\
\end{cases}\end{aligned}$$ where $\text{Score}$ is a function $(\mathbb{R}^M \times \mathbb{R}^N) \rightarrow \mathbb{R}$, $M$ is the number of hidden units for the encoder, $N$ is the number of hidden units for the decoder, and both $W_s$ and $V_s^{\intercal}$ are weight matrices. Finally, the decoder predicts the probability of target sequence $\mathbf{y}$ at time $t$ based on the previous output $y_{<t}$ and $c_t$, which can be formulated as:
$$\log{p(\mathbf{y}|\mathbf{x})} = \sum_{t=1}^{T}\log{p(y_t|y_{<t}, c_t)} .$$
The previous outputs $c_t$ can obtained with greedy decoding, i.e. by taking the output with the highest probability for each time step (e.g in [@kim2016sequence]). It is also possible to perform *beam search* to obtain more than one $c_t$ [@sutskever2014sequence].
Seq2Seq can handle virtually any sequence related tasks[@sutskever2014sequence], such as NMT and speech recognition. For speech recognition, the input $\mathbf{x}$ is a sequence of feature vectors derived from audio waveform such as Short-time Fourier Transform (STFT) spectrogram or Mel-frequency cepstral coefficients (MFCC). Therefore, $ \mathbf{x} $ is a real vector with ${S \times D}$ dimension where D is the number of the features and S is the total length of the utterance in frames. The output $\mathbf{y}$ can be a sequence of phonemes or graphemes (characters).
![Architecture of Seq2Seq with Attention module[]{data-label="fig:seq2seq"}](figs/seq2seq-diag.png){width="0.3\linewidth"}
Knowledge Distillation {#sec:pagestyle}
----------------------
In knowledge distillation (KD), a teacher model’s probability distribution $q(\theta_T | x; \theta)$ is trained by using a given dataset, where $\theta_t$ is a set of its parameters. Using the same dataset, a student model $p(\theta | x; \theta)$ is trained by minimizing cross-entropy with the teacher model’s probability distribution $q(\theta_T | x; \theta)$ instead of the ground truth labels from the dataset.
Let an input-label pairs be $(x, y)$ and ${\mathcal{V}}$ is a set of possible classes. Then, the loss for KD is given as:
$${\mathcal{L}}_{\text{KD}}(\theta;\theta_T) = - \sum_{k=1}^{|{\mathcal{V}}|} q(y = k {\,|\,}x;\theta_T)\log p(y = k {\,|\,}x;\theta) .$$
In KD training, the teacher produces softmax probabilities (*soft targets*), which reveal teacher’s confidences for what classes it predicts given an input. With this additional knowledge, the student can model the data distribution better than learning directly from the ground truth labels consisting of one-hot vectors (*hard targets*) [@hinton2015distilling].
Sequence-Level Knowledge Distillation
-------------------------------------
While it is possible to use the original KD method to train autoregressive models such as RNN, it only gives non-significant accuracy gains [@kim2016sequence], or simply degrade the performance compared to training with the dataset directly [@sak2015acoustic].
To adapt KD to autoregressive models, [@kim2016sequence] suggested to use the approximation of a teacher’s sequence-level distribution instead of the teacher’s single time step frame-level distribution, so as to capture the time-dependencies between the inputs. Consider the sequence-level distribution specified by the model over all possible teacher label sequences ${\mathbf{t}}\in {\mathcal{T}}$, given the input sequence **s** : $$p({\mathbf{t}}{\,|\,}{\mathbf{s}}) = \prod_{j=1}^J p(t_j {\,|\,}{\mathbf{s}}, {\mathbf{t}}_{<j}),$$ for any length $J$. The sequence-level loss for Seq2Seq involves matching the input **s** with the one-hot distribution of all the complete sequences: $$\begin{aligned}
&&{\mathcal{L}}_{\text{SEQ-NLL}} = -\sum_{{\mathbf{t}}\in {\mathcal{T}}} \mathbbm{1}\{{\mathbf{t}}={\mathbf{y}}\} \log p({\mathbf{t}}{\,|\,}{\mathbf{s}}) \\
&= & -\sum_{j=1}^J \sum_{k=1}^{|{\mathcal{V}}|} \mathbbm{1}\{y_j=k\} \log p(t_{j}=k {\,|\,}{\mathbf{s}}, {\mathbf{t}}_{<j}),
$$ where $\mathbbm{1}\{\cdot\}$ is the indicator function and ${\mathbf{y}}= [y_1, \dots, y_J]$ is the observed sequence. To formulate the sequence-level KD, we use $q({\mathbf{t}}{\,|\,}{\mathbf{s}})$ to represent the teacher’s sequence distribution over the sample space of all possible sequences, $${\mathcal{L}}_{\text{SEQ-KD}} = -\sum_{{\mathbf{t}}\in {\mathcal{T}}} q({\mathbf{t}}{\,|\,}{\mathbf{s}}) \log p ({\mathbf{t}}{\,|\,}{\mathbf{s}})$$
Different from previously stated ${\mathcal{L}}_\text{KD}$, ${\mathcal{L}}_\text{SEQ-KD}$ minimizes the loss on the whole-sequence level. However, this loss is intractable. An approximation to calculate it is required.
There are many ways to approximate the loss. The sequence-level KD for NMT [@kim2016sequence] uses a single hypothesis with the best score as the pseudo label per input. For CTC-based speech recognition [@takashima2018ctc], this loss was *k*-best hypotheses from beam search per input (Figure \[fig:seq-kd\]). The loss is then $$\begin{aligned}
{\mathcal{L}}_{\text{SEQ-KD}} &\approx& - \sum_{{\mathbf{t}}\in {\mathcal{T}}} \mathbbm{1}\{{\mathbf{t}}= \hat{{\mathbf{y}}} \} \log p ({\mathbf{t}}{\,|\,}{\mathbf{s}}) \\
&=& - \log p ({\mathbf{t}}= \hat{{\mathbf{y}}} {\,|\,}{\mathbf{s}}),\end{aligned}$$ where $\hat{{\mathbf{y}}}$ is the output hypothesis obtained from running beam search with the teacher model. In this work, we investigated on how to apply these for Seq2Seq-based speech recognition models, which will be discussed in the next sections.
Multiple Hypotheses from Beam search for Knowledge Distillation
===============================================================
For Seq2Seq-based speech recognition, the sequence-level KD with the loss approximation can be done by the following three steps: (1) Pre-train a teacher model with a dataset, (2) With the same dataset, generate *k*-best sequences with beam search, and save them as pseudo labels, (3) Train a student model with the same dataset but replace the ground truth labels with generated pseudo labels (Figure \[fig:seq-kd\]). In this procedure, the size of beam search and the value of *k* are adjustable hyperparameters. The dataset size increases with factor of *k* from the method using the 1-best [@kim2016sequence]. The pseudo labels are analogous to *soft targets* in the original KD. Even if the pseudo labels are not fully accurate, the student is expected to achieve better performance with this training method since the student tries to imitate the teacher’s distribution instead of trying to model the training data distribution directly. Training with these pseudo labels can be seen as a form of regularization similar to the original KD [@hinton2015distilling], because the slightly inaccurate transcriptions from pseudo labels can prevent the trained models from overfitting to training data distribution (Figure \[fig:seq-kd\]).
Implementation and Experiments
==============================
Seq2Seq Model Architecture
--------------------------
The input speech audio waveform is sampled at 16kHz, then transformed into STFT spectrogram with Hanning window of 20ms and step size of 10ms. Then the STFT spectrograms are fed into 2D convolutional neural network (CNN) with two layers, as described by [@maas2017building; @amodei2016deep], as this STFT and CNN combination can further improve accuracy compared to using MFCC alone. 2D CNN is configured with filters=32, kernelsize=(5, 8), stride=(2,2).
Bidirectional Gated Recurrent Unit (GRU) [@chung2014empirical] is used for the encoder and the decoder. The feature vector from 2D CNN is first fed into the encoder. Then, the hidden representation made by the encoder is fed to an embedding layer of size 32. Next, the output of the embedding layer is fed into the decoder.
The output of the decoder is fed into the softmax layer consisting of 31 classes (26 English alphabet plus 5 classes for start-of-sentence (sos), end-of-sentence (eos), space, apostrophe and period). The decoder’s attention module consists of 1D CNN layer with kernel=128, kernelsize=15, stride=1, padding=7 followed by a fully-connected layer, as proposed by [@bahdanau2016end]. The models are trained with Dropout set to 0.4.
The model configurations are summarised in Table 1. We have designed two kinds of student models with different sizes, Student-mid and Student-small.
\[tab:arch\]
Model Encoder bi-GRU Decoder bi-GRU
--------------- -------------------- -------------------- -- --
Teacher 5 layers 384 cells 3 layers 384 cells
Student-mid 4 layers 256 cells 1 layer 256 cells
Student-small 3 layers 128 cells 1 layer 128 cells
: Model configurations. 2D CNN and Attention module configuration is the same for all models.
Dataset and Software
--------------------
Wall Street Journal (WSJ) corpus (available at the Linguistic Data Consortium as LDC93S6B and LDC94S13B) is used as the training and testing datasets. From the corpus, trainsi284 (81 hours of 37k sentences) is used for training, dev93 is used for validation, and eval92 is used for testing. Data is extracted as according to WSJ recipe from Kaldi toolkit [@povey2011kaldi]. STFT spectrogram extraction, implementation of the models, training and testing are conducted using Python 3.6, Scipy 1.0.1 and Pytorch 0.3.1.
Experimental Setup
------------------
First, the teacher model and student models are trained directly with trainsi284 and tested with eval92 to serve as the baselines. To perform Sequence-level KD, the teacher model pre-trained with trainsi284 is used for pseudo labels generation. This is done by extracting *k*-best hypotheses from beam search with combinations of beam size of 5, 10 and *k*-best of 1, 5, 10.
For each model the training is done using Adam optimizer, with learning rate of 2e-4 exponential decay rate of 0.99 per epoch, and mini-batch size set at 16. Seq2Seq teacher forcing rate [@bahdanau2016end] is set at 0.4. Training is done up to 200 epochs, or until no improvement can be seen in validation or testing set.
\[tab:results\]
Model CER (%) WER (%)
--------------------------------- --------- ---------- -- --
Teacher
(Params: 16.8M; $ 100 \%$ size)
Baseline 4.6 15.3
Student-mid
(Params: 6.1M; $ 37 \%$ size)
Baseline 7.0 21.8
topK=1, beamSize=5 6.4 20.5
topK=1, beamSize=10 6.5 20.5
topK=5, beamSize=5 **6.0** 20.1
topK=5, beamSize=10 6.5 21.2
topK=10, beamSize=10 6.1 **19.7**
Student-small
(Params: 1.7M; $ 10 \%$ size)
Baseline 9.2 28.7
topK=1, beamSize=5 7.6 26.1
topK=1, beamSize=10 7.7 25.3
topK=5, beamSize=5 **6.5** **22.3**
topK=5, beamSize=10 6.9 23.3
topK=10, beamSize=10 7.4 24.7
: Results on WSJ “eval92” (trained on “train\_si284”). beamSize (shorthand for size of beam search) and topK (shorthand for top *k*-best hypotheses) are hyperparameters. We measured the accuracy using the character-error rate (CER) and word-error rate (WER).
Results
=======
The results are shown in Table 2. The teacher model serves the reference baseline for student models. We benchmarked the character-error rate (CER) and word-error rate (WER). With KD training, the student models managed to achieve better CER and WER than the case when training directly with the dataset.
For Student-mid model, the number of parameters is $37\%$ of the teacher ($2.7\times$ reduction). It achieved the best CER (6.0%) with beamSize=5 and topK=5, and the best WER (19.7%) with beamSize=10 and beamSize=10.
For Student-small model, the number of parameters is $10\%$ of the teacher ($9.8\times$ reduction). As expected, the model suffered higher CER and WER compared to Student-mid model. The model achieved the best CER (6.5%) and WER (22.3%) with beamSize=5 and topK=5. The effect of sequence-level KD is more obvious in Student-small, where it achieve 6.4% reduction in WER with KD training compared to directly training with the dataset.
Training was done using a server with Intel Xeon E5-2680 2.4GHz and NVIDIA Tesla P100 16GB. We did not found significant improvement in training and testing time. Student-mid and Student-small achieved speed-up of $1.4\times$ and $1.7\times$ respectively, relative to the teacher model. This relatively small improvement may be due to inherent sequential computations using RNN where some operations simply cannot be paralellized.
To summarise, generally we found that using beamSize=5 and topK=5 are sufficient for reasonable WER and CER reduction. Increasing beamSize and/or topK further do not necessarily improve the performance.
Conclusion
==========
In this work, we successfully performed sequence-level KD training for Seq2Seq speech recognition models. Using beam search to approximate the teacher’s distribution, we extracted *k*-best hypotheses to be used as pseudo labels to train the student on the same dataset. We managed to train the students with reduction of $9.8\times$ parameter size of the teacher, with increase of WER of $7.0\%$ relative to the teacher.
There are many problems left for future work. Since RNN is limited in inference speed due to its sequential operations, we plan to investigate the feasibility of sequence-level KD for other highly parallelizable attention-based architectures which are not based on RNN, such as, Transformer [@vaswani2017attention] and S2SConv [@gehring2017convolutional].
Acknowledgment
==============
This work was supported by JST CREST Grant Number JPMJCR1687 and JSPS KAKEN Grant Number 16H02845.
|
---
abstract: 'This paper presents a study of circumstellar debris around Sun-like stars using data from the *Herschel* DEBRIS Key Programme. DEBRIS is an unbiased survey comprising the nearest $\sim$90 stars of each spectral type A-M. Analysis of the 275 F-K stars shows that excess emission from a debris disc was detected around 47 stars, giving a detection rate of 17.1$^{+2.6}_{-2.3}$ per cent, with lower rates for later spectral types. For each target a blackbody spectrum was fitted to the dust emission to determine its fractional luminosity and temperature. The derived underlying distribution of fractional luminosity versus blackbody radius in the population showed that most detected discs are concentrated at $f\sim10^{-5}$ and at temperatures corresponding to blackbody radii 7-40 AU, which scales to $\sim40$AU for realistic dust properties (similar to the current Kuiper belt). Two outlying populations are also evident; five stars have exceptionally bright emission ($f>5\times10^{-5}$), and one has unusually hot dust $<4$AU. The excess emission distributions at all wavelengths were fitted with a steady-state evolution model, showing these are compatible with all stars being born with a narrow belt that then undergoes collisional grinding. However, the model cannot explain the hot dust systems - likely originating in transient events - and bright emission systems - arising potentially from atypically massive discs or recent stirring. The emission from the present-day Kuiper belt is predicted to be close to the median of the population, suggesting that half of stars have either depleted their Kuiper belts (similar to the Solar System), or had a lower planetesimal formation efficiency.'
author:
- '\'
bibliography:
- 'biblio.bib'
date: 'Accepted YYYY MMMM DD. Received YYYY MMMM DD; in original form YYYY MMMM DD'
title: 'Analysis of the *Herschel* DEBRIS Sun-like star sample'
---
\[firstpage\]
stars: circumstellar matter, infrared: stars
Introduction
============
Debris discs are belts of dusty circumstellar material created during the on-going collision of orbiting planetesimals throughout a star’s lifetime [@Wyatt2008]. The requirement for planetesimals, themselves a step in the planet formation process, makes debris discs a fundamental component of planetary systems. Therefore, by studying the incidence, properties and evolution of debris discs it is possible to obtain a greater understanding of planetary systems [@Matthews2014a; @Moro2015].
Whilst the term incidence is often used in the literature, it in fact describes the disc detection fraction, and not the true incidence of debris. The detection fraction is the fraction of stars with clear (typically $>3\sigma$) emission in excess of the stellar photosphere at the location of the star, attributed to a debris disc. The sensitivity to such an excess varies significantly as a function of many stellar and disc parameters, as well as the depth, wavelength, angular resolution and calibration accuracy of the survey data. Consequently, it is difficult to directly compare ‘incidence’ rates from various surveys of different stellar samples.
For consistency with previous works the term incidence is used in this paper to describe the detection fractions. However, it must be borne in mind that these are just a rough comparison since surveys utilize observations at different wavelengths, depths and resolutions, and in turn have very different sensitivities, even to the same disc or to the disc parameters around different stars. Knowledge of the biases and completeness thresholds of the data presented are used in an attempt to correct for these factors, and provide correct incidence rates, but still only within the range of the probed parameter space.
In recent years infrared data from the MIPS camera [@Rieke2004] on-board the *Spitzer* Space Telescope [@Werner2004] has been used to investigate the properties of discs around ‘Sun-like’ stars [e.g. @Bryden2006; @Trilling2008; @Carpenter2009; @Kains2011]. Disc incidence rates of between 10 and 15 per cent have been reported for F, G and K spectral type stars, substantially lower than the $\sim$32 per cent found around A stars [@Su2006]. More recently, however, the Dust Around Nearby Stars (DUNES) survey team [@Eiroa2013] reported a higher rate of 20.2$\pm$2 per cent. This result was derived from new infrared data obtained with the PACS camera [@Poglitsch2010] on-board the *Herschel*[^1] Space Observatory [@Pilbratt2010].
The variation in disc incidence as a function of observed wavelength makes it difficult to perform direct comparisons between these data sets. Moreover, various biases, including variation in mean distance to different spectral types, makes it difficult to draw fundamental conclusions about these sources using such statistics, as a function of spectral type. However, a trend of disc incidence with stellar age has been observed [@Su2006; @Wyatt2007a; @Carpenter2009]. In practice, disc incidence is a combination of the disc fractional luminosity ($f=L_{\rm{disc}}/L_{\rm{star}}$, a wavelength independent property), the detection limits of the data, and the distance of the source. Therefore, the biases described can be characterised and accounted for using a large unbiased dataset, down to the lowest detection limit for the sample.
This paper presents an analysis of debris disc properties around such an unbiased sample of 275 stars of spectral types F, G and K. A description of the target sample, their observation and data reduction is given in Section \[sec2\]. This section includes information on source measurement and determination of the disc model parameters when a significant disc excess is detected. Section \[sec3\] presents the derived disc model parameters and discusses the incidence rates of discs within the fractional luminosity vs disc radius parameter space. The steady-state evolution of these sources is then presented in Section \[sec4\], followed by a discussion of the results and summary of this work in Sections \[sec5\] and \[sec6\] respectively.
Sample and Observations {#sec2}
=======================
The DEBRIS survey
-----------------
The work presented in this paper is based on data from the Disc Emission via a Bias Free Reconnaissance in the Infrared and Submillimetre [DEBRIS; @Matthews2010] *Herschel* Key Programme (KPOT\_bmatthew\_1). DEBRIS is a survey of 446 nearby stars of spectral types A-M. All targets were observed at 100 and 160$\mu$m using the PACS photometer, with additional 70$\mu$m follow-up of interesting sources. Particularly bright sources were also followed-up with SPIRE photometry [@Griffin2010] at 250, 350 and 500$\mu$m. The detection of infrared discs excesses in these data is limited by instrument noise and confusion with background objects. In no cases is excess detection purely limited by uncertainties in the instrument calibration or stellar photospheric flux.
DEBRIS was executed in partnership with the DUNES *Herschel* Key Programme, with sources common to both teams being observed only once by either team, and the resulting data shared. As a result, 98 DEBRIS targets were observed by the DUNES team (project code KPOT\_ceiroa\_1). By design DEBRIS was a flux-limited survey. However, due to the data sharing arrangement with the DUNES programme, and their need to often go beyond the nominal DEBRIS flux density limit, many of the DEBRIS observations are deeper than this limit. This is also the case for DEBRIS-only targets when follow-up observations of interesting sources was performed [e.g. @Lestrade2012]. The analysis presented here is designed to make the maximum use of the available data, consequently where additional data are available they are included and used. As a result some targets have flux limits lower than the nominal DEBRIS level. Since targets are at a range of distances and around varied stellar spectral types, disc sensitivity inherently varies between individual targets, even within a flux-limited data set. Therefore including deeper observations where possible does not adversely impact on the unbiased nature of this sample.
The DEBRIS sample is drawn from the Unbiased Nearby Stars catalogue [UNS; @Phillips2010], with omissions being made only when the predicted 1$\sigma$ cirrus confusion noise level was considered too high to provide a useful debris disc detection ($\ge1.2$mJy at 100$\mu$m). Two further source in the DEBRIS catalogue, $\epsilon$ Eridani (K001) and $\tau$ Ceti (G002), observed as part of the *Herschel* guaranteed time programme (KPGT\_golofs01\_1) have been included in this sample. Whilst these targets were not observed by DEBRIS, they were in the original DEBRIS target list, and therefore do not bias the sample by their inclusion. Had DEBRIS not been prevented from observing these sources due to duplication with the guaranteed time programme they would have been including in the submitted target list.
The sample used is volume limited, being made up of the nearest $\sim$90 targets of each spectral type that passed the cirrus confusion limit cut. As a result, this sample is free of bias towards any particular stellar parameters, or any prior knowledge of the disc or planetary system. The varied frequency of different spectral types, however, means that different volume limits are used for each spectral type star, with the more common M-types having the smallest limit, and the rarer A-types having the largest limit. Since the disc detection limit varies as a function of target distance, and stellar luminosity, disc detection biases do exist across the range in spectral types. Even so, the variable distance limits do not necessarily imply that this sample is biased against finding large numbers of discs around early type star sub-sample, and vice versa for late types. The data from this survey provide a robust statistical data set from which to study debris discs.
The FGK sample
--------------
The DEBRIS sample contains 94, 88 and 91 F, G and K-type stars (hereafter FGKs) respectively, and includes stars observed by both DEBRIS and DUNES. The sample is volume limited, with the largest distance to an F, G and K star being 24, 21 and 16 pc respectively (Figure \[fig1\]). In cases where multiple star systems are present [@Rodriguez2015], only the primary star is included in this work. Since these are all field stars, and not generally members of clusters or associations, the stellar ages are uncorrelated.
As the methods used to determine the ages of stars typically have different systematic errors associated with them, it was decided to use a single age determination method for all sources. This means that any systematic error in the stellar ages is common to all sources, and therefore could be discounted when assessing trends in the data. The ages are determined using chromospheric activity as an indicator and all ages were taken from the work of [@Vican2012]. Whilst the ages in [@Vican2012] have been disputed, the benefits of the uniform approach to stellar ages provided by this work, and applicability to the DEBRIS sample, makes them a good choice for this analysis. Age trends are used only in the modelling work presented here, and the age uncertainties do not have a significant impact on the conclusions drawn. The range of stellar ages in this sample is 1Myr–11Gyr with a median sample age of 3.3Gyr; a histogram of stellar ages is given in Figure \[fig2\].
![Distribution of stellar distance and effective temperature for the DEBRIS FGK star sample. Targets within the DEBRIS sample, observed by DUNES, are also shown for comparison.[]{data-label="fig1"}](fig1.eps){width="8.5cm"}
![Histogram of stellar ages within the DEBRIS FGK star sample.[]{data-label="fig2"}](fig2.eps){width="8.5cm"}
Observations and data reduction
-------------------------------
The ‘mini scan-map’ observing mode was used for all PACS observations. Two scans of each target were performed with a relative scanning angle of 40 deg to mitigate striping artifacts associated with low frequency noise. Scan-maps used a scanning rate of 20 arcsec per second and were constructed of 3 arcmin long scan legs with a separation of 4 arcsec between legs. The nominal DEBRIS observations used 8 scan-legs per map and performed 2 map repeats per scanning direction. DUNES led observations typically used 10 scan-legs per map, and performed 2 or more map repeats, depending on the specific source. The only practical impact of these different observing parameters is a change in the noise level in the images, and hence the disc detection threshold.
The data were reduced using Version 10.0 of the Herschel Interactive Pipeline Environment [HIPE; @Ott2010]. The standard pipeline processing steps were used and maps were made using the photProject task. The time-ordered data were high-pass filtered, passing scales smaller than 66 arcsec at 70 and 100$\mu$m and 102 arcsec at 160$\mu$m (equivalent to a filter radius of 16 and 25 frames respectively), to remove low frequency noise in the scan direction. Sources $\ge$2$\sigma$ were then identified in this first stage ‘dirty’ map to create a filter mask. The original data are then filtered a second time, using the derived mask to exclude bright sources which would otherwise result in ringing artifacts, and a final map produced. To maximise the signal to noise ratio of the output maps data from the telescope turn-around phase, at the end of each scan leg, were included in these reductions. Data were included for telescope scan rates down to 5 arcsec per second.
Source extraction and photometry
--------------------------------
Flux density measurements of the targets were made using a combination of point spread function (PSF) fitting and aperture photometry. *Herschel* calibration observations were used for PSFs, and because the maps were created in sky coordinates, the PSFs were rotated to the angle appropriate for a given observation. PSF fitting was used by default, with apertures used where the PSF fitting residuals revealed resolved sources. For clean unresolved sources it was found that PSF fitting and aperture photometry yielded very similar results; the preference for PSF fitting is largely to allow mitigation of the effects of confusion from additional point sources.
As PACS observes either 70 or 100$\mu$m simultaneously with 160 $\mu$m, both wavelengths were fitted for an observation at the same time, with the source location the same in each image. That is, for a single PACS observation there are four parameters to fit; the x/y position and the flux density at 70/100 and 160 $\mu$m. The same approach was used for SPIRE, but the three wavelengths, 250/350/500 $\mu$m, were fitted simultaneously. In the case of aperture measurements, the location was determined from the PSF fitting, and the appropriate aperture size chosen by hand.
As the locations of the target stars were well known, the PSF fitting routine was initialised at the expected star location (or locations in the case of multiple systems), and then the MPFIT least-squares minimisation routine was used to find the best fitting point source model for each observation. In cases where additional sources were visible (e.g. background galaxies) additional point sources were added to avoid biasing the fluxes. In complex cases where both a resolved disc and confusion were present, a more individually tailored approach was taken [e.g. @Wyatt2012; @Lestrade2012].
As noted in the Herschel documentation (PICC-ME-TN-037), the calibration depends somewhat on the data reduction, in particular the filter scale, masking, drizzling and frame selection. Therefore, in order to calibrate photometry for the specific data reduction pipeline used, aperture corrections were independently derived. These use all observations with 70/110 deg cross-linked scans used in the aforementioned calibration document. Aperture photometry with 4, 5, and 8 arcsec radius apertures (for maximal S/N) was made at the position found by fitting a 2 dimensional Gaussian. These fluxes were colour corrected by dividing by colour corrections of 1.016, 1.033, and 1.074. The conversion for these aperture sizes was then derived by dividing each flux by the expected photospheric fluxes for each target at each wavelength. The corrections obtained are 0.4859, 0.5275, and 0.5321, with standard deviations of 1.6 to 1.9 per cent. These compare favourably with the supplied values of 0.476, 0.513, and 0.521.
The PSF fitting results are then compared to aperture photometry using these corrections. The PSF fitted values are systematically low by $\sim$20 per cent due to flux lost in the wings of the PSF by filtering the images. The typical aperture/PSF-fit flux ratio for a large number of targets is derived to be 1.19 at 100$\mu$m, and 1.21 at 160$\mu$m, with uncertainties of about 0.05 [@Kennedy2012a]. At 70$\mu$m deriving this factor is more difficult since there are fewer high S/N observations of point sources, since DEBRIS and DUNES preferentially targeted resolved discs for observations in this band. PSF fitting to very high S/N calibration sources was found to be only marginally useful, since the PSF fits are generally poor due to variation in the PSF shape at 70$\mu$m. A tentative value of 1.16 was found by [@Kennedy2012a], and there is as yet no evidence that this value is incorrect, so we retain this value.
Estimates of the uncertainty in the measured fluxes were made by measuring the flux density dispersion from apertures placed at several hundred locations in high coverage regions of the maps. Uncertainties are also returned directly from the PSF fitting, but because the chosen pixel size and the drizzling method result in correlated noise, these uncertainties are underestimated by a factor of about 3.6 [@Fruchter2002; @Kennedy2012a]. It was found that these two methods yielded comparable results, with the main difference being that the PSF fitting results are not necessarily sensitive to larger scale variations in the local background. Thus, the final uncertainties used were the larger of these two methods.
Stellar photosphere and disc modelling {#sec2.5}
--------------------------------------
Debris discs can be discovered by infrared (IR) excesses or resolved images. In nearly all cases where a disc is resolved in thermal emission an IR excess is also seen, though this is not always the case for mid-IR imaging [@Moerchen2010]. Thus, to detect debris discs around DEBRIS stars a model for the stellar photospheric emission is needed. Optical photometry and near/mid-IR from a variety of sources yields a stellar model, and the extrapolation of this model to longer wavelengths allows a comparison with the *Herschel* photometry and a test of whether an IR excess is present.
The spectral energy distribution (SED) modelling method implemented has been used successfully for DEBRIS and other surveys [@Kennedy2012b; @Kennedy2012c]. Synthetic photometry of stellar atmosphere models yields flux densities that are compared with observations, and the best fitting model found by a combination of brute-force grid searches and least squares minimisation. The primary goal is to make the best predictions of the stellar fluxes in the IR, so the approach has not been fine tuned in an attempt to provide precise effective temperatures, surface gravities or metallicities. The former typically agree with other results to within a few hundred K, sufficient for our purposes here. Distances are known to all of our target stars, so the main stellar parameters the SED fitting yields are the effective temperature, radius, and luminosity.
Where available, data from the *Spitzer* MIPS at 24 and 70$\mu$m and *Spitzer* InfraRed Spectrograph [IRS; @Houck2004] spectra from [@Lebouteiller2011] were also used in the SED modelling for the DEBRIS FGK sample. As described in Wyatt et al. (2012), synthetic photometry of these spectra in seven artificial bands between 5 and 33 $\mu$m was derived, and then used in the SED fitting in the same way as all other photometry.
Once the best fitting stellar model is found, the ratio of the observed flux, $F_{\lambda}$, to the stellar photospheric flux, $P_{\lambda}$, can be determined ($R_{\lambda}=F_{\lambda}/P_{\lambda}$), which is used in the modelling below. The value of $R_{\lambda}$ does not indicate the significance of any excess however. The significance metric for an IR excess in some band $B$ with observed flux $F_{\rm B}$ and photospheric flux $P_{\rm B}$ is $$\chi_{\rm B} = \frac{F_{\rm \lambda} - P_{\rm B}}
{\sqrt{\sigma_{F_{\rm \lambda}}^2 + \sigma_{P_{\rm B}}^2}},$$ where $\sigma_{\rm F_{\rm B}}$ and $\sigma_{\rm P_{\rm B}}$ are the uncertainty on the photometry and photosphere model respectively. If the photometry is very precise and $\sigma_{F_{\rm \lambda}}$ has reached a minimum possible value, set at some fraction of the total flux due to the accuracy of the calibration of the instrument, then the measurement can become “calibration limited”. This is generally the case with MIPS 24$\mu$m for example. If the photometry is less precise then it is said to be “sensitivity limited”. In this work a detection significance threshold of 3 is used, meaning that a measurement must be 3 standard deviations above the photosphere to be considered a real excess.
Because an individual star typically has several mid- to far-IR measurements with which the presence of an IR excess could be detected, it is possible potentially to look for faint excesses where the significance in no individual band exceeds 3, but collectively an excess appears significant (e.g. two $\chi_{\rm B}=2.9$ excesses). There are some cases like this among DEBRIS stars, however, such an approach was found to be problematic and the excesses discovered this way to be implausible in many cases. Thus, for a star to be deemed to possess an IR excess it is required that at least one band have $\chi_{\rm B}>3$.
If an excess is present, then a model is fit to the star-subtracted infrared photometry to derive some basic properties of the excess emission. The disc model used is simply a blackbody, with a modification to allow for inefficient emission from small grains at (sub)mm wavelengths. The disc model is therefore a Planck function at temperature $T_{\rm disc}$ with some solid angle $\Omega_{\rm disc}$, multiplied by $(\lambda_0/\lambda)^\beta$ beyond the “turnover” wavelength $\lambda_0$. The disc temperature, $T_{\rm disc}$ can be converted into a blackbody radius for a given stellar luminosity, $L_\star$, with $r_{\rm bb,disc}=(278.3/T_{\rm disc})^2 \sqrt{L_\star}$. It should be noted that whilst a blackbody model provides a useful representative radius, it can underestimate the true disc radius by up to a factor of 2.5 [@Booth2013; @Pawellek2014]. Nonetheless, it provides a simple and useful reference disc radius of this general analysis. In some cases the excess emission is poorly modelled by a single blackbody, and for these a second blackbody component is added. The $\lambda_0$ and $\beta$ cannot be constrained for each component individually, so these are the same for both. For a detailed study of the identification and interpretation of these so-called “two-temperature” discs, including those identified in the DEBRIS sample, see [@Kennedy2014], [@Chen2009], [@Morales2011] and [@Ballering2013]. Here the cooler of the two components is used if two are found to be present. A two temperature fit was required for only three of the discs detected, so this approach has no significant impact on the results presented here.
Uncertainties on the fitted models are estimated in two ways. The first is simply the result of the least squares fitting. However, in many cases the parameters are poorly constrained and degeneracies mean that the least squares uncertainties are both underestimated and not representative. Uncertainties obtained by computing the $\Delta_{\chi^2}$ (relative to the best fitting $\chi^2$) from brute force grids are therefore also derived. For the stellar parameters these grids simply show that the least squares results provide useful uncertainties, hence the latter are used.
For the disc parameters we compute grids over the four disc parameters ($T_{\rm disc}$, $\Omega_{\rm disc}$, $\lambda_0$, and $\beta$), or over only the first two if there are insufficient IR photometry to constrain the latter two. Of particular interest here are the constraints on the disc fractional luminosity ($f=L_{\rm disc}/L_\star$) and disc radius and/or temperature, so a grid with these parameters is also computed. An overall disc significance metric is also used, which is simply $\chi_{\rm{tot}}=\sqrt{\Delta_{\chi^2}}$, where $\Delta_{\chi^2}$ here is the difference between $\chi^2$ for the best fitting $\Omega_{\rm disc}$ and that for $\Omega_{\rm disc}=0$ (i.e. no disc). The output distributions have been checked and the $\chi_{\rm{tot}}$ histograms are consistent with a Gaussian with unity dispersion, plus a positive population attributable to the discs.
Two independent analyses were performed using these data, the first uses the disc detections and physical parameters derived from the SED fitting described in this section to investigate debris disc incidence rates and the distribution of discs within the fractional luminosity vs disc radius parameter space. The second uses the raw flux density measurements obtained from the *Herschel* maps, as well as data from MIPS at 24 and 70$\mu$m, to constrain a disc evolution model, and thereby understand the disc population in a general way.
Both analyses focus on a general characterisation of debris discs around stars of F, G and K spectral types. However, there is great diversity in the range of parameters of each debris system; some sources are known to harbour multiple discs [e.g. @Wyatt2012; @Gaspar2014; @Matthews2014], whilst in other cases there are insufficient data to uniquely constrain the system architecture [e.g. @Churcher2011]. In order to perform a general analysis of these sources this work makes the assumption that all systems are composed of a single temperature disc. For those SEDs for which two temperatures were required, it is only the cooler component which is considered in this analysis.
Disc incidence around F, G and K type stars {#sec3}
===========================================
A total of 47 debris discs were identified in the DEBRIS FGK sample of 275 stars. The stars hosting discs are listed in Table \[table1\], along with the derived parameters for the fitted modified blackbody disc model. A break-down of detected discs by host star spectral type and associated incidence rates are given in Table \[table2\], whilst a complete list of results including measured photometry for all targets is given in Table \[tableA1\]. The detection significance, $\chi_{\rm tot}$, is given in Table \[table1\].
Of the 31 disc hosting stars identified by the DUNES team 25 are included in the DEBRIS sample. The remaining 6 were excluded as they either lay beyond the distance limits, or the cirrus confusion was predicted to be above the cut-off, for the DEBRIS survey. This analysis, however, finds discs around only 19 of these 25 sources. No disc is detected around HD 224930 (HIP 171), HD 20807 (HIP 15371), HD 40307 (HIP 27887), HD 43834 (HIP 29271), HD 88230 (HIP 49908) and HD 90839 (HIP 51459); data for these sources are given in Appendix \[ApB\]. Five of these sources are identified by [@Eiroa2013] as new discs discovered by *Herschel*. An explanation of how this analysis came to a different conclusion to that of the DUNES team for each of these sources, using the same dataset, is given below:
**HD 224930 (HIP 171):**
: Following subtraction of photospheric flux no significant emission remained at the location of the star. It should be noted, however, that there is a second confusing 3$\sigma$ source nearby at 160$\mu$m which could account for the DUNES detection (which is only significant at 160$\mu$m).
**HD 20807 (HIP 15371, zet02 Ret):**
: Data at 70 and 100$\mu$m show signs of three distinct sources, one at the position of the star and two nearby. The nearby sources were regarded as confusion and fitted and subtracted separately. This is likely the source of the difference with the DUNES detection.
**HD 40307 (HIP 27887):**
: Following subtraction of photospheric flux no significant emission remained at the location of the star. A potentially confusing cirrus can be seen at 160$\mu$m, however.
**HD 43834 (HIP 29271):**
: Following subtraction of photospheric flux no significant emission remained at the location of the star. It should be noted that the DUNES detection is based on an excess at 160$\mu$m.
**HD 88230 (HIP 49908):**
: Here two sources were fit, the star and a second confusing point source nearby. Following subtraction of photospheric flux no significant emission remained in either PACS band at the location of the star. However, it should be noted that there is significant cirrus emission within the field, although not close to the star position, at 160$\mu$m.
**HD 90839 (HIP 51459):**
: Following subtraction of photospheric flux no significant emission remained at the location of the star. In addition, the uncertainty calculated in this work is almost twice that found by [@Eiroa2013].
[@Montesinos2016] subsequently presented a further analysis on behalf of the DUNES team which included an additional 54 sources originally observed by DEBRIS. All of these targets are included in this work, and excess detections are in agreement with one exception, HD 216803. [@Montesinos2016] identify an excess for this source based on data at 160$\mu$m; no excess is detected in this analysis due to a larger uncertainty found for this source at this wavelength in this work, taking it below the detection threshold.
The disc incidence for both the full DEBRIS FGK sample, and individual spectral types, is given in Table \[table2\]; uncertainties are calculated in a way suitable for small number statistics using the tables in [@Gehrels1986].
For the combined FGK star sample the incidence is 17.1$^{+2.6}_{-2.3}$ per cent, consistent with the excess rate found by [@Trilling2008] using 70$\mu$m *Spitzer* data ($16.3^{+2.9}_{-2.8}$). The trend for smaller incidence rates for later spectral types seen by @Trilling2008 is also reproduced, with incidence rates within 3, 3 and 7 per cent for the F, G and K spectral types respectively. This trend is really only significant, however, between F and G/K populations. It should be noted that this comparison is based on the 70$\mu$m excess incidence reported by @Trilling2008. The DEBRIS FGK incidence rates are obtained from model fits to data at all available wavelengths, including 70$\mu$m. The differences seen here are attributed primarily to the greater depth of the PACS data and constraints provided by the multi-wavelength analysis.
---------- ----------- ----------- ----------------------- -----------------------
Spectral Completeness
type No. stars No. discs Incidence adjusted
incidence
F 92 22 23.9$^{+5.3}_{-4.7}$% 37.4$^{+6.1}_{-5.1}$%
G 91 13 14.3$^{+4.7}_{-3.8}$% 24.6$^{+5.3}_{-4.9}$%
K 92 12 13.0$^{+4.5}_{-3.6}$% 22.5$^{+5.6}_{-4.2}$%
Total 275 47 17.1$^{+2.6}_{-2.3}$% 27.7$^{+2.9}_{-2.9}$%
---------- ----------- ----------- ----------------------- -----------------------
: Summary of debris disc detections and associated disc incidence as a function of the host star spectral type. The incidence adjusted for incompleteness in this sample is also given. Uncertainties are calculated in a way suitable for small number statistics using the tables in [@Gehrels1986].[]{data-label="table2"}
A more direct comparison can be made with the results of [@Eiroa2013], who use a 3$\sigma$ excess detection at any PACS wavelength as their disc detection requirement. [@Eiroa2013] find incidence rates for their 20pc limited sub-sample of 20$^{+13}_{-9.3}$, 22$^{+7.4}_{-6.2}$ and 18.5$^{+6.8}_{-5.5}$ per cent for their F, G and K spectral types respectively, and a combined rate of 20$^{+4.3}_{-3.7}$ per cent. Equivalent results from [@Montesinos2016] are 24$^{7.5}_{-6.3}$, 20$^{6.1}_{-5.1}$, 18$^{6.4}_{-5.2}$ and 20$^{9.6}_{-8.3}$ per cent. Whilst these rates are, with the exception of the F-types, higher than those found within the DEBRIS sample, the difference is less than 1$\sigma$ in each case, meaning that they are generally in agreement.
When the six sources with contested excess detections (Section \[sec3\]) are removed from the DUNES 20pc sub-sample, their F, G, and K star incidences decrease to $15^{+12}_{-8.0}$, $16^{+6.9}_{-5.3}$ and $15^{+6.5}_{-5.0}$ per cent, with a combined incidence of $15^{+3.9}_{-3.3}$ per cent, with consistent agreement for the results of [@Montesinos2016]. This change largely accounts for the difference in incidence between DUNES and DEBRIS, bringing them well within the associated uncertainties of the two measurements.
It is interesting to note that the raw incidence for A-stars in the DEBRIS survey (24 per cent) is similar to that of the F-star sample [@Thureau2014]. Similarly, the G and K incidence rates are very close in value. This suggests a possible link between the A and F, and G and K stars, which is distinct for these two subgroups. In addition, the incidence found for M-stars within the DEBRIS survey is $2.2^{+3.4}_{-2.0}$ per cent, just over 2$\sigma$ below that of the K-star sample found here (Lestrade et al. in prep.). Furthermore, within the FGK DEBRIS sample, we also note a difference in raw incidence rates among F0–F4 and F5–F9 subsamples, significant at the 97.8 per cent confidence level (see Figure \[fig3\]). No significant difference is observed between early- and late-G stars, while a tentative difference is observed between early- and late-K stars. Overall, despite limited sample sizes, this suggests a gradual decline of incidence rate towards lower stellar mass.
![Incidence rate of debris disks within the DEBRIS survey as a function of spectral type. Filled and open squares indicate raw and completeness-corrected incidence rates respectively shown in Table \[table2\]. Horizontal errorbars indicate the range of spectral type associated with each estimate. The A- and M-type incidence rates, shown in black, are from Thureau et al. (2014) and Lestrade et al. (in prep.), respectively. For each of the F-, G- and K-type subsamples (red, green and blue symbols, respectively), further incidence rates are computed for earlier and later-type stars, shown by filled circles; i.e., splitting F stars in F0–F4 and F4.5-F9, G stars in G0–G4 and G4.5–G9 and K stars in K0-4 and K4.5–9 sub-samples.[]{data-label="fig3"}](fig3.eps){width="8.5cm"}
Completeness corrected incidences {#sec3.1}
---------------------------------
The ability to detect excess emission from a debris disc around a star varies for each target, depending on the disc and stellar physical parameters (including distance), as well as the range and depth of available data. Assuming a single component disc system that emits as a blackbody (Section \[sec2.5\]), it is possible to determine in which regions of $f$ vs $r_{\rm{bb,disc}}$ parameter space a disc could have been detected for any individual source. The variable detection limits from star-to-star within this parameter space mean that, for any specific combination of $r_{\rm{bb,disc}}$ and $f$, a disc might be detectable for only a fraction of the full DEBRIS FGK sample. This fraction provides a measure of the known completeness, for a given combination of $r_{\rm{bb,disc}}$ and $f$, for this sample. Combining the detection limits provides a function giving a measure of the completeness within a 2-dimensional parameter space.
Figure \[fig4\]a shows the $f$ vs $r_{\rm{bb,disc}}$ parameter space, with the location of the confirmed discs plotted therein. The grey contours on Figures \[fig4\]b and \[fig4\]c show the fraction of the sample for which a disc could have been detected if it existed in this region of parameter space, for the entire FGK sample, based on the all the available data. This is taken to be the sample completeness at this point in parameter space. The region at the top of the figures is 100 per cent complete, meaning that a disc with parameters within this space could have been detected around all of the stars in the sample. The completeness contours decrease in steps of 10 per cent down to the shaded region of parameter space, in which no discs could have been detected around any of the sample (cross-hatched region).
Using this completeness function, it is possible to adjust the raw incidence rates given in Table \[table2\] in an attempt to account for the known incompleteness of these data. It should be noted that this is only applicable in regions of parameter space wherein at least one source is detectable; no conclusions can be drawn for parameter space with zero completeness. Therefore, any results from this adjustment remain lower limits to the potential true disc incidence.
To calculate the completeness adjusted incidence rates in Table \[table2\], the completeness at the location of each detected source is first calculated. The number of detected sources, adjusted for completeness, is then given by one over the derived completeness at that point in parameter space. For example, if a disc is detected in a region of $f$ vs $r_{\rm{bb,disc}}$ parameter space wherein only 50 per of the sample would have yielded a detection, the completeness fraction is 0.5. Thus, the number of detected discs, adjusted for completeness, would be 2. This is replicated for all detected sources to estimate the number of discs detected, adjusted for sample completeness. This number is then divided by the sample size, to obtain the adjusted incidence rate given in Table \[table2\]. The errors are equally scaled by completeness, with scaling only applied when the completeness is greater than 10 per cent, to avoid extremely large adjustments, with equally large uncertainties. The full dataset is broken into three separate samples so as to determine a completeness function for each spectral type separately. These adjustments are illustrative of the effects of incompleteness within this sample, but should not be regarded as fully correcting for completeness.
The trend for smaller incidence rates for later spectral types is maintained, even after attempting to correct for incompleteness. This suggests that the relative incidence between spectral types is reasonably robust, and this trend is real. The similar completeness levels for the three spectral types means that this correction effectively acts as a positive uniform scaling, increasing the average incidence rate to $27.7_{-2.9}^{+2.9}$ per cent. It should be noted that the specific make-up of each sample impacts on the completeness of the sample, and therefore on the completeness correction applied and 10 per cent threshold cut-off. It is this effect that results in a completeness adjusted incidence for the entire sample being lower than what might naively be calculated from the mean of the completeness adjusted incidence rates for each of the three sub-samples from which it is composed. The larger sample size for the combined FGK sample also provides greater statistical robustness to the influence of discs in regions of low completeness regions, which can otherwise bias the reported incidence towards higher values. Such biases are more common in the individual spectral type samples due to their lower levels of completeness at higher fractional luminosity and blackbody disc radius.
Disc fractional luminosity vs radius distribution {#sec3.2}
-------------------------------------------------
The DEBRIS FGK star sample is a large and unbiased dataset. These two properties make it possible to study the parameter space of the disc properties, determined by SED model fitting (Section \[sec2.5\]), in a more general way than has been possible before.
Figure \[fig4\] shows the process by which we estimated the completeness adjusted incidence throughout the range of fractional luminosity vs disc radius parameter space probed in this sample. This process starts with the discs for which significant emission was detected, which are shown on Figure \[fig4\]a at the radius and fractional luminosity of the best fit from the SED modelling. However, this modelling also quantified the uncertainties in these parameters, and the same figure also shows the 1$\sigma$ uncertainty contours for the detected discs. These contours are typically asymmetric, with a diagonal ‘banana’ shape running from the top left to bottom right, illustrating the degeneracy inherent in the SED model. The SED model fit information is then used in Figure \[fig4\]b to determine the fraction of stars for which a disc is detected in a given region of parameter space. The colour scale gives the disc incidence per log fractional luminosity per log AU, and so is indicative of the number of discs that have been found in different pixels in the image. To make this image, the uncertainties in the parameters for the detected discs were accounted for by spreading each disc across the allowed range of those parameters, weighted according to the probability of the disc having those parameters (which was achieved using 1000 realisations for each disc). This image is then corrected for completeness in Figure \[fig4\]c, which is the same as Figure \[fig4\]b but divided by the fraction of the sample for which discs could have been detected at this point in parameter space (which is shown by the contours on these figures). The resulting completeness-corrected disc incidence is only shown for regions of parameter space for which completeness is $>10$ per cent, since below this point the uncertainties and associated completeness correction become too large to be useful.
The completeness adjusted incidence rate, for the parameter space above the 10 per cent completeness level in Figure \[fig4\]c, is 28 per cent, the same as that given in Table \[table2\]. The only practical difference between this estimate and the one in Table \[table2\] is that here the uncertainties of the derived fractional luminosity and blackbody radius for each disc are free to vary within their uncertainties. This introduces a variation in the completeness adjustment applied to each disc. The same incidence obtained by both methods shows that the impact of completeness variability is negligible for this sample.
![(a) Location of detected debris discs (open black circles) within the fractional luminosity vs blackbody radius parameter space. The line around each detected disc shows the 1$\sigma$ uncertainty for each parameter. The cross-hatched region shows the region of parameter space in which no discs could have been detected with this sample, i.e. zero completeness. (b) The colour scale shows the disc incidence, per log AU per log unit fractional luminosity, as determined from a Monte-Carlo simulation of this sample, with the associated 1$\sigma$ uncertainty contours in fitted disc radius and fractional luminosity used shown in (a). The contour lines show levels of completeness from zero (cross-hatched region) to 100 per cent, in steps of 10 per cent. (c) The colour scale shows the completeness adjusted disc incidence, per log au per log unit fractional luminosity. As with (b), this is calculated from a Monte-Carlo simulation of this sample and the associated 1$\sigma$ uncertainty contours in fitted disc radius and fractional luminosity used shown in (a).[]{data-label="fig4"}](fig4.eps)
### Distribution of observed disc properties {#sec3.2.1}
The data in Figure \[fig4\] show that the disc population can be split into three categories: a smooth ‘normal’ disc population, and two outlying ‘island’ populations, one characterised by small radii (hot) discs and the other by bright discs. The two island populations can be most clearly identified as distinct from the normal disc population in Figures \[fig4\]b and \[fig4\]c. The members of the two island populations are individually labeled in Figure \[fig4\]a. It should be noted when studying this plot that it is assumed that discs emit as a blackbody, which can lead to an underestimate of the true physical radius when realistic dust grain emission is considered.
The small radii population contains only one disc with a radius smaller than 4AU, HD 69830. No excess is detected in the *Herschel* data, with the disc only detected at 8–35$\mu$m with *Spitzer* and ground-based mid-IR observations. This excess is attributed to very small dust grain emission [@Beichman2005], potentially from a recent single large cometary collision, or interaction with a planetary system [@Lovis2006]. This dust is therefore likely to be transient in nature, and therefore have abnormal properties within the context of the wider sample.
The bright disc population consists of five discs with $r_{\rm{bb,disc}}>4$AU and $f>5\times10^{-5}$: HD 166, HD 10647 (q1 Eri), HD 207129, HIP 1368 and HD 48682. With the exception of HD 166, with an age of $\sim$200Myr, these five bright discs are all fairly old systems, with a mean age of $\sim$2.5Gyr. This is contrary to what might be expected when considering steady-state disc evolution [@Wyatt2008]. Work by [@Lohne2012] finds that the disc around HD 207129 can be explained by steady-state evolution alone, and [@Gaspar2013] concluded that this might too be possible for the other four members of this sample. Even so, recent dynamical interaction with a planetary system or other transient events, such as a collision between two particularly large planetesimals, cannot be discounted as an explanation for their late period disc brightness [@Wyatt2008]. However, only HD 10647 is known to harbour an exoplanet system [@Butler2006].
The remaining disc detections fall within the ‘normal’ disc population, occupying the $f<5\times10^{-5}$ parameter space. This population shows a generally smooth completeness adjusted incidence rate distribution between $r_{\rm{bb,disc}}=4-300$AU, with a clear concentration of debris discs in the $r_{\rm{bb,disc}}=$7-40AU range, and a peak rate at $\sim 12$AU.
The upper envelope of the disc incidence of the normal disc population resembles an upside-down ‘V’ shape similar to that expected for a population of discs that have been evolving by steady state collisional erosion (Wyatt et al. 2007a). The peak in this V occurs at radii of 10-30AU, and the fractional luminosity of this envelope decreases with increasing radius. In the steady state model this results from the long collision timescale at large radii, which means that the fractional luminosity of such large discs is simply a reflection of the amount of mass that they were born with, and how much light that mass can intercept when ground into dust; e.g., if disc masses are independent of their radii then this envelope would decrease $\propto r^{-2}$ [@Wyatt2008]. The low disc detection rate in the $r_{\rm{bb,disc}}=$1-10AU and $f>10^{-5}$ range suggests that there may be a genuine decrease in disc rates in this region of parameter space. However, the lack of sensitivity in these data to discs at small radii with $f\ll 5 \times10^{-5}$ makes it difficult to assess the population at radii much below 7AU. Nevertheless, the short collision timescale for discs with small radii could have resulted in a high decay rate for discs in this region, resulting in their fractional luminosity being reduced to a level at which no discs are detectable within this dataset. Indeed, the steady-state evolution model of @Wyatt2007a [see Section \[sec4\]] predicts that the upper envelope in the disc incidence should turn over at some radius (causing the aforementioned upside-down V shape). Based on these data, this turn-over appears to occur within the region of peak disc detection rate, i.e. at $r_{\rm{bb,disc}}=$7-40AU. Though it is not possible to determine if $<7$AU discs have been collisionally depleted, or simply never existed in the first place.
One point to note from Figure \[fig4\]c is that, even after adjusting for completeness, the disc incidence decreases toward lower fractional luminosities; i.e., there are appear to be more discs per log AU per log fractional luminosity at fractional luminosities of $\sim 10^{-5}$ than close to the 10 per cent completeness cut-off limit. This could be a result of truly decreased incidence, which could be indicative of a bimodal disc population, or point to an insufficient quantity of discs to accurately apply this correction method across such a broad parameter space, even given the Monte-Carlo implementation. In any case, the disc incidence in these low completeness regions is an important indicator of the incidence in this region.
The interpretation of the black body radius parameter requires consideration of the fact that this is expected to underestimate the disc’s true radius. A study of discs around A-type stars [@Booth2013] finds that the blackbody radius underestimates the true radius by a factor of between 1 and 2.5, with tentative evidence for an increase in this factor for later spectral types [see @Pawellek2014]. Adopting the upper limit of this range, as is most applicable for this F, G and K star sample, gives a typical radius of approximately 30AU, and a range of $\sim$17-100AU, based on the data in Figure \[fig4\]. This spans the current estimated radius for the Kuiper belt [@Vitense2010], making it typical within this sample. The depth of these data are insufficient to accurately characterise the measured disc population down to the fractional luminosity of the Kuiper belt, however, but do place the Kuiper belt within the range of the ‘typical’ disc radius.
Steady-state evolution of debris around F, G and K stars {#sec4}
========================================================
While Figure \[fig4\] provides a valuable guide to the underlying debris disc population, it is appropriate when fitting a model to this population (which is the purpose of this section) to compare the model more directly with the observations. The excess ratio, i.e. the ratio of the disc to stellar photospheric flux density ($R_{\lambda}=F_{\rm{\lambda,disc}}/F_{\rm{\lambda,star}}$), is a fundamental measurable parameter of debris discs in the infrared. It is a function of disc temperature/radius and fractional luminosity, $f=L_{\rm{disc}}/L_{\rm{star}}$, and is different for each observed wavelength. By studying the distributions of excesses within a sample of stars across multiple wavelengths it is possible to constrain model disc distributions, and thereby better understand the underlying disc population.
Figure \[fig5\] shows with black dots the fraction of DEBRIS FGK targets with an excess greater than $R_{\lambda}$ as a function of $R_{\lambda}$. The top two panels show results for the 100 and 160$\mu$m DEBRIS data, while the lower two panels show the same plot for the 235 stars (86 per cent of the total sample) for which MIPS 24 and 70$\mu$m data are available. The main plot in each panel shows the positive excesses on a log-log scale for clarity, with the sub-plot in each panel showing the distribution with linear axes, truncated to show excesses in the range $R_{\lambda}=$ -1 to 1.
![Fraction of FGK star sample with fractional disc excess greater than or equal to $R_{\lambda}$ as a function of $R_{\lambda}$ (black dots) at 24, 70, 100 and 160$\mu$m. The red dashed line shows the mean model fit to these data, and the grey shaded contours show the 1, 2 and 3$\sigma$ limits for the model fit. The uncertainty in model fit results from the finite size of the DEBRIS FGK star sampled used in this analysis. The main plots show only the positive region wherein the disc population resides, whilst the inserts show the model fit truncated to $R_{\lambda}=-1$ to $+1$, and plotted with linear axes.[]{data-label="fig5"}](fig5.eps){width="1.0\columnwidth"}
Since the majority of stars do not have detectable discs, these cumulative excess fraction plots all intercept the y-axis at a value close to 0.5, which represents the mean excess of the measured population. The negative excesses are the result of the negative half of the Normal noise distribution, when observing targets hosting faint, or non-existent discs.
Disc evolution model {#sec4.1}
--------------------
The work of [@Wyatt2007a] provides a simple model for the steady-state evolution of debris discs. The model assumes that all stars are born with a planetesimal belt, and that some of the properties of those belts are common among all stars; that is, all belts have the same maximum planetesimal size, those planetesimals have the same strength, and are stirred to the same level as defined by a mean eccentricity. The planetesimal belts of different stars have different initial masses and radii, and evolve after formation by steady state collisional erosion. Here we use this model to interpret the measured disc excesses in the PACS 100 and 160$\mu$m bands, and also in the MIPS 24 and 70$\mu$m band for the same targets when available.
A disc is modelled as a single belt of planetesimals at a radius $r$, with width $dr$, in collisional cascade. The size distribution is given by $n(D)\propto D^{2-3q}$, where $q=11/6$ [@Dohnanyi1969], and applies from the largest planetesimal, $D_{\rm{c}}$, down to the blow-out dust grain size, $D_{\rm{bl}}$. All particles are assumed to be spherical and to act as blackbodies. Given these assumptions, the fractional luminosity is given by $f = \sigma_{\rm{tot}}/4\pi r^2$, where $\sigma_{\rm{tot}}$ is the cross-sectional area of the particles in AU$^2$. Therefore, with the planetesimal size distribution defined above $f\propto M_{\rm{mid}}D_{\rm c}^{-0.5}$. The blackbody assumption also makes it possible to define the disc temperature, $T=278.3 L_{\rm{star}}^{0.25} r^{-0.5}$ in K, and flux density, $F_{\rm{\nu,disc}}=2.35\times10^{-11}B_\nu(T)\sigma_{\rm{tot}}d^{-2}$ in Jy, where $d$ is the distance to the star in pc and $B_\nu$ is the Planck function in Jysr$^{-1}$.
The long-term evolution of a disc in a steady-state collisional cascade depends only on the collisional lifetime, $t_{\rm{c}}$ of the largest planetesimals, given by, $$t_{\rm{c}} = \frac{3.8\rho r^{3.5}(dr/r)D_{\rm{c}}}{M_{\rm{star}}^{0.5}M_{\rm{tot}}}\frac{8}{9G(X_{\rm{c}})},
\label{tc}$$ where $t_{\rm{c}}$ is in Myr, $\rho$ is the particle density in kgm$^{-3}$, $D_{\rm{c}}$ is in km, $M_{\rm{star}}$ is the stellar mass in units of $M_{\rm{\odot}}$, $M_{\rm{tot}}$ is the solid disc mass (i.e. excluding gas) in units of $M_{\rm{\oplus}}$, $G(X_{\rm{c}})$ is a factor defined in Equation 9 of [@Wyatt2007a], and $X_{\rm{c}}=D_{\rm{cc}}/D_{\rm{c}}$, where $D_{\rm{cc}}$ is the diameter of the smallest planetesimal that has sufficient energy to destroy a planetesimal of size $D_{\rm{c}}$. This value can be calculated from the dispersal threshold, $Q_{\rm{D}}^{*}$, defined as the specific incident energy required to catastrophically destroy a particle [@Wyatt2002], given by $$X_{\rm{c}} = 1.3\times10^{-3}\left(\frac{Q_{\rm{D}}^{*}rM_{\rm{star}}^{-1}}{2.25e^2}\right)^{1/3},$$ where $Q_{\rm{D}}^{*}$ has units of Jkg$^{-1}$, and $e$ is the particle eccentricity.
This is a simplified formalism of the equation used in [@Wyatt2007a], in which it is assumed that particle eccentricities and inclinations, $I$, are equal. This assumption is used throughout the modeling presented in this work.
The time dependence of the disc mass can then be calculated by solving the differential equation $dM_{\rm{tot}}/dt = -M_{\rm{tot}}/t_{\rm{c}}$, which gives $M_{\rm{tot}}(t) = M_{\rm{tot}}(0)/(1+t/t_{\rm{c}}(0))$. This result accounts for the mass evolution resulting from collisional processes, which through the assumed size distribution also sets the evolution of the discs’ fractional luminosities and fractional excesses.
For full details of this model see [@Wyatt2007a], and also [@Wyatt2007b], [@Kains2011] and [@Morey2014] for additional useful examples of its implementation.
Model implementation and fitting {#sec4.2}
--------------------------------
The model described in Section \[sec4.1\] was implemented for all stars in the DEBRIS FGK sample. The initial disc parameters are defined by $M_{\rm{tot}}(0)$, $r$, $dr$, and $\rho$, and the disc evolution by $Q_{\rm{D}}^*$, $e$, $I$ and $D_{\rm{c}}$. To simplify the modeling the following parameters were fixed: $\rho=2700$kgm$^{-3}$, $e/I=1$, $q=11/6$ and $dr=r/2$ , following [@Wyatt2007b] and [@Kains2011]. All stars were assumed to harbour a disc, and a log-normal distribution was used to define the initial disc masses of the model population. This follows [@Andrews2005], who found such a distribution in a sub-millimeter study of young protoplanetary discs in the Taurus-Auriga star forming region. This distribution was parameterised by the distribution centre, $M_{\rm{mid}}$, and the distribution width. The width was set to 1.14 dex, the value found by [@Andrews2005]. The model disc radii are defined by a power law distribution with exponent $\gamma$, between minimum ($R_{\rm{min}}$) and maximum ($R_{\rm{max}}$) radii. Radii of 1 and 1000AU were adopted for $R_{\rm{min}}$ and $R_{\rm{max}}$ respectively, based on the data in presented in Section \[sec3.2.1\].
Consequently, there are five remaining parameters in this model: $M_{\rm{mid}}$, $\gamma$, $Q_{\rm{D}}^*$, $e$ and $D_{\rm{c}}$. However, as explained in @Wyatt2007b, the parameters only affect the observable properties of the disks in certain combinations. Thus without loss of generality we can reduce the number of free parameters to three: $A=D_{\rm{c}}^{1/2}{Q_{\rm{D}}^*}^{5/6}e^{-5/3}$, $B=M_{\rm{mid}}D_{\rm{c}}^{-1/2}$ and $\gamma$. Fixing the combination of parameters given by $A$ ensures that a disc’s collisional evolution timescale is constant, which also sets its fractional luminosity at late times. Fixing the combination of parameters given by $B$ ensures that the disc population is born with the same distribution of fractional luminosity.
The aim of this modelling was to generate simulated datasets, at all wavelengths simultaneously, that could be compared with the observed FGK star data shown in Figure \[fig5\]. To ensure that the model dataset matched the DEBRIS FGK star sample as well as possible the estimated stellar parameters for this sample were used as an input to the model. This differs from previous implementations of the model, wherein the stellar parameters were drawn from a given distribution. The use of the known stellar parameters for this sample, including stellar distance, luminosity, mass, age and effective temperature means that this model dataset reproduces the unavoidable observational biases and sample size limitations of the final dataset. To create a fully representative simulated dataset, the appropriate source measurement and stellar flux density uncertainties were then applied to the output model data, along with a calibration uncertainty for each waveband. This provides a dataset which can be analysed in a self-consistent way and directly compared to the observed data.
The limited size of this stellar sample leads to potentially significant statistical variation in the model output for the same input parameters. The model was therefore run 1000 times for each set of input free parameters, and the cumulative fractional excess plots constructed in the same way as was done for the real measured data. The mean of the runs was then taken as the representative output for the given input parameters, and compared to the observed data.
The degenerate nature of the model makes constraining the free parameters difficult. To fully investigate the parameter space a three dimensional parameter grid was created and the model run for all combinations of input parameters within this grid. The grid was filled with the output $\chi^2$ measurement for each combination of parameters: $$\chi^{2}=\sum_{i,k}{\Bigg(\frac{f(>R_{\lambda_i})_{\rm obs}-f(>R_{\lambda_i})_{\rm mod}}{\sigma_{\rm{s}_{i,k}}}\Bigg)^2}.$$ Here $f(>R_{\lambda_i})_{\rm obs}$ is the observed fraction of stars with fractional excess at wavelength $\lambda_i$ above a given level, which is measured at the $k$ values of $R_{\lambda_i}$ given by those of the discs observed in the sample. The equivalent distribution for the model is given by $f(>R_{\lambda_i})_{\rm mod}$, which is calculated as the mean of many runs performed for each set of input parameters, while $\sigma_{\rm{s}_{i,k}}$ is the standard deviation of the model distribution (on the basis that this is indicative of the level of uncertainty in the observed distribution due to the small sample size).
This model fitting approach exploits the full dataset, including sources with large negative excesses, rather than implementing an arbitrary $\sigma$ threshold cut. As a result, it is possible to better constrain the model, and determine more representative values for the free parameters, and thus the underlying disc population. Using the data in this way requires a good understanding of the uncertainties, including knowledge that the uncertainties follow a Normal distribution. The smooth curves in the plots in Figure \[fig5\] highlight this, following the curve expected from a Normal distribution, with deviations occurring due only to the disc population. This is with the exception of the *Spitzer* 70$\mu$m data, which shows a minor deviation from the expected smooth curve below $R_{70\mu\rm{m}}\sim-0.25$. This deviation was regarded as sufficiently small not to significantly adversely influence the derived best fit model, and therefore was not excluded.
Best fit parameters {#sec4.3}
-------------------
The dashed line in Figure \[fig5\] shows the mean best fit model output compared to the measured data (black dots), and the grey shaded contours show the 1, 2 and 3$\sigma$ variations in the simulated output from each of the model implementations. Note that these uncertainties are applicable to the output of any single model run using these data due to the small sample size, and are not representative of the uncertainties in the observed disc data (although they should be indicative of the uncertainty expected due to the small sample size if the model is an accurate representation of the underlying disc population). The parameters for this best fit model are $A=D_{\rm{c}}^{1/2}{Q_{\rm{D}}^*}^{5/6}e^{-5/3}=5.5\times10^5$ $\rm{km}^{1/2}\rm{J}^{5/6}\rm{kg}^{-5/6}$, $B=M_{\rm{mid}}D_{\rm c}^{-1/2}=0.1$$\rm{M}_{\oplus}\rm{km}^{-1/2}$ and $\gamma=-1.7$, with $R_{\rm{min}}$ and $R_{\rm{max}}$ set to 1 and 1000AU respectively. The $\Delta \chi^2$ for the parameter space investigated is shown in Figure \[fig6\] for each parameter. The minimum $\chi^2$ found is shown by the white circle and the intersection of the dashed lines which span the panels. For reference, the output model parameters found by @Kains2011 are also indicated in each panel by the white triangle.
![$\chi^{2}$ model fits for all combinations of the fitted model parameters. The panels include the minimum $\chi^2$ found in the output model grid. The colour images show the $\chi^2$ output in a log scale, and the plotted white circle shows the location of the minimum within the given parameter space. The best fit parameters obtained by @Kains2011 are also shown as a white solid triangle on each panel for comparison. The absolute levels in this figure are unimportant, therefore a colour scale is regarded as unnecessary.[]{data-label="fig6"}](fig6.eps)
While the model was constrained by a fit to the fractional excess distributions of Figure \[fig5\], to illustrate how the model parameters ($\gamma$ and the combinations of parameters $A$ and $B$) were constrained, it is helpful to refer to the fractional luminosity vs disc radius plot of Figure \[fig7\]. This is because any model that fits the fractional excess distributions must result in a fractional luminosity vs radius distribution that is not far from that observed, and changes in the model parameters translate directly into changes in the distribution of model discs shown in the colour scale and red stars of Figure \[fig7\]. As already mentioned in Section \[sec3.2.1\], the model population in this figure always looks like an upside-down ‘V’. On the right-hand side of the V, the fractional luminosity decreases with radius $\propto r^{-2}$ at a level that scales with the parameter $B$. This is because $B$ sets the initial fractional luminosity of the disc population, and the long collision timescales at these large radii mean that this part of the population shows little evidence for collisional depletion. On the left-hand side of the V, the fractional luminosity increases with radius $\propto r^{7/3}$ at a level that scales with the parameter $A$. This is because the short collisional lifetime at small radii means that all such discs are collisionally depleted and so tend to a fractional luminosity that depends only on their age and radius [@Wyatt2007a]. The parameter $\gamma$ sets the ratio of discs in the left- and right-hand sides ($\gamma>-1$ means a population dominated by large radii discs in terms of number per log radius, and $\gamma<-1$ means a population dominated by small radii discs).
This helps to explain the shape of the $\chi^2$ distribution in Figure \[fig6\], and the ‘L’-shaped degeneracy in the $B$ vs $A$ plot. Starting from the best fit model, this shows that a reasonable fit can be found by increasing $B$ and so the initial masses of the discs. This would result in more bright cold discs, but this can be counteracted by increasing the relative number of small discs by decreasing $\gamma$. Likewise a reasonable fit can be found by decreasing $B$ and increasing $\gamma$. The value of $A$ is reasonably well constrained by the small radii disc population for a value of $B$ close to the best fit model (albeit with the caveats discussed in Section \[sec5.1\] about how the small radii disc population might be biased by any warm or hot disc components). However, as $B$ is decreased, eventually $A$ becomes unconstrained and can be arbitrarily large. This is both because the population becomes dominated by large radii discs as $\gamma$ is increased, and also because discs can never have a higher luminosity than their initial value which is set by $B$ irrespective of how large $A$ is.
While Figure \[fig6\] makes it appear that these extremes in parameter space provide a reasonable fit to the observations, the fit is clearly improved with the best fit model parameters. The best-fit model readily explains the relatively uniform distribution of discs with large radii, as well as the cluster of discs seen at 7-40AU. In the model this cluster arises at the apex of the upside-down ‘V’, which occurs at radii for which the largest planetesimals come into collisional equilibrium on a timescale of the average age of stars in the population.
The best fit model parameters are, with the exception of $\gamma$, close to those found by @Kains2011. The likely cause of the difference is that @Kains2011 did not have access to longer wavelength data from Herschel which now provides improved constraints on the disc radius distribution. Indeed, using the model parameters of @Kains2011 (which also assumed $r_{\rm max}=160$AU) provides a significant overestimate of discs with large excess at 100 and 160$\mu$m. This is corrected for in the best fit model, whilst maintaining a good fit to shorter wavelength data, by decreasing the disc radial distribution exponent, $\gamma$, from -0.6 to -1.7.
Discussion {#sec5}
==========
Accuracy of Model fit to Observations {#sec5.1}
-------------------------------------
Figure \[fig5\] shows that the DEBRIS FGK star observables (i.e., the fractional excess distributions) are in general well fit by the simulated model data created using the disc evolution model described in Section \[sec4\]. As noted in Section \[sec4.2\], the observed distribution at 70$\mu$m departs from the smooth curve expected for Gaussian noise for negative values of $R_{\lambda}$. However, the fit is good for positive values of $R_{\lambda}$. The model also slightly over-predicts the number of discs at high values of $R_{\lambda}$ in the 100 and 160$\mu$m bands. One cause for this could be the assumption that the spectrum resembles a black body at all wavelengths, whereas a faster fall-off in the spectrum is expected due to the lower emission efficiencies of small grains at long wavelengths. Such a fall-off was included in the SED fitting (Section \[sec2.5\]), which found several discs with $\lambda_{\rm{0}}<100$$\mu$m, which would lead to a lower $\geq 100$$\mu$m flux in the disc evolution model had this effect been included in the modelling.
Given the good fit to the observed fractional excess distributions, it is perhaps unsurprising that the model also provides an accurate prediction for the incidence rate for the sample. At the 3$\sigma$ sensitivity limit for these data the incidence rate for the model output is $\sim$19 per cent (i.e., the model predicts that 19 per cent of observed stars should show an excess in at least one waveband), close to the rate of $17.1_{-2.3}^{+2.6}$ per cent found in Section \[sec3\].
The success of the model at fitting the observed fractional excess distributions suggests that the population can be explained by all stars having a single temperature component debris belt. However, while it is not necessary to invoke multiple components to explain the observations, e.g. the warm and cool components that [@Gaspar2013] modelled as contributing independently to the 24 and 70/100$\mu$m excesses, it is likely that some stars do indeed have two (or more) independent components [@Chen2009; @Kennedy2014]. Indeed, in three cases a two temperature component fit was required to satisfactorily explain the data, as described above; the warmer component had already subtracted in this analysis. This may bias the distribution of planetesimal belt radii inferred from this model, since the model would require a disc with a relatively small radius to explain a 24$\mu$m excess that arises from a warm component belt, even if that belt resides within a system with a cold outer belt. In other words, there is no guarantee that the model will completely reproduce the inferred fractional luminosity vs disc radius distribution seen in Figure \[fig4\] which was derived for cold outer belts, i.e., ignoring any warm belt component.
Nevertheless, Figure \[fig7\] shows that the fractional luminosity vs disc radius distribution is reasonably well reproduced by the model (which also justifies the use of this figure in Section \[sec4.3\] to explain how the model parameters were constrained in the fit). This figure shows the same parameter space as in Figure \[fig4\], but with the incidence shown with the colour scale being derived from a Monte-Carlo run of the disc evolution model, using the obtained best fit parameters as input. The red filled and unfilled stars show the detected and non-detected discs output for a single run of this model respectively. Detection of a disc is determined randomly, weighted by the known completeness level for the DEBRIS FGK star sample within the given region of parameter space. The zero completeness contour used in Figure \[fig4\] is included for reference, along with the location of the original measured debris disc detections. The model population maintains the same stellar parameters as the real sample, but with random initial disc properties. The success of the model is evident in that the example model output shown in Figure \[fig7\] classed 49 discs as detected, close to the 47 detected in the real sample. It also reproduces the clustering of ‘detected’ sources at approximately 7-40AU seen in Figure \[fig4\].
These successes aside, the regions where the model population provides a poor fit to the observed population are also informative. For example, the model does not accurately reproduce the two outlying island populations discussed in Section \[sec3.2.1\]. That the model cannot reproduce the bright hot dust systems like HD 69830 is perhaps unsurprising, given that they are thought to be a transient feature that cannot be fitted by steady state models [@Wyatt2007a]. However, the 24$\mu$m emission from such transient hot dust systems is still reproduced in the population statistics, and likely contributes (along with the warm component debris belts discussed above) to biasing the model to discs with small radii. This explains, to some extent, why the model predicts a relatively large number of bright discs in the 4-20AU size range.
The model also does not provide an accurate estimate of the number of bright ($f>5\times10^{-5}$) outer ($r>4$AU) discs in the other outlier island discussed in Section \[sec3.2.1\]. In the example shown in Figure \[fig7\] the model predicts 9 discs in this region with a median age of $\sim$0.8Gyr (with an average of 12 and mean age of 1.8Gyr over 100 model runs); this is nearly twice the number observed in DEBRIS FGK sample. However, the model discs in this population are at smaller radii than those observed, perhaps because the model had to compromise to fit the mid-IR and far-IR data simultaneously. This suggests that the observed bright outer discs might be the result of the steady-state evolution of the most massive discs in the underlying population, although their unusually high dust levels could also be the result of recent stochastic events or planet interactions.
It is perhaps interesting to note that the model predicts that there are few discs with a fractional luminosity below the detection limit of these data within the peak 7-40AU. This could imply that we have already discovered most of the discs that have radii in this range, and that most of the non-detections correspond to much smaller (collisionally depleted) discs (which is the case for the model in Figure \[fig7\]), or much larger (intrinsically faint) discs. However, any conclusions from the modelling on the properties of the discs of stars without detected emission are in part inherited from the assumptions about the underlying distributions of disc radii and masses (which are independent, and a power law and log-normal distribution, respectively). That is, no strong conclusions can be reached on the non-detections without further testing of the model predictions.
Underlying distributions {#sec5.2}
------------------------
Once the best fit parameters were identified, the model was run again. On this occasion no noise was added to the simulated data before determining the fractional excess. This provides an estimate of the cumulative fractional excess as a function of $R_{\lambda}$ for the underlying disc population (Figure \[fig8\]). The 1$\sigma$ uncertainties in the model output arising from the finite sample size are again given in Figure \[fig8\] by the grey shaded region. These distributions can be compared with those derived using an alternative method, providing further corroboration of the model. Moreover these distributions quantify some of the model predictions for future observations that probe to lower levels of $R_\lambda$ than the present data.
The alternative measurement of the underlying disc fractional excess distributions applies the method of @Wyatt2014 [hereafter W14] to the input FGK sample data. In this case, to calculate the fraction of stars in a subsample of size $N$ that have an excess level above, say $R_{\lambda}=1$, then a subset of $N^\prime$ stars within that sub-sample for which an excess could have been detected at that level are identified. The fraction of stars with an excess above that level is then the number of detections within that subset $N^\prime_{\rm{det}}$ divided by $N^\prime$. Here a 3$\sigma$ threshold was used to identify whether a detection was possible. The 1$\sigma$ uncertainties associated with this method are also shown by the diagonal line shaded region in Figure \[fig8\]. These uncertainties are again calculated for small number statistics using the tables in [@Gehrels1986]. This method can be used down to an excess whereupon $N^\prime=0$. This cutoff is typically just above three times the calibration uncertainty, and can be seen in Figure \[fig8\]. For more details see [@Wyatt2014].
![This figure shows the same best fit model output as Figure \[fig5\], but without instrumental, calibration and stellar photosphere uncertainties added (connected dotted line). The grey shaded region shows the 1$\sigma$ uncertainties of this model output, arising from the small size of the DEBRIS FGK star sampled used in this work. The blue line shows the alternative method described in W14 applied to the same data, with the associated 1$\sigma$ uncertainties shown by the diagonal filled blue shaded regions. []{data-label="fig8"}](fig8.eps){width="1.0\columnwidth"}
From the comparison of the distribution of fractional excesses in the underlying best fit model population, with the estimate of these distributions taken directly from the observations (W14), it can be seen that the two distributions agree well at all wavelengths for the range of $R_\lambda$ probed by the observations. The 100 and 160$\mu$m model data are at the upper limit of the incidence estimates derived using the W14 method, probably for the same reason that the model slightly over-predicts the fractional excess distributions at these wavelengths in Figure \[fig5\] which was discussed at the beginning of Section \[sec5.1\]. These two methods for getting the underlying disc fractional excess distributions are complementary: the W14 method provides a measure which requires no assumptions whatsoever, but provides an uneven output with large uncertainties close to the sensitivity threshold and no information below that threshold; the model approach provides a smooth distribution and can also be used to extrapolate to the disc population below the sensitivity threshold, but requires assumptions to be made about the underlying population. The model curves show how the incidence might increase as observations become sensitive to lower $R_{\lambda}$, approaching a detection rate of 100 per cent as $R_{\lambda} \rightarrow 0$ for the model assumption that all stars host a disc.
The Solar System in Context {#sec5.3}
---------------------------
The analyses and results described in Sections \[sec3\] and \[sec4\] characterise the physical properties and evolution (within the limits of the applied evolutionary model) of the solar-type stars within the DEBRIS sample. These results, however, are generally applicable to the solar-type star population, and can therefore be used to better understand this population as a whole. Also, since the Solar System is near the middle of the age distribution for the DEBRIS FGK star sample, which has a median value of 3.3Gyr and an interquartile range of 4.5Gyr, with approximately 40 per cent of stars having an age estimate $\ge$4.5Gyr [@Vican2012], it is also appropriate to consider the position of our own Kuiper belt within this sample.
For example, the extrapolation of the model population in Section \[sec5.2\] can be used to consider how the fractional excess of the present day Kuiper belt compares with those of nearby stars. For reference, the predicted fractional excess from the Kuiper belt is less than $\sim 1$per cent at wavelengths 70-160$\mu$m, peaking at $\sim50\,\mu$m (Vitense et al. 2012). Thus, the Kuiper belt is more than an order of magnitude below the threshold of detectability around nearby stars. However, it is fairly average compared with the nearby disc population with this extrapolation, since its thermal emission has fractional excess close to 50 per cent point in the distribution of Figure \[fig8\]. This makes it less extreme than the extrapolation of Greaves & Wyatt (2010), which put the Kuiper belt in the bottom 10 per cent of the distribution.
However, the Kuiper belt is thought to have followed a different evolution to that in the model of Section \[sec4.1\]. Rather than the belt mass evolving solely through steady state collisional erosion, the interaction of planetesimals from the belt with the giant planets caused those planets to migrate eventually leading to a system-wide instability that depleted the Kuiper belt [@Gomes2005] and resulted in the late heavy bombardment of the inner Solar System [@Tera1974]. Between the onset of the LHB and the present day the mass of the Solar-System’s planetesimal belt dropped by nearly three orders of magnitude [@Gladman2001; @Bernstein2004; @Levison2008], with 90 per cent of the disc mass being lost within the first 100Myr [Equation 1 of @Booth2009].
To illustrate the consequence of an LHB-like depletion, the two solid lines in Figure \[fig7\] show the predicted fractional luminosity, as a function of disc radius, for two different initial disc masses ($35M_\oplus$ and $0.035M_\oplus$) after 4.5Gyr of steady state evolution. These use the best fit model parameters from Section \[sec4\]. However, to determine the initial luminosity of the disc it was also necessary to assume a maximum planetesimal size. This was assumed to be $D_{\rm{c}}=$5000km, meaning that the median disc mass in the population is $7M_\oplus$, and results in an initial fractional luminosity for both cases, before any evolution takes place, that is shown by the dashed lines. Note that the fractional luminosities of the discs that have undergone evolution from their initial values are independent of the assumptions about the maximum planetesimal size, and depend only on the parameter $A$. Figure \[fig7\] shows how the pre-LHB Kuiper belt, which is thought to have a mass of $35M_\oplus$ and would have had black body radius of $\sim$10AU (assuming scaling factor of 2.5 as before), would have been readily detectable prior to LHB, as also noted in [@Booth2009], but not detectable following LHB depletion.
If such system-wide instabilities and disc clearing are common amongst nearby stars then their belt masses might be expected to exhibit a bimodal distribution. This possibility was not explored in the modelling, which assumed a log-normal distribution of masses for the underlying population moreover with a fixed width. Thus it is not possible to tell if the stars without detected discs are those with close-in belts that have undergone significant collisional erosion (as in the model presented in this paper), or if they are instead those that underwent LHB-like depletions.
Summary {#sec6}
=======
1. This paper has presented a study of debris discs around a sample of 275 F, G and K spectral type stars. This sample is drawn from the DEBRIS *Herschel* open time key programme and is unbaised towards any stellar property.
2. The SED of each source was modelled using a modified blackbody function. These fits were made to the DEBRIS data obtained at 100 and 160$\mu$m, as well as other ancillary data. All of the data were used in combination to determine the significance of a disc detection. A threshold of 3$\sigma$ was set for a positive detection of a debris disc.
3. A total of 47 discs were detected. The mean raw disc incidence was $17.1^{+2.6}_{-2.3}$ per cent for fractional luminosities greater than $\sim5\times10^{-6}$, and ranged from 23-13 per cent from spectral types F-K. The measured incidence is in-keeping with previous results from *Spitzer* studies. After adjusting for completeness within the probed disc parameter space the incidence becomes 27.7$^{+2.9}_{-2.9}$ per cent for the whole sample.
4. The disc incidence as a function of radius and fractional luminosity was mapped out within the spread of disc properties identified in this sample. The incidence map was adjusted for incompleteness, and showed a high concentration of debris discs at blackbody radii between 7 and 40AU, and fractional luminosities in the range $(0.4-4)\times10^{-5}$.
5. Two outlying populations of discs were also identified: hot discs with a radius smaller than 4AU, and bright discs with fractional luminosities larger than $5\times10^{-5}$. The median age of the 5 bright discs is 2.5Gyr, suggesting that these cannot be explained by youth.
6. A steady-state disc evolution model was fitted simultaneously to the MIPS 24/70, and PACS 100/160$\mu$m data for this sample. The steady-state model was found to provide a reasonable fit at all bands. A best fit model was produced with the disc radii defined by a power law ranging from 1-1000AU with an exponent of -1.7, and other model parameters constrained to have $M_{\rm mid}D_{\rm c}^{-1/2}=0.1\,M_\oplus$km$^{-1/2}$ (where $M_{\rm mid}$ is the median disc mass in the population and $D_{\rm c}$ is the maximum planetesimal size), and $D_{\rm c}^{1/2}Q_{\rm D}^{\star{5/6}}e^{-5/3}=10^4$km$^{1/2}$J$^{5/6}$kg$^{-5/6}$ (where $Q_{\rm D}^\star$ is the dispersal threshold and $e$ the mean eccentricity of the planetesimal orbits).
7. The success of the steady-state model shows that all stars could be born with a belt of planetesimals that then evolves by collisional erosion. However, it is worth noting that the model still cannot explain the hot dust systems, which likely originate in transient events. Also, the best fit parameters are affected to some extent by the assumption that all stars host just one belt, whereas we know that some stars have mid-IR emission from an additional warmer inner component.
8. Moreover, the population model is based on the $\sim20$ per cent of stars with detected discs, and thus its predictions for discs below the detection threshold are to a large extent a reflection of the assumptions made about the functional forms for the distributions of radii and masses. Nevertheless, these predictions are valuable, since the model can be readily used to predict the observable properties of the discs of populations of stars with different age, distance and spectral type distributions. Any predictions that future observations show to be incorrect can be used to refine the distributions of disc radii and masses that stars are born with in the population model.
9. The Kuiper belt was found to be a typical, albeit relatively low mass, example of a debris disc within the sample population. The typical blackbody disc radius in the sample was found to be $\sim$10AU, which translates to a true disc radius of $\sim25$AU when scaled by a factor of 2.5 to account for realistic grain optical properties, which is only slightly smaller than the nominal present day Kuiper belt radius of $\sim$40AU.
10. The fractional luminosity of the current Kuiper belt is an order of magnitude too faint to have been detected. Its far-IR flux is, however, close to the median of that of the steady state population model. The detected discs have fractional luminosities close to that of the primordial Kuiper belt. This suggests that the majority of stars either had a low planetesimal formation efficiency, or depleted their planetesimal belts in a similar manner to the Solar System (e.g., through dynamical instability in their planetary systems).
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the European Union through ERC grant number 279973 (MCW, GMK). GMK was also supported by the Royal Society as a Royal Society University Research Fellow.
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Results table {#tableA1}
=============
Confused sources {#ApB}
================
In Section 3, six sources are identified as having no disc, contradicting the existing publications. This Appendix contains the *Herschel*-PACS data for these sources and the PSF model subtracted residuals for each image. The asterisk shows the expected source location, and the plus sign shows the fitted location. All primary sources are specified by the UNS designation. Any background sources identified are also fit to provide clean photometry, these are identified by the same ID, with a X1, X2 etc. suffix, up to the number of background sources included in the fitting. The black dashed circle shows the beam size for these data. In all cases both pairs of data are shown, i.e. 70 or 100$\mu$m and the associated 160$\mu$m images.
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
\[lastpage\]
[^1]: Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
|
---
author:
- 'H. Makino$^1$, N. Minami$^2$, and S. Tasaki$^3$'
title: Statistical properties of spectral fluctuations for a quantum system with infinitely many components
---
Introduction {#sect1}
============
Study of energy level statistics has played an important role in elucidating the universal properties of quantum systems, which in the semiclassical limit, reflect regular and chaotic features of classical dynamics. Berry and Tabor conjectured that the eigenenergy levels of a quantum system, whose classical dynamical system is integrable, have the same fluctuation property as uncorrelated random numbers from a Poisson process, and thus are characterized by Poisson statistics[@BT77]. This conjecture is in contrast with the conjecture of Bohigas, Giannoni, and Schmit, which assert that GOE or GUE statistics based on random matrix theory (RMT) are applicable to the fluctuation property of energy levels of a quantum system whose classical dynamical system is fully chaotic[@BGS84]. These contrasting conjectures have been examined using various statistical observables, e.g., the nearest-neighbor level-spacing distribution (NNLSD) and the level number variance (LNV)[@ME91].\
The NNLSD $P(S)$ is the observable most commonly used to study short-range fluctuations in a spectrum. For eigenenergy levels on the unfolded scale[@Bo89], this observable is defined as the probability density of finding a distance $S$ between adjacent levels. In Poisson statistics, it is characterized by the exponential distribution $P(S)=e^{-S}$; on the other hand, in GOE/GUE statistics, it approximates the Wigner distribution very well[@ME91].\
The LNV $\Sigma^2(L)$ is the observable commonly used to study correlations between pairs of levels, which characterizes long-range fluctuations in a spectrum. It is defined as the average variance of the number of levels in an energy interval containing an average of $L$ levels. On the unfolded scale, this interval is equivalent to an interval of length $L$, and the LNV is thus defined as $$\Sigma^2(L)=\langle\left({\mathcal{N}}(\epsilon+L)-{\mathcal{N}}(\epsilon)-L\right)^2\rangle, \label{EQ.(1.1)}$$ where ${\mathcal{N}}(\epsilon)$ is the number of eigenenergy levels below $\epsilon$, and the brackets $\langle\cdots\rangle$ represent the average over the value of $\epsilon$. In Poisson statistics, the LNV is equal to the number itself $\left[\mbox{i.e., }\Sigma^2_{\mbox{\tiny Poisson}}(L)=L\right]$; on the other hand, in RMT, in which $\beta=1$ and $2$ correspond to the GOE and GUE level statistics, respectively, the LNV increases logarithmically with $L$ $\left[\mbox{i.e., } \Sigma^2_{\beta}(L)\sim(2/\beta\pi^2)\ln(2\pi L) \right]$[@ME91]. According to Berry’s semiclassical theory[@Be85], the LNV of a quantum system with $f$ degrees-of-freedom should display these universal statistics in the region $L<<L_{\mbox{\tiny max}}\sim \hbar^{-(f-1)}$ as a consequence of the uniform distributed property of periodic orbits in the phase space[@HO84]. In the semiclassical limit, in which the Planck constant tends to zero $(\hbar\to 0)$, one expects to observe the universalities for all $L\geq0$ in systems with $f \geq 2$.\
There are many works examining the Berry-Tabor conjecture in classically integrable quantum systems[@Be85; @BT76; @Mo81; @Bl90; @Sa97; @CK97; @Ma98; @Ma01; @Es; @CCG85; @Fe85; @SV86; @RV98]. Although the mechanism that supports this conjecture remains to be clarified; the statistical property of eigenenergy levels to be characterized by Poisson statistics is now widely accepted as a universal property of generic integrable quantum systems in the semiclassical limit.\
One possible mechanism producing Poisson statistics has been proposed by Makino [*[et al]{}*]{}[@MT03], based on the Berry-Robnik approach[@BR84]. We briefly review the outline below: For an integrable system, individual orbits are confined in each inherent torus whose surface is defined by holding its action variable constant, and the whole region of the phase space is densely covered with invariant tori. In other words, the phase space consists of infinitely many regions, which have infinitesimal volumes in Liouville measure. Because of the suppression of quantum tunneling in the semiclassical limit, the Wigner functions of quantal eigenstates are expected to be localized in the phase space region explored by a typical trajectory[@Be77; @Ro98], and to form independent components. For a classically integrable quantum system, the Wigner function localizes on the infinitesimal region in $\hbar\to0$ and tends to a $\delta$ function on a torus that is designated by quantum numbers[@Be1977]. Then, the eigenenergy levels can be represented as a statistically independent superposition of infinitely many components, each of which gives an infinitesimal contribution to the level statistics. Therefore, if the individual spectral components are sparse enough, one would expect Poisson statistics to apply here, as a result of the law of small numbers[@Fe57].\
The statistical independence of spectral components is assumed to be justified by the principle of uniform semiclassical condensation of eigenstates in the phase space and by the lack of their mutual overlap, and thus, can only be expected in the semiclassical limit[@Ro98; @LR94]. This mechanism was introduced as a basis for the Berry-Robnik approach to investigate the relation between the phase space geometry and the formation of energy level sequence in a generic mixed quantum system, whose classical dynamics is partly regular and partly chaotic[@BR84]. The validity of the Berry-Robnik approach is confirmed by numerical computations for the mixed quantum systems in the extremely deep semiclassical region[@PR93; @Pr96], which is called the [*[Berry-Robnik regime]{}*]{}[@PR97].\
Based on this view, Makino and Tasaki investigated the short-range spectral statistics of classically integrable systems[@MT03]. They derived the cumulative function of NNLSD, i.e., $M(S)=\int_0^S P(x) dx$, is derived in the limit of infinitely many components, which is characterized by a single monotonically increasing function $\bar{\mu}(0;S)$ of the nearest level spacing $S$ as $$M_{\bar{\mu}}(S)=1-\left[1-\bar{\mu}(0;S)\right]
\exp{\left(-\int_0^S\left[ 1-\bar{\mu}(0;x)\right]dx\right)}.
\label{EQ.(1.2)}$$ The function $\bar{\mu}(0;S)$ classifies $M_{\bar{\mu}}(S)$ into three cases: Case 1, Poisson distribution $M_{\bar{\mu}}(S)=1-e^{-S}$ for ${}^\forall S\geq 0$ if $\bar{\mu}(0;+\infty)=0$; Case 2, asymptotic Poisson distribution, which converges to the Poisson distribution for $S\to+\infty$, but possibly not for small spacings $S$ if $0<\bar{\mu}(0;+\infty)<1$; and Case 3, sub-Poisson distribution, which converges to $1$ for $S\to+\infty$ more slowly than does the Poisson distribution, if $\bar{\mu}(0;+\infty)=1$. Therefore, the Berry-Robnik approach when applied to classically integrable quantum systems, admits deviations from Poisson statistics.\
Cases 2 and 3 are possible when the eigenenergy levels of individual components show a singular NNLSD result from strong accumulation[@MT03] (see also Section [\[sect3\]]{} of the present paper), which is expected when the classical dynamical systems have a spatial symmetry[@BT77; @CK97; @Es; @SV86; @RV98; @RB81; @BW84; @Sh89; @BAL91] or a time-reversal symmetry[@Sh75; @CS95; @FS97]. One possible example is a rectangular billiard with a rational ratio of squared sides[@BT77; @CK97; @RV98; @Sh89; @BAL91]. These results suggest the existence of a new statistical law different from Poisson statistics in the strongly-degenerate quantum systems, and raises the question as to whether similar behaviors appear in the two-point spectral correlation.\
Deviation from Poisson statistics, resulting from the symmetry, is also exhibited within the framework of the periodic-orbit theory[@Gu70]. Based on the semiclassical theory of Berry and Tabor[@BT76], Biswas, Azam, and Lawande investigated the LNV for classically integrable systems with degeneracy in orbit actions[@BAL91]. They showed that the slope $g_{\mbox{\tiny{av}}}(L)$ of the LNV $\Sigma^2(L)=g_{\mbox{\tiny{av}}}(L) L$ is described by the average degeneracy of actions of the periodic orbits, which for the degenerated quantum systems, is greater than $1$, and it is the slope of the variance of Poisson statistics. This result leads to the possibility that the degeneracy of actions, induced by the symmetry, could be an essential factor for the singularity of individual spectral components that yields Cases 2 and 3. This possibility is confirmed by examining the properties of LNV for Cases 2 and 3.\
In this study, extending the theory of Makino and Tasaki[@MT03], we investigate the LNV $\Sigma^2(L)$ of quantum systems whose energy levels consist of infinitely many independent components, and show that the non-Poisson limits (Cases 2 and 3) are possibly observed also in the long-range spectral fluctuations.\
Based on the Berry-Robnik approach, the overall LNV is derived as follows: We consider a system whose classical phase space is decomposed into $N$ disjoint regions that give the distinct spectral components. The Liouville measures of these regions are denoted by $\rho_n (n=1,2,3,\cdots,N)$, which satisfy $\sum_{n=1}^N\rho_n=1$. Let $E(k;L),k=0,1,2,\cdots$ be the distribution function, which denotes the probability to find $k$ levels in an interval $(0,L)$[@ME91; @MC72; @ABS97]. The LNV $\Sigma^2(L)$ is expressed by $E(k;L)$ as $$\Sigma^2(L) = \sum_{k=0}^{+\infty}(k-L)^2 E(k;L).
\label{EQ.(1.3)}$$ Let $P(k;S),k=0,1,2,\cdots,$ be the level-spacing distribution, which denotes the probability density to find $k$ levels in an interval of length $S$ beginning at an arbitrary level $\epsilon_i$ and ending at the level $\epsilon_{i+k+1}$. $P(k;S)$ is related to $E(k;L)$ as $$P(k;L) = \frac{\partial^2}{\partial L^2}\sum_{j=0}^k (k-j+1) E(j;L),
\label{EQ.(1.5)}$$ and $$E(k;L) = \int_L^{+\infty}dx
\int_x^{+\infty}\left[ P(k;S)-2P(k-1;S)+P(k-2;S)\right]dS,
\label{EQ.(1.4)}$$ where $P(j<0;S)=0$[@ME91; @MC72; @ABS97]. Eqs.(\[EQ.(1.5)\]) and (\[EQ.(1.4)\]) are known as the formulae in the theory of point process, and are derived as corollaries of the Palm-Khinchin formula[@MI07]. The NNLSD, which was expressed by $P(S)$ at the beginning of this section, is the special case with $k=0$.\
When the entire sequence of energy levels is a product of statistically independent superposition of $N$ subsequences, $E(k;L)$ is decomposed into those of subsequences, $e_n(k;L)$ as[@ME91; @Po60] $$E_N(k;L)=\sum_{\sum_{n=1}^N k_n=k}\prod_{n=1}^N e_n(k_n;\rho_n L),
\label{EQ.(1.6)}$$ where $e_n$ satisfies the normalization conditions $$\sum_{k=0}^{+\infty}e_n(k;\rho_n L)=1
\label{EQ.(1.7)}$$ and $$\sum_{k=0}^{+\infty}k e_n(k;\rho_n L)=\rho_n L.
\label{EQ.(1.8)}$$ In terms of normalized level-spacing distribution $p_n(k;S)$ of the subsequence, $e_n(k;L)$ is specified as $$e_n(k;L)=\rho_n\int_L^{+\infty}dx\int_x^{+\infty}\left[ p_n(k;S)-2p_n(k-1;S)+p_n(k-2;S) \right]dS,
\label{EQ.(1.9)}$$ where $p_n(j<0;S)=0$, and $p_n(k;S)$ satisfies the normalizations $$\int_0^{+\infty}p_n(k;S)dS=1
\label{EQ.(1.10)}$$ and $$\int_0^{+\infty}S p_n(k;S)dS=\frac{k+1}{\rho_n}.
\label{EQ.(1.11)}$$ Note that in general, the individual components are not always unfolded automatically even when the overall spectrum is unfolded. However, in a sufficiently small interval $[\epsilon,\epsilon+\Delta\epsilon ]$, each spectral component obeys the same scaling law (see Appendix of Makino [*[et al]{}*]{}[@MT03]) and thus is unfolded automatically by an overall unfolding procedure. In the Berry-Robnik approach, Eq.(\[EQ.(1.6)\]) relates the level statistics in the semiclassical limit with the phase space geometry.\
In most general cases, the individual components might have degeneracy of levels that lead to singular level spacing distributions. In such a case, it is convenient to use its cumulative distribution function $\mu_n(k;S)$, $$\mu_n(k;S)=\int_0^S p_n(k;x)dx.
\label{EQ.(1.12)}$$ This function satisfies $$\mu_n(k-1;S)\geq \mu_n(k;S)\quad\mbox{for all}\quad k\geq 1\ \mbox{and for all}\quad S\geq0,
\label{EQ.(1.13)}$$ and the normalization condition $$\sum_{k=0}^{+\infty}\left[\mu_n(k-1;S)-\mu_n(k;S)\right] = 0,
\label{EQ.(1.14)}$$ with $\mu_n(j=-1;S)=0$, since $f_n(k;S)$, $$f_n(k;S)= \left\{
\begin{array}{lc}
\mu_n(k-1;S)-\mu_n(k;S),&\qquad k\geq1\nonumber\\
1-\mu_n(0;S),&\qquad k=0\nonumber
\end{array}\right.\label{EQ.(1.15)}$$ denotes the probability to find $k$ levels in an interval of length $S$ beginning at an arbitrary level, and obviously satisfies, from the definition of this function, the conditions $f_n(k;S)\geq0$ and $\sum_{k=0}^{+\infty}f_n(k;S)=1$.\
In addition to Eq.(\[EQ.(1.6)\]), we introduce the following two assumptions that were introduced in Ref.[@MT03]:\
(i). The statistical weights of independent regions vanish uniformly in the limit of infinitely many regions: $$\max_n \rho_n \rightarrow 0\quad \mbox{as}\quad N\rightarrow +\infty.
\label{EQ.(1.16)}$$ (ii). For $k=0,1,2,\cdots$, the weighted mean of the cumulative distribution of energy spacing, namely, $$\mu(k;x)=\sum_{n=1}^N \rho_n\mu_n(k;x)
\label{EQ.(1.17)}$$ converges as $N\to +\infty$ to $\bar{\mu}(k;x)$, $$\lim_{N\to +\infty}\mu(k;x) = \bar{\mu}(k;x),
\label{EQ.(1.18)}$$ where the convergence is uniform on each closed interval: $0\leq x \leq S$. In the Berry-Robnik approach, the statistical weights of individual components are the phase volumes (Liouville measures) of the corresponding invariant regions.\
Under the Assumptions (i) and (ii), Eqs.(\[EQ.(1.3)\]) and (\[EQ.(1.6)\]), we obtain the overall LNV in the limit $N\to +\infty$: $$\Sigma_{\bar{\mu}}^2(L)= L +2 \int_0^L \sum_{k=0}^{+\infty} \bar{\mu}(k;S)dS.
\label{EQ.(1.19)}$$ When $\bar{\mu}(k;S)=0$ for all $k$, the LNV of the whole energy sequence reduces to $\Sigma^2_{\mbox{\tiny Poisson}}(L)=L$. This condition is expected when the individual components are sufficiently sparse. In general, we expect $\bar{\mu}(k;S) > 0$, which corresponds to a certain accumulation of the levels of individual components. In this case, the LNV of the whole energy sequence deviates from $\Sigma^2_{\mbox{\tiny Poisson}}(L)$ in such a way that the slope of the variance is greater than $1$.\
The present paper is organized as follows. The limiting LNV (\[EQ.(1.19)\]) is derived from Eqs.(\[EQ.(1.3)\]) and (\[EQ.(1.6)\]), and Assumptions (i) and (ii) in Section \[sect2\]. In Section \[sect3\], the property of the limiting LNV is analyzed for the Cases 1–3, wherein the deviations from Poisson statistics are clearly observed in Cases 2 and 3. We present a numerical analysis for the rectangular billiard that shows deviations from Poisson statistics in Section \[sect4\]. In the concluding section, we discuss some relationships between our results and other related works.
limiting level number variance {#sect2}
==============================
In this section, by using Eqs.(\[EQ.(1.6)\]) and (\[EQ.(1.3)\]), and Assumptions (i) and (ii) introduced in Section\[sect1\], we derive the limiting LNV, $$\Sigma^2_{\bar{\mu}}(L)=L+2\int_0^L \sum_{k=0}^{+\infty}\bar{\mu}(k;S)dS,
\label{EQ.(2.1)}$$ in the limit of infinitely many components $N\to+\infty$.\
First, we rewrite Eq.(\[EQ.(1.3)\]) in terms of the function $e_n$, and decompose it into the LNV $\sigma_n^2$ of individual components: $$\begin{aligned}
\Sigma_N^2(L)&=&\sum_{k=0}^{+\infty}(k-L)^2 \sum_{\sum_{n=1}^N k_n=k}\prod_{n=1}^N e_n(k_n;\rho_n L)\label{EQ.(2.2)}\\
&=& \sum_{n=1}^N \sum_{k=0}^{+\infty}(k-\rho_n L)^2 e_n(k;\rho_n L) \equiv \sum_{n=1}^N \sigma_n^2
(\rho_n L),\label{EQ.(2.3)}\end{aligned}$$ where we have used the properties $\sum_{k=0}^{+\infty}E_N(k;L)=1$, $\sum_{k=0}^{+\infty}k E_N (k;L)=L$, and $$\sum_{k=0}^{+\infty}k^2 E_N(k;L)= \sum_{n=1}^N \sum_{k_n=0}^{+\infty}k_n^2 e_n(k_n;\rho_n L) + L^2-\sum_{n=1}^N(\rho_n L)^2,
\label{EQ.(2.4)}$$ which follow from Eqs.(\[EQ.(1.7)\]) and (\[EQ.(1.8)\]), and the relationships $k=\sum_{n=1}^N k_n$ and $\sum_{n=1}^N\rho_n =1$ (see also Appendix A). The functions $e_n(k;\rho_n L)$ in the above equation are rewritten in terms of the cumulative level-spacing distribution functions $\mu_n(k;S)$ of individual components. $$\begin{aligned}
e_n(k;\rho_n L)
&=&\left\{
\begin{array}{lc}
-\rho_n \int_L^{+\infty} dS \left[\mu_n(k;S)-2\mu_n(k-1;S)+\mu_n(k-2;S)\right],&\qquad k\geq 2 \\
-\rho_n \int_L^{+\infty} dS \left[\mu_n(1;S)-2\mu_n(0;S)+1\right],&\qquad k=1\\
\rho_n \int_L^{+\infty} dS \left[1-\mu_n(0;S) \right],&\qquad k=0
\end{array}\right.\label{EQ.(2.5)}\\
&=&\left\{
\begin{array}{lc}
\rho_n \int_0^L dS \left[\mu_n(k;S)-2\mu_n(k-1;S)+\mu_n(k-2;S)\right],&\qquad k\geq 2 \\
\rho_n \int_0^L dS \left[\mu_n(1;S)-2\mu_n(0;S)+1 \right],&\qquad k=1 \\
1 - \rho_n\int_0^L dS \left[1-\mu_n(1;S) \right],&\qquad k=0
\end{array}\right.
\label{EQ.(2.6)}\end{aligned}$$ where Eq.(\[EQ.(2.6)\]) follows from Eq.(\[EQ.(1.11)\]), integration by parts and the following limits which result from the existence of the average: $$\begin{aligned}
&&\lim_{S\to+\infty}S\left[\mu_n(k;S)-2\mu(k-1;S)+\mu_n(k-2;S)\right]=0,\qquad k\geq2\nonumber\\
&&\lim_{S\to+\infty}S\left[\mu_n(1;S)-2\mu(0;S)+1\right]=0,\qquad k=1\label{EQ.(2.7)}\\
&&\lim_{S\to+\infty}S\left[1-\mu_n(0;S)\right]=0,\qquad k=0\nonumber\end{aligned}$$ Then, $\sigma_n^2(\rho_n L)$ is described in terms of $\mu_n(k;S)$ as $$\begin{aligned}
\sigma_n^2(\rho_n L)&=&\sum_{k=0}^{+\infty} k^2 e_n(k;\rho_n L)-\rho_n^2 L^2\nonumber\\
&=&\rho_n\int_0^L dS \sum_{k=1}^{+\infty}(2k+1)\left[ \mu_n(k-1;S)-\mu_n(k;S) \right]
+\rho_n\int_0^L\left[1-\mu_n(0;S)\right]dS -\rho_n^2 L^2\nonumber\\
&=&\rho_n L + 2\int_0^L dS \sum_{k=0}^{+\infty}\rho_n\mu_n(k;S) -\rho_n^2 L^2,
\label{EQ.(2.8)}\end{aligned}$$ where we have used Eq.(\[EQ.(1.14)\]) and the relation $$\sum_{k=0}^{+\infty}\left[(k-1)\mu_n(k-1;S)-k\mu_n(k;S)\right] = 0
\label{EQ.(2.9)}$$ with $\mu_n(-1;S)=0$. Therefore, in the limit $N\to+\infty$, we have the convergence $$\Sigma_{\bar{\mu}}^2(L) =
\lim_{N\to+\infty} \sum_{n=1}^N \sigma_n^2(\rho_n L) =
L + 2\int_0^L dS \sum_{k=0}^{+\infty}\bar{\mu}(k;S),
\label{EQ.(2.10)}$$ where the limit (\[EQ.(2.10)\]) follows from Assumption (ii) and the following property results from Assumption (i): $$\sum_{n=1}^N\rho_n^2 \leq \max_n \rho_n
\sum_{n=1}^N\rho_n =\max_n\rho_n\to 0.
\label{EQ.(2.11)}$$ We note the spectral rigidity $\Delta_3(L)$, which was introduced by Dyson and Mehta[@DM63]. In combination with the LNV, this quantity has played a major role in the study of the long-range spectral statistics. According to Pandey[@Pa79] and Eq.(\[EQ.(2.10)\]), the spectral rigidity is described in terms of $\bar{\mu}(k;S)$ as $$\begin{aligned}
\Delta_{3,\bar{\mu}}(L)
&=&\frac{2}{L^4}\int_0^L dx(L^3-2 L^2 x +x^3)\Sigma^2_{\bar{\mu}}(x)\nonumber\\
&=&\frac{L}{15}+\frac{4}{L^4}\int_0^L dx(L^3-2L^2x+x^3)\int_0^x dS \sum_{k=0}^{+\infty
}\bar{\mu} (k;S)
\label{EQ.(2.12)}.\end{aligned}$$
Properties of limiting level number variance {#sect3}
============================================
We analyze the slope of the limiting LNV $$g(L) = \frac{1}{L} \Sigma^2_{\bar{\mu}}(L)
= 1 + \frac{2}{L}\int_0^L dS \sum_{k=0}^{+\infty}\bar{\mu}(k;S),
\label{EQ.(3.2)}$$ which has the following convergence: $$\lim_{L\to+\infty}g(L)= 1 + 2\sum_{k=0}^{+\infty}\bar{\mu}(k;+\infty)\geq 1+2\bar{\mu}(0;+\infty).$$ Since $\mu_n(k;S)$ is monotonically increasing for $S\geq0$, monotonically decreasing for $k=0,1,2,\cdots$, and $0\leq\mu_n(k;S)\leq1$, $\bar{\mu}(k;S)$ has the same properties and is bounded by $\bar{\mu}(0;+\infty)$ as $$0\leq \bar{\mu}(k;S)\leq \bar{\mu}(0;S)\leq \bar{\mu}(0;+\infty) \leq 1,
\label{EQ.(3.1)}$$ where $\bar{\mu}(0;+\infty)$ classifies the cumulative NNLSD (\[EQ.(1.2)\]) into three cases: Case 1, the Poisson distribution if $\bar{\mu}(0;+\infty)=0$; Case 2, the asymptotic Poisson distribution if $0<\bar{\mu}(0;+\infty)<1$; and Case 3, the sub-Poisson distribution if $\bar{\mu}(0;+\infty)=1$.\
Then, the property of the limiting LNV is evaluated for Cases 1—3 as follows:\
Case 1, $\bar{\mu}(0;+\infty)=0$: The limiting LNV $\Sigma_{\bar{\mu}}^2(L)$ agrees with the LNV of Poisson statistics: $\Sigma_{\mbox{\tiny Poisson}}^2(L)=L$. Note that this condition is equivalent to $\sum_{k=0}^{+\infty}\bar{\mu}(k;S)=0$ since $\bar{\mu}(k;S)$ is monotonically increasing for $S$ and decreasing for $k$.\
Case 2, $0<\bar{\mu}(0;+\infty)<1$: $\Sigma_{\bar{\mu}}^2(L)$ possibly deviates from $\Sigma_{\mbox{\tiny Poisson}}^2(L)$ in such a way that the slope of the limiting LNV is $1$ at $L=0$, increases monotonically with $L$, and approaches a number $1+2\bar{\mu}(0;+\infty)$ or more as $L\to+\infty$.\
Case 3, $\bar{\mu}(0;+\infty)=1$: $\Sigma_{\bar{\mu}}^2(L)$ possibly deviates from $\Sigma_{\mbox{\tiny Poisson}}^2(L)$ in such a way that the slope of the limiting LNV is $1$ at $L=0$, increases monotonically with $L$, and approaches a number $3$ or more as $L\to+\infty$.\
One has Case 1 if the NNLSD of individual components are derived from the scaled distribution functions $\varphi_n(0;S)$ as $$\mu_n(0;S)=\rho_n\int_0^S \varphi_n(0;\rho_n x)dx,
\label{EQ.(3.5)}$$ where $\varphi_n(0;\rho_n S )=p_n(0;S)/\rho_n$ and satisfy $$\int_0^{+\infty}\varphi_n(0;x)dx=1,\quad\int_0^{+\infty}x \varphi_n(0;x)dx= 1,
\label{EQ.(3.6)}$$ and are uniformly bounded by a positive constant $D$ : $\left| \varphi_n(0;S)\right|\leq D$ ( $1\leq n\leq N$ ). Indeed, the following holds: $$\left| \mu(0;S)\right|\leq \sum_{n=1}^N \rho_n^2\int_0^S\left|
\varphi_n(0;\rho_n x)\right| dx
\leq DS \sum_{n=1}^N \rho_n^2\leq DS\max_{n}\rho_n
\sum_{n=1}^N\rho_n\to 0\equiv\bar{\mu}(0;S).
\label{EQ.(3.7)}$$ Such a bounded condition is possible when the individual spectral components are sparse enough.\
In general, one may expect Cases 2 or 3 with $\bar{\mu}(0; S)>0$, which corresponds to strong accumulation of energy levels, leading to a singular NNLSD of the individual components. Such accumulation is expected to arise from the symmetry of the system. In the next section, we will analyze the LNV of the rectangular billiard systems which is known to deviate from Poisson statistics.
Rectangular billiard system {#sect4}
===========================
We present our results on the various statistical measures discussed in the previous section for a rectangular billiard system whose spectral statistics has been precisely analyzed in a number of works[@BT77; @Be85; @CK97; @CCG85; @RV98; @MT03; @Sh89; @BAL91]. The eigenenergy levels of this system are given by $$\epsilon_{n,m}=n^2+\alpha m^2,\label{EQ.(4.1)}$$ where $n$ and $m$ are positive integers, and $\alpha$ is denoted by the lengths of two sides $a$ and $b$ as $\alpha=a^2/b^2$. The unfolding transformation $\{\epsilon_{n,m}\} \to \{\bar{\epsilon} _{n,m}\}$ is carried out by using the leading Weyl term of the integrated density of states, $\mathcal{N}(\epsilon)$, as $$\bar{\epsilon}_{n,m}={\mathcal{N}}(\epsilon_{n,m})
=\frac{\pi}{4\sqrt{\mathstrut\alpha}}\epsilon_{n,m}.\label{EQ.(4.2)}$$ Berry and Tabor observed that the NNLSD of this system possibly deviates from Poisson statistics when $\alpha$ is rational[@BT77]. In this paper, we study irrational cases in addition to a rational case($\alpha=1$) that are described by a finite continued fraction of the golden mean $(\sqrt{\mathstrut 5}+1)/2$, $$\alpha = 1+\frac{1}{1+}\frac{1}{1+}\cdots
\frac{1}{1+}\frac{1}{1+\delta}=\left[1;1,1,\cdots,1,1+\delta\right],
\label{EQ.(4.3)}$$ with an irrational truncation parameter $\delta\in [0,1)$.\
Fig.1 shows the plots of the LNV $\Sigma^2(L)$ for $\alpha$ corresponding to the (a) 25th, (b) 8th, and (c) 4th approximations of the golden mean, and (d) $\alpha=1$. Our analysis is valid in the region $L << L_{\mbox{\tiny max}}$, where $L_{\mbox{\tiny max}}=\sqrt{\mathstrut\pi\bar{\epsilon}_{nm}}\alpha^{-1/4}$ for the rectangular billiard[@Be85; @CCG85]. We used energy levels $\bar{\epsilon}_{n,m}\in [4000\times10^7,4001\times10^7]$, which correspond to $L_{\mbox{\tiny max}}\sim 3.1\times 10^4$. The numerical computation was carried out using a double precision real number operation. When the continued fraction is close to the golden mean, $\Sigma^2(L)$ is well approximated by $\Sigma^2_{\mbox{\tiny Poisson}}(L)=L$ \[plot (a)\]. On the other hand, in cases in which the continued fractions are far from the golden mean, $\Sigma^2(L)$ clearly deviates from $\Sigma^2_{\mbox{\tiny Poisson}}(L)$ \[plots (b) – (d)\].\
Fig.2 shows the slope $g(L)$ of the LNV for the four values of $\alpha$ corresponding to the plots (a)–(d) in Fig.1. When $\Sigma^2(L)$ is well approximated by $\Sigma^2_{\mbox{\tiny Poisson}} (L)$, the slope $g(L)$ is $1$ \[plot (a)\]. Since $\sum_{k=0}^{+\infty}\bar{\mu}(k;S)=0$ is equivalent to $\bar{\mu}(0;S)=0$, this result corresponds to Case 1 given in the previous section(see also Eq.(\[EQ.(3.2)\])). In the case in which $\Sigma^2(L)$ deviates from $\Sigma^2_{\mbox{\tiny Poisson}}(L)$, the slope $g(L)$ is $1$ at $L=0$ and increases monotonically with $L$ \[plots (b) – (d)\]. In the limit of $L\to+\infty$, plot (b) approaches a number less than $3$ and this result clearly corresponds to Case 2, while plots (c) and (d) approach numbers greater than $3$ and these results correspond to Case 2 or 3.\
In order to clarify the attribute of plots (c) and (d), we consider $$\tilde{\mu}(S) = 1 - \frac{1-M(S)}{1-\int_0^S (1-M(x))dx}.\label{EQ.(4.4)}$$ that is obtained by the cumulative NNLSD $M(S)=\int_0^S P(0;x)dx$. The function $\tilde{\mu}(S)$ is equivalent to $\bar{\mu}(0;S)$ in the semiclassical limit $\epsilon\to+\infty$.\
Figs.3(a)–3(d) show $-\ln{[1-M(S)]}$ for the four values of $\alpha$ corresponding to the plots (a)–(d) in Figs.1 and 2, respectively. The dotted line in each figure corresponds to the cumulative Poisson distribution $M_{\mbox{\tiny Poisson}}(S)=1-\exp{(-S)}$. Plots (a)–(d) in Fig.4 show $\tilde{\mu}(S)$ for the four values of $\alpha$ corresponding to Figs.3(a)–3(d), respectively. When $\Sigma^2(L)$ approximates $\Sigma^2_{\mbox{\tiny Poisson}}(L)$, $M(S)$ fits $M_{\mbox{\tiny Poisson}}(S)$ very well\[Fig.3(a)\]. In this case, $\tilde{\mu}(S)$ is obviously $0$ \[plot (a) in Fig.4\]. In case that $\Sigma^2(L)$ deviates from $\Sigma^2_{\mbox{\tiny Poisson}}(L)$, $M(S)$ for small value of $S$ clearly deviates from $M_{\mbox{\tiny Poisson}}(S)$\[Figs.3(b)–3(d)\]. However, for large value of $S$, it approaches a line whose slope is $1$(see the dashed line in Figs. 3(b)–3(d)). In these cases, $\tilde{\mu}(S)$ in $S\to+\infty$ approaches a number $\tilde{\mu}(+\infty)$ such that $0<\tilde{\mu}(+\infty)<1$ \[plots (b) – (d) in Fig.4\]. Therefore, plots (b)–(d) correspond to Case 2.\
For the rectangular billiard in the finite energy region, we have not yet succeeded in observing Case 3. This case is expected to arise in a square billiard ($\alpha=1$) in the high energy limit where a stronger accumulation of levels is generated. Based on the number-theoretical result of Landau (1908)[@Landau1908], Connors and Keating have proved that the eigenenergy levels $\epsilon_{n,m}$ of square billiard show a logarithmic increase in the mean degeneracy of levels as $\epsilon\to+\infty$, which is described as $$1-M(+0) \simeq \frac{4}{\pi}\frac{C_2}{\sqrt{\mathstrut\ln \epsilon}}\rightarrow 0,
\label{EQ.(4.5)}$$ where $C_2$ converges to give $C_2\simeq 0.764$[@CK97]. The above limit corresponds to a delta function of NNLSD, $P(S)=\delta(S)$, and is consistent with $\bar{\mu}(0;S=+0)=\lim_{\epsilon \to +\infty}M(+0)= 1$, which indicates an extremely slow approach to Case 3 in $\epsilon\to+\infty$.\
Fig.5 shows the cumulative NNLSD $M(S)$ for $\alpha=1$. This function is not smooth at the level spacings separated by a step $\pi/4$[@BT77], which correspond to accumulation of levels. Note that $M(+0)>0$ due to the degeneracy at $S=0$. Since $\tilde{\mu}(+0)=M(+0)>0$, this degeneracy at $S=0$ is identified also in Fig.4. As the eigenenergy levels become higher, $M(+0)$ increases monotonically and approaches $1$. In the limit $\epsilon \to +\infty$ where $M(+0)=\bar{\mu}(0,S=+0)=1$, all steps except the step at $S=0$ are suppressed since $M(S)$ is monotonically increasing for $S>0$.\
Fig.6 shows $1-M(+0)$ vs $4C_2/\pi\sqrt{\mathstrut\ln \epsilon}$ for various energy ranges. Although we are not yet far enough in the high energy region where $1-M(+0)=1-\tilde{\mu}(+0)<<1$, the agreement between them is very good, and is better as $\epsilon \to +\infty$. Therefore, the extremely slow convergence to Case 3 is well reproduced by a numerical computation. The almost same results for $\alpha=22/21$ have already been reported by Robnik and Veble[@RV98].
Conclusion and Discussion {#sect5}
=========================
Based on the approach of Berry and Robnik, we have investigated the energy level statistics of classically integrable quantum system and discussed its deviations from Poisson statistics. In the Berry-Robnik approach, individual eigenstates localizing on the different phase space regions provide mutually independent contributions to the statistics of energy levels in the semiclassical limit. Since the phase space of integrable system is densely covered with invariant tori, the eigenfunctions of the classically integrable quantum system are localized on the regions that have infinitesimal volumes in Liouville measure. Therefore, we have considered the situation in which the eigenenergy sequence is a superposition of infinitely many independent components, and each of which gives an infinitesimal contribution to the level statistics. Moreover, by developing the approach of Makino [*[et al]{}*]{}[@MT03] into the statistics of higher order (long-range) spectral fluctuations, the LNV of systems consisting of infinitely many spectral components are obtained. The LNV is characterized by the monotonically increasing functions $\bar{\mu}(k;S),k=0,1,2,\cdots$, of the level-spacing $S$, where the lowest order term $\bar{\mu}(0;S)$ is associated with the NNLSD. The property of the LNV is classified as follows: Case 1, $\bar{\mu}(0;+\infty)=0$ where the cumulative NNLSD is the Poisson distribution, the LNV is the Poissonian $\Sigma^2_{\mbox{\tiny Poisson}}(L)=L$; Case 2, $0<\bar{\mu}(0;+\infty)<1$ where the cumulative NNLSD is the asymptotic Poisson distribution, the LNV deviates from $\Sigma^2_{\mbox{\tiny Poisson}}(L)$ in such a way that the slope is greater than $1$ and approaches a number $\geq 1+2\bar{\mu}(0;+\infty)$ as $L\to+\infty$; Case 3, $\bar{\mu}(0;+\infty)=1$ where the cumulative NNLSD is the sub-Poisson distribution, the LNV deviates from $\Sigma^2_{\mbox{\tiny Poisson}}(L)$ in such a way that the slope is greater than $1$ and approaches a number $\geq 3$ as $L\to+\infty$. Therefore, we have shown that deviations from Poisson statistics (Cases 2 and 3) are possibly observed, not only in the property of NNLSD characterizing short-range spectral fluctuation as shown in the work of Makino [*[et al]{}*]{}[@MT03], but also in the property of the LNV characterizing the fluctuations of all ranges.\
Note that Cases 2 and 3 may arise when there is a strong accumulation of levels, which is characterized by the singular NNLSD of individual components. Such accumulation would be expected when there is a spatial symmetry or time-reversal symmetry. One example is a rectangular billiard shown in Section \[sect4\] where the result shows Case 2 in addition to Case 1, and an extremely slow approach to Case 3 . Similar results due to a spatial symmetry are also shown in the equilateral-triangular billiard[@BW84; @Ag08], torus billiard[@RB81; @RV98], and integrable Morse oscillator[@Sh89]. Another example is a certain type of systems with the time-reversal symmetry studied by Shnirelman[@Sh75], Chirikov and Shepelyansky[@CS95], and Frahm and Shepelyansky[@FS97]. In this system, the strong accumulation of energy levels, resulting from the time reversibility, is reflected in the NNLSD as a sharp Shnirelman peak at small level-spacings.\
Rigorous results confirming the Berry–Tabor conjecture (Case 1) are also reported for a certain classically integrable system. In the work of Marklof[@Ma01] and Eskin [*[et al]{}*]{}[@Es], eigenvalue problems are reformulated as lattice point problems, and it is exactly proved under explicit diophantine conditions that a two-point spectral correlation exhibits Poisson statistics.\
The periodic-orbit theory, applied for an integrable system with degeneracy in orbit actions, gives a similar result. In the work of Biswas [*[et al]{}*]{}[@BAL91], it was shown that the slope $g_{\mbox{\tiny{av}}}(L)$ of the LNV $\Sigma^2(L)=g_{\mbox{\tiny{av}}}(L) L$ represents the average degeneracy of actions of periodic orbits, which is greater than $1$ for the rectangular billiard with a rational ratio $\alpha$ of squared sides. Since this property is qualitatively consistent with the property of the limiting LNV (\[EQ.(2.10)\]) in Cases 2 and 3, one might expect to have the relation: $$g_{\mbox{\tiny{av}}}(L) = 1 + \frac{2}{L}\int_0^L \sum_{k=0}^{+\infty}\bar{\mu}(k;S)dS,
\label{EQ(5.1)}$$ which enables us to discuss the properties of the overall NNLSD from the periodic-orbit theory. From Eqs.(\[EQ(5.1)\]) and (\[EQ.(3.1)\]), we expect that the Case 1 corresponds to the non-degeneracy of actions: $g_{\mbox{\tiny{av}}}(L)=1$, while Cases 2 and 3 correspond to the degeneracy of actions:$g_{\mbox{\tiny{av}}}(L)>1$. Therefore, we have confirmed that Cases 2 and 3 are closely related to the degeneracy in orbit actions. At the same time, we have only superficially examined the possibility that the degeneracy of actions, induced by the symmetry, could be an essential factor for the singularity of individual spectral components that yields Cases 2 and 3. This part should be investigated in detail in a future work.\
There is another approach similar to the one presented in this paper, in which the LNV is derived in the limit of infinitely many components[@MT05]. The LNV is described by the Dyson two-level cluster function $Y_2$ as $$\Sigma^2(L) = L -2\int_0^L (L-S) Y_2(S)dS.$$ For spectral superposition, $Y_2$ is described in terms of the cluster function of individual components $y_{2,n}$ as in Pandey’s work[@Pa79] $$Y_2(S)=\sum_{n=1}^N \rho_n^2 y_{2,n}(\rho_n S).$$ Then, for the overall LNV, one has the convergence $$\begin{aligned}
\lim_{N\to+\infty}\Sigma^2(L)= L + 2\int_0^L \bar{c}(S)dS
\label{limit5-1}\end{aligned}$$ with $$\bar{c}(S)=- \lim_{N\to+\infty}\sum_{n=0}^N \rho_n \int_0^{\rho_n S} y_{2,n}(x) dx,$$ where $\bar{c}(S)=0$ corresponds to Poisson statistics and $\bar{c}(S)\not=0$ indicates deviations from Poisson statistics. Since $y_{2,n}$ is associated with the level-spacing distribution of individual components as $y_{2,n}(x)=1- \sum_{k=0}^{+\infty}p_n(k;x)$, $\bar{c}(S)$ is rewritten as $$\bar{c}(S)= \lim_{N\to+\infty}\sum_{n=0}^N \rho_n \int_0^{\rho_n S}\sum_{k=0}^{+\infty}p_n(k;x)dx.$$ When $p_n(k;S), k=0,1,2,\cdots$, are assumed to satisfy the commutation relation $\int_0^{\rho_n x}\sum_{k=0}^{+\infty}p_n(k;S)dS=\sum_{k=0}^{+\infty}\int_0^{\rho_n x}p_n(k;S)dS$, $\bar{c}(S)$ is described as $\bar{c}(S)=\sum_{k=0}^{+\infty}\bar{\mu}(k;S)$ and limit (\[limit5-1\]) is consistent with the limit (\[EQ.(2.10)\]).\
\
Acknowledgment\
H.M. is grateful to Prof. M. Robnik, Prof. A. Shudo, and Prof. Y. Aizawa for helpful suggestions. This work is partially supported by KAKENHI(No.18740241 to H.M., No.17540100 to N.M., No.17540365 to S.T.), and by a Grant-in-Aid for the “Academic Frontier” Project at Waseda University from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
Derivation of Eqs.(\[EQ.(2.3)\]) and (\[EQ.(2.4)\]) {#appendix}
===================================================
We briefly show the detailed process to derive Eqs.(\[EQ.(2.3)\]) and (\[EQ.(2.4)\]) from Eqs.(\[EQ.(1.5)\]), (\[EQ.(1.7)\]), and (\[EQ.(1.8)\]). First, we rewrite Eq.(\[EQ.(1.3)\]) as $$\sum_{k=0}^{+\infty}(k-L)^2E_N(k;L)=\sum_{k=0}^{+\infty}k^2 E_N(k;L) -L^2,
\label{(A1)}$$ where we have used the normalization conditions $\sum_{k=0}^{+\infty}E_N(k;L)=1$ and $\sum_{k=0}^{+\infty}k E_N(k;L)=L$, which follow from Eqs.(\[EQ.(1.7)\]) and (\[EQ.(1.8)\]), and the relations $k=\sum_{n=1}^N k_n$ and $\sum_{n=1}^N\rho_n =1$ as $$\sum_{k=0}^{+\infty}E_N(k;L)=\prod_{n=1}^N \sum_{k_n=0}^{+\infty}e_n(k_n;\rho_n L)=1,
\label{(A2)}$$ $$\begin{aligned}
\sum_{k=0}^{+\infty}kE_N(k;L)
&=&\sum_{n=1}^N\sum_{k_n=0}^{+\infty}k_n e_n(k_n;\rho_n L)\prod_{i=1,i\not=n}^N\sum_{k_i=0}^{+\infty}e_i(k_i;\rho_iL)\nonumber\\
&=&\sum_{n=1}^N\rho_n L\cdot 1^{N-1}=L.
\label{(A3)}\end{aligned}$$ By using Eqs.(\[(A2)\]) and (\[(A3)\]), $\sum_{k=0}^{+\infty}k^2 E_N(k;L)$ in the right hand side of Eq.(\[(A1)\]) is calculated in a similar manner: $$\begin{aligned}
&& \sum_{k=0}^{+\infty}k^2 E_N(k;L) \nonumber\\
&=&\sum_{n=1}^N \sum_{\sum_{i=1,i\not=n}^N k_i=0}^{+\infty}\left(\prod_{i=1,i\not=n}^N e_i
(k_i;\rho_i L)\right)
\sum_{k_n=0}^{+\infty}k_n^2e_n(k_n;\rho_n L)\nonumber\\
&&+\sum_{n\not=i}^N\sum_{k_n=0}^{+\infty}\sum_{k_i=0}^{+\infty}k_n e_n(k_n;\rho_n L)k_i e_i
(k_i;\rho_i L)
\nonumber\\
&=&\sum_{n=1}^N\left(\sum_{k_n=0}^{+\infty}k_n^2 e_n(k_n;\rho_n L)\right)\left(\sum_{k_i=0}^
{+\infty}e_{i\not=n}
(k_i;\rho_i L)\right)^{N-1}\label{(2.4)}\\
&&+ \left( \sum_{n=1}^N\sum_{k_n=0}^{+\infty}k_n e_n(k_n;\rho_n L)\right)^2 -\sum_{n=1}^N\left
(\sum_{k_n=0}^
{+\infty}k_n e_n(k_n;\rho_n L) \right)^2\nonumber\\
&=&\sum_{n=1}^N \sum_{k_n=0}^{+\infty}k_n^2 e_n(k_n;\rho_n L) + L^2-\sum_{n=1}^N(\rho_n L)^2.
\label{(A5)}\end{aligned}$$ Since the off-diagonal terms $n\not=i$ vanish in the second equality, we have $$\begin{aligned}
\sum_{k=0}^{+\infty}k^2E_N(k;L)-L^2&=&\sum_{n=1}^N\left(\sum_{k_n=0}^{+\infty}k_n^2 e_n(k_n;\rho_n L)-(\rho_n L)^2\right)\nonumber\\
&=&\sum_{n=1}^N\sum_{k_n=0}^{+\infty}(k_n-\rho_n L)^2 e_n(k_n;\rho_n L).
\label{(A6)}\end{aligned}$$ Therefore, the overall LNV of the system with $N$ independent components is described as $$\Sigma_N^2(L)=\sum_{n=1}^N \sigma^2_n(\rho_n L),
\label{(A7)}$$ where $\sigma^2_n$ is the LNV of the spectral component: $$\sigma^2_n(\rho_n L)
=\sum_{k=0}^{+\infty}(k-\rho_n L)^2 e_n(k;\rho_n L).
\label{(A8)}$$
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{width="8"}
FIG.1 Level number variance $\Sigma^2(L)$ of the rectangular billiard systems for (a) $25$th, (b) $8$th, (c) $4$th approximations of $\alpha=(\sqrt{\mathstrut 5}+1)/2$, and (d) $\alpha=1$. The solid line denotes the LNV of Poisson statistics, $\Sigma_{\mbox{\tiny Poisson}}^2(L)=L$. We used energy levels $\bar{\epsilon}_{n,m}\in [4000 \times 10^7,4001 \times 10^7]$. The total numbers of levels are (a) 10000457, (b) 10000428, (c)10000266, and (d) 10000162. The truncation parameters $\delta$ are (a) $\pi\times 10^{-9}$ (b) $\pi/3\times 10^{-7}$ and (c) $\pi\times10^{-9}$. \[fig\_1\]
{width="8"}
[FIG.2 Slope of the level number variance, $g(L)=\Sigma^2(L)/L$, for (a) $25$th, (b) $8$th, (c) $4$th approximations of $\alpha=(\sqrt{\mathstrut 5}+1)/2$, and (d) $\alpha=1$. The solid line, $g(L)=1$, corresponds to Poisson statistics.]{} \[fig\_2\]
{width="15"}
[FIG.3 Function $-\ln{[1-M(S)]}$ for (a) $25$th, (b) $8$th, (c) $4$th approximations of $\alpha=(\sqrt{\mathstrut 5}+1)/2$, and (d) $\alpha=1$. The dotted line corresponds to the cumulative Poisson distribution $M(S)=1-e^{-S}$.]{} \[fig\_3\]
{width="8"}
[FIG.4 Parameter function $\tilde{\mu}(S)$ for (a) $25$th, (b) $8$th, (c) $4$th approximations of $\alpha=(\sqrt{\mathstrut 5}+1)/2$, and (d) $\alpha=1$. The solid line, $\tilde{\mu}(S)=0$, corresponds to the Poisson statistics.]{} \[fig\_4\]
{width="8"}
[FIG.5 Cumulative NNLSD $M(S)$ for $\alpha=1$. We used $999844$ levels $\bar{\epsilon}_{n,m}\in [40\times 10^6,41\times 10^6]$ for plot (A), $10001872$ levels $\bar{\epsilon}_{n,m}\in [4000000\times 10^7,4000001\times 10^7]$ for plot (B). We observe $M(+0)\simeq 0.767$ for plot (A) and $M(+0)\simeq 0.826$ for plot (B). The dotted curve is the cumulative Poisson distribution $M(S)=1-e^{-S}$]{} \[fig\_5\]
{width="8"}
[FIG.6 Numerical test of Eq.(\[EQ.(4.5)\]) for a square billiard ($\alpha=1$). The solid line exhibits the theoretical prediction, which is valid in the semiclassical limit $\epsilon\to+\infty$. In each plot, we used $10^7$ eigenenergy levels obtained from the unfolded energy range. The error bar exhibits the energy range acquiring numerical data.]{} \[fig\_6\]
|
---
abstract: 'If colloidal solute particles are suspended in a solvent close to its critical point, they act as cavities in a fluctuating medium and thereby restrict and modify the fluctuation spectrum in a way which depends on their relative configuration. As a result effective, so-called critical Casimir forces (CCFs) emerge between the colloids. The range and the amplitude of CCFs depend sensitively on the temperature and the composition of the solvent as well as on the boundary conditions of the order parameter of the solvent at the particle surfaces. These remarkable, moreover universal features of the CCFs provide the possibility for an active control over the assembly of colloids. This has triggered a recent surge of experimental and theoretical interest in these phenomena. We present an overview of current research activities in this area. Various experiments demonstrate the occurrence of thermally reversible self-assembly or aggregation or even equilibrium phase transitions of colloids in the mixed phase below the lower consolute points of binary solvents. We discuss the status of the theoretical description of these phenomena, in particular the validity of a description in terms of effective, one-component colloidal systems and the necessity of a full treatment of a ternary solvent-colloid mixture. We suggest perspectives on the directions towards which future research in this field might develop.'
author:
- 'A. Macio[ł]{}ek'
- 'S. Dietrich'
title: Collective behavior of colloids due to critical Casimir interactions
---
Introduction {#sec:intro}
============
Finite-size contributions to the free energy of a spatially confined fluid give rise to an effective force per area acting on the confining surfaces [@Evans:1990]. Fisher and de Gennes [@Fisher-et:1978] made the crucial observation that this fluid-mediated interaction acquires a universal, long-ranged contribution $f_C$ if the bulk critical point of the fluid is approached. This is due to *critical fluctuations*, and hence the notion ‘critical Casimir force’, in analogy with quantum-mechanical Casimir forces which are due to quantum fluctuations of confined electromagnetic fields [@Casimir:1948; @Kardar-et:1999]. In the case of colloidal suspensions with near-critical suspending fluids (referred to as solvents), the typically micrometer-sized colloidal particles act as cavities inside the critical solvent. At the colloid surfaces these cavities impose boundary conditions for the fluctuating order parameter of the solvent and perturb the order parameter field on the length scale of the bulk correlation length $\xi$. Such modifications of the order parameter and the restrictions of its fluctuation spectrum depend on the spatial configuration of the colloids. Following the argument by Fisher and de Gennes [@Fisher-et:1978], this gives rise to CCFs between the colloids, which is attractive for identical particles and has a range set by the bulk correlation length $\xi$ of the solvent. Since $\xi(t=(T_c^{(s)}-T)/T_c^{(s)}\to 0)\sim |t|^{-\nu}$, where $\nu$ is a standard bulk critical exponent, this range diverges upon approaching the bulk critical temperature $T_c^{(s)}$ of the pure [*s*]{}olvent.
The collective behavior of colloids dissolved in the near-critical solvent is determined by the interplay between the CCFs and other interactions acting between the constituents. In general, in colloidal suspensions the dissolved particles interact directly via van der Waals interactions; these are attractive and lead to irreversible aggregation (called coagulation) [@VarweyOverbeek:1948]. In charge-stabilized suspensions, the colloids acquire surface charges due to dissociation of the surface groups in water or due to chemical functionalization of the surface of the particles. This causes the formation of electric double layers around the colloids and results in electrostatic repulsion between them. In sterically-stabilized suspensions, the short polymer chains grafted onto the surface of colloidal particles give rise to a repulsive interaction which is of entropic origin. Additionally, the presence of other smaller solute particles or macromolecular additives such as polymers, surfactants, or micelles [@Likos:2001], induces effective entropic interactions between the colloids, called depletion forces, which are predominantly attractive and short-ranged [@Asakura-et:1954; @Vrij:76]. If the CCFs between colloidal particles are attractive and sufficiently strong to overcome the direct repulsive forces, one may expect the occurrence of a thermodynamically stable colloid-rich liquid or solid phase - even in the absence of any direct attractive interactions. If the resulting attractive potential is sufficiently strong, the condensation transition from a colloid-poor (’gas’) to a colloid-rich (’liquid’) phase may be preempted (on the characteristic time scales of the observations) by the formation of [*non-equilibrium*]{} aggregates in which the colloidal particle stick together. In general, such aggregates may grow or shrink and their structure varies from loose fractals through gels and glasses to crystals, depending on the packing fraction of the colloidal particles in the aggregates and on the strength of the attraction among the colloidal particles.
Beysens and Est[è]{}ve [@Beysens-et:1985] were the first to study experimentally aggregation phenomena for colloids suspended in binary solvents. These authors studied silica spheres immersed in a water-lutidine mixture by using light scattering. They found the formation of aggregates which sediment upon approaching the bulk coexistence region of demixing from the one-phase region of the binary liquid mixture at constant composition of the solvent. Strikingly, the observed aggregation was thermally reversible; moving back the thermodynamic state deeply into the one-phase region the sediments dissolved again. In the decade following this pioneering work, quite a number of further experiments were performed leading to a similar behavior for various binary solvents and a variety of colloids. The structure of the aggregates as well as the the kinetics of aggregation and the reverse process of fragmentation have been investigated. Silica, quartz powder, and polystyrene particles immersed in water-lutidine mixtures were studied in Refs. [@Gurfein-et:1989; @gallagher:92; @Broide-et:1993; @narayanan:95; @Kurnaz], whereas in Refs. [@narayanan:93; @Jayalakshmi-et:1997; @Kline-et:1994; @Koehler-et:1997; @grull:97] other solvents were employed; for corresponding reviews see Refs. [@Beysens-et:1994; @Beysens-et:1999; @Law:2001]. These experiments, which were performed mostly in the one-phase region of a binary liquid mixture, revealed that reversible aggregation (termed flocculation) is accompanied by a strong adsorption phenomenon in the vicinity of the bulk two-phase coexistence curve. Generically, colloidal particles have a preference for one of the two components of the binary solvent. At the surface of the colloid this preference gives rise to an effective surface field conjugate to the order parameter at the surface and thus leads to an adsorption layer rich in this preferred component. The measurements demonstrated that the temperature - composition $(T,c)$ region in which colloidal aggregation appears is not symmetric about the critical composition $c_c$ of the binary solvent. Strong aggregation occurs on that side of the critical composition which is rich in the component not preferred by the colloids.
Various mechanisms were put forward for strong adsorption giving rise to attraction which in turn could explain the occurrence of (non-equilibrium) flocculation. For example, capillary condensation and/or wetting can occur when particles come close together via diffusion, even far off the critical composition of the solvent. In this case a liquid ’bridge’ can form which induces attractive solvation forces [@Bauer-et:2000]. Another possibility is that the presence of an adsorption layer around the colloidal particles increases the strength of the direct attractive dispersion interactions. However, in the close vicinity of the bulk critical point of the solvent, in line with the predictions of Fisher and de Gennes, effective attraction induced by critical fluctuations is expected be dominant. In their original paper, Beysens and Est[è]{}ve have identified an “aggregation line” in the temperature-composition phase diagram of the solvent with a prewetting line. However, they have observed that this aggregation line extends to temperatures below the lower demixing critical point of a solvent. However, in general such an extent of prewetting lines has not been found up to now, neither for planar nor for spherical substrates. Actually, positive curvature even shortens prewetting lines [@Bieker-et:1998]. (Regrettably, the wetting behavior of this system in planar geometry has never been investigated.) The authors did not comment on this difference nor did they admit the role of critical fluctuations and Casimir forces for aggregation in the critical region. They have presumed that the aggregation process results from the attractive forces between the colloids due to the presence of the adsorption layer and have referred to de Gennes [@deGennes] as the one who had proposed such fluctuation induced interactions at and near the bulk critical point of the solvent, but they have not put their results into the proper context of de Gennes’s predictions. According to another interpretation of the experimental findings mentioned above, the observed phenomenon is regarded as a precursor of a bona fide phase transition in the ternary mixture rather then a non-equilibrium flocculation of colloidal particles [@Kline-et:1994; @Jayalakshmi-et:1997; @Koehler-et:1997]. A few theoretical [@Sluckin:1990] and simulation [@Loewen:1995; @Netz:1996] attempts have been concerned with such an interpretation. The status of knowledge about reversible aggregation of colloids in binary solvents up to the late 1990s has been reviewed by Beysens and Narayanan (see Ref. [@Beysens-et:1999] and references therein). More recently, the scenario of ’bridging’ transitions has again been studied theoretically in Refs. [@Archer-et:2005; @Okamoto-et:2011a; @Okamoto-et:2013; @Labbe-Laurent-et:2017].
In spite of the relevance of aggregation phenomena for the stability of colloidal suspensions, the basic understanding of the collective behavior of colloids dissolved in a near-critical solvent has started to emerge only recently. This progress had to await the advances made during the last decade concerning the statistical mechanical theory and computer simulations of CCFs. The accumulated theoretical knowledge of two-body CCFs has triggered also an increase of experimental activities in this field. This renewed interest is driven by application perspectives, in particular concerning the buildup of nanostructured materials of well defined structure by using self-assembly of colloidal particles. In order to achieve a desirable morphology of aggregates, one has to be able to control colloidal self-assembly and to manipulate the particles. The remarkable features of CCFs offer such possibilities. The range and the strength of the CCFs, which depend sensitively on temperature via the bulk correlation length $\xi$, can be tuned reversibly and [*continuously*]{} by moving the thermodynamic state of the solvent around its critical point. The sign of $f_C$ can be manipulated as well by suitable surface treatments of the colloids [@Hertlein-et:2008; @Gambassi-et:2009; @Nellen-et:2009]. Additional interest is sparked by the potential relevance of CCFs for lipid membranes. These are two-dimensional (2d) liquids consisting of two (or more) components, such as cholesterol and saturated and unsaturated lipids, which can undergo phase separation into two liquid phases, one being rich in the first two components and the other rich in the third [@membranes1]. Lipid membranes serve as model systems for cell plasma membranes [@LS]. Recent experiments suggest that cell membranes are tuned to the miscibility critical point of the $2d$ Ising model [@membranes; @Machta-et:2016] so that CCFs may arise between macromolecules embedded in the membrane [@sehtna; @Machta-et:2012; @Benet-et:2017].
Other mechanisms, which similar to the ones generating the critical Casimir effect also induce solvent-mediated long-ranged interactions, occur inter alia in a chemical sol upon approaching its percolation transition [@Gnan-et:2014], in a binary liquid mixture subjected to a steady temperature gradient due to the concomitant nonequilibrium concentration fluctuations [@Kirkpatrick-et:2015], in driven noncohesive granular media due to hydrodynamic fluctuations [@Cattuto-et:2006], or if the solvent comprises active matter such as bacteria or self-propelled colloidal particles [@Ray-et:2014; @Ni-et:2015].
Compared with other effective forces between colloid particles or macromolecules, CCFs have two advantages. First, due to the concept of universality for critical phenomena, to a large extent CCFs do not depend on microscopic details of the system. Second, whereas adding depletion agents or ions changes the resulting effective forces de facto irreversibly, the tuning of $f_C$ via temperature is fully and easily reversible.
The present article discusses current theoretical and numerical approaches towards the description of the static, equilibrium properties of colloidal suspensions with a near-critical binary solvent. The related experimental body of research is put into the corresponding context and a number of intriguing possible developments are highlighted. Recent developments concerning colloidal assembly due to CCFs with a focus on the experimental observations is reviewed in Ref. [@Schall_review].
Effective one-component approach {#sec:eff}
================================
A common approach to the statistical mechanics description of colloidal suspensions follows the ideas developed for multi-component molecular liquids, such as ionic solutions, by considering the colloidal particles as ’supramolecules’ [@VNFA:78; @Hansen:93; @Likos:2001]. Within this approach, the degrees of freedom of the solvent and the ions, in the case of charged-stabilized suspensions, are traced out in order to construct an effective one-component system of colloidal particles interacting via state- and configuration-dependent forces. For most cases, carrying out the integration over microscopic degrees of freedom can be done only approximately, leading to additive pairwise interactions between the colloidal particles (see the corresponding discussion below). For any binary mixture with pair interactions for which the volume integral is finite, a formal expression for an effective Hamiltonian, describing particles of one species only but in the presence of the particles of the other species, has been given in Ref. [@Dijkstra-et:1999]. This effective Hamiltonian consists of zero-body, one-body, two-body, three-body, and higher-body interactions, which depend on the density of the second species and have to be determined one by one. For additive hard-sphere mixtures with a large size asymmetry, a comparison with direct simulations of true binary mixtures has shown that the pairwise (depletion) potential approximation of the effective Hamiltonian between two large particles accounts remarkably well for the phase equilibria, even in limits for which one might expect that higher-body terms cannot be neglected. This success encourages one to use an effective one-component approach (with the approximation of an additive pairwise potential) to colloids suspended in a near-critical solvent, despite the fact that the CCFs are inherently non additive. The critical Casimir interaction between two colloidal particles depends on the (instantaneous) spatial configuration of [*all*]{} colloids [@Mattos-et:2013; @Mattos-et:2015; @Hobrecht-et:2015; @Volpe-et]. Only for dilute suspensions or for temperatures sufficiently far away from the bulk critical temperature $T_c^{(s)}$ of the pure solvent, such that the range of the critical Casimir interaction between the colloids is much smaller than the mean distance between them, the assumption of pairwise additive CCFs is expected to be reliable.
Effective interactions {#subsubsection:eff_inter}
----------------------
In most of the experimentally studied systems, the solvent is a binary mixture of molecular liquids and the colloidal particles are micro-sized spheres of a radius $R$. For such a sizewise highly asymmetric multi-component system, one can ignore the discrete nature of the solvent and use a simplified pair potential model for the background interaction potential between the colloids, which is present also away from the critical temperature $T_c^{(s)}$ of the solvent. Such a model is supposed to capture only the essential features of a stable suspension on the relevant, i.e., mesoscopic, length scale. Besides the van der Waals contribution, which will be discussed below, these features are the hard core repulsion for center-to-center distances $r<2R$ and a soft repulsive contribution, for which one can employ the Yukawa potential. This leads to the screened Coulomb model of suspensions which are charge-stabilized against flocculation [@Russel-et:1989; @Hansen_Loewen:2000; @Barrat-et:2003]: $$\label{eq:1}
V_{rep}(D)/(k_BT) = U_{rep}(D) = \left( U_0 /(\kappa D) \right)\exp(-\kappa D), \qquad D = r -2R >0,$$ where $D$ is the surface-to-surface distance and $k_B$ is the Boltzmann constant. The range $\kappa^{-1}$ of the repulsion is the Debye screening length $\kappa^{-1} = \sqrt{\epsilon\epsilon_0 k_B T/(e^2\sum_i\rho_i)}$ (see, e.g., Ref. [@vdW]), where $e$ is the elementary charge, $\epsilon$ the permittivity of the solvent relative to the vaccum, $\epsilon_0 = 8.854\times 10^{-12}$ C$^2$/(Jm) is the vacuum permittivity, and $\{\rho_i\}$ the number densities of all ions (regardless of the sign of their charges). A simplified, purely exponential form of the repulsive pair potential, $$\label{eq:2}
U_{rep}(D) = A \exp(-\kappa D),$$ is often used for suspensions in which $\kappa^{-1 }\ll R$ for distances $2R> D > R+\kappa^{-1}$, for which all curvature effects associated with the spherical geometry of the colloidal particles effectively drop out [@Israelachvili:1998; @Levin:2002]. The corresponding condition $\kappa^{-1 }\ll R$ is practically satisfied for the experimentally relevant systems for which the Debye length is of the order of 10 nm and the colloidal size of the order of 1$\mu m$. For the effective Coulomb interaction screened by counterions the amplitude $A$ is given by [@Russel-et:1989] $$\label{eq:3}
A= 2 \pi (\epsilon \epsilon_0 )^{-1} \Upsilon^{2}\kappa^{-2}R/(k_B T),$$ where $\Upsilon$ is the surface charge density of the colloid. The purely exponential form of repulsion (Eq. (\[eq:2\])) can also describe sterically stabilized suspensions beyond the hard-sphere model [@Israelachvili:1998]. In that case the range $\kappa^{-1}$ of the repulsion is associated with the length of the grafted polymers and the strength $A$ of the repulsion depends on the surface coverage of grafted polymers.
Upon approaching the bulk critical point $(T_c^{(s)}, c_c$) of the solvent, CCFs between the particles emerge and the corresponding pair potential $V_C/(k_BT)=U_C$ adds to the background contribution. In the well defined scaling limit of all length scales of the system being large on molecular scales, $U_C$ attains a scaling form [@Barber:1983; @privman; @Krech-et:1992] in terms of suitable dimensionless scaling variables describing the distance between the colloids, the dependence on the thermodynamic state of the solvent, and the shape of the colloidal particles. For example, for the spherical particles one has $$\label{eq:4}
\frac{V_C(D)}{k_BT}= U_C(D) \simeq \frac{R}{D} \Theta\left({\mathcal Y} = {\mathrm sgn}(t)\frac{D}{\xi_t},\; \Delta = \frac{D}{R},\; \Lambda = {\mathrm sgn}(h_b)\frac{D}{\xi_h} \right), \quad D = r -2R >0.$$ In Eq. (\[eq:4\]), $\Theta$ is a universal scaling function. Here $\xi_t(t\gtrless 0)=\xi_{t,\pm}^{(0)}|t|^{-\nu}$, with $t=\pm(T-T_c^{(s)})/T_c^{(s)}$ for an upper ($+$) and a lower ($-$) critical point, respectively, is the true correlation length governing the exponential decay of the solvent bulk two-point order parameter (OP) correlation function for $t\to0^{\pm}$ and $h_b=0$ where $h_b$ is the bulk ordering field conjugate to the OP. The amplitudes $\xi_{t,\pm}^{(0)}$ (with $\pm$ referring to the sign of $t$) are non-universal but their ratio $\xi_{t,+}^{(0)}/\xi_{t,-}^{(0)}$ is universal. The correlation length $\xi_h =\xi^{(0)}_h|h_b|^{-\nu/(\beta\delta)}$ governs the exponential decay of the solvent bulk two-point OP correlation function for $t=0$ and $h_b\to0$, where $\xi^{(0)}_h$ is a non-universal amplitude related to $\xi_{t,\pm}^{(0)}$ via universal amplitude ratios; $\nu$, $\beta$, and $\delta$ are standard bulk critical exponents [@Pelissetto-et:2002]. For the demixing phase transition of a binary liquid mixture, the OP $\phi$ is proportional to the deviation of the concentration of species, say $a$, $$\label{eq:con}
c_a = {\varrho}_{a}/({\varrho}_{a} + {\varrho}_{b})$$ from its value $c_{a,c}$ at the critical point, i.e., $\phi \sim c_a - c_{a,c}$; here ${\varrho}_{\alpha}$, $\alpha\in\left\{a,b\right\}$, are the number densities of the particles of species $a$ and $b$, respectively. The bulk ordering field, conjugate to this order parameter, is proportional to the deviation of the difference $\Delta\mu=\mu_a-\mu_b$ of the chemical potentials $\mu_{\alpha}$, $\alpha\in\left\{a,b\right\}$, of the two species from its critical value, i.e., $h_b\sim \Delta\mu-\Delta\mu_c$. We note, that the actual scaling fields of fluids are linear combinations of $h_b$ and of the reduced temperature $t$.
The minimal model for a pair potential describing the effects of a critical solvent on dissolved colloids due to CCFs, i.e., corresponding to the sum of Eqs. (\[eq:2\]) and (\[eq:4\]), has been used in Refs. [@Bonn-et:2009; @Gambassi-et:2010; @Mohry-et:2012a; @Mohry-et:2012b; @Mohry-et:2014; @Nguyen-et:2013; @Dang-et:2013]. In Ref. [@Zvyagolskaya-et:2011], instead of a soft repulsive potential of the form as in Eqs. (\[eq:2\]) and (\[eq:3\]), the electrostatic repulsion has been modeled via a hard disc repulsion with an effective diameter. In some of the studies cited above, a simplified functional form of the universal scaling function $\Theta$ has been employed [@Bonn-et:2009; @Gambassi-et:2010; @Nguyen-et:2013; @Dang-et:2013], such as using a form valid only asymptotically for large values of the temperature scaling variable neglecting the dependence on the other, also relevant scaling variables. Since, however, the shape of the total pair potential depends sensitively on details of the CCFs pair potential, there is a need to discuss them (see Subsec. \[subsection:CCFpot\] below).
In order to be able to describe certain experimental systems, one has to consider also the interaction which accounts for effectively attractive dispersion forces. For two spheres, the (nonretarded) van der Waals forces contribute to the total potential through a term (see, e.g., Refs. [@Chen:1996; @vdW]) $$\label{eq:5}
V_{\rm vdW}(\Delta)/(k_BT) = U_{\rm vdW}(\Delta) = - \frac{A_H/(k_BT)}{6} \left[\frac{2}{\Delta(\Delta + 4)} + \frac{2}{(2
+ \Delta)^2} + \ln\left( \frac{1+4/\Delta}{(1+2/\Delta)^2}\right)\right],$$ where $A_H$ is the Hamaker constant. As $\Delta = D/R$ increases, this term crosses over from the behavior $U_{\rm vdW}(D\ll R) \simeq
-(A_H/6) (R/D)^2$ to $U_{\rm vdW}(D\gg R) \simeq
-(2A_H/3) (R/D)^3$. The Hamakar constant depends on the dielectric properties of the materials involved in the experiment under consideration [@vdW]. By using index-of-refraction-matched colloidal suspensions [@Israelachvili:1998] its value can be strongly reduced. This way dispersion forces can be effectively switched off. (A detailed discussion of the Hamakar constant for a polystyrene colloid near a silica glass substrate immersed in a mixture of water and 2,6-lutidine can be found in Ref. [@Gambassi-et:2009].)
Additionally, in the presence of small co-solutes such as free polymer coils or smaller colloids in a stericly stabilized colloidal suspension, one has to consider depletion interactions which arise between large colloidal particles due to entropic effects caused by the small solutes. This so-called depletion interaction is mainly attractive and has a range proportional to the size of the depletant [@Asakura-et:1954; @Vrij:76]. There are several theories and approximations for the depletion potential, which are summarized in Refs. [@Goetzelmann-et:1998; @Roth-et:2000]. For example, for a fluid of large hard spheres of radius $R$ and a small spherical depletant of diameter $\sigma$, which on its own behaves as an ideal gas, the depletion potential is [@Vrij:76] $$\label{eq:6}
V_d(D)/(k_BT) = U_d(D)=
\begin{cases}
-n_bV_{ov}(D), & 0 \le D \le \sigma \\
0, & D\ge \sigma,
\end{cases}$$ where $V_{ov}(D)= (\pi/6)(\sigma -D)^2(3R+\sigma+D/2)$. In Eq. (\[eq:6\]) $n_b$ is the bulk number density of the small spheres. Note that $|V_d(D)|$ in Eq. (\[eq:6\]) is equal to the pressure ($p=n_bk_BT$) times the overlap volume $V_{ov}$ between the excluded volumes denied to the centers of the small spheres around each big sphere. Taking into account hard-core repulsion among the depletants produces a repulsive contribution to the depletion interaction as well as an oscillatory decay at large distances [@Goetzelmann-et:1998; @Roth-et:2000]. For many actual colloidal suspensions the depletion attraction is strong enough to induce colloidal aggregation and, despite of its rather short range, a gas-liquid-like phase separation, because even a small degree of polydispersity or non-sphericity of the particles causes the fluid phase to be not preempted by crystallization.
If the depletant is only a co-solute in the suspension in which the solvent becomes critical, the resulting depletion potential simply adds to the background potential. It may, however, occur that due to an effective depletant-depletant interaction the depletant itself exhibits a phase transition with a critical point. This means that the solvent, which is common to both the big spheres and the depletant, does not display a critical point of its own as the one discussed above. A system which realizes this interesting scenario was studied experimentally in Refs. [@Buzzaccaro-et:2010; @Piazza-et:2011] and via Monte Carlo simulations [@Gnan-et:2012a] (see Secs. \[subsubsec:pp\] and \[subsec:eff\_dep\] below where we shall review the results of these studies). In such a case one expects that the depletant produces one unique effective pair potential between the big particles. Far away from the critical point of the depletant, this unique pair potential has the character of a depletion interaction, whereas close to the critical region of the depletant it should display the features of the CCFs pair potential. As has been shown in Refs. [@Buzzaccaro-et:2010; @Piazza-et:2011], the framework of density functional theory (DFT) [@Evans:1979], which is commonly adopted in colloidal science, can provide both forms for the effective pair potential in the corresponding limits (see Sec. \[subsection:CCFpot\] below for details). In Ref. [@Gnan-et:2012a], the effective pair potential $V_{eff}(r)$ between two big hard-sphere colloids has been determined numerically for two models of the depletant particles within a wide range of state points, including the critical region. In the first model, the spherical depletant particles interact via a pairwise square-well potential (SW): $$\label{eq:7}
V^{SW}(r)=
\begin{cases}
\infty, & r < \sigma \\
-d_w, & \sigma \le r \le (1 + r_w \sigma) \\
0, & r \ge (1 + r_w )\sigma,
\end{cases}$$ where $r$ is the center-to-center distance between two depletant particles, $r_w$ is a dimensionless well width, and $d_w$ is the well depth. The second depletant model is an anisotropic three-patches (3P) Kern-Frenkel system which consists of hard-sphere particles decorated with three attractive sites [@KF]. The critical packing fraction of these patchy particles is very small - as in the experimental system studied in Refs. [@Buzzaccaro-et:2010; @Piazza-et:2011]. For this kind of depletant, the numerically determined effective pair potential $V_{eff}(r)$ between two big spheres has subsequently been used in a grand canonical off-lattice MC simulation in order to analyze the stability of the colloidal suspension for a system of colloidal particles interacting via the pairwise additive interaction $ V_{HS} (r) + V_{eff}(r) $, where $V_{HS} (r)$ is a hard-sphere potential. The results of this study will be discussed below in Sec. \[subsubsec:pp\].
Critical Casimir pair potential {#subsection:CCFpot}
-------------------------------
A crucial ingredient for an effective one-component approach is to have an accurate critical Casimir potential (CCP) between two colloidal particles. This is not only relevant for the investigation of aggregation or the bulk phase behavior of colloids, but it is also of intrinsic scientific interest. In recent years an experimental technique (total internal reflection microscopy (TIRM) [@Walz-97; @Prieve-99; @h-th]) has been developed which allows one to measure directly and with fN resolution the effective potential of the CCF between a colloidal particle, suspended in a near-critical binary mixture, and a fixed object such as a planar wall [@Hertlein-et:2008; @Nellen-et:2009; @Gambassi-et:2009]. Video microscopy has also been used in order to determine the potential of the CCF between two spherical colloids [@Nguyen-et:2013; @Dang-et:2013; @Shelke-et:2013; @Marcel-et]. Although the resolution and the sophistication of such experiments increase, it is difficult to interpret the data of these measurements mainly due to the inevitable, simultaneous presence of various contributions to the effective pair potential. Therefore reliable theoretical results are required in order to improve the interpretations.
The solvent-mediated force between two spherical particles, a surface-to-surface distance $D$ apart, is defined as the negative derivative of the excess free energy ${\cal F}^{ex}={\cal F} -V f_b$, $$\begin{aligned}
\label{eq:8}
f_s= -\frac{\partial {\cal F}^{ex}}{\partial D}
=-\frac{\partial( {\cal F} -V f_b)}{\partial D},\end{aligned}$$ where $f_b$ is the bulk free energy density of the solvent and ${\cal F}$ is the free energy of the [*s*]{}olvent in the macroscopically large volume $V$ excluding the volume of two suspended colloids. The CCF $f_C$ is the long-ranged universal contribution to $f_s$ which emerges upon approaching the bulk critical point of the solvent. The associated critical Casimir potential (CCP) is $$\begin{aligned}
\label{eq:9}
V_C(D,R,T,h_b) \equiv \int_D^\infty dz f_C(z,R,T,h_b)\end{aligned}$$ so that the critical Casimir force (CCF) is given by $f_C(D,R,T,h_b)=-\frac{\partial}{\partial D}V_C(D,R,T,h_b)$. According to finite-size scaling theory [@Barber:1983; @privman], the CCP exhibits scaling described by a universal scaling function $\Theta$ as given by Eq. (\[eq:4\]). This scaling function is determined solely by the so-called universality class of the continuous phase transition occurring in the bulk, the geometry of the setup, and the surface universality classes of the confining surfaces [@Diehl:1986; @Krech:1990:0; @dantchev; @gambassi:2009]. The relevant bulk universality class for colloidal suspensions is the Ising universality class in spatial dimension $d=3$ or $d=2$. For the CCF one has $$\begin{aligned}
\label{eq:9a}
f_C/(k_BT) \simeq \frac{R}{D^2} \vartheta\left({\mathcal Y}, \Delta, \Lambda \right)=\frac{R}{D^2}\left[ \Theta - {\mathcal Y}\frac{\partial}{\partial {\mathcal Y}}\Theta - \Delta\frac{\partial}{\partial\Delta}\Theta - \Lambda\frac{\partial}{\partial\Lambda}\Theta\right].\end{aligned}$$
The main difficulty in determining theoretically the scaling function of CCFs and their potential lies in the character of the critical fluctuations; an adequate treatment has to include non-Gaussian fluctuations. Usually colloidal particles exert a potential on the surrounding fluid which is infinitely repulsive at short distances and attractive at large distances. Such potentials give rise to pronounced peaks in the density profile of the fluid near the surface of the particle, i.e., there is strong adsorption. In terms of a field-theoretical description this corresponds to the presence of a strong (dimensionless) [*s*]{}urface field $h_s\gg 1$. For binary liquid mixtures one has $h_s \sim (\delta \Delta\mu_s)/(k_BT)$ so that there is a local increment at the surface of the chemical potential difference between the two species. It determines which species of the solvent is preferentially adsorbed at the surface of the colloid. The preference for one component of a binary liquid mixture may be so strong as to saturate the surface of the colloids with the preferred component, which corresponds to $h_s = +\infty (-\infty)$. For two colloids this gives rise to symmetry-breaking boundary conditions (denoted by $(+, +)$ or $(-, -)$) for the solvent order parameter. The resulting spatial variation of the order parameter poses a significant complication for obtaining analytic results. Moreover, non-planar geometries lower the symmetry of the problem. Apart from a few exceptions and limiting cases, the presently available analytical results for CCFs are of approximate character.
At first, for two spherical colloidal particles, the corresponding CCP has been studied theoretically right at the bulk critical point. De Gennes [@deGennes] has proposed the singular effective interaction potential between two widely separated spheres by using a free energy functional, which goes beyond mean-field in the sense that it incorporates the non-classical bulk critical exponents, but neglects the critical exponent $\eta$; $\eta$ is the standard bulk critical exponent for the two-point correlation function at criticality with $\eta \sim 0.04$ in $d=3$ and $\eta =1/4$ in $d=2$ [@Pelissetto-et:2002]. Within this approach he has found that for $d=3$ the energy of interaction in the limit $D\gg R$ is $-21.3\times k_BT_c^{(s)}R/D$. For that interaction potential, the so-called protein limit, which corresponds to $D/R$, $\xi/D \gg 1$ and which is based on exact arguments using conformal invariance, renders, within a small-sphere expansion [@Burkhardt-et:1995; @Eisenriegler-et:95], $$\begin{aligned}
\label{eq:10}
U_C(D;T = T_c^{(s)},h_b=0,R)\sim R^{d-2+\eta}D^{-(d-2 +\eta)} & \stackrel{d=3}{=}& (R/D)^{-(1 +\eta)} \\
&\stackrel{d=2}{=}& (R/D)^{\eta}, \nonumber \end{aligned}$$ which implies for the scaling function in Eq. (\[eq:4\]) $\Theta\left(\mathcal{Y}=0, \Delta=D/R \to \infty, \Lambda = 0\right) \sim \Delta^{-(d-3+\eta})
\stackrel{d=3}{=} \Delta^{-\eta} $. In Ref. [@Burkhardt-et:1995] the amplitude of $U_C$ in $d=3$ has been estimated to be slightly larger than $\sqrt{2}$, which is a factor of ca 15 smaller than the above prediction given by de Gennes [@deGennes]. In the opposite, the so-called Derjaguin limit $D \ll R$, one has [@Burkhardt-et:1995; @Eisenriegler-et:95] $$\begin{aligned}
\label{eq:11}
U_C(D;T=T_c^{(s)},h_b=0,R) \sim R^{(d-1)/2}D^{-(d-1)/2} &\stackrel{d=3}{=}& R D^{-1}=\Delta^{-1} \\
&\stackrel{d=2}{=}& (R/D)^{1/2}=\Delta^{-1/2}, \nonumber
\end{aligned}$$ which in turn implies for the scaling function in Eq. (\[eq:4\]) $\Theta\left(\mathcal{Y}=0, \Delta = \frac{D}{R}\to 0, \Lambda = 0\right) = const$. These results confirm that the CCFs can indeed successfully compete with direct dispersion [@Dantchev-et:2007; @Dantchev-et:2017] or electrostatic forces in determining the stability and phase behavior of colloidal systems.
### Mean field theory {#subsubsec:mf}
Concerning the full range of parameters, theoretical predictions for the universal scaling function of the CCP between spheres are available only within mean field theory [@Hanke-et:1998; @Schlesener-et:2003]. Within this Landau - Ginzburg - Wilson approach, the CCF is conveniently calculated using the stress tensor ${\cal T}(\phi({\bf r}))$ in terms of the mean field profile $\phi({\bf r})$ [@Eisenriegler-et:1994]: $$\begin{aligned}
\label{eq:12}
{\bf f}_C = k_BT \int_{\cal A}d^{d-1} r {\cal T}(\phi({\bf r})) \cdot {\bf n} \end{aligned}$$ where ${\cal A}$ is an arbitrary $(d-1)$-dimensional surface enclosing a colloid, ${\bf n}$ is the outward normal of this surface, and ${\bf f}_C = f_C {\bf e}$ is the force between two colloids, where ${\bf e}$ is a unit vector along the line connecting their centers. The orientation of ${\bf e}$ is such that $f_C< 0 \;(>0)$ corresponds to attraction (repulsion). In most cases the mean-field profile $\phi({\bf r})$ is determined by numerical minimization of the Landau-Ginzburg-Wilson Hamiltonian encompassing suitable surface contributions from the colloid surfaces in order to account for symmetry-breaking boundary conditions there. In $d=3$ spatial dimensions such a theory is approximate. In the spirit of a systematic expansion in terms of $\epsilon=4-d$, it is exact in $d=4$ for four-dimensional spheres. Recently, the same approach has been employed in order to calculate the scaling function of the CCFs for three-dimensional spheres posing as hypercylinders $(H_{d=4,d^*=3})$ in spatial dimension $d=4$ [^1] [@Mohry-et:2014], where $H_{d,d^*}=\{\bf{r}=(\bf{r}_{\perp},\bf{r}_{||})\in \mathbb{R}^{d^*}\times \mathbb{R}^{d-d^*}\mid |\bf{r}_{\perp}|\le R \}$. The obtained results differ from the ones for four-dimensional spherical particles $H_{d=4,d^*=4}$ in $d=4$. This raises the question whether $H_{d=4,d^*=3}$ or $H_{d=4,d^*=4}$ renders the better mean-field approximation for the physically relevant case of three-dimensional spheres $H_{d=3,d^*=3}$ in $d=3$. Due to this uncertainty more accurate theoretical approaches are highly desirable. Recent MC simulations of a sphere near a wall constitute a first step in this direction [@Hasenbusch].
The mean field Landau-Ginzburg-Wilson theory is a versatile approach for calculating CCFs. In the case of the simple film geometry, this approximate approach reproduces correctly the qualitative behavior of CCFs (i.e., their sign, functional form, and structure) for various combinations of surface universality classes. Therefore it has been used to calculate CCFs for various shapes of colloidal particles and for other geometries. Non-spherical, i.e., highly ellipsoidal or sphero-cylindrical colloids, or elongated particles such as cylindrical micelles [@cyl_mic], block copolymers [@block_copol], the mosaic tobacco virus [@mos_tob_vir], and carbon nanotubes [@carbon_nan] are experimentally available and widely used in the corresponding current research efforts, with application perspectives towards new materials in mind. The orientation dependent CCP for ellipsoidal particles near a planar wall at vanishing bulk field $h_b = 0$ has been studied in Ref. [@Kondrat-et:2009]. In this case, due to the anisotropy of the particles, there is not only a force but also a torque acting on the particle. This may lead to additional interesting effects such as the orientational ordering of nonspherical colloids in a critical solvent. The behavior of hypercylinders $H_{d=4,d^*=3}$ and $H_{d=4,d^*=2}$ near planar, chemically structured substrates have been studied in Ref. [@Troendle-et:2009-10]. In Ref. [@Mattos-et:2013], the scaling function associated with CCFs for a system consisting of two spherical particles facing a planar, homogeneous substrate has been calculated. This allows one to determine the change of the lateral CCF between two colloids upon approaching a wall which acts like a large third body. Within the applied mean field theory, this many body contribution can reach up to 25$\%$ of the pure pair interaction. As one would expect, the many-body effects were found to be more pronounced for small distances, as well as for temperatures close to criticality. This trend has been confirmed by studying three parallel hypercylinders within mean field theory [@Mattos-et:2015] and three discs in $d=2$ within MC simulations [@Hobrecht-et:2015]. Three-body interactions in $d=3$ for spherical colloids have been determined experimentally [@Volpe-et]. However, at the present stage these data cannot yet be compared quantitatively with theoretical results. In Ref. [@Labbe-Laurent] the CCP for and the ensuing alignment of cylindrical colloids near chemically patterned substrates has been determined within mean field theory. The case in which the particles exhibit spatially inhomogeneous surface properties, forming so-called Janus particles which carry two opposing boundary conditions, has also been considered [@Labbe-Laurent-et:2016]. The experimental fabrication of such particles is of research interest in itself [@fabrication; @Labbe-Laurent-et:2016] as is the theoretical understanding of the interactions between spherical [@spherical_Janus; @Labbe-Laurent-et:2016] or non-spherical Janus particles [@Labbe-Laurent-et:2016; @nonspherical_Janus], because they are considered to be promising building-blocks for self-assembling materials [@materials; @Iwashita-et:2013; @Iwashita-et:2014].
### Beyond mean field theory {#subsubsec:bmf}
Beyond mean field theory, theoretical predictions for the CCP for two spheres immersed in a critical fluid are available from conformal field methods, which, however, are restricted to the bulk critical point of the solvent and to $d=2$. As mentioned in passing, the exact analytical results for the limiting behavior of spheres which nearly touch (Eq. (\[eq:11\])) and spheres which are widely separated (Eq. (\[eq:10\])) have been obtained by using conformal invariance of the free energy and taking the Derjaguin limit in the first case and applying the small-sphere expansion in the second case, respectively [@Burkhardt-et:1995]. For the $2d$ Ising universality class, the CCP has been calculated numerically [@Burkhardt-et:1995] within the full range of distances $D$ between two spheres. More recently this was achieved also analytically [@Machta-et:2012] via the partition function of the critical Ising model on a cylinder, using a conformal mapping onto an annulus. For non-spherical colloidal particles with their dummbell or lens shapes being small compared to the correlation length and to the interparticle distances, exact results for the orientation-dependent CCFs have been obtained by using a small-particle operator expansion and by exploiting conformal invariance for $d \lesssim 4$ and $d=2$ [@Eisenriegler:2004]. In fact, in $d=2$ conformal field theory provides a general scheme for critical Casimir interactions between two (or more) objects of arbitrary shape [@Bimonte-et:2013]. This can be achieved by using the local conformal mappings of the exterior region of two such objects onto a circular annulus for which the stress tensor is known. Assuming the availability of a simple transformation law for the stress tensor under any such (local) conformal mappings, the CCFs are obtained from the contour integral of the transformed stress tensor along a contour surrounding either one of the two objects [@Bimonte-et:2013].
Another nonperturbative approach, which allows one to calculate CCFs directly at a fixed specific spatial dimension potentially an advantage over field-theoretic approaches based on a systematic expansion in terms of $\epsilon = 4-d$ is the use of semi-empirical free energy functionals for critical inhomogeneous fluids and Ising-like systems. They have been developed by Fisher and Upton [@Fisher-et:1990] in order to extend the original de Gennes-Fisher critical-point ansatz [@Fisher-et:1978; @Fisher-et:1980]. Upon construction, these functionals fulfill the necessary analytic properties as a function of $T$ and a proper scaling behavior for arbitrary $d$. The only input needed is the bulk Helmholtz free energy and the values of the critical exponents. (However, the available functional is valid only for symmetry breaking boundary conditions.) The predictions of this functional for films with $(+,+)$ boundary conditions are in very good agreement with previous results obtained $\epsilon$-expansion and conformal invariance for the scaling function of the order parameter and for the critical Casimir amplitudes [@Borjan-et:1998]. Also the predictions for the full scaling functions of the CCFs at $h_b =0$ [@Borjan-et:2008] are in good agreement with results from Monte Carlo simulations. These predictions have been obtained from a linear parametric model, which in the neighborhood of a critical point provides a simple scaled representation of the Helmholtz free energy in terms of “polar” coordinates $(r,\theta)$ centered at the critical point $(t,h_b,\phi)=(0,0,0)$, where $r$ is a measure of the distance from the critical point, and which assumes a linear relationship between the OP $\phi$ and $\theta$. A similar local-functional approach proposed by Okamoto and Onuki [@Okamoto-et:2012] uses a form of the bulk Helmholtz free energy which differs from the one employed in Ref. [@Borjan-et:2008], in that it is a field-theoretic expression for the free energy with (in the sense of renormalization group theory) renormalized coefficients. Such a version does not seem to produce more accurate results for the Casimir amplitudes [@Okamoto-et:2012]. Within this renormalized local-functional approach, the CCFs between two spherical particles immersed in a near-critical binary mixture (consisting of $a$ and $b$ particles) has been calculated as a function of both scaling fields, $T$ and $h_b$ [@Okamoto-et:2013]. The focus of the study is the situation that the $b$-rich phase forms on the colloid surfaces in thermal equilibrium with the subcritical $a$-rich bulk solvent ($T <T_c^{(s)}$ and $h_b < 0$). This gives rise to a bridging transition between two spherical particles and the resulting effective forces are of different nature than the CCFs. We note that the validity of the extended de Gennes-Fisher or renormalized local-functional approach in the presence of bulk ordering fields has not yet been tested, not even for the simple film geometry.
### Computer simulations {#subsubsec:cs}
In cases in which one cannot obtain analytical results, Monte Carlo simulations offer a highly welcome tool in order to overcome the shortcomings of approximate theoretical approaches and to study CCPs within the whole temperature range and also in the presence of the bulk ordering field $h_b$. In the case that the solvent is a simple fluid and in the spirit of the universality concept of critical phenomena, one can study the simplest representative of the corresponding universality class of the critical solvent, e.g., spin models such as the Ising lattice gas model. In the case of a lattice, the derivative in Eq. (\[eq:8\]) is replaced by a finite difference $\Delta {\cal F}^{ex}$ associated with a single lattice spacing. This also requires to introduce a lattice version of a spherical particle. In general, Monte Carlo methods are not efficient to determine quantities, such as the free energy, which cannot be expressed in terms of ensemble averages. Nevertheless, free energy differences can be cast into such a form via, e.g., the so-called “coupling parameter approach” (see, e.g., Ref. [@Mon]). This approach can be applied for systems characterized by two distinct Hamiltonians ${\cal H}_0$ and ${\cal H}_1$ but the same configurational space. In such a case, one can introduce the [*cr*]{}ossover Hamiltonian ${\cal H}_{cr}(\lambda) = (1-\lambda){\cal H}_0 + \lambda {\cal H}_1$, which interpolates between ${\cal H}_0$ and ${\cal H}_1$ as the crossover parameter $\lambda \in [0,1]$ increases from 0 to 1. The difference between the free energies of two systems characterized by these different Hamiltonians can be conveniently expressed as an integral with respect to $\lambda$ over canonical ensemble averages of ${\cal H}_1-{\cal H}_0$ (with the averages taken by using the corresponding crossover Hamiltonian for a given value of the coupling parameter). Alternatively, the free energy difference can be determined by integrating the corresponding difference of internal energies over the inverse temperature. The drawback of both methods is that they usually require knowledge of the corresponding [*bulk*]{} free energy density (as in the case of CCFs for a slab geometry). The accurate computation of the [*bulk*]{} free energy density poses a numerical challenge by itself and extracting it from finite-size data requires a very accurate analysis. Moreover, the internal energy differences, when determined by using standard MC algorithms, are affected by huge variances, especially for non-planar geometries such as the sphere - planar wall or the sphere - sphere geometry. In such geometries, the differences between local energies of the two systems with different (by one lattice spacing) sphere - wall or sphere - sphere distances are large only close to the sphere or the wall. This implies that the variance is dominated by the remaining part of the system, for which the local energy difference is very small. Recently a more sophisticated algorithm [@Hasenbusch] and new approaches [@Hobrecht_Hucht:14] have been developed in order to overcome these problems. The numerical method proposed in Ref. [@Hobrecht_Hucht:14] is analogous to the experimental one used by Hertlein [*et al.*]{} [@Hertlein-et:2008] according to which the CCP is inferred directly from the Boltzmann distribution function of the positions of the two interacting objects. (In this experiment a sphere performs Brownian motion near a planar substrate.) So far, the improved algorithm of Ref. [@Hasenbusch] and the dynamic method of Ref. [@Hobrecht_Hucht:14] have been applied only to the slab geometry and to the geometry of a single sphere (or disc) near a planar wall. The only available simulation data for two quasi-spheres in the $d=3$ Ising model have been obtained by using a method based on the integration of the local magnetization over the applied local magnetic field [@Vasilyev:2014]. In that study, the CCP has been calculated at fixed distances between the two spheres as a function of the temperature scaling variable (related to ${\mathrm sgn}(t)(D/\xi_t)$) for a few values of $h_b$, or as a function of the bulk field scaling variable (related to ${\mathrm sgn}(h_b)(D/\xi_h) $) for several temperatures. Results of this calculation have been obtained only for small separations $D\le 2R$, where $R=3.5$ (in units of the lattice spacing $a$) is the radius of the particle, because the strength of the potential decreases rapidly upon increasing $D$. The accuracy of these data deteriorates at low temperatures.
Alternative ways of computing the sphere - sphere CCP via MC simulations have been used for two-dimensional lattice models. One of them [@Machta-et:2012] uses Bennet’s method [@Bennet:76], according to which one can efficiently estimate the free energy difference between two canonical ensembles, characterized by two different energies $E_0$ and $E_1$ with the same configuration space, provided that these two ensembles exhibit a significant overlap of common configurations. This method has been employed for the $2d$ Ising model, with frozen spins forming two quasi-spheres with an effective radius $R$; in the reference ensemble (with energy $E_0$) the quasi-spheres are separated by a distance $D$ whereas in the second ensemble (with energy $E_1$), they are separated by a distance $D+a$ [@Machta-et:2012]. In order to estimate the corresponding free energy difference, within this approach one considers a trial move which keeps the configuration space the same but switches the energy from $E_0$ to $E_1$, e.g., by mimicking the move of one of the particles from a distance $D$ to a distance $D+a$ by suitably changing and exchanging spins. The estimate of the free energy difference is given by $F_1 - F_0 = -(1/k_BT)\ln\langle \exp(E_0-E_1)/(k_BT)\rangle_0$, where the canonical ensemble average $\langle \; \cdot \; \rangle_0$ is taken with respect to the “reference” ensemble with the energy $E_0$. Integrating this free energy difference up to infinity, one obtains the CCP. Reference [@Machta-et:2012] provides the results for the CCP as a function of distance $D/R \gtrsim 25$ for four values of the temperature in the disordered phase of the solvent, i.e., $T\ge T_c^{(s)}$ and $h_b=0$. At the critical temperature, these MC simulation results agree well with the analytical ones obtained from conformal field theory. Still another route towards determining the CCP from MC simulations has been followed in Refs. [@bob-et:2014; @Tasios-et:2016] for two quasi-discs immersed in a $2d$ lattice model of a binary liquid mixture [@Rabani-et:2003]. Here, the pair potential $V_C(x,y)$ has been obtained from the numerically determined probability $P(x,y)$ of finding one colloid at position $(x,y)$ provided that another one is fixed at the origin: $V_C(x,y) = -k_BT \ln(P(x,y)/P(\infty,\infty))$. In order to determine $P(x,y)$ accurately at fixed values of $h_b\sim \Delta\mu-\Delta\mu_c$ and $T$, the so-called Transition Matrix Monte Carlo technique [@Errington:2003] has been employed. This technique relies on monitoring the attempted transitions between macrostates, as defined, for example, by specific positions of the two quasi-discs, and on using this knowledge in order to infer their relative probability $P(Y)/P(Y')$, where $Y$ and $Y'$ are two distinct states of the system; once sufficient transition data are collected, the entire probability distribution $P(Y)$ can be constructed. The studies in Refs. [@bob-et:2014; @Tasios-et:2016] have been focused on the case that the b-rich phase adsorbs on the colloid surfaces in thermal equilibrium with a supercritical a-rich bulk solvent ($T \ge T_c^{(s)}$ and $h_b \le 0$). It has been found that data for $U_C = V_C/(k_BT)$ obtained for $h_b=0$ and $D \ll R$ collapse on a common master curve if multiplied by $(D/R)^{1/2}$ and plotted as a function of $D/\xi_t$, i.e., $(D/R)^{1/2}U_C = \Theta\left(\mathcal{Y}= {\mathrm sgn}(t)D/\xi_t, \Delta = D/R =0,\Lambda = {\mathrm sgn}(h_b)D/\xi_h = 0 \right)$. The scaling exponent 1/2 for the prefactor agrees with the prediction in Eq. (\[eq:11\]) for $d=2$, which is obtained based on the conformal invariance in the Derjaguin limit $D\ll R$, and which implies $\Theta(y \to 0, \Delta \to 0, \Lambda \to 0) = const$.
The standard algorithms used for the MC simulation studies described above are based on trial moves generating a trial configuration, which are local (Metropolis-type flips of spins) and hence become very slow near the critical point. This critical slowing down effect can be weakened by using so-called cluster algorithms such as the Swendsen and Wang [@Swendsen-et:1987] or the Wolff algorithm [@Wolff:1989] in which instead of a single spin a whole cluster of spins is flipped simultaneously. The Wolff algorithm constructs clusters which consist of spins that are aligned and connected by bonds. The proof that the Wolff algorithm obeys detailed balance, and hence generates the Boltzmann distribution, hinges on the spin inversion symmetry of the Hamiltonian. (The Wolff algorithm can be generalized to systems which contain bulk or surface fields [@Binder_Landau].) One can exploit also other symmetries in order to to develop a cluster method, for example, by using geometric operations on the spin positions such as, e.g., point reflection or rotation with respect to a randomly chosen “pivot”. Hobrecht and Hucht [@Hobrecht-et:2015] have extended the geometric cluster algorithm (GCA), introduced by Heringa and Blöte for bulk Ising models in the absence of external fields [@Heringa-et:98], to the case of Ising systems containing areas of spins with a fixed orientation, facilitated by infinitely strong bonds, which mimic a colloidal suspension. The GCA makes use of the invariance of the Hamiltonian with respect to a point inflection in order to construct two symmetric clusters of spins which are then exchanged. Contrary to the Wolff cluster algorithm, the GCA conserves the order parameter. The modification due to Hobrecht and Hucht consists of including into the clusters not only spins but also the bond configuration between them. This way, the particles encoded into the bond configuration can be moved and the configuration of a solvent represented by the Ising spins can be updated within one cluster step. In the cluster exchange the neighboring lattice sites $i$ and $j$ as well as the connecting bond $\langle ij\rangle$ are mapped via point reflection with respect to a pivot onto the sites $i'$ and $j'$ and the bond $\langle i'j'\rangle$, respectively. Using this apparently very efficient MC cluster algorithm, the authors of Ref. [@Hobrecht-et:2015] have studied two-dimensional systems with a fixed number of identical, disclike particles defined as regions of fixed positive spins, which thus effectively impose symmetry-breaking (+) BCs onto the surrounding free spins. Within this scheme, they have calculated the two- and three-body CCP at the bulk critical point of the Ising model. The authors report strong finite-size effects: for periodic simulation boxes with a fluctuating total magnetization, the presence of a nonzero number density of colloidal particles with a non-neutral surface preference for up and down spins shifts the system away from the critical point. As a result, their MC results for the CCP do not exhibit the form expected to hold at the critical point for a single pair of particles in solution (Eqs. (\[eq:10\]) and (\[eq:11\]). In order to suppress this effect, the authors of Ref. [@Hobrecht-et:2015] have proposed to use a fixed total magnetization $M=0$ or to insert in addition the same number of particles but with the opposite surface preference.
### Derjaguin approximation {#subsubsec:DA}
Within the so-called Derjaguin approximation curved, smooth surfaces are approximated by surfaces which are a steplike sequence of parallel planar pieces [@Derjaguin:1934]. Between two vis-à-vis, flat pieces of the opposing surfaces partitioned this way, locally a force acts like in the slab geometry and the total force is taken to be the sum of the forces between each individual pair with the appropriate areal weight. The Derjaguin approximation is widely used to estimate CCFs forces between colloidal particles, because the forces between parallel surfaces are much easier to calculate. Indeed, for the slab geometry the CCFs are known even beyond mean field theory (see below). Therefore, by using this approximation one can account even for non-Gaussian critical fluctuations, albeit at the expense of not fully considering the shape of the particles. The Derjaguin approximation is valid for temperatures which correspond to $\xi_t \lesssim R$, because under this condition the CCFs between the colloids act only at surface-to-surface distances $D$ which are small compared with $R$. In many cases this approximation is surprisingly reliable even for $D\lesssim R$ [@Gambassi-et:2009; @Troendle-et:2009-10]. In the absence of a bulk ordering field, i.e., for $h_b=0$ and strongly adsorbing $(+,+)$ confining surfaces the results for the critical Casimir interactions in the film geometry have been provided by field-theoretical studies [@Krech:1997], the extended de Gennes - Fisher local functional method [@Borjan-et:2008; @Okamoto-et:2012; @Mohry-et:2014], and Monte Carlo simulations of the Ising model [@Vasilyev-et:2007; @Vasilyev-et:2009; @Hasenbusch:2012] or improved models which offer the benefit that the amplitude of the leading bulk correction to scaling vanishes [@Hasenbusch:2012; @Hasenbusch:2010; @Francesco-et:2013; @Hasenbusch:2015]. Within MC simulations also the case of weakly adsorbing surfaces has been considered [@Vasilyev-et:2011; @Hasenbusch:2011]. In this latter case the corresponding surface field $h_s$ might be so small that upon approaching the critical point one effectively observes a crossover of the type of boundary condition imposed on the order parameter from symmetry preserving to symmetry breaking boundary conditions. In this case there appears to be no effective enhancement of the order parameter upon approaching the confining wall. The CCP reflects such crossover behaviors; depending on the film thickness, the CCP can even change sign [@Vasilyev-et:2011; @Hasenbusch:2011; @Mohry-et:2010]. On the basis of scaling arguments one expects that for moderate adsorption preferences the scaling function of the CCP in film geometry additionally depends on the dimensionless scaling variables $y^{(i)}_{s} = a_ih^{(i)}_{s}L^{\Delta_1/\nu}, i= 1 , 2$, where $h^{(1)}_{s}$ and $h^{(2)}_{s}$ are the effective surface fields at the two confining surfaces, $a_1$ and $a_2$ are nonuniversal amplitudes, $\Delta_1 =0.45672(5)$ [@Hasenbusch:2011] is the surface crossover exponent at the so-called ordinary surface phase transition.
The knowledge of the dependence of CCPs on the bulk ordering field $h_b$ is rather limited, although it is crucial for understanding the aggregation of colloids near the bulk critical point of their solvent. The presently available MC simulations for Ising films provide such results only along the critical isotherm $T=T_c^{(s)}$ [@Vasilyev-et:2013]. In Refs. [@Schlesener-et:2003; @Mohry-et:2012a; @Mohry-et:2012b; @Mohry-et:2014] the variation of CCFs with $h_b$ has been approximated by adopting the functional [ *form*]{} obtained within mean field theory (MFT, $d=4$) by using a field-theoretical approach within the framework of the Landau - Ginzburg theory, but by keeping the actual critical exponents in $d=3$ for the scaling variables. Within this “dimensional” approximation, the scaling function $\vartheta^{(d=3)}_{||} = f_C D^3/(k_BT\mathcal{S})$ of the CCF for the film geometry $(||)$ with macroscopically large surface area $\mathcal{S}$ of one wall (see the text below Eq. (\[eq:9\])) is taken to be $\vartheta^{(d=3)}_{||}(\mathcal{Y} = \mathrm{sgn}(t)D/\xi_t, \mathsf{\Sigma}=\Lambda/\mathcal{Y}= \mathrm{sgn}(th_b )\xi_t/\xi_h )\simeq \vartheta^{(d=3)}_{||}(\mathcal{Y},\mathsf{\Sigma}=0)\vartheta^{(d=4)}_{||}(\mathcal{Y},\mathsf{\Sigma})/\vartheta^{(d=4)}_{||}(\mathcal{Y},\mathsf{\Sigma}=0)$, where $\vartheta^{(d=3)}_{||}(\mathcal{Y},\mathsf{\Sigma}=0)$ has been adopted from MC simulation data [@Vasilyev-et:2007; @Vasilyev-et:2009]. (The normalization by $\vartheta^{(d=4)}_{||}(\mathcal{Y},\mathsf{\Sigma}=0)$ eliminates a nonuniversal prefactor carried by the Landau - Ginzburg expression for the scaling function of CCFs.) In Ref. [@Mohry-et:2014] the scaling functions $\vartheta^{(d=3)}_{||}$ resulting from the “dimensional” approximation have been compared with those obtained within the extended de Gennes - Fisher local functional. This allows us to compare in Fig. \[fig:1\] (unpublished) results for the scaling function $\Theta_{\oplus\oplus}^{(d=3,Derj)}(\mathcal{Y},\mathsf{\Sigma})$ of the sphere - sphere CCP (see Eq. (\[eq:4\])) within the Derjaguin approximation, i.e., $\Theta_{\oplus\oplus}^{(d=3,Derj)}(\mathcal{Y},\mathsf{\Sigma})\simeq (D/R) U_c = \pi \int_1^{\infty} \mathrm{d}x(x^{-2}-x^{-3})\vartheta^{(d=3)}_{||}(x\mathcal{Y},\mathsf{\Sigma})$ [@Mohry-et:2014], where $\vartheta^{(d=3)}_{||}$ is obtained by using these two aforementioned approaches. Only for weak bulk fields the curves in Fig. \[fig:1\] for both approaches compare well. Otherwise there are clear quantitative discrepancies, the origin of which is not clear. The “dimensional” approximation might deteriorate upon increasing $h_b$. On the other hand, as already mentioned in passing, the reliability of the local functional approach for $h_b \ne 0$ has not yet been tested systematically. For example, the result of the local functional approach for the scaling function of the CCFs along the critical isotherm differs substantially from the one obtained from MC simulations [@Mohry-et:2014]. In Refs. [@Buzzaccaro-et:2010; @Piazza-et:2011] results of a long wavelength analysis of density functional theory for the scaling function of CCF have been reported (see also Sec. \[subsubsec:pp\] below ); they are in qualitative agreement with the results obtained using the extended de Gennes - Fisher local functional [@Mohry-et:2014]. The first microscopic off-lattice results for the scaling function of CCF, both along the critical isochore and in the off-critical regime, have been worked out in Ref. [@Anzini-et:2016]. The authors of Ref. [@Anzini-et:2016] have studied a hard core Yukawa model of a fluid by using density functional theory (DFT) within a specific weighted density approximation [@Leidl-et:1993], coupled with hierarchical reference theory [@Parola-et:1995]. This kind of DFT weighted density approximation, related to the one proposed early on by Tarazona and, independently, by Curtin and Ashcroft [@Tarazona:1985; @Curtin-et:1985], captures the short-ranged correlations of the underlying hard sphere fluid rather accurately, whereas the hierarchical reference theory is able to account also for the critical properties of a homogeneous fluid. In Ref. [@Anzini-et:2016], this technique has been applied in order to determine the effective interaction between two hard walls immersed in that fluid. This approach facilitates to investigate the crossover between depletion-like roots of such effective forces at high temperatures and the critical Casimir effect upon approaching the critical point of the fluid. It appears that for hard core Yukawa fluids the universal features of CCF emerge only in close neighborhood of the critical point. The predictions obtained in Ref. [@Anzini-et:2016] for the scaling function of the CCF for various temperatures along the critical isochore ($h_b=0$) differ significantly from the Ising model MC simulation results in Ref. [@Vasilyev-et:2009]. Moreover, the data along various isotherms do not collapse as expected from scaling theory. The authors of Ref. [@Anzini-et:2016] interpret these deviations as an indication of strong corrections to scaling occurring in the hard core Yukawa fluid.
As far as experimental data for the [*film*]{} geometry are concerned, they are available only indirectly from measurements of the thickness of wetting films; they correspond to opposing $(+,-)$ boundary conditions [@Fukuto-et:2005; @Rafai-et:2007]. The scaling function of the CCF determined by Fukuto [*et al.*]{} [@Fukuto-et:2005] compare well with the theoretical predictions. The measurements performed by Rafai [*et al.*]{} [@Rafai-et:2007] have provided data which show stronger deviations from the theoretical curves, in particular around the maximum of the CCF. The scaling function of CCFs for wetting films corresponding to $(+,+)$ boundary conditions could not be determined due to the collapse of the incomplete wetting film in the course of the measurements [@Rafai-et:2007]. Based on exact results in $d=2$ [@Evans_Stecki; @Abraham_Maciolek:2010; @Abraham_Maciolek:2013] and in $d=4$ [@Krech:1997], and based on the predictions from the local-functional approach in $d=3$ [@Borjan-et:2008], it is expected that the CCF $f_C(D)$ between two planar walls with symmetry-breaking boundary conditions at separation $D$ varies as $\exp(-D/\xi_t)$ for $D\gg \xi_t$ and $t >0$, i.e., in the one-phase region of the solvent. This implies that the asymptotic behavior of the scaling function is given by $\vartheta^{(d=3)}_{||}(\mathcal{Y}= \mathrm{sgn}(t)D/\xi_t\gg 1,\mathsf{\Sigma} = \mathrm{sgn}(th_b )\xi_t/\xi_h=0)=\mathcal{A}_+\mathcal{Y}^d\exp(-\mathcal{Y})$, where $\mathcal{A}_+=1.2-1.5$ is a universal number valid for $(+,+)$ boundary conditions [@Gambassi-et:2009]. By applying the Derjaguin approximation to this functional form of the CCFs, one obtains the following asymptotic behavior of the scaling function $\Theta_{\oplus\oplus}^{(d=3,Derj)}(\mathcal{Y},\mathsf{\Sigma}=0)$ for the sphere - sphere CCP : $$\label{eq:13}
\Theta_{\oplus\oplus}^{(d=3,Derj)}(\mathcal{Y}\to\infty,\mathsf{\Sigma}=0) = \pi \mathcal{A}_+\mathcal{Y}e^{-\mathcal{Y}}, \qquad D\gg \xi_t, \qquad t>0,$$ so that in this limit $U_C^{(d=3,Derj)}=\pi\mathcal{A}_+\frac{R}{\xi_t}e^{-D/\xi_t}$ (Eq. (\[eq:4\])).
### Pair potential in the presence of depletants {#subsubsec:pp}
In order to calculate the effective pair potential between two large hard spheres immersed into a fluid of depletants close to their gas-liquid critical point, the authors of Refs. [@Buzzaccaro-et:2010; @Piazza-et:2011] have used density functional theory (DFT), which is a powerful tool for describing equilibrium properties of colloidal suspensions (see the last two paragraphs in Sec. \[subsubsection:eff\_inter\]). The hard spheres have been analyzed within the Derjaguin approximation, which is inherent in recent approaches to depletion forces acting in hard-sphere mixtures. In this context, the Derjaguin approximation relates the force between the two big objects to the integral of the solvation force $f_{solv}(D) = -(1/\mathcal{S}) (\partial\Omega/\partial D)_{\mu,T,\mathcal{S}} -p)$ of the small particles (i.e., the depletant agents) confined between two parallel planar walls with cross-sectional area $\mathcal{S}$. (The film solvation force per surface area $\mathcal{S}$, $f_{solv}$, is an excess pressure over the bulk value $p$ of the confined fluid described by the grand potential $\Omega$.) In the limiting case of hard walls exposed to an ideal gas of depletants, this relation reproduces the well known Asakura-Oosawa result for depletion forces [@Asakura-et:1954]. This scheme can be fruitfully applied also to interacting systems. In fact such a relation is equivalent to the general formula obtained by Derjaguin [@Derjaguin:1934] relating the force between two convex bodies to the free energy, in excess of its bulk value, of a fluid confined between planar walls. Within an actual DFT approach this formula holds even for approximations of the excess part of an intrinsic Helmholtz free energy functional of a fluid. The authors of Refs. [@Buzzaccaro-et:2010; @Piazza-et:2011] have employed the square gradient local density approximation for the intrinsic Helmholtz free energy functional entering into DFT, which is valid for spatially slowly varying depletant number densities $\rho$ such that $\nabla\rho/\rho \ll 1/\xi_t$. They have simplified the Derjaguin approximation by replacing the two big spheres by two planar parallel walls. Within the square gradient local density approximation, the effective (solvation) force per area $f_{solv}$ between two parallel walls is expressed solely in terms of the bulk free energy density $f(\rho)$ of the depletant host fluid and in terms of the value of the equilibrium number density profile $\rho(z=z_{mid})$ of such a fluid at the midpoint $z=z_{mid}$ between the two walls. The solution of the extremum condition for the free energy functional provides the second equation relating, albeit in an implicit way, $f_{solv}$ and $\rho(z=z_{mid})$ to the density at the wall $\rho(z=0)$, which in turn depends on the wall-fluid interactions. In order to incorporate the non-Gaussian behavior near the gas-liquid critical point of the depletant, in the spirit of the Fisk - Widom or local-functional approach [@Fisk:1969; @Fisher-et:1990; @Okamoto-et:2012], the authors of Refs. [@Buzzaccaro-et:2010; @Piazza-et:2011] have used the scaling form for the singular part of the bulk free energy density of the fluid $\left(f(\rho)-f(\rho_c)\right)/(k_BT) = a_{11}t^{2-\alpha} \Psi(x)$ with the scaling variable $x=b_1\phi t^{-\beta}$, where $\phi = \rho -\rho_c$ is the bulk order parameter OP, $a_{11}$ and $b_1$ are non-universal, dimensional constants, and $\beta$ and $\alpha$ are bulk critical exponents [@Pelissetto-et:2002]. In the scaling limit, the general expression for the effective force between two parallel walls can be expressed in terms of $\Psi(x)$, the rescaled midpoint OP $x_0=b_1\left(\rho(z_{mid})-\rho_c\right)t^{-\beta}$, and the universal amplitude ratio $g_4^+$ [@Pelissetto-et:2002]. This expression takes the scaling form characteristic of CCFs. By using a parametric expression for $\Psi$ [@Pelissetto-et:2002] and the critical exponents of $3d$ Ising universality class, the authors have obtained very good agreement between their analytic theory and the scaling function of CCFs obtained from MC simulations [@Vasilyev-et:2009] at $h_b=0$. This is not surprising because, within the square gradient local-density approximation, DFT reduces to the local-functional approach which proved to capture correctly critical fluctuations [@Fisher-et:1990; @Borjan-et:1998; @Borjan-et:2008], at least for $h_b=0$.
This theoretical approach has been followed up by MC simulation studies [@Gnan-et:2012a] aiming at the computation of the CCP between two hard-sphere colloids suspended in an implicit solvent in the presence of interacting depletant particles. In this off-lattice MC simulations of fluctuation induced forces, the effective potential between two hard spheres has been determined upon approaching the gas-liquid critical point of the depletant for two different depletant models, one for SW and one for 3P particles (see Eq. (\[eq:7\]) and thereafter). Given the computational limitations, the authors have considered the size ratio $q$ between the hard-sphere depletant and the hard-sphere colloid within the range $ 0.05 < q < 0.2$. The resulting effective colloid-colloid force has been evaluated by using canonical Monte Carlo simulations for various fixed values of the depletant concentration in the reservoir. The method consists of performing virtual displacements of each colloid from its fixed position and of computing the probability of encountering at least one collision with a depletant particle. The effective colloid-colloid potential follows from integrating the corresponding force. These MC results show that upon cooling the effective potential between two colloidal particles gradually looses its high-temperature, pure hard-sphere depletion character of exhibiting oscillations and transforms into a completely attractive potential with a progressive and significant increase of its range, signaling the onset of critical Casimir forces. For large distances between the surfaces of the two colloids, the MC data for the CCP fit well to the asymptotic form given by Eq. (\[eq:13\]). This numerical study has been extended to the case in which colloids interact with SW depletant particles; this interaction has been continuously modified from hard-core repulsion to strong attraction, thus changing from $(-,-)$ boundary conditions (i.e., preference for the gas phase) to $(+,+)$ boundary conditions (i.e., preference for the liquid phase) [@Gnan-et:2012c]. For strong colloid-depletant attraction, the effective colloid-colloid potential exhibits oscillations, as they occur for the high-temperature depletion potential, modulating its exponentially decaying attractive tail. The variation of the colloid-colloid effective potential upon crossing over from $(-,-)$ to $(+,-)$ and from $(+,+)$ to $(+,-)$ boundary conditions has been determined, too. In the asymptotic spatial range these two crossovers are the same. However, in the numerical study of Ref. [@Gnan-et:2012c] only the behavior at short distances has been probed, where the effective potential is dominated by non-universal aspects of the solvent-colloid interaction so that the aforementioned pairs of boundary conditions are no longer equivalent.
### Main properties of CCFs {#subsubsec:sumCCP}
We close this section by listing the main properties of critical Casimir interactions between two colloidal particles in very dilute suspensions. The results for spheres in $d = 3$, obtained from the variety of methods described above, tell that they share the same qualitative features of the CCFs with “spheres” in $d=2$ and $d=4$ as well as with planar walls. This refers to the property that for like boundary conditions CCFs are attractive and it refers to the position of a force maximum along various thermodynamic paths. Around the consolute point of a binary solvent the main features of the CCFs between two planar walls are summarized in Fig. \[fig:2\] in terms of the force scaling function $\vartheta^{(d=4)}_{||}(\mathsf{Y},\mathsf{\Sigma})$ obtained from Landau theory (see Fig. 1 in Ref. [@Mohry-et:2014]). The main message conveyed by Fig. \[fig:2\] is the asymmetry of the CCFs around the critical point of the solvent with the maximum strength occurring at $h_b<0$. This asymmetry is due to the presence of surface fields, which in films lead to capillary condensation [@Evans:1990], whereas between spherical colloids to bridging [@Bauer-et:2000; @Archer-et:2005; @Okamoto-et:2013]. Near these phase transitions, the effective force acting between the confining surfaces is attractive, exhibits (within Landau theory) a cusp, and is very strong: the depth of the corresponding CCPs can reach a few hundred $k_BT$. The concomitant strong increase of the absolute value of the force is reflected by its corresponding universal scaling function and extends to the thermodynamic region above the capillary condensation or bridging critical point, even to temperatures $T >T_c^{(s)}$.
Stability {#subsec:stab}
---------
Knowing the effective pair potential for the colloids one can investigate the stability of colloidal suspensions and the aggregation of colloids. These phenomena are related to kinetic processes (see Ref. [@Russel-et:1989] and references therein), which are based on the diffusion of single particles in the presence of other particles of the same kind, interacting with them via interaction potentials which contain both attractive and repulsive contributions. Aggregation occurs if the attractive interactions of the particles dominate over their thermal kinetic energy, which is responsible for the Brownian motion of the particles. Hydrodynamic interactions may also play a role, e.g., by slowing down the aggregation process for solvents with high viscosity. In order to quantify the behavior of interacting particles, which irreversibly stick together once their surfaces touch each other, Fuchs has introduced the concept of a stability ratio [@Fuchs:1934]. It is defined as the ratio $W=J_0/J$ between the Brownian motion induced pair formation rate $J_0$ in the absence of other than excluded volume interactions between the particles, and the corresponding formation rate $J$ of particle pairs in the presence of such interactions. $W$ can be calculated by extending Smoluchowski’s diffusion equation for the radially symmetric relative motion of two coagulating spherical particles of radius $R$ in order to account for their interaction potential [@Fuchs:1934]: $$\label{eq:15}
W = 2R \int_{2R}^{\infty}\frac{\exp\left\{U(r)\right\}}{r^2} \text{ d}r.$$ The analysis, which leads to this expression, is valid only in the early stages of coagulation before triplets etc. are formed. However, it neither deals with the very beginning of the coagulation process but considers only the steady-state situation, which is established quickly. In the analysis based on Smoluchowski’s equation hydrodynamic interactions are neglected. From the definition of $W$ it follows that for hard spheres $W=1$, while for $W>1$ ($W<1$) the repulsive (attractive) part of the pair potential $(k_BT)U$ dominates. In the case of a potential barrier, i.e., if $U(r)\gg 1$ for a certain range of distances $r>2R$, which leads to $W>1$, one can expect that on intermediate time scales the suspension will be in a (meta)stable homogeneous state. The cluster formation will set in only on a very large time scale. This time scale is proportional to the ratio between the characteristic times for diffusion of a single particle, $t_{diff}$, and for the formation of a pair of particles, $t_{pair}$ [@Russel-et:1989]: $$\label{eq:16}
t_{diff}/t_{pair}=3 \eta/W$$ with the packing fraction $\eta=(4\pi/3)R^{3}\rho$ where $\rho$ is the number density of the colloidal particles. For a near-critical solvent, the pair potential can be taken as (see Eqs. (\[eq:2\]) and (\[eq:4\]); $\mathsf{\Sigma} = \Lambda/\mathcal{Y}$)
$$\label{eq:17}
U(r)=
\begin{cases}
\infty, & D<0 \\
U_{rep}+U_C^{(d=3)}=
A\exp(-\kappa D)+(1/\Delta)\Theta^{(d=3)}(\mathcal{Y},\Delta, \mathsf{\Sigma}), & D>0.
\end{cases}$$
For the mixed phase of the solvent the ratio $W$ has been calculated in Ref. [@Mohry-et:2012b]. The results of this calculation show that the CCF can lead to a rapid coagulation, setting in within a narrow temperature interval, and that the extent of the coagulation region reflects the fact that the CCFs are stronger for compositions of the solvent slightly poor in the component preferred by the colloids. The effective pair potential given by Eq. (\[eq:17\]) is applicable only for sufficiently large distances $D\gtrsim\kappa^{-1}$, because it takes into account only the electrostatic repulsive interactions and neglects possible short-ranged contributions to the effective van-der-Waals interactions. Furthermore, the CCP attains its universal form as given by the scaling function $\Theta^{(d=3)}$ only in the scaling limit, i.e., for distances $D$ which are sufficiently large compared with the correlation length amplitude $\xi_{t,+}^{(0)}\approx 0.25nm$. Analogously, also $\xi_t$ and $\xi_h$ must be sufficiently large compared with microscopic scales. From the behavior of the scaling function $\Theta^{(d=3)}$ (discussed in Ref. [@Mohry-et:2012b] within the Derjaguin approximation) it follows that along the typical thermodynamic paths realized experimentally, the [*range*]{} of attraction due to the CCFs grows steadily upon increasing the bulk correlation length $\xi_t$, but the [*amplitude*]{} of the CCFs is a nonmonotonic function of $\xi_t$ with its maximal strength attained for an intermediate value of $\xi_t$. Depending on the values of $A$ in Eq. (\[eq:17\]) and of $\xi_t$, the CCFs compensate the repulsion for all values of $D$ or only within certain ranges of $D$, i.e., a secondary attractive minimum of $U(r)$ can occur at a certain distance $D_{min}$ (in addition to the primary, global minimum at $D \to 0^+$) while for small distances $D < D_{min}$ and for $D<0$ the potential maintains a repulsive part. The presence of the repulsive barrier and of the attractive secondary minimum in the effective potential, and thus the occurrence of coagulation, depends on temperature. But also in the case, in which for [*all*]{} values of $\xi_t$ a repulsive barrier remains, coagulation can appear, due to a deep secondary minimum. The nonmonotonic dependence of the maximal strength of the CCF on temperature results in a nonmonotonic behavior of the ratio $W$, which is shown in Figs. 2 and 3 of Ref. [@Mohry-et:2012b].
In Ref. [@Mohry-et:2012b], the analysis of the stability ratio for the diversity of possible shapes of the effective pair potential given in Eq. (\[eq:17\]) has been complemented by the analysis of the bulk structure of colloidal suspension. To this end the radial distribution function $g(r)$ has been calculated within the integral equation approach by using the hypernetted-chain and the Percus-Yevick closure [@Hansen-et:1976]. The results for both types of closure are almost the same. (The applicability and reliability of this integral equation approach is discussed in detail in Ref. [@Caccamo:1996].) For temperatures far away from the critical temperature of the solvent, the colloids behave effectively as hard spheres with an effective diameter $\sigma > 2R$ due to the soft repulsive background contribution $U_{rep}$. Accordingly, for such values of $\xi_t$, $g(r)$ has the corresponding characteristics of a fluid of hard spheres, such as the rather broad first peak for small values of $D$. Due to the emerging attractive CCFs, for increasing $\xi_t$ the radial distribution function $g(r)$ is enhanced close to the surfaces of the colloids. This implies an enhanced short-ranged order and that the formation of colloidal dimers is favored. The way in which the shape of the radial distribution function $g(r)$ changes upon increasing the temperature reveals whether the effective potential exhibits a repulsive barrier at small values of $D$ and is attractive throughout large distances or whether an attractive minimum develops at intermediate values of $D$ upon increasing temperature while repulsion remains at small and large values of $D$.
One can address the issue whether a relationship can be established between the onset of aggregation and the behavior of a certain quantity, which is accessible both theoretically and experimentally, such as the second virial coefficient $B_2$. For dilute suspensions, the second virial coefficient [@Hansen-et:1976] provides information about the strength of the radially symmetric attraction between spherical particles: $$\label{eq:18}
B_2 = \frac{1}{2}\int \text{d}^3r \left(1-e^{-U(r)}\right) = 2\pi \int_{0}^{\infty}\text{d}r\hspace*{1pt} r^{2} \left(1-e^{-U(r)}\right) = 2\pi \int_{0}^{\infty}\text{d}r\hspace*{1pt} r^{2} \left(1-g(r)\right).$$ Beyond the ideal gas contribution it determines the leading non-trivial term in the expansion of the pressure $p(\rho)/(k_BT\rho)=1+B_2\rho+\ldots$ in terms of powers of the number density $\rho$. In Ref. [@Mohry-et:2012b] $B_2$ has been calculated for the potential given by Eq. (\[eq:17\]) at the state points for which, according to the experiments reported in Ref. [@gallagher:92], aggregation sets in. It turns out that at these states of aggregation onset the values of $B_2$ are close to each other and that to a certain extent those $B_2$-isolines, which emerge by belonging to these experimental data points, agree with each other and with the possible shape of the aggregation onset line (see Fig. \[fig:3\]). However, based on this analysis one cannot state definitely that $B_2$ can serve as a quantitative indicator for the onset of aggregation. Further efforts, both theoretically and experimentally, are needed to provide more precise values of the relevant quantities.
Phase behavior {#subsec:therm}
--------------
In suspensions with sufficiently large packing fractions of colloids, the attraction among them due to CCFs can induce a so-called “liquid-gas” phase separation of the colloids, i.e., the separation of two phases which differ with respect to their colloidal number density. Adopting the effective one-component approach allows one to use standard liquid state theory in order to determine the onset of phase separation. Within this effective approach feedback mechanisms of the colloids, acting on the solvent and changing its critical behavior, are neglected. Therefore this approximation does not allow one to describe reliably all details of the full many-component system. However, one can identify certain regions of the thermodynamic phase space for which this approach is applicable. One expects the effective one-component model to work well for temperatures corresponding to the one-phase region of the pure solvent and for an intermediate range of values of the colloid number density $\rho$. The latter should be large enough so that the competition between the configurational entropy and the potential energy due to the effective forces can induce a phase separation, but small enough so that the approximation of using an effective pair potential between the colloids is valid and the influence of the colloids on the phase behavior of the solvent is secondary.
In Ref. [@Mohry-et:2012a], for particles interacting via the effective pair potential given in Eq. (\[eq:17\]), the phase coexistence curve $T^{(\mathrm{eff})}_{cx}(\eta=(4/3)\pi R^3\rho\,\vert\, c_a)$ with the critical point temperature $T_c^{(eff)}$ and the spinodal (i.e., the loci where, within mean-field theory, the isothermal compressibility $\chi_T$ diverges) have been calculated using density functional theory within the so called random phase approximation [@Evans:1979] and the integral equation approach [@Hansen-et:1976]. The results obtained from these two approaches, which are presented in Fig. \[fig:4\] (Fig. 2 in Ref. [@Mohry-et:2012a]), differ only slightly. The loci of the phase separation depends sensitively on the strength $A$ of the repulsive part of the effective pair potential (Eq. (\[eq:3\])) and the solvent compositions. For solvent compositions, which are somewhat poor in the component preferred by the colloids, even short correlation lengths suffice to bring about phase separation. The critical value $\eta_c$ of the colloidal packing fraction $\eta=(4\pi/3)R^3\rho$ (in terms of number density $\rho$) is rather small, i.e., $\eta_c\approx 0.07$, because the effective hard sphere diameter $\sigma$, which results from the soft repulsion $U_{rep}$, is larger than $2R$. Furthermore, the binodals shown in Fig. \[fig:4\] are rather flat compared with, e.g., the ones for hard spheres interacting via a short-ranged, attractive, and temperature independent potential. In the present system, the deviation of $T$ from the critical temperature $T_c^{(s)}$ which corresponds to a range of $\eta$ for the coexisting phases as large as the one shown in Fig. \[fig:4\], is about $1$, whereas for a system of hard spheres with an additional attractive interaction the corresponding temperature deviation is a few percent. The critical temperature $T_c^{(eff)}$ obtained within DFT agrees with the simple prediction $B^*_2 = B^*_{2,c}$, acting as an implicit equation for $T_c$, as suggested by Vliegenthart and Lekkerkerker [@Vliegenthart-et:2000] and Noro and Frenkel [@Noro-et:2000] (VLNF), where $B_2^*(T) \equiv B_2(T)/B_2^{(HS)}$ is the reduced second virial coefficient and $B^*_{2,c}$ is the critical value of $B_2^*$ for Baxter’s model of adhesive hard spheres [@Baxter:1968]. $B_2^{(HS)} =\frac{2\pi}{3}\sigma^3$ is the second virial coefficient of a suitable reference system of hard spheres (HS) with diameter $\sigma$. The effective hard-sphere diameter can be taken as $\sigma=\int_0^{r_0}(1-\exp (-U(r))\mathrm{d}r $, with $U(r=r_0)=0$. One can adopt also other definitions of $\sigma$ (for a discussion see Ref. [@Andersen-et:1971]). According to VLNF, $B_2^*$ is a useful indicator of the occurrence of a phase separation into a colloidal-rich (“liquid”) and a colloidal-poor (“gas”) phase. An extended law of corresponding states proposed by VLNF predicts that the value of the reduced second virial coefficient $B_2^*$ at the critical point is the same for all systems composed of particles with short-ranged attractive interactions, regardless of the details of these interactions. This (approximate) empirical rule is supported by experimental data [@Vliegenthart-et:2000] and theoretical results [@thN].
Multicomponent mixture {#sec:ter}
======================
General discussion {#subsec:gd}
------------------
For the kind of systems considered here, the determination of phase equilibria is rather subtle because due to the concomitant adsorption phenomena, which are state dependent the effective potential between the colloids depends on the thermodynamic state itself. Experiments have revealed [@gallagher:92; @grull:97] that for suspensions very dilute in colloids and with a phase separated solvent, basically all colloidal particles are populating the phase rich in the component, say $a$, preferred by the colloids (this phase has concentration $c_a^{(1)}$). This implies that in coexisting distinct phases the effective potential acting between the particles is different. This is not captured by the effective potential approach discussed above. Besides the colloid-solvent also the solvent-solvent interactions can influence the effective potential and, accordingly, the phase behavior of the effective colloidal system. This has been demonstrated by recent MC studies in which various kinds of model solvents have been used [@Gnan-et:2012a]. On the other hand, the presence of colloidal particles may alter the phase behavior of the solvent. In the case of molecular fluids, it is well established that the phase diagrams of ternary mixtures, as they emerge from those of binary mixtures by adding a third component, are distorted and become more complex relative to the underlying original binary ones [@Andon-et:1952; @Prafulla-et:1992]. Similar distortions and complex features of phase diagrams are observed experimentally [@Kline-et:1994; @Jayalakshmi-et:1997; @Koehler-et:1997] upon adding colloidal particles to binary solvents. In particular, one finds a decrease of the lower critical temperature. These observations tell that for reliably determining such phase diagrams the full, many component mixture has to be considered. The importance of considering the colloidal suspension with a binary solvent as a truly ternary mixture has been already pointed out in Ref. [@Sluckin:1990]. Taking into account the features of the CCP one can try to predict the expected topology of the phase diagrams for ternary solvent-solvent-colloid mixtures [@Mohry-et:2012a]. Upon adding colloids, the two-phase region of the demixed phases of the pure solvent extends into the three-dimensional thermodynamic space of the actual colloidal suspension. At fixed pressure this space is spanned by $T$, the concentration $c_a$ of the component $a$ of the binary solvent, and, e.g., by the colloidal number density $\rho$. The actual shape of the two-phase region forming a tube-like manifold is expected to depend sensitively on all interactions present in the ternary mixture (see Fig. \[fig:5\]). The foremost difficulty in describing such mixtures theoretically or in dealing with them via computer simulations consists of the fact that the sizes of their constituents differ by, a few, orders of magnitude. This property distinguishes them significantly from mixtures of hard spheres, needles, and polymers [@Schmidt-et:2002; @Schmidt:2011], for which the constituents are of comparable size, i.e., none of them is larger by a factor of ten or more than the others. In contrast to molecular ternary mixtures as modeled, e.g., in terms of a lattice gas [@terMixTheory], in colloidal suspensions the colloidal particles influence the other two components not only by direct interactions but also via strong entropic effects. These occur because their surfaces act as confinements to fluctuations of the concentration of the solvent and they also generate a sizeable excluded volume for the solvent particles.
Monte Carlo studies {#subsec:mcm}
-------------------
MC simulations for a lattice model of ternary mixtures, suitably mimicking colloidal particles suspended in a near-critical binary solvent, offer the advantage over other available approaches that they account for both fluctuations and nonadditivity of the emerging CCFs. Due to the large size difference between the colloid and solvent particles and due to the critical slowing down of the ternary mixture upon approaching its critical point, studying the corresponding three-dimensional systems by MC simulations is computationally challenging. As a computationally cheaper substitute, two-dimensional lattice models have been treated by this method in Refs. [@bob-et:2014; @Tasios-et:2016; @Hobrecht-et:2015]. In a first of such a MC simulation study [@bob-et:2014], the colloids have been modeled as discretized hard discs of radius $R$ (in units of the lattice spacing) occupying a fraction $\eta$ of sites on the square lattice. The remaining lattice sites have been taken to be occupied by a solvent molecule of either species $a$ or $b$ with no empty sites left. A similar model of solvent-solvent-colloid mixtures has been considered earlier by Rabani [*et al.*]{} [@Rabani-et:2003] in order to study drying-mediated self-assembly of nanoparticles. In Refs. [@bob-et:2014; @Tasios-et:2016], only the nearest neighbor repulsive interactions between the components of a binary solvent have been considered. This drives phase segregation, which in the absence of colloids has a critical point belonging to the $2d$ Ising universality class. In order to mimic the preference of the colloids for component $b$ of the binary solvent, a nearest-neighbor attractive interaction between the colloid and component $b$ has been taken into account. It has been assumed that there is only a nearest-neighbor repulsion between the colloid and component $a$. In the limit $\Delta\mu_s = \mu_a - \mu_b \to \infty$ of the chemical potential difference between the components $a$ and $b$, the ternary mixture reduces to a binary mixture of $2d$ hard-discs and solvent component $a$, which exhibits coexistence between a fluid and a solid phase. In the study by Edison [*et al.*]{} [@bob-et:2014], the preclusion of solvent-mediated colloidal aggregation, arising from complete wetting and capillary condensation, has been implemented by focusing on colloids, which are immersed in a supercritical mixed phase of components $a$ and $b$, such that $b$-rich critical adsorption layers (for critical adsorption profiles at spheres and cylinders see Refs. [@Hanke-et:1999; @Yubanaka-et:2017] and references therein) form on the colloid surfaces in equilibrium with a supercritical $a$-rich solvent in the bulk. The careful simulation study in Ref. [@Tasios-et:2016], in which the solvent has been treated grand canonically at constant pressure, has revealed the existence of gas-liquid and fluid-solid transitions, occurring in a region of the thermodynamic variables $(\eta, t=(T-T_c^{(s)})/T_c^{(s)}, \Delta\mu_s = \mu_a - \mu_b)$ of ternary mixtures which extends well away from the critical point of the solvent reservoir (especially concerning the fluid-solid transitions). It has been found that in all phases the local solvent composition is strongly correlated with the local colloid density. The coexisting colloidal liquid and solid phases are poor in component $a$, whereas in the gas phases coexisting with the liquid or the solid phases the solvent composition is very close to the composition of the solvent reservoir, which, however, is far from its critical composition. These features agree with experimental observations [@gallagher:92; @grull:97] and cannot be captured by an effective one-component approach as discussed in the previous section. As expected [@Evans-et:1994], all pair correlation functions decay exponentially on the scale of the same correlation length. Strikingly, the correlation length found in the homogeneous supercritical state of the ternary mixture was much larger than the colloid radius ($R=6$), which in turn clearly exceeded the correlation length of the solvent reservoir. Upon adding a small volume fraction of colloids, the gas-liquid critical point of a ternary mixture shifts continuously from that of the colloid-free solvent to the negative values of $\Delta\mu_s$, which decrease upon increasing $t$. In addition, the MC simulation data indicate the existence of a second, lower metastable gas-liquid critical point located at larger values of $\eta$. The gas-liquid-solid triple point is also expected to occur near the fluid-solid coexistence of pure hard discs shown as the vertical dashed lines in Fig. \[fig:6\]. The authors of Ref. [@Tasios-et:2016] have checked to which extent a description in terms of an effective pair potential can account for the phase behavior observed in their MC simulations. They have concluded that those approaches, which exclusively employ effective pair potentials, as obtained from, e.g., planar slit studies combined with the Derjaguin approximation, overestimate the extent of gas-liquid coexistence and underestimate how much the critical point of the ternary mixture is shifted relative to that of the solvent reservoir.
In Ref. [@Hobrecht-et:2015], the two-dimensional Ising model, with embedded colloids represented as disclike clusters of spins with fixed orientation, has been treated by the geometric cluster algorithm (see Sec. \[subsubsec:cs\]). However, the focus of this study has not been the phase behavior of a ternary mixture but rather the two- and three-body CCP. Employing the geometric cluster algorithm facilitated the study of the same lattice model as in Refs. [@bob-et:2014; @Tasios-et:2016] but in $d=3$ [@Tasios-et:2017]. As anticipated in Refs. [@bob-et:2014; @Tasios-et:2016], the phase diagram displays the same qualitative features as in the two-dimensional case.
Mean field theory {#subsec:mftm}
-----------------
The mean field approximation of the model considered in Ref. [@Tasios-et:2016] was studied in more detail in Ref. [@Edison-et:2015b]. It is based on free-volume arguments, according to which the Helmholtz free energy is taken as the sum of three contributions, describing the direct interactions among the pure colloids, the pure binary mixture of solvent $a$ and solvent $b$, and the colloid - solvent $b$ mixture. For the pure-colloid contribution the hard-disc free energy has been employed, with its distinct expressions for the fluid [@Santos-et:1995] and the solid [@Young-et:1979] phase. In the free space between the colloids, the mean field free energy for the $a-b$ mixture has been taken; the solvent is excluded from the volume of the colloids. Concerning the colloid - solvent $b$ contribution, the mean adsorption energy of component $b$ at the colloid surface has been taken. As can be inferred from Fig. \[fig:6\], the topology of the resulting phase diagram and its temperature dependence agree qualitatively with the one obtained from simulations (see Fig. S2 in the Supplemental Material for Ref. [@Tasios-et:2016] and Fig. 1 in Ref. [@Edison-et:2015b]). In particular, a lower critical point has been found. However, as already expected on the basis of the different spatial dimensions, there is no quantitative agreement; the parameter values used in order to obtain phase diagrams, which are similar to those from simulations, were chosen empirically. It seems that in these studies the underlying mean field approximation severely underestimates the effect of the hard-core repulsion of the colloids. In addition to the colloidal gas and colloidal liquid, this theory predicts the occurrence of two crystal phases which have the same (hexagonal) structure but with different lattice spacings. In the 3d thermodynamic state space of a ternary mixture spanned by $\Delta\mu_s$, $\eta$, and $t$ one finds also upper and (metastable) lower colloidal gas-liquid critical lines and a colloidal solid-solid critical line. (The critical points in Fig. \[fig:6\](a) lie in the planes cutting the full phase diagram at three constant temperatures.) The fluid-solid transition corresponds to a first-order freezing transition. The critical point of the colloid-free solvent ($t=0$, $\Delta\mu_s = 0$, $\eta = 0$) is shifted considerably upon adding a small amount of colloids (see the red dots in Fig. \[fig:6\](a)). The authors of Ref. [@Edison-et:2015b] have pointed out that experiments on colloidal aggregation are typically performed by suspending a fixed number of colloids in a solvent at a fixed (pure) solvent composition, which we denote by $c_a^{(s)}$, and then by adjusting the temperature of the system to reversibly induce aggregation. They have identified the so-called aggregation line, i.e., the loci of the points $(t,c_a^{(s)})$ at fixed $\eta$ and $c_a^{(s)}$ at which aggregation is observed first, as the line which demarcates the one-phase region of the ternary mixture, which lies at the outside of the line, and the region at its inside where colloidal phase separation can be found. Such a line (see Fig. \[fig:7\]) ends at an $\eta$-dependent critical point of the ternary mixture, which is shifted from the critical point of the pure solvent towards a higher ($\eta$-dependent) temperature. In the vicinity of their end points these lines are slightly bent, which indicates reentrant dissolution. Such lines qualitatively resemble the experimental results for strong aggregation reported by Beysens and Estève in Ref. [@Beysens-et:1985] (see the solid line in Fig. 3 therein). This tends to support the view on the aggregation lines as the line of onset of colloidal phase separation in the full ternary mixture. The above mean field calculation has been performed in $d=2$ [@bob-et:2014] and then extended to $d=3$ [@Edison-et:2015b] but no new features of the phase behavior have been found. Although the uncertainties generated by the large differences of length scales prevent a quantitative comparison with the available experimental data, the studies reported in Refs. [@bob-et:2014; @Edison-et:2015b] are very valuable because they reveal the possible scenarios for the phase behavior, which can occur in such ternary mixture, highlighting the physical mechanisms behind it.
Experiments {#sec:exp}
===========
Phase transitions and aggregation in bulk {#subsec:ag_ph}
-----------------------------------------
The view, that aggregation, as observed in pioneering experiments by Beysens and Estève [@Beysens-et:1985], is in fact a reversible phase transition in a ternary suspension, has been tested via early experimental studies by Wegdam, Schall [*et al.*]{} [@Guo-et:2008]. In these studies, charge-stabilized polystyrene spheres of radius $R=105$ nm suspended in a mixture of 3-methylpyridine (3MP), water, and heavy water near its lower critical point have been considered. The mass fractions of the components of this latter mixture have been chosen such that the mass density of the solvent mixture closely matches that of the colloidal particles in the region of the parameter space where phase transitions have occurred. If the system is density-matched, the growth of the nucleated liquid or solid phases is not perturbed by gravity at a very early stage and can be followed until macroscopically large coexisting phases are formed. In order to characterize the phase behavior of the system the authors have used small angle $X$-ray scattering (SAXS). Transition temperatures were also determined by measuring the sample turbidity. Moreover, the samples have been observed directly with a CCD camera. The only information, which has been provided by the authors about the phase diagram of the 3MP-water-heavy water mixture at the mass fractions actually used, is the [*coe[*x*]{}istence*]{} temperature $T^{(s)}_{cx}\approx 65$ at which this (chemically in fact binary) solvent mixture demixes. The schematic phase diagram shown as Fig. 1(a) in Ref. [@Guo-et:2008] suggests that the considered mass fraction $\omega_{\mathrm{3MP}}$ of 3-methylpyridine investigated in this study (and therein denoted as $c_{\mathrm{3MP}}$) has been smaller than the critical one. Instead of $t=(1-T/T^{(s)}_c)$, the authors of Ref. [@Guo-et:2008] have used the deviation $\Delta T = T-T^{(s)}_{cx}$ as the actual control parameter. Upon increasing the temperature from a value in the one-phase region of the pure solvent towards $T^{(s)}_{cx}$, the onset of colloid aggregation, at which the system separates into a colloid-rich and a colloid-poor phase as indicated by regions of high and low turbidity and by the appearance of peaks in the structure factor, has been observed at a sharply defined temperature $T_a$. Depending on the particle volume fraction of the colloids in the suspension (which is the ratio between the volume of all colloids in the suspension and the volume of the system), two or three phases of the colloidal suspension have been found: at low volume fractions this is a fluid phase in equilibrium with a fcc crystal, whereas at larger volume fractions there is a gas phase in equilibrium with a liquid phase and this liquid phase is coexisting with a fcc solid. Interestingly, upon increasing temperature quickly the measured structure factor indicated the formation of a glassy state as in a molecular system. On the basis of the experimental results reported in Ref. [@Guo-et:2008], it is not possible to infer uniquely the physical origin of the attractive potential which gives rise to the observed reversible aggregation. The authors concluded that all mechanisms discussed in Sec. \[sec:eff\] can play a role. Concerning the role of CCFs in this experiment, one finds that the temperature $T_a$ of the onset of aggregation is ca. $8-4$ K (depending on the volume fraction of the colloids) below $T^{(s)}_{cx}$, which corresponds to $(T^{(s)}_{cx}-T_a)/T^{(s)}_{cx} \simeq 0.02 -0.01$, which seems to lie outside the critical region. However, in Ref. [@Guo-et:2008] the actual critical temperature $T^{(s)}_c$ of the solvent is not given; if $T^{(s)}_c$ is lower, it is possible that the corresponding values of the reduced temperature actually lie within the critical region. The bulk correlation length of the solvent at $T=T_a$ has been estimated from light scattering to be ca. 8 nm, which is somewhat short. However, the experiments were performed at an off-critical composition of 3MP such that along the corresponding thermodynamic path the phase is relatively poor in the solvent component preferred by the colloids. If $T_a < T_c^{(s)}$, this composition precludes that colloidal aggregation arises from complete wetting and capillary condensation [@Evans:1990]. On the other hand, it is this thermodynamic region where the CCFs are expected to be strongest although their range, governed by bulk correlation length, is smaller than the one at the critical composition. Thus it is rather plausible that CCFs are playing a crucial role in this experiment.
In a subsequent study by Wegdam, Schall [*et al.*]{} [@Bonn-et:2009], the aggregation of colloidal particles suspended in the same (quasi-) binary mixture of 3MP, water, and heavy water has been observed directly by using confocal microscopy. The fluorescent fluorinated latex colloids of radius $R=200$ nm used in this study exhibit affinity for water. In this experiment the refractive indices between the colloids and the solvent have been matched closely. Two different compositions of 3MP have been studied, one below (mass fraction $\omega_{3MP}^{(1)} = 0.24$) and one above (mass fraction $\omega_{3MP}^{(2)} = 0.37$) the critical composition $\omega_{3MP,c} = 0.31$. (In Ref. [@Bonn-et:2009] the critical mass fraction of 3MP is denoted as $X_c$.) The colloid volume fraction has been kept fixed at the rather small value 0.002 and the D$_2$O/H$_2$O mass ratio was chosen as $0.25$. For the suspension poor in 3MP no aggregation has been observed upon increasing temperature within the homogeneous one-phase region of the solvent until phase separation of the whole mixture has taken place. This is expected to occur for 3MP mass fractions so far off the critical value that the CCFs are negligibly small. Above $\omega_{3MP,c}$, within a rather wide temperature range of $T^{(s)}_{cx} - T_a = 8$ K, reversible aggregation has been observed. For the very small volume fraction of colloids used, the formation of clusters rather than a colloid phase transition has been observed and the kinetics of this aggregation process has been studied. The clusters have been a few particle diameters wide. The sizes of the clusters, as inferred from the maximum of the confocal image intensity $I(q)$ in Fourier space, have been found to to increase linearly as function of time. The fractal dimension of these aggregates has indicated that the growth process is diffusion limited. Because the particles have desorbed from the clusters at a certain finite rate, the authors have been able to estimate the energy scale of the attraction between the particles by comparing the observed escape frequency with the corresponding attempt frequency, which is the inverse of the Brownian time, i.e., the time in which a particle diffuses a distance equal to its radius, and by assuming that this whole process is thermally activated; this renders 3$k_BT$ as the estimate of for the energy scale of attraction. The authors rightfully argued that the CCFs alone are sufficient to induce aggregation in charged-stabilized colloids. In order to show that this is indeed the case for their experiments, they analyzed their data using a simple expression for the effective pair potential between the colloidal particles. As in Eq. (\[eq:17\]), it consists of two competing contributions with the repulsive part decaying exponentially on the scale of the Debye screening length $\kappa^{-1}$ (denoted as $\ell_D$ in Ref. [@Bonn-et:2009].) Concerning the CCP contribution, a rather simple expression has been adopted. First, although the solvent in the actual experiment has not been at the critical composition, the occurrence of the corresponding relevant scaling variable $\Lambda$ (Eq. (\[eq:4\])) or, equivalently, $\Sigma$ (Fig. \[fig:1\]) has been neglected. Second, the Derjaguin approximation (Eq. (\[eq:13\])) has been employed with $\mathcal{A}_+$ set to 2. As both contributions decay exponentially, the hypothesis was put forward that aggregation should occur when both decay lengths, $\xi_t$ and $\kappa^{-1}$, become comparable. This hypothesis has been tested by adding salt in order to vary the range of the repulsive Coulomb interactions and by changing temperature in order to vary the critical Casimir attraction. The Debye screening length $\kappa^{-1}$ follows from the formula $\kappa^{-1} =\sqrt{\epsilon\epsilon_0 k_BT/(e^2\sum_i\rho_i)}$ (see below Eq. (\[eq:1\])) with $\sum_i\rho_i = 2N_A\ell$, where $\{\rho_i\}$ are the number densities of all ions (regardless of the sign of their charges), $N_A$ is the Avogadro number, and $\ell$ is the ionic strength (i.e., the molar density of positively charged ions) of the solvent. The bulk correlation length $\xi_t$ has been determined independently using X-ray and light scattering. The data representing the onset of aggregation for various values of $\kappa^{-1}$ fit well to the anticipated relation $\xi_t(T) = \kappa^{-1}$.
This data analysis has been objected and redone in Ref. [@Gambassi-et:2010] with the conclusion that, in fact, most of the data reported in Ref. [@Bonn-et:2009] for the onset of aggregation have been located within ranges of values of $\xi$ and $\kappa$ for which the proposed pair potential does not apply or, in other words, the model proposed in Ref. [@Bonn-et:2009] does not predict aggregation to occur at $\xi_t = \kappa^{-1}$. The comment [@Gambassi-et:2010] linked to Ref. [@Bonn-et:2009] pointed out the following weaknesses of the theoretical analysis in Ref. [@Bonn-et:2009]. The majority of the experimental data correspond to rather small bulk correlation lengths, i.e., $\xi_t \lesssim 5 \xi_{t,+}^{(0)}$, for which the adopted form of the CCP is not valid and should be complemented by corrections to scaling. Moreover, the effect of screening of the surface charge upon adding salt to the solution, which changes the amplitude $A$ of the repulsive contribution to the pair potential (Eq. (\[eq:17\])), has been neglected. Therefore, as a function of $\xi_t$ and $\kappa^{-1}$ the total pair potential exhibits richer forms than the two forms (a purely repulsive potential above the line $\xi_t(T) = \kappa^{-1}$ and in addition an attractive potential well below the line $\xi_t(T) = \kappa^{-1}$) shown in Ref. [@Bonn-et:2009] (see Fig. \[fig:8\]). The model pair potential actually used in Ref. [@Bonn-et:2009] is valid in the range where the distances $D$ between the colloids (denoted as $\ell$ in Refs. [@Bonn-et:2009] and [@Gambassi-et:2010]), which are close to the position $D_{min}$ of the minimum of the total potential, satisfy $R\gg D_{min} \gtrsim \xi_t, \kappa^{-1}$. Together with the requirement that the potential depth should be at least $-3k_BT$, this leads to a range of applicability of the model which is much more stringent than $\xi_t(T) < \kappa^{-1}$ as indicated in Refs. [@Bonn-et:2009]. However, the reanalysis presented in Ref. [@Gambassi-et:2010] does not exclude the possibility that the observed aggregation is not solely due to the competing effects of repulsive electrostatic and attractive critical Casimir interactions.
Effective colloid-colloid pair potential {#subsec:pp}
----------------------------------------
In Refs. [@Nguyen-et:2013; @Dang-et:2013] the effective pair potential between colloidal particles for a dilute (volume fraction 2%) suspension of poly-$n$-isopropyl acrylamid microgel (PNIPAM) particles with a radius of $R=250$ nm has been inferred from the pair correlation function $g(r)$. For a sufficiently dilute suspension of solute, the potential of mean force $V_{mf}(r) \sim - \ln g(r)$ can be identified with the effective pair potential $V(r)$; $g(r)$ has been determined from the $2d$ images obtained by confocal microscopy. The spatial resolution has been estimated to be ca. $0.03\mu m$ in the image plane and ca. $ 0.05\mu m$ in the direction perpendicular to it. The solvent was, as before, a (quasi) binary mixture of 3MP - water - heavy water with various compositions, including the critical one and compositions slightly off the critical one (towards the 3MP poor phase). The measurements have been performed for various temperatures upon approaching the solvent two-phase coexistence curve from the homogeneous mixed phase. The obtained pair potential displayed a very soft repulsion at small separations and developed an increasingly deep minimum as $T$ approaches the coexistence temperature $T^{(s)}_{cx}$ of the solvent. At low temperatures, at which $V(r)$ is purely repulsive, the authors have been able to fit $V(r)/(k_BT)$ to the screened electrostatic, exponentially decaying potential $U_{rep}$ (see Eq. (\[eq:2\])) with plausible values of the parameters. From this the CCP has been then determined by the best fit to the exponential form given in Eq. (\[eq:13\]), but with the amplitude and the decay length treated as fit parameters, assuming that the total $U(r)$ is the sum of $U_C$ and $U_{rep}$ (as in Eq. (\[eq:16\])). This assumption is justified for small salt concentrations [@ions] which has been the case for the samples studied there. (There is experimental [@Nellen-et:2011] and theoretical [@ions] evidence that at larger salt concentrations the coupling between the charge density and the order parameter can alter significantly the standard critical adsorption and CCFs.) Moreover, it has been assumed that for the temperatures studied, within a range $\Delta T < -1\mathrm{K}$, the ‘background’ repulsive contribution is de facto temperature independent. This data analysis has rendered the length scale for the experimental decay of the CCP, which, however, differs from the bulk correlation length. This discrepancy as well as the softness of the repulsion at small separations have been attributed to the fluffiness of the colloidal particles. The choice of PNIPAM particles has been motivated by their special properties when dissolved in the solvent described above. They swell, which prevents sedimentation, and in this swollen state their refractive index matches that of the solvent.
In a subsequent paper [@Dang-et:2013], the fitting procedure has been improved in the sense that the bulk correlation length $\xi_t$ has not been anymore an adjustable parameter. Assuming the standard scaling law for $\xi_t$ as a function of temperature, the amplitude $\xi_{t,+}^{(0)}$ has been used as a fit parameter in addition to the amplitude and the decay length of $U_{rep}$. Using one set of these three fit parameters, Dang and coworkers [@Dang-et:2013] have fitted all experimental data to the sum of two exponentially decaying functions. The fitted analytic expression for the potential has been further used for performing Monte Carlo (MC) simulations of the colloidal sample. The colloidal gas-liquid coexistence has been investigated by using Gibbs Ensemble MC [@Frenkel-et:2001] whereas colloidal liquid-solid coexistence has been determined by using Kofke’s Gibbs-Duhem integration technique [@Kofke:1993]. The conclusiveness of these MC simulation results for the actual colloidal phase behavior may be questioned because the fits of the experimental data to the analytic expression for the potential are not completely satisfactory. The strongest deviations have occurred at small separations, where the potential has been finite even for center-to-center separations smaller than the diameter of the particles. Moreover, the pair potentials obtained from fitting have not followed the data around the minimum of the total potential, especially close to $T_c$. Also these deviations have been attributed to the softness of the PNIPAM particles and to the fact that the scaling variable $\Lambda$ describing off-critical compositions has been neglected. It has been tested if all these deviations may significantly affect the observed phase behavior by calculating the reduced second virial coefficient $B^*_2=B_2/B_2^{(HS)}$. Empirically, gas-liquid phase transitions are expected to occur for $B^*_2 \lesssim -1.5$ [@Vliegenthart-et:2000; @Noro-et:2000]. Applying this criterion for experimentally determined and fitted potentials has led to an estimate for an upper bound for the deviation $\Delta T= T_{cx} - T^{(s)}_{cx}(c_a) < - 0.3$K of the temperature $T_{cx}$, at which a colloidal gas-liquid transition takes place, from that of the pure solvent phase separation (i.e., $T^{(s)}_{cx}(c_a)$), which is in agreement both with the experiments and the simulations reported in Ref. [@Dang-et:2013]. The comparison between computed and measured phase diagrams at fixed composition of a pure solvent in the plane spanned by $\Delta T = T_{cx} - T^{(s)}_{cx}(c_a)$ and by the colloidal volume fraction (in Ref. [@Dang-et:2013] denoted as $\phi$) is shown in Fig. \[fig:9\] (Fig. 2 in Ref. [@Dang-et:2013]). For solutions with off-critical compositions of a (pure) solvent, a reasonable agreement has been found, given the large uncertainties in the experimental determinations of the volume fraction and of $\Delta T$, as well as given the simplified functional form taken for $U_C$. On the other hand, for solutions with the (supposedly) critical composition, the agreement is less good. Deviations occur concerning the shape of the colloidal gas-liquid coexistence curve, which in simulations is shifted towards values of $\Delta T$ lower than those for the experimental data. Such a shift suggests that the fitted potentials underestimate the attractions between the colloids. According to the authors, this is due to many-body interactions and too small simulation boxes; both are particularly relevant for systems at the critical composition. Another possibility is that the effective one-component approach is not adequate to describe the considered experimental system. Concerning the colloidal liquid-solid phase coexistence in this system, there are no experimental data.
The experimental data for $g(r)$ at a 3MP mass fraction of $0.28$, which is close to the critical value, have been re-analyzed in Ref. [@Mohry-et:2014]. The main improvement over the earlier analyses has been due to a better treatment of the CCP contribution $U(r) = U_{bck} + U_C$ to the total pair potential, where $U_{bck}$ is a background term. Within this approach, also the dependence of $U_C$ on the solvent mass fraction has been taken into account. The scaling function of $U_C$ as a function of the scaling variable $\Sigma =\mathrm{sgn}(th_b)\xi_t/\xi_h$ (see Eq. (\[eq:17\]) and Fig. \[fig:1\]) has been calculated within the Derjaguin approximation by using a local functional approach. In order to minimize the number of fit parameters, the amplitude $\xi_{t,+}^{(0)}$ has been set to the value extracted from the experimental data presented in Ref. [@Sorensen-et:1985]. However, the value of the critical mass fraction $\omega_{3MP,c}$ of the 3MP-heavy water binary liquid mixture is not well established. The inaccuracy of the value for $\omega_{3MP,c}$ enters into the reduced order parameter $\widetilde\phi = (\omega_{3MP,c}-\omega_{3MP})/{\cal B}$, where ${\cal B}$ is the non-universal amplitude of the bulk coexistence curve $\omega_{3MP,cx}(t<0) = \omega_{3MP,c} \pm {\cal B}{|t|}^{\beta}$. Via the equation of state, which close to the critical point attains its scaling form, this determines the scaling variable $\Sigma = \mathrm{sgn}(th_b)\xi_t/\xi_h = \mathrm{sgn}(th_b)\mathcal{E}_{\pm}(|t|^{\beta}\widetilde{\phi})$, where $\pm$ refers to the sign of $t$. (In Ref. [@Mohry-et:2014] the scaling functions $\mathcal{E}_{\pm}$ have been obtained by using the equation of state within a linear parametric model.) Because $U_C$ depends sensitively on $\Sigma$, it has been natural to use the reduced order parameter $\widetilde\phi$ as a fitting parameter for achieving the weakest variation of the background potential $U_{bck}$ with temperature. A fair agreement with the experimental data has been obtained by allowing the value of $\omega_{3MP,c}$ to differ significantly from the value cited by the authors of the experiment (see Fig. 5 in Ref. [@Mohry-et:2014]). The fit has rendered the background potential which varies slightly with temperature and has an attractive part. This might be due to the the coupling between critical fluctuations and electrostatic interactions and thus deserves further studies.
This detailed knowledge of the CCFs as function of $T$ and $h_b$ has been used by the authors of Ref. [@Mohry-et:2014] to re-analyze the experimental data for the phase segregation, obtained in Ref. [@Dang-et:2013] and shown in Fig. \[fig:9\](a). The colloidal gas-liquid coexistence has been calculated within the effective one-component DFT approach using the pair potential $U(r) = U_{bck} + U_C$ fitted to the experimental data as described above, where $U_C$ has the scaling form as in Eq. (\[eq:17\]). The random-phase (RPA) approximation has been employed for the free energy together with the Percus-Yevick expression for the hard-sphere reference contribution. For the off-critical composition of the solvent, which renders the best expression (i.e., the least temperature dependent one) for $U_{bck}$ extracted from the experimental data of Ref. [@Dang-et:2013] (with, however, $U_{bck}$ still varying slightly with temperature), the phase diagram has been calculated by taking the mean curve of those $U_{bck}$ which correspond to the various temperatures considered. The corresponding results together with the experimental and MC simulation data from Ref. [@Dang-et:2013] are shown in Fig. \[fig:10\]. At high colloidal densities, the RPA is in surprisingly good agreement with the experimental data. On the other hand, at low densities the RPA agrees well with the Monte Carlo simulations, while for these densities both approaches underestimate the experimental values which, in turn, agree well with the RPA spinodal. While this latter ‘agreement’ might be accidental, it nevertheless raises the question whether the experimental system had actually been fully equilibrated at the time of the measurements. As pointed out in Refs. [@Nguyen-et:2013; @Dang-et:2013], the phase diagrams shown in Fig. \[fig:9\] are analogous to those of molecular fluids modeled, e.g., by Lennard-Jones or square-well fluids, but with a lower critical point. The peculiarity of the CCP, i.e., the strong temperature dependence of the shape of the potential and of its range, is mirrored by the small temperature range over which colloidal gas-liquid coexistence extends and in the shift of the critical point and of colloidal liquid-solid coexistence to lower volume fractions. Experience gained from the studies summarized above tells that a meaningful comparison of experimental data and theoretical predictions for the CCP requires accurate experimental knowledge of the solvent bulk phase diagram and of the solvent bulk correlation length. Such corresponding dedicated measurements have been reported in Ref. [@Marcel-et] for PNIPAM particles suspended in a binary liquid mixture of 3MP and heavy water with addition of 1mM KCl salt. The solvents have been prepared with 3MP mass fractions $\omega_{3MP}$ (and therein denoted as $c$ and given in units of weight percentage wt% 3MP$ = 100 \omega_{3MP}$) ranging from 23.5% to 33% around the critical value $\omega_{3MP,c}^{(s)}\approx 28.0$%, and for each of them the temperature has been varied from the homogeneous, mixed phase towards the solvent two-phase coexistence. In order to achieve intrinsic consistency of the experimental data and to link them to theoretical predictions, the measurements of the pair potentials of the particles in terms of radial distribution functions $g(r)$ have been complemented with dynamic light scattering measurements of the solvent phase diagram and of the bulk correlation length. Below the lower critical temperature $T_c^{(s)}$, the solvent correlation lengths have been inferred from the temperature-dependent correlation functions determining the scattered intensity, which are well-described by single-exponential decays with a characteristic time scale $\mathsf{t}_{d}$ related to the effective diffusion coefficient $D=(q^2\mathsf{t}_{d})^{-1}$, where $q$ is the wave vector of the incident wave (equal to 19$\mu m^{-1}$ in these specific measurements). This effective diffusion coefficient depends on the linear extent $\xi$ of the correlated regions via a relation analogous to the Stokes-Einstein relation for Brownian particles. However, close to the critical point one has to take into account that $D=D_\text{c} + D_\text{bg}$ decomposes into a [*c*]{}ritical and a [*b*]{}ack[*g*]{}round part [@Sorensen-et:1985], where [@Sorensen-et:1985; @Burstyn-et] $
D_\text{c} = \frac{\mathcal{R}\,k_B T}{6\pi\,\eta\,\xi}\,K(q\,\xi)\left(1+b^2(q\,\xi)^2\right)^{z/2}$ and $D_\text{bg} = \frac{k_B T}{16\,\eta_\text{bg}\,\xi}\,\frac{1+(q\,\xi)^2}{q_c\,\xi}.
$ Here, $\mathcal{R}\approx 1.05$ is a universal dynamic amplitude ratio [@Das-et:2007; @Burstyn-et], $K(x)=3/(4 x^2)[1+x^2+(x^3-x^{-1})\arctan x]$ is the Kawasaki function [@Kawasaki], $b=0.55$ [@Burstyn-et] is a correction to scaling amplitude and $\eta_\text{bg}$ is the background viscosity. The latter has been obtained as a function of $T$ and $\omega_{3MP}$ by extrapolating the available viscosity data [@Oleinikova-et:1999] to the critical region. This detailed analysis has allowed one to determine the correlation length for various compositions and to extract the amplitude $\xi^{(0)}_{t,+}=0.44$nm. The phase separation temperatures have been defined as those which correspond the minimum of the diffusion constant for for various values of $\omega_{3MP}$. They are fitted to the relation $\omega_{3MP}-\omega_{3MP,c}=\mathcal{B}|t|^{\beta}$ with the fixed critical exponent $\beta=0.3265$ leading to the estimate $\mathcal{B}=0.6$ for the OP amplitude. The diameter $2R=2.12\mu$m of the particles has been deduced by using confocal microscopy whereas their surface charge density $\Upsilon \simeq -0.17$e nm$^{-2}$ has been obtained from electrophoresis. The experimental radial distribution function $g(r)$, determined by particle tracking, reveals a very soft repulsion, according to which the measured effective potential fulfills $U(r<2R) > 0$ and is very large but not infinite as one would expect for hard core repulsion. As already mentioned in the present section, in the previous studies [@Nguyen-et:2013] this softness has been attributed to the fluffiness of the colloidal particles. A plausible alternative explanation for this softness consists of the inaccuracy associated with determining a three-dimensional $g(r)$ from two-dimensional images and, in addition, a certain polydispersity. In Ref. [@Marcel-et] this inaccuracy of the experimental radial distribution function has been compared with the projected theoretical function $g_\text{proj}(r'=\sqrt{x'^2+y'^2})=\int_{-\infty}^\infty\mathrm{d}z\int_{-\infty}^\infty\mathrm{d}y\int_{-\infty}^\infty\mathrm{d}x\, f_{\{0,\sigma_z\}}(z)f_{\{y',\sigma\}}(y)f_{\{x',\sigma\}}(x) g(\sqrt{x^2+y^2+z^2})$, where $f_{\{x',\sigma\}}(x)$, $f_{\{y',\sigma\}}(y)$, and $f_{\{0,\sigma_z\}}(z)$ are the probability distributions, which account for the uncertainty in the two horizontal directions and the vertical direction, respectively, with the in-plane spreads $\sigma=\sigma_x=\sigma_y$ being equal. In Ref. [@Marcel-et] these distributions have been taken to be the normal ones. Building upon the consistent description of the bulk properties of the binary solvent, the comparison between the measured pair potentials and the theoretical model, given by Eq. (\[eq:17\]) within the Derjaguin approximation for the CCP, would have not required any fitting parameter if the surface of the colloids had exhibited strong preferential adsorption. However, the PNIMP particles are only weakly hydrophilic. In such a case the CCP depends also on the surface field $h_s$ via the scaling variable $\hat h_s = h_s |t|^{-\Delta_1}$, where $\Delta _1$ is the surface counterpart of the bulk gap exponent [@Diehl:1986] (see Secs. \[subsection:CCFpot\] and \[subsubsec:DA\]). For the film geometry, within mean field theory the dependence on $\hat h_s$ at the critical concentration ($\Sigma=0$) can be reduced to a re-mapping $\tilde\vartheta^{(d)}_{||}(\mathcal{Y}, \mathsf{\Sigma}; \hat h_s) = s^{d}\,\vartheta^{(d)}_{||}(s^{-1}\mathcal{Y}, \mathsf{\Sigma})$ with a rescaling parameter $s$, which depends on $\hat h_s$ [@Mohry-et:2010]. The authors of Ref. [@Marcel-et] have found that within the experimentally accessible range of the CCP scaling function, which usually consists of only its exponential tail, one may mimic such a rescaling by using an effective temperature offset $t_{off}$ which shifts the reduced temperature according to $t'=(T_c-T+\Delta T_{off})/T_c=t+t_{off}$. Assuming that such a rescaling is valid also beyond mean-field theory, the authors of Ref. [@Marcel-et] have been able to fit all pair correlation functions, for a variety of temperatures and even of off-critical compositions, by taking the single parameter $t_{off}$ to vary smoothly with composition. Based on these fits the predictions for the pair potential are also given (see Fig. \[fig:11\]). Remarkably, any distorting influence on $g(r)$, which can be described by probability distribution functions which are symmetric about their argument, leaves the second virial coefficient unchanged. This holds because $B_2$ is determined by the integrated pair potential (see Eq. (\[eq:18\])). Taking advantage of this property, the authors of Ref. [@Marcel-et] could compare virial coefficients computed from the raw data of $g(r)$ directly with theoretical predictions, without any need to account for experimental inaccuracies and particle polydispersity. The experimental and the theoretical values of $B_2^*$ show very good agreement in the entire temperature-composition plane (see Fig. \[fig:12\]). The comparison based directly on the raw data provides clear evidence that in the investigated solvent composition range it is indeed the critical Casimir interaction which underlies the colloidal attraction. Hence, this direct comparison suggests that not only at the critical composition, but also at these off-critical compositions, the attraction is described in terms of a critical Casimir force rather than by wetting effects. Yet, at even larger off-critical compositions, wetting effects are expected to eventually take over and dominate the attraction as observed clearly in Ref. [@Hertlein-et:2008].
Using the same experimental imaging and particle tracking techniques, in Ref. [@Newton-et:2017] the effective interaction potentials for non-spherical dumbbell particles have been extracted from observed radial and angular distributions. The colloidal patchy dumbbell particles have been suspended in heavy water and 3MP with a mass fraction $\omega_{3MP} = 0.25$ (therein denoted as $c_{3MP}$). At this subcritical composition, the hydrophobic spherical ends prefer 3MP, while the neck joining these two spherical ends (called by the authors “shell”) prefers water. While the one-to-one mapping between radial distribution function and the effective, angularly averaged pair potential still holds for the anisotropic particles, the simple procedure of inferring the effective potential from the radial distribution function is not valid anymore. In order to find an optimal effective potential the authors of Ref. [@Newton-et:2017] have assumed that dumbbells are the rigid construction of two isotropic spheres each of which interacts via an isotropic pair potential. They have used the following three distinct distribution functions which facilitate the comparison of simulations, theory, and experiment this way determining that set of potential parameters which renders the best match: (1) the minimum distance radial distribution, for which only the minimum center-to-center distance $r_{i\alpha,j\gamma}$ between two spheres $\alpha$ and $\gamma$ belonging to dumbbell $i$ and $j$, respectively, contributes, (2) the, “site-site” radial distribution between sphere ( called site) $\alpha =1,2$ on dumbbell $i$ and a sphere (site) $\gamma =1,2$ on a different dumbbell $j$, and (3) the bond-angle distribution, which measures the mutual orientation of the particles forming dumbbells. As a first approximation, the observed site-site radial distribution function has been matched to an underlying potential between the dumbbell particles using the so-called reference interaction site model (RISM) in the context of integral equation theory for molecular fluids [@Chandler-et:1972]. Within the latter theory, a molecule is taken to contain several interaction sites so that the total interaction between two molecules is the sum of the site-site interactions. The RISM equation is a generalization of the Ornstein-Zernike equation [@OZ] for hard-core site-site interactions. The authors of Ref. [@Newton-et:2017] have used the hypernetted chain approximation as a closure. All three measured distributions have been compared with the ones computed using MC simulations of the dumbbell system in which every pair of spheres, apart from the one on the same dumbbell interacts via an effective potential as given by Eq. (\[eq:17\]). Because the corresponding experimental system has not been density matched, the particles sedimented, which was taken into account by including the gravitational potential and a surface field in order to describe the interaction with the bottom wall. Concerning the CCP, the authors of Ref. [@Newton-et:2017] adopted an oversimplified expression as the one given by Eq. (\[eq:13\]), in line with Refs. [@Bonn-et:2009; @Nguyen-et:2013; @Dang-et:2013]. As discussed above, such a form is valid only in the limit $D/\xi \to \infty$ and at the critical composition. Both conditions are not met in the experiment under consideration. Moreover, the amplitude of the bulk correlation length of the solvent and the surface charge, which determines the strength and the decay length of the electrostatic repulsion, respectively, have not been measured. As a consequence, the numerous adjustable parameters have been used to fit the theoretically proposed effective potentials to the measured ones. Following Ref. [@Marcel-et], the authors of Ref. [@Newton-et:2017] have used the projected theoretical distribution functions in order to mimic the experimental uncertainties. Nevertheless, the extent of agreement between the measured and the theoretically proposed effective potentials depends on temperature and is not satisfactory. The results for the dumbbell effective pair potentials, based on so many crude approximations and numerous fit parameters, are not conclusive and cannot be predictive. In particular, the quantitative modeling carried out in Ref. [@Newton-et:2017] is not able to provide the explanation for the observed two ranges of temperatures featuring distinct aggregation behaviors (see Sec. \[subsec:applic\] below).
Applications of CCF for self-assembly and aggregation of colloids {#subsec:applic}
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The possibility of controlled aggregation by exploiting CCFs has been used, near the lower critical point of the suspension, for the self-assembly of cadium telluride quantum dots in water - 3MP liquid mixtures with NaCl salt [@Marino:2016]. By measuring the intensity of scattered light as a function of time and by following the time evolution of the intensity correlation function, it has been found that 1 K below the phase separation temperature of the suspension the hydrophilic quantum dots with a size of ca. 2.6 nm form aggregates with an average radius of ca. 700 nm. As expected, for a composition with the 3MP volume fraction being larger than its critical value, i.e., corresponding to the phase poor in the component preferred by the quantum dots, aggregation takes place on shorter time scales and within a larger temperature interval than for a mixture with a 3MP volume fraction being smaller than the critical one.
Critical Casimir interactions can be utilized for aggregation of colloids being induced by a substrate such that the colloids follow the chemical pattern designed on the surface of a substrate which characterizes the boundary conditions. This is because CCFs respond sensitively to the chemical properties of the confining surfaces. As already discussed in Sec. \[sec:eff\], depending on whether the surfaces of the colloid and of the substrate have the same or opposite preferences for the species of the solvent (symmetric or asymmetric boundary conditions), attractive or repulsive CCFs arise. In Refs. [@Soyka-et:2008; @Troendle-et:2011], charged polystyrene (PS) spheres with $R=1.2\mathrm{\mu m}$ have been suspended in a water-2,6-lutidine (WL) mixture at its critical composition, i.e., a lutidine mass fraction of $\omega_{L,c} \cong 0.286$. The suspension has been exposed to a chemically patterned substrate with well defined, spatially varying adsorption preferences. This has been achieved by first coating the glass surface with a monolayer of hexamethyldisilazane (HMDS) which rendered the glass surface hydrophobic with a preferential adsorption of lutidine, corresponding to a ($+$) boundary condition. Spatial patterning of the boundary conditions has been obtained by using a focused ion beam (FIB) of positively charged gallium (Ga) ions which created well-defined hydrophilic ($-$) areas with a lateral resolution on the order of several tens of nanometers extending over an area of approximately $400 \times 400$ $\mathrm{\mu m}^2$. Close to the lower critical demixing point $T_c \approx 307\mathrm{K}$ of WL [@Beysens-et:1985], normal and lateral CCFs lead to a strongly temperature dependent attraction between the hydrophilic ($-$) PS particles and the hydrophilic squares (forming a $2d$ square lattice of locally symmetric boundary conditions) and to a repulsion from the hydrophobic regions ($+$) (locally asymmetric boundary conditions). This gives rise to the formation of highly ordered colloidal self-assemblies, the structure of which is controlled by the underlying chemical pattern (see Fig. \[fig:13\]). At higher particle concentrations, additional CCFs between neighboring particles arise and eventually lead to the formation of three-dimensional, facetted colloidal islands on the substrate. In order to quantify lateral CCFs, substrates with a periodic one-dimensional chemical pattern have been created forming hydrophilic ($-$) and hydrophobic ($+$) stripes with widths of 24.6 $\mathrm{\mu m}$ and 5.2 $\mathrm{\mu m}$, respectively. For these one-dimensional surface patterns the particle distribution has been measured by digital video microscopy and from the two-dimensional projection of this distribution the effective one-dimensional CCP has been determined by resorting to the Boltzmann factor. In Ref. [@Troendle-et:2011] the experiment has been repeated for substrates with the chemical pattern created by micro-contact printing, which provided sharper, chemical steplike, onedimensional interfaces between alternating regions of antagonistic adsorption preferences than the FIB technique does. The reason for the disadvantage of FIB is that the ion beam charges the substrate surface and thus deflects the incoming beam which gives rise to fuzzy chemical steps. It turns out that the agreement between theory and the experimental data is so sensitive (Fig. \[fig:13\]), that the CCP can be used to probe the geometry of the chemical structures, which at present cannot be achieved by other experimental techniques. Accordingly CCFs can be used as a novel surface sensitive probe.
Reversible aggregation of spherical Janus particles has been studied experimentally for micro-sized silica particles half covered by a layer of gold and suspended in the WL mixture at the critical concentration [@Iwashita-et:2013]. The gold caps of the particles have been modified chemically by sulfonic groups, which have bestowed a large charge density on their surfaces, leading to an adsorption property which differs from that of the bare silica surfaces. Due to this anisotropy, the particles assembled to clusters with a very specific structure, followed by a hierarchical growth of the clusters. These structures depend on the “valence” of the Janus particle, i.e., the maximum possible number of bonded nearest neighbors. In two spatial dimensions, which applies to the dispersion of the sedimented particles considered in the experiment, the valence is equal to six. Direct visual observation has revealed that particles, randomly dispersed at low temperatures, have started to form micellar structures upon increasing $T$ towards the lower critical point of the WL mixture. Most of the clusters have been trimers and tetramers, with gold-coated hemispheres inside the micelle and silica hemispheres always facing outwards. No “inverted” micelles have been observed, which indicates that the effective attraction between golden patches of colloids has been much stronger than the attraction between the silica parts of the particles. At higher temperatures small clusters, mostly tetramers, have assembled into chain-like structures, finally forming a percolating network. The reversibility of the aggregation, the value of the onset temperature, and the apparent increase of the strength of attraction upon approaching $T_c^{(s)}$, have supported [ the]{} expectation that this aggregation is due to CCFs. The structures of the clusters observed in the experiments are similar to those obtained within MC simulations of hard discs with a pairwise square-well potential acting between the semi-circular patches of the particles. However, the strength of attraction could not be determined from the experimental data. The analysis of the cluster structures corresponding to the lowest internal energy and of the hierarchical clustering suggested the possibility that the self-assembly of Janus particles is governed by the valence structure of the clusters and not by that of a single particle.
Recent progress in synthesizing colloidal building blocks allowed the authors of Ref. [@Nguyen-et:2017; @Nguyen-et:2017a] to fabricate particles with complex shapes and surface patch properties which act as analogues of molecular valence. Multivalent particles, such as dimers, trimers, and tetramers have been produced by swelling and polymerizing clusters of PMMA spheres with a methylmethacrylate/methacrylic acid shell, resulting in geometrically well-defined patches. The specific solvent affinity of the particle patches has been achieved by grafting a polyhydroxy stearic acid-copolymer (PHSA) onto the surface patch, which renders it hydrophobic. The central part of the patchy particles is made hydrophilic by using the ionic initiator potassium persulfate. The particles have been suspended in the homogeneous phase of a binary solvent of heavy water and 3-methylpyridine (3MP) at temperatures below the lower critical temperature $T_c=38.55$ , which has been determined by light scattering and microscopy from solvent phase separation at the critical composition. The solvents were prepared with 3MP mass fractions $\omega_{3MP} = 0.25$ and 0.31 (therein denoted as $c_{3MP}$), i.e., slightly below and above the critical composition $\omega_{3MP,c} = 0.28$, respectively. The hydrophobic patches have a strong affinity for the non-aqueous component 3MP of the binary solvent, while the hydrophilic central part has an affinity for water. It has been observed that, in solvents poor in the component preferred by the particle patches, the patches approach each other at temperatures close to $T^{(s)}_c$, and that dimer particles assemble into directed, chain-like structures. The bending stiffness of the chains has been measured directly by monitoring thermally activated bending fluctuations. In contrast, in 3MP-rich solvents, the particles approach each other sideways resulting in distinct parallel structures. For trimers, the patch-to-patch binding in 3MP-poor solvents leads to staggered chains, while the side-by-side binding in 3MP-rich solvents leads to bent filaments associated with the dense alternating stacking of trimers. In all cases, the assembly is fully reversible as confirmed by the break-up of aggregates upon lowering the temperature several degrees below $T^{(s)}_c$. Interestingly, in the case that the chain structure is formed by dimers, upon further approaching $T^{(s)}_c$, the authors of Ref. [@Nguyen-et:2017] have observed that the chain spontaneously collapses into a compact state, the dimer particles approach each other sideways, and eventually form a close-packed arrangement. It has been argued that in this close-packed state, a particle has more bonding neighbors, and hence a more negative bond energy. The authors of Ref. [@Nguyen-et:2017] regard this as being similar to the collapse transition of a polymer, which occurs if the solvent conditions go from good to poor. For polymers, too, the reduction of the conformational entropy of the chain is offset by the stronger interparticle interaction energy. Using MC simulations with the effective pair potential determined in Ref. [@Newton-et:2017] (see the preceding Sec. \[subsec:pp\]), it has been argued in Ref. [@Nguyen-et:2017], that the colloidal chain collapse results from the enlarged interaction range due to the increase of the solvent correlation length upon approaching the solvent critical point. In Ref. [@Nguyen-et:2017a], the effect of patch width on the topology of colloidal aggregates has been experimentally investigated.
Colloidal mixtures {#subsec:cm}
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The way the unique properties of CCFs can be harnessed to manipulate colloidal suspensions has been demonstrated by experiments on colloidal mixtures [@Zvyagolskaya-et:2011]. This experimental system has been composed of a binary mixture of micro-sized latex particles with slightly different diameters which were suspended in the WL mixture. For this small size difference of the colloids the inherent depletion interaction cannot induce demixing. However, the CCFs can accomplished this, if the two types of particles carry opposite adsorption preferences. The $a$-type particles were functionalized with silane rendering them hydrophobic, i.e., $(+)$ BC, whereas the $b$-type particles had a strong adsorption preference for water, i.e., $(-)$ BC. For such BCs the CCFs among the $a$-type particles and among the $b$-type particles are attractive, whereas between $a$-particles and $b$-particles they are repulsive. At the same temperature and the same distance, the repulsive CCF is stronger than the attractive CCF. In the system under consideration, the van der Waals forces have been eliminated by index matching so that besides the CCFs the only remaining forces have been the screened electrostatic interactions. In the suspension, this mixture of colloids sedimented at the bottom of the cell forming a dense monolayer. One species has been labeled with a fluorescent dye and traced by using video microscopy. As expected, upon approaching the (lower) critical point of the binary solvent at its critical composition, large structural changes in the colloidal mixture have been observed, signaling the demixing process. It was found that the morphology of this process depends strongly on the mixing ratio $x_{a,b}= \rho_{a,b}/(\rho_a+\rho_b)$ of $a$- and $b$-particles, where $\rho_{a,b}$ are the number density of $a$ and $b$ particles (see Fig. \[fig:16\], which has been taken from Ref. [@Zvyagolskaya-et:2011]). For $x_a = 0.54$ and at low temperatures an initially random distribution of particles transforms into a bicontinuous network, which is coarsening further upon increasing the temperature. For $x_a = 0.28$, no bicontinuous structure has been observed; demixing has proceeded via the growth of small clusters of the minority phase (here rich in $a$-particles).
In the theoretical part of this study [@Zvyagolskaya-et:2011], the effective approach has been employed in order to construct an approximation for the Helmholtz free energy of the colloid mixture. It has been assumed that the effective pair potential $U_{i,j}, i,j = a, b$, can be split into a hard-disc part and into a tail, which for identical particles is attractive and for the distinct ones repulsive. For the hard-disc part of the free energy the scaled particle approximation has been used, whereas the attractive or repulsive tail has been treated within the mean field van der Waals approximation. The predictions of this simple theory for the locations of the colloid demixing transitions and their critical point agree fairly well with the experimental data.
Effects of depletants on colloidal phase separations {#subsec:eff_dep}
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Piazza and collaborators [@Buzzaccaro-et:2010; @Piazza-et:2011] have studied experimentally the colloidal phase separation in suspensions with a depletion agent, which occurred near the critical point of the depletant enriched solvent. They have used the so-called Hyflon$^{TM}$ MFA latex particles (a copolymer of tetrafluoroethylene (TFE) and perfluoromethylvinylether (PF-MVE); the producer does not provide the meaning of the acronym MFA) of average size $R \sim 90$ nm suspended in water with the nonionic surfactant C$_{12}$E$_8$. Salt (NaCl) has been added in order to screen electrostatic interactions between the colloids, caused by their surfaces carrying a negative charge. In addition, this surfactant has provided also steric stabilization of the suspension due to its spontaneous adsorption on the colloid surfaces. What makes this system interesting is, that at concentrations above the critical micellar concentration, C$_{12}$E$_8$ in water forms globular micelles with a radius $\sigma = 3.5$ nm. These micells act as a depletant for the MFA particles. Moreover, the surfactant-water mixture exhibits a liquid-liquid phase transition terminating at a lower critical point $(T_c^{(s)}, c^{(c)}_s)$ with a very small value of the [*c*]{}ritical concentration $c^{(c)}_s$ of the [*s*]{}urfactant (1.8% mass fraction $c_s=m_s/m_{tot}$, where $m_s$ is the mass of the surfactant and $m_{tot}$ is the total mass of the sample). (We use the superscript $s$ as an acronym for the solvent and the subscript $s$ as the one for the surfactant.) Above this critical point two liquid phases exist, one rich and the other poor in micelles. The authors have determined, as a function of temperature, the [*m*]{}inimum concentration $c^{(m)}_s$ of surfactants required to induce colloidal gas-liquid phase separation. The onset of this phase separation has been assumed to manifest itself via the sudden increase of turbidity followed by fast sedimentation of the colloidal particles. The results of these measurements (see Fig. \[fig:15\], taken from Ref. [@Piazza-et:2011]) show a drastic decrease of $c^{(m)}_s$ upon increasing temperature towards the consolute point $(T^{(s)}_c,c^{(c)}_s)$ of the surfactant-water mixture, such that $c^{(m)}_s$ approaches $c^{(c)}_s$ as $T \to T^{(s)}_c$. Far below the surfactant-water miscibility gap, the colloidal phase separation has been obtained due to the action of depletion forces, provided that a sufficient amount of surfactant has been added. The range of the depletion interaction is set by the ratio $\approx 0.03$ between the micelle and the particle size and is very short, but the strength depends on the concentration of the micelles. In Fig. \[fig:17\], the full dots $(T^{(m)}_s,c^{(m)}_s)$ show the minimum amount $c^{(m)}_s$ of surfactant required to induce colloidal phase separation at the temperature $T^{(m)}_s$. These points are expected to correspond to the thermodynamic states of the surfactant-water solvent for which the strength of the effective attractive potential between the colloidal particles is roughly constant. The authors have interpreted the reduction in $c^{(m)}_s$ observed at higher temperatures as an increase of depletant “efficiency” suggesting that the emerging critical Casimir interactions near the critical point of the surfactant-water mixture can be considered as an extreme case of depletion forces in which the correlated regions of micelles act as a depletant of “renormalized” size, i.e., bigger than that of a micelle, which acts as a bare depletant. However, this interpretation is overstreched because depletion and Casimir forces originate from very different physical mechanisms. In particular, upon approaching the critical point the correlated regions of the emerging phases do not form bubbles but self-similar, bicontinuous structures.
Inspired by these experiments, Sciortino and co-workers [@Gnan-et:2012a] have performed a numerical study of the phase separation of hard spheres dispersed in an implicit solvent (i.e., there are no direct interactions, beyond the hard-core ones, between the solvent molecules and these spheres) in the presence of interacting depletant particles. The particles have been taken to interact via the corresponding effective potential $V_{eff}$ as determined by MC simulations for SW and 3P models of depletants (see Eq. (\[eq:7\]) as well as Secs. \[sec:eff\] and \[subsec:eff\_dep\]). The phase separation of the colloids as a function of the depletant concentration for various depletant-colloid size ratios has been determined within grand canonical MC simulations. The numerically determined loci of the onset of colloidal gas-liquid phase separation has been displayed within the phase diagram of the depletant which exhibits an upper critical point. This revealed that almost for all parameters studied in the MC simulations, the colloidal phase separation induced by CCFs is preempted by the one driven by standard depletion forces. The important message of this study is that only by weakening the attractive depletion interactions, either by lowering the critical depletant concentration $c^{(c)}_s$ or by introducing a repulsive interaction between the colloids, it is possible to exploit CCFs for the fine tuning of the self-assembly of colloids in solvents with interacting depletant agents.
Aggregation kinetics and structures of aggregates {#subsec:kin_aggreg}
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Aggregation of colloids in a near-critical binary solvent has been studied experimentally on the ground and under microgravity conditions [@Bonn-et:2009; @Veen-et:2012; @Shelke-et:2013; @Potenza-et:2014]. The advantage of the critical Casimir effect of providing the ability to tune the effective interactions between the colloids by varying the temperature has been used in order to study systematically the internal structure of the aggregates as a function of their interparticle attraction. Direct visual observations via confocal microscopy of a dilute suspension of latex particles of radius $R$ in a $\mathrm{D_2O/H_2O - 3MP}$ mixture have revealed the formation of fractal clusters of colloidal particles near the critical point of the solvent [@Bonn-et:2009]. The analysis of the time-dependent intensity $I(q,\mathsf{t})$ revealed that $I(q,\mathsf{t})$ exhibits a maximum at $q=q^*(\mathsf{t})$, which corresponds to the inverse of the mean cluster size. It has been found that in the course of time $q^*$ decreases as $q^*(\mathsf{t}) \propto \mathsf{t}^{-1}$ and that this linear growth of the size of the clusters with time $\mathsf{t}$ is self-similar. The latter property has been inferred from the data collapse of $I(q,\mathsf{t})$ plotted as a function of the rescaled wave vector $q' = q/q_0 = (\mathsf{t}/\mathsf{t}_0)qR$ with $\mathsf{t}_0 = 33$ min[^2] The fractal dimensions $d_f$ determined from $2d$ projection images have indicated that this system exhibits diffusion-limited cluster aggregation rather than diffusion-limited particle aggregation, for which $d_f$ is larger [@Dutcher-et:2004]. The fractal dimension reflects the internal structure of an aggregate and relates its radius $R_g$ of gyration to the number of particles $N$ according to $R_g \propto N^{1/d_f}$; $d_f \lesssim 3$ corresponds to a close packed structure. $R_g$ measures the mean size of the fractal aggregate. It is defined in terms of the mass density $\rho(r)$ of the particles at a distance $r$ from the center of mass of the aggregate as $R_g^2 = \int_{0}^{\infty} r^4 \rho(r) dr/\int_{0}^{\infty} r^2 \rho(r) dr$ or, alternatively, based on the pair correlation function $g(r)$ of colloidal particles $R_g^2 = (1/2) \int_{0}^{\infty} r^4 g(r) dr/\int_{0}^{\infty} r^2 g(r) dr$ (see, e.g., Ref. [@vanSaarloos:1987]). For fractal structures consisting of spheres with spherically symmetric mass density distribution $\rho(r)$, one has $R_g = R_c$, where $R_c$ is the largest radius beyond which the aggregate has zero mass. The images from confocal microscopy have shown that the width of the branches of the clusters is a few particle sizes and that the particles escape from the aggregate at a finite rate. The latter observation and the assumption that this process is thermally activated, allows one to estimate the energy scale of attraction between the particles as ca. $\approx 3\times k_BT$. A more detailed investigation of the structure of the clusters and its evolution was impeded by sedimentation.
In order to avoid this complication, using near field light scattering (NFS), new measurements for the same type of particles have been performed under microgravity conditions on board of the International Space Station and, simultaneously, on the ground [@Veen-et:2012; @Potenza-et:2014]. (For these experiments, the composition of the solvent and the volume fraction of the colloidal particles have been taken to be different from those in Ref. [@Bonn-et:2009]. Moreover, different amounts of salt (NaCl) have been added. Thus the results of both studies cannot be compared directly.) The light scattering intensity $I(q,\mathsf{t})$ has provided information about the time evolution of the structure and of the mean size of the aggregates. Its normalized variance, defined as $\langle I^2(q,\mathsf{t}) \rangle/\langle I(q,\mathsf{t}) \rangle^2 -1 $, with the angular brackets $\langle \cdots \rangle$ denoting the time average, has revealed the onset and the time scale of the aggregation process. The time dependence of the normalized variance results from variations of the number density of the scatters. Accordingly, the start of [*a*]{}ggregation has been marked by the time $\mathsf{t}_a$ at which this quantity starts to increase from zero. From the slope in the log-log plot of $I(q,\mathsf{t})$ at large momentum transfer $q$, which has been temporally constant during the growth process (which is characteristic of scattering off fractal structures), a fractal dimension has been inferred. It has been found that, upon varying the temperature towards $T_c^{(s)}$, $d_f$ has decreased from a value, which is close to the theoretical one of 2.5 for diffusion-limited aggregation, to about 1.8. This indicates that more open structures of clusters are formed if - as expected - the strength of the attraction increases. This has been interpreted to the effect that for an attractive interaction potential with a well depth of ca. $1 \times k_BT$ the restructuring of the aggregates into more compact objects can proceed, whereas for deeper attraction wells restructuring gradually stops and the resulting structure is more open. Such more open structures have been observed in the presence of gravity with $d_f \simeq 1.6 - 1.8$ for all temperatures studied. In both cases, i.e., in the presence and absence of gravity, only a slight variation with the salt concentration has been found. Moreover, it has been observed that independent of temperature, and thus the strength of the attraction, the aggregates grow similarly. Specifically, the scattered intensity of the growing aggregates as obtained at different times has been [*red*]{}uced to a scaling function $I(q,\mathsf{t}) \approx \left(q_{red}\right)^{-d_f}F(q/q_{red}(\mathsf{t}))$, where the time dependent characteristic momentum transfer $q_{red}(\mathsf{t})$ has been determined from the best fit in the sense of $I(q,\mathsf{t})\left(q_{red}\right)^{d_f}$ yielding data collapse to a function $F$ of a single variable $q/q_{red}(\mathsf{t})$. Based on this apparent scaling behavior, an analogy has been drawn between the observed aggregation process and spinodal decomposition processes. For the latter, a similar scaling holds with $d_f$ replaced by the spatial dimension $d$ [@Carpineti:1992]. As for spinodal decomposition, $I(q)$ exhibits a maximum at $q\simeq q_{red}$ corresponding to the inverse of the characteristic length in the system. In the course of time the momentum $q_{red} $ shifts to smaller values indicating the growth of this characteristic length. The decrease of $I(q)$ for small values of $q$ reflects the fact that a region around the cluster is depleted of colloids [@Carpineti:1992]. The growth rate of the characteristic length scale $q_{red}^{-1} \propto R_g\propto N^{1/d_f}$ of the aggregates has been found to be a power law with exponent $1/d_f$, regardless of the salt concentration. This is characteristic of diffusion-limited models of aggregation [@WittenMeakin]. Under gravity, the growth rate has been observed to be significantly faster and nonmonotonic (see Fig. \[fig:18\], which is taken from Ref. [@Veen-et:2012]). At early stages it has followed the purely diffusive behavior. After that, aggregation has been influenced strongly by convection of the solvent and by sedimentation. The aggregation among clusters has been found to set in rather fast, resulting in an exponential growth rate, which is characteristic of reaction limited aggregation with a strong dependence on temperature and thus on the strength of the attraction. At still later times, a sudden drop in $q_{red}^{-1}$ has been observed, which corresponds to the sedimentation of the largest clusters.
Whereas in Ref. [@Veen-et:2012] the static properties of the aggregates, i.e., their structure factor $S(q)$, have been measured as aggregation proceeds, in the subsequent NFS study of this system [@Potenza-et:2014] the dynamical counterpart of $S(q)$, the intermediate scattering function $\mathfrak{S}(q,\mathsf{t})$ has been measured under microgravity conditions. The aim has been to determine the ratio $\beta \equiv R_h/R_g$ of the hydrodynamic radius $R_h$ of the aggregate and its gyration radius $R_g$. This ratio provides information about the density distribution within an aggregate as function of its fractal dimension, which in turn depends on the strength of the particle attraction. The hydrodynamic radius $R_h$ of the aggregate is defined through the translational diffusion coefficient $D = k_BT/(6\pi\eta R_h)$, where $\eta $ is viscosity of the solvent. For a densely packed spherical aggregate of radius $R_c$ and with $d_f = 3$ one expects $R_h$ to be very close to $R_c$. The intermediate scattering function $\mathfrak{S}(q,\mathsf{t})$ is the spatial (three-dimensional) Fourier transform of the van Hove distribution function $G(r,\mathsf{t})$, which is the dynamical counterpart of the radial distribution function $g(r)$ [@Hansen-et:1976]. The NSF technique enables one to measure $\mathfrak{S}(q,\mathsf{t})$ instantaneously, on the time scale of the much slower aggregation and diffusion processes. This provides the concurrent measurement of $R_h$ from the dynamic and $R_g$ from the static structure factor, simultaneously for all accessible wave vectors. $R_g$ follows from the Fisher-Burford expression [@FB] $S(q; R_g) = (1 + (2/3d_f)q^2R^2_g)^{-d_f/2}$, which is valid for monodisperse fractal aggregates. $R_h$ has been determined via the effective diffusion constant $D_{eff}$ obtained from the measured decay time $\mathsf{t}_d$ of $\mathfrak{S}(q,\mathsf{t})\propto e^{-\mathsf{t}/\mathsf{t}_d}$ via $\mathsf{t}_d = 1/(D_{eff}q^2)$ followed by relating $D_{eff}$ to $D$ by using $S(q)$ [@Lin]. It has been concluded that in order to obtained reliable data for $R_h$, one has to consider the late stages of aggregation, in which the clusters reach sizes large enough to exhibit a well-defined fractal dimension and internal structure and in which the actual polydispersity of the aggregates does not play a significant role. The analysis of the data has shown, that upon approaching the critical point of demixing (i.e., for stronger attraction) the ratio $\beta$ has varied between 0.76 for $d_f = 1.8$ and 0.98 for $d_f = 2.5$ at the onset of aggregation (i.e., for weaker attraction). The authors of Ref. [@Potenza-et:2014] have assumed that the fractals are spherically symmetric. Accordingly they have defined the radial density distribution of the fractal objects as $\rho(r) = r^{d_f-3}f_{cut}(r)$, where $r$ is the distance from the center of mass of the cluster and $f_{cut}$ is a cutoff function which accounts for the finite size of the aggregates. (The exponent follows from the fact that $\rho(r) = \mathcal{N}(r)/V(r) \sim r^{d_f-3}$, where $\mathcal{N}(r)\sim r^{d_f}$ is the number of particles within a sphere of radius $r$ from the center of the cluster and $V(r)\propto r^{-3}$ is the volume of such sphere.) The expected behavior of the ratio for various forms of a cutoff function has been then compared with the experimental data. It has been observed (see Fig. \[fig:19\], which is taken from Ref. [@Potenza-et:2014]) that the data for the ratio $\beta$ have been closest to those values which correspond to the assumption of fully compact objects with a Heaviside step function for $f_{cut}$, independent of the strength of the attraction and of the fractal dimension.
The way to avoid sedimentation for experiments performed on the ground [@Shelke-et:2013] has been to use poly-n-isopropyl acrylanid particles (PNIPAM), which swell in solution. This swelling adjusts their buoyancy, preventing particle sedimentation. This also has allowed the authors to observe directly individual particles (after labeling them with a fluorescent dye) even deep in the bulk of the suspension. Using confocal microscopy, the compactness of aggregates of PNIPAM particles, suspended at small volume fractions in the 3MP - D$_2$O/H$_2$O mixture, has been studied by determining the number of particles $\mathcal{N}(r)$ within a sphere of radius $r$ from the center of the cluster for quenches with various temperature deviations $\Delta T$ from and below the lower critical point $T_c^{(s)}$ of the demixing transition of the solvent. These data have confirmed the fractal character of the structures, i.e., the relation $\mathcal{N}(r) \sim r^{d_f}$, with $d_f$ decreasing continuously upon increasing temperature towards $T_c^{(s)}$ from $d_f \approx 3$ to $d_f\approx 2.1$. This indicates the formation of more compact structures for weaker attraction (see Fig. \[fig:19\](h) which is taken from Ref. [@Shelke-et:2013]), in agreement with the behavior observed for the latex particles under microgravity (see the above paragraph). Monte Carlo simulations of diffusion-limited particle aggregation in $d=2$ and $d=3$ have been used in order to relate $d_f$ to the attractive effective interaction potential of the particles. Experimentally, the pair potential has been determined from the radial distribution function of the colloids in a way similar to that described in Ref. [@Nguyen-et:2013]. The depth $V_0 < 0$ of the attractive well of this pair potential has been related to the quantity $\alpha \propto \exp (V_0/(k_BT))$ used in the simulations for the probability with which the particles can detach from the growing cluster. The experimental variations of $d-d_f$ as a function of $V_0$ and of $\alpha$ in simulations exhibit a common curve after suitable rescaling. The authors have interpreted this result as a manifestation of a certain type of universality which occurs although the corresponding simulations have been carried out for different spatial dimensions ($d=2$ and $d=3$) and although the mechanisms of aggregation differ, i.e., diffusion-limited cluster aggregation in the actual experiments, in which clusters aggregate to form a fractal, and diffusion-limited particle aggregation in simulations, in which a fractal grows by aggregation of single particles.
Perspectives {#sec:persp}
============
It remains as a challenge to accurately determine the critical Casimir pair potential and its scaling function within the range of the relevant parameters and boundary conditions. This applies to chemically homogeneous or Janus particles of spherical or anisotropic shapes such as cylinders, ellipsoids, cubes, or even more complex shapes such as dummbells or $L$-like ones. Quantitative reliability demands to go beyond mean field theory and the Derjaguin approximation. This requires the development of new theoretical approaches and simulation algorithms. Janus particles have already been used in experiments on aggregation in near-critical binary solvents [@Iwashita-et:2013]. Colloids with chemically homogeneous or inhomogeneous surfaces, forming patchy colloids [@Schall_review; @Garcia-et:2017], with various shapes as well as their mixtures are attractive building blocks for generating a large variety of self-assembled structures by using CCFs. In order to make further progress in understanding the interplay between CCFs and other forces, such as electrostatic ones, for aggregation phenomena, more elaborate and combined experimental and theoretical investigations are required. Concerning the thermodynamic properties of colloidal suspensions with near-critical solvents, a multi-component theory beyond the effective approach and beyond mean field theory is needed. One of the challenges along this line is to construct a suitable theory, which accounts for all relevant degrees of freedom of the solvent and of the solute particles and masters the size differences of the various species. An interesting line of research would be to study aggregation of colloids in lipid membranes close to the corresponding lipid phase segregation or in a liquid analogue of such quasi two-dimensional systems.
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![Scaling function $\Theta_{\oplus\oplus}^{(d=3,Derj)}(\mathcal{Y},\mathsf{\Sigma}) \simeq (D/R) U_c(D;t,h_b,R)$ (see Eq. (\[eq:4\]) and the main text) of the sphere - sphere CCP, as obtained within the Derjaguin approximation by using the extended de Gennes-Fisher functional (solid lines) and the “dimensional” approximation (dashed lines) as a function of the surface-to-surface distance $D$ (in units of $\xi_t$) for several values of the scaling variable $\mathsf{\mathsf{\Sigma}} = \Lambda/\mathcal{Y}= \mathrm{sgn}(th_b)\xi_t/\xi_h$ related to the bulk ordering field $h_b$. At fixed temperature in the one-phase region of the solvent ($t>0$), the CCP is shorter-ranged and much weaker for $h_b>0$ (i.e., $\Sigma > 0$), which favors the same $(+)$ phase of the solvent as the one preferred by the colloid surfaces, whereas $h_b < 0$ (i.e., $\Sigma < 0$) favors the other phase. []{data-label="fig:1"}](fig1){width="0.8\linewidth"}
![ Normalized mean-field CCF scaling function $\vartheta^{(d=4)}_{||}(\mathsf{Y},\mathsf{\Sigma}) = D^4f_{C}^{||}/(k_BT{\cal S})$, where $ \mathsf{Y}=\mathrm{sgn}(t)D/\xi(t,h_b)$ and $\mathsf{\Sigma}=\mathrm{sgn}(th_b)\xi_t/\xi_h$, for films (of thickness $D$ and macroscopically large cross-sectional area ${\cal S}$) along isolines of constant scaling variable $\mathsf{Y}=4,5,\ldots, 10$ (from the inner to the outermost ring) in the thermodynamic state space of the solvent spanned by $\hat t = (D/\xi_{t,+}^{(0)})^{1/\nu}t$ and $\hat{h}_b = (D/\xi_h^{(0)})^{\beta\delta/\nu}h_b$ (see Fig. 1 in Ref. [@Mohry-et:2014]); $\xi(t,h_b)$ is the bulk correlation length of the solvent with $\xi_t=\xi(t,h_b=0)$ and $\xi_h=\xi(t=0,h_b)$. (Note that depending on the particular thermodynamic path under consideration, representations of the scaling function of the critical Casimir force can be more convenient in terms of other scaling variables, such as in Fig. \[fig:1\] where $\mathcal{Y}=\mathrm{sgn}(t)D/\xi_t$ is chosen. The color along the lines of constant $\mathsf{Y}$ indicates the absolute value $|\vartheta^{(d=4)}_{||}/\Delta^{(d=4)}_{||}|$. The *bulk* critical point of the solvent $(\hat t,\hat{h}_b)=(0,0)$ is indicated by $\bullet$. The region shown here lies [*above*]{} the capillary transition critical point, where the *film* coexistence line ends. For $(+,+)$ boundary conditions as considered here, the capillary condensation transition occurs for $\hat t<0$ and $\hat{h}_b<0$. The dashed line indicates the path of constant order parameter $\phi$ of the solvent $\hat\phi=\frac{D}{\xi_{t,+}^{(0)}}\phi/\mathcal{B}=-5$, where $\mathcal{B}$ is the non-universal amplitude of the bulk OP $\phi=\mathcal{B}t^{\beta}$. Within mean-field theory $\nu=\beta=1/2$ and $\nu/(\beta\delta)=1/3$. $\Delta_{||}^{(d=4)}=\vartheta^{(d=4)}_{||}(\mathsf{Y}=0,\mathsf{\Sigma}=0)$.[]{data-label="fig:2"}](fig2)
![ The experimentally determined coexistence points \[$\odot$\] of the binary liquid mixture lutidine-water [@gallagher:92] which exhibits a lower, continuous demixing phase transition. $T_c^{(s)}$ is the critical temperature of this demixing transition and $\omega_L$ is the mass fraction of lutidine. These coexistence points on the binodal of phase segregation agree well with the relation $|\omega_L-\omega_{c,L}|=B_{\omega}|t|^{\beta}$ , where $\beta = 0.3265$ and $B_{\omega} = 0.765$ (dark gray line). Squares denote the experimentally obtained state points of the onset of aggregation (the straight black dotted lines in between are a guide to the eye) taken from the middle of Fig. 1 in Ref. \[11(b)\]. Each isoline of constant $B_2$ (full, colored lines; for visibility of the blue line the red one is dashed, both lines nearly coincide) corresponding to one of the state points (squares), is calculated by using the effective potential given by Eq. (\[eq:17\]); their values are $B_2/(\frac{4\pi}{3}R^3) = -67$ (red square), -65 (blue square), -23 (yellow square), and 5.4 (green square) (Fig. 8 in Ref. [@Mohry-et:2012b]). Each $B_2$-isoline can capture some qualitative trends of the possible shape of the line of onset of aggregation. The blue and the red lines reveal agreement, but the yellow and the green one not. []{data-label="fig:3"}](fig3)
![In the plane spanned by temperature and colloidal packing fraction $\eta$: (a) colloidal gas-liquid phase coexistence curves $T^{(\mathrm{eff})}_{cx}(\eta=(4/3)\pi R^3\rho\,\vert\,c_a)$ (full lines, see, c.f., Fig. \[fig:5\]), spinodals (i.e., loci of mean-field divergence of the isothermal compressibility $\chi_T$, dotted lines), and the critical points $T^{(\mathrm{eff})}_{c}$ (crosses) of an effective, one-component system of large colloidal particles as obtained by [*density functional theory*]{} (Fig. 2 in Ref. [@Mohry-et:2012a]). These particles of radius $R$ interact via an effective potential given by Eq. (\[eq:17\]) with parameters $\kappa R=10$ and $A=1000$ [@Mohry-et:2012a] taken at $T^{(s)}_{c}$). The curves correspond to a [*s*]{}olvent with a lower critical temperature $T_c^{(s)}$ and with various fixed solvent compositions $c_a$ (Eq. (\[eq:con\])) represented by the variable $m_0=\mathrm{sgn}(\phi)(\zeta_0)^{1/\nu}|\mathcal{B}/\phi|^{1/\beta} = -100$ (red), $-20$ (green), $-6$ (blue). $\mathcal{B}$ is defined via the shape of the solvent binodal $\phi=\mathcal{B}t^{\beta}$ and $\zeta_0=\kappa\xi^{(0)}_{t,+}$ (see below). The bulk critical exponents used here are $\nu=1/2$ and $\beta=1/2$. Close to the phase separation of the solvent the dominant temperature dependence within the effective approach described by Eq. (\[eq:18\]) is that of the critical Casimir forces (CCFs), encoded in $\zeta(t=1-T/T_c^{(s)})=\mathrm{sgn}(t)\kappa\xi_t(t)=\mathrm{sgn}(t)\zeta_0|t|^{-\nu}$. Therefore, if the temperature (see panel (a)) is expressed in terms of $\zeta$ (see panel (b)) the members of each set of curves with equal color in (a), corresponding to various values of $\zeta_0$ ($\zeta_0=0.01\; (\Box)$, $0.05\; (\circ)$, and $0.1\; (\vartriangle)$) fall de facto on top of each other and are characterized by $m_0$. The dashed lines in (b) correspond to the spinodals determined within the [*integral equation approach*]{}. (Due to $m_0\sim(c_a-c^{(s)}_{a,c})^{-1}$, the quantities $m_0= \pm \infty$ correspond to the critical composition $c_{a,c}^{(s)}$). For solvent compositions which are somewhat poor in the component preferred by the colloids, i.e., for intermediate negative values such as $m_0 \simeq -20 $, the critical Casimir forces are strongly attractive. Therefore, for them short correlation lengths suffice to bring about phase separation; accordingly the binodals occur at small values of $\zeta$. Here only thermodynamic states of the solvent which are in the one-phase region, i.e., for $t>0$, are considered. []{data-label="fig:4"}](fig4){width="\textwidth"}
![ Sketch of the phase diagram for colloids immersed in a binary liquid mixture at fixed pressure corresponding to a liquid state of the system (Fig. 1 in Ref. [@Mohry-et:2012a]). Upon adding colloids the phase separation curve $T^{(s)}_{cx}(c_a)$ of a pure solvent in the $(T,c_a,\rho=0)$ plane extends to a tube-like two-phase region $T_{cx}(c_a,\rho)$ in the $3d$ thermodynamic (td) space spanned by the temperature $T$, the concentration $c_a$, and the colloidal number density $\rho$. The lower critical point $\boxast$ ($T_c^{(s)},c_{a,c}^{(s)},\rho=0$) of a pure solvent extends to a line $\mathcal{C}_c$ (black curve) of critical points (some of which are shown as black squares). Its shape reflects the fact that the $2d$ manifold $T_{cx}(c_a,\rho)$ of coexisting states is not straight but bent and twisted due to the specific properties of the critical Casimir potential. The red dashed curves denote the projections of $\mathcal{C}_c$ onto the planes $(\rho,c_a)$ and $(T,c_a)$. Within the effective one-component approach the coexistence curves are explicit functions of $\rho$ only and depend parametrically on the overall concentration $c_a$. All three panels show that in general for $T=const$ the coexisting phases (i.e., the points connected by a tie-line) differ with respect to both $\rho$ and $c_a$. Thus the effective one-component approach has a limited applicability for determining the phase diagram. Experimentally or within suitable, sufficiently rich models, upon increasing temperature (along thermodynamic paths indicated by vertical arrows) one is able to determine a coexistence curve (black line in (c)) in the $3d$ td space. For a description of further details see the caption of Fig. 1 in Ref. [@Mohry-et:2012a]. []{data-label="fig:5"}](fig5)
![ Phase diagrams of the full ternary colloid - solvent $a$ - solvent $b$ $2d$ lattice model (Fig. S2 in Ref. [@bob-et:2014]) for three values of $t=(1-T/T_c^{(s)})$ (in Ref. [@bob-et:2014] denoted by $\tau$): 0.025 (black), 0.05 (dark red), and 0.075 (blue) in the $(\Delta\mu_s,\eta)$ plane of the solvent chemical potential difference and the colloidal packing fraction. The quantity $\Delta\mu_s =\mu_a - \mu_b$ is the chemical potential difference between species $a$ and $b$ (in units of the solvent-solvent interaction strength). The upper panel shows results obtained within mean field theory. The grey, pale red, and pale blue curves correspond to metastable colloidal gas-liquid (G-L) coexistence, which also terminates at the critical point. For each $\tau$ the upper (stable) and lower (metastable) gas-liquid critical points are indicated by red dots. X denotes the solid phase. (The various phases are inferred from monitoring their free energies.) The lower panel shows the corresponding phase diagrams as determined by Monte Carlo simulations. The diamonds and dots denote the phase boundaries as obtained from grand canonical staged insertion MC simulations and $(\Delta\mu_s/(k_BT), \eta, \tau)$ ensemble MC simulations, respectively. Their color corresponds to the value of $\tau$ given in the upper panel. The blue area corresponds to the two-phase region for $\tau = 0.075$; for $\tau = 0.05$ and 0.025 the two-phase regions encompass the previous regions and have added (colored) slices. The vertical dashed lines denote fluid-solid coexistence for pure colloidal hard discs. $T_c^{(s)}=T^{MFT}_c$ and $T_c^{(s)}=T^{MC}_c)$ is the critical temperature of the binary ($ab$) solvent within mean field theory and MC simulations, respectively. []{data-label="fig:6"}](fig05_a "fig:"){width="43.00000%"} ![ Phase diagrams of the full ternary colloid - solvent $a$ - solvent $b$ $2d$ lattice model (Fig. S2 in Ref. [@bob-et:2014]) for three values of $t=(1-T/T_c^{(s)})$ (in Ref. [@bob-et:2014] denoted by $\tau$): 0.025 (black), 0.05 (dark red), and 0.075 (blue) in the $(\Delta\mu_s,\eta)$ plane of the solvent chemical potential difference and the colloidal packing fraction. The quantity $\Delta\mu_s =\mu_a - \mu_b$ is the chemical potential difference between species $a$ and $b$ (in units of the solvent-solvent interaction strength). The upper panel shows results obtained within mean field theory. The grey, pale red, and pale blue curves correspond to metastable colloidal gas-liquid (G-L) coexistence, which also terminates at the critical point. For each $\tau$ the upper (stable) and lower (metastable) gas-liquid critical points are indicated by red dots. X denotes the solid phase. (The various phases are inferred from monitoring their free energies.) The lower panel shows the corresponding phase diagrams as determined by Monte Carlo simulations. The diamonds and dots denote the phase boundaries as obtained from grand canonical staged insertion MC simulations and $(\Delta\mu_s/(k_BT), \eta, \tau)$ ensemble MC simulations, respectively. Their color corresponds to the value of $\tau$ given in the upper panel. The blue area corresponds to the two-phase region for $\tau = 0.075$; for $\tau = 0.05$ and 0.025 the two-phase regions encompass the previous regions and have added (colored) slices. The vertical dashed lines denote fluid-solid coexistence for pure colloidal hard discs. $T_c^{(s)}=T^{MFT}_c$ and $T_c^{(s)}=T^{MC}_c)$ is the critical temperature of the binary ($ab$) solvent within mean field theory and MC simulations, respectively. []{data-label="fig:6"}](fig05_b "fig:"){width="45.00000%"}
![ Aggregation lines (black and green) for a ternary colloid - solvent $a$ - solvent $b$ mixture (interpreted as a colloidal condensation transition), as determined within mean field theory for the lattice model considered in Ref. [@bob-et:2014] (Fig. 4 in Ref. [@Edison-et:2015b]), plotted in the reduced temperature - composition of the pure solvent in the [*r*]{}eservoir $(t=(1-T/T_c^{(s)}), c_a)$ representation (in Ref. [@Edison-et:2015b] denoted by $(\tau=(T-T^{MFT}_c)/T^{MFT}_c,x_r))$. A point on the aggregation line at $x_r$ gives the temperature at which phase separation of colloids is observed first, upon cooling the suspension at a fixed packing fraction $\eta$ of colloids and for fixed solvent composition $x_r$. The $2d$ calculations have been performed for a parameter $v_c$ (approximately equal to the area of the colloidal disc) chosen to be equal to 1000${\it a}^2$ (in order to reproduce most closely the MC simulations results of Ref. [@bob-et:2014]), where ${\it a}$ is the lattice spacing, for the coupling strength between colloid and solvent species $b$ equal to 32 (in units of the solvent - solvent interaction strength), and for two fixed values $\eta = 0.1$ and $\eta = 0.05$ of the colloid packing fraction. The colloids interact with the members of solvent species $a$ and with each other via a hard core repulsion. The dotted orange line is the binodal of the colloid-free $ab$ solvent. Each aggregation line ends at a critical point of the ternary mixture, which is removed from the binodal of the solvent reservoir (see the main text). The origin of the break in slopes of the aggregation curves is not discussed in Ref. [@Edison-et:2015b]. The bent of the aggregation lines near their critical points implies the occurrence of reentrant dissolution upon cooling. []{data-label="fig:7"}](fig7){width="\textwidth"}
![Shapes of the effective total potential $V(D)= V_{el}(D) + V_{C}(D)$ of the force acting on two identical colloids of radius $R$ for various values of the bulk correlation length $\xi=\xi(t,h_b=0)=\xi_t$ and the Debye screening length $\kappa^{-1}$, where $D$ is the surface-to-surface distance of the two colloids (Fig. 1 in Ref. [@Gambassi-et:2010]). Six different regions of distinct shapes of $V(D)$ are limited by the thin solid lines. These lines meet at $\kappa^{-1}=\xi_t \equiv \xi^*_c$. $D_{min}$ is the position $V_{min}$ of the minimum of $V(D)$. Within the hatched area enclosed by the thick dashed lines the condition $R\gg D_{min} \gtrsim \xi_t,\kappa^{-1}$ is satisfied. The additional requirement that $V_{min} \lesssim -3 k_BT$ is fulfilled in the cross-hatched part of the hatched are. The black dots mark the experimentally determined [@Bonn-et:2009] aggregation line. For further details see Ref. [@Gambassi-et:2010] and Ref. 1 therein. []{data-label="fig:8"}](fig8)
![ Phase diagram in terms of $\Delta T = T- T^{(s)}_{cx}$ and volume fraction, in Ref. [@Nguyen-et:2013] denoted as $\phi =V_s/V_{tot}$, where $V_s$ is the volume of colloids and $V_{tot}$ is the total volume of the sample. These data have been obtained from MC simulations (colored symbols) for the effective one-component colloidal system governed by a pair potential which is the sum of repulsive and attractive, exponentially decaying, functions describing screened electrostatic and critical Casimir interactions (Fig. 2 in Ref. [@Nguyen-et:2013]). G, L, F, and C denote colloidal gas, liquid, fluid, and crystal phases, respectively. G+L, G+L, and L+C stand for the gas-liquid, gas-crystal, and the liquid-crystal coexistence regions. Black squares with error bars are experimental data for PNIPAM particles suspended (a) in the 3MP - heavy water mixture at the critical 3MP mass fraction $ \omega_{3MP}= 0.28 \simeq \omega_{3MP,c}$ and (b) in the 3MP - water - heavy water mixture at the off-critical 3MP mass fraction $\omega_{3MP} = 0.25$. The authors of Ref. [@Nguyen-et:2013] interpret the snapshots of confocal microscopy images for the mixture in (b) as the coexistence of colloidal gas and crystal at $\Delta T = - 0.2$ (c) and colloidal gas and liquid at $\Delta T = - 0.3$ (d). Stars indicate the experimental position of the colloidal gas-liquid critical point as estimated by using the law of rectilinear diameters. In (a) the star appears as being placed at a too small value of $|\Delta T|$. []{data-label="fig:9"}](fig9)
![ Segregation phase diagrams obtained from theory (RPA), experiment, and simulations (MC) (Fig. 6 in Ref. [@Mohry-et:2014]). (a) The phase diagram obtained within RPA using the four available background potentials $U_{bck}$ extracted from effective potentials inferred from experimental data [@Mohry-et:2014] at $\Delta T/\textrm{K}=0.6$ (magenta), $0.5$ (green), $0.4$ (orange), and $0.3$ (blue), and their mean curve (thick dark red curve). The solid lines show the phase boundaries in terms of the packing fraction $\eta$ of the colloids, whereas the dashed lines correspond to the spinodals, and dots represent critical points. (b) Comparison of the theoretical predictions for the phase boundaries (based on the mean $U_{bck}$) with MC simulation data ($\boxdot$) and experimental data ($\boldsymbol{\times}$, with error bars) from Ref. [@Dang-et:2013]. On the temperature axis $\Delta{}T= T^{(s)}_c - T$ increases from top to bottom in order to mimic the visual impression of a lower critical point $T^{(s)}_c$ (of the solvent) as observed experimentally. $\widetilde\phi = (\omega_{3MP,c}-\omega_{3MP})/{\cal B} \quad = -0.088$, where ${\cal B}$ is the non-universal amplitude of the bulk coexistence curve (see the main text). []{data-label="fig:10"}](fig10)
(a)\
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{width="\textwidth"}
(a)(b)\
![ Reduced second virial coefficient $B_2^*=B_2/B_2^{(HS)}$ as function of $T$ and $c$ for the same system as in Fig. \[fig:11\] (see Fig. 9 in Ref. [@Marcel-et]). The color of the shading provides the theoretically predicted values of $B_2^*(T,c)$, while the colored symbols provide the experimental value of $B_2^*(T,c)$ as obtained by numerically integrating the measured radial distribution function $g(r)$ (Eq. (\[eq:18\])). The weak color contrast between the colors of the symbol and the corresponding underlying shading indicates agreement between the experimental and theoretical data. The values $B_2^* \approx 1$ depicted in blue indicate a significant repulsion, while the values $B_2^* \approx -1$ depicted in red indicate strong attraction. Yellow marks the crossover. The critical temperature $T^{(s)}_c$ (denoted as $T_c$) and the critical mass fraction $c_{c}$ are indicated by arrows and the critical point is indicated by $+$. In (a), via the color, $B_2^*$ is shown in the entire $(T,c)$ plane. The black dashed lines correspond to those five concentrations for which there are experimental data; each dashed line corresponds to a certain symbol type. In (c) the values of $B_2^*$ are shown as functions of temperature for the five values of $c$ as introduced in (a) via the corresponding symbol type. The horizontal dashed yellow and red lines indicate the isolines for $B_2^*=0$ and $B_2^*=-1.2$, marking the crossover from repulsion to attraction and the critical value of the sticky spheres model, respectively. The value of $B_2^*$ can be read off both from the vertical axis and from the color of the symbols. Panel (b) shows the same data but the curves corresponding to distinct values of $c$ are shifted up vertically in order to gain visual clarity. The shifts are chosen such that the shifted values of $B_2^*$ at $T=35$ are separated equally. This generates a perspective view of the $B_2^*$ values above the $(T,c)$ plane, revealing a folded curtain-like shape of the surface $B_2^*(T,c)$. The color shading is discretized due to the shifts of the curves. The yellow and red dashed lines emerge from the corresponding ones in (c) by the shifts. The vertical axis in (b) is $B_2^*$, but shifted. In the upper left corner of (b), the five dashed lines are labeled both by the symbol types and the corresponding numerical values of the concentration. []{data-label="fig:12"}](fig12 "fig:")\
(c)
![Mean particle density distribution (represented by different colors ranging from black to yellow for minimal to maximal density, respectively) of a dilute colloidal suspension of spherical particles with $R= 2.4 \mu m$ dissolved in a critical WL mixture in the presence of a chemically patterned substrate (Fig. 1 in Ref. [@Soyka-et:2008]). Particle positions were determined by digital video microscopy with a spatial resolution of ca. $50\mathrm{nm}$. $T_c^{(s)}-T=0.72\mathrm{K}$ (a), $0.25\mathrm{K}$ (b), $0.23\mathrm{K}$ (c), and $0.14\mathrm{K}$ (d). []{data-label="fig:13"}](fig13.jpg){width="70.00000%"}
![Total effective potential $\delta\hat{V}(x)=\hat{V}(x)-\hat{V}(x=P/2)$ of the forces acting on a hydrophilic polystyrene spheres of radius $R=1.2\mathrm{\mu m}$ above a chemically striped pattern of periodicity $P$ with alternating ($-$) and ($+$) boundary conditions and immersed in a water - lutidine mixture at its critical concentration, as function of its lateral position $x$ for various temperatures $T^{(s)}_c - \Delta T$ below the experimental value $T^{(s)}_c$ of the lower critical temperature of the solvent (Fig. 5 in Ref. [@Troendle-et:2011]). The potential is given by $\hat{V}(x)=-k_BT\ln(\hat{\rho}(x))$, where $\hat{\rho}(x)$ is the effective number density of the colloids at $x$, obtained by projecting the actual number density onto the $x$ axis. The width of ($-$) and ($+$) stripes is $2.25\mathrm{\mu m}$. Symbols indicate experimental data, whereas the lines are the corresponding theoretical predictions for sharp (dashed lines) and fuzzy (solid lines) chemical steps. From top to bottom the measured temperature deviations $\Delta T$ are 0.175 (0.165), 0.16 (0.152), 0.145 (0.143), 0.13, 0.115, 0.10 K. If indicated, the values in paranthesis are corrected values of temperature (but compatible within the experimental inaccuracy) which have been used for evaluating the theoretical predictions. []{data-label="fig:14"}](fig14){width="90.00000%"}
![Snapshots of the twodimensional configurations of binary colloidal systems (black and orange discs, Fig. 1 in Ref. [@Zvyagolskaya-et:2011]) for three different compositions, expressed in terms of the concentration $x_a=\rho_a/(\rho_a+\rho_b)$, where $ \rho_a$ and $\rho_b$ are (areal) number densities of the particles of type $a$ and $b$, respectively: (a) $x_a=0.28$, (b) $x_a = 0.32$, and (c) $x_a =0.54$, with $a=$ orange, exhibiting distinct structures formed after 1 hour at a temperature deviation $\Delta T = T_c^{(s)} - T = 0.01$K from the lower critical point of the solvent. The horizontal scales range from 20 to 120 in (a) and from 0 to 100 in (b) and (c), whereas the vertical scales range from 20 to 80 in (a) and (b) and from 0 to 80 in (c); the authors have not provided the units. []{data-label="fig:15"}](fig15.jpg){width="\textwidth"}
![Experimental phase diagram of aqueous suspensions of MFA latex particles in the presence of a nonionic surfactant of mass fraction $c_s=m_s/m_{tot}$ (in units of mass percentage, sometimes called weight percentage, % w/w $ = 100\times c_s$), where $m_s$ is the mass of all surfactant particles and $m_{tot}$ is the total mass of the sample (Fig. 8 in Ref. [@Piazza-et:2011]). For state points of the surfactant - water mixtures to the left of the line of dots (denoted as “stable”) the dissolved colloidal particles with a volume fraction $V_c/V_{tot} = 0.03$ form a homogeneous colloidal phase whereas for state points to the right of the line of dots (denoted as “separated”) the colloids phase separate into a colloidal gas and a colloidal liquid. ($V_c$ is the volume taken by colloidal particles and $V_{tot}$ is the total volume of the sample). At a given temperature, the full black dots represent the [*m*]{}inimum amount $c^{(m)}_s$ of [*s*]{}urfactant required to induce colloidal gas-liquid phase separation at a given temperature. The rather shallow coexistence curve of the surfactant-water mixture with a lower critical point (located around 1.8 % of $c_s$ - not marked in the plot) is shown by open blue dots. The data correspond to 250 mM added NaCl salt. The inset shows (in our present notation) $c^{(m)}_s$ in units of its critical value $c^{(c)}_s$ (i.e., $c_s$ at the critical point) as a function of reduced temperature $t =(T^{(s)}_c-T)/T^{(s)}_c$ (note that - as in Ref. [@Piazza-et:2011] - in this figure $c^{(m)}_s/c^{(c)}_s$ and $t$ are denoted by $c_s/c_c$ and $\varepsilon$, respectively.) []{data-label="fig:16"}](fig16){width="60.00000%"}
![The evolution of the characteristic size scale of the aggregating colloidal suspension studied in Ref. [@Veen-et:2012] (see Fig. 3 therein) as given by the quantity $q_{red}^{-1}(\mathsf{t})$, which renders a [*red*]{}uced description of the light scattering intensity $I(q,\mathsf{t})$ of the growing aggregates in terms of a scaling function $F(q/q_{red}(\mathsf{t}))$ of a single variable (see the main text). Results both from microgravity (full symbols) and from ground experiments (open symbols) are shown. (The sample contains 1.5 mmol/liter NaCl.) The curves correspond to various temperatures $T$ ramped up beyond the aggregation temperature $T_{agg}$, up to $T_{agg} + 0.4°$C (from bottom to top as indicated by the arrow; the caption to Fig. 3 in Ref. [@Veen-et:2012] does not provide the temperature values corresponding to the various symbols). The temperature $T_{agg}$ at which the aggregation starts is identified (somewhat loosely) as the onset of the rapid increase of the normalized variance of $I(q)$ (see the main text). The sudden drop of $q^{-1}_{red}$ at late times is due to the massive sedimentation of the aggregates so that the suspension becomes poor in large aggregates and rich in small ones which shifts down $q^{-1}_{red}$. The inset provides an enlarged view of the data concerning the onset of the aggregation process, at which the characteristic length scale $q_{red}^{-1}$ starts to increase.[]{data-label="fig:17"}](fig17.png){width="80.00000%"}
![Ratio $\beta = R_h/R_g$ of the hydrodynamic to the gyration radius of aggregates as a function of their fractal dimension $d_f$ for various temperature deviations $\Delta T$ above the aggregation temperature $T_{agg}$ (Fig. 4 in Ref. [@Potenza-et:2014]) characterized by the colors of the symbols: $\Delta T =0$ (black), 0.1 K (red), 0.2 K (blue), 0.3 K (green), and 0.4 K (violet). $T_{agg}$ is determined as in Fig. \[fig:18\], i.e., from the behavior of the normalized variance of the scattered intensity $I(q)$. The types of symbols indicate the salt concentrations of 0.31 mmol/liter (squares), 1.5 (circles), and 2.7 (triangles), corresponding to the Debye screening length $\kappa^{-1}$ (denoted as $\lambda_D$ in Ref. [@Potenza-et:2014]) of 14, 6.4, and 4.8 nm, respectively. Concerning the values of $\beta$, from top to bottom the lines indicate the dependence on $d_f$ as expected for steplike, Gaussian, and exponentially decaying cut functions of the density distribution of the aggregates, respectively (see the main text). The insets show holographic reconstructions of the the real-space shape of the aggregates (white regions) grown at $T=T_{agg} +0.4$K (closest to $T^{(s)}_c$ and hence strongest attraction between the colloids, top) and $T=T_{agg}$ (furthest from $T_c^{(s)}$ and hence weakest attraction between the colloids, bottom). The length of the scale bar is 25 $\mu$m.[]{data-label="fig:18"}](fig18){width="80.00000%"}
![Tuning the morphology of colloidal aggregates by critical Casimir forces (Fig. 1 in Ref. [@Shelke-et:2013]). Confocal microscopy images and the threedimensional reconstructions of the aggregates formed upon temperature quenches to $\Delta T = T_c^{(s)}-T = 0.2$K (a,d), 0.14K (b,e), and 0.12K (c,f) below the lower critical phase separation temperature $T_c^{(s)}$ of the binary solvent. (g) Scaling of the number of particles $\mathcal{N}(r)$ within a sphere of radius $r$ from the center of the cluster at $\Delta T = 0.2$K (triangles), $\Delta T = 0.14$K (squares), and $\Delta T = 0.12$K (circles); $r_0$ is the radius of the particles. (h) Fractal dimension as a function of $\Delta T$ determined from $\mathcal{N}(r) \sim r^{d_f}f_{cut}(r)$ and thus from the slopes in (g). The dashed line is a guide to the eye. Upon increasing $\Delta T$ a continuous increase of the fractal dimension to the space-filling limit $d_f=3$ is observed (see the main text). []{data-label="fig:19"}](fig19){width="\textwidth"}
[^1]: In the present notation $d$ and $d^*$ correspond to $D$ and $d$ used in Ref. [@Mohry-et:2014], respectively.
[^2]: In the original paper [@Bonn-et:2009] $q'$ is given as $q'= (\mathsf{t}_0/\mathsf{t})qR$, which we consider as a typo.
|
---
abstract: 'We predict a new mechanism to induce collective excitations of a fermionic superfluid via sudden switch-on of two-body loss, for which we extend the BCS theory to fully incorporate quantum jumps. We find that such dissipation induces an amplitude oscillation of the superfluid order parameter accompanied by chirped phase rotation, which highlights the role of dissipation in a superfluid as a consequence of particle loss. We demonstrate that when the dissipation is introduced to one of the two superfluids coupled via a Josephson junction, it gives rise to a relative-phase mode analogous to the Leggett mode, which can be detected from time evolution of the Josephson current. We find that the coupled system exhibits a nonequilibrium dissipative phase transition characterized by the vanishing dc Josephson current. The dissipation-induced collective modes can be realized with ultracold fermionic atoms undergoing inelastic collisions.'
author:
- Kazuki Yamamoto
- Masaya Nakagawa
- Naoto Tsuji
- Masahito Ueda
- Norio Kawakami
bibliography:
- 'OpenBCS.bib'
title: Collective Excitations and Nonequilibrium Phase Transition in Dissipative Fermionic Superfluids
---
*Introduction*.—Collective excitations of superconductors and superfluids have been widely studied in condensed matter physics [@Volkov74; @Pekker15; @Shimano19]. Recent experimental progress in ultracold atoms has enabled the studies of out-of-equilibrium dynamics of superfluid order parameters beyond the linear-response regime [@Barankov04; @Barankov06A; @Yuzbashyan062; @Yuzbashyan15; @Andreev04; @Barankov06; @Yuzbashyan06]. For example, a sudden quench of an attractive interaction or a periodic modulation of the amplitude of the order parameter excites the Higgs amplitude mode, which has been observed with ultracold fermions [@Barankov06; @Yuzbashyan06; @Kohl18] and also in solid-state systems by light illumination on BCS superconductors [@Matsunaga13; @Shimano14; @Tsuji15]. As for collective phase modes, the Nambu-Goldstone mode exists in neutral superfluids, and the relative-phase Leggett mode has been predicted for multiband superfluids [@Leggett66; @Sharapov02; @Burnell10; @Bittner15; @Krull16; @Cea16; @Tsuji17]. Especially, ultracold atoms allow for a dynamical control of various system parameters, offering an ideal playground to investigate collective modes. However, they inevitably suffer from atomic loss due to inelastic scattering, which has received little attention in literature.
In dissipative open quantum systems, the dynamics, after environmental degrees of freedom are traced out, is nonunitary and described by a completely positive and trace-preserving map [@Lindblad76; @Daley14]. Such nonunitary dynamics is relevant for atomic, molecular, and optical systems, drastically changing various aspects of physics such as quantum critical phenomena [@Ashida17; @Nakagawa18], quantum phase transitions [@Vidanovi14], quantum transport [@Damanet19] and superfluidity [@Han09]. In particular, high controllability of parameters in ultracold atoms has enabled observation of non-equilibrium quantum dynamics induced by dissipation [@Witthaut08; @Tomita17; @Ott13; @Ott15; @Schneider17; @Nagerl12; @Tomita19; @Yamamoto19; @Nakagawa20; @Nakagawa20arxiv; @Takasu20]. The effect of particle loss in fermionic superfluids has been studied in the framework of the non-Hermitian BCS theory [@Yamamoto19]; however, it does not take account of the change in the number of particles due to quantum jumps. Thus, it is necessary to go beyond the non-Hermitian framework to describe the long-time dynamics of a superfluid and associated collective modes of order parameters.
In this Letter, we theoretically investigate collective excitations of fermionic superfluids driven by sudden switch-on of two-particle loss due to inelastic collisions between atoms. By formulating a dissipative BCS theory that fully incorporates quantum jumps, we find that dissipation fundamentally alters the superfluid order parameter and induces collective oscillations in its amplitude and phase. In particular, we elucidate that a coupling between the order parameter and dissipation leads to a chirped phase rotation, which is in sharp contrast to the case of interaction quench in closed systems \[see Fig. \[fig\_Schematic\](a)\].
To experimentally observe the chirped phase rotation unique to dissipative systems, we propose that a particle loss leads to a relative-phase oscillation analogous to the Leggett mode [@Leggett66; @Sharapov02; @Burnell10; @Bittner15; @Krull16; @Cea16; @Tsuji17] when weak dissipation is introduced to one of two coupled superfluids \[see Fig. \[fig\_Schematic\](b)\]. The phase mode causes an oscillation of a Josephson current around a nonvanishing dc component. Remarkably, when dissipation becomes strong, the coupled system undergoes a dissipative phase transition characterized by the vanishing dc Josephson current due to a monotonic increase in the phase difference. Our findings can experimentally be tested in ultracold atoms through introduction of dissipation via a photoassociation process [@Tomita17; @Takasu20].
*Dissipative BCS theory*.—We consider ultracold fermionic atoms described by the three-dimensional attractive Hubbard model $$\begin{aligned}
H=\sum_{\bm{k}\sigma}\epsilon_{\bm{k}}c_{\bm{k}\sigma}^\dagger{c}_{\bm{k}\sigma}-U_{\mathrm{R}}\sum_{i}c_{i\uparrow}^{\dagger}c_{i\downarrow}^{\dagger}c_{i\downarrow}c_{i\uparrow},
\label{eq_Hubbard}\end{aligned}$$ where $U_{\mathrm{R}}>0$ (the subscript R indicates that the original interaction is real), $\epsilon_{\bm{k}}$ is the single-particle energy dispersion, and $c_{\bm k\sigma}$ ($c_{i\sigma}$) denotes the annihilation operator of a spin-$\sigma$ fermion with momentum $\bm{k}$ (at site $i$). When the system is subject to inelastic collisions, scattered atoms are lost into a surrounding environment, resulting in dissipative dynamics as observed experimentally [@Tomita17; @Nagerl12; @Tomita19]. Here, we study the time evolution of the density matrix $\rho$ which is described by the Lindblad equation [@Lindblad76; @Daley14] $$\begin{aligned}
\frac{d\rho}{dt}=\mathcal{L}\rho=-i[H,\rho]-\frac{\gamma}{2}\sum_i (\{L_i^\dag {L}_i, \rho\} -2L_i \rho L_i^\dagger),
\label{eq_Lindblad}\end{aligned}$$ where $L_i = c_{i\downarrow} c_{i \uparrow}$ is a Lindblad operator that describes two-body loss of Cooper pairs with loss rate $\gamma>0$. The last term on the right-hand side of Eq. represents a quantum jump process.
![(a) Schematic illustration of the amplitude and phase modes in a Mexican-hat free-energy potential as a function of the complex order parameter $\Delta$. A sudden quench of the interaction $U_{\mathrm R}$ and dissipation $\gamma$ kicks $\Delta$ in a direction parallel and perpendicular to the radial direction, respectively. (b) Two superfluids coupled via a Josephson junction, where one superfluid (system 2) is subject to two-body loss.[]{data-label="fig_Schematic"}](SchematicFig.pdf){width="8.5cm"}
We first study how the standard BCS theory is generalized in an open dissipative system by formulating a time-dependent mean-field theory in terms of a closed-time-contour path integral [@Diehl16; @Kamenev11]. We start with a generating functional defined as $$\begin{aligned}
Z=\mathrm{tr}\rho=\int\mathcal D[c_-, \bar c_-, c_+, \bar c_+]e^{iS}=1,\label{eq_pf}\end{aligned}$$ with an action $$\begin{aligned}
&S=\int_{-\infty}^\infty dt \Bigg[\sum_{\bm k \sigma}(\bar c_{\bm k \sigma +} i\partial_t c_{\bm k \sigma +} -\bar c_{\bm k \sigma -}i\partial_t c_{\bm k \sigma -})-H_+\notag\\
&+ H_- + \frac{i \gamma}{2}\sum_i(\bar L_{i+}L_{i+}+\bar L_{i-}L_{i-}-2L_{i+}\bar L_{i-})\Bigg],
\label{eq_action}\end{aligned}$$ where the subscripts $+$ and $-$ denote forward and backward paths, $H_\alpha= \sum_{\bm{k}\sigma}\epsilon_{\bm{k}}\bar c_{\bm{k}\sigma\alpha} c_{\bm{k}\sigma\alpha}-U_{\mathrm{R}}\sum_{i}\bar c_{i\uparrow\alpha}\bar c_{i\downarrow\alpha} c_{i\downarrow\alpha}c_{i\uparrow\alpha}$, $L_{i\alpha}=c_{i\downarrow\alpha} c_{i \uparrow\alpha}$, and $\bar L_{i\alpha}=\bar c_{i \uparrow\alpha}\bar c_{i\downarrow\alpha}$ ($\alpha=+$, $-$). Note that the action has U(1) symmetry under $c_{i\sigma\alpha}\to e^{i\theta}c_{i\sigma\alpha}$ despite the fact that the particle number is not conserved [@Buca12; @Albert14]. By introducing auxiliary fields via the Hubbard-Stratonovich transformation, we rewrite the action in the quadratic form of fermions as [@Yamamoto19; @Supple] $$\begin{aligned}
S=\int dt\Bigg\{&\sum_{\bm k}\Bigg[
\bar \psi_{\bm k +}^t
\left(
\begin{matrix}
i\partial_t-\epsilon_{\bm{k}}&-\Delta\\
-\Delta^*&-i\partial_t+\epsilon_{\bm k}
\end{matrix}
\right)
\psi_{\bm k +}\notag\\
&-
\bar \psi_{\bm k -}^t
\left(
\begin{matrix}
i\partial_t-\epsilon_{\bm k}&-\Delta\\
-\Delta^*&-i\partial_t+\epsilon_{\bm k}
\end{matrix}
\right)
\psi_{\bm k -}\Bigg]\Bigg\},
\label{eq_action2}\end{aligned}$$ where $\bar \psi_{\bm k \alpha}=\left(
\begin{matrix}
\bar c_{\bm{k}\uparrow\alpha},&c_{-\bm{k}\downarrow\alpha}
\end{matrix}
\right)^t$ and $\psi_{\bm k \alpha}=\left(
\begin{matrix}
c_{\bm{k}\uparrow\alpha},&\bar c_{-\bm{k}\downarrow\alpha}
\end{matrix}
\right)^t$ ($\alpha=+, -$). Here $\Delta$ is the superfluid order parameter which can be determined from the requirement that the action be extremal as [@Supple] $$\begin{aligned}
\Delta=-\frac{U}{N_0}\sum_{\bm k}\mathrm{tr}(c_{-\bm k\downarrow}c_{\bm k \uparrow}\rho)\equiv-\frac{U}{N_0}\sum_{\bm k}\langle c_{-\bm k\downarrow}c_{\bm k \uparrow}\rangle,\end{aligned}$$ where $U=U_R+i\gamma/2$ is an effective complex coupling constant including a contribution from atomic loss [@Yamamoto19], $N_0$ is the number of sites, and the center-of-mass momentum of Cooper pairs is neglected. Importantly, the order parameter includes the loss rate $\gamma$. This leads to dissipation-induced collective modes as discussed below. The action describes the mean-field time-evolution equation of the density matrix as $$\begin{gathered}
\frac{d\rho}{dt}=-i[H_{\mathrm{eff}},\rho],
\label{eq_unitary}\\
H_{\mathrm{eff}}=\sum_{\bm{k}}
\Psi_{\bm k}^\dag
\left(
\begin{matrix}
\epsilon_{\bm{k}}&\Delta\\
\Delta^*&-\epsilon_{\bm{k}}
\end{matrix}
\right)
\Psi_{\bm k},
\label{eq_HamiltonianEff}\end{gathered}$$ where $\Psi_{\bm k}=\left(\begin{matrix}c_{\bm k \uparrow},&c_{-\bm k \downarrow}^\dag \end{matrix}\right)^t$ is the Nambu spinor. In the Supplemental Material [@Supple], we show that Eq. can also be derived from two different methods, i.e. the mean-field theory for the Lindblad equation and the time-dependent Bogoliubov-de Gennes analysis. While Eq. appears to describe unitary evolution, it is consistent with the original Lindblad equation as a consequence of the time-dependent BCS ansatz [@Supple].
We use Anderson’s pseudospin representation [@Anderson58; @Barankov04; @Tsuji15; @Barankov06A; @Yuzbashyan062; @Yuzbashyan06; @Yuzbashyan15; @Andreev04; @Barankov06] defined by $\bm \sigma_{\bm k}=\frac{1}{2}\Psi_{\bm k}^\dag\cdot\bm \tau\cdot\Psi_{\bm k}$ and $H_{\mathrm{eff}}=2\sum_{\bm k}\bm b_{\bm k}\cdot\bm \sigma_{\bm k}$, where $\bm \tau=\left(\begin{matrix}\tau_x,&\tau_y,&\tau_z\end{matrix}\right)$ is the vector of the Pauli matrices. The pseudospins satisfy the commutation relations $[\sigma_{\bm k}^j,\sigma_{\bm k}^k]=i\epsilon_{jkl}\sigma_{\bm k}^l$. For simplicity of notation, we omit the bracket $\langle\cdots\rangle$ and regard $\bm \sigma_{\bm k}$ as the expectation value of the pseudospin operator. By using the commutation relation of the pseudospins, Eq. is mapped to the Bloch equation $$\begin{gathered}
\frac{d\bm \sigma_{\bm k}}{dt}
=2\bm b_{\bm k}\times\bm \sigma_{\bm k},\label{eq_bloch}\\
\bm b_{\bm k}=\left(\begin{matrix}\mathrm{Re}\Delta,&-\mathrm{Im}\Delta,&\epsilon_{\bm k}\end{matrix}\right),\end{gathered}$$ and the superfluid dynamics is characterized by a precession of pseudospins in an effective magnetic field $\bm b_k$. Here, the order parameter is determined self-consistently from the pseudospin expectation value as $$\begin{aligned}
\Delta=|\Delta|e^{i\theta}=-\frac{U}{N_0}\sum_{\bm k}\left(\sigma_{\bm k}^x-i\sigma_{\bm k}^y\right).
\label{eq_order}\end{aligned}$$ It is noteworthy that the norm of the pseudospin is conserved by the Bloch equation . The time evolution of the particle number due to particle loss is obtained from Eq. as $$\begin{aligned}
\frac{1}{N_0}\frac{dN}{dt}=-\frac{2\gamma|\Delta|^2}{|U|^2},
\label{eq_Ndt}\end{aligned}$$ which varies in time in a manner depending on the superfluid order parameter.
*Collective excitations: phase and amplitude modes*.—We numerically solve the Bloch equation self-consistently under the condition . As an initial state, we prepare a BCS ground state with $\gamma=0$, whose pseudospin representation is given by $\sigma_{\bm k}^x(0)=-\Delta_0/\sqrt{{\epsilon_{\bm k}}^2+\Delta_0^2}$, $\sigma_{\bm k}^y(0)=0$ and $\sigma_{\bm k}^z(0)=-\epsilon_{\bm k}/\sqrt{{\epsilon_{\bm k}}^2+\Delta_0^2}$ with $\Delta_0\in\mathbb{R}$; we then switch on the atomic loss $\gamma$ at $t=0$. The energy $\epsilon_{\bm k}$ is measured from the Fermi energy of the initial state. The results are shown in Fig. \[fig\_lossquench\], which are obtained by using the second-order Runge-Kutta method with a constant density of states.
![Dynamics of a superfluid after the atomic loss with $\gamma=2.81\Delta_0$ is switched on for the initial state with $U_{\mathrm{R}}=12.2\Delta_0$ and bandwidth $W=46.8\Delta_0$, where $\Delta_0$ is the superfluid order parameter in the absence of the atomic loss. (a) Real parts (light green), imaginary parts (blue), and the amplitude (violet) of the order parameter. (b) Angular velocity (pink) and particle number (yellow) plotted against time.[]{data-label="fig_lossquench"}](LossQuench1.pdf){width="8.5cm"}
In the long-time limit, the amplitude of the superfluid order parameter $\Delta$ is suppressed due to dissipation, indicating a decay of superfluidity \[see Fig. \[fig\_lossquench\](a)\]. This is a direct consequence of the reduction of the particle number of the system during the time evolution shown in Fig. \[fig\_lossquench\](b). Remarkably, after the dissipation $\gamma$ is introduced, the U(1) phase of the order parameter rotates and shows chirping, i.e. its angular velocity increases with time \[see Fig. \[fig\_lossquench\](a), (b)\] as a consequence of the dynamical shift of the Fermi level [@Supple]. This property is unique to the dissipative superfluid and completely different from the usual dynamics in isolated systems where the U(1) phase stays constant [@Barankov06; @Yuzbashyan06]. The phase rotation is understood from a Mexican-hat free-energy potential as a function of the complex order parameter $\Delta$ \[see Fig. \[fig\_Schematic\](a)\]. When dissipation is introduced, the sudden quench of the imaginary part of $U$ in Eq. pushes the order parameter towards the direction perpendicular to the radial direction. Thus, the phase of the order parameter revolves in a way analogous to a ball rotating in a parabolic vessel. Another way to understand the phase rotation is to introduce an effective chemical potential as $\Delta(t)=\exp(-2i\int_0^t \mu_{\mathrm{eff}}(t)dt)\Omega(t)$ $(\Omega(t)\in\mathbb R)$. By performing a global gauge transformation from $\Delta(t)$ to $\Omega(t)$, the Bloch equation is mapped to that in the Larmor frame in which the energy dispersion is replaced by $\xi_{\bm k}(t) = \epsilon_{\bm k} - \mu_{\mathrm{eff}}(t)$. This gauge transformation indicates that the phase rotation corresponds to a decrease of the effective chemical potential, which is consistent with the behaviors of $\dot \theta$ and $N$ in Fig. \[fig\_lossquench\](b).
We also find amplitude oscillations in $|\Delta|$ as shown in Fig. \[fig\_lossquench\](a). The amplitude oscillations are more pronounced when the interaction and the dissipation are simultaneously quenched [@Supple]. The mechanism behind the oscillations is that the quench of the imaginary part of $U$ changes the absolute value of $\Delta$ (see Fig. \[fig\_Schematic\](a)), and induces an oscillation in the time derivative of the particle number due to Eq. [@Supple]. Contrary to the case of isolated systems where the particle number does not change, the dissipation-induced dynamics can provide a unique way to observe the amplitude mode of superfluid order parameters from the measurement of the time-dependent particle number.
{width="18cm"}
*Collective excitations: Leggett mode*.—To observe the chirped phase rotation of the superfluid order parameter which is a unique feature of dissipative superfluids, we propose that the phase rotation induced by dissipation can be detected when two superfluids are connected via a Josephson junction [@Spun07; @Roati15; @Roati18; @Moritz19]. As the phase difference in the two superfluid order parameters is gauge-invariant, it leads to an observable Josephson current. We introduce dissipation to one of the superfluids as schematically illustrated in Fig. \[fig\_Schematic\](b) and assume that the two superfluids are coupled via a tunneling Hamiltonian [@Tsuji17; @Leggett66] $$\begin{aligned}
H_{\mathrm{tun}}=-\frac{V}{N_0}\sum_{\bm k \bm{k}'}\left( c_{1\bm{k} \uparrow}^\dag c_{1-\bm{k} \downarrow}^\dag c_{2-\bm k'\downarrow}c_{2\bm k' \uparrow}+\mathrm{H.c.}\right),
\label{eq_tunnel}\end{aligned}$$ where $V>0$ is the amplitude of Cooper-pair tunneling between the system $1$ without dissipation and the system $2$ with dissipation. By performing a mean-field analysis, we can write the system Hamiltonian as $$\begin{aligned}
H_{\mathrm{sys}}
=H_1 + H_2 + H_{\mathrm{tun}}
\equiv\tilde H_1+\tilde H_2,\end{aligned}$$ where $H_i=\sum_{i\bm k \sigma}\epsilon_{\bm k \sigma} c_{i\bm k \sigma}^\dag c_{i\bm k \sigma} + \sum_{\bm k}(\Delta_i c_{i \bm k \uparrow}^\dag c_{i -\bm k \downarrow}^\dag + \mathrm{H.c.})$ ($i=1,2$) is the mean-field pairing Hamiltonian of the system $i$ and $\tilde H_i = H_i -V/N_0\sum_{\bm k \bm k'} (\langle c_{j -\bm k'\downarrow}c_{j\bm k' \uparrow}\rangle c_{i\bm k \uparrow}^\dag c_{i-\bm k \downarrow}^\dag + \mathrm{H.c.})$ \[$(i, j) = (1, 2)$ or $(2, 1)$\]. In the pseudospin respresentation, the Hamiltonian is written as $\tilde H_i=2\sum_{\bm k}\bm b_{i\bm k}\cdot\bm \sigma_{i\bm k}$ with an effective magnetic field $\bm b_{i\bm k}
=\left(
\begin{matrix}
\mathrm{Re}\tilde \Delta_i,&-\mathrm{Im}\tilde \Delta_i,&\epsilon_{i\bm k}
\end{matrix}
\right)$, which yields the Bloch equation $d\bm \sigma_{i\bm k}/dt=2\bm b_{i\bm k}\times\bm \sigma_{i\bm k}$. The self-consistent conditions for the order parameters read $$\begin{gathered}
\Delta_1
=|\Delta_1|e^{i\theta_1}=-\frac{U_{\mathrm{R}}}{N_0}\sum_{\bm k}\left(\sigma_{1 \bm k}^x-i\sigma_{1 \bm k}^y\right), \label{eq_OP1}\\
\Delta_2
=|\Delta_2|e^{i\theta_2}=-\frac{U}{N_0}\sum_{\bm k}\left(\sigma_{2 \bm k}^x-i\sigma_{2 \bm k}^y\right), \label{eq_OP2}\end{gathered}$$ where $N_0$ is the number of sites of each system. Here, the relations $\tilde \Delta_i=\Delta_i-V/N_0\sum_{\bm k}(\sigma_{j \bm k}^x-i\sigma_{j \bm k}^y)$ ($(i, j) = (1, 2)$ or $(2, 1)$) are satisfied. Then, the Josephson current between the two superfluids is given by a rate of change in the particle number of system 1: $$\begin{aligned}
\frac{1}{N_0}\frac{dN_1}{dt}
=-\frac{4V|\Delta_1||\Delta_2|}{U_{\mathrm{R}}|U|}\sin\left(\theta_2-\theta_1+\delta\right),
\label{eq_JosephsonCurrent}\end{aligned}$$ where $\delta=\tan^{-1}(-\gamma/2U_{\mathrm{R}})$ is the phase shift due to the sudden switch-on of the atomic loss.
We numerically solve the coupled Bloch equations for $\bm \sigma_{i\bm k}$. We assume that dissipation $\gamma$ and tunneling $V$ are turned on at $t=0$ for the BCS ground state. The numerical results for weak dissipation are shown in Fig. \[fig\_multiquench\](a1)-(d1). In Figs. \[fig\_multiquench\](a1) and (b1), the dynamics of two superfluids almost synchronize with each other because the time scale of particle loss is sufficiently longer than the tunneling time. In the pseudospin picture, the dynamics of particle numbers shown in Fig. \[fig\_multiquench\](c1) can be interpreted as the nutation of pseudospins. As inferred from Fig. \[fig\_multiquench\](d1), the Josephson current oscillates around its dc component. Such behavior is reminiscent of Shapiro steps in a Josephson junction under irradiation of a microwave [@Tinkham04]; however, in the present case, it *occurs spontaneously without any external field*. Moreover, from Fig. \[fig\_multiquench\](d1), the frequency of the oscillation of the phase difference between the two systems shows good agreement with that of the relative-phase mode known as the Leggett mode [@Leggett66; @Tsuji17] $$\begin{aligned}
\omega_{\mathrm L}^2=4\left(\frac{\lambda_{12}+\lambda_{21}}{\mathrm{det}\lambda}\right)|\Delta_1||\Delta_2|,
\label{eq_leggett}\end{aligned}$$ where $\lambda_{11}=\lambda_{22}=U_{\mathrm{R}}/W$, $\lambda_{12}=\lambda_{21}=V/W$ and $\mathrm{det} \lambda=\lambda_{11}\lambda_{22}-\lambda_{12}\lambda_{21}$. The Leggett mode with the frequency has been discussed in the context of a collective mode in a multiband superconductor irradiated by light [@Tsuji17]. The agreement between the frequencies of the relative-phase modes in seemingly different situations can be understood as follows. When the atomic loss is weak, the time evolution of the order parameters is given by $\Delta_i(t) = \exp(-2i\int_0^t dt\mu_{i\mathrm{eff}}(t))|\Delta_i(t)|$ with effective chemical potentials $\mu_{i\mathrm{eff}}(t)$. Then, by performing a global gauge transformation from $\epsilon_{i\bm k}$ to $\epsilon_{i\bm k}-\sum_i\mu_{i\mathrm{eff}}/2$ and assuming that the variation of the particle number is sufficiently small, we can linearize the Bloch equation with respect to the relative phase difference between $\Delta_i$’s, which can be dealt with by an analysis similar to that under the electric field that induces the Leggett mode [@Tsuji17].
![(a) DC component of the Josephson oscillation defined by ($\max_{0\le t \le t_{\mathrm f}}\{\sin\left(\theta_2(t)-\theta_1(t)+\delta\right)\} + \min_{0\le t \le t_{\mathrm f}}\{\sin\left(\theta_2(t)-\theta_1(t)+\delta\right)\})/2$ with $t_{\mathrm f} = 97.9 / \Delta_0$. (b) Phase difference between the two systems (blue) and particle numbers of system 1 (red) and system 2 (yellow) after a sufficiently long time evolution ($t_{\mathrm f}=97.9/\Delta_0$). The parameters are set to $U_{\mathrm{R}}=3.06\Delta_0$, $V=0.02\Delta_0$, and $W=5.11\Delta_0$.[]{data-label="fig_transition"}](Transition.pdf){width="8.5cm"}
*Nonequilibrium phase transition*.—By contrast, when dissipation becomes strong, the order parameter of system 2 starts to show an oscillation faster than that of system 1 \[see Fig. \[fig\_multiquench\](a2), (b2)\] and the phase difference $\theta_2 - \theta_1$ monotonically increases in time \[see Fig. \[fig\_multiquench\](d2)\]. This is because the dissipation rate larger than the tunneling rate makes system 1 fail to follow the decay of system 2, resulting in the dynamics similar to a single superfluid shown in Fig. \[fig\_lossquench\]. In particular, the chirped phase rotation of the superfluid order parameter in system 2 can be observed as the Josephson current \[Fig. \[fig\_multiquench\](d2)\]. As the superfluidity of system 2 is suppressed, the Josephson current also decays, and the particle number in system 1 stays constant after a long time \[see Fig. \[fig\_multiquench\](c2)\]. The latter behavior is attributed to the continuous quantum Zeno effect [@Syassen08; @Garcia09; @Zhu14; @Daley09; @Yan13], which states that an effective decay rate of system 1 is given by $\gamma_{\mathrm{eff}} \equiv |V_{\mathrm{eff}}|^2 / \gamma$ with the effective tunneling rate $V_{\mathrm{eff}} = V\Delta_2 / U_{\mathrm R}$ from Eq. , leading to the suppression of decay $\gamma_{\mathrm{eff}} \to 0$ for $\Delta_2 \to 0$.
The qualitative change in the superfluid behaviors with respect to the dissipation strength highlights a dissipative phase transition characterized by the vanishing dc Josephson current \[Fig. \[fig\_transition\](a)\], where the dc component of the Josephson oscillation is defined by ($\max_{0\le t \le t_{\mathrm f}}\{\sin\left(\theta_2(t)-\theta_1(t)+\delta\right)\} + \min_{0\le t \le t_{\mathrm f}}\{\sin\left(\theta_2(t)-\theta_1(t)+\delta\right)\})/2$ \[see Eq. \] after a sufficiently long time evolution with $t_{\mathrm f} = 97.9 / \Delta_0$. From Fig. \[fig\_transition\](b), we see that the phase difference $\theta_2-\theta_1$ starts to increase monotonically at the critical point and that the difference in particle number $(N_2-N_1)/N_0$ becomes much larger. The behavior of the phase difference is reminiscent of the localization-diffusion transition of a quantum-mechanical particle moving in a washboard potential in the presence of frictional forces [@Caldeira81; @Schdmit83; @Guinea85]. However, the origin of the transition shown in Fig. \[fig\_transition\] is essentially different from frictional forces, since they cannot change the particle number. In fact, as shown in the Supplemental Material [@Supple], the dissipative phase transition in Fig. \[fig\_transition\] is triggered by the competition between the Josephson coupling and particle loss. The transition shown in Fig. \[fig\_transition\] thus exemplifies a novel nonequilibrium phase transition in lossy systems.
*Conclusions.*—We have investigated the out-of-equilibrium dynamics of fermionic superfluids in the presence of two-body loss of Cooper pairs. We have demonstrated that the dynamics exhibits amplitude and phase modes with chirped oscillations, the latter of which is a salient feature of a dissipative superfluid. To observe the chirped phase rotation, we have proposed a Josephson junction comprised of dissipative and nondissipative superfluids. In the case of weak dissipation, we have shown that the relative-phase Leggett mode can be detected from the Josephson current. We have also found that, when dissipation becomes strong, the superfluids exhibit the unique dissipative phase transition triggered by particle loss. Our prediction can be tested with ultracold atomic systems of $^6$Li [@Roati15; @Roati18], for example, by introducing dissipation using photoassociation processes [@Tomita17; @Takasu20]. In ultracold atomic systems, the Josephson junction is realized by bisecting a trapped superfluid with an optical laser [@Roati15].
We are grateful to Yuto Ashida, Philipp Werner, Shuntaro Sumita, and Yoshiro Takahashi for fruitful discussions. This work was supported by KAKENHI (Grants No. JP18H01140, No. JP18H01145, and No. JP19H01838) and a Grant-in-Aid for Scientific Research on Innovative Areas (KAKENHI Grant No. JP15H05855) from the Japan Society for the Promotion of Science. K.Y. was supported by WISE Program, MEXT and JSPS KAKENHI Grant-in-Aid for JSPS fellows Grant No. JP20J21318. M.N. was supported by KAKENHI (Grant No. JP20K14383). N.T. acknowledges support by JST PRESTO (Grant No. JPMJPR16N7) and KAKENHI (Grant No. JP20K03811).
\
**“Collective Excitations and Nonequilibrium Phase Transition in Dissipative Fermionic Superfluids”**
Detailed calculations of the time-dependent dissipative BCS theory
==================================================================
We explain the details of the Hubbard-Stratonovich transformation that is used for the derivation of the time-dependent dissipative BCS theory in the path-integral formalism. The action in the main text is rewritten as $$\begin{aligned}
S=\int_{-\infty}^\infty dt\Big[&\sum_{\bm k \sigma} \Big( \bar c _{\bm k \sigma +}(i\partial_t-\epsilon_{\bm k}) c_{\bm k \sigma +}-\bar c _{\bm k \sigma -}(i\partial_t-\epsilon_{\bm k}) c_{\bm k \sigma -}\Big)\nonumber\\
&+\sum_{\bm k \bm k'}\left(U\bar c_{\bm{k}\uparrow+}\bar c_{-\bm{k}\downarrow+}c_{-\bm{k}'\downarrow+}c_{\bm{k}'\uparrow+}-U^*\bar c_{\bm{k}\uparrow-}\bar c_{-\bm{k}\downarrow-}c_{-\bm{k}'\downarrow-}c_{\bm{k}'\uparrow-}-i\gamma c_{-\bm{k}\downarrow+}c_{\bm{k}\uparrow+}\bar c_{\bm{k}'\uparrow-}\bar c_{-\bm k'\downarrow-}\right)\Big].
\label{eq_actionS1}\end{aligned}$$ We perform the Hubbard-Stratonovich transformation for each term in the second line of Eq. with auxiliary fields $\Delta_\alpha$ ($\alpha=+$, $-$, $\pm$) as $$\begin{gathered}
iU\bar c_{\bm{k}\uparrow+}\bar c_{-\bm{k}\downarrow+}c_{-\bm{k}'\downarrow+}c_{\bm{k}'\uparrow+}\to-i\Delta_+\bar c_{\bm{k}\uparrow+}\bar c_{-\bm{k}\downarrow+}-i\bar \Delta_+ c_{-\bm{k}\downarrow+}c_{\bm{k}\uparrow+}+\frac{\bar \Delta_+\Delta_+}{iU}\label{eq_U},\\
-iU^*\bar c_{\bm{k}\uparrow-}\bar c_{-\bm{k}\downarrow-}c_{-\bm{k}'\downarrow-}c_{\bm{k}'\uparrow-}\to i\Delta_-\bar c_{\bm{k}\uparrow-}\bar c_{-\bm{k}\downarrow-}+i\bar \Delta_-c_{-\bm{k}\downarrow-}c_{\bm{k}\uparrow-}-\frac{\bar \Delta_-\Delta_-}{iU^*},\\
\gamma c_{-\bm{k}\downarrow+}c_{\bm{k}\uparrow+}\bar c_{\bm{k}'\uparrow-}\bar c_{-\bm k'\downarrow-}\to -\Delta_\pm\bar c_{\bm k\uparrow-}\bar c_{-\bm k \downarrow-}-\bar \Delta_\pm c_{-\bm{k}\downarrow+}c_{\bm{k}\uparrow+}-\frac{\bar \Delta_\pm\Delta_\pm}{\gamma},
\label{eq_Ustar}\end{gathered}$$ which yield $$\begin{aligned}
S=\int dt\Bigg\{\sum_{\bm k}\Bigg[
\bar \psi_{\bm k +}^t
\left(
\begin{matrix}
i\partial_t-\epsilon_{\bm{k}}&-\Delta_+\\
-\bar \Delta_+ + i\bar \Delta_\pm&-i\partial_t+\epsilon_{\bm k}
\end{matrix}
\right)
\psi_{\bm k +}
-
\bar \psi_{\bm k -}^t
&\left(
\begin{matrix}
i\partial_t-\epsilon_{\bm k}&-\Delta_- -i\Delta_\pm\\
-\bar \Delta_-&-i\partial_t+\epsilon_{\bm k}
\end{matrix}
\right)
\psi_{\bm k -}\Bigg]\notag\\
&+
\frac{\bar \Delta_+\Delta_+}{iU}-\frac{\bar \Delta_-\Delta_-}{iU^*}-\frac{\bar \Delta_\pm\Delta_\pm}{\gamma}\Bigg\},
\label{eq_actionS2}\end{aligned}$$ where $\bar \psi_{\bm k \alpha}=\left(
\begin{matrix}
\bar c_{\bm{k}\uparrow\alpha},&c_{-\bm{k}\downarrow\alpha}
\end{matrix}
\right)^t$ and $\psi_{\bm k \alpha}=\left(
\begin{matrix}
c_{\bm{k}\uparrow\alpha},&\bar c_{-\bm{k}\downarrow\alpha}
\end{matrix}
\right)^t$ ($\alpha=+, -$). Then, from the saddle-point condition $\left\langle \partial S/\partial\Delta_\alpha \right\rangle=\left\langle \partial S/\partial\bar\Delta_\alpha \right\rangle=0$ ($\alpha=+$, $-$, $\pm$), we obtain $$\begin{gathered}
\Delta_+=-\frac{U}{N_0}\sum_{\bm k}\langle c_{-\bm k\downarrow+}c_{\bm k\uparrow+}\rangle,
\quad\bar \Delta_+=-\frac{U}{N_0}\sum_{\bm k}\langle c_{\bm k\uparrow+}^\dag c_{-\bm k\downarrow+}^\dag\rangle,\label{eq_deltap}\\
\Delta_-=-\frac{U^*}{N_0}\sum_{\bm k}\langle c_{-\bm k\downarrow-}c_{\bm k\uparrow-}\rangle,
\quad\bar \Delta_-=-\frac{U^*}{N_0}\sum_{\bm k}\langle c_{\bm k\uparrow-}^\dag c_{-\bm k\downarrow-}^\dag\rangle,\label{eq_deltam}\\
\Delta_\pm=\frac{\gamma\Delta_+}{U},
\quad\bar \Delta_\pm=\frac{\gamma\bar \Delta_-}{U^*}\label{eq_deltapm},\end{gathered}$$ where $N_0$ is the number of lattice sites and $\langle\cdots\rangle$ is the expectation value for fixed $\Delta_\alpha$ and $\bar \Delta_\alpha$. Then, we can reduce the number of the auxiliary fields by using $\langle c_{-\bm k\downarrow\alpha}c_{\bm k \uparrow\alpha}\rangle=\mathrm{tr}(c_{-\bm k\downarrow}c_{\bm k \uparrow}\rho)$ ($\alpha=+,-$) and $\mathrm{tr}(A^\dag\rho)=[\mathrm{tr}(A\rho)]^*$ [@Kamenev11], giving $$\begin{aligned}
\Delta_+&=\bar \Delta_-^*,\label{eq_Delta1}\\
\Delta_-&=\bar\Delta_+^*.\label{eq_Delta2}\end{aligned}$$ Finally, the action is rewritten as $$\begin{aligned}
S=\int dt\sum_{\bm k}\Bigg\{
\bar \psi_{\bm k +}^t
\left(
\begin{matrix}
i\partial_t-\epsilon_{\bm{k}}&-\Delta\\
-\Delta^*&-i\partial_t+\epsilon_{\bm k}
\end{matrix}
\right)
\psi_{\bm k +}
-
\bar \psi_{\bm k -}^t
\left(
\begin{matrix}
i\partial_t-\epsilon_{\bm k}&-\Delta\\
-\Delta^*&-i\partial_t+\epsilon_{\bm k}
\end{matrix}
\right)
\psi_{\bm k -}\Bigg\},\end{aligned}$$ where the superfluid order parameter of the system is given by $$\begin{aligned}
\Delta=-\frac{U}{N_0}\sum_{\bm k}\mathrm{tr}(c_{-\bm k\downarrow}c_{\bm k \uparrow}\rho)\equiv-\frac{U}{N_0}\sum_{\bm k}\langle c_{-\bm k\downarrow}c_{\bm k \uparrow}\rangle.\end{aligned}$$
Operator formalism of the BCS theory for a dissipative superfluid
=================================================================
Here, we explain the operator formalism of the BCS theory for a dissipative superfluid and clarify that this leads to Eq. in the main text which describes the time evolution of the density operator. First, we note that an operator $\ket{\psi_+}\bra{\psi_-}$ acting on the Hilbert space of the system can be mapped to a tensor-product state $\ket{\psi_+}\otimes\ket{\psi_-}$ in the doubled Hilbert space $\mathbb H_+\otimes\mathbb H_-$ [@Shibata19a; @Shibata19b]. Using this mapping, we can rewrite the Liouvillian \[see Eq. in the main text for definition\] as $$\begin{aligned}
i\mathcal{L}&= H_+ - H_- +i\sum_i \gamma_i(L_{i+}L_{i-}^\dag -\frac{1}{2}L_{i+}^\dag L_{i+}-\frac{1}{2} L_{i-}^\dag L_{i-})\notag\\
&=\mathcal H_+ - \mathcal H_-^* + i\gamma\sum_{\bm k \bm{k}'}c_{-\bm k\downarrow+}c_{\bm k \uparrow+}c_{\bm{k}' \uparrow-}^\dag c_{-\bm{k}' \downarrow-}^\dag,
\label{eq_LiouvilS}\end{aligned}$$ where $L_{i\alpha} = c_{i\downarrow\alpha} c_{i \uparrow\alpha}$, and $c_{i\sigma\alpha}$ ($c_{\bm k\sigma\alpha}$) with $\alpha=+, -$ is the fermion annihilation operator in the real (momentum) space that acts on the Hilbert space $\mathbb{H}_\alpha$. The fermion operator with subscript $+$ ($-$) corresponds to the fermion field in the forward (backward) path in the path-integral formalism. The BCS Hamiltonian equivalent to Eq. is given by $$\begin{aligned}
H_\alpha=\sum_{\bm{k}\sigma}\epsilon_{\bm{k}}c_{\bm{k}\sigma\alpha}^\dagger{c}_{\bm{k}\sigma\alpha}-U_{\mathrm{R}}\sum_{\bm{k}\bm{k}'}c_{\bm{k}\uparrow\alpha}^{\dagger}c_{-\bm{k}\downarrow\alpha}^\dagger{c}_{-\bm{k}'\downarrow\alpha}c_{\bm{k}'\uparrow\alpha},
\label{eq_HubbardS}\end{aligned}$$ and $\mathcal H_\alpha$ is defined as $$\begin{aligned}
\mathcal H_\alpha&=H_\alpha-\frac{1}{2}i\gamma \sum_i L_{i\alpha}^\dag L_{i\alpha}\notag\\
&=\sum_{\bm{k}\sigma}\epsilon_{\bm{k}}c_{\bm{k}\sigma\alpha}^\dagger{c}_{\bm{k}\sigma\alpha}-U\sum_{\bm{k}\bm{k}'}c_{\bm{k}\uparrow\alpha}^{\dagger}c_{-\bm{k}\downarrow\alpha}^\dagger{c}_{-\bm{k}'\downarrow\alpha}c_{\bm{k}'\uparrow\alpha}.
\label{eq_HubbardS}\end{aligned}$$ By applying the mean-field approximation to $\mathcal L$, we obtain the mean-field Liouvillian as $$\begin{aligned}
i\mathcal{L}_{\mathrm{MF}}
=&\sum_{\bm k\sigma}\epsilon_{\bm k}c_{\bm k\sigma+}^\dag c_{\bm k\sigma+}+\Delta_+\sum_{\bm k}c_{\bm k\uparrow+}^\dag c_{-\bm k\downarrow+}^\dag+(\bar \Delta_+ -\frac{i\gamma}{U^*}\bar\Delta_-)\sum_{\bm k}c_{-\bm k\downarrow+}c_{\bm k\uparrow+}\nonumber\\
&-\sum_{\bm k\sigma}\epsilon_{\bm k}c_{\bm k\sigma-}^\dag c_{\bm k\sigma-} -(\Delta_- +\frac{i\gamma}{U}\Delta_+)\sum_{\bm k}c_{\bm k\uparrow-}^\dag c_{-\bm k\downarrow-}^\dag-\bar\Delta_-\sum_{\bm k}c_{-\bm k\downarrow-}c_{\bm k\uparrow-}\notag\\
=&\sum_{\bm{k}}
\Psi_{\bm k+}^\dag
\left(
\begin{matrix}
\epsilon_{\bm{k}}&\Delta_+\\
\bar{\Delta}_+-\frac{i\gamma}{U^*}\bar\Delta_-&-\epsilon_{\bm{k}}
\end{matrix}
\right)
\Psi_{\bm k+}
-\sum_{\bm{k}}
\Psi_{\bm k +}^\dag
\left(
\begin{matrix}
\epsilon_{\bm{k}}&\Delta_- +\frac{i\gamma}{U}\Delta_+\\
\bar\Delta_-&-\epsilon_{\bm{k}}
\end{matrix}
\right)
\Psi_{\bm k -},
\label{eq_meanfieldL}\end{aligned}$$ where $\Psi_{\bm k}=\left(\begin{matrix}c_{\bm k \uparrow},&c_{-\bm k \downarrow}^\dag \end{matrix}\right)^t$ is the Nambu spinor. As we can see from Eq. and Eq. , the Liouvillian is invariant under the U(1) gauge transformation $c_{\bm k\sigma+}\to e^{i\theta}c_{\bm k\sigma+}$ and $c_{\bm k\sigma-}\to e^{i\theta}c_{\bm k\sigma-}$. Moreover, under the exchange of forward and backward operators, $Pc_{\bm k \sigma+}P^{-1}=c_{\bm k \sigma-}$, $Pc_{\bm k \sigma+}^\dag P^{-1}=c_{\bm k \sigma-}^\dag$, and $P^2=1$, the Liouvillian has the following symmetry $$\begin{aligned}
P(i\mathcal L)^\dag P^{-1}=-i\mathcal{L}.\end{aligned}$$ By imposing the same symmetry on the mean-field Liouvillian as $P(i\mathcal{L}_{\mathrm{MF}})^\dag P^{-1}=-i\mathcal{L}_{\mathrm{MF}}$, we obtain the relations for the order parameters as $$\begin{aligned}
\Delta_+^*&=\bar \Delta_-,\\
\Delta_-^*&=\bar\Delta_+,\end{aligned}$$ which coincide with those obtained in the path-integral formalism \[see Eqs. and \]. Finally, by rewriting the superfluid order parameter $\Delta_+$ as $$\begin{aligned}
\Delta=-\frac{U}{N_0}\sum_{\bm k}\mathrm{tr}(c_{-\bm k\downarrow}c_{\bm k \uparrow}\rho)\equiv-\frac{U}{N_0}\sum_{\bm k}\langle c_{-\bm k\downarrow}c_{\bm k \uparrow}\rangle,\end{aligned}$$ we obtain the equations for the density matrix as $$\begin{gathered}
\dot{\rho}=-i[H_{\mathrm{eff}},\rho],
\label{eq_unitaryS}\\
H_{\mathrm{eff}}=\sum_{\bm{k}}
\Psi_{\bm k}^\dag
\left(
\begin{matrix}
\epsilon_{\bm{k}}&\Delta\\
\Delta^*&-\epsilon_{\bm{k}}
\end{matrix}
\right)
\Psi_{\bm k},
\label{eq_HamiltonianEffS}\end{gathered}$$ which are the same as Eqs. and in the main text.
Generalization of the Bogolibov-de Gennes analysis with a time-dependent BCS state
==================================================================================
We explain that Eq. (Eq. in the psedouspin representation) in the main text is equivalent to the Bogoliubov-de Gennes equation with a time-dependent BCS state [@Barankov06A; @Volkov74], which describes the unitary evolution of the density matrix.
We introduce the time-dependent BCS state of the effective Hamiltonian $H_{\mathrm{eff}}=\sum_{\bm{k}}
\Psi_{\bm k}^\dag
\left(
\begin{matrix}
\epsilon_{\bm{k}}&\Delta\\
\Delta^*&-\epsilon_{\bm{k}}
\end{matrix}
\right)
\Psi_{\bm k}
\label{eq_HamiltonianEffS}$ as follows: $$\begin{gathered}
|\Psi_{\mathrm{BCS}}(t)\rangle=\prod_{\bm k} (u_{\bm k}(t) + v_{\bm k}(t) c_{\bm k \uparrow}^\dag c_{-\bm k \downarrow}^\dag)|0\rangle\label{eq_BCS},\\
|u_{\bm k}|^2+|v_{\bm k}|^2=1,\end{gathered}$$ where $|0\rangle$ is the vacuum of fermions. Here, the superfluid order parameter $\Delta$ is rewritten as $$\begin{aligned}
\Delta=-\frac{U}{N_0}\sum_{\bm k}\langle c_{-\bm k \downarrow} c_{\bm k \uparrow}\rangle=-\frac{U}{N_0}\sum_{\bm k} u_{\bm k}^*(t)v_{\bm k}(t).\end{aligned}$$ Suppose that the density matrix is given by $$\begin{aligned}
\rho(t)=\ket{\Psi_{\mathrm{BCS}}(t)}\bra{\Psi_{\mathrm{BCS}}(t)}.
\label{eq_rhoBCS}\end{aligned}$$ Then, the time-evolution equation $$\begin{aligned}
\frac{d\rho(t)}{dt}=-i[H_{\mathrm{eff}},\rho(t)]
\label{eq_unitaryS1}\end{aligned}$$ \[Eq. in the main text\] is equivalent to the Bogoliubov-de Gennes equation with the time-dependent BCS state $$\begin{aligned}
i\partial_t
\left(
\begin{matrix}
u_{\bm k}\\
v_{\bm k}
\end{matrix}
\right)
=
\left(
\begin{matrix}
-\epsilon_{\bm k}&\Delta^*\\
\Delta&\epsilon_{\bm k}
\end{matrix}
\right)
\left(
\begin{matrix}
u_{\bm k}\\
v_{\bm k}
\end{matrix}
\right).
\label{eq_BogoliubovS}\end{aligned}$$ By defining $f_{\bm k}(t)$ and $g_{\bm k}(t)$ as $$\begin{gathered}
f_{\bm k}=u_{\bm k}^* v_{\bm k},\\
g_{\bm k}=\frac{1}{2}(|u_{\bm k}|^2 - |v_{\bm k}|^2),\end{gathered}$$ Eq. is rewritten as $$\begin{gathered}
\frac{d f_{\bm k}}{dt}=-2i\epsilon_{\bm k}f_{\bm k}-2i\Delta g_{\bm k},\\
\frac{d g_{\bm k}}{dt}=i\Delta f_{\bm k}^*-i\Delta^*f_{\bm k}.\end{gathered}$$ These equations take the same forms as those for closed systems [@Barankov06A; @Volkov74]. Finally, by defining the psedouspins as $$\begin{gathered}
f_{\bm k}=\sigma_{\bm k}^x -i \sigma_{\bm k}^y,\\
g_{\bm k}=-\sigma_{\bm k}^z,\end{gathered}$$ we obtain the same Bloch equation as discussed in the main text: $$\begin{gathered}
\frac{d\bm \sigma_{\bm k}}{dt}
=2\bm b_{\bm k}\times\bm \sigma_{\bm k},\label{eq_blochS}\\
\bm b_{\bm k}=\left(\begin{matrix}\mathrm{Re}\Delta,&-\mathrm{Im}\Delta,&\epsilon_{\bm k}\end{matrix}\right).\end{gathered}$$ We note that the dynamics described by Eq. conserves the purity $\mathrm{tr}[\rho^2]$ as $$\begin{aligned}
\frac{d \mathrm{tr}[\rho^2]}{d t}=2\mathrm{tr}\left[\rho \frac{d\rho}{dt}\right]=-2i\mathrm{tr}(\rho[H_{\mathrm{eff}}, \rho])=0.\end{aligned}$$
In general, the purity should decrease during the time evolution described by the quantum master equation \[Eq. in the main text\]. This fact is consistent with the time-dependent BCS ansatz as follows. Since $|\Psi_{\mathrm{BCS}}(t)\rangle$ can be expanded in terms of an $N$-particle state $$\begin{aligned}
|\Psi_N\rangle=\sum_{\bm k_1\cdots\bm k_{N/2}}a_{\bm k_1}\cdots a_{\bm k_{N/2}}c_{\bm k_1 \uparrow}^\dag c_{-\bm k_1 \downarrow}^\dag \cdots c_{\bm k_{N/2} \uparrow}^\dag c_{-\bm k_{N/2} \downarrow}^\dag|0\rangle\end{aligned}$$ as $$\begin{aligned}
|\Psi_{\mathrm{BCS}}(t)\rangle=\sum_N c_N|\Psi_N\rangle,
\label{eq_pureS}\end{aligned}$$ the density matrix is written as $$\begin{aligned}
\rho&=|\Psi_{\mathrm{BCS}}(t)\rangle\langle\Psi_{\mathrm{BCS}}(t)|\notag\\
&=\sum_N |c_N|^2|\Psi_N\rangle\langle\Psi_N| + \sum_{N\neq N^\prime} c_{N^\prime}^* c_N|\Psi_N\rangle\langle\Psi_{N^\prime}|.
\label{eq_rhoS}\end{aligned}$$ Then, for a gauge-invariant observable $\mathcal O$, its expectation value is given by $$\begin{aligned}
\langle\mathcal O\rangle\equiv\mathrm{tr}[\mathcal O \rho]=\sum_N|c_N|^2\langle\Psi_N|\mathcal O|\Psi_N\rangle=\mathrm{tr}[\mathcal O \rho^\prime],\end{aligned}$$ where $$\begin{aligned}
\rho^\prime=\sum_N |c_N|^2|\Psi_N\rangle\langle\Psi_N|
\label{eq_rhomixed}\end{aligned}$$ is a mixed state of different particle numbers. Therefore, concerning gauge-invariant observables, the time-dependent BCS state is indistinguishable from the mixed state with $\mathrm{tr}[\rho^{\prime 2}] < 1$. Since any physically observable quantity should be gauge invariant, the time-dependent BCS ansatz can describe the time evolution of the density matrix consistently with the quantum master equation .
Dynamics of pseudospins after the switch-on of dissipation
==========================================================
The physical origin of the chirping of the U(1) phase can be understood from the pseudospin picture. As shown in Fig. \[fig\_psedou\], when the sign of $\sigma_{\bm k}^z$ changes from positive to negative, the magnitudes of $\sigma_{\bm k}^x$ and $\sigma_{\bm k}^y$ increase because of the norm conservation of pseudospins. This indicates that the Cooper-pair amplitude at specific momenta rapidly changes when atoms at those momenta are lost from the system. Since Cooper pairs are formed near the Fermi surface, a loss of Cooper pairs leads to a downward shift of the Fermi level. Now the angular velocity of the order parameter is written as $$\begin{aligned}
\frac{d\theta}{dt} = U_{\mathrm{R}}(1-\frac{N}{N_0})-\frac{2|U|^2}{|\Delta|^2N_0^2}\sum_{\substack {\bm k \bm k' \\ \alpha=x,y}}\sigma_{\bm k}^\alpha\cdot\epsilon_{\bm k'}\sigma_{\bm k'}^\alpha,
\label{eq_thetadot}\end{aligned}$$ which is derived from differentiation of Eq. in the main text with respect to time, and the last term in Eq. increases due to the shift of the Fermi level, leading to chirping of the U(1) phase. Here, we note that the first term on the right-hand side of Eq. also increases as the particle number decreases; however, it is much smaller than the second term.
![Dynamics of pseudospins \[$\sigma_{\bm k}^x$ (light green), $\sigma_{\bm k}^y$ (blue), $\sigma_{\bm k}^z$ (orange), $|\bm \sigma_{\bm k}|$ (violet)\] after the atomic loss with $\gamma=2.81\Delta_0$ is switched on for the initial state with $U_{\mathrm R}=12.2\Delta_0$ and the Fermi energy $\epsilon_{\mathrm F}=0$ for $\epsilon=-18.7\Delta_0$ (left), $-9.35\Delta_0$ (center), and $0$ (right) in the energy band $-23.4\Delta_0\le\epsilon\le23.4\Delta_0$.[]{data-label="fig_psedou"}](Psedou.pdf){width="15cm"}
Dynamics after sudden change of both the interaction and the dissipation
========================================================================
Figure \[fig\_intloss\] shows the dynamics after the dissipation $\gamma$ is introduced at $t=0$ and the interaction strength is simultaneously changed from $U_{\mathrm R}=8.4\Delta_0$ to $U_{\mathrm R}=16.8\Delta_0$. In Figs. \[fig\_intloss\](a) and (b), we see an amplitude oscillation larger than that in the loss quench dynamics shown in Figs. \[fig\_lossquench\](a) and (b) in the main text. This behavior is due to the fact that the change in the real part of $U$ causes a large initial shift of the amplitude of the order parameter \[see Fig. \[fig\_Schematic\](a) in the main text\]. The U(1) phase rotates with an increasing angular velocity due to dissipation as in Figs. \[fig\_intloss\](a) and (b). The amplitude oscillation of the order parameter can be detected through monitoring of the particle number and from Eq. in the main text. As shown in Figs. \[fig\_intloss\](c) and (d), the amplitude of the particle-number oscillation is a few percent of the initial particle number, which can be detected with current experimental techniques [@Roati15]. Since the oscillation in the particle number cannot appear in isolated systems, the dissipation-induced dynamics can be used as a unique signature for the amplitude mode of the superfluid order parameters.
![Dynamics of a superfluid after the interaction and the atomic loss are suddenly changed. (a) Real parts (light green) and imaginary parts (blue) of the order parameter. The amplitudes of the order parameter are shown as violet curves in both figures. (c) The particle number of the system normalized by the initial particle number $N_0$ (red) and the rate of change of the particle number (yellow). (d) An enlarged view of the particle number near $t=0$. The parameters are suddenly changed from $U_{\mathrm R}=8.4\Delta_0$ to $U=(16.8+0.45i)\Delta_0$ at $t=0$ and the bandwidth is set to $W=28\Delta_0$.[]{data-label="fig_intloss"}](IntLossQuench1.pdf){width="15cm"}
Understanding the dissipative phase transition from a simplified model
======================================================================
Here, we explain how the dissipative phase transition associated with the vanishing dc Josephson current can be understood by considering a simplified model of the Josephson junction with particle loss. We consider two fermionic superfuids coupled via a Josephson junction, where two-body loss is introduced to one of them, as shown in Fig. \[fig\_Schematic\](b) in the main text. The Josephson current flowing between the two systems is given by $$\begin{aligned}
I=\frac{1}{N_0}\frac{dN_1}{dt}=-I_0\sin(\Delta\theta),\end{aligned}$$ where, from Eq. in the main text, $\Delta\theta=\theta_2-\theta_1$, $I_0=4V|\Delta_1||\Delta_2|/U_{\mathrm R}|U|$, and we have neglected the phase $\delta$ because $\delta\ll\Delta\theta$. Taking into account the particle loss \[see Eq. in the main text\], we obtain the rate of change of the particle number of system 1 and that of system 2 as $$\begin{gathered}
\frac{1}{N_0}\frac{dN_1}{dt}=-I_0\sin(\Delta\theta),\label{eq_N1S}\\
\frac{1}{N_0}\frac{dN_2}{dt}=I_0\sin(\Delta\theta)-\frac{2\gamma|\Delta_2|^2}{|U|^2}.\label{eq_N2S}\end{gathered}$$ We assume that the time evolution of the phase difference $\Delta\theta$ is given by the effective chemical potential difference $\Delta\mu_{\mathrm {eff}}$ between the two systems as $$\begin{aligned}
\frac{d\Delta\theta}{dt}=-2\Delta\mu_{\mathrm{eff}}=-\frac{W}{N_0}(N_2-N_1),\label{eq_thetaS}\end{aligned}$$ where $W$ is a bandwidth and we assume a constant density of states for simplicity. We obtain the equation of motion for $\Delta\theta$ by using Eqs. , , and as $$\begin{aligned}
\frac{d^2\Delta\theta}{dt^2}=-2WI_0\sin(\Delta\theta)+\frac{2\gamma W|\Delta_2|^2}{|U|^2}.
\label{eq_Deltatheta}\end{aligned}$$ This system is regarded as a Josephson junction with shunt resistance $R=+\infty$, capacitance $C=1/2W$ and an external force $F=\gamma|\Delta_2|^2/|U|^2$, which is described as $$\begin{aligned}
C\frac{d^2\Delta\theta}{dt^2}+\frac{1}{R}\frac{d\Delta\theta}{dt}+I_0\sin(\Delta\theta)=F.
\label{eq_washboard}\end{aligned}$$ That is, the time evolution of $\Delta\theta$ is equivalent to that of a particle moving in a washboard potential $$\begin{aligned}
V_{\mathrm{wash}}=-2WI_0\cos(\Delta\theta)-\frac{2\gamma W|\Delta_2|^2\Delta\theta}{|U|^2}.\end{aligned}$$ The condition for the extremum of $V_{\mathrm {wash}}$ is given by $dV_{\mathrm {wash}}/d\Delta\theta=0$, giving $$\begin{aligned}
\sin(\Delta\theta)=\frac{\gamma|\Delta_2|^2}{I_0|U|^2}.\end{aligned}$$ The solution to this equation does not exist for $\gamma|\Delta_2|^2/I_0|U|^2> 1$ and the time evolution of $\Delta\theta$ becomes unstable. If we assume $|\Delta_1|\simeq|\Delta_2|$ when the time evolution is sufficiently slow, we obtain the critical strength of the atomic loss as $$\begin{aligned}
\gamma_c\simeq4V,\end{aligned}$$ which gives the same order of magnitude as that in the main text ($\gamma_c\simeq3V$). Thus, the system exhibits a dissipative phase transition from where $\Delta\theta$ oscillates around the extremal of $V_{\mathrm{wash}}$ for $\gamma<\gamma_c$ to where $\Delta\theta$ slips down the washboard potential for $\gamma>\gamma_c$. We note that the particle loss $\gamma$ acts as an external force rather than friction in Eq. . Thus, the dissipative phase transition caused by the particle loss is the one between a trapped state and a running state, which is essentially different from the localization-delocalization transition induced by frictional forces [@Caldeira81; @Schdmit83; @Guinea85].
These features are also obtained from a phenomenological introduction of two-body loss [@Syassen08; @Garcia09], under which the rate of change of the particle number of system 1 and that of system 2 are given by $$\begin{gathered}
\frac{1}{N_0}\frac{dN_1}{dt}=-I_0\sin(\Delta\theta),\label{eq_N1Sa}\\
\frac{1}{N_0}\frac{dN_2}{dt}=I_0\sin(\Delta\theta)-\kappa_2\left(\frac{N_2}{N_0}\right)^2,\label{eq_N2Sa}\end{gathered}$$ where $\kappa_2$ is the two-body loss rate. By using Eqs. , , and , we obtain the equation of motion for $\Delta\theta$ as $$\begin{aligned}
\frac{d^2\Delta\theta}{dt^2}=-2WI_0\sin(\Delta\theta)+W\kappa_2\left(\frac{N_2}{N_0}\right)^2.
\label{eq_Deltatheta1}\end{aligned}$$ This system is regarded as a Josephson junction with resistance $R=+\infty$, capacitance $C=1/2W$ and an external force $F=\frac{\kappa_2}{2}\left(\frac{N_2}{N_0}\right)^2$. As the system has the washboard potential $$\begin{aligned}
V_{\mathrm{wash}}=-2WI_0\cos(\Delta\theta)-W\kappa_2\left(\frac{N_2}{N_0}\right)^2\Delta\theta,\end{aligned}$$ the condition for the extremal of $V_{\mathrm{wash}}$ is given by $$\begin{aligned}
\sin(\Delta\theta)=\frac{\kappa_2}{2I_0}\left(\frac{N_2}{N_0}\right)^2.\end{aligned}$$ If we assume that the variation of the parameters is sufficiently slow and approximate them as constant, the critical strength of the loss rate where the solution of $\Delta\theta$ becomes unstable is given by $$\begin{aligned}
\kappa_{2c}\simeq2I_0.\end{aligned}$$ We can numerically solve Eq. , , and , and the results are shown in Fig. \[fig\_phenomenology\]. We see that the results shown in Fig. \[fig\_phenomenology\](a1) and (b1) \[Fig. \[fig\_phenomenology\](a2) and (b2)\] are qualitatively the same as those in Fig. \[fig\_multiquench\](c1) and (d1) \[Fig. \[fig\_multiquench\](c2) and (d2)\] in the main text, respectively.
![Numerical solution of $N_1/N_0$ (red), $N_2/N_0$ (yellow), and $\Delta\theta$ (light blue) in Eqs. , , and with $W=5.11\Delta_0$ and $I_0=0.009\Delta_0$. The loss rate $\kappa_2$ is set to $\kappa_2=0.006\Delta_0$ in (a1) and (b1), and $\kappa_2=0.02\Delta_0$ in (a2) and (b2).[]{data-label="fig_phenomenology"}](TransitionPhenomenology.pdf){width="15cm"}
|
---
author:
- 'Stanislav Stefanik[^1]'
- Dalibor Nosek
title: 'Variability of VHE $\gamma$–ray emission from the binary PSR B1259–63/LS 2883'
---
Introduction {#intro}
============
[PSR B1259–63/LS 2883]{} is one of few $\gamma$–ray binaries detected in the very high energetic (VHE) regime. It is known to consist out of a Be star with an equatorial disk tilted relative to the movement of the pulsar which orbits around the star with a period $T = 1237~\mathrm{days}$ [@Johnston]. The variability of the source emission has been clearly observed in multiple wavebands from radio to VHE with the most prominent changes occurring around the periastron passages [@hess_2013]. The cause of this variations is believed to be the interaction of the pulsar with the stellar disk. Significant variability of the VHE $\gamma$–ray flux around periastron [@hess_2005; @hess_2009] was observed by the H.E.S.S. telescope array. Although a considerable overlap of the pre– and post–periastron data has not been obtained so far, a stronger evidence for the repetitive behaviour stemming from the orbital motion was claimed based on the correspondence of the newest data with the older lighcurve [@hess_2013]. Recently, detection of a GeV flare by the *Fermi*–LAT [@Abdo; @Tam] and a lack of evidence for an attributable VHE flare [@hess_2013] lead to the conclusion that the high energy (HE) photons have a different origin than the VHE ones. In this paper, we reexamine the changes of VHE activity of [PSR B1259–63/LS 2883]{} during campaigns targeted at periods of periastron passages.
We deal with publicly available VHE data gathered by Cherenkov telescopes in the on/off observation mode [@Berge]. In this approach, the numbers of detected events ${N_{\mathrm{on}}}$ and ${N_{\mathrm{off}}}$ are extracted from the observed signal (on) and background (off) regions, respectively. They are related by the on–off parameter $\alpha$ which is the ratio of exposures of signal and background regions. Traditionally, the level of significance of the source presence in the on–source region is determined using the Li–Ma method [@LiMa]. In this work, we use its modification for the investigation of changes in the source intensity (see [@Nosek; @Stefanik] for details). A source parameter $\beta > 0$ is introduced to quantify the signal in the on–source region when compared to the background ascertained from the reference regions. The core of the modified on–off method is the statement of the null hypothesis that a previously identified source of interest attains intensity of chosen value $\beta$, i.e. ${N_{\mathrm{on}}}= \alpha \beta {N_{\mathrm{off}}}$ [@Nosek]. A level of significance for the rejection of the assumption of constant source activity is given by [@Nosek; @Stefanik]
$$\label{eq:lima}
{S_{\mathrm{LM}}}= s\sqrt{2}\left\lbrace {N_{\mathrm{on}}}\ln{ X_{1} } + {N_{\mathrm{off}}}\ln{ X_{2} } \right\rbrace^{\frac{1}{2}},$$
for the asymptotic Li–Ma statistics. Here, the logarithmic arguments are $X_{1} = \frac{1+\alpha\beta}{\alpha\beta} \frac{{N_{\mathrm{on}}}}{{N_{\mathrm{on}}}+ {N_{\mathrm{off}}}}$ and $X_{2} = (1+\alpha\beta) \frac{{N_{\mathrm{off}}}}{{N_{\mathrm{on}}}+ {N_{\mathrm{off}}}} $. The $s$–term in Eq. (\[eq:lima\]) is either $+1$ or $-1$ for observations of an excess (${S_{\mathrm{LM}}}>0$) or deficit (${S_{\mathrm{LM}}}<0$) of events, respectively.
Li–Ma significance given in Eq. (\[eq:lima\]) asymptotically follows the normal distribution with zero mean and unit variance [@Nosek]. Therefore, any inconsistency between the sample distributions of ${S_{\mathrm{LM}}}$ and the reference standard Gaussian distribution should be regarded as a sign of change in the tested $\gamma$–ray intensity. It is also worth noting, that the test of the null hypothesis can be advantageously inverted to derive confidence intervals for the source parameter $\beta$ at a given level of significance. Sequence of such confidence intervals then carries information about changes of the observed $\gamma$–ray activity.
The correct treatment of Poisson observables is inherent in the modified on–off method which straightforwardly exploits just the numbers of detected photons. Its other benefit is that systematic uncertainties are fully accounted for and are already included in the confidence intervals for the source parameter. The application of the method is thus very simple yet the range of possible utilizations and the interpretations of results remain utmost general.
Data analysis {#sec:analysis}
=============
Data sets used in our study comprise results of observations of [PSR B1259–63/LS 2883]{} by the H.E.S.S. telescope array during periastron passages in 2004 [@hess_2005], 2007 [@hess_2009] and 2011 [@hess_2013]. Presented numbers of on– and off–source counts together with values of the parameter $\alpha$ were used as input. Measured counts in each studied observational period ranged between over one hundred to several thousands. Data were taken in configurations with different numbers of active telescopes.
We examined changes of the observed source intensity deduced from the H.E.S.S. data by the means of confidence intervals for the source parameter $\beta$. These intervals were constructed at a $3\sigma$ ($\approx 99.7\%$) level of significance for the sequence of monthly triplets $({N_{\mathrm{on}}},~{N_{\mathrm{off}}},~\alpha)$ using Eq. (\[eq:lima\]) such that $|{S_{\mathrm{LM}}}({N_{\mathrm{on}}},~{N_{\mathrm{off}}},~\alpha;~\beta)|<3$. Furthermore, we combined the monthly data into larger sets according to calendar years and calculated the corresponding confidence intervals. The results are visualized in Fig. \[fig:CI\] where the intervals are depicted as MJD–ordered time series. Several non–overlapping monthly confidence intervals (black segments) indicate variations of the intensity observed from the source. Annual observations provide intervals (hatched bands) which are consistent altogether. Confidence interval for 2011 campaign depicted in Fig. \[fig:CI\] was obtained with the modified data set (see Tab. \[tab:CI\] and Section \[sec:discussion\]).
![ MJD–ordered sequence of confidence intervals at a $3\sigma$ level for the source parameter $\beta$ obtained for the $\gamma$–ray binary [PSR B1259–63/LS 2883]{} [@hess_2005; @hess_2009; @hess_2013]. Segments with points represent monthly intervals, hatched bands denote yearly intervals. The horizontal dashed line indicates the average value of the source parameter taken over all measurements. Interval for 2011 observations was estimated by accounting for the energy spectrum presented by the H.E.S.S. collaboration, for details see Tab. \[tab:CI\] and corresponding text. []{data-label="fig:CI"}](fig1){width="\columnwidth"}
![ Confidence intervals at a $1\sigma$ level as a function of observation date with respect to the periastron occurrence which is marked by vertical dotted line. The horizontal dashed line denotes the reference value corresponding to the background–only dominated on–source region. Observations in 2011, $\langle \beta_{-},~\beta_{+} \rangle = \langle 3.57,~4.44 \rangle$, are indicated by the arrow. For further details see caption to Fig. \[fig:CI\]. []{data-label="fig:CI_phase"}](fig2){width="\columnwidth"}
The observed data can be further depicted in a plot for confidence intervals at a $1\sigma$ level arranged according to the proximity of observations to the periastron date. In Fig. \[fig:CI\_phase\], indications of a double–peak structure can be recognized with a local minimum around the date of periastron.
In order to quantify the trends of the source activity emerging in the sequence of confidence intervals we also performed test with the 2004 and 2007 data for the assumption of constant intensity. The source activity to be tested was chosen as an average value of the ratio of observed and expected on–source events over the observational periods in 2007, i.e. $\beta_{07} = \langle {N_{\mathrm{on}}}/ \alpha {N_{\mathrm{off}}}\rangle = 1.21$. Our results of testing changes of the source activity are presented as a quantile–quantile (QQ) plot. For this purpose, Li–Ma significances were calculated according to Eq. (\[eq:lima\]) and then assigned to the quantiles of the standard normal distribution $N(0,1)$. The quantiles were chosen to be where $m$ denotes the number of observed events and $k = 1 \dots m$. The observational time sequence is indicated as increasing sizes of markers.
![ QQ–plot of asymptotic Li–Ma significances for the $\gamma$–ray events detected from the direction of [PSR B1259–63/LS 2883]{} [@hess_2005; @hess_2009]. Average source parameter $\beta_{07} = 1.21$ calculated from 2007 measurements was tested. The chronological sequence of observations is given by increasing sizes of markers. The dashed line denotes the reference $45^{\circ}$ line through the origin. []{data-label="fig:QQ"}](fig3){width="\columnwidth"}
In Fig. \[fig:QQ\], a QQ–plot of the distribution of Li–Ma significances is shown. The dashed line denotes the reference $45^{\circ}$ line through the origin on which the data points lie provided the hypothesis of constant source intensity of given value is true. Most of the measurements conducted in 2007 (squares) are approximately consistent with the reference line apart from the minor downward shift which indicates that the value of intensity during these periods was slightly lower than the chosen one. The only exceptional result in 2007 is obtained for the June data set which yields significances for the rejection of a steady source activity assumption ${S_{\mathrm{LM}}}> 3.6$. On contrary, the 2004 sample significances (circles) exhibit considerable dispersion from the reference line. Different skewness of the distributions of sample significances compared to the reference standard normal distribution hints that the source intensity changed. At two occasions a significant excess of source activity is recognized with the significances ${S_{\mathrm{LM}}}> 3.6$. In one case a considerable deficit of events is observed with ${S_{\mathrm{LM}}}\approx -3$.
Discussion {#sec:discussion}
==========
Period ${N_{\mathrm{on}}}$ ${N_{\mathrm{off}}}$ $\alpha$ $\langle \beta_{-},~\beta_{+} \rangle_{3\sigma}$
-------- --------------------- ---------------------- ---------- --------------------------------------------------
2011 112 365 0.077 $\langle 2.85,~5.46 \rangle$
2011\* 80 365 0.077 $\langle 1.93,~4.07 \rangle$
: Numbers of detected on– and off–source events ($E>500$ GeV) and the on–off parameter $\alpha$ for the H.E.S.S. observations of [PSR B1259–63/LS 2883]{} during 2011 [@hess_2013]. $\langle \beta_{-},~\beta_{+} \rangle_{3\sigma}$ are confidence intervals for the source parameter $\beta$ constructed at a $3\sigma$ level. Asterisk denotes the modified data set. []{data-label="tab:CI"}
The H.E.S.S. collaboration stated that there is evidence for variability of integral flux above the energy threshold 380 GeV from [PSR B1259–63/LS 2883]{} in 2004 [@hess_2005] and also above 620 GeV in 2007 [@hess_2009]. Moreover, the overall flux level was found to correspond to the one measured in 2004. Observations in 2005 and 2006, i.e. during the epoch between two consecutive periastron passages, did not provide significant detection of the source thus indicating its variable character [@hess_2009]. No variability on the time scale of individual nights during 2011 was proved [@hess_2013]. It was further stated that the 2011 flux above the energy threshold 1 TeV is consistent with the flux levels obtained in 2004 and 2007.
In the framework of the modified on–off method, we arrive at similar results with regard to the evidence for variability of observed $\gamma$–ray intensity in 2004 and 2007, see Figs. \[fig:CI\] and \[fig:QQ\]. Likewise, increase of the source emission before and after the periastron visible in Fig. \[fig:CI\_phase\] agrees with the results of the lightcurve analysis [@hess_2009; @hess_2013]. Interestingly, our results suggest that the source activity observed in 2011 was not consistent with the overall intensity in previous campaigns. First row in Tab. \[tab:CI\] shows that the $99.7\%$ confidence interval for 2011 measurements is lying considerably higher than any other one deduced from earlier observations, compare with Fig. \[fig:CI\]. This translates also in the significance ${S_{\mathrm{LM}}}= 9.6$ with which one can reject the assumption of the constant source intensity. Here, the source parameter $\beta_{47} = 1.25$ to be tested was chosen as an average value taken from the 2004 and 2007 campaigns.
These results are not contradictory, however. Firstly, due to the smaller data set the results of the 2011 measurements are less constraining than the previous ones [@hess_2013]. Secondly, we worked also with events with energies below 1 TeV, the threshold above which H.E.S.S. determined the integral flux [@hess_2013]. Finally, it is worth noting that the H.E.S.S. experiment measured considerably increased flux of photons at energies around 500 GeV, see Fig.1 in [@hess_2013].
In an attempt to explain the inconsistencies we changed the number of photons observed in the first interval of the energy spectrum. To this end, we assumed the same simple power law spectral shape with the index $\Gamma = 2.92$ as reported in the H.E.S.S. analysis [@hess_2013]. The modified number of photons in the lowest energy bin was set as consistent with the power law derived from higher energy bins. The resulting 2011 confidence interval for the source parameter $\beta$ stemming from this modification is given in the second row in Tab. \[tab:CI\]. This interval still indicates that the observed source activity above the energy threshold of roughly 500 GeV was in 2011 above the intensities deduced from previous campaigns provided the background remained stable throughout all observation periods.
In view of the 2011 detection of a HE $\gamma$–ray flare by the *Fermi*–LAT [@Abdo; @Tam], it may be suggestive to relate this flare to the higher level of the VHE activity as seen in our analysis. However, when testing the ‘Pre–flare’ and ‘Flare’ data sets of H.E.S.S. [@hess_2013] for deviations of the source intensity from its average value, the test significances for both periods are $|{S_{\mathrm{LM}}}|<1$. Thus, we cannot reject the assumption of constant intensity during 2011 based on these data and, accordingly, the connection with the GeV flare is questionable.
The increase of the observed VHE activity may be attributable to the truly intrinsically variable emission or it may be some effect of the detection process. We cannot exclude the latter possibility without further details on the used data, particularly on the exact detection thresholds in each observational campaign. Nevertheless, we have no firm indication of inconsistency of the source intensity above 1 TeV in 2011 when compared to the previous observations. Hence, we comply with the H.E.S.S. conclusions about the overall agreement of integral fluxes above this energy.
Conclusions
===========
We used the modified on–off method to judge whether significant changes of the $\gamma$–ray intensity from the direction of binary [PSR B1259–63/LS 2883]{} were observed during its three periastron passages. Within our method, the problem of variable source activity can be treated using solely the numbers of detected events.
Temporal evolution of the observed source intensity is clearly recognized on the account of the modified on–off technique. The results of our analysis of the data gathered by the H.E.S.S. experiment clearly show that changes of the source activity occurred during 2004 and 2007 between individual months while the overall intensities in both seasons stayed in agreement. Our findings on the data obtained above approximately 500 GeV in 2011 show that the observed source intensity was enhanced in this season with respect to earlier measurements.
This work was supported by the Czech Science Foundation grant 14-17501S.
S. Johnston et al., ApJ **387**, L37–L41 (1992) H.E.S.S. Collaboration, A&A **551**, A94 (2013) F. Aharonian et al., A&A **442**, 1–10 (2005) F. Aharonian et al., A&A **507**, 389–396 (2009) A. A. Abdo et al., ApJ Lett. **736**, L11 (2011) P. H. T. Tam et al., ApJ Lett. **736**, L10 (2011) D. Berge, S. Funk, J. Hinton, A&A **446**, 1219–1229 (2007) T.P. Li, Y.Q. Ma, ApJ **272**, 317–324 (1983) D. Nosek, S. Stefanik, J. Noskova, ICRC 2013, arXiv:1309.6476 S. Stefanik, D. Nosek, Nucl. Phys. B-Proc. Sup. **256**, 258–263 (2014)
[^1]:
|
---
abstract: 'Electronic health records (EHR) are increasingly being used for constructing disease risk prediction models. Feature engineering in EHR data however is challenging due to their highly dimensional and heterogeneous nature. Low-dimensional representations of EHR data can potentially mitigate these challenges. In this paper, we use global vectors (*GloVe*) to learn word embeddings for diagnoses and procedures recorded using 13 million ontology terms across 2.7 million hospitalisations in national UK EHR. We demonstrate the utility of these embeddings by evaluating their performance in identifying patients which are at higher risk of being hospitalized for congestive heart failure. Our findings indicate that embeddings can enable the creation of robust EHR-derived disease risk prediction models and address some the limitations associated with manual clinical feature engineering.'
author:
- |
Spiros Denaxas\
Institute of Health Informatics\
University College London, UK\
`s.denaxas@ucl.ac.uk`\
Pontus Stenetorp, Sebastian Riedel\
Department of Computer Science\
University College London, UK\
`{S.Riedel,P.Stenetorp}@cs.ucl.ac.uk`\
Maria Pikoula, Richard Dobson, Harry Hemingway\
Institute of Health Informatics\
University College London, UK\
`{m.pikoula, r.dobson, h.hemingway}@ucl.ac.uk`\
bibliography:
- 'references.bib'
title: Application of Clinical Concept Embeddings for Heart Failure Prediction in UK EHR data
---
Introduction
============
Risk prediction models are statistical tools which are used to predict the probability that an individual with a given set of characteristics (e.g. smoking, blood pressure, family history of cancer) will experience a health outcome (e.g. heart attack, type 2 diabetes, death). They are a cornerstone of modern clinical medicine [@moons2009prognosis] as they enable clinicians to intervene earlier or chose the optimal therapeutic strategy for a patient. Electronic health records (EHR), data generated during routine patient interactions with healthcare providers [@hemingway2017big; @jensen2012mining], offer the opportunity to create risk prediction models in larger sample populations and higher clinical resolution [@rapsomaniki2013prognostic] than previously available. Utilizing EHR data however is challenging [@morley2014defining; @hripcsak2012next; @albers2018estimating; @paxton2013developing] and a recent review [@goldstein2017opportunities] illustrated that EHR-derived predictive models used a median of only 27 clinical features, mostly engineered in a cross-sectional fashion.
Clinical concept embeddings, i.e. multi-dimensional vector representations of medical concepts, can potentially enable the creation of risk prediction models that make use of a patient’s entire EHR (e.g. diagnoses, procedures) and reduce the need for manual feature engineering. Contemporary approaches for learning word embeddings are influenced by the neural language model developed by Bengio et al. [@bengio2003neural]. Word embeddings are a very popular way of representing high-dimensional and high-sparsity data in the field of natural language processing and have demonstrated a significant improvement in classification accuracy when combined with existing labelled data [@turian2010word] . Popular approaches include *word2vec*[@mikolov2013efficient], which includes the continuous bag of words and skip-gram models, and *GloVe* [@pennington2014glove], which produces word embeddings by fitting a weighted log-linear model to aggregated global word-word co-occurrence statistics.
Previous research and contribution
----------------------------------
Word embedding approaches have been previously used to create low-dimensional representations of heterogeneous clinical concepts (e.g. diagnoses, prescriptions, procedures, laboratory findings) from raw EHR data for various supervised and unsupervised learning tasks [@beam2018clinical; @shickel2017deep]. Previous research has evaluated the use of clinical concept embeddings in US data for predicting the risk of developing heart failure using recurrent [@choi2016medical; @che2017exploiting] or convolutional [@che2017exploiting] neural network architectures. In other disease areas, embeddings have been evaluated for predicting events in critical care patients [@farhan2016predictive; @johnson2016mimic], length of stay and associated costs [@feng2017patient], suicide in mental health patients [@tran2015learning] and hierarchical regularities in dependencies between health states [@miotto2016deep].
A cornerstone of building risk prediction tools is external replication of findings [@collins2014external]. Here, we attempt to replicate and compare, to a certain degree, findings obtained using US data from single hospital care providers [@choi2016medical] using UK EHR from multiple providers, and illustrate the application of embeddings for risk prediction. This is the first study, to our knowledge, to utilize UK EHR data in this context. The UK and US healthcare systems are significantly different in terms of planning, delivery and reimbursement. This in turn directly influences what data are recorded in a patient’s electronic health record. Additionally, in contrast with previous research, we evaluate the predictive performance of different clinical information (e.g. diagnoses, procedures) independently as well since including *all* available data might potentially be counterproductive given the noisy and heterogeneous nature of EHR data.
Methods
=======
Data sources and population
---------------------------
We used secondary care EHR from the UK Biobank [@sudlow2015uk], a population-based research study comprising 502,629 individuals in the UK. The study contains extensive phenotypic and genotypic information and longitudinal follow-up for health-related outcomes is through linkages to national EHR from hospital care and mortality registers. Diagnoses and procedures were recorded using controlled clinical terminologies, i.e. hierarchical ontologies enables clinicians to systematically record information about a patient’s health and treatment and enable the subsequent use of data for reimbursement [@o2005measuring; @bright1989medicaid] and research [@bhaskaran2014body; @rapsomaniki2014blood]. Diagnoses were recorded using ICD-9 and ICD-10 [@world2004international] and procedures using OPCS-4 [@de2001survey]. Admitted patients are assigned a primary and up to 15 secondary causes of admission.
We defined incident and prevalent HF cases using a previously-validated phenotyping algorithm from the CALIBER resource [@denaxas2012data; @koudstaal2017prognostic; @gho2018electronic]. HF cases were identified using ICD-9 and ICD-10 terms occurring at any position during a patient admission (i.e. primary or otherwise) in patients aged 40-85 years old at the time of admission. For patients with multiple HF diagnoses, the date of onset was defined as the earliest date of admission. HF cases were matched with four eligible controls on assessment centre, year of recruitment, sex and year of birth. Controls were assigned an index date, which was the date of HF diagnosis of the matched case. We excluded prevalent HF cases from our analyses.
Learning concept and patient embeddings
---------------------------------------
We created four corpuses (Table 1) using: a) primary diagnosis terms (PRIMDX), b) primary diagnosis terms and procedure terms (PRIMDX-PROC), c) using primary and secondary diagnosis terms (PRIMDX-SECDX) and, d) primary and secondary diagnosis terms and procedure terms (PRIMDX-SECDX-PROC). We learned *concept-level embeddings* using the *GLoVe* model on the four corpuses and evaluated combinations of embedding dimension (50, 100, 150, 250, 500, 1000) and window sizes (50, 10, 20). Models were trained using Adagrad [@duchi2011adaptive] and 150 epochs. We created *patient-level embeddings* (Figure 1.) by: a) extracting all terms from a patients EHR record from the start of follow up to six months (to exclude features very strongly correlated with a subsequent diagnosis [@kourou2015machine]) prior to date of HF diagnoses for cases or the index date for matched controls, b) looking up the vector representations for each embedding, c) creating a vector composed of the mean, max and min of all concept vector representations and, d) normalizing to zero mean and unit variance. For comparisons purposes, we additionally created one-hot representations of EHR data where the feature vector had the same size as the entire vocabulary and only one dimension is on.
------------------- ------------ ---------- ---------- ------------
Corpus Tokens Tokens Tokens Vocabulary
(total) (unique) (median) size
PRIMDX 2,766,487 10,606 4 5,581
PRIMDX-SECDX 7,699,930 13,883 7 7,797
PRIMDX-PROC 7,904,942 18,608 11 10,949
PRIMDX-SECDX-PROC 12,838,385 21,885 15 13,165
------------------- ------------ ---------- ---------- ------------
: Information on the corpuses used as sources for training the clinical concept embeddings.[]{data-label="tcorpus"}
Risk prediction
---------------
We evaluated each set embeddings by applying a linear support vector machine (SVM) classifier to predict HF onset as a supervised binary classification task using the normalized patient-level embeddings as input. We split the data into a training dataset and a test dataset (ratio 3:1) and performed six-fold cross-validation in all modeling iterations on the training data to find the optimal hyper-parameters. We evaluated predictive performance using the area under the weighted receiver operating characteristic curve (AUROC) and the weighted F1 score computed on the test dataset which was unseen.
Implementation
--------------
The SVM was implemented using scikit-learn [@pedregosa2011scikit]. *GloVe* embeddings were trained using binaries from pennington2014glove. The documented source code using sample synthetic data for our experiments is available at https://github.com/spiros. EHR data used in our experiments are available by applying to the UK Biobank [@sudlow2015uk]. UK Biobank ethical approval ref. 9922.
Experimental Results
====================
We used raw EHR data from 502,639 participants and identified 4,581 HF cases (30.52% female) and matched them as previously described to 13,740 controls. The mean age at HF diagnosis was 63.397 (95% CI 63.174-63.619). We trained the clinical concept embeddings from 2,447 ICD-9, 10,527 ICD-10 and 6,887 OPCS-4 terms across 2,779,598 hospitalizations. We observed that similarly with previous research studies using clinical concept embeddings, diseases which are biologically or contextually closely related across the entire corpus are located close to each other in the vector space (Figure 2).
We observed similar predictive performance across both one-hot and clinical concept embedding prediction experiments. Clinical concept embeddings performed marginally better than one-hot encoded data. The highest performing models were the ones using information combining all diagnoses and surgical procedures (Table 2), obtained with a vector size of 250 and a context window size of five with embeddings. For models using the less extensive corpuses, the best performing results were observed with vectors of smaller size (50 dimensions) and larger context windows (ranging from 10-20). Although, counter-intuitively, the PRIMDX best embedding outperformed PRIMDX-PROC (using procedures and primary diagnoses), PRIMDX-PROC performed better than PRIMDX on average across all vector size and context window combinations. This suggests that clinical concept vectors could be beneficial for risk prediction in absence of a domain ontology or in a semi-supervised fashion combined with labelled data to boost performance [@mencia2016medical].
------------------- ------------ -------- ------------ --------
Embedding AUROC F1 AUROC F1
PRIMDX 0.6543 0.7558 0.6720 0.7390
PRIMDX-PROC 0.6445 0.7362 0.6662 0.7341
PRIMDX-SECDX 0.6697 0.7527 0.6878 0.7568
PRIMDX-SECDX-PROC **0.6815** 0.7664 **0.6965** 0.7500
------------------- ------------ -------- ------------ --------
: Best performing embeddings in test dataset with optimal hyper-parameters.[]{data-label="sample-table3"}
Direct comparison with previous studies is challenging due to the use of different underlying populations, study designs and incomplete definitions of cohorts and outcomes [@walsh2014effects; @rajkomar2018scalable]. When comparing our results with previous studies which used clinical concept embeddings to predict HF onset in a similar experimental setup, our approach achieved broadly similar (but slightly worse) overall performance and followed similar patterns: Choi [@choi2016medical] et al. utilized clinical concept vectors trained using *word2vec* skip-gram and reported an AUROC of 0.711 with one-hot encoded input and AUROC of 0.743 using embeddings with a SVM classifier. Interestingly, the fact that we observed similar (albeit slightly worse) results when using data from multiple hospitals compared to a study sourcing data from a single hospital indicates that embedding approaches can potentially be a very useful tool for scaling analyses across large heterogeneous data source and are insensitive to source variations.
Conclusion
==========
Our work evaluated the use of word embeddings trained using *GloVe* for creating low-dimensionality representations of heterogeneous clinical concepts in UK EHR data. The use of clinical embeddings produced marginally improved predictive performance compared to conventional one-hot models and thus potentially has has numerous applications in healthcare settings where complex, heterogeneous information requires succinct representation or a domain ontology is not fit for purpose. Further research is required to evaluate performance across different prediction windows and increase model interpretability to enable their rapid translation into clinical care.
|
[**Cayley-Dickson Algebras and Finite Geometry**]{}
Metod Saniga,$^{1,2}$ Frédéric Holweck$^{3}$ and Petr Pracna$^{4}$
$^{1}$Institute for Discrete Mathematics and Geometry, Vienna University of Technology\
Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria\
(metod.saniga@tuwien.ac.at)
$^{2}$Astronomical Institute, Slovak Academy of Sciences\
SK-05960 Tatransk' a Lomnica, Slovak Republic\
(msaniga@astro.sk)
$^{3}$Laboratoire IRTES/M3M, Université de Technologie de Belfort-Montbéliard\
F-90010 Belfort, France\
(frederic.holweck@utbm.fr)
$^{4}$J. Heyrovsk' y Institute of Physical Chemistry, v.v.i.\
Academy of Sciences of the Czech Republic\
Dolejškova 3, CZ-18223 Prague, Czech Republic\
(pracna@jh-inst.cas.cz)
[**Abstract**]{}
Given a $2^N$-dimensional Cayley-Dickson algebra, where $3 \leq N \leq 6$, we first observe that the multiplication table of its imaginary units $e_a$, $1 \leq a \leq 2^N -1$, is encoded in the properties of the projective space PG$(N-1,2)$ if one regards these imaginary units as points and distinguished triads of them $\{e_a, e_b, e_c\}$, $1 \leq a < b <c \leq 2^N -1$ and $e_ae_b = \pm e_c$, as lines. This projective space is seen to feature two distinct kinds of lines according as $a+b = c$ or $a+b \neq c$. Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG$(N-1,2)$, the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a binomial $\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configuration ${\cal C}_N$; in particular, ${\cal C}_3$ (octonions) is isomorphic to the Pasch $(6_2,4_3)$-configuration, ${\cal C}_4$ (sedenions) is the famous Desargues $(10_3)$-configuration, ${\cal C}_5$ (32-nions) coincides with the Cayley-Salmon $(15_4,20_3)$-configuration found in the well-known Pascal mystic hexagram and ${\cal C}_6$ (64-nions) is identical with a particular $(21_5,35_3)$-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. We also draw attention to a remarkable nesting pattern formed by these configurations, where ${\cal C}_{N-1}$ occurs as a geometric hyperplane of ${\cal C}_N$. Finally, a brief examination of the structure of generic ${\cal C}_N$ leads to a conjecture that ${\cal C}_N$ is isomorphic to a combinatorial Grassmannian of type $G_2(N+1)$.
[**Keywords:**]{} Cayley-Dickson Algebras – Veldkamp Spaces – Finite Geometries
Introduction
============
As it is well known (see, e.g., [@jac; @scha]), the Cayley-Dickson algebras represent a nested sequence $A_0, A_1, A_2,\ldots, A_N, \ldots$ of $2^N$-dimensional (in general non-associative) $\mathbb{R}$-algebras with $A_N \subset A_{N+1}$, where $A_0 = \mathbb{R}$ and where for any $N \geq 0$ $A_{N+1}$ comprises all ordered pairs of elements from $A_{N}$ with conjugation defined by $$(x,y)^{\ast} = (x^{\ast}, -y)
\label{eq1}$$ and multiplication usually by $$(x,y)(X,Y) = (xX - Yy^{\ast}, x^{\ast}Y + Xy).
\label{eq2}$$ Every finite-dimensional algebra (see, e.g., [@scha]) is basically defined by the multiplication rule of its basis. The basis elements (or units) $e_0, e_1, e_2, \ldots, e_{2^{N+1} -1}$ of $A_{N+1}$, $e_0$ being the real basis element (identity), can be chosen in various ways. Our preference is the [*canonical*]{} basis $$\begin{aligned}
e_0 &= (e_0,0), & e_1 &= (e_1,0), & e_2 &= (e_2,0), && \ldots, && e_{2^N - 1} &= (e_{2^N - 1},0), &~~~~~& \\
e_{2^N} &= (0,e_0), & e_{2^N+1} &= (0,e_1), & e_{2^N+2} &= (0,e_2), && \ldots, && e_{2^{N+1} - 1} &= (0,e_{2^{N} - 1}),&~~~~~~&\end{aligned}$$ where, by abuse of notation (see, e.g., [@mor]), the same symbols are also used for the basis elements of $A_N$. This is because the paper essentially focuses on [*multiplication*]{} properties of basis elements and the canonical basis seems to display most naturally the inherent symmetry of this operation. For, in addition to revealing the nature of the Cayley-Dickson recursive process, it also implies that for both $a$ and $b$ being non-zero we have $e_a e_b = \pm e_{a \oplus b},$ where the symbol ‘$\oplus$’ denotes ‘[*exclusive or*]{}’ of the binary representations of $a$ and $b$ (see, e.g., [@arndt]). From the above expressions and Eqs. (\[eq1\]) and (\[eq2\]) one can readily find the product of any two distinct units of $A_{N+1}$ if the multiplication properties of those of $A_{N}$ are given. Such products are usually expressed/presented in a tabular form, and we shall also follow this tradition here. All the multiplication tables we made use of were computed for us by Jörg Arndt; the computer program is named [cayley-dickson-demo.cc]{} and is freely available from his web-site [http://jjj.de/fxt/demo/arith/index.html]{}.
Employing such a multiplication table of $A_N$, $N \geq 2$, it can be verified that the $2^N-1$ imaginaries $e_a$, $1 \leq a \leq 2^N -1$, form ${2^N-1 \choose 2}/3$ distinguished sets each of which comprises three different units $\{e_a, e_b, e_c\}$ that satisfy equation $$e_a e_b = \pm e_c,
\label{eq4}$$ and where each unit is found to belong to $2^{N-1} - 1$ such sets.[^1] Regarding the imaginaries as points and their distinguished triples as lines, one gets a point-line incidence geometry where every line has three points and through each point there pass $2^{N-1} - 1$ lines and which is isomorphic to PG$(N-1,2)$, the $(N-1)$-dimensional projective space over the smallest Galois field $GF(2)$ (see also [@baez] for $N=3$ and [@culb] for $N=4$). Let us assume, without loss of generality, that the elements in any distinguished triple $\{e_a, e_b, e_c\}$ of $A_N$ are ordered in such a way that $a < b < c$. Then, for $N \geq 3$, we can naturally speak about two different kinds of triples and, hence, two distinct kinds of lines of the associated $2^N$-nionic PG$(N-1,2)$, according as $a+b = c$ or $a+b \neq c$; in what follows a line of the former/latter kind will be called ordinary/defective. This stratification of the line-set of the PG$(N-1,2)$ induces a similar partition of the point-set of the latter space into several types, where a point of a given type is characterized by the same number of lines of either kind that pass through it.
Obviously, if our projective space PG$(N-1,2)$ is regarded as an [*abstract*]{} geometry [*per se*]{}, every point and/or every line in it has the same footing. So, to account for the above-described ‘refinement’ of the structure of our $2^N$-nionic PG$(N-1,2)$, it turns out to be necessary to find a representation of this space where each point/line is ascribed a certain ‘internal’ structure, which at first sight may seem to be quite a challenging task. To tackle this task successfully, we need to introduce a few notions/concepts from the realm of finite geometry.
We start with a finite [*point-line incidence structure*]{} $\mathcal{C} = (\mathcal{P},\mathcal{L},I)$ where $\mathcal{P}$ and $\mathcal{L}$ are, respectively, finite sets of points and lines and where incidence $I \subseteq \mathcal{P} \times \mathcal{L}$ is a binary relation indicating which point-line pairs are incident (see, e.g., [@shult]). Here, we shall only be concerned with specific point-line incidence structures called [*configurations*]{} [@grun]. A $(v_r,b_k)$-configuration is a $\mathcal{C}$ where: 1) $v = \vert \mathcal{P} \vert$ and $b = \vert \mathcal{L}\vert$, 2) every line has $k$ points and every point is on $r$ lines, and 3) two distinct lines intersect in at most one point and every two distinct points are joined by at most one line; a configuration where $v=b$ and $r=k$ is called symmetric (or balanced), and usually denoted as a $(v_r)$-configuration. A $(v_r,b_k)$-configuration with $v = {r+k-1 \choose r}$ and $b = {r+k-1 \choose k}$ is called a $binomial$ configuration. Next it comes a [*geometric hyperplane*]{} of $\mathcal{C} = (\mathcal{P},\mathcal{L},I)$, which is a proper subset of $\mathcal{P}$ such that a line from $\mathcal{C}$ either lies fully in the subset, or shares with it only one point. If $\mathcal{C}$ possesses geometric hyperplanes, then one can define the [*Veldkamp space*]{} of $\mathcal{C}$, $\mathcal{V}(\mathcal{C})$, as follows [@buec]: (i) a point of $\mathcal{V}(\mathcal{C})$ is a geometric hyperplane of $\mathcal{C}$ and (ii) a line of $\mathcal{V}(\mathcal{C})$ is the collection $H'H''$ of all geometric hyperplanes $H$ of $\mathcal{C}$ such that $H' \cap H'' = H' \cap H = H'' \cap H$ or $H = H', H''$, where $H'$ and $H''$ are distinct geometric hyperplanes. If each line of $\mathcal{C}$ has three points and $\mathcal{C}$ ‘behaves well,’ a line of $\mathcal{V}(\mathcal{C})$ is also of size three and can equivalently be defined as $\{H', H'', \overline{H' \Delta H''}\}$, where the symbol $\Delta$ stands for the symmetric difference of the two geometric hyperplanes and an overbar denotes the complement of the object indicated. From its definition it is obvious that $\mathcal{V}(\mathcal{C})$ is well suitable for our needs because its points, being themselves [*sets*]{} of points, have different ‘internal’ structure and so, in general, they can no longer be on the same par; clearly, the same applies to the lines $\mathcal{V}(\mathcal{C})$. Our task thus basically boils down to finding such $\mathcal{C}_N$ whose $\mathcal{V}(\mathcal{C}_N)$ is isomorphic to PG$(N-1,2)$ and completely reproduces its $2^N$-nionic fine structure. This will be carried out in great detail for the first four non-trivial cases, $3 \leq N \leq 6$, which, when combined with the two trivial cases ($N=1,2$), will provide us with sufficient amount of information to guess a general pattern.
The paper is organized as follows. In Sect. 2 it is shown that ${\cal C}_3$ (octonions) is isomorphic to the Pasch $(6_2,4_3)$-configuration, which plays a key role in classifying Steiner triple systems. In Sect. 3 one demonstrates that ${\cal C}_4$ (sedenions) is nothing but the famous Desargues $(10_3)$-configuration. In Sect. 4 our ${\cal C}_5$ (32-nions) is shown to be identical with the Cayley-Salmon $(15_4,20_3)$-configuration found in the well-known Pascal mystic hexagram. In Sect. 5 we find that ${\cal C}_6$ corresponds to a particular $(21_5,35_3)$-configuration encompassing seven distinct copies of the Cayley-Salmon $(15_4,20_3)$-configuration as geometric hyperplanes. In Sect. 6 some rudimentary properties of the generic ${\cal C}_N \cong \left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configuration are outlined and its isomorphism to a combinatorial Grassmannian of type $G_2(N+1)$ is conjectured. Finally, Sect. 7 is reserved for concluding remarks.
Octonions and the Pasch $(6_2,4_3)$-configuration
=================================================
From the nesting property of the Cayley-Dickson construction of $A_N$ it is obvious that the smallest non-trivial case to be addressed is $A_3$, the algebra of octonions, whose multiplication table is presented in Table 1.
$*$ 1 2 3 4 5 6 7
----- ------ ------ ------ ------ ------ ------ ------
1 $-$0 $-$3 +2 $-$5 +4 +7 $-$6
2 +3 $-$0 $-$1 $-$6 $-$7 +4 +5
3 $-$2 +1 $-$0 $-$7 +6 $-$5 +4
4 +5 +6 +7 $-$0 $-$1 $-$2 $-$3
5 $-$4 +7 $-$6 +1 $-$0 +3 $-$2
6 $-$7 $-$4 +5 +2 $-$3 $-$0 +1
7 +6 $-$5 $-$4 +3 +2 $-$1 $-$0
: The multiplication table of the imaginary unit octonions $e_a$, $1 \leq a \leq 7$. For the sake of simplicity, in what follows we shall employ a short-hand notation $e_a \equiv a$; likewise for the real unit $e_0 \equiv 0$. There are also delineated multiplication tables corresponding to the distinguished nested sequence of sub-algebras of complex numbers ($a=1$, the upper left corner) and quaternions ($1 \leq a \leq 3$, the upper left $3 \times 3$ square).
The above-given multiplication table implies the existence of the following seven [*distinguished*]{} trios of imaginary units:
$$\{1,2,3\}, \{1,4,5\}, \{1,6,7\},$$ $$\{2,4,6\}, \{2,5,7\},$$ $$\{3,4,7\}, \{3,5,6\}.$$
Regarding the seven imaginary units as points and the seven distinguished triples of them as lines, we obtain a point-line incidence structure where each line has three points and, dually, each point is on three lines, and which is isomorphic to the smallest projective plane PG$(2,2)$, often called the Fano plane, depicted in Figure \[f1\].
![An illustration of the structure of PG$(2,2)$, the Fano plane, that provides the multiplication law for octonions (see, e.g., [@baez]). The points of the plane are seven small circles. The lines are represented by triples of circles located on the sides of the triangle, on its altitudes, and by the triple lying on the big circle. The three imaginaries lying on the same line satisfy Eq.(\[eq4\]).[]{data-label="f1"}](f1.eps){width="4.0truecm"}
It is then readily seen that we have six ordinary lines, namely $$\{1,2,3\}, \{1,4,5\}, \{1,6,7\},$$ $$\{2,4,6\}, \{2,5,7\},$$ $$\{3,4,7\},$$ and only single defective one, viz. $$\{3,5,6\}.$$ Similarly, our octonionic PG$(2,2)$ features two distinct types of points. A type-one point is such that two lines passing through it are ordinary, the remaining one being defective; such a point lies in the set $$\{3,5,6\} \equiv \alpha.$$ A type-two point is such that every line passing through it is ordinary; such a point belongs to the set $$\{1,2,4,7\} \equiv \beta,$$ which is highlighted by gray color in Figure \[f1\].
A configuration $\mathcal{C}_3$ whose Veldkamp space reproduces the above-described partitions of points and lines of PG$(2,2)$ is, as we will soon see, nothing but the well-known [*Pasch*]{} $(6_2,4_3)$-configuration, $\mathcal{P}$. This configuration, which plays a very important role in classifying Steiner triple systems (see, e.g., [@enc]), is depicted in Figure \[f2\] in a form showing an automorphism of order three; it also lives in the Fano plane and, as it is readily seen by comparing Figures \[f1\] and \[f2\], it can be obtained from the latter by removal of any of its seven points and all the three lines passing through it.
In order to see that $\mathcal{V}(\mathcal{P}) \cong$ PG$(2,2)$ we shall first show, using our diagrammatical representation of $\mathcal{P}$, all seven geometric hyperplanes of $\mathcal{P}$ — Figure \[f3\]. We see that they are indeed of two different forms, of cardinality three and four. A member of the former set comprises two points at maximum distance from each other. Such geometric hyperplane corresponds to a type-one (or $\alpha$-) point of PG$(2,2)$. A member of the latter set features three points on a common line; such a geometric hyperplane of $\mathcal{P}$ corresponds to a type-two (or $\beta$-) point of our PG$(2,2)$. The seven lines of $\mathcal{V}(\mathcal{P})$ are illustrated in a compact diagrammatic form in Figure \[f4\]; as it is easily discernible, each of the six ordinary lines is of the form $\{\alpha, \beta, \beta\}$, whilst the remaining defective one has the $\{\alpha, \alpha,
\alpha\}$ shape.
![An illustrative portrayal of the Pasch configuration: circles stand for its points, whereas its lines are represented by triples of points on common straight segments (three) and the triple lying on a big circle.[]{data-label="f2"}](f2.eps){width="2.5truecm"}
![The seven geometric hyperplanes of the Pasch configuration. The hyperplanes are labelled by imaginary units of octonions in such a way that — as it is obvious from the next figure — the seven lines of the Veldkamp space of the Pasch configuration are identical with the seven distinguished triples of units, that is with the seven lines of the PG$(2,2)$ shown in Figure \[f1\].[]{data-label="f3"}](f3.eps){width="5truecm"}
![A unified view of the seven Veldkamp lines of the Pasch configuration. The reader can readily verify that for any three geometric hyperplanes lying on a given line of the Fano plane, one is the complement of the symmetric difference of the other two.[]{data-label="f4"}](f4.eps){width="7.5truecm"}
Sedenions and the Desargues $(10_3)$-configuration
==================================================
Our next focus is on $A_4$, the sedenions, whose basic multiplication properties are summarized in Table 2.
An inspection of this table yields as many as 35 distinguished triples, namely: $$\{1,2,3\}, \{1,4,5\}, \{1,6,7\}, \{1,8,9\}, \{1,10,11\}, \{1,12,13\}, \{1,14,15\},$$ $$\{2,4,6\}, \{2,5,7\}, \{2,8,10\}, \{2,9,11\}, \{2,12,14\}, \{2,13,15\},$$ $$\{3,4,7\}, \{3,5,6\}, \{3,8,11\}, \{3,9,10\}, \{3,12,15\}, \{3,13,14\},$$ $$\{4,8,12\}, \{4,9,13\}, \{4,10,14\}, \{4,11,15\},$$ $$\{5,8,13\}, \{5,9,12\}, \{5,10,15\}, \{5,11,14\},$$ $$\{6,8,14\}, \{6,9,15\}, \{6,10,12\}, \{6,11,13\},$$ $$\{7,8,15\}, \{7,9,14\}, \{7,10,13\}, \{7,11,12\}.$$
Regarding the 15 imaginary units as points and the 35 distinguished trios of them as lines, we obtain a point-line incidence structure where each line has three points and each point is on seven lines, and which is isomorphic to PG$(3,2)$, the smallest projective space — as depicted in Figure \[fgr1\]. The latter figure employs a diagrammatical model of PG$(3,2)$ built, after Polster [@pol], around the pentagonal model of the generalized quadrangle of type GQ$(2,2)$ whose 15 lines are illustrated by triples of points lying on black line-segments (10 of them) and/or black arcs of circles (5). The remaining 20 lines of PG$(3,2)$ comprise four distinct orbits: the yellow, red, blue and green one consisting, respectively, of the yellow ($\{1,10,11\}$), red ($\{1,8,9\}$), blue ($\{3,13,14\}$) and green ($\{3,12,15\}$) line and other four lines we get from each by rotation through 72 degrees around the center of the pentagon.
![An illustration of the structure of PG$(3,2)$ that provides the multiplication law for sedenions. As in the previous case, the three imaginaries lying on the same line are such that the product of two of them yields the third one, sign disregarded.[]{data-label="fgr1"}](fgr1.eps){width="9truecm"}
It is not difficult to verify that we have 25 ordinary lines, namely $$\{1,2,3\}, \{1,4,5\}, \{1,6,7\}, \{1,8,9\}, \{1,10,11\}, \{1,12,13\}, \{1,14,15\},$$ $$\{2,4,6\}, \{2,5,7\}, \{2,8,10\}, \{2,9,11\}, \{2,12,14\}, \{2,13,15\},$$ $$\{3,4,7\}, \{4,8,12\}, \{4,9,13\}, \{4,10,14\}, \{4,11,15\},$$ $$\{3,8,11\}, \{5,8,13\}, \{6,8,14\}, \{7,8,15\},$$ $$\{3,12,15\}, \{5,10,15\}, \{6,9,15\},$$ and 10 defective ones, namely $$\{3,5,6\}, \{3,9,10\}, \{3,13,14\},$$ $$\{5,9,12\}, \{5,11,14\},$$ $$\{6,10,12\}, \{6,11,13\},$$ $$\{7,9,14\}, \{7,10,13\}, \{7,11,12\}.$$ Similarly, our sedenionic PG$(3,2)$ features two distinct types of points. A type-one point is such that four lines passing through it are ordinary, the remaining three being defective; such a point lies in the set $$\{3,5,6,7,9,10,11,12,13,14\} \equiv \alpha.$$ A type-two point is such that every line passing through it is ordinary; such a point belongs to the set $$\{1,2,4,8,15\} \equiv \beta,$$ being illustrated by gray shading in Figure \[fgr1\]. We see that all defective lines are of the same form, namely $\{\alpha, \alpha, \alpha \}$. The 25 ordinary lines split into two distinct families. Ten of them are of the form $\{\alpha, \beta, \beta\}$, namely $$\{1,2,3\}, \{1,4,5\}, \{1,8,9\}, \{1,14,15\},$$ $$\{2,4,6\}, \{2,8,10\}, \{2,13,15\},$$ $$\{4,8,12\},\{4,11,15\},$$ $$\{7,8,15\},$$ and the remaining 15 are of the form $\{\alpha, \alpha, \beta\}$, namely $$\{1,6,7\}, \{1,10,11\}, \{1,12,13\},$$ $$\{2,5,7\}, \{2,9,11\}, \{2,12,14\},$$ $$\{3,4,7\}, \{4,9,13\}, \{4,10,14\},$$ $$\{3,8,11\}, \{5,8,13\}, \{6,8,14\},$$ $$\{3,12,15\}, \{5,10,15\}, \{6,9,15\}.$$
A configuration $\mathcal{C}_4$ whose Veldkamp space reproduces the above-described partitions of points and lines of PG$(3,2)$ is, as demonstrated below, the famous [*Desargues*]{} $(10_3)$-configuration, $\mathcal{D}$, which is one of the most prominent point-line incidence structures (see, e.g., [@piser]). Up to isomorphism, there exist altogether ten $(10_3)$-configurations. The Desargues configuration is, unlike the others, flag-transitive and the only one where for [*each*]{} of its points the three points that are not collinear with it lie on a line. This configuration, depicted in Figure \[fgr2\] in a form showing its automorphism of order three, also lives in our sedenionic PG$(3,2)$; here, its points are the ten $\alpha$-points and its lines are all the defective lines. In order to see that $\mathcal{V}(\mathcal{D}) \cong$ PG$(3,2)$ we shall first introduce, using our diagrammatical representation of $\mathcal{D}$, all 15 geometric hyperplanes of $\mathcal{D}$ — Figure \[fgr3\]. We see that they are indeed of two different forms, and of required cardinality ten and five. A member of the former set comprises a point and three points not collinear with it. Such geometric hyperplane corresponds to a type-one (or $\alpha$-) point of PG$(3,2)$. A member of the latter set features six points located on four lines, with two lines per each point; this is nothing but the Pasch configuration we introduced in the previous section. Such a geometric hyperplane of $\mathcal{D}$ corresponds to a type-two (or $\beta$-) point of our PG$(3,2)$. It is also a straightforward task to verify that $\mathcal{V}(\mathcal{D})$ is endowed with 35 lines splitting into the required three families; those that correspond to defective lines of our sedenionic PG$(3,2)$ are shown in Figure \[fgr4\], while those that correspond to ordinary lines are depicted in Figure \[fgr5\] (of type $\{\alpha, \beta, \beta\}$) and Figure \[fgr6\] (of type $\{\alpha, \alpha, \beta\}$). Figure \[fgr7\] offers a ‘condensed’ view of the isomorphism $\mathcal{V}(\mathcal{D}) \cong$ PG$(3,2)$.
We shall finalize this section by pointing out that the existence of two different kinds of geometric hyperplanes of the Desargues configuration is closely connected with two well-known views of this configuration. The first one is as a pair of triangles that are in perspective from both a point and a line (Desargues’ theorem), the point and the line forming a geometric hyperplane. The other view is as the incidence sum of a complete quadrangle (i.e., a $(4_3,6_2)$-configuration) and a Pasch $(6_2,4_3)$-configuration [@bogepi].
![An illustrative portrayal of the Desargues configuration, built around the model of the Pasch configuration shown in Figure \[f2\]: circles stand for its points, whereas its lines are represented by triples of points on common straight segments (six), arcs of circles (three) and a big circle.[]{data-label="fgr2"}](fgr2.eps){width="4.5truecm"}
![The fifteen geometric hyperplanes of the Desargues configuration. The hyperplanes are labelled by imaginary units of sedenions in such a way that — as we shall verify in the next three figures — the 35 lines of the Veldkamp space of the Desargues configuration are identical with the 35 distinguished triples of units, that is with the 35 lines of the PG$(3,2)$ shown in Figure \[fgr1\].[]{data-label="fgr3"}](fgr3.eps){width="12truecm"}
![A compact graphical view of illustrating the bijection between 15 imaginary unit sedenions and 15 geometric hyperplanes of the Desargues configuration, as well as between 35 distinguished triples of units and 35 Veldkamp lines of the Desargues configuration. []{data-label="fgr7"}](fgr7.eps){width="14truecm"}
32-nions and the Cayley-Salmon $(15_4,20_3)$-configuration
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Our next case is $A_5$, or the 32-nions, whose multiplication properties are encoded in Table 3.
-- --
-- --
: The multiplication table — for better readability split into two parts — of the imaginary unit 32-nions $e_a$, $1 \leq a \leq 31$, in short-handed notation $e_a \equiv a$ (and $e_0 \equiv 0$).
From this table we infer the existence of 155 distinguished triples of imaginary units. Regarding the 31 imaginary units of $A_5$ as points and the 155 distinguished triples of them as lines, we obtain a point-line incidence structure where each line has three points and each point is on 15 lines, and which is isomorphic to PG$(4,2)$. We next find that 65 lines of this space are defective and 90 ordinary. However, unlike the preceding two cases, there are [*three*]{} different types of points in our 32-nionic PG$(4,2)$: ten $\alpha$-points, $$\alpha \equiv \{7,11,13,14,19,21,22,25,26,28\},$$ each of which is on nine defective and six ordinary lines; fifteen $\beta$-points, $$\beta \equiv \{3,5,6,9,10,12,15,17,18,20,23,24,27,29,30\},$$ each of which is on seven defective and eight ordinary lines; and six $\gamma$-points, $$\gamma \equiv \{1,2,4,8,16,31\},$$ each of them being on fifteen ordinary (and, hence, on zero defective) lines. This stratification of the point-set of PG$(4,2)$ leads, in turn, to two different kinds of defective lines and three distinct kinds of ordinary lines. Concerning the former, there are 45 of them of type $\{\alpha,\alpha,\beta\}$, namely $$\{3,13,14\}, \{3,21,22\}, \{3,25,26\},$$ $$\{5,11,14\}, \{5,19,22\}, \{5,25,28\},$$ $$\{6,11,13\}, \{6,19,21\}, \{6,26,28\},$$ $$\{7,9,14\}, \{7,10,13\}, \{7,11,12\}, \{7,17,22\}, \{7,18,21\}, \{7,19,20\},$$ $$\{7,25,30\}, \{7,26,29\}, \{7,27,28\},$$ $$\{9,19,26\}, \{9,21,28\},$$ $$\{10,19,25\}, \{10,22,28\},$$ $$\{11,17,26\}, \{11,18,25\}, \{11,19,24\}, \{11,21,30\}, \{11,22,29\}, \{11,23,28\},$$ $$\{12,21,25\}, \{12,22,26\},$$ $$\{13,17,28\}, \{13,19,30\}, \{13,20,25\}, \{13,21,24\}, \{13,22,27\}, \{13,23,26\},$$ $$\{14,18,28\}, \{14,19,29\}, \{14,20,26\}, \{14,21,27\}, \{14,22,24\}, \{14,23,25\},$$ $$\{15,19,28\}, \{15,21,26\}, \{15,22,25\},$$ and 20 of type $\{\beta,\beta,\beta\}$, namely $$\{3,5,6\}, \{3,9,10\}, \{3,17,18\}, \{3,29,30\},$$ $$\{5,9,12\}, \{5,17,20\}, \{5,27,30\},$$ $$\{6,10,12\}, \{6,18,20\}, \{6,27,29\},$$ $$\{9,17,24\}, \{9,23,30\},$$ $$\{10,18,24\}, \{10,23,29\},$$ $$\{12,20,24\}, \{12,23,27\},$$ $$\{15,17,30\}, \{15,18,29\}, \{15,20,27\}, \{15,23,24\}.$$ As per the latter, one finds 15 of them of type $\{\beta,\beta,\beta\}$, namely $$\{3,12,15\}, \{3,20,23\}, \{3,24,27\},$$ $$\{5,10,15\}, \{5,18,23\}, \{5,24,29\},$$ $$\{6,9,15\}, \{6,17,23\}, \{6,24,30\},$$ $$\{9,18,27\}, \{9,20,29\},$$ $$\{10,17,27\}, \{10,20,30\},$$ $$\{12,17,29\}, \{12,18,30\},$$ 60 of type $\{\alpha,\beta,\gamma \}$, namely $$\{1, 6, 7\}, \{1, 10, 11\}, \{1, 12, 13\}, \{1, 14, 15\}, \{1, 18, 19\}, \{1, 20, 21\},$$ $$\{1, 22, 23\}, \{1, 24, 25\},
\{1, 26, 27\}, \{1, 28, 29\},$$ $$\{2, 5, 7\}, \{2, 9, 11\}, \{2, 12, 14\}, \{2, 13, 15\}, \{2, 17, 19\}, \{2, 20, 22\},$$ $$\{2, 21, 23\}, \{2, 24, 26\},
\{2, 25, 27\}, \{2, 28, 30\},$$ $$\{3, 4, 7\}, \{3, 8, 11\}, \{3, 16, 19\}, \{3, 28, 31\},$$ $$\{4, 9, 13\}, \{4, 10, 14\}, \{4, 11, 15\}, \{4, 17, 21\}, \{4, 18, 22\},$$ $$\{4, 19, 23\},
\{4, 24, 28\}, \{4, 25, 29\},
\{4, 26, 30\},$$ $$\{5, 8, 13\}, \{5, 16, 21\}, \{5, 26, 31\},$$ $$\{6, 8, 14\}, \{6, 16, 22\}, \{6, 25, 31\},$$ $$\{7, 8, 15\}, \{7, 16, 23\}, \{7, 24, 31\},$$ $$\{8, 17, 25\}, \{8, 18, 26\}, \{8, 19, 27\}, \{8, 20, 28\}, \{8, 21, 29\}, \{8, 22, 30\},$$ $$\{9, 16, 25\}, \{9, 22, 31\},$$ $$\{10, 16, 26\}, \{10, 21, 31\},$$ $$\{11, 16, 27\}, \{11, 20, 31\},$$ $$\{12, 16, 28\}, \{12, 19, 31\},$$ $$\{13, 16, 29\}, \{13, 18, 31\},$$ $$\{14, 16, 30\}, \{14, 17, 31\},$$ and, finally, 15 of type $\{\beta, \gamma, \gamma \}$, namely
$$\{1, 2, 3\}, \{1, 4, 5\}, \{1, 8, 9\}, \{1, 16, 17\}, \{1, 30, 31\},$$ $$\{2, 4, 6\}, \{2, 8, 10\}, \{2, 16, 18\}, \{2, 29, 31\},$$ $$\{4, 8, 12\}, \{4, 16, 20\}, \{4, 27, 31\},$$ $$\{8, 16, 24\}, \{8, 23, 31\},$$ $$\{5, 16, 31\}.$$
A point-line configuration $\mathcal{C}_5$ whose Veldkamp space accounts for these stratifications of both the point- and line-set of our 32-nionic PG$(4,2)$ is of type $(15_4,20_3)$.[^2] This configuration is formed within our PG$(4,2)$ by 15 $\beta$-points and 20 defective lines of $\{\beta,\beta,\beta\}$ type and its structure is sketched in Figure \[fig1\]. It is a rather easy task to verify that this particular $(15_4,20_3)$-configuration possesses 31 distinct geometric hyperplanes that fall into three different types. A type-one hyperplane consists of a pair of skew lines at maximum distance from each other; there are, as depicted in Figure \[fig2\], ten hyperplanes of this type and they correspond to $\alpha$-points of PG$(4,2)$. A type-two hyperplane features a point and all the points not collinear with it, the latter forming — not surprisingly — the Pasch configuration; there are, as shown in Figure \[fig3\], fifteen hyperplanes of this type and their counterparts are $\beta$-points of PG$(4,2)$. A type-three hyperplane is identical with the Desargues configuration; we find, as portrayed in Figure \[fig4\], altogether six guys of this type, each standing for a $\gamma$-point of PG$(4,2)$.
![An illustration of the structure of the $(15_4,20_3)$-configuration, built around the model of the Desargues configuration shown in Figure \[fgr2\]. The five points added to the Desargues configuration are the three peripheral points and the red and blue point in the center. The ten lines added are three lines denoted by red color, three blue lines, three lines joining pairwise the three peripheral points and the line that comprises the three points in the center of the figure, that is the ones represented by a bigger red circle, a smaller blue circle and a medium-sized black one.[]{data-label="fig1"}](fig1.eps){width="9truecm"}
![The ten geometric hyperplanes of the $(15_4,20_3)$-configuration of type one; the number below a subfigure indicates how many hyperplane’s copies we get by rotating the particular subfigure through 120 degrees around its center.[]{data-label="fig2"}](fig2.eps){width="14truecm"}
![The fifteen geometric hyperplanes of the $(15_4,20_3)$-configuration of type two.[]{data-label="fig3"}](fig3.eps){width="14truecm"}
![The six geometric hyperplanes of the $(15_4,20_3)$-configuration of type three.[]{data-label="fig4"}](fig4.eps){width="14truecm"}
![The five types of Veldkamp lines of the $(15_4,20_3)$-configuration. Here, unlike Figures 8 to 10, each representative of a geometric hyperplane is drawn separately and different colors are used to distinguish between different hyperplane types: red is reserved for type one, yellow for type two and blue for type three hyperplanes. As before, black color denotes the core of a Veldkamp line, that is the elements common to all the three hyperplanes comprising it.[]{data-label="fig5"}](fig5.eps){width="14truecm"}
We also find that our $(15_4,20_3)$-configuration yields 155 Veldkamp lines that are, as expected, of five different types. A type-I Veldkamp line, shown in Figure \[fig5\]a, features two hyperplanes of type one and a type-two hyperplane and its core consists of two points that are at maximum distance from each other; there are ${10 \choose 2} = 15 \times 6/2 = 45$ Veldkamp lines of this type and they correspond to defective lines of PG$(4,2)$ of type $\{\alpha,\alpha,\beta\}$. A type-II Veldkamp line, featured in Figure \[fig5\]b, is composed of three hyperplanes of type two that share three points on a common line; there are, obviously, 20 Veldkamp lines of this type, having for their counterparts defective lines of PG$(4,2)$ of type $\{\beta, \beta, \beta\}$. A type-III Veldkamp line, portrayed in Figure \[fig5\]c, also consists of three hyperplanes of type two, but in this case the three common points are pairwise at maximum distance from each other; a quick count leads to 15 Veldkamp lines of this type, these being in a bijection with 15 ordinary lines of PG$(4,2)$ of type $\{\beta, \beta, \beta\}$. Next, it comes a type-IV Veldkamp line, depicted in Figure \[fig5\]d, which exhibits a hyperplane of each type and whose core is composed of a line and a point at the maximum distance from it; since for each line of our $(15_4,20_3)$-configuration there are three points at maximum distance from it, there are $20 \times 3 = 60$ Veldkamp lines of this type, having their twins in ordinary lines of PG$(4,2)$ of type $\{\alpha,\beta,\gamma \}$. Finally, we meet a type-V Veldkamp line, sketched in Figure \[fig5\]e, which is endowed with two hyperplanes of type three and a single one of type two, and whose core is isomorphic to the Pasch configuration; hence, we have ${6 \choose 2} = 15$ Veldkamp lines of this type, being all representatives of ordinary lines of PG$(4,2)$ of type $\{\beta, \gamma, \gamma \}$.
Before embarking on the final case to be dealt with in detail, it is worth having a closer look at our $(15_4,20_3)$-configuration and pointing out its intimate relation with famous Pascal’s Mystic Hexagram. If six arbitrary points are chosen on a conic section and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet in three points that lie on a straight line, the latter being called the Pascal line. Taking the permutations of the six points, one obtains 60 different hexagons. Thus, the so-called complete Pascal hexagon determines altogether 60 Pascal lines, which generate a remarkable configuration of 146 points and 110 lines called the [*hexagrammum mysticum*]{}, or the complete Pascal figure (for the most comprehensive, applet-based representation of this remarkable geometrical object, see [@norma]). Both the points and lines of the complete Pascal figure split into several distinct families, usually named after their discoverers in the first half of the 19-th century. We are concerned here with the 15 Salmon points and the 20 Cayley lines (see, e.g. [@lord; @cory]) which form a $(15_4,20_3)$-configuration. This configuration is discussed in some detail in [@bogepi], where it is also depicted (Figure 6) and called the [*Cayley-Salmon*]{} $(15_4,20_3)$-configuration. And it is precisely this Cayley-Salmon $(15_4,20_3)$-configuration which our 32-nionic $(15_4,20_3)$-configuration is isomorphic to. The same configuration is also portrayed in Figure 8 of [@norma]. In the latter work, two different views/interpretations of the configuration are also mentioned. One is as three pairwise-disjoint triangles that are in perspective from a line, in which case the centers of perspectivity are guaranteed by Desargues’ theorem to also lie on a line; we just stress here that these two lines form a geometric hyperplane (of type one, see Figure \[fig2\]). The other view of the figure takes any point of the configuration to be the center of perspectivity of two quadrangles whose six pairs of corresponding sides meet necessarily in the points of a Pasch configuration; again, the point and the associated Pasch configuration form a geometric hyperplane (of type two, see Figure \[fig3\]). Obviously, we can offer one more view of the configuration, that stemming from the existence of type-three hyperplanes, namely as the incidence sum of a Desargues configuration and three triangles on a commmon side (see Figure \[fig4\]).
64-nions and a $(21_5,35_3)$-configuration
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The final algebra we shall treat in sufficient detail is $A_6$, or the 64-nions. From the corresponding multiplication table, which due to its size we do not show here but which is freely available at [http://jjj.de/tmp-zero-divisors/mult-table-64-ions.txt]{}, we infer the existence of 651 distinguished triples of imaginary units. Regarding the 63 imaginary units of 64-nions as points and the 651 distinguished triples of them as lines, we obtain a point-line incidence structure where each line has three points and each point is on 31 lines, and which is isomorphic to PG$(5,2)$. Following the usual procedure, we find that 350 lines of this space are defective and 301 ordinary. Likewise the preceding case, we encounter [*three*]{} different types of points in our 64-nionic PG$(5,2)$: 35 $\alpha$-points, each of which is on 21 defective and 10 ordinary lines; 21 $\beta$-points, each of which is on 15 defective and 16 ordinary lines; and seven $\gamma$-points, each of them being on 31 ordinary (and, hence, on zero defective) lines. This stratification of the point-set of PG$(5,2)$ leads, in turn, to three different kinds of defective lines and four distinct kinds of ordinary lines. Out of 350 defective lines, we find 105 of type $\{\alpha, \alpha, \alpha\}$, 210 of type $\{\alpha, \alpha, \beta\}$ and 35 of type $\{\beta, \beta, \beta\}$. On the other hand, 301 ordinary lines are partitioned into 105 guys of type $\{\alpha, \beta, \beta\}$, 70 of type $\{\alpha, \alpha, \gamma\}$, 105 of type $\{\alpha, \beta, \gamma\}$ and 21 of type $\{\beta, \gamma, \gamma\}$.
The Veldkamp space mimicking such a fine structure of PG$(5,2)$ is that of a particular $(21_5,35_3)$-configuration, $\mathcal{C}_6$, that also lives in our PG$(5,2)$ and whose points are the 21 $\beta$-points and whose lines are the 35 defective lines of $\{\beta, \beta, \beta\}$ type. To visualise this configuration, we build it around the model of the Cayley-Salmon $(15_4,20_3)$-configuration of 32-nions shown in Figure \[fig1\]. Given the Cayley-Salmon configuration, there are six points and 15 lines to be added to yield our $(21_5,35_3)$-configuration, and this is to be done in such a way that the configuration we started with forms a geometric hyperplane in it. As putting all the lines into a single figure would make the latter look rather messy, in Figure \[f64\] we briefly illustrate this construction by drawing six different figures, each featuring all six additional points (gray) but only five out of 15 additional lines (these lines being also drawn in gray color), namely those passing through a selected additional point (represented by a doubled circle). Employing this handy diagrammatical representation, one can verify that our $(21_5,35_3)$-configuration exhibits 63 geometric hyperplanes that fall into three distinct types. A type-one hyperplane consists of a line and its complement, which is the Pasch configuration; there are 35 distinct hyperplanes of this form, each corresponding to an $\alpha$-point of our PG$(5,2)$. A type-two hyperplane comprises a point and its complement, which is the Desargues configuration; there are 21 hyperplanes of this form, each having a $\beta$-point for its PG$(5,2)$ counterpart. Finally, a type-three hyperplane is isomorphic to the Cayley-Salmon configuration; there are seven distinct guys of this type, each answering to a $\gamma$-point of the PG$(5,2)$. We leave it with the interested reader to verify by themselves that the Veldkamp space of our $(21_5,35_3)$-configuration indeed features 651 lines that do fall into the above-mentioned seven distinct kinds.
![An illustration of the structure of the $(21_5,35_3)$-configuration, built around the model of the Cayley-Salmon $(15_4,20_3)$-configuration shown in Figure \[fig1\].[]{data-label="f64"}](f64a.eps "fig:"){width="4truecm"}![An illustration of the structure of the $(21_5,35_3)$-configuration, built around the model of the Cayley-Salmon $(15_4,20_3)$-configuration shown in Figure \[fig1\].[]{data-label="f64"}](f64b.eps "fig:"){width="4truecm"}![An illustration of the structure of the $(21_5,35_3)$-configuration, built around the model of the Cayley-Salmon $(15_4,20_3)$-configuration shown in Figure \[fig1\].[]{data-label="f64"}](f64c.eps "fig:"){width="4truecm"}
![An illustration of the structure of the $(21_5,35_3)$-configuration, built around the model of the Cayley-Salmon $(15_4,20_3)$-configuration shown in Figure \[fig1\].[]{data-label="f64"}](f64d.eps "fig:"){width="4.4truecm"}![An illustration of the structure of the $(21_5,35_3)$-configuration, built around the model of the Cayley-Salmon $(15_4,20_3)$-configuration shown in Figure \[fig1\].[]{data-label="f64"}](f64e.eps "fig:"){width="4truecm"}![An illustration of the structure of the $(21_5,35_3)$-configuration, built around the model of the Cayley-Salmon $(15_4,20_3)$-configuration shown in Figure \[fig1\].[]{data-label="f64"}](f64f.eps "fig:"){width="4truecm"}
As in the previous two cases, we shall briefly describe a couple of interesting views of our $(21_5,35_3)$-configuration, both related to type-one hyperplanes. The first one is as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration, the line and the Pasch configuration comprising a geometric hyperplane (compare with the first view of both the Desargues and the Cayley-Salmon configuration). This is sketched in Figure \[f64view\], where the four triangles are denoted, in boldfacing, by green, red, yellow and blue color, the line of perspectivity by boldfaced gray color, and the points of perspectivity of pairs of triangles (together with the corresponding lines they lie on and that are also boldfaced) by black color. The other view is as three complete quadrangles that are pairwise in perspective in such a way that the three points of perspectivity lie on a line and where the six triples of their corresponding sides meet at points located on a Pasch configuration, again the line and the Pasch configuration forming a geometric hyperplane (compare with the second view of both the Desargues and the Cayley-Salmon configuration).
![A ‘generalized Desargues’ view of the $(21_5,35_3)$-configuration.[]{data-label="f64view"}](f64view.eps){width="6truecm"}
$2^N$-nions and a $\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configuration
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At this point it is quite easy to spot the general pattern. If one also includes the trivial cases of complex numbers ($N=1$), where the relevant geometry is just a single point i.e. the $(1_0,0_3)$-configuration, and quaternions ($N=2$), whose geometry is a single line i.e. the $(3_1,1_3)$-configuration, we obtain the following nested sequence of configurations whose Veldkamp spaces capture the stratification/partition of the point- and line-sets of the $2^N$-nionic PG$(N-1,2)$, $N$ being a positive integer, $$(1_0,0_3),$$ $$(3_1,1_3),$$ $$(6_2,4_3),$$ $$(10_3,10_3),$$ $$(15_4,20_3),$$ $$(21_5,35_3),$$ $$\ldots,$$ $$\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right),$$ $$\ldots.$$ It is curious to notice that the first entry represents a [*triangular*]{} number, while the second one is a [*tetrahedral*]{} number, or triangular pyramidal number. In other words, we get a nested sequence of [*binomial*]{} $({r+k-1 \choose r}_r, {r+k-1 \choose k}_k)$-configurations with $r = N-1$ and $k=3$, whose properties have very recently been discussed in a couple of interesting papers [@prap; @pepr]. The first few configurations are shown, in a form where the configurations are [*nested*]{} inside each other, in Figure \[nested2\].
![A nested hierarchy of finite $\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configurations of $2^N$-nions for $1 \leq N \leq 5$ when embedded in the Cayley-Salmon configuration ($N=5$).[]{data-label="nested2"}](nested2.eps){width="14truecm"}
A particular character of this nesting is reflected in the structure of geometric hyperplanes. Denoting our generic $\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configuration by ${\cal C}_N$, we can express the types of geometric hyperplanes of the above-discussed cases in a compact form as follows
--------------- ---------------- --------------------------------- --------------------------------- -- --
${\cal C}_1$: $\varnothing$,
${\cal C}_2$: ${\cal C}_1$,
${\cal C}_3$: ${\cal C}_2$, ${\cal C}_1 \sqcup {\cal C}_1$,
${\cal C}_4$: ${\cal C}_3$, ${\cal C}_2 \sqcup {\cal C}_1$,
${\cal C}_5$: ${\cal C}_4$, ${\cal C}_3 \sqcup {\cal C}_1$, ${\cal C}_2 \sqcup {\cal C}_2$,
${\cal C}_6$: ${\cal C}_5$, ${\cal C}_4 \sqcup {\cal C}_1$, ${\cal C}_3 \sqcup {\cal C}_2$,
--------------- ---------------- --------------------------------- --------------------------------- -- --
which implies the following generic hyperplane compositions
--------------- ------------------- ------------------------------------- ------------------------------------- ---- -----------------------------------------------------------
${\cal C}_N$: ${\cal C}_{N-1}$, ${\cal C}_{N-2} \sqcup {\cal C}_1$, ${\cal C}_{N-3} \sqcup {\cal C}_2$, …, ${\cal C}_{\frac{N}{2}} \sqcup {\cal C}_{\frac{N}{2}-1}$,
--------------- ------------------- ------------------------------------- ------------------------------------- ---- -----------------------------------------------------------
or
--------------- ------------------- ------------------------------------- ------------------------------------- ---- -------------------------------------------------------------------------------------
${\cal C}_N$: ${\cal C}_{N-1}$, ${\cal C}_{N-2} \sqcup {\cal C}_1$, ${\cal C}_{N-3} \sqcup {\cal C}_2$, …, ${\cal C}_{\lfloor\frac{N}{2}\rfloor} \sqcup {\cal C}_{\lfloor\frac{N}{2}\rfloor}$,
--------------- ------------------- ------------------------------------- ------------------------------------- ---- -------------------------------------------------------------------------------------
according as $N$ is even or odd, respectively; here, the symbol ‘$\sqcup$’ stands for a disjoint union of two sets.
In the spirit of previous sections, let us also have a closer look at the nature of our generic ${\cal C}_N$. To this end, we first recall the following observations. ${\cal C}_4$, the Desargues configuration, can be viewed as ($4-2=$) two triangles in perspective from a line which are also perspective from a point, that is ${\cal C}_1$; the line and the point form a geometric hyperplane of ${\cal C}_4$. Next, ${\cal C}_5$, the Cayley-Salmon configuration, admits a view as ($5-2=$) three triangles in perspective from a line where the points of perspectivity of three pairs of them are on a line, [*aka*]{} ${\cal C}_2$; the two lines form a geometric hyperplane of ${\cal C}_5$. Finally, ${\cal C}_6$, our $(21_5,35_3)$-configuration, can be treated as ($6-2=$) four triangles in perspective from a line where the points of perspectivity of six pairs of them lie on a Pasch configuration, [*alias*]{} ${\cal C}_3$; the line and the Pasch configuration form a geometric hyperplane of ${\cal C}_6$. Generalizing these observations, we conjecture that for [*any*]{} $N \geq 4$, ${\cal C}_N$ can be regarded as $N-2$ triangles that are in perspective from a line in such a way that the points of perspectivity of ${N-2 \choose 2}$ pairs of them form the configuration isomorphic to ${\cal C}_{N-3}$, where the latter and the axis of perspectivity form a geometric hyperplane of ${\cal C}_N$.
Next, we invoke the concept of combinatorial Grassmannian (see, e.g., [@pra1; @pra2]). Briefly, a combinatorial Grassmannian $G_k(|X|)$, where $k$ is a positive integer and $X$ is a finite set, is a point-line incidence structure whose points are $k$-element subsets of $X$ and whose lines are $(k + 1)$-element subsets of $X$, incidence being inclusion. It is known [@pra1] that if $|X| =
N+1$ and $k=2$, $G_2(N+1)$ is a binomial $\left({N+1 \choose 2}_{N-1}, {N+1 \choose
3}_{3}\right)$-configuration; in particular, $G_2(4)$ is the Pasch configuration, $G_2(5)$ is the Desargues configuration and $G_2(N+1)$’s with $N \geq 5$ are called [*generalized*]{} Desargues configurations. Now, from our detailed examination of the four cases it follows that ${\cal C}_3$, ${\cal C}_4$, ${\cal C}_5$, ${\cal C}_6$, …, ${\cal C}_N$ are endowed with 1, 5, 15, 35, …, ${N+1 \choose 4}$ Pasch configurations. And as ${N+1 \choose 4}$ is also the number of Pasch configurations in $G_2(N+1)$, $N \geq 3$, we are also naturally led to conjecture that ${\cal C}_N \cong G_2(N+1)$. From what we have found in the previous sections it follows that this property indeed holds for $1 \leq N \leq 6$, being illustrated for $N=5$ and $N=6$ in Figure \[grass\].
![[*Left*]{}: – A diagrammatical proof of the isomorphism between ${\cal C}_5$ and $G_2(6)$. The points of ${\cal C}_5$ are labeled by pairs of elements from the set $\{1,2,\ldots, 6\}$ in such a way that each line of the configuration is indeed of the form $\{\{a,b\}, \{a,c\}, \{b,c\}\}$, $a \neq b \neq c \neq a$. [*Right*]{}: – A pictorial illustration of ${\cal C}_6 \cong G_2(7)$. Here, the labels of six additional points are only depicted, the rest of the labeling being identical to that shown in the left-hand side figure.[]{data-label="grass"}](grassl.eps "fig:"){width="7.0truecm"}![[*Left*]{}: – A diagrammatical proof of the isomorphism between ${\cal C}_5$ and $G_2(6)$. The points of ${\cal C}_5$ are labeled by pairs of elements from the set $\{1,2,\ldots, 6\}$ in such a way that each line of the configuration is indeed of the form $\{\{a,b\}, \{a,c\}, \{b,c\}\}$, $a \neq b \neq c \neq a$. [*Right*]{}: – A pictorial illustration of ${\cal C}_6 \cong G_2(7)$. Here, the labels of six additional points are only depicted, the rest of the labeling being identical to that shown in the left-hand side figure.[]{data-label="grass"}](grassr.eps "fig:"){width="6truecm"}
Conclusion
==========
An intriguing finite-geometrical underpinning of the multiplication tables of Cayley-Dickson algebras $A_N$, $3 \leq N \leq 6$, has been found that admits generalization to any higher-dimensional $A_N$. This started with an observation that the multiplication properties of imaginary units of the algebra $A_N$ are encoded in the structure of the projective space PG$(N-1,2)$. Next, this space was shown to possess a refined structure stemming from particular properties of triples of imaginary units forming its lines. To account for this refinement, we employed the concept of Veldkamp space of point-line incidence structure and found out the latter to be a binomial $\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configuration ${\cal C}_N$; in particular, ${\cal C}_3$ (octonions) was found to be isomorphic to the Pasch $(6_2,4_3)$-configuration, ${\cal C}_4$ (sedenions) to the famous Desargues $(10_3)$-configuration, ${\cal C}_5$ (32-nions) to the Cayley-Salmon $(15_4,20_3)$-configuration found in the well-known Pascal mystic hexagram and ${\cal C}_6$ (64-nions) was shown to be identical with a particular $(21_5,35_3)$-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. These configurations are seen to form a remarkable nested pattern, where ${\cal C}_{N-1}$ is embedded in ${\cal C}_N$ as its geometric hyperplane, that naturally reflects the spirit of the Cayley-Dickson recursive construction of corresponding algebras.
It is a well-known fact that the only first four algebras $A_N$, $0 \leq N \leq 3$, are ‘well-behaving’ in the sense of being normed, alternative and devoid of zero-divisors — the facts that are frequently offered as an explanation why a relatively little attention has been paid so far to their higher-dimensional cousins, these latter being even regarded by some scholars as ‘pathological.’ It may well be that our finite-geometric, Veldkamp-space-based approach will be able to shed a novel, unexpected light at this issue as it is only starting with $N=4$ when ${\cal C}_N$ is found to feature a ‘generalized Desargues property’ in the sense that it can be interpreted as $N-2$ triangles that are in perspective from a line in such a way that the points of perspectivity of ${N-2 \choose 2}$ pairs of them form the configuration isomorphic to ${\cal C}_{N-3}$. Or, in a slightly different form, it is only for $N \geq 4$ when ${\cal C}_N$ contains [*Desargues*]{} configurations, these occurring as components of its geometric hyperplanes at that.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by the VEGA Grant Agency, Project 2/0003/13, as well as by the Austrian Science Fund (Fonds zur Förderung der Wissenschaftlichen Forschung (FWF)), Research Project M1564–N27. We would like to express our sincere gratitude to Hans Havlicek (Vienna University of Technology) for a number of critical remarks on the first draft of the paper. Our special thanks go to Jörg Arndt who computed for us multiplication tables of octonions, sedenions, 32-nions and 64-nions. We are also very grateful to Boris Odehnal (University of Applied Arts, Vienna) for creating nice computer versions of several figures.
[10]{} =-2pt N. Jacobson, Basic Algebra I, W. H. Freeman, San Francisco, 1974, 417–427. R. D. Schafer, Introduction to Non-Associative Algebras, Dover, New York, 1995. G. Moreno, Hopf construction map in higher dimensions, Bol. Soc. Mat. Mexicana 10 (2004) 383–397. J. Arndt, Matters Computational: Ideas, Algorithms, Source Code, Springer, Heidelberg, 2011, Chapter 39. J. C. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002) 145–205. C. Culbert, Cayley-Dickson algebras and loops, J. Gen. Lie Theory Appl. 1 (2007) 1–17. E. E. Shult, Points and Lines: Characterizing the Classical Geometries, Springer, Berlin – Heidelberg, 2011. B. Grünbaum, Configurations of Points and Lines, Graduate Studies in Mathematics 103, American Mathematical Society, Providence, 2009. F. Buekenhout and A. M. Cohen, Diagram Geometry: Related to Classical Groups and Buildings, Springer, Berlin – Heidelberg, 2013, Sec. 8.2. M. J. Grannell and T. S. Griggs, The Pasch configuration, Encyclopaedia of Mathematics, available at http://www.encyclopediaofmath.org/index.php/Pasch$\underline{~~}$configuration. B. Polster, A Geometrical Picture Book, Springer Verlag, New York, 1998, Chapter 5. T. Pisanski and B. Servatius, Configurations from a Graphical Viewpoint, Birkhäuser, Basel, 2013, Chapter 5.3.4. M. Saniga, M. Planat, P. Pracna and H. Havlicek, The Veldkamp space of two-qubits, SIGMA 3 (2007) 075 (7 pages). J. Ch. Fisher and N. Fuller, The complete Pascal figure graphically presented, available at http://www.math.uregina.ca/$\widetilde{~}$fisher/Norma/paper.html. E. Lord, Symmetry and Pattern in Projective Geometry, Springer, Heidelberg, 2013, Chapter 4.9. J. Conway and A. Ryba, The Pascal mysticum demystified, Mathematical Intelligencer 34 (2012) 4–8. M. Boben, G. Gévay and T. Pisanski, Danzer’s configuration revisited, preprint arXiv:1301.1067. M. Prażmowska and K. Prażmowski, Binomial configurations which contain complete graphs: $K_n$-subgraphs of a $\left(\binom{n+1}{2}_{n-1} \binom{n+1}{3}_3\right)$-configuration, preprint arXiv:1404.4064. K. Petelczyc, M. Prażmowska and K. Prażmowski, Complete classification of the $(15_4 20_3)$-configurations with at least three $K_5$-graphs, preprint arXiv:1404.4352. M. Prażmowska, Multiple perspectives and generalizations of the Desargues configuration, Demonstratio Math. 39 (2006) 887–906. M. Prażmowska and K. Prażmowski, A generalization of Desargues and Veronese configurations, Serdica Math. J. 32 (2006) 185–208.
[^1]: Although these properties will explicitly be illustrated only for the cases $3 \leq N \leq 6$, due to the nested structure of the algebras they must be exhibited by any other higher-dimensional $A_N$.
[^2]: It is worth mentioning here that there exists another remarkable configuration whose Veldkamp space does the same job for us, namely the generalized quadrangle of order two, also known as the Cremona-Richmond $(15_3)$-configuration (see, e.g., [@twoq]). However, this configuration is [*triangle-free*]{} and so it can$not$ contain the Desargues configuration as dictated by the nesting property of the Cayley-Dickson algebras.
|
---
abstract: 'Next-generation high-power laser systems that can be focused to ultra-high intensities exceeding $10^{23}$ W/cm$^2$ are enabling new physics regimes and applications. The physics of how these lasers interact with matter is highly nonlinear, relativistic, and it involves non-classical processes such as radiation reaction and quantum effects. The current tool of choice for modeling these interactions is the particle-in-cell (PIC) method. In the presence of strong electromagnetic fields, the motion of charged particles and their spin is effected by radiation reaction (either the semi-classical or quantum limit). Standard (PIC) codes usually use Boris or similar operator-splitting methods to advance the particle in standard phase space. These methods have been shown to fail in the strong field regime and it is not straightforward to extend them to include radiation reaction. In addition, some problems require tracking the spin of the particle which means that the phase space of the particle is now nine-dimensional, i.e., $(\bm{x},\bm{u},\bm{s})$ (9D). Therefore, numerical algorithms that enable high-fidelity modeling of the 9D phase space in the strong-field regime (where both the spin and momentum evolution is effected by radiation reaction) are required. We present a new particle pusher that works in 6D and 9D phase space based on the analytical and not leapfrog solutions to the momentum and spin advance due to the Lorentz force, together with the semi-classical form of radiation reaction in Landau-Lifshitz, and the Bargmann-Michel-Telegdi equation for the evolution of the spin. Analytic solutions for the position advance are also obtained but these are not amenable to the staggering of space and time in the PIC code. These analytical solutions are obtained by only assuming a locally uniform and constant electromagnetic field during a time step. The solutions can provide the 9D phase space advance in terms of the duration of the particles’ proper time and then a mapping is used to determine the proper time step duration for each particle as a function of the lab frame time step. Owing to the analytical integration of particle trajectory and spin orbit, the constraint on the time step needed to resolve the trajectory in ultra-high fields can be greatly reduced. The time step required in a PIC code for accurately advancing the fields may provide additional constraints. We present examples of the motion of single particle simulations to show the proposed particle pusher can greatly improve the accuracy of the particle trajectory in 6D or 9D phase space for given laser fields. We also implemented the new pusher into the state-of-art PIC code <span style="font-variant:small-caps;">Osiris</span> to include the effects of the field solver for lasers in vacuum. Examples are presented that show the proposed pusher provides improvement for a given time step. A discussion on the numerical efficiency of the proposed pusher is also provided.'
address:
- 'Department of Electrical Engineering, University of California Los Angeles, Los Angeles, CA 90095, USA'
- 'Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA'
- 'GOLP/Instituto de Plasma e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal'
- 'ISCTE - Instituto Universitário de Lisboa, 1649–026, Lisbon, Portugal'
author:
- Fei Li
- 'Viktor K. Decyk'
- 'Kyle G. Miller'
- Adam Tableman
- 'Frank S. Tsung'
- Marija Vranic
- 'Ricardo A. Fonseca'
- 'Warren B. Mori'
bibliography:
- 'refs.bib'
title: 'Accurately simulating nine-dimensional phase space of relativistic particles in strong field'
---
particle pusher ,laser-plasma interaction ,radiation reaction ,Landau-Lifshitz equation ,Bargmann-Michel-Telegdi equation ,spin precession ,particle-in-cell algorithm
Introduction
============
With the recent availability of petawatt class lasers and roadmap for multi-petawatt class laser systems [@ELI; @XCELS; @SULF], laser intensities exceeding $10^{23}$ W/cm$^{-2}$ will soon become accessible. This will open a new door for research avenues in plasma physics, including plasma-based acceleration [@tajima1979; @chen1985; @joshi2003; @lu2007] in the strong-field regime, the coupling of laser-plasma interactions and quantum electrodynamic [@vranic2016a], and the ability to mimic some astrophysical phenomena (e.g., gamma-ray bursts and supernova explosions) in the laboratory. The physics of how ultra-high intensity lasers interact with matter is highly nonlinear, relativistic, and it involves non-classical processes such as radiation reaction and quantum effects. Simulations will be a critical partner with experiments to unravel this physics. Electromagnetic particle-in-cell (PIC) algorithm [@dawson1983; @hockney1988; @birdsall2005] has been successfully applied to the research of plasma or charged particle beams interacting with radiations for nearly half a century. With moderate radiation (laser) parameters, e.g., $eA/(m_ec^2)\gtrsim1$ where $A$ is the vector potential of laser, PIC simulations have been proved a reliable tool. However, in the strong-field regime where $eA/(m_ec^2)\gg1$ accurate modeling becomes much more challenging. Developing high-fidelity PIC simulation algorithms requires a comprehensive and deep understanding of each aspect of the numerical algorithm and the physical problem itself. To improve the simulation accuracy and reliability, much effort has already been undertaken to mitigate various numerical errors including improper numerical dispersion, errors to the Lorentz force for a relativistic particle interacting with a laser, numerical Cerenkov radiation and an associated instability [@godfrey2013; @xu2013; @yu2015a; @yu2015b; @li2017], finite-grid instability [@langdon1970; @okuda1972; @meyers2015; @huang2016] and spurious fields surrounding relativistic particles [@xu2020].
In this article, we address inaccuracies and challenges for the particle pusher used as part of a PIC code. The pusher has been found to be one of the major factors that preventing high-fidelity PIC simulations in strong-field regime. Most PIC codes use the standard Boris scheme [@boris1972] or its variants [@vay2008; @higuera2017] for the particle push. These later variants correct a shortcoming of the Boris push when a particle moves relativistically where $\bm E + \bm v \times \bm B \approx 0$. In the standard Boris split algorithm, the velocity can change even when the Lorentz force vanishes.
Gordon et al. [@gordon2017] showed that it is possible to construct an analytic or exact covariant non-splitting pusher. This method assumes the fields (forces) are constant during an interval of the proper time and then advances the particle momentum using the analytic solutions. Since this method pushes particle in proper time rather than in the observer’s time, it cannot be directly applied to PIC simulations but only be used for single-particle tracking. Gordon et al. also discussed how to include radiation reaction (RR). However, they used a form for RR that is challenging to incorporate. Pétri [@petri2019] (who does not seem to be aware of the work of Gordon et al.) recently proposed a different implementation of the exact pusher that included a mapping between the proper and observer time step, allowing the pusher to be applicable for PIC simulations. However, he did not consider RR. Although the motivation for developing an analytic pushers was to handle ultra-high fields, it will also accurately model the motion of a relativistic particle when $\bm E + \bm v \times \bm B \approx 0$.
In strong fields, the motion of charged particles will be significantly impacted by the radiative reaction force and its accompanied energy loss. Therefore, determining how to accurately model the RR effect is also of crucial importance for the strong-field regime. In the semi-classical perspective, the most prominent equation to describe radiation reaction is Lorentz-Abraham-Dirac (LAD) equation [@jackson2007]. However, the expression for RR provided by Landau-Lifshitz (LL) [@landau2013] is more practical than that of LAD because it avoids unphysical solutions. There exist various semi-classical models appropriate for numerical simulations based on the LL model and Vranic et al. [@vranic2016b] have made a careful comparison of them. The usual way to implement the additional radiative force in PIC codes is to first integrate the particle trajectory using a pusher (splitting or exact) solely for the Lorentz force and then to add an impulse from the radiative reaction force separately [@tamburini2010; @gordon2017]. This splitting process is simple to implement but it can lead to the accumulation of errors in simulation with a large number of time steps even though the radiation reaction effect is perturbative. The particle pusher proposed in this article extends the analytical solution to include the LL equation. The only assumption is that the fields are constant within each simulation time step. Therefore, the proposed pusher is free of numerical errors caused by splitting the operator for Lorentz force. The analytic pusher (or any sub-cycling approach [@arefiev2015]) will have errors to a particles trajectory if assuming constant fields during a time step. Therefore, the time step must properly resolve the evolution of the fields as well.
The PIC method is also beginning to be used to study the production of spin-polarized particle beams. Furthermore, there is also a growing interest [@li2019; @song2019; @geng2020] in how RR effects the particle spin dynamics in strong fields. Particle spin precession follows the Bargmann-Michel-Telegdi (BMT) equation [@jackson2007] in which the phase space $(\bm{x},\bm{u})$ is used to evaluate the spin $\bm{s}$. Therefore the effect of radiation reaction on modifying the phase space trajectories $(\bm{x},\bm{u})$ will be also manifested in the behavior of spin dynamics. However, there is far less literature directly related to the numerical schemes of the spin “push” than that of momentum “push”. A typical numerical method [@vieira2011] is similar to the Boris scheme. This method approximates the motion of spin orbit to be a pure rotation with a frequency which is evaluated with the time-centered values of the electromagnetic fields and particle momentum. This Boris-like scheme is subject to large numerical errors in the strong-field regime as will be shown later. In this article, we also derive semi-analytic solutions to the BMT equation by utilizing the analytic expressions of particle momentum without radiation reaction to advance the spin within an interval of time. The RR is then included as an impulse. In the next interval of time, the initial conditions for the analytical update of the momentum are thus different which then changes the spin evolution in the next interval of time. Obtaining a fully analytic solution to BMT in the presence of radiation reaction is extremely difficult and probably impossible; however, the radiative force can still be nearly accurately included via the aforementioned splitting correction method.
Although we are focusing on finding analytical solutions for both the momentum and position advance during intervals of time for where the fields are constant, it is still important to relate this to the leapfrog time indices in a standard PIC code. In most PIC codes (with the exception of the pseudo-spectral analytic time-domain (PSATD) codes [@vay2013]), the $\bm E$ and $\bm B$ are staggered in time. The momentum (proper velocity) and position are also staggered in time such that $\bm E$ and $\bm x$ are known on half-integer values of time and $\bm B$ and $\bm u$ are known at integer values (in the PSATD algorithm $\bm E$, $\bm B$ and $\bm x$ are known at half-integer values of time). In the push the fields are assumed constant during an interval of time between $n\Delta t$ and $(n+1)\Delta t$ and the fields are assumed to be given by their values at time $(n+1/2)\Delta t$ so that the particle positions are assumed known at $(n+1/2)\Delta t$. This implicitly assumes that the particle’s position does not change during this time. Under these conditions, we look for analytically advancing the momentum forward to $(n+1)\Delta t$. Although we can then advance the particle position analytically forward in time assuming ($d\bm x/dt=\bm u/\gamma$), this can only be done during intervals in time for which $\bm u$ is known, which in this case is until $(n+1)\Delta t$. Therefore, the analytical pusher is really doing an analytical advance of the particle momentum. However, this can still lead to significant improvements in accuracy as it is the momentum and not the position advance which can lead to large errors. This is easy to see by noting that the particle’s speed is limited to the speed of light, from which it follows that during a time step a particle can only move a fraction of a cell for any field strength. On the other hand, for ultra-high fields the change in the proper velocity during a time step can be many orders of magnitude, i.e., $\Delta u/u \gg1$ during a time step. In a later section, we show examples where the errors associated with the leapfrog advance in the position leads to noticeable errors. The use of the analytic solutions together with the PSATD or new concepts where the position and momentum are defined at the same time, may lead to new PIC time indexing algorithms.
The remaining of the paper is organized as follows: In section \[sec:soln\_eq\_motion\] and in \[sec:eigensys\], we derive the equations for an analytic push of the Lorentz force and introduce the mathematical formalism that can be extended to include the LL and BMT equations. In section \[sec:soln\_LL\], we use the mathematical formalism in ref[sec:eigensys]{} to obtain an analytic particle pusher for the 6D phase space including the LL equation. These solutions are exact if the fields are constant during an interval of proper time. In both sections, we also show how to obtain a mapping between the time step in the lab frame and the proper time step. In section \[sec:soln\_BMT\], we derive the analytic solutions to the BMT equation employing the analytical solutions of momentum obtained in section \[sec:soln\_LL\]. The workflow and implementation of the proposed pusher for the 9D phase space are described in section \[sec:algorithm\]. In section \[sec:sample\], we first compare the simulation results for a single particle in an ultra-intense laser field in vacuum using the proposed particle pusher with the standard Boris, Higuera-Cary schemes and the Boris-like scheme for the spin push. It is shown that the conventional numerical methods lead to large errors in the advance of 9D phase space while the proposed method provides accurate results. Then full PIC simulations using <span style="font-variant:small-caps;">Osiris</span> [@fonseca2002; @hemker2015] are conducted to investigate the difference of particle collective behavior using the proposed and conventional pushers. A summary and directions for future work are given in section \[sec:summary\].
Particle motion in constant and uniform fields without radiation reaction {#sec:soln_eq_motion}
=========================================================================
In this section, we will present a derivation of exact solutions to both the momentum and position updates for constant fields. Analytic expressions can be obtained in various ways. Pétri [@petri2019] introduced a Lorentz boosted frame where the $\bm E$ and $\bm B$ fields are parallel, for which analytic solutions are possible. These results then need to be transformed back to the lab frame. He also provided a mapping between the boosted (proper) and lab frame time steps. Gordon et al. [@gordon2017] showed that the momentum update can be solved analytically in a covariant form and described a matrix representation of the analytic solution. However, Gordon et al. did not provide a mapping between the proper and lab frame time steps nor address special cases that need to be considered. Although the underlying mathematics for obtaining solutions is different, each of the above approaches yields the same net result for the cases considered. However, the forms for the solutions can have different degrees of algorithmic complexity. Here, we will present another method for finding an analytic expression that is more compact and easier to implement into a PIC code. We use the covariant form for the equations of motion. More importantly, the mathematical formalism we use will be extended to include the LL and the BMT equations in later sections.
The covariant form of the equation of motion without radiation reaction is $${\frac{\text{d}u^\mu}{\text{d}\tau}} = \frac{q}{mc} F^\mu_{~\nu} u^\nu$$ where $u^\mu$ is the four-velocity, $q$ is the particle charge and $m$ is the particle mass. The field tensor is written as $$\label{eq:field_tensor}
F^{\mu}_{~\nu}=
\begin{pmatrix}
0 & E_1 & E_2 & E_3 \\
E_1 & 0 & B_3 & -B_2 \\
E_2 & -B_3 & 0 & B_1 \\
E_3 & B_2 & -B_1 & 0
\end{pmatrix}.$$ To avoid rewriting constant factors, we use normalized physical quantities, i.e. $\tau\rightarrow\omega_0\tau$, $q\rightarrow \frac{q}{e}$, $m\rightarrow \frac{m}{m_e}$ and $F^\mu_{~\nu}\rightarrow \frac{eF^\mu_{~\nu}}{m_e\omega_0c}$, where $e$ is the elementary charge, $m_e$ is the rest mass of electron and $\omega_0$ is a characteristic reference frequency which, for instance, can be chosen to be the electron plasma frequency or the frequency of the laser. We also absorb the charge-mass ratio into $F^\mu_{~\nu}$, i.e. $F^\mu_{~\nu}\rightarrow \frac{q}{m}F^\mu_{~\nu}$, to further simplify the expressions. If not specified in the remaining of the paper, we will use $F$ to denote the tensor $F^{\mu}_{~\nu}$. The normalized equation of motion is thus given by $$\label{eq:eq_motion}
{\frac{\text{d}u}{\text{d}\tau}}=Fu.$$
If the elements of $F$ are all constant in $\tau$ during an interval in time, then it is clear that this equation is easy to solve if we know the eigenvalues ($\lambda$) and eigenvectors ($\hat{e}$) of $F$. In \[sec:eigensys\], it is shown that the field tensor has four eigenvalues which come in pairs. One pair is real where $\lambda= \pm\kappa$ and the other pair is purely imaginary $\lambda =\pm{\text{i}}\omega$, where $$\label{eq:eigenval}
\kappa = \frac{1}{\sqrt{2}}\sqrt{\mathcal{I}_1+\sqrt{\mathcal{I}_1^2+4\mathcal{I}_2^2}},\quad
\omega = \frac{1}{\sqrt{2}}\sqrt{-\mathcal{I}_1+\sqrt{\mathcal{I}_1^2+4\mathcal{I}_2^2}}.$$ It is also shown that $$\label{eq:invariant}
\mathcal{I}_1 = |\bm E|^2-|\bm B|^2,\quad
\mathcal{I}_2 = \bm{E}\cdot\bm{B}.$$ are Lorentz invariants. The vector space of $F$ can be expanded into a set of unit base vectors $\hat{e}$ and that this can be used to construct a set of orthogonal base vectors, $b$ which are given by $$\label{eq:base_vector}
\begin{aligned}
b_\kappa^+ = \left(0, \kappa^2\bm{E}+\mathcal{I}_2\bm{B} \right), &\quad
b_\kappa^- = \kappa\left(|\bm E|^2+\omega^2, \bm{E}\times\bm{B} \right) \\
b_\omega^+ = \left(0, -\omega^2\bm{E}+\mathcal{I}_2\bm{B} \right), &\quad
b_\omega^- = {\text{i}}\omega \left( |\bm E|^2-\kappa^2, \bm{E}\times\bm{B} \right).
\end{aligned}$$ In \[sec:eigensys\] it is also shown that the four-vectors $b_\kappa^\pm$ and $b_\omega^\pm$ can be expressed as linear combinations of the eigenvectors $\hat{e}$ which are associated with the eigenvalues $\pm\kappa$ and $\pm{\text{i}}\omega$, respectively. We call the subspace expanded by $b_\kappa^\pm$ and that expanded by $b_\omega^\pm$, the “$\kappa$-space” and “$\omega$-space” respectively. By construction these two subspaces are also mutually orthogonal.
We hereafter adopt the notation $(\cdot|\cdot)$ to represent the inner product of two four-vectors. The left vector in the parentheses is covariant while the right is contravariant. The modulus or length of a four-vector $V$ is thus defined as $|V|\equiv\sqrt{(V|V)}$. Direct calculation shows that the modulus of these orthogonal base vectors are $|b_\kappa^\pm|^2=\mp\kappa^2 (\omega^2+\kappa^2)(|\bm E|^2+\omega^2)$ and $|b_\omega^\pm|^2=\mp\omega^2 (\omega^2+\kappa^2)(|\bm E|^2-\kappa^2)$, and thus a set of normalized unit base vectors can be defined as $\hat{b}_\kappa^\pm=b_\kappa^\pm/|b_\kappa^\pm|$ and $\hat{b}_\omega^\pm=b_\omega^\pm/|b_\omega^\pm|$. The initial components of $u$ in $\kappa$- and $\omega$-space, i.e., $u_{\kappa0}$ and $u_{\omega0}$, can thus be obtained by projecting $u_0$ along unit base vectors, $$\label{eq:u0_components}
u_{\kappa0} = (u_0|\hat{b}_\kappa^-)\hat{b}_\kappa^- + (u_0|\hat{b}_\kappa^+)\hat{b}_\kappa^+, \quad
u_{\omega0} = (u_0|\hat{b}_\omega^-)\hat{b}_\omega^- + (u_0|\hat{b}_\omega^+)\hat{b}_\omega^+.$$ In practice, we determine the projections by calculating either $u_{\kappa0}$ or $u_{\omega0}$ with Eq. (\[eq:u0\_components\]) and then calculating the other component using $u_0=u_{\kappa0}+u_{\omega0}$.
Next, we separately explore the evolution of $u_\kappa$ and $u_\omega$. Taking the proper time derivative of both sides of Eq. (\[eq:eq\_motion\]) and using the properties $F^2u_\kappa=\kappa^2 u_\kappa$ and $F^2u_\omega=-\omega^2 u_\omega$ (see \[sec:eigensys\]), the equation of motion can be decomposed into $${\frac{\text{d}^2 u_\kappa}{\text{d}\tau^2}} = \kappa^2 u_\kappa$$ and $${\frac{\text{d}^2 u_\omega}{\text{d}\tau^2}} = -\omega^2 u_\omega.$$ The solutions are given by $$\label{eq:uk}
u_\kappa(\tau) = u_{\kappa0}\cosh(\kappa\tau) + \kappa^{-1} Fu_{\kappa0}\sinh(\kappa\tau)$$ and $$\label{eq:uo}
u_\omega(\tau) = u_{\omega0}\cos(\omega\tau) + \omega^{-1} Fu_{\omega0}\sin(\omega\tau),$$ where $u_{\kappa0}=u_\kappa(\tau=0)$ and $u_{\omega0}=u_\omega(\tau=0)$. The four-position can be also obtained by directly integrating the expressions for the proper velocity over the proper time $$\begin{aligned}
\label{eq:xk}
x_\kappa(\tau)-x_{\kappa0} &= \kappa^{-1}\left[ u_{\kappa0}\sinh(\kappa\tau) + \kappa^{-1} F u_{\kappa0}(\cosh(\kappa\tau)-1)\right] \\
\label{eq:xo}
x_\omega(\tau)-x_{\omega0} &= \omega^{-1}\left[ u_{\omega0}\sin(\omega\tau) - \omega^{-1} F u_{\omega0}(\cos(\omega\tau)-1)\right]\end{aligned}$$ where the initial position components $x_{\kappa0}$ and $x_{\omega0}$ can be likewise obtained as how we decompose $u_0$ \[Eq. (\[eq:u0\_components\])\].
Until now we have obtained the analytic expressions for particle four-velocity and four-position in absence of radiation reaction. Given the proper time step $\Delta\tau$, the 6D phase space evolution could be advanced in proper time. However, in a PIC simulation, the fields are advanced using the lab frame time step $\Delta t$. We therefore need to advance forward the phase space for fixed lab frame steps rather than a fixed proper time step for each particle. Although $\Delta\tau$ changes for each simulation (observer) time step, a mapping between the lab and proper time intervals for fixed fields can be found using the time-like component of Eqs. (\[eq:xk\]) and (\[eq:xo\]) $$\label{eq:dt_mapping}
\Delta t = x^0_{\kappa}(\Delta\tau) + x^0_{\omega}(\Delta\tau) - x^0_{\kappa0} - x^0_{\omega0}$$ This is a transcendental equation consisting of trigonometric and hyperbolic functions. We usually need to resort to some root-finding algorithms such as Newton-Raphson method with second-order precision or Householder method with higher precision to seek the solution. We discuss this later when we compare the computational efficiency of the proposed pusher.
The above analytic solutions (and those obtained in sections to follow) are mathematically well-behaved even when the eigenvalues vanish. However, when the the eigenvalues are very small, it can be difficult to computationally evaluate these expressions. Therefore, it is important to show how the solutions behave when the eigenvalues become degenerate.
Special case: $\mathcal{I}_1\neq0,~\mathcal{I}_2=0$ {#sec:spe_case1}
---------------------------------------------------
In this case, $\bm{E}$ and $\bm{B}$ are orthogonal to one another and either $\kappa$ or $\omega$ vanishes depending on whether $|\bm{E}|<|\bm{B}|$ or $|\bm{E}|>|\bm{B}|$. In this case, there are only two non-vanishing eigenvectors $\hat{b}$ and the rank of the subspace for $u$ reduces from 2 to 1. However, this subspace is still perpendicular to the other subspace even in the limit that the eigenvalue goes to zero. Therefore, the sub-space decomposition and above discussion are still valid except we need to carefully treat the singularity appearing in the expression of $u$ and $x$ (Eqs. (\[eq:uo\]) and (\[eq:xo\])).
If $|\bm E|>|\bm B|$, we have $\omega=0$. In this case, $u_{\omega0}$ cannot be directly obtained by projecting $u_0$ according to Eq. (\[eq:u0\_components\]) due to the singularity in the denominator but the projection of $u_{\kappa0}$ is still valid. Therefore, we can obtain $u_{\omega0}$ through $u_{\omega0}=u_0-u_{\kappa0}$. The expression of $u_\omega$ can be obtained from Eq. (\[eq:uo\]) by simply taking the limit $\omega\rightarrow0$ $$u_\omega(\tau) = u_{\omega0} + \tau F u_{\omega0}.$$ Integrating this result over the proper time provides the expression for four-position $$x_\omega(\tau)-x_{\omega0} = u_{\omega0}\tau+\frac{1}{2}Fu_{\omega0}\tau^2$$ Analogously, if $|\bm B|>|\bm E|$, we have $\kappa=0$ and $u_{\kappa0}$ is obtained by $u_{\kappa0}=u_0-u_{\omega0}$. The expressions for $u_\kappa$ and $x_\kappa$ are then given by $$u_\kappa(\tau) = u_{\kappa0} + \tau F u_{\kappa0}$$ and $$x_\kappa(\tau)-x_{\kappa0} = u_{\kappa0}\tau+\frac{1}{2}Fu_{\kappa0}\tau^2.$$
Special case: $\mathcal{I}_1=\mathcal{I}_2=0$ {#sec:spe_case2}
---------------------------------------------
In the limit that the electric and magnetic fields on a particle are both orthogonal and equal in magnitude, $\mathcal{I}_1=\mathcal{I}_2=0$. In this case, all the eigenvalues of the field tensor vanish, and thus the sub-space decomposition is no longer valid. We note that in this special case, for any integer $n\ge3$, $F^n=0$. Using this property, we can construct a solution as the linear combination of the base $\{u_0, Fu_0, F^2u_0\}$ $$\label{eq:expansion_spe}
u(\tau)=c_1(\tau)u_0 + c_2(\tau)Fu_0 + c_3(\tau)F^2u_0.$$ Inserting the solution into Eq. (\[eq:eq\_motion\]) and comparing the coefficients, we obtain a set of ODEs $$\label{eq:coef}
\dot{c}_1 = 0, \quad
\dot{c}_2 = c_1, \quad
\dot{c}_3 = c_2.$$ Utilizing the initial conditions $c_1(0)=1$ and $c_2(0)=c_3(0)=0$, we can obtain a solution for $u(\tau)$ $$u(\tau)
=u_0+\tau Fu_0+\frac{1}{2}\tau^2F^2u_0,$$ and the four-position can be obtained by integrating the equation for $u(\tau)$, $$x(\tau)-x_0 = u_0\tau + \frac{1}{2}\tau^2 Fu_0 + \frac{1}{6}\tau^3 F^2u_0.$$
Particle motion in constant and uniform fields with radiation reaction {#sec:soln_LL}
======================================================================
In this section, we will derive the exact solutions to the LL equation by utilizing the orthogonality of $\kappa$- and $\omega$-space introduced in the previous section. We will see that by splitting the four-velocity $u$ into components belonging to the two sub-spaces, i.e., $u_\kappa$ and $u_\omega$, the integration of the LL equation is greatly simplified. The covariant form for the LL equation can be written as $${\frac{\text{d}u^\mu}{\text{d}\tau}} = \frac{q}{mc}F^\mu_{~\nu}u^\nu + \frac{2q^3}{3m^2c^3}\left(
{\frac{\partialF^\mu_{~\nu}}{\partialx^i}}u^\nu u^i
- \frac{q}{mc^2}F^\mu_{~\nu}F_i^{~\nu}u^i
+ \frac{q}{mc^2}(F^i_{~j}u^j)(F_i^{~k}u_k)u^\mu
\right),$$ where $x^i$ is the four-position. We are investigating cases where the fields are assumed constant in the proper time during a time step. Furthermore, it has been shown by others [@tamburini2010] that the first term in the parentheses that has the partial derivatives of $x^i$ can be neglected. This is referred to as the reduced Landau-Lifshitz model. After the normalizations described above, the reduced LL equation can be written as $$\label{eq:reduced_LL}
{\frac{\text{d}u}{\text{d}\tau}} = Fu + \sigma_0\frac{q^2}{m}\left[ F^2u - (u|F^2 u)u \right]$$ where $\sigma_0$ is a dimensionless parameter defined as $\sigma_0=\frac{2e^2\omega_0}{3m_ec^3}$.
By utilizing the subspace decomposition for $u$, i.e., $u = u_\kappa + u_\omega$, and recalling the relations $F^2 u_\kappa=\kappa^2 u_\kappa$ and $F^2 u_\omega=-\omega^2 u_\omega$, it can be shown that the contraction $(u|F^2u)$ becomes $(u|F^2u)=\kappa^2(u|u_\kappa)-\omega^2(u|u_\omega)=\kappa^2 |u_\kappa|^2 - \omega^2 |u_\omega|^2$. Substituting this result into the reduced LL equation (\[eq:reduced\_LL\]) and using the fact that the four-velocity has unity length, i.e., $|u|^2=|u_\kappa|^2+|u_\omega|^2=1$, we obtain two decoupled nonlinear differential equations $$\begin{aligned}
\label{eq:ode_kappa}
{\frac{\text{d}u_\kappa}{\text{d}\tau}} &= F u_\kappa + \alpha_0( 1 - |u_\kappa|^2 ) u_\kappa \\
\label{eq:ode_omega}
{\frac{\text{d}u_\omega}{\text{d}\tau}} &= F u_\omega - \alpha_0( 1 - |u_\omega|^2 ) u_\omega\end{aligned}$$ where $\alpha_0\equiv\sigma_0\frac{q^2}{m}(\kappa^2+\omega^2)$. To solve these nonlinear ordinary differential equations (\[eq:ode\_kappa\]) (Eq. (\[eq:ode\_omega\]) can be solved in an analogous manner), we first construct a trial solution as the product of the amplitude for $u_{\kappa}$ and a four-vector $w_\kappa$, i.e., $u_\kappa=|u_\kappa(\tau)| w_\kappa$. This implies that $w_\kappa$ is also enforced to have unity length. With this assumption Eq. (\[eq:ode\_kappa\]) can be separated into to two ODEs as $$\begin{aligned}
\label{eq:pha_kappa}
{\frac{\text{d}w_\kappa}{\text{d}\tau}} &=F w_\kappa, \\
\label{eq:amp_kappa}
{\frac{\text{d}|u_\kappa|}{\text{d}\tau}} &=\alpha_0(1-|u_\kappa|^2)|u_\kappa|.\end{aligned}$$ The first ODE is exactly the unperturbed Lorentz equation. It implies that the modulus of $w_\kappa$ does not change, which can be justified by left multiplying $w_\kappa$ on both sides of the equation and using the property $(w_\kappa|Fw_\kappa)=0$ as described in \[sec:eigensys\]. As the discussion in section \[sec:soln\_eq\_motion\], the unperturbed Lorentz equation (\[eq:pha\_kappa\]) has the solution $$w_\kappa(\tau) = w_{\kappa0}\cosh(\kappa\tau) + \kappa^{-1} F w_{\kappa0}\sinh(\kappa\tau).$$ where $w_{\kappa0}=w_\kappa(\tau=0)$.
Eq. (\[eq:amp\_kappa\]) can be directly integrated to obtain a solution to the amplitude equation $$|u_\kappa(\tau)|=\frac{|u_{\kappa0}|}{ \sqrt{|u_{\kappa0}|^2+|u_{\omega0}|^2 e^{-2\alpha_0\tau}} }$$ Combining the solutions for $w_\kappa$ and $|u_\kappa|$ yields $$\label{eq:uk_rr}
u_\kappa(\tau) = \frac{1}{ \sqrt{|u_{\kappa0}|^2+|u_{\omega0}|^2 e^{-2\alpha_0\tau}} }
\left[ u_{\kappa0}\cosh(\kappa\tau) + \kappa^{-1} F u_{\kappa0}\sinh(\kappa\tau)\right].$$ The solution to $u_\omega$ can be obtained in an analogous way $$\label{eq:uo_rr}
u_\omega(\tau) = \frac{1}{ \sqrt{|u_{\omega0}|^2+|u_{\kappa0}|^2 e^{2\alpha_0\tau}} }
\left[ u_{\omega0}\cos(\omega\tau) + \omega^{-1} F u_{\omega0}\sin(\omega\tau)\right].$$
Although there is no simple and closed-form expression for the four-position if radiation reaction is included, it is still possible to obtain approximate expressions with sufficiently high accuracy as long as the “friction” coefficient $\alpha_0$ is much less than $\Delta\tau$. Simple estimations can show this premise is often true for problems of interest. According to its definition, we know that $\alpha_0=(\frac{4\pi}{3}\frac{r_e}{\lambda_0})\frac{q^4}{m^3}\sqrt{\mathcal{I}_1^2+4\mathcal{I}_2^2} \le (\frac{4\pi}{3}\frac{r_e}{\lambda_0})\frac{q^4}{m^3}(|\bm E|^2+|\bm B|^2)$, where $r_e$ is classical electron radius. The equality is true if and only if $\bm{E}\cdot\bm{B}=|\bm E||\bm B|$. For example, assuming the characteristic length $\lambda_0\sim1~\mu\text{m}$ and the normalized field strengths $E$ and $B$ are on the order of $10^3$, we get $\alpha_0\sim10^{-2}$. In simulations, the time step must be properly selected to well resolve the characteristic time scales, say $\Delta t\sim0.1$, and thus $\Delta\tau\sim\Delta t/\gamma\le1$. Therefore the upper limit of $\alpha_0\tau$ is on the order of $10^{-3}$ when $0<\tau<\Delta \tau$, and keeping only the first term in the Taylor expansions of the denominator in Eqs. (\[eq:uk\_rr\]) and (\[eq:uo\_rr\]) is consequently valid. The four-position $x_\kappa$ can be approximately given by integrating the lowest order expansion of Eq. (\[eq:uk\_rr\]), $$\label{eq:xk_rr}
\begin{aligned}
x_\kappa(\tau)-x_{\kappa0}
&= \kappa^{-1}\left[u_{\kappa0}\sinh(\kappa\tau) + \kappa^{-1} F u_{\kappa0}\cosh(\kappa\tau)\right](1+\alpha_0|u_{\omega0}|^2\tau)-\kappa^{-2}Fu_{\kappa0} \\
&- \alpha_0|u_{\omega0}|^2\kappa^{-2}\left[u_{\kappa0}(\cosh(\kappa\tau)-1) + \kappa^{-1} F u_{\kappa0}\sinh(\kappa\tau)\right] + \mathcal{O}(\alpha_0^2\tau)
\end{aligned}$$ Similarly, we have $$\label{eq:xo_rr}
\begin{aligned}
x_\omega(\tau)-x_{\omega0}
&= \omega^{-1}\left[u_{\omega0}\sin(\omega\tau) - \omega^{-1} F u_{\omega0}\cos(\omega\tau)\right](1-\alpha_0|u_{\kappa0}|^2\tau)+\omega^{-2}Fu_{\omega0} \\
&- \alpha_0|u_{\kappa0}|^2\omega^{-2}\left[u_{\omega0}(\cos(\omega\tau)-1) + \omega^{-1} F u_{\omega0}\sin(\omega\tau)\right] + \mathcal{O}(\alpha_0^2\tau)
\end{aligned}$$ These expressions can then be used to approximately obtain the time step mapping using Eq. (\[eq:dt\_mapping\]), and the fast root finding algorithms mentioned previously in section \[sec:soln\_eq\_motion\] are still applicable.
Special case: $\mathcal{I}_1\neq0,~\mathcal{I}_2=0$ {#sec:spe_case1_rr}
---------------------------------------------------
As in section \[sec:spe\_case1\], the treatment of special cases then $\kappa$ and/or $\omega$ vanish must be considered with care. The singularity appearing in the solution when $\bm E$ and $\bm B$ are perpendicular is identical to that in section \[sec:spe\_case1\]. Again, for the $|\bm E|>|\bm B|$ case, $u_\omega$ can be obtained from Eq. (\[eq:uo\_rr\]) by simply taking the limit $\omega\rightarrow0$ $$u_\omega(\tau) = \frac{u_{\omega0} + \tau F u_{\omega0}}{ \sqrt{|u_{\omega0}|^2+|u_{\kappa0}|^2 e^{2\alpha_0\tau}} }.$$ Conducting a Taylor expansion in terms of $\alpha_0$ and integrating, the four-position becomes $$x_\omega(\tau)-x_{\omega0} =
u_{\omega0}\tau+\frac{1}{2}(Fu_{\omega0}-\alpha_0|u_{\kappa0}|^2 u_{\omega0})\tau^2
-\frac{1}{3}\alpha_0|u_{\kappa0}|^2Fu_{\omega0}\tau^3 + \mathcal{O}(\alpha_0^2\tau)$$ Similarly, if $|\bm B|>|\bm E|$, we have $\kappa=0$ and the expressions for $u_\kappa$ and $x_\kappa$ are given by $$u_\kappa(\tau) = \frac{u_{\kappa0} + \tau F u_{\kappa0}}{ \sqrt{|u_{\kappa0}|^2+|u_{\omega0}|^2 e^{-2\alpha_0\tau}} }.$$ and $$x_\kappa(\tau)-x_{\kappa0} =
u_{\kappa0}\tau+\frac{1}{2}(Fu_{\kappa0}+\alpha_0|u_{\omega0}|^2 u_{\kappa0})\tau^2
+\frac{1}{3}\alpha_0|u_{\omega0}|^2Fu_{\kappa0}\tau^3 + \mathcal{O}(\alpha_0^2\tau).$$
Special case: $\mathcal{I}_1=\mathcal{I}_2=0$ {#sec:spe_case2_rr}
---------------------------------------------
For the situation where $\mathcal{I}_1=\mathcal{I}_2=0$, the sub-space decomposition is invalid as discussed previously and we can follow the same procedure and construct the solution for the four-velocity in terms of $\{u_0, Fu_0, F^2u_0\}$ as Eq. (\[eq:expansion\_spe\]). Inserting the trial solution into Eq. (\[eq:reduced\_LL\]) and comparing the coefficients, we obtain a set of ODEs $$\label{eq:coef_rr}
\dot{c}_1 = -by(\tau)c_1, \quad
\dot{c}_2 = c_1 - by(\tau)c_2, \quad
\dot{c}_3 = c_2 + bc_1 - by(\tau)c_3$$ where $b\equiv\sigma_0\frac{q^2}{m}$ and $y(\tau)\equiv(u|F^2u)$. Using Eq. (\[eq:reduced\_LL\]) and taking the time derivative of $y(\tau)$, we can obtain the following differential equation $$\dot{y}(\tau) = -2by^2(\tau)$$ With the initial condition $y(\tau=0)=y_0$ we have $$y(\tau) = \frac{y_0}{1+2y_0b\tau},$$ Substituting this expression into Eq. (\[eq:coef\_rr\]) and utilizing the initial conditions $c_1(0)=1$ and $c_2(0)=c_3(0)=0$, we can eventually get the solution to $u(\tau)$ $$u(\tau)
=\frac{1}{\sqrt{1+2y_0b\tau}}\left[u_0+\tau Fu_0+\left(\frac{1}{2}\tau^2+b\tau\right)F^2u_0\right].$$
The above equation can be analytically integrated to obtain $x$ but the result is rather lengthy and therefore should be avoided for code implementation. For the purpose of obtaining a more efficient computation that is still accurate perform a Taylor expansion in the small parameter $b$ to obtain the expression, $$x(\tau)-x_0 = \left(u_0\tau + \frac{1}{2}\tau^2 Fu_0 + \frac{1}{6}\tau^3 F^2u_0\right)
- \left( \frac{1}{2}y_0 u_0 + \frac{1}{3} y_0\tau Fu_0 + \frac{1}{2}\left(\frac{1}{4}y_0\tau^2 - 1\right)F^2u_0 \right)b\tau^2 + \mathcal{O}(b^2)$$
Spin precession in uniform and constant fields {#sec:soln_BMT}
==============================================
In this section, we will derive the semi-analytic solutions to the particle four-spin vector in uniform and constant fields by utilizing the analytic expression of four-velocity in absence of radiation reaction. The basic idea is that the evolution of the spin depends on the evolution of the proper four velocity. We obtain analytic solutions for the evolution of the spin based on the analytic evolution of $u$ without radiation reaction. Radiation reaction is then included as two half impulse split operators at the beginning and end of each time step. As in the previous sections, we will first discuss the solutions for the general case and then two special cases for which the two or more of the eigenvalues vanish.
The spin precession of a single charged particle is described by the BMT equation. The covariant form of BMT equation, according to [@bargmann1959], is $${\frac{\text{d}s^\mu}{\text{d}\tau}} = \frac{q}{mc}\left[ \frac{g}{2}F^\mu_{~\nu}s^\nu - \frac{1}{c^2}
\left(\frac{g}{2}-1\right)(u_i F^i_{~j} s^j)u^\mu \right]$$ where $g$ is the Landé g-factor and is dimensionless.
The four-spin $s^\mu$ here is described in the observer frame and hence its time-like component may is nonzero. However, as an intrinsic property, it is more conventional to investigate the spin precession dynamics in the particle rest frame. Therefore, we need to transform $s^\mu$ to the particle rest frame after solving the BMT equation. Using normalized units and absorbing the $\frac{q}{m}$ factor into $F$ as done in the previous two sections, the BMT equation can be written as $$\label{eq:bmt}
{\frac{\text{d}s}{\text{d}\tau}} = (1+a)Fs - a(u|Fs)u$$ where $a\equiv\frac{g}{2}-1$ is the anomalous magnet moment and $a\simeq0.0011614$ for electrons. Eq. (\[eq:bmt\]) is a set of four coupled linear ODEs for spin with variable coefficients due to the presence of the proper velocity terms. If the analytic solutions for the four-velocity in the presence of radiation reaction are used, there is no analytic solution to the four-spin. However, we show next that if the analytic solutions for for the four velocity without radiation reaction is used then an analytic solution for the spin can be found.
We first explore the time evolution of the scalar $f\equiv (u|Fs)$ and show that it can be analytically solved even without knowing how $s$ evolves. We define $f(\tau)=(u_\kappa|Fs_\kappa) + (u_\omega|Fs_\omega) \equiv f_\kappa(\tau) + f_\omega(\tau)$, and then split Eq. (\[eq:bmt\]) into the $\kappa$- and $\omega$-components based on the eigenvalues of F as was done for the proper velocity, $$\begin{aligned}
\label{eq:ode_sk}
{\frac{\text{d}s_\kappa}{\text{d}\tau}} &= (1+a)Fs_\kappa - a f(\tau)u_\kappa, \\
\label{eq:ode_so}
{\frac{\text{d}s_\omega}{\text{d}\tau}} &= (1+a)Fs_\omega - a f(\tau)u_\omega.\end{aligned}$$ Combining Eqs. (\[eq:ode\_kappa\]) and (\[eq:ode\_sk\]), and using the fact that $F^2s_\kappa=\kappa^2 s_\kappa$ and $(u_\kappa|Fu_\kappa)=0$, it follows that the time derivative of $f_\kappa$ is $$\label{eq:ode_fk}
{\frac{\text{d}f_\kappa}{\text{d}\tau}} = a\kappa^2 (u_\kappa|s_\kappa).$$ Similarly, the time derivative of $f_\omega$ is $$\label{eq:ode_fo}
{\frac{\text{d}f_\omega}{\text{d}\tau}} = -a\omega^2 (u_\omega|s_\omega).$$ The quantity $\mathcal{I}_3\equiv \omega^2f_\kappa-\kappa^2 f_\omega$ is an invariant. This can be readily verified by taking the appropriate linear combination Eqs. (\[eq:ode\_fk\]) and (\[eq:ode\_fo\]), $${\frac{\text{d}\mathcal{I}_3}{\text{d}\tau}}=a\omega^2\kappa^2
\left[(u_\kappa|s_\kappa) + (u_\omega|s_\omega)\right]
=a\omega^2\kappa^2 (u|s) = 0.$$ Here we have also used the fact $(u|s)=0$ which follows from the fact that the time-like component of four-spin in the particle rest frame is zero, i.e., according to the Lorentz transform $s'^0=\gamma s^0 - \bm{u}\cdot\bm{s}\equiv (u|s)=0$. Taking the time derivative of Eq. 4.5 and substituting Eqs. (\[eq:ode\_kappa\]) and (\[eq:ode\_sk\]) gives $${\frac{\text{d}^2f_\kappa}{\text{d}\tau^2}} = a^2\kappa^2 \left( |u_\omega|^2f_\kappa - |u_\kappa|^2 f_\omega \right).$$ We can similarly get the second order ODE for $f_\omega$ $${\frac{\text{d}^2f_\omega}{\text{d}\tau^2}} = -a^2\omega^2 \left( |u_\kappa|^2f_\omega - |u_\omega|^2 f_\kappa \right).$$ Adding these two ODEs together and using the relations $|u_\kappa|^2+|u_\omega|^2=1$ and $f=f_\kappa+f_\omega$, we finally arrive at $$\label{eq:ode_f}
{\frac{\text{d}^2 f}{\text{d}\tau^2}}=-a^2\Omega^2 f + a^2\mathcal{I}_3$$ where $\Omega=\sqrt{\omega^2|u_\kappa|^2-\kappa^2|u_\omega|^2}=\sqrt{\omega^2|u_{\kappa0}|^2-\kappa^2|u_{\omega0}|^2}$ (note that $|u_\kappa|$ and $|u_\omega|$ are constant without radiation reaction). The solution is $$\label{eq:f}
f(\tau)=-\Omega^{-2}h_0\cos(a\Omega\tau)
+ (a\Omega)^{-1}\dot f_0\sin(a\Omega\tau) + \Omega^{-2}\mathcal{I}_3$$ where $h_0\equiv\mathcal I_3-f_0\Omega^2$ and we have used the initial conditions $f_0 = (u_{\kappa0}|Fs_{\kappa0}) + (u_{\omega0}|Fs_{\omega0})$ and $\dot f_0 = a[\kappa^2(u_{\kappa0}|s_{\kappa0})-\omega^2(u_{\omega0}|s_{\omega0})]$. After obtaining the solution to $f(\tau)$, we insert it back into Eqs. (\[eq:ode\_sk\]) and (\[eq:ode\_so\]) to solve for $s_\kappa$ and $s_\omega$. Eqs. (\[eq:ode\_sk\]) and (\[eq:ode\_so\]) now can be treated as inhomogeneous ODE equations and the homogeneous parts of the solutions satisfy $$\ddot{\bar s}_\kappa = (1+a)^2 \kappa^2 \bar s_\kappa, \quad
\ddot{\bar s}_\omega = -(1+a)^2 \omega^2 \bar s_\omega.$$ The homogeneous components then take the form $$\begin{aligned}
\label{eq:sk_bar}
\bar{s}_\kappa(\tau) &= A_\kappa \cosh[(1+a)\kappa\tau] + \kappa^{-1}B_\kappa \sinh[(1+a)\kappa\tau] \\
\label{eq:so_bar}
\bar{s}_\omega(\tau) &= A_\omega \cos[(1+a)\omega\tau] + \omega^{-1}B_\omega \sin[(1+a)\omega\tau]\end{aligned}$$ where $A_\kappa$, $B_\kappa$, $A_\omega$ and $B_\omega$ are constant four-vectors to be determined.
It can be shown (see \[sect:part\_soln\_app\]) that the inhomogeneous solutions to $s$ can be expressed by the linear combination of $u$ and $\dot u$, i.e. $$\tilde s_\kappa = C_\kappa(\tau)u_\kappa + D_\kappa(\tau)\dot u_\kappa$$ and $$\tilde s_\omega = C_\omega(\tau)u_\omega + D_\omega(\tau)\dot u_\omega.$$ The complete solutions are the sum of homogeneous and inhomogeneous solutions, i.e., $s_\kappa=\bar s_\kappa + \tilde s_\kappa$ and $s_\omega=\bar s_\omega + \tilde s_\omega$. Once given the coefficients $C_\kappa$, $D_\kappa$, $C_\omega$ and $D_\omega$ in the inhomogeneous solutions, the constant four-vectors in the homogeneous solutions can be determined according to the initial condition. \[sect:part\_soln\_app\] also gives the relationships between the initial values of these coefficients $\dot C_{\kappa0}=a\kappa^2 D_{\kappa0}-a f_0$ and $\dot D_{\kappa0}=aC_{\kappa0}$. Combining these relationships gives $$\label{eq:AB_kappa}
\begin{aligned}
A_\kappa &= s_{\kappa0} - C_{\kappa0}u_{\kappa0} - D_{\kappa0}\dot u_{\kappa0} \\
B_\kappa &= Fs_{\kappa0} - \kappa^2 D_{\kappa0}u_{\kappa0} - C_{\kappa0}\dot u_{\kappa0}.
\end{aligned}$$ Similarly, using the relations $\dot C_{\omega0}=-a\omega^2 D_{\omega0}-a f_0$ and $\dot D_{\omega0}=aC_{\omega0}$ given in \[sect:part\_soln\_app\], $A_\omega$ and $B_\omega$ can be written as $$\label{eq:AB_omega}
\begin{aligned}
A_\omega &= s_{\omega0} - C_{\omega0}u_{\omega0} - D_{\omega0}\dot u_{\omega0} \\
B_\omega &= Fs_{\omega0} + \omega^2 D_{\omega0}u_{\omega0} - C_{\omega0}\dot u_{\omega0}.
\end{aligned}$$ The detailed process for generating the coefficients $C_\kappa$, $D_\kappa$, $C_\omega$ and $D_\omega$ is tedious and can be found in \[sect:part\_soln\_app\]. Here, we directly list the final results. For the general case, we have $$\label{eq:CD_k}
C_\kappa(\tau) = \frac{\Omega\dot f_0\cos(a\Omega\tau)+ah_0\sin(a\Omega\tau)}{a\Omega(\kappa^2+\Omega^2)}, \quad
D_\kappa(\tau) = \frac{\Omega\dot f_0\sin(a\Omega\tau)-ah_0\cos(a\Omega\tau)}{a\Omega^2(\kappa^2+\Omega^2)}$$ and $$\label{eq:CD_o}
C_\omega(\tau) = \frac{\Omega\dot f_0\cos(a\Omega\tau)+ah_0\sin(a\Omega\tau)}{a\Omega(\Omega^2-\omega^2)}, \quad
D_\omega(\tau) = \frac{\Omega\dot f_0\sin(a\Omega\tau)-ah_0\cos(a\Omega\tau)}{a\Omega^2(\Omega^2-\omega^2)}.$$
It can be seen that a singularity appears in the above expression when $\Omega=0,~{\text{i}}\kappa$ or $\omega$. Although none of these special cases occurs in practice, it is still necessary to carefully treat the singularity because the solutions become very sensitive to numerical noise when the singularities nearly arise. We include a tedious discussion on the singularity treatment into \[sect:singularity\_1\].
Special case: $\mathcal{I}_1\neq0$ and $\mathcal{I}_2=0$
--------------------------------------------------------
As discussed in section \[sec:spe\_case1\], when the electric and magnetic field are perpendicular to each other and their amplitudes are unequal, we know that the subspace decomposition is still valid and the solutions can be obtained by directly taking the limit of general solutions when $\kappa$ or $\omega$ approaches zero. When $|B|>|E|$, $\kappa=0$ and thus $\Omega=\omega|u_{\kappa0}|$. According to Eq. (\[eq:sk\_bar\]), the general solution to $s_\kappa$ reduces to $$\bar s_\kappa(\tau) = A_\kappa + (1+a)B_\kappa\tau$$ and the constant four-vectors $A_\kappa$ and $B_\kappa$ still follow Eq. (\[eq:AB\_kappa\]). When $\Omega\neq0$, Eq. (\[eq:CD\_k\]) is still valid for $C_\kappa$ and $D_\kappa$ by simply setting $\kappa=0$. The solution to $s_\omega$ is unchanged since there is no singularity in the $\omega$-component.
Similarly, when the electric field becomes dominant, $|E|>|B|$, $i.e.$, $\omega=0$ and $\Omega={\text{i}}\kappa|u_{\omega0}|$, the general solution to $s_\omega$ reduces to $$\bar s_\omega(\tau) = A_\omega + (1+a)B_\omega\tau,$$ and Eq. (\[eq:CD\_o\]) is still valid if $\Omega\neq0$.
The treatment of the singularity in $C_\kappa,~D_\kappa,~C_\omega$ and $D_\omega$ differs from that of the general case ($\mathcal I_1\neq0$ and $\mathcal I_2\neq0$), and it can be found in \[sect:singularity\_2\].
Special case: $\mathcal{I}_1=0$ and $\mathcal{I}_2=0$
-----------------------------------------------------
As discussed in section \[sec:spe\_case2\], for this special case the subspace decomposition fails for the LL equation as well as for BMT equation. Explicit calculating the second derivative of $f(\tau)$, and utilizing Eq. (\[eq:bmt\]) and the relationships $\dot u=Fu$ and $(u|Fu)=0$, it can be shown $f(\tau)$ is evolves as $$\ddot f(\tau) + a^2 y(\tau)f(\tau) = 0.$$ Note that in the absence of radiation reaction $y(\tau)=y_0$, so we have $$f(\tau) = f_0 \cos(\lambda_0\tau) + \lambda_0^{-1}\dot f_0\sin(\lambda_0\tau)$$ where $\lambda_0\equiv a\sqrt{y_0}$. Upon substituting the solution for $f(\tau)$, Eq. (\[eq:bmt\]) can be viewed as an inhomogeneous ODE and the complete solution $s(\tau)$ can be written as the sum of the homogeneous solution $\bar s(\tau)$ and the inhomogeneous solution $\tilde s(\tau)$. Expanding $\bar s(\tau)$ in terms of the base vectors $s_0,~Fs_0$ and $F^2s_0$ as $$\bar s(\tau) = A_1(\tau) s_0 + A_2(\tau) Fs_0 + A_3(\tau) F^2 s_0$$ and substituting this into the homogeneous counterpart of Eq. (\[eq:bmt\]) gives $$\dot A_1=0,\quad
\dot A_2=(1+a)A_1,\quad
\dot A_3=(1+a)A_2.$$
The inhomogeneous solution can be assumed to be of the form, $$\tilde s(\tau) = B_1(\tau)u(\tau) + B_2(\tau)Fu(\tau) + B_3(\tau)F^2u(\tau).$$ Inserting this into Eq. (\[eq:bmt\]) and comparing the coefficients of $u,~Fu$ and $F^2u$ leads to $$\dot B_1=-af(\tau),\quad
\dot B_2=aB_1,\quad
\dot B_3=aB_2.$$
Next, we seek a set of coefficients $A_i$ and $B_i$ ($i=1,2,3$) such that the initial conditions $\bar s(0)=s_0$ and $\tilde s(0)=0$ are satisfied. This indicates these coefficients must be subject to the initial conditions $A_1(0)=1,~A_2(0)=A_3(0)=0$ and $B_1(0)=B_2(0)=B_3(0)=0$. A solution that meets these conditions is given by $$A_1=1,\quad
A_2=(1+a)\tau,\quad
A_3=\frac{1}{2}(1+a)^2\tau^2,$$ and $$\begin{aligned}
B_1 & = y_0^{-1/2}\left[ \lambda_0^{-1}\dot f_0 (\cos(\lambda_0\tau)-1)-f_0\sin(\lambda_0\tau)\right],\\
B_2 & = y_0^{-1}\left[ \lambda_0^{-1}\dot f_0 (\sin(\lambda_0\tau)-\lambda_0\tau)+f_0(\cos(\lambda_0\tau)-1)\right], \\
B_3 & = -y_0^{-3/2}\left[ \lambda_0^{-1}\dot f_0 \left(\cos(\lambda_0\tau)+\frac{1}{2}\lambda_0^2\tau^2-1\right) - f_0(\sin(\lambda_0\tau)-\lambda_0\tau)\right].
\end{aligned}$$
Algorithm implementation {#sec:algorithm}
========================
![Numerical workflow of four algorithm implementation options. The relevant equation numbers are summarized in each block.[]{data-label="fig:workflow"}](./figure/workflow.pdf){width="70.00000%"}
In this section, we will introduce the algorithm workflow using the analytical expressions of 9D phase space obtained in previous sections. Depending on the purposes and program implementations of PIC codes, we provide four distinct algorithms which are characterized by different choices from the subset of the analytical solutions. Figure \[fig:workflow\] shows the numerical workflow of the four algorithm implementations. The numbers for the requisite equations for each algorithm have been summarized in each block.
For existing PIC codes the momentum and position of the particles are staggered in time and the fields are known at the same time as the position. As a result only the red and blue paths in fig. \[fig:workflow\] are possible without significant rework of the algorithm. Keeping the leapfrog of the position then permits only modifying the momentum update, and the field solve and current deposit do not have to be modified. If updating the spin is not important then the red path is desirable. This uses the analytic updates for the momentum even if RR is included. This implementation employs the analytic solution to momentum and the leap frog advance for position. For PIC codes defining position and momentum at the same time meshgrid, when the evolution of the full 9D phase space is important then the algorithm based on the yellow path should be used, otherwise the green path should be selected. The yellow path utilizes the analytical solutions to $(\bm x, \bm u, \bm s)$ without RR and the effects of RR are incorporated as by breaking the change in momentum from RR into two half impulses before and after the analytic solution without RR is used. In the time interval $n\Delta t<t<(n+1)\Delta t$, the first half impulse can be applied to $\bm u$ via $$\bm u^- = \bm u^n + \frac{\Delta t}{2}\bm f_\text{RR}(\bm u^n)$$ and then $\bm u^-$ is pushed to $\bm u^+$ along with $\bm x$ and $\bm s$ using the analytical solutions for a full time step. The other half impulse is then applied via $$\bm u^{n+1} = \bm u^+ + \frac{\Delta t}{2}\bm f_\text{RR}(\bm u^+).$$ Here the radiative reaction force is evaluated as follows $$\bm f_\text{RR}(\bm u) = \sigma_0\frac{q^2}{\gamma m}[F^2 u-(u|F^2u)u]_\text{spatial}$$ and the subscript “spatial” refers to space-like component of a four-vector.
If the position and momentum are staggered in time, the analytical expressions of $\bm x$ cannot be used any more because in the time interval $n\Delta t<t<(n+1)\Delta t$ where the solution to $\bm u$ is known, $\bm x$ is known only within $n\Delta t<t<(n+1/2)\Delta t$. Therefore, the positions need to be advanced in the conventional leapfrog manner, i.e. $$\bm x^{n+\frac{3}{2}} = \bm x^{n+\frac{1}{2}} + \bm u^{n+1} \Delta t/\gamma^n.$$ Two algorithm implementation shown by the red and blue paths in fig. \[fig:workflow\] use the leap frog method to update the position. We point out that $\bm s$ can be defined on the identical time grid points as $\bm u$ so that the analytical solutions still applies.
Sample simulations {#sec:sample}
==================
In this section, we will compare different particle pushers through a series of particle tracking simulations where a single particle interacts with an ultra-intense laser pulse in prescribed fields or in fields obtained in a self-consistent PIC simulation using <span style="font-variant:small-caps;">Osiris</span>. As we have multiple options to advance the particle position, momentum and spin, the following schemes (SC1-SC6) will be investigated to see how well they can advance the $(\bm x, \bm u)$ phase space:
1. [**SC1**]{} – The Boris pusher is used to advance the particle momentum and the position is advanced in a leapfrog manner with a second order accuracy in $\Delta t$. The radiative reaction force is added according to the splitting method addressed in sec. \[sec:algorithm\]. Vieira’s scheme is used to advance spin precession.
2. [**SC2**]{} – The set-up is identical to SC1 except the HC pusher is used to advance particle momentum.
3. [**SC3**]{} to [**SC6**]{} refer to the blue, red, yellow and green paths in fig. \[fig:workflow\] respectively.
Single particle motion in ultra-intense laser fields
----------------------------------------------------
In this section, we compare the various pushers using a particle tracking code in which the fields are prescribed. This then permits using the analytic position update as well. We consider a one-dimensional case in which a lasers pulse propagates in vacuum. Test particles are initialized in front of the laser pulse. The plane-wave laser is linearly polarized in the $\hat 2$-direction and moves in the $\hat 1$-direction. The normalized vector potential is given by $$\label{eq:analytic_field}
\bm A = a_0 \cos^2\left(\frac{\pi\phi}{2\omega_0\tau_\text{FWHM}}\right) \cos\phi\ \hat{\bm e}_2$$ when the phase $\phi\equiv \omega_0 t-k_0x_1$ is within $[-\omega_0\tau_\text{FWHM},~\omega_0\tau_\text{FWHM}]$ or otherwise vanishes. Here, $\tau_\text{FWHM}$ is the defined as the full-width-at-half-maximum of the field envelope and $a_0$ is the strength parameter which is connected with the peak intensity via $a_0=0.86\sqrt{I_0 [10^{18}\text{W/cm}^2]}\lambda_0[\mu\text{m}]$. In all the following comparisons, a pulse duration of $\tau_\text{FWHM}=50~\omega_0^{-1}$ is chosen and the field is expressed analytically according to Eq. (\[eq:analytic\_field\]). We assume the laser wavelength is 0.8 $\mu\text{m}$ and set the reference frequency to be the laser frequency $\omega_0$ so that the dimensionless radiative damping parameter $\sigma_0\approx 1.474\times10^{-8}$.
In the first set of simulations, the test particle has an initial momentum of $p_{10}=-30\ m_ec$ (the negative sign means it counter-propagates relative to the laser) and the initial spin is along the positive $\hat 1$-direction. We tracked the transverse momentum $p_2$, phase $\phi$ and transverse spin $s_2$ during the particle-wave interaction under situations where various $\Delta t$ are selected. For relatively weak laser intensities where $a_0$ is on the order of unity, it is found that all the aforementioned numerical schemes provide identical and correct phase space trajectories. This is not the case for higher intensities. Figure \[fig:a0\_300\_u0\_m30\] shows the results for $a_0=300$ ($I_0\sim 2.2\times10^{23}$ W/cm$^2$) but with different $\Delta t$. The black dashed line is obtained using a fourth-order Runge-Kutta integrator with sufficiently small time step that it can be viewed as the “correct” result. It can be seen that the schemes that use the split operator, i.e., standard particle pushers (SC1 and SC2), lead to incorrect results for both $\Delta t = 0.2\omega_0^{-1}$ and $\Delta t = 0.1\omega_0^{-1}$. The phase shift of particles pushed by SC1 and SC2 are severely miscalculated (see fig. \[fig:a0\_300\_u0\_m30\](b) and (e)), which leads to a large deviation in the phase space trajectories. According to our tests, SC1 and SC2 do not converge until reducing $\Delta t$ to $\sim0.02\omega_0^{-1}$. For SC3 and SC4 which advance the position in a leapfrog manner and the momentum with the analytical pusher (SC3 also analytically advances spin), the momentum and spin oscillation and phase shift are qualitatively correct although they are quantitatively inaccurate for $\Delta t=0.2\omega_0^{-1}$ (see fig. \[fig:a0\_300\_u0\_m30\](a)-(c)). When the time step is reduced to $\Delta t=0.1\omega_0^{-1}$, both SC3 and SC4 converge well to the “correct” results as shown in fig. \[fig:a0\_300\_u0\_m30\](d)-(f). Since SC5 and SC6 advance both position and momentum analytically (SC5 also advances spin analytically), they give good agreement with the “correct” results for the two time steps as expected.
We also tested how well these numerical schemes work with zero initial momentum as shown in fig. \[fig:a0\_300\_u0\_0\]. According to Vranic et al. [@vranic2016b], this situation is more sensitive to numerical noise. Due to the energy loss during the laser-particle interaction, the particle will stay much longer in phase, and thus the interaction duration is significantly longer. We tested two time steps $\Delta t=0.2\omega_0^{-1}$ and $\Delta t=0.05\omega_0^{-1}$, and again SC1 and SC2 lead to a large deviation from the correct results. For $\Delta t=0.2\omega_0^{-1}$, SC3 and SC4 can lead to the correct phase space trajectory results for the first few cycles, but clear deviations appear at later times due to the accumulation of numerical error over longer durations. Good agreement can be reached when the time step is reduced to $\Delta t=0.05\omega_0^{-1}$. As before, SC5 and SC6 lead to excellent agreement with the correct results even for a time step typically used to accurately solve for the fields in laser-plasma-interaction simulations ($\Delta t=0.2\omega_0^{-1}$).
![Single-particle motion in the head-on collision with an ultra-intense laser ($a_0=300$) using various numerical schemes. The test particle has an initial longitudinal momentum $p_{10}=-30m_ec$. Evolution of (a) transverse momentum $p_2$, (b) phase in laser field and (c) transverse spin $s_2$ are compared. The field felt by the test particle is analytic.[]{data-label="fig:a0_300_u0_m30"}](./figure/a0_300_u0_m30.pdf){width="\textwidth"}
![Single-particle motion in an ultra-intense laser ($a_0=300$) using various numerical schemes. The test particle is initialized at rest. Evolution of (a) transverse momentum $p_2$, (b) phase in laser field and (c) transverse spin $s_2$ are compared. The field felt by the test particle is analytic.[]{data-label="fig:a0_300_u0_0"}](./figure/a0_300_u0_0.pdf){width="\textwidth"}
Full PIC simulation of beam-laser interaction
---------------------------------------------
As shown in the last section, the single particle motion in strong laser fields varies significantly when different particle pushers are used even when prescribed (analytical) fields are used for the laser. In this section, we will show that the collective behavior of a bunch of particles can also be significantly different depending on the choice of the pusher unless very small time steps are used. We have implemented the proposed particle pusher into <span style="font-variant:small-caps;">Osiris</span>. The aforementioned SC3 is adopted because of the time-staggering layout for particle position and momentum and the resulting need for a leap frog advance of the particle position. In the full 2D PIC simulations, a $0.8\ \mu\text{m}$ wavelength plane wave laser pulse with $a_0=500$ and 50 $\omega_0^{-1}$ FWHM duration for the envelope collides head-on with an electron beam. The electron beam has a bi-Gaussian density distribution with an rms radius $\sigma_2=10k_0^{-1}$ and rms length $\sigma_1=15k_0^{-1}$ ($\hat 1$ is the propagation direction), and the initial momentum $p_{10}=-10m_ec$. The cell sizes are $\Delta x_1=0.2k_0^{-1}$ and $\Delta x_2=2k_0^{-1}$, and the time step is $\Delta t=0.1\omega_0^{-1}$. To accurately simulate the particle motion in the laser field, we have used a Maxwell solver with an extended stencil [@li2020] to reduce the numerical errors arising from the numerical dispersion and interlacing of $E$ and $B$ fields in time.
Figure \[fig:collision-sketch\] shows the laser field and beam density distribution. As shown in fig. \[fig:collision-sketch\](a), the electron beam initially moves toward the laser pulse from right to left. The the bunch length is then compressed by the extremely strong radiation pressure of the leading edge of the laser. The propagation direction of the beam is eventually reversed so that it co-moves with the laser pulse as shown in fig. \[fig:collision-sketch\](b). There are significant differences in phase space between the “standard” and the proposed numerical schemes. Figure \[fig:x1p1p2\] shows the $x_1$-$p_1$-$p_2$ space phase for SC1 \[fig. \[fig:x1p1p2\](b) and (d)\] and SC3 \[fig. \[fig:x1p1p2\](a) and (c)\] at $t=60\omega_0^{-1}$ and $t=200\omega_0^{-1}$. At $t=60\omega_0^{-1}$ there is only slight differences between the two schemes. At $t=200\omega_0^{-1}$ the phase space distribution begins to broaden for SC1 while it remains narrow for SC3. Since the macro-particles only have negligible charge (they are test particles) and thus the interaction between particles is very weak, the physical quantities including $p_1$ and $p_2$ should only be a function of only $x_1$ for given $t$ in the 1D limit (plane-wave laser). Therefore, the narrow distribution in fig. \[fig:x1p1p2\](c) is what is expected. We also conducted convergence tests using SC1 with ten-fold higher resolution in space and time. The results converged with that in fig. \[fig:x1p1p2\](c) for the larger time step.
![2D PIC simulations results using <span style="font-variant:small-caps;">Osiris</span>. Snapshots of beam density (green) and laser field (red and blue) taken at (a) $t=0$ and (b) $t=38\omega_0^{-1}$.[]{data-label="fig:collision-sketch"}](./figure/collision-sketch.pdf){width="70.00000%"}
![Particle distributions in $x_1$-$p_1$-$p_2$ phase space of (a)(c) scheme 3 and (b)(d) scheme 1.[]{data-label="fig:x1p1p2"}](./figure/x1p1p2.pdf){width="70.00000%"}
We also compared the how the evolution of the spin precession is modified from the different schemes. The spin of the electron beam is initially polarized along the positive $\hat 1$-direction with a small divergence as shown in fig. \[fig:spin\_space\](a). In fig. \[fig:spin\_space\] we plot the spin in the rest frame so that all the particles move on the surface of a sphere of radius in $\hbar/2$ in $s_1$-$s_2$-$s_3$ space. When the beam starts to interact with the laser field, the particles move down the negative $\hat 1$-direction along the longitude. Significant differences can be seen at $t=200\omega_0^{-1}$. The particles advanced in momentum space in SC3 and in spin space with the exact spin pusher (see fig. \[fig:spin\_space\](b)) fully congregated at the negative $\hat 1$-direction while the particles advanced by SC1 and the Vieira scheme (see fig. \[fig:spin\_space\](c)) spread over a much wider region around the pole.
![Particle distributions in $s_1$-$s_2$-$s_3$ space at (a) $t=0$ and (b)(c) $t=200\omega_0^{-1}$. SC3 and SC1 were used to obtain the results in (b) and (c) respectively.[]{data-label="fig:spin_space"}](./figure/spin_space.pdf){width="\textwidth"}
Performance
===========
In this section, we compare the performance of the implementation of the red (SC4) and blue path (SC3) workflows into <span style="font-variant:small-caps;">Osiris</span> against each other and against the standard Boris push. We carried out two-dimensional simulations with a thermal plasma with 4 macro-particles per cell and fixed ions. The particle shapes corresponded to linear weighting or interpolation. The box used $512\times512$ cells and the cell size was 0.3 Debye length in each direction. <span style="font-variant:small-caps;">Osiris</span> was compiled using GNU Fortran 8.3 with -O3 optimization. We did compare vectorized versions. For each time step, the time cost of different procedures within the full push momentum, spin, and position advance, and field interpolation and current update tare summarized in fig. \[fig:performance\]. The computational cost of momentum advance in SC3 is 2.4 times that of SC1 (Boris scheme), and the spin advance in SC3 is about 5.4 times slower than that in SC1 (Vieira scheme). When taking RR into account, the computational cost of SC3 is 1.7 times that of SC1. SC4 is even slightly faster than SC3 and only 1.5 times that of SC1. The additional cost of updating the positions, interpolating fields from grid onto particle positions, and depositing the current onto the grids is shown by the yellow blocks in fig. \[fig:performance\]. Therefore, in simulations where the spin is not considered, the schemes employing exact momentum pushers can provide competitive performance on a per time step basis to the schemes using regular pushers even in weak/moderate field scenarios where the regular pushers are still accurate at conventionally selected time steps. However, in moderate to strong-field regimes, time steps 10 to 100 times smaller are needed for regular pushers to obtain the accuracy of the analytical pushers. Therefore, these new pushers can significantly reduce the computational time needed for high fidelity simulations. We note that the performance differences will become even smaller as the order of the particle shape increases as the parts parts of the code encapsulated in the orange will take longer per particle and these parts are the same for each scheme.
![Performance of various pushers implemented in <span style="font-variant:small-caps;">Osiris</span>.[]{data-label="fig:performance"}](./figure/performance.pdf){width="60.00000%"}
Conclusion {#sec:summary}
==========
In this article, we derived the analytic solutions to the change in the four vector of momentum including a reduced form of the radiation reaction. When written in covariant form it is clear that if the electric and magnetic fields are constant (in both space and time) that there are analytic solutions. We obtained forms of the solutions to both the momentum (proper velocity) and position ($(\bm x,\ \bm u)$ phase space) using basis vectors amenable to PIC codes. The trajectory of $(\bm x,\ \bm u)$ can be accurately computed in the strong regime with these explicit close-form expressions using much larger time steps than would be required for standard pushers. These expressions are analytic so the errors arise only from the assumption of a constant and uniform field at each time step. When the radiation reaction is involved, these expressions still highly accurate until situations where classical theory fails.
With the analytical solution to $\bm u$ and remaining the fields constant and uniform, the Bargmann-Michel-Telegdi equation can also be analytically solved and the closed-form solutions can be used to simulate spin precession in strong fields. Although these expressions are only perfectly accurate without radiation reaction, the radiative effect can still be properly taken into account by separately including impulse corrections from radiative correction to $\bm u$.
The computational efficiency defined as the computational time to solution of the proposed 9D phase space pusher over existing schemes was demonstrated through a series of single-particle simulations. It is shown that, for typical time steps that resolve the laser period and evolution of the laser fields, the proposed pusher can yield correct or sufficiently accurate phase space trajectories for $a_0$ on the order of $10^2$ whereas the results given by the regular pushers deviate significantly from analytic results. The standard pushers usually need a much smaller time step to converge in this scenario. We note that for problems where the fields vary very slowly in time (including large imposed magnetic fields) the proposed pusher will be even more efficient than standard schemes.
We implemented the analytic solution for the momentum update while keeping the leap frog position advanced into the code <span style="font-variant:small-caps;">Osiris</span>. Therefore, only the momentum update needed to be modified while the fields solver, position update, and current deposit did not need be changed. Future work might involve developing PIC algorithms that define the position and momentum at the same time or embedding predictor corrector algorithms. We found that the proposed scheme without the spin advance is only tens of percent slower per particle and with the spin advance only 70 percent slower than standard pushers. However, as the proposed scheme can provide accurate solutions with time steps much larger depending on the field strength and configuration it can lead to significant speed ups. For example, for some of the laser plasma interaction examples presented here time steps as much as 10 times larger can be used.
Using <span style="font-variant:small-caps;">Osiris</span>, PIC simulations were also conducted to compare the proposed numerical scheme against standard schemes. The head-on collision of an electron beam and an ultra-intense laser pulse was modeled. The results showed significant differences in the phase space (including the spin precession) between the proposed and the standard schemes. As the time step was reduced the standard pusher simulations converged to that of the analytical pusher case with the larger time steps. Although these sample simulations are all conducted in the context of laser-plasma interaction, the proposed algorithm itself is general and can apply to many other research fields.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in parts by the US Department of Energy contract number DE-SC0010064, DE-SC0019010 and SciDAC FNAL subcontract 644405, Lawrence Livermore National Laboratory subcontract B634451, and US National Science Foundation grant numbers 1806046. Simulations were carried out on the Cori Cluster of the National Energy Research Scientific Computing Center (NERSC).
Eigensystem and relevant properties {#sec:eigensys}
===================================
The eigenvalues of field tensor $F$ is determined by the characteristic equation $$\text{det}(F-\lambda I_4)=0.$$ where $I_4$ is the $4\times4$ identity tensor. Explicit calculation shows there are two pairs of eigenvalues $\lambda=\pm\kappa$ and $\lambda=\pm{\text{i}}\omega$ where $$\kappa = \frac{1}{\sqrt{2}}\sqrt{\mathcal{I}_1+\sqrt{\mathcal{I}_1^2+4\mathcal{I}_2^2}},\quad
\omega = \frac{1}{\sqrt{2}}\sqrt{-\mathcal{I}_1+\sqrt{\mathcal{I}_1^2+4\mathcal{I}_2^2}}.$$ Here $\mathcal{I}_1$ and $\mathcal{I}_2$ are the well-known Lorentz invariants [@jackson2007] given by $$\mathcal{I}_1 = |\bm E|^2-|\bm B|^2,\quad
\mathcal{I}_2 = \bm{E}\cdot\bm{B}.$$
After knowing the eigenvalues, the eigenvector $e_\lambda$ associated to the eigenvalue $\lambda$ ($\lambda$ can be taken to anyone among $\pm\kappa$ and $\pm{\text{i}}\omega$) can be generated from an arbitrary four-vector $V$ using $$\label{eq:eigvec_gen}
e_\lambda=\left(\prod_{\lambda'\neq\lambda} (F-\lambda'I_4)\right)V,$$ For simplicity, we choose $V=(1,0,0,0)$ and after tedious algebra the four eigenvectors can be written as $$\begin{aligned}
e_{\pm\kappa} &= \left(\pm\kappa(|\bm E|^2+\omega^2),\kappa^2\bm{E}+\mathcal{I}_2\bm{B}\pm\kappa(\bm{E}\times\bm{B})\right) \\
e_{\pm\omega} &= \left(\pm{\text{i}}\omega(|\bm E|^2-\kappa^2),-\omega^2\bm{E}+\mathcal{I}_2\bm{B}\pm{\text{i}}\omega(\bm{E}\times\bm{B})\right),
\end{aligned}$$ where $e_{\pm\kappa}$ are the eigenvectors associated to eigenvalues $\pm\kappa$ and $e_{\pm\omega}$ are the ones associated to eigenvalues $\pm{\text{i}}\omega$.
In this article we have introduced the denotation $(U|V)$ to represent the inner product of two four-vectors $U$ and $V$, i.e., $(U|V)=U_\mu V^\mu$. The left vector in the parentheses is covariant and the right is contravariant. It is easy to verify that the eigenvectors satisfy $(e_{\pm\kappa}|e_{\pm\omega})=0$, which indicates the “$\kappa$-space” expanded by $\hat{e}_{\pm\kappa}$ is orthogonal to the “$\omega$-space” expanded by $e_{\pm\omega}$. The two eigenvectors belonging to the same sub-space are not orthogonal. Therefore, we perform the following orthogonalization $$\label{eq:orth_base}
b_\lambda^+ = \frac{1}{2}(e_\lambda + e_{-\lambda}),\quad
b_\lambda^- = \frac{1}{2}(e_\lambda - e_{-\lambda}),\quad
(\lambda = \kappa, \omega)$$ to get a set of orthogonal bases and they can be written explicitly as $$\begin{aligned}
b_\kappa^+ = \left(0, \kappa^2\bm{E}+\mathcal{I}_2\bm{B} \right), &\quad
b_\kappa^- = \kappa \left(|\bm E|^2+\omega^2, \bm{E}\times\bm{B} \right) \\
b_\omega^+ = \left(0, -\omega^2\bm{E}+\mathcal{I}_2\bm{B} \right), &\quad
b_\omega^- = {\text{i}}\omega \left(|\bm E|^2-\kappa^2, \bm{E}\times\bm{B} \right).
\end{aligned}$$
Now we list several important properties related to the four-vector inner product and the orthogonal base vectors.
1. The inner product is subject to commutative law, i.e., $(U|V)=(V|U)$.
2. The field tensor has the property $F^\mu_{~\nu}=-F^{~\mu}_\nu$. Therefore we have $(FU|V)=-(U|FV)$ where the vector $FU$ appearing on the left side in the parentheses should be read as $F_\mu^{~\nu}U_\nu$ while it represents $F^\mu_{~\nu}U^\nu$ if appearing on the right side.
3. The base vectors have the following property according to Eqs. (\[eq:orth\_base\]) $$\label{eq:property1}
F b_\lambda^\pm = \lambda b_\lambda^\mp,\quad
(\lambda=\kappa,~\omega),$$ which implies both the $\kappa$-space and $\omega$-space are closed under $F$-mapping. In another word, multiplying a vector that originally residing in either $\kappa$- or $\omega$-space by $F$ still maps it to the same sub-space.
4. For arbitrary vector $V$ belonging to either $\kappa$- or $\omega$-space, the inner product $(V|FV)=0$. The relation can readily verified by expanding it as $V=a_1 b_\lambda^+ + a_2 b_\lambda^-$ ($\lambda=\kappa,~\omega$) and using Eqs. (\[eq:orth\_base\]), the orthogonality $(b_\lambda^+|b_\lambda^-)=0$ and the fact that the eigenvectors have null length.
Inhomogeneous solutions to Eqs. (\[eq:ode\_sk\]) and (\[eq:ode\_so\]) {#sect:part_soln_app}
=====================================================================
In this appendix, we will seek the inhomogeneous solutions to Eq. (\[eq:ode\_sk\]) and (\[eq:ode\_so\]). Notice that the inhomogeneous term of Eq. (\[eq:ode\_sk\]) and (\[eq:ode\_so\]) contain $u_\kappa$ and $u_\omega$ respectively, so the trial solutions can be fabricated in such a form $$\begin{aligned}
\label{eq:trial_sk_part}
\tilde s_\kappa &= C_\kappa(\tau)u_\kappa + D_\kappa(\tau)\dot u_\kappa, \\
\label{eq:trial_so_part}
\tilde s_\omega &= C_\omega(\tau)u_\omega + D_\omega(\tau)\dot u_\omega,\end{aligned}$$ Inserting the trial solution Eq. (\[eq:trial\_sk\_part\]) back into Eq. (\[eq:ode\_sk\]) and comparing the coefficients of terms $u_\kappa$ and $\dot u_\kappa$, we get two ODEs for $C_\kappa(\tau)$ and $D_\kappa(\tau)$ $$\begin{aligned}
\label{eq:ode_CD1}
\dot C_\kappa(\tau)-a\kappa^2 D_\kappa(\tau) &= -af(\tau) \\
\label{eq:ode_CD2}
\dot D_\kappa(\tau) &= aC_\kappa(\tau)\end{aligned}$$ Taking time derivative of Eq. (\[eq:ode\_CD1\]) and substituting Eq. (\[eq:ode\_CD2\]) to eliminate $D_\kappa(\tau)$ yields $$\label{eq:ode_Ck}
\ddot C_\kappa(\tau)-a^2\kappa^2 C_\kappa(\tau) = -a\dot f(\tau).$$ According to Eq. (\[eq:f\]) we know $\dot f(\tau)=\dot f_0\cos(a\Omega\tau)+a\Omega^{-1}h_0\sin(a\Omega\tau)$ where $h_0\equiv\mathcal{I}_3-f_0\Omega^2$, so we can get one of the solutions to $C_\kappa(\tau)$ $$\label{eq:Ck_app}
C_\kappa(\tau) = \frac{\Omega\dot f_0\cos(a\Omega\tau)+ah_0\sin(a\Omega\tau)}{a\Omega(\kappa^2+\Omega^2)}.$$ Then $D_\kappa(\tau)$ is obtained by integrating Eq. (\[eq:ode\_CD2\]) $$\label{eq:Dk_app}
D_\kappa(\tau) = \frac{\Omega\dot f_0\sin(a\Omega\tau)-ah_0\cos(a\Omega\tau)}{a\Omega^2(\kappa^2+\Omega^2)}$$ The ODEs of $C_\omega$ and $D_\omega$ can be similarly established by inserting Eq. (\[eq:trial\_so\_part\]) into Eq. (\[eq:ode\_so\]) and comparing the coefficients $$\begin{aligned}
\label{eq:ode_CD3}
\dot C_\omega(\tau)+a\omega^2 D_\omega(\tau) &= -af(\tau) \\
\label{eq:ode_CD4}
\dot D_\omega(\tau) &= aC_\omega(\tau)\end{aligned}$$ The solutions are $$\begin{aligned}
\label{eq:Co_app}
C_\omega(\tau) &= \frac{\Omega\dot f_0\cos(a\Omega\tau)+ah_0\sin(a\Omega\tau)}{a\Omega(\Omega^2-\omega^2)}, \\
\label{eq:Do_app}
D_\omega(\tau) &= \frac{\Omega\dot f_0\sin(a\Omega\tau)-ah_0\cos(a\Omega\tau)}{a\Omega^2(\Omega^2-\omega^2)}.\end{aligned}$$
Singularity treatment in particular solutions for $\mathcal I_1\neq0,~\mathcal I_2\neq0$ {#sect:singularity_1}
========================================================================================
For general case where $\mathcal I_1\neq0$ and $\mathcal I_2\neq0$, or equivalently $\kappa\neq0$ and $\omega\neq0$, singularity appears in Eqs. (\[eq:Ck\_app\]), (\[eq:Dk\_app\]), (\[eq:Co\_app\]) and (\[eq:Do\_app\]) when $\Omega=0$, $\Omega^2+\kappa^2=0$ or $\Omega^2-\omega^2=0$ is satisfied.
For $\Omega=0$, according to Eq. (\[eq:ode\_f\]), we have $$f(\tau)= f_0 + \dot f_0\tau + \frac{1}{2}a^2\mathcal{I}_3\tau^2.$$ Substituting into Eqs. (\[eq:ode\_CD1\]) and (\[eq:ode\_CD2\]) we can get one of the possible solutions $$\begin{aligned}
C_\kappa(\tau) &= a^{-1}\kappa^{-2}(\dot f_0+a^2\mathcal{I}_3\tau) \\
D_\kappa(\tau) &= \kappa^{-2}\left(\dot f_0\tau+\frac{1}{2}a^2\mathcal{I}_3\tau^2\right).\end{aligned}$$ Similarly, $C_\omega$ and $D_\omega$ are given by $$\begin{aligned}
C_\omega(\tau) &= -a^{-1}\omega^{-2}(\dot f_0+a^2\mathcal{I}_3\tau) \\
D_\omega(\tau) &= -\omega^{-2}\left(\dot f_0\tau+\frac{1}{2}a^2\mathcal{I}_3\tau^2\right).\end{aligned}$$
When $\Omega^2+\kappa^2=0$ which implies $|u_{\kappa0}|=0$ and $|u_{\omega0}|=1$ according to the definition of $\Omega$, the original expressions for $C_\kappa$ and $D_\kappa$ present singularity. According to Eq. (\[eq:f\]), note that $\mathcal I_3=-\kappa^2 f_{\omega0}$, $h_0=0$ and $\dot f_0=0$ (it can be verified by using $|u_{\kappa0}|=1$ and $(u_{\kappa0}|s_{\kappa0})+(u_{\omega0}|s_{\omega0})=0$) in this situation, $f(\tau)$ becomes a constant $f(\tau)=f_{\omega0}$. One of the solutions to Eqs. (\[eq:ode\_CD1\]) and (\[eq:ode\_CD2\]) is $$\label{eq:CD_k1_app_s1}
C_\kappa=0,\quad D_\kappa=\kappa^{-2}f_{\omega0}.$$ The coefficients $C_\omega$ and $D_\omega$ still follow Eqs. (\[eq:ode\_CD3\]) and (\[eq:ode\_CD4\]), which gives $$\label{eq:CD_o1_app_s1}
C_\omega=0,\quad D_\omega=-\omega^{-2}f_{\omega0}.$$
When $\Omega^2-\omega^2=0$ implying $|u_{\kappa0}|=1$ and $|u_{\omega0}|=0$, we can similarly verify that $\mathcal I_3=\omega^2 f_{\kappa0}$, $h_0=0$ and $\dot f_0=0$, which yields $f(\tau)=f_{\kappa0}$. Therefore, the coefficients can be selected as one of the solutions to Eqs. (\[eq:ode\_CD1\]), (\[eq:ode\_CD2\]), (\[eq:ode\_CD3\]) and (\[eq:ode\_CD4\]) $$\label{eq:CD_k2_app_s1}
C_\kappa=0,\quad D_\kappa=\kappa^{-2}f_{\kappa0},$$ and $$\label{eq:CD_o2_app_s1}
C_\omega=0,\quad D_\omega=-\omega^{-2}f_{\kappa0}.$$
Singularity treatment in particular solutions for $\mathcal I_1\neq0$, $\mathcal I_2=0$ {#sect:singularity_2}
=======================================================================================
For the situation $\mathcal I_1\neq0$ and $\mathcal I_2=0$, or equivalently either $\kappa$ or $\omega$ vanishes, the singularity treatment is different. When $\kappa=0$ and $\omega\neq0$ there is singularity at $\Omega=0$ and $\Omega=\omega$. According to the definition $\Omega=0$ implies $|u_{\kappa0}|=0$ and $|u_{\omega0}|=1$, from which we know $\mathcal I_3=0$, $f_0=f_{\omega0}$ and $\dot f_0=\dot f_{\omega0}$. The solution to $f(\tau)$ derived from Eq. (\[eq:ode\_f\]) is thus given by $f(\tau)=f_{\omega0}+\dot f_{\omega0}\tau$. Inserting into Eqs. (\[eq:ode\_CD1\]), (\[eq:ode\_CD2\]), (\[eq:ode\_CD3\]) and (\[eq:ode\_CD4\]) the coefficients can be selected as $$C_\kappa = -a\tau\left( f_{\omega0}+\frac{1}{2}\dot f_{\omega0}\tau \right), \quad
D_\kappa = -\frac{1}{2}a^2 \tau^2 \left( f_{\omega0}+\frac{1}{3}\dot f_{\omega0}\tau \right),$$ and $$C_\omega = -a^{-1}\omega^{-2}\dot f_{\omega0}, \quad
D_\omega = -\omega^{-2} \dot f_{\omega0}\tau.$$ When $\Omega=\omega$ is satisfied, which implies $|u_{\kappa0}|=1$ and $|u_{\omega0}|=0$, we have $\mathcal I_3=\omega^2 f_{\kappa0}$, $f_0=f_{\kappa0}$ and $\dot f_0=0$. Eq. (\[eq:ode\_f\]) gives out $f(\tau)=f_{\kappa0}$. Inserting into Eqs. (\[eq:ode\_CD1\]), (\[eq:ode\_CD2\]), (\[eq:ode\_CD3\]) and (\[eq:ode\_CD4\]) the coefficients can be given by $$C_\kappa = -a f_{\kappa0}\tau, \quad
D_\kappa = -\frac{1}{2}a^2 f_{\kappa0} \tau^2$$ and $$C_\omega = 0, \quad
D_\omega = -\omega^{-2}f_{\kappa0}$$
For $\omega=0$ and $\kappa\neq0$ we can similarly obtain these coefficients. When $\Omega=0$, $\mathcal I_3=0$, $f_0=f_{\kappa0}$ and $\dot f_0=\dot f_{\kappa0}$, which gives $f(\tau)=f_{\kappa0}+\dot f_{\kappa0}\tau$ according to Eq. (\[eq:ode\_f\]). Combining with Eqs. (\[eq:ode\_CD1\]), (\[eq:ode\_CD2\]), (\[eq:ode\_CD3\]) and (\[eq:ode\_CD4\]), it can be shown that $$C_\omega = -a\tau\left( f_{\kappa0}+\frac{1}{2}\dot f_{\kappa0}\tau \right), \quad
D_\omega = -\frac{1}{2}a^2 \tau^2 \left( f_{\kappa0}+\frac{1}{3}\dot f_{\kappa0}\tau \right),$$ and $$C_\kappa = a^{-1}\kappa^{-2}\dot f_{\kappa0}, \quad
D_\kappa = \kappa^{-2} \dot f_{\kappa0}\tau.$$ When $\Omega={\text{i}}\kappa$, we have $\mathcal I_3=-\kappa^2 f_{\omega0}$, $f_0=f_{\omega0}$ and $\dot f_0=0$. Eq. (\[eq:ode\_f\]) gives out $f(\tau)=f_{\omega0}$. Inserting into Eqs. (\[eq:ode\_CD1\]), (\[eq:ode\_CD2\]), (\[eq:ode\_CD3\]) and (\[eq:ode\_CD4\]) the coefficients can be given by $$C_\omega = -a f_{\omega0}\tau, \quad
D_\omega = -\frac{1}{2}a^2 f_{\omega0}\tau^2,$$ and $$C_\kappa = 0, \quad
D_\kappa = \kappa^{-2} f_{\omega0}.$$
|
---
abstract: 'The addition of nuclear and neutrino physics to general relativistic fluid codes allows for a more realistic description of hot nuclear matter in neutron star and black hole systems. This additional microphysics requires that each processor have access to large tables of data, such as equations of state, and in large simulations the memory required to store these tables locally can become excessive unless an alternative execution model is used. In this work we present relativistic fluid evolutions of a neutron star obtained using a message driven multi-threaded execution model known as ParalleX. These neutron star simulations would require substantial memory overhead dedicated entirely to the equation of state table if using a more traditional execution model. We introduce a ParalleX component based on Futures for accessing large tables of data, including out-of-core sized tables, which does not require substantial memory overhead and effectively hides any increased network latency.'
author:
-
bibliography:
- 'IEEEabrv.bib'
- 'pxBib.bib'
title: |
Neutron Star Evolutions using Tabulated Equations of State\
with a New Execution Model
---
Astrophysics applications, ParalleX, HPX, Futures
Introduction
============
Future achievements in leading-edge science demand innovations in parallel computing models and methods to improve efficiency and dramatically increase scalability. One controversial issue is the relative value of global address space models and management versus more conventional distributed memory structure. This paper demonstrates one important use of global address space in the context of the advanced ParalleX execution model that has enabled simulation improvements to be achieved in the domain of astrophysics that is not feasible using conventional practices.
Accurate treatment of hot nuclear matter in astrophysical compact object simulation is becoming increasingly important in the search for coincident detection of gravitational radiation and electromagnetic or neutrino emissions originating from the same source. Recent simulations have shown that the gravitational wave signature from a binary neutron star merger or a neutron star–black hole merger may even reveal important empirical details about the neutron star equation of state itself [@Read2009; @Lackey2011].
The addition of nuclear and neutrino physics to general relativistic fluid codes allows for a more realistic description of hot nuclear matter in neutron star and black hole systems. Unfortunately, most of these and other microphysics routines cannot be computed in place; they must be precomputed in a large table which is then read and interpolated as the relativistic hydrodynamics simulation proceeds. Accurate neutron star simulations will increasingly rely upon ever larger tables of microphysics data which must be read into memory, searched, and interpolated [@Ott2010]. As the memory requirements grow for these tables, approaching and often exceeding the size of the physical memory of a single node, the need for an alternative to the traditional MPI-OpenMP hybrid approach is increasingly evident.
In this work we explore the experimental execution model called ParalleX [@gao; @scaling_impaired_apps; @tabbal; @Dekate2012] as a means of addressing the critical computational requirement of dealing with very large tables containing microphysics. ParalleX provides many other performance benefits to a distributed relativistic hydrodynamics simulation but the focus of this work will be on matters related to equation of state tables.
ParalleX is a synthesis of complementing semantic constructs delivering a dynamic adaptive framework for message-driven multi-threaded computing in a global address space context with constraint-based synchronization to exploit locality and manage asynchrony. The result is introspective runtime alignment of computing requirements and computing resources while permiting asynchronous operation across physically distributed resources. ParalleX has been first implemented in the form of the HPX runtime system [@hpx_svn; @amr1d], developed to support the semantics and mechanisms comprising ParalleX targeting conventional SMP and commodity cluster computing platforms. This experimental software package is developed to test the semantics of ParalleX, to measure the overhead costs of software implementation, and to provide a prototype of the next generation runtime system for extreme scale applications.
The outline of this paper as follows: section \[sec:fluid\] describes the relativistic fluid evolution, initial data, numerical methods, and equation of state details; section \[sec:hpx\] gives a brief overview of the HPX runtime system implementation of ParalleX; section \[sec:eos\] describes how the equation of state is distributed across nodes and how Futures are used to hide network latency in table access; section \[sec:results\] gives neutron star evolution performance results using the Shen equation of state comparing the Futures approach of accessing the equation of state table with that of reading in the table for every core (referred to hereafter as the conventional approach); section \[sec:conclusion\] gives our conclusions and implications for future work.
Relativistic Hydrodynamics {#sec:fluid}
==========================
The work presented here adopts the flux-conservative formulation of the relativistic magnetohydrodynamics equations presented in [@Anderson2006] and includes high-resolution shock capturing (HRSC) methods. To calculate the numerical fluxes, we use the Piecewise Parabolic Method (PPM) [@ppm] for reconstructing fluid variables. The approximate Riemann solver employed is Harten-Lax-vanLeer-Einfeldt (HLLE) [@hlle]. While the code is capable of also evolving magnetic fields, no magnetic fields were added to the initial data at this stage. Neutron star simulations presented in section \[sec:results\] were conducted using the Cowling approximation.
The tabulated equation of state used for generating the initial neutron star and all subsequent fluid evolution is the Shen equation of state [@1998NuPhA.637..435S]. A tabulated Shen equation of state was provided by C. Ott and is available for download at [@stellarcollapse_webpage]. The publicly available tabulated Shen equation of state used for tests here is 288 MB; however, an updated Shen equation of state has since been released [@Shen2011] and we have created a table based on this update which is 5.9 GB in size. Consequently, results from both tables will be presented in the results section. The neutron star initial data was generated using the Lorene libraries [@lorene_webpage]; the neutron star has a mass of 1.4 solar masses and radius of 15.947 km. Tables related to neutrino transport, including neutrino opacity, were not included in this work but are part of future work.
The entire relativistic magnetohydrodynamics solver has been implemented in C++ using the HPX runtime system for parallelism. The application has been modularized for future incorporation into the HPX Adaptive Mesh Refinement (AMR) toolkit although simulations for this work performed all computations in unigrid. The next section will give a brief overview of HPX and introduce the concepts crucial for asynchrony management in tabular access and parallel computation in general.
The HPX Runtime System {#sec:hpx}
======================
{width="0.99\linewidth" height="0.4\textheight"}
The C++ prototype runtime implementation of ParalleX is called High Performance ParalleX (HPX). A walkthrough description of the HPX architecture is found in Figure \[fig:hpxarch\]. An incoming parcel (delivered over the interconnect) is received by the parcel port. One or more parcel handlers are connected to a single parcel port, optionally allowing to distinguish different parts of the system as the parcel’s final destination. An example for such different destinations is to have both normal cores and special hardware (such as a GPGPU) in the same locality (ParalleX term identifying synchronous domain of computation, such as a single compute node in a cluster). The main task of the parcel handler is to buffer incoming parcels for the action manager. The action manager decodes the parcel, which contains an action bundled with relevant operands. An action is either a global function call or a method call on a globally addressable object. The action manager creates a HPX-thread based on the encoded information.
All HPX-threads are managed by the thread manager, which schedules their execution on one of the OS-threads allocated to it. HPX threads are implemented as user level threads, which decreases the costs associated with their creation, destruction, and state updates by minimizing the number of interactions with the OS kernel. HPX creates one worker OS-thread for each available core, whose purpose is to carry out the majority of computations in an application. Several scheduling policies have been implemented for the thread manager, such as the global queue scheduler, where all cores pull their work from a single global queue, or the local queue scheduler, where each core pulls its work from a separate queue. The latter supports work stealing for better load balancing. In the local scheduler, a queue is created for each of the worker OS-threads. When a worker thread is searching for work, it first checks its own queue. If there is no work there, the OS-thread begins to steal work by searching for work in other queues, first from its own non-uniform memory access (NUMA) domain, then from cores located on different NUMA domains.
If a possibly remote action has to be executed by a HPX-thread, the action manager queries the active global address space (AGAS) to determine whether the target of the action is local or remote to the locality that the HPX-thread is running on. If the target happens to be local, a new HPX-thread is created immediately and passed to the thread manager. This thread encapsulates the work (function) and the corresponding arguments for that action. If the target is remote, the action manager creates a parcel encoding the action (i.e. the function and its arguments). This parcel is handed to the parcel handler, which makes sure that it gets sent over the interconnect, causing a new HPX-thread to be created at the target locality.
The Active Global Address Space (AGAS) provides global address resolution services that are used by the parcel port and the action manager. AGAS addresses are 128bit unique global identifiers (GIDs). AGAS maps these global identifiers to local addresses, and additionally provides symbolic mappings from strings to GIDs. The local addresses that GIDs are bound to are typed, providing a degree of protection from type errors. Any object that has been registered with a GID in AGAS is addressable from all localities in an instance of the HPX runtime. AGAS also provides a powerful reference counting system which implements transparent and automatic global garbage collection.
Lightweight Control Objects (LCOs) are the synchronization primitives upon which HPX applications are built. LCOs provide a means of controlling parallelization and synchronization of HPX-threads. Semaphores, mutexes and condition variables [@mutex] are all available in HPX as LCOs. Futures [@future1] are another type of LCO provided by HPX, and are discussed in greater detail later in this paper.
Local memory management, performance counters (a generic monitoring framework), LCOs and AGAS are all implemented on top of an underlying component framework. Components are the main building blocks of remotely executable actions and can encapsulate arbitrary, possibly application specific functionality. Actions are special types which expose the functionality of a (possibly remote) function. An action can be invoked on a component instance using only its GID, which allows any locality to invoke the exposed methods of a component. In the case of the aforementioned components, the HPX runtime system implements its own functionality in terms of this component framework. Typically, any application written using HPX extends the set of existing components based on its requirements.
The relativistic hydrodynamics simulations make use of all the key features of HPX. The most crucial feature for contention and network latency hiding in tabulated equation of state access is the Future [@future1; @future2]. The next section will describe the strategy adopted in HPX for distributing a large table and hiding network latency.
Using the Shen Equation of State Tables {#sec:eos}
=======================================
The Shen equation of state (EOS) tables of nuclear matter at finite temperature and density with various electron fractions within the relativistic mean field (RMF) theory are a set of three dimensional data arrays enabling high precision interpolation of 19 relevant parameters required for neutron star simulations. As noted in section \[sec:fluid\], the publicly available Shen equation of state table is relatively small in size (288MB); however, the most recent Shen table created for the neutron star evolutions presented here is 5.9GB in size. Results using both tables will be presented in section \[sec:results\]. In the case of the larger table, loading the whole data set into main memory on each locality is not feasible. In conventional MPI based applications the full tables would have to be either loaded into each MPI process or a distributed partitioning scheme would have to be implemented. These options are either not viable or difficult to implement using MPI.
Interpolation Technique and Characteristics {#subsec:interpolation}
-------------------------------------------
The values of each of 19 variables describing the Shen EOS are contained in individual 3-D tables stored in memory in a row-major fashion. Single table data are arranged as samples of a single physical quantity computed at coordinates laying on a regularly spaced grid. The sizes of each grid vary from $220\times180\times50$ for the smaller 288MB set to $440\times360\times130$ for the large, 5.9GB set. The dimensions correspond to baryon mass density, temperature, and electron fraction, respectively.
To obtain the value of a variable at an arbitrary point within the 3-D domain, a $2\times2\times2$ cube of double-precision floating numbers must be accessed. The result is computed using a simple tri-linear interpolation from these values. Our neutron star simulations require only 8 out 19 tabulated quantities, which helps reduce the memory pressure. However, even when using a single instance of the smaller set of tables on each node to permit the efficent sharing of table data among all cores, the aggregate size of accessed data volume still significantly exceeds the combined size of L3 processor caches. Furthermore, given that in each interpolation request effectively at most $2\times8=16$ out of 64 bytes per cache line (x86 architecture) are used and the coordinate stream generated by the application is random, the performance of interpolation function is memory bound.
The Overhead of Futures {#subsec:futures}
-----------------------
Many HPX applications, including the relativistic hydrodynamics simulation detailed here, utilize Futures for ease of parallelization and synchronization. In HPX, Futures are implemented as two types of LCOs, Eager Futures and Lazy Futures, differing in the evaluation mode of produced value. In Eager Futures the computation providing the result value is triggered as soon as the Future object is instantiated, whereas for Lazy Futures it remains dormant until at least one consumer of the value references it. The neutron star simulation detailed in this paper makes extensive use of Eager Futures (we will refer to them simply as Futures for brevity). For this reason, the overheads of these constructs are a large factor in the total overhead of the HPX runtime in our code. In this subsection, we give a description of Futures, outline a performance test for measuring the overhead of Futures, and present the results of the test.
![ []{data-label="fig:future_schematics"}](figures/future_schematics){width="0.99\linewidth"}
As shown in Figure \[fig:future\_schematics\], a Future encapsulates a delayed computation. It acts as a proxy for a result initially not known, most of the time because the computation of the result has not completed yet. The Future synchronizes the access of this value by optionally suspending HPX-threads requesting the result until the value is available. When a Future is created, it spawns a new HPX-thread (either remotely with a parcel or locally by placing it into the thread queue) which, when run, will execute the action associated with the Future. The arguments of the action are bound when the Future is created. Once the action has finished executing, a write operation is performed on the Future. The write operation marks the Future as completed, and optionally stores data returned by the action.
When the result of the delayed computation is needed, a read operation is performed on the Future. If the Future’s action hasn’t completed when a read operation is performed on it, the reader HPX-thread is suspended until the Future is ready.
Our benchmark for Future overhead created a fixed number of Futures, each of which had a fixed workload. Then asynchronous read operations were performed on the Futures until all of the Futures had completed. A high resolution timer measured the wall-time of the aforementioned operations. The test was run on an 8-socket HP ProLiant DL785 (each socket sports a 6-core AMD Opteron 8431) with 96 GB of RAM (533 MHz DDR2). Trials were done with varying workloads and OS-threads. Five runs were performed for each combination of the parameters and the results were averaged to produce a final dataset. The numbers are presented in Figure \[fig:figure\_3\].
![ []{data-label="fig:figure_3"}](figures/figure_3){width="0.99\linewidth"}
On the locality we used for this benchmark, the amortized overhead of a Future is approximately *17 microseconds*. This overhead includes the time required to create the Future instance, start the evaluator thread, perform synchronization with the accessors, and destruct the Future object. This number was extrapolated from the data presented in Figure \[fig:figure\_3\]. We multiplied the workload by the number of Futures used in each run, and then subtracted that from the average wall-time of the trial. We divided that number by the number of Futures invoked in the trial to get the overhead per Future for each set of parameters.
$$\begin{aligned}
& \mbox{\textit{overhead}} = \frac{\mbox{\textit{avg. wall-time}}\,-\,(\mbox{\textit{workload}}\,*\,\mbox{\textit{futures invoked}})}{\mbox{\textit{futures invoked}}}\end{aligned}$$
The scaling results in Figure \[fig:figure\_3\] call for some explanation. The parabolic curves are formed primarily by contention in the thread queue scheduler. As the number of OS-threads is increased, the contention on the thread queue scheduler also increases, due to a higher number of concurrent searches for available work. This increased contention occurs in both global queue schedulers (where all OS-threads poll the same work queue, and must obtain exclusive access to said queue for some period of time) and to some extent in local queue schedulers (where work stealing occurs, which causes queue contention, albeit to a lesser degree than in the global queue scheduler). As we increase the workload in each Future, OS-threads spend more time executing the workloads and less time searching for more work. This decreases contention on the queues. Adding a new OS-thread is beneficial as long as the contention overhead that it causes is not greater than the parallel speedup that it provides.
The Overhead of the Shen EOS Table Partitioning {#subsec:sheneos}
-----------------------------------------------
We created an HPX component encapsulating the minimally overlapping partitioning (ghost zone of single element width) and distribution of the Shen EOS tables to all available localities, thus reducing the required memory footprint per locality [@sheneos_svn]. A special client side object ensures the transparent dispatching of interpolation requests to the appropriate partition corresponding to the locality holding the required part of the tables (see Figure \[fig:sheneos\]). The client side object exposes a simple API for easy programmability.
![ []{data-label="fig:sheneos"}](figures/sheneos){width="0.99\linewidth"}
The second part of this section describes the setup and results of the measurements we performed in order to estimate the overheads introduced by distributing the Shen EOS tables across all localities. To evaluate the scalability and associated overheads of the distributed implementation of the Shen EOS tables, a number of tests have been performed, all of them with a fixed number of total data accesses (measuring strong scaling). The tests have been run on a different number of localities and with varying numbers of OS-threads per locality. The current HPX implementation supports only a centralized AGAS server that may be invoked in two configurations: either as a standalone task on a dedicated locality or as a part of one of the user application tasks. Our tests used a standalone AGAS server, firstly to avoid interfering with the user workload and secondly to eliminate the generation of asymmetric AGAS traffic on localities hosting data tables. Unlike the client applications, the AGAS server used a fixed number of OS-threads throughout the testing to ensure that sufficient processing resources are available to the incoming resolution requests.
The tests were performed on a small heterogeneous cluster. The cluster consists of 18 localities (excluding the head node) connected by Gigabit Ethernet network. Two of the machines are 8-socket HP ProLiant DL785s, with 6-core AMD Opteron 8431s and 96 GB of RAM (533 MHz DDR2). The other 16 localities are single-socket HP ProLiant DL120s, with Intel Xeon X3430s and 4 GB of RAM (1332 MHz DDR3). All machines run x86-64 Debian Linux. Torque PBS was used to run multi-locality tests.
Figure \[fig:sheneos\_execution\_time\] shows the execution times collected for the data access phase with a special test application executed on up to 16 localities and 1, 2, and 4 OS-threads per locality. The total number of distributed partitions was fixed at 32 to preserve the AGAS traffic pattern when run on a different number of localities; all partitions were uniformly distributed across the test localities. The number of separate, non-bulk queries to the distributed Shen EOS partitions was set to a fixed number of 16K. Each of these queries created a Future encapsulating the whole operation of sending the request to the remote partition, schedule and execute a HPX-thread, perform the interpolation based on the supplied arguments for the Shen EOS data, sending back the resulting values to the requesting HPX-thread, and resuming the HPX-thread which was suspended by the Future in order to wait for the results to come back.
![ []{data-label="fig:sheneos_execution_time"}](figures/sheneos_execution_time){width="0.99\linewidth"}
The graph demonstrates that the overhead of distributed table implementation does not increase significantly over the entire range of available localities. While the scaling is much better when the number of localities remains small (up to 4), the overall time required to service the full 16K data lookup requests remains roughly constant. The test application itself does not execute any work besides querying and interpolating the distributed tables, which does not leave much room to overlap the significant network traffic generated with useful computation. This causes the scaling to flatten out beyond 8 localities. Using the distributed tables in real applications doing much more work will allow to further amortize the introduced network overheads. The results also imply that a single AGAS server is quite capable of servicing at least 16 client localities, especially considering the intensity of request traffic over Ethernet interconnect deployed in out testbed. We plan to further evaluate this aspect of distributed table implementation using faster interconnect networks, such as Infiniband.
Results {#sec:results}
=======
Accessing a single, potentially distributed Shen equation of state table using multiple threads for converting conservative variables to primitives and vice versa as required for the flux-conservative HRSC method results in a slowdown when compared with using multiple independent copies of the table. There is also some additional overhead in using futures in the tabular access. In Figure \[fig:ws\_shen\] the table access slowdown relative to single core on a shared memory machine is presented. The results are a weak scaling test where the results for each workload have been normalized to the corresponding single core performance. In this test, each core accesses and interpolates 64k unique values in the table as a single bulk operation. Using HPX on a single core of an Intel Xeon X5660 processor, this test takes 0.0728 seconds; for comparison, using the Fortran codes provided at [@stellarcollapse_webpage] access and interpolation of the exact same 64k values takes 0.0549 seconds, reflecting the increased overhead in using HPX. As the number of cores accessing the same table increases, the access performance degrades. The primary reason for that is the competition of hardware threads executing on the same processor for access to memory, since most of the interpolation requests cannot be satisfied solely from processor caches (for the machine used in test, the ratio of utilized fraction of EOS dataset to the aggregate size of L3 caches was about 5). This is compounded even further by the fact that accesses are sparse and random in nature, and therefore result in decrease of the effective memory bandwidth. When no additional work is overlapped with the table access, the table access slowdown relative to a single core can reach as high as a factor of 3, an unacceptably high number for neutron star simulations. However, by overlapping work with the table access futures, the slowdown relative to a single core becomes much more reasonable.
![[]{data-label="fig:ws_shen"}](figures/figure_6_ws){width="0.99\linewidth"}
For relatively small tables like the 288 MB table explored here, the memory cost of reading in a table for each core might be a manageable strategy in order to avoid any higher overhead costs associated with sharing the table using futures. For very large tables, however, there is no other viable option: the table would have to be distributed across several nodes and shared using futures. Using the recently released improved Shen equation of state [@Shen2011], we have created Shen tables 5.9 GB in size for neutron star simulations. The increased table resolution contributes to improving the accuracy and robustness of neutron star evolutions as well as includes the most recently improved RMF results.
In Figure \[fig:ws\_distrshen\] the table access slowdown relative for the larger table is presented. This figure presents a weak scaling table access test where each core accesses and interpolates 64K different table queries. Because of the size of the table and the need for memory dedicated to the fluid field meshes and evolution, this table has to be distributed. The distributed results presented have been normalized to the corresponding two node performance. We use 8 cores in each node, consisting of two quad-core Intel Nehalem (2.8 GHz) processors with gigabit ethernet interconnect. The only workload provided was that of table access and interpolation; no additional workload was added. For this larger table, the relative slowdown in distributing the table is, at worst, a factor of 1.2 if distributed across 5 nodes with a benefit of saving enormous amounts of memory. There is no toolkit or routine at present based on a different execution model against which to compare this result.
![[]{data-label="fig:ws_distrshen"}](figures/figure_7_ws){width="0.99\linewidth"}
In Figure \[fig:ns\] we evolve a neutron star with the Shen equation of state on a shared memory machine (Intel Xeon X5660, 12 cores) for 10 iterations using a grid with $50^3$ points across the computational domain and compare performance between the Futures based table access approach and the conventional approach of reading in a separate table for each core. By necessity this comparison had to use the smaller (288 MB) Shen table because using the larger table would have exceeded the memory available when testing the conventional approach. The Futures based table access performs extremely well compared to the conventional approach with negligible slowdown on up to 4 cores. At worst, when the table is shared across 12 cores, the slowdown is a factor of 1.13, consistent with the numbers presented in Figure \[fig:ws\_shen\] for the 14$\mu$s workload case.
![[]{data-label="fig:ns"}](figures/figure_8_ws){width="0.99\linewidth"}
Conclusion {#sec:conclusion}
==========
We have examined finite temperature tabulated equation of state access in the context of neutron star simulations using a relatively new execution model called ParalleX. Using Futures to manage asynchrony, amortize contention, and hide network latency, we have presented a strategy for performing neutron star evolutions using extremely large tabulated equations of state with minimal performance and memory cost. Using the Futures based partitioned Shen EOS table access the slowdown compared to the conventional way of accessing these tables is less than $\sim$15%, and often much less than that. This added cost is justifiable, since as larger tables become available in simulation efforts, astrophysics simulations can then achieve a more realistic description of hot nuclear matter and incorporate more microphysics, including neutrino transport. Managing large tables in this asynchronous way would be difficult to implement when using conventional programming models, such as MPI.
Several key improvements to the results presented here are currently underway. As the HPX runtime system becomes NUMA aware, much of the memory contention observed here in both OpenMP and HPX runs can be eliminated [@heller]. Ways to reduce the Futures overhead reported in Fig. \[fig:figure\_3\] even further are currently under investigation. All distributed runs presented here used gigabit Ethernet interconnect; however, HPX support for the native Verbs interface for Infiniband is also underway. Hardware support for AGAS translation, whose first-cut implementation could utilize FPGA (Field-Programmable Gate Array) technology, promises to reduce key overheads, both in execution and storage, related to the software implementation. OpenCL support via percolation in HPX is also under development and could subtantially impact the capability to perform neutrino transport in neutron star simulations.
As a final note, we point out that switching from a message passing to message driven style computation for neutron star simulations has significant performance impacts beyond just those discussed in this paper involving the finite temperature equation of state tables. While those improvements are outside the scope of this work, the key concepts of managing asynchrony, amortizing contention, and hiding network latency make a significant positive impact in the scalability of neutron star simulations.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Steven Brandt, Luis Lehner, and David Neilsen for stimulating discussions. We also thank Lloyd Brown for his technical assistance. Computations were done at Brigham Young University (Marylou5), Lousiana State University (Hermione), and Indiana University. This work was supported by the NSF under grants CNS-1117470 and CCF-1142905.
|
---
abstract: '[We study the black body radiation in cavities of different geometry with different boundary conditions, and the formulas of energy spectral densities in films and rods are obtained, also the approximate energy spectral densities in cubic boxes are calculated. We find that the energy spectral densities deviate from Planck’s formula greatly when the length(s) at some directions of the cavity can be compared to the typical wavelengths of black body radiation in infinite volume and the boundary conditions also affect the results. We obtain an asymptotic formula of energy spectral density for a closed cavity with Dirichlet boundary condition and find it still fails when the size of the cavity is too small.]{}'
author:
- Zheng Zhang
- Tong Zhao
- 'Y.-H Xia'
- Hongshi Zong
bibliography:
- 'reference.bib'
title: Geometry Effect on Black Body Radiation with Different Boundary Conditions
---
Introduction {#sec:Introduction}
============
In early days of quantum mechanics, black body radiation played an important role. In 1901, Planck first proposed the concept of energy quantization when he studied black body radiation [@planck], and this is just the origin of quantum mechanics. It is well known that the Planck’s formula (PF) for blackbody radiation states that the energy spectral density only depends on temperature, regardless of the shape and the size of the cavity. The derivation of this formula assumes that the size of the cavity is large enough and thus the density of state (the number of modes in a frequency interval) is independent of the geometry of it. Now it’s natural to ask: Can PF correctly describe the blackbody radiation when the cavity size is no longer large enough? To illustrate this clearly, let us consider the following case: at 300K, more than 99% energy is carried by the modes whose wavelengths are larger than 3$\times 10^{-6}$ m according to Planck’s formula. However, for example, in a cube whose length is shorter than 3$\times 10^{-6}$ m, the modes larger than 3$\times 10^{-6}$ m can’t exist, so the spectral density in this cube will deviate greatly from the PF. Just as pointed out in Ref. [@rytov] that the PF is only applicable if $(\lambda/L)^3\ll\Delta\lambda/\lambda\ll 1$, where $L$ is the length of the cube and $\Delta\lambda$ is the wavelength interval. This clearly shows that when the size of the cavity is small enough, the PF is no longer suitable, so we need to develop new methods to study the blackbody radiation in small cavities.
After Planck’s publication, the wave equation’s eigenvalue distribution was investigated in Refs. [@weyl; @asy; @cou; @asy1; @asy2; @asy3], which is related to the correction of Planck’s formula. By their method of asymptotic expansion, we can get some correction terms which relate to the geometry properties of the cavity as well as the boundary conditions. However, as indicated in Ref. [@pra], we will show in this paper, this asymptotic expansion method has its inherent limitations, that is, the size of the cavity is limited small but still large enough compared to the wavelength of black body radiation. When the size of the cavity is small enough, more exact but laborious method is needed. In this work, we intend to study how geometry and boundary conditions affect blackbody radiation when the size of the cavity is small enough. This paper is organized as follows: In Sec II A, we study the energy spectral density in film and rod with periodic, antiperiodic and Dirichlet boundary conditions respectively. In Sec II B, we calculate the approximate energy spectral density in cubic box with periodic, antiperiodic and Dirichlet boundary conditions respectively. In addition, we compare the differences in energy spectral densities caused by geometry and boundary conditions. We also obtain an asymptotic formula of energy spectral density for a closed cavity with Dirichlet boundary condition and show in what range it is better than the PF and in what range it fails.
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black body radiation in different cavities with different boundary conditions {#sec:film}
=============================================================================
Film and Rod
------------
First, let’s review the derivation of PF. Consider an ensemble of oscillators in thermal equilibrium, by assuming the quantized values of an oscillator’s energy, we obtain an oscillator with frequency $\omega$ has average energy $\overline{\epsilon}(\omega)=\hbar\omega[\mathrm{exp}(\hbar\omega/k_BT)-1]^{-1}$, where $k_B$ is the Boltzmann constant. Then we calculate the density of states (DOS) $g(\omega)\mathrm{d}\omega$ in the cavity, which means the number of modes in the interval $(\omega,\omega+\mathrm{d}\omega)$. If we assume that the size of the cavity is large enough, $\omega$ becomes continuous and we can get $g(\omega)={V}\omega^2/{\pi^2c^3}$, which is independent of the geometry of the cavity. Finally, the spatial energy spectral density (energy spectral density in unit volume, we will call it energy spectral density for short in this paper) can be written as $$u ( \omega , T ) \mathrm { d } \omega = \frac{\overline { \varepsilon } g ( \omega ) \mathrm { d } \omega}{V} = \frac { 1} { \pi ^2 c ^ { 3 } } \frac { \hbar \omega ^ { 3 } \mathrm { d } \omega } { \mathrm { e } ^ { \hbar \omega / k T } - 1 },$$ which is the famous PF. It’s notable here that Eq. (1) is independent of the geometry of the cavity, which is caused by the geometric independence of $\overline{\epsilon}(\omega)$ and DOS. $\overline{\epsilon}(\omega)$ is derived from statistical considerations, and DOS is derived from geometry considerations, which is a good approximation only for a cavity with infinite volume. A proof for the validity of this approximation is given by Courant and Hilbert [@cou].
Similar to Planck’s derivation, in this work we assume that the small size of the cavity does not affect the Bose statistics of the photons, *i.e*., we suppose the average energy formula $\overline{\epsilon}(\omega)=\hbar\omega[\mathrm{exp}(\hbar\omega/k_BT)-1]^{-1}$ still works. This means that our correction of PF concentrates on the correction of DOS.
Now, Let’s derive the energy spectral density in a thin film, which can be regarded as two infinite plates at a distance. For the simplicity in later discussions, we first consider the eigenmodes of black body radiation in a box. $L_1, L_2$ and $L_3$ are the lengths of the box at $x, y$ and $z$ directions. If we take the periodic boundary conditions, the wave vector is given by $$\begin{aligned}
k&=&\sqrt{k_x^2+k_y^2+k_z^2}\nonumber\\
&=& 2\pi \sqrt{\frac{n_1^2}{L_1^2}+\frac{n_2^2}{L_2^2}+\frac{n_3^2}{L_3^2}}, \nonumber \\
&& n_1,n_2,n_3=0,\pm1,\pm2,\dots ,\end{aligned}$$ where $k_x, k_y$ and $k_z$ are the three components of the wave vector. In the case of film, we let $L_2$ and $L_3$ approach to infinite but $L_1$ be finite, thus $k_y$ and $k_z$ become continuous. It’s easy to write the DOS $$g(k)\mathrm{d}k=\frac{L_2L_3}{\pi}\sum_{k_i}k\mathrm{d}k, \ \ \ |k_i|=\frac{2\pi |n_1|}{L_1}<k,\ n_1\in Z,$$ which can be written as $$g(k)\mathrm{d}k=\frac{L_2L_3}{\pi}(2[\frac{kL_1}{2\pi}]+1)k\mathrm{d}k,$$ where $[x]$ means integer-valued function. The energy spectral density is $$u(\omega,T)\mathrm{d}\omega=\frac{1}{\pi c^2L_1}\frac { \hbar \omega^2 } { \mathrm { e } ^ { \hbar\omega / k T } - 1 }(2[\frac{\omega L_1}{2c\pi}]+1)\mathrm{d}\omega.$$ It depends on the distance between the two plates. When $L_1$ approaches to infinite, $2[{\omega L_1}/{2c\pi}]+1 \approx {\omega L_1}/{c\pi}$, Eq. (5) reduces to the PF.
If the antiperiodic boundary condition is taken, we can get $$u(\omega,T)\mathrm{d}\omega=\frac{2}{\pi c^2L_1}\frac { \hbar \omega^2 } { \mathrm { e } ^ { \hbar\omega / k T } - 1 }[\frac{\omega L_1}{2c\pi}+\frac{1}{2}]\mathrm{d}\omega.$$ Similarly, in Dirichlet boundary condition, we have $$u(\omega,T)\mathrm{d}\omega= \frac{1}{\pi c^2L_1}\frac { \hbar \omega^2 } { \mathrm { e } ^ { \hbar\omega / k T } - 1 }[\frac{\omega L_1}{c\pi}]\mathrm{d}\omega.$$ It’s easy to find the results with different boundary conditions are different, especially when $L_1$ is small compared to the typical wavelengths. Fig. 1 shows the energy spectral densities of films at different $L_1$ with different boundary conditions. For antiperiodic and Dirichlet boundary conditions, there’s a cut-off frequency. But for periodic boundary condition, there’s not such a cut-off frequency and thus the energy density tends to infinite as $L_1$ tends to 0. For a slim rod with periodic boundary condition, there is also such “infrared divergence”. The authors of Ref. [@pra] dealt with the problem by claiming that every experimental apparatus has a resolution $\Delta E$ and we need not to count the modes whose $\omega<\Delta E/\hbar $. For Dirichlet boundary condition, energy densities at every frequency are not higher than that of the PF’s results, which can be easily seen from Eq. (7), because $[x]\leq x$. But for periodic and antiperiodic boundary conditions, energy densities at some frequencies are higher than PF’s results. When $L_1$ is large enough compared to the typical wavelengths, the energy spectral densities with different boundary conditions all coincide with the PF.
Similar to the derivation above, we derive the energy spectral density of a slim rod. In Eq. (2), we let $L_3$ approach to infinite but $L_1, L_2$ be finite, it’s also easy to write the DOS $$\begin{split}
g(k)\mathrm{d}k=\frac{2L_3}{\pi}\sum_{k_1^2+k_2^2<k^2}\frac{k\mathrm{d}k}{\sqrt{k^2-(k_1^2+k_2^2)}},\\
(k_1,k_2)=(\frac{2n_1\pi}{L_1},\frac{2n_2\pi}{L_2}),\ n_1,n_2 \in Z.
\end{split}$$ The energy spectral density is $$\begin{aligned}
u(\omega,T)\mathrm{d}\omega &=& \frac{2}{\pi c^2L_1L_2}\frac { \hbar \omega^2 } { \mathrm { e } ^ { \hbar\omega / k T } - 1 }\times \nonumber\\
&&\sum_{k_1^2+k_2^2<\frac{\omega^2}{c^2}}\frac{\mathrm{d}\omega}{\sqrt{\frac{\omega^2}{c^2}-(k_1^2+k_2^2)}},\nonumber\\
(k_1,k_2)&=&(\frac{2n_1\pi}{L_1},\frac{2n_2\pi}{L_2}),\ n_1,n_2 \in Z.\end{aligned}$$ In antiperiodic boundary condition case, we just need to replace $(k_1, k_2)$ by $$\begin{aligned}
(k_1,k_2)&=&(\frac{2(n_1+1/2)\pi}{L_1},\frac{2(n_2+1/2)\pi}{L_2}),\ n_1,n_2 \in Z.
\nonumber\\\end{aligned}$$ in Eq. (9). Also, in Dirichlet boundary condition case, we replace $(k_1, k_2)$ by $$\begin{aligned}
(k_1,k_2)&=&(\frac{n_1\pi}{L_1},\frac{n_2\pi}{L_2}),\ n_1,n_2 \in N^*.\end{aligned}$$ in Eq. (9). Fig. 2 shows the energy spectral densities in rods of different $L_1$ ($L_2=L_1$) with different boundary conditions. It is found that the energy spectral densities reduce to PF when $L_1$ is large enough. Similar to the case of film, when $L_1$ is small compared to the typical wavelengths, the energy spectral densities deviate from PF greatly. The boundary conditions also influence the results. It is noticed for all these three boundary conditions, the energy densities at some frequencies are higher than PF’s results. We also notice for a rod, the curves of spectrums oscillate more violent than that of a film.
An important thing we note is that so long as the length(s) in the constrained direction(s) is small enough, the spectrum will deviate from the PF greatly even at high frequencies where the PF is supposed to be exact.
Just as shown above we discussed the effect of small size on energy spectral density in the case of film and rod, in the next section, we will study the effect of small size on the blackbody radiation in a cubic box.
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Box {#sec:Box}
---
Different from the cases of film and rod, we can not define DOS for a cavity with finite volume because of the discrete values of frequencies, so we also can’t define the energy spectral density for such systems. But we can define an approximate energy spectral density. If we divide the frequency into many small intervals with the length of $\Delta\omega$, the energy spectral density on $(\omega,\omega+\Delta\omega)$ can be defined as $$u(\omega,T)=\sum_{\omega<\omega_i<\omega+\Delta\omega}\frac{N(\omega_i)}{V\Delta\omega}\frac { \hbar \omega_i } { \mathrm { e } ^ { \hbar\omega_i / k T } - 1 },$$ where $\omega_i$ are the allowed discrete values of $\omega$ and $N(\omega)$ is the number of modes of frequency $\omega$. When the number of modes is great enough in each interval, the approximate energy spectral density appears to be continuous and inert to the change of $\Delta\omega$, as long as $\Delta\omega$ is still small enough. This definition is similar to the definition given by Ref. [@pra], but our $\Delta \omega$ is a constant once specified.
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{width="1\linewidth"}\
For a cubic box with the periodic boundary condition, the wave vector takes the values given in Eq. (2). We run over all integers to get $N(\omega)$ and then calculate the approximate energy spectral density. Antiperiodic boundary condition and Dirichlet boundary condition cases are obtained in the same way. Fig. 3 shows the approximate energy spectral densities of a cubic box at different length $L$ with periodic and antiperiodic boundary conditions. We can see that when $L$ is large enough, the approximate energy spectral densities coincide with PF but when $L$ is small enough, they deviate from the PF greatly. When $L$ is small, the oscillation of the curves indicates that there are only a few modes in each interval $\Delta\omega$, which means that the spectral density cannot be considered as continuous anymore, whereby the concept of energy spectral density is not suitable. It is amazing to find that if the length of the cubic box is small enough, the energy density in the box will become very small compared with that in infinite volume.
Now we want to know when the asymptotic formula (correction formula by the method of asymptotic expansion) will work well and fail. From the result given in Ref. [@math], we obtain an asymptotic formula with Dirichlet boundary condition for a closed cavity with an arbitrary shape, $$u_{asy}(\omega,T)\mathrm{d}\omega=(\frac{\hbar\omega^3}{\pi^2c^3}-\frac{A}{V}\frac{\hbar\omega^2}{4\pi c^2}+\frac{M}{V}\frac{\hbar\omega}{3\pi^2c})\frac{1}{{ \mathrm { e } ^ { \hbar\omega / k T } - 1 }}\mathrm{d}\omega,$$ where $A$ is the surface area of the cavity, $M$ is the surface integral of mean curvature. $M=\int_S{\frac{1}{2}(\kappa_1+\kappa_2)\mathrm{d}S}$ with $\kappa_1$ and $\kappa_2$ the principle curvatures at the surface. Fig. 4 shows the comparison between the PF, Eq. (13) and the approximate energy spectral density by Eq. (12) for a cubic box and a sphere with Dirichlet boundary condition. We can see when the length of the cavity is so small that the spectral density becomes discrete, both PF and (13) fail and (13) even gives meaningless minus results. When the cavity is large enough, the three coincide each other. When $L$ is neither extremely small nor large, Eq. (13) is more close to the real energy spectral density than PF.
As shown in Figs. (2-4), when the cavity size is large enough, the blackbody radiation spectrums obtained by three different boundary conditions tend to the standard PF, which we can expect in advance. However, when the cavity size is small enough, the blackbody radiation spectrums obtained with different boundary conditions are completely different. Then, a natural problem arises, that is, which boundary condition is true? Regarding this issue, our views are as follows: When we consider the blackbody radiation in a small cavity , since the photons cannot escape from the cavity, it is theoretically necessary to use Dirichlet boundary condition. Of course, this requires further testing by the experiment.
Conclusion {#sec:Conclusion}
==========
In this paper, we investigated the geometry effect on blackbody radiation with different boundary conditions. Here we summarize the main conclusions of this study (include some common knowledge):
\(1) We derived the formula of blackbody radiation for a film and a rod with different boundary conditions.
\(2) The geometry effect of black body radiation is great when the length(s) at constrained direction(s) of the cavity is not large compared to the typical wavelengths of black body radiation.
\(3) Planck’s formula can not describe black body radiation properly in a very small cavity, neither does asymptotic formula, which is better than PF when the volume is finite and not too small.
\(4) The boundary conditions also influence blackbody radiation, especially when in small cavities.
\(5) The concept of energy spectral density is not suitable for a closed cavity, especially when the size of the cavity is close to the wavelengths of blackbody radiation.
Finally, we stress that in this paper we still assume that the small size does not affect the Bose statistics, which obviously needs to be tested experimentally, and deserves further research. In fact, some experiments have recently suggested that small volume effect may affect quantum statistical distribution. For example, as shown in Ref. [@BEC], no more than ten photons show Bose-Einstein condensation in a small cavity. This indicates that new physics may exist in the extremely small volume.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported in part by the National Natural Science Foundation of China (under Grants No. 11475085, No. 11535005, and No. 11690030) and by Nation Major State Basic Research and Development of China (2016YFE0129300).
[4]{}
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A. Reiser and L. Sch" achter, Phys. Rev. A 87, 033801 (2013). S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol 3: Elements of Random Fields (Springer-Verlag, Berlin, 1989). H. Weyl, Math. Ann. 71, 441 (1912).
R. T. Waechter, Proc. Cambridge Philos. Soc. 72, 439-447 (1972).
R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989), Chap. VI, Sec. 4.
T. Carleman, in Proceedings of the Eighth Scandinavian Mathematics Congress, Stockholm (Ohlsson, Lund, 1935), p.34. Å. Pleijel, Commun. Pure Appl. Math. 3, 1 (1950). F. H. Brownell, Pacific J. Math. 5, 483 (1955). H. P. Baltes and E. R. Hilf, Spectra of Finite Systems (Bibliographisches Institut, Mannheim, 1976). B. T. Walker, L. C. Flatten, H. J. Hesten, F. Mintert, D. Hunger, Aurélien A. P. Trichet, J. M. Smith, and R. A. Nyman, Nat. Phys. 14, 1173 (2018).
|
---
author:
- Florenc Demrozi
- Graziano Pravadelli
- Azra Bihorac
- Parisa Rashidi
bibliography:
- 'biblio.bib'
title: 'Human Activity Recognition using Inertial, Physiological and Environmental Sensors: a Comprehensive Survey'
---
Introduction {#sec:intro}
============
Existing Surveys {#sec:surv}
================
Selection Criteria {#sec:sel}
==================
Background {#sec:back}
==========
Human Activity {#sec:humact}
==============
Data Source Devices In HAR {#sec:device}
==========================
Data {#sec:data}
====
Classification Model and Evaluation {#sec:eval}
===================================
Discussion {#sec:disc}
==========
Conclusion {#sec:conc}
==========
|
---
abstract: 'Atomic magnetometry was performed at Earth’s magnetic field over a free-space distance of ten meters. Two laser beams aimed at a distant alkali-vapor cell excited and detected the $^{87}$Rb magnetic resonance, allowing the magnetic field within the cell to be interrogated remotely. Operated as a driven oscillator, the magnetometer measured the geomagnetic field with 3.5pT precision in a $\sim$2s data acquisition; this precision was likely limited by ambient field fluctuations. The sensor was also operated in self-oscillating mode with a 5.3pT/$\sqrt{\textrm{Hz}}$ noise floor. Further optimization will yield a high-bandwidth, fully remote magnetometer with sub-pT sensitivity.'
address:
- 'Department of Physics, University of California, Berkeley, CA 94720-7300'
- 'University of Groningen, Kernfysisch Versneller Instituut, NL-9747 AA Groningen, The Netherlands'
- 'Southwest Sciences, Inc., Cincinnati, OH 45244'
- 'Department of Physics, University of California, Berkeley, CA 94720-7300'
- 'Department of Physics and Astronomy, Bucknell University, Lewisburg, PA 17837'
- 'Department of Physics, University of California, Berkeley, CA 94720-7300'
- 'Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720'
author:
- 'B. Patton'
- 'O. O. Versolato'
- 'D. C. Hovde'
- 'E. Corsini'
- 'J. M. Higbie'
- 'D. Budker'
title: 'A Remotely Interrogated All-Optical $^{87}$Rb Magnetometer'
---
Shortly after the inception of atomic magnetometry, alkali-vapor magnetometers were being used to measure the Earth’s magnetic field to unprecedented precision. During the same era, Bell and Bloom first demonstrated all-optical atomic magnetometry through synchronous optical pumping [@Bell_PRL61]. In this approach, optical-pumping light is frequency- or amplitude-modulated at harmonics of the Larmor frequency $\omega_{\textrm{L}}$ to generate a precessing spin polarization within an alkali vapor at finite magnetic field. Although this technique received considerable attention from the atomic physics community for its applicability to optical pumping experiments, Earth’s-field alkali-vapor atomic magnetometers continued to rely on radiofrequency (RF) field excitation for several decades. Upon the advent of diode lasers at suitable wavelengths, synchronously pumped magnetometers experienced a revival beginning the late 1980s. In recent years, advances in all-optical magnetometers using amplitude-modulated[@Gawlik_APL06] and frequency-modulated[@Acosta_PRA06] light have resulted in applications such as nuclear magnetic resonance detection[@Bevilacqua_JMR09], quantum control experiments[@Pustelny_PRA11], and chip-scale devices intended for spacecraft use[@Korth_JHAPL10].
All-optical magnetometers possess several advantages over devices which employ RF coils. RF-driven magnetometers can suffer from cross-talk if two sensors are placed in close proximity, since the AC magnetic field driving resonance in one vapor cell can adversely affect the other. All-optical magnetometers are free from such interference. When operated in self-oscillating mode [@Bloom_AO62], RF-driven magnetometers require an added $\pm90^{\circ}$ electronic phase shift in the feedback loop to counter the intrinsic phase shift between the RF field and the probe-beam modulation. In an all-optical magnetometer this same phase shift can be achieved simply by varying the relative orientations of the pump and probe beam polarizations [@Higbie_RSI06]. Most importantly, all-optical magnetometers require no physical connection between the driving electronics and the alkali-vapor cell. This allows completely remote interrogation of the magnetic resonance in a faraway atomic sample. Here we describe a demonstration of remotely interrogated all-optical magnetometry
A schematic of the remote-detection magnetometer is shown in Fig. \[RemoteSetup\]. The unshielded sensor was similar to that described in Ref. [@Higbie_RSI06] in that the pump and probe beams were derived from a single laser whose frequency was stabilized by a dichroic atomic vapor laser lock (DAVLL) [@Yashchuk_RSI00]. The atomic sample consisted of an antirelaxation-coated[@Balabas_PRL10] alkali-vapor cell containing enriched $^{87}$Rb and no buffer gas; the longitudinal spin relaxation time of atoms within the cell was 1.2 s. The laser beams were carried from an optics and electronics rack to a launcher assembly via polarization-maintaining optical fibers; the pump beam amplitude was modulated with a fiberized Mach-Zender electro-optic modulator (EOM). At the launcher, the collimated output beams were linearly polarized and aimed at a sensor head placed ten meters away. This assembly contained the $^{87}$Rb cell within an enclosure heated to 34.5 $^{\circ}$C by a 1.7 kHz alternating current flowing through counter-wrapped heating wires. These wires comprised the only physical contact between the experimental apparatus and the atomic sample. In principle such heating is not necessary, but it was employed here to boost the optical-rotation signal above electronic interference from AM radio stations.
The probe beam traveled horizontally through the optical cell in a double-pass configuration, reflecting off a mirror behind the cell and propagating back toward the launcher. There a balanced polarimeter split the probe beam into orthogonal polarizations which were projected onto two photodiodes, allowing optical rotation to be measured. Synchronous optical pumping at 2$\omega_{\textrm{L}}$ created atomic alignment[@Yashchuk_PRL03; @Note1] within the $^{87}$Rb vapor which produced time-varying optical rotation of the probe polarization at $\omega_{\textrm{L}}$ and 2$\omega_{\textrm{L}}$. The $\omega_{\textrm{L}}$ harmonic arises when the field is tilted away from the direction of the probe beam propagation vector[@Pustelny_PRA06]. In the current experiment the ambient geomagnetic field pointed 32$^{\circ}$ from the vertical, leading to an optical rotation component at $\omega_{\textrm{L}}$ which was several times larger than the 2$\omega_{\textrm{L}}$ harmonic. Note that this field orientation is far from optimal, since both harmonics exhibit zero amplitude when the magnetic field is perpendicular to the probe beam propagation[@Pustelny_PRA06] (a configuration known as a “dead zone”).
The Zeeman shifts of the alkali ground-state sublevels (total electron spin $J = 1/2$) at a magnetic field $B$ can be calculated from the Breit-Rabi equation[@OPA_Ed1]. $$\begin{aligned}
E(F, m_F) & = & -\frac{{\mathcal{A}_{\textrm{hfs}}}}{4} - g_I {\mu_{\textrm{B}}}m_F B \nonumber \\ & \pm & \frac{{\mathcal{A}_{\textrm{hfs}}}(I+\tfrac{1}{2})}{2}\sqrt{1+\frac{4 m_F x}{2I+1}+x^2},
\label{BR01}\end{aligned}$$ where $E$ is the energy of the ground state sublevel with quantum numbers $F$ and $m_F$ ($F = I
\pm \frac{1}{2}$ being the total angular momentum of the ground state and $I$ the nuclear spin), ${\mathcal{A}_{\textrm{hfs}}}$ is the hyperfine structure constant, and the perturbation parameter $x$ is given by: $$x \equiv \frac{(g_J + g_I){\mu_{\textrm{B}}}B}{{\mathcal{A}_{\textrm{hfs}}}(I+\tfrac{1}{2})}. \label{xequals}$$ Here $g_J$ and $g_I$ are the electron and nuclear $g$ factors, respectively[@Note2]. At low magnetic fields, a linear approximation to Eq. (\[BR01\]) predicts a single resonance at $\omega_{\textrm{L}}$ for all transitions with $\Delta m_F = 1$ and another resonance at 2$\omega_{\textrm{L}}$ for all $\Delta m_F = 2$ transitions. At Earth’s field ($B_E$), these resonances split into sets of resolved transitions due to higher-order corrections. In the present study the laser was tuned to address the $F=2$ ground-state hyperfine manifold of $^{87}$Rb, yielding four resonances with $\Delta m_F = 1$ and three with $\Delta m_F = 2$. The magnetometer is nominally designed to probe $\Delta m_F = 2$ resonances in order to reduce systematic errors[@Patton_HE], but in this study it could also be operated near $\omega_{\textrm{L}}$ due to the field configuration.
In driven-oscillation mode, a function generator was used to drive the EOM and the probe beam optical rotation was detected with a lock-in amplifier (SR844, Stanford Research Systems, Inc.). The modulation frequency was swept around 2$\omega_{\textrm{L}}$ in order to map out the $\Delta
m_F = 2$ magnetic-resonance curve, which was recorded on an oscilloscope. An example data set is shown in Fig. \[FO\]. For these data, the pump beam power was set to 260 $\mu$W peak with a 50% duty cycle and the probe beam was 55 $\mu$W continuous. Three resonances separated by the $\sim$70 Hz nonlinear Zeeman splitting are clearly visible.
![\[FO\] (Color online) Driven-oscillation data recorded with the remote magnetometer. A spurious background has been subtracted from the data, which were then fit to the spectrum predicted by Eq. (\[BR01\]). The data and the fit have been re-phased to portray purely absorptive and dispersive quadratures.](Fig02.eps)
Due to electrical interference, a phase-coherent signal was picked up by the lock-in amplifier; this produced a frequency-dependent offset even when the pump and probe beams were blocked. This spurious baseline was subtracted from the data in Fig. \[FO\] and the data were fit to the three-Lorentzian magnetic-resonance spectrum predicted by Eq. (\[BR01\]). According to this fit, the central ($m_F = -1 \rightarrow m_F = +1$) magnetic resonance occurs at 682 504.318 $\pm$ 0.050 Hz (1$\sigma$ uncertainty). Converting the best-fit frequency uncertainty into a field uncertainty yields a magnetic sensitivity of 3.5 pT. The lock-in time constant was 10 ms and the frequency sweep rate was 200 Hz/s, such that most of the spectrum was recorded within a span of $\sim$2 seconds. For the fitting procedure, the data were averaged in 10 ms bins in order to reduce correlations in point-to-point noise which would erroneously reduce the estimated frequency uncertainty. As a cross-check of this sensitivity figure, many sets of simulated data were generated with random noise which was statistically equivalent to the off-resonant noise measured in the experiment. Repeated least-squares fitting of this simulated data yielded a root-mean-square scatter of 0.046 Hz in best-fit frequency (equivalent to 3.3 pT) when all other fitting parameters were held fixed. The off-resonant noise indicates that if the EOM driving frequency were set to the zero-crossing of the dispersive trace shown in Fig. \[FO\] and a steady-state experiment performed, fluctuations of 9.6 pT could be detected with a 1 Hz noise bandwidth.
This magnetometric sensitivity was achieved in spite of several sub-optimal experimental conditions: high pump and probe beam powers, excessive pump duty cycle, analog data transmission, and sub-optimal field orientation[@Note3]. In a reference sensor consisting of identical components interrogated non-remotely[@Corsini_PhD], optimization of the pump power and duty cycle yielded an optical rotation signal 14 times larger than that shown in Fig. \[FO\]. This implies that optimization of the pump beam characteristics could immediately yield sub-pT sensitivity in the remote scheme. Moreover, these signals were recorded in an unshielded environment and subject to fluctuations in the ambient field which were often larger than 10 pT/s. Most likely the sensitivity demonstrated here is limited by genuine field fluctuations. Improved sensitivities can therefore be expected in future experiments, particularly if a gradiometric scheme is employed.
In self-oscillating mode the polarimeter output was conditioned by a triggering circuit to drive the EOM directly, generating a positive feedback loop and causing the system to oscillate spontaneously at the magnetic-resonance frequency. A passive band-pass filter of width 10 kHz centered around $\omega_{\textrm{L}}$ was included in the loop to reduce broadband noise fed into the triggering circuit. (Oscillation at 2$\omega_{\textrm{L}}$ was also possible, but less robust due to AM radio interference and the smaller signal amplitude.) The probe beam power was 50 $\mu$W leaving the launcher; the pump beam power was 10 $\mu$W time-averaged with a low (10-20%) duty cycle. To quantify the magnetometer’s performance we mixed down its self-oscillation signal with that of the reference sensor using the lock-in amplifier, with the reference sensor acting as the external frequency reference and the remote sensor as the signal input. Helmholtz coils near the test sensor generated a field offset seen by the two magnetometers. This gradient was tuned to generate a self-oscillation beat frequency of $\sim$275 Hz and the lock-in time constant set to 1 ms. The output of the lock-in amplifier was recorded with a data acquisition card and saved to a computer.
![\[PSD\] (Color online) PSD of the gradiometer signal. The beat frequency of the sensors was calculated as a function of time, converted into a field difference, and Fourier transformed to yield the magnetic noise as a function of frequency. The thin blue trace is the Fourier transform; the thicker red trace is the same data smoothed into 1 Hz bins. The mean noise floor between 1 Hz and 50 Hz is 5.3 pT/$\sqrt{\textrm{Hz}}$. Ambient 60-Hz magnetic field noise can be clearly seen.](Fig03.eps)
Figure \[PSD\] shows an analysis of the magnetic-field noise observed by the sensors. The SR844 output was digitally filtered with a 125 Hz band-pass filter about the intermediate frequency, then fit with a running sine wave in 8 ms segments to calculate the beat frequency as a function of time. This frequency was then converted into a fluctuation about Earth’s field using Eq. (\[BR01\]), assuming that both magnetometers were oscillating on the same $\Delta m_F
= 1$ resonance[@Note4]. A Fourier transform of the field difference yielded the power spectral density (PSD) of the reported magnetic noise. The average noise floor from 1 Hz to 50 Hz was 5.3 pT/$\sqrt{\textrm{Hz}}$. It is not clear how much of the PSD noise floor arises from sensor noise and how much can be attributed to current noise in the power supply driving the Helmholtz coils or fluctuations in the ambient field gradient. A principal advantage of the self-oscillating scheme is its high bandwidth – AC magnetic fields of frequency 1 kHz and magnitude 1 nT have been detected with high SNR using a version of the reference sensor described here.
Although a flat mirror was used in these experiments, such a configuration is not practical for remote magnetometry because the mirror must be aligned at the cell in order to reflect the probe beam back to the polarimeter. Replacing this mirror with a polarization-preserving retroreflector would allow for truly adjustment-free, long-baseline magnetometry. Corner-cube reflectors can significantly alter the polarization properties of an interrogating laser. An omnidirectional retroreflecting sphere is a promising choice, with a graded-index Luneberg-like sphere being ideal. Polarimetry tests were conducted using a surrogate 1 cm diameter sphere with index of refraction $n\approx2$, silver coated on its distal surface. Initial tests using a 633 nm laser and a digital polarimeter showed that the retroreflector (O’Hara Corp) preserved linear polarization of a probe beam to within a few degrees of ellipticity, which represented the measurement error of the polarimeter. Future tests will incorporate this retroreflector in the magnetometer design. In addition, the free-space baseline of the magnetometer will be increased, with the expectation that this technique can be extended to distances of several hundred meters before atmospheric seeing becomes a significant noise source[@Buck_AO67]. Beyond this distance scale, adaptive optics techniques may become necessary to mitigate the effects of atmospheric turbulence and retain magnetometric sensitivity.
A sensitive remote magnetometer capable of being interrogated over several kilometers of free space would be highly desirable in several applications, including ordnance detection, perimeter monitoring, and geophysical surveys. Inexpensive manufacturing of the cell/retroreflector package would allow many such sensors to be widely distributed and interrogated by a single optical setup. Further research in remote magnetometry will also contribute to recently proposed efforts to measure the Earth’s magnetic field using mesospheric sodium atoms and laser guide-star technology[@Higbie_PNAS11].
The authors would like to thank collaborators Charles Stevens and Joseph Tringe (LLNL) for initiating this project and providing the retroreflector, as well as Mikhail Balabas for the antirelaxation-coated $^{87}$Rb cells. We also thank Mark Prouty, Ron Royal, and Lynn Edwards at Geometrics, Inc. for experimental assistance and use of magnetometric facilities. We acknowledge support from Victoria Franques through the Department of Energy’s National Nuclear Security Agency (NNSA-NA-22)NA 22, Office of Nonproliferation Research and Development. This work was also supported in part by the Navy (contract N68335-06-C-0042), by the Department of Energy Office of Nuclear Science (Award DE-FG02-08ER84989), and by NSF (ARRA 855552). Parts of this work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
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---
address: |
Math. Institut\
Ernst-Zermelo-Str. 1\
79104 Freiburg\
Germany
author:
- Annette Huber
bibliography:
- 'periods.bib'
title: Galois theory of periods
---
Introduction
============
We are interested in the period algebra ${\mathcal{P}}$, a countable subalgebra of ${\mathbb{C}}$. Its elements are roughly of the form $$\int_\sigma\omega$$ where $\omega$ is a closed differential form over ${\mathbb{Q}}$ or ${{\overline{{\mathbb{Q}}}}}$ and $\sigma$ a suitable domain of integration also defined over ${\mathbb{Q}}$ or ${{\overline{{\mathbb{Q}}}}}$. All algebraic numbers are periods in this sense. Moreover, many interesting transcendental number like $2\pi i$, $\log(2)$, or $\zeta(3)$ are also period numbers.
In this mostly expository note, we want to explain how the work of Nori extends Galois theory from the field extension ${{\overline{{\mathbb{Q}}}}}/{\mathbb{Q}}$ to ${\mathcal{P}}/{\mathbb{Q}}$, at least conjecturally. This realises and generalises a vision of Grothendieck.
As explained in more detail below, we should formulate Galois theory as saying that the natural operation $${\mathrm{Gal}}({{\overline{{\mathbb{Q}}}}}/{\mathbb{Q}})\times {\mathrm{Spec}}({{\overline{{\mathbb{Q}}}}})\to {\mathrm{Spec}}({{\overline{{\mathbb{Q}}}}})$$ is a torsor, see Section \[ssec:galois\] for more details. By the period conjecture (in Kontsevich’s formulation in [@kontsevich]), we also have a torsor structure $$G_{\mathrm{mot}}({\mathbb{Q}})\times {\mathrm{Spec}}({\mathcal{P}})\to{\mathrm{Spec}}({\mathcal{P}})$$ where $G_{\mathrm{mot}}({\mathbb{Q}})$ is Nori’s motivic Galois group of ${\mathbb{Q}}$. This is a pro-algebraic group attached to the rigid tensor category of mixed motives over ${\mathbb{Q}}$. By Tannaka theory, rigid tensor subcategories of the category of motives correspond to quotients of $G_{\mathrm{mot}}({\mathbb{Q}})$. The case of Artin motives gives back ordinary Galois theory. In particular, there is a natural exact sequence $$0\to G_{\mathrm{mot}}({{\overline{{\mathbb{Q}}}}})\to G_{\mathrm{mot}}({\mathbb{Q}})\to{\mathrm{Gal}}({{\overline{{\mathbb{Q}}}}}/{\mathbb{Q}})\to 0.$$ It corresponds to the inclusion ${{\overline{{\mathbb{Q}}}}}\subset{\mathcal{P}}$.
It is a nice feature of Nori’s theory (and a key advantage over the older approach) that it works for any abelian category of motives, even if it is not closed under tensor product. We get a weaker structure that we call semi-torsor, see Definition \[defn:semi\]. Hence, we also have a (conjectural) generalisation of Galois theory to sub-vector-spaces of ${\mathcal{P}}$. It is unconditonal in the case of $1$-motives.
While not bringing us closer to a proof of the period conjecture in general, the theory has structural consequences: for example the period conjectures for intermediate fields ${\mathbb{Q}}\subset{\mathbb{K}}\subset {{\overline{{\mathbb{Q}}}}}$ are equivalent. Another direct consequence is the fully faithfulness of the functor attaching to a Nori motive $M$ the pair $(H_{\mathrm{dR}}(M),H_{\mathrm{sing}}(M),\phi)$ of its realisations linked by the period isomorphism. This observation connects the period conjecture to fullness conjectures in the spirit of the Hodge or Tate conjecture. One such was recently formulated by Andreatta, Barbieri-Viale and Bertapelle, see [@ABB]. As explained below, their conjecture has overlap with the period conjecture, but there is no implication in either direction.
Actual evidence for the period conjecture is weak. The $0$-dimensional case amounts to Galois theory. The $1$-dimensional case has now been completed by the author and Wüstholz in [@huber-wuestholz], generalising work by Lindemann, Gelfond, Schneider, Siegel, Baker and Wüstholz.
The idea of the present paper was conceived after my talk at the 2017 workshop ”Galois Theory of Periods and Applications” at MSRI. It does not aim at proving new results, but rather at making key ideas of the long texts [@period-buch] and [@huber-wuestholz] more accessible. We also take the opportunity to discuss the relation between the various period conjectures in the literature.
Structure of the paper {#structure-of-the-paper .unnumbered}
----------------------
We start by giving several, equivalent definitions of the period algebra. We then turn to the period conjecture. We explain Kontsevich’s formal period algebra ${\tilde{{\mathcal{P}}}}$ in terms of explicit generators and relations. The period conjecture simply asserts that the obvious surjective map ${\tilde{{\mathcal{P}}}}\to{\mathcal{P}}$ is also injective. We then give a reinterpretation of ${\tilde{{\mathcal{P}}}}$ as a torsor. Thus the main aim of the note is achieved: a (conjectural) Galois theory of periods.
The following sections make the connections to Grothendieck’s version of the period conjecture and the more recent cycle theoretic approach of Bost/Charles and Andreatta/Barbieri-Viale/Bertapelle.
The last section inspects some examples of special period spaces studied in the literature: Artin motives (${{\overline{{\mathbb{Q}}}}}$), mixed Tate motives (multiple zeta-values) and $1$-motives (periods of curves).
Acknowledgements {#acknowledgements .unnumbered}
----------------
I thank my coauthors Stefan Müller-Stach and Gisbert Wüstholz for many insights and stimulating discussions. My ideas on the period conjecture were also heavily influenced by a discussion with Yves André during a visit to Freiburg. The aspect of a Galois theory of periods is also stressed in his survey articles [@andre1], [@andre3].
Period spaces
=============
When we talk about periods, we mean a certain countable subalgebra ${\mathcal{P}}$ of ${\mathbb{C}}$. It has several, surprisingly different, descriptions. Roughly a period is a number of the form $$\int_\sigma\omega$$ where both $\omega$ and $\sigma$ are something defined over ${\mathbb{Q}}$ or ${{\overline{{\mathbb{Q}}}}}$.
There are many possible ways to specify what $\omega$ and $\sigma$ to use. We will go through a list of possible choices: nc-periods (or just periods) in Section \[ssec:first\], Kontsevich-Zagier periods in Section \[ssec:KZ\], cohomological periods in Section \[ssec:coh\], and periods of motives in Section \[ssec:mot\_period\]. In most cases, we will define the subalgebra ${\mathcal{P}}^{\mathrm{eff}}$ of effective periods and then pass to ${\mathcal{P}}:={\mathcal{P}}^{\mathrm{eff}}[\pi^{-1}]$. In particular, $\pi$ will be an element of ${\mathcal{P}}^{\mathrm{eff}}$.
Good to know:
All definitions describe the same set.
Throughout, we fix an embedding ${{\overline{{\mathbb{Q}}}}}\to{\mathbb{C}}$. Let ${\tilde{{\mathbb{Q}}}}={{\overline{{\mathbb{Q}}}}}\cap {\mathbb{R}}$. We denote by ${\mathbb{K}}$ a subfield of ${{\overline{{\mathbb{Q}}}}}$.
First definition {#ssec:first}
----------------
Let $X$ be a smooth algebraic variety over ${\mathbb{K}}$ of dimension $d$, $D\subset X$ a divisor with normal crossings (also defined over ${\mathbb{K}}$), $\omega\in\Omega^d_X(X)$ a global algebraic differential form, $\sigma=\sum a_i\gamma_i$ a differentiable relative chain, i.e., $a_i\in{\mathbb{Q}}$, $\gamma_i:\Delta^d\to X^{\mathrm{an}}$ a $C^\infty$-map on the $d$-simplex such that $\partial \sigma$ is a chain on $D^{\mathrm{an}}$. Then $$\int_\sigma\omega:=\sum a_i\int_{\Delta^d}\gamma_i^*\omega^{\mathrm{an}}$$ is a complex number.
The set of these complex numbers is called the set of effective periods ${\mathcal{P}}^{\mathrm{eff}}$.
They appear as nc (for normal crossings) periods in [@period-buch Section 11.1].
The algebraic number $\alpha$ satisfying the non-trivial polynomial $P\in {\mathbb{K}}[t]$ is an effective period as the value of the integral $$\int_0^\alpha dt$$ with $X={\mathbb{A}}^1$, $D=V(tP)$, $\omega=dt$, $\sigma=\gamma$ the standard path from $0$ to $\alpha\in D({\mathbb{C}})$ in ${\mathbb{A}}^1({\mathbb{C}})={\mathbb{C}}$.
The number $2\pi i$ is an effective period as the value of the integral $$\int_{S^1}\frac{dz}{z}$$ with $X={\mathbb{G}}_m$, $D=\emptyset$, $\omega= dz/z$, $\sigma=\gamma:[0,1]\to {\mathbb{C}}^*$ given by $\gamma(t)=\exp(2\pi i t)$.
The set ${\mathcal{P}}^{\mathrm{eff}}$ is a ${{\overline{{\mathbb{Q}}}}}$-algebra. It is independent of the choice of ${\mathbb{K}}$.
[@period-buch Proposition 11.1.7].
In particular, $\pi \in {\mathcal{P}}^{\mathrm{eff}}$.
\[defn:first\] $${\mathcal{P}}:={\mathcal{P}}^{\mathrm{eff}}[\pi^{-1}].$$
1. In the above definition, it even suffices to restrict to affine $X$.
2. We may also relax the assumptions: $X$ need not be smooth, $D$ only a subvariety, $\omega\in\Omega^n(X)$ not necessarily of top degree but only closed.
3. We may also allow $\omega$ to have poles on $X$. We then need to impose the condition that the integral is absolutely convergent. This is the version given by Kontsevich and Zagier in [@kontsevich_zagier].
These implications are not obvious, but see [@period-buch Theorem 12.2.1].
Semi-algebraic sets {#ssec:KZ}
-------------------
The easiest example of a differential form is $dt_1\wedge\dots\wedge dt_n$ on ${\mathbb{A}}^n$ with coordinates $t_1,\dots,t_n$. Actually, this is the only one we need.
\[defn:KZ\]An *effective Kontsevich-Zagier period* is complex number whose real and imaginary part can be written as difference between the volumes of ${\mathbb{Q}}$-semi-algebraic subsets of ${\mathbb{R}}^n$ (of finite volume).
Note that a subset of ${\mathbb{R}}^n$ is ${\mathbb{Q}}$-semi-algebraic if and only if it is ${\tilde{{\mathbb{Q}}}}$-semi-algebraic.
\[prop:all\_KZ\] The set of effective Kontsevich-Zagier periods agrees with ${\mathcal{P}}^{\mathrm{eff}}$.
[@period-buch Prop. 12.1.6], combined with [@period-buch Theorem 12.2.1].
Cohomological periods {#ssec:coh}
---------------------
The definition becomes more conceptual when thinking in terms of cohomology. Chains define classes in (relative) singular homology and differential forms define classes in (relative) algebraic de Rham cohomology. The definition of algebraic de Rham cohomology is very well-known in the case of a single smooth algebraic variety. There are different methods of extending this to the singular and to the relative case, see [@period-buch Chapter 3]. Singular cohomology and algebraic de Rham cohomology become isomorphic after base change to the complex numbers. By adjunction this defines the *period pairing*.
Let $X/{\mathbb{K}}$ be an algebraic variety, $Y\subset X$ a subvariety. Let $$\int: H^i_{\mathrm{dR}}(X,Y)\times H_i^{\mathrm{sing}}(X,Y;{\mathbb{Q}})\to {\mathbb{C}}$$ be the period pairing.
\[defn:coh\_period\]The numbers in the image of $\int$ (for varying $X$, $Y$, $i$) are called *effective cohomological periods*.
\[prop:all\_coh\]The set of effective cohomological periods agrees with ${\mathcal{P}}^{\mathrm{eff}}$.
Periods of motives {#ssec:mot_period}
------------------
The insistance that we need to invert $\pi$ in order to get a good object becomes clear from the motivic point of view.
Two kinds of motives are relevant for our discussions:
1. Voevodsky’s triangulated categories of (effective) geometric motives ${{\mathsf{DM}}_{{\mathrm{gm}}}}^{\mathrm{eff}}({\mathbb{K}},{\mathbb{Q}})$ and ${{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}},{\mathbb{Q}})$;
2. Nori’s abelian category ${\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}},{\mathbb{Q}})$ of motives over ${\mathbb{K}}$ with respect to singular cohomology and its subcategory of effective motives.
There is a natural contravariant trianguled tensor-functor $${{\mathsf{DM}}_{{\mathrm{gm}}}}^{\mathrm{eff}}({\mathbb{K}},{\mathbb{Q}})\to D^b({\mathcal{MM}_{\mathrm{Nori}}}^{\mathrm{eff}}({\mathbb{K}},{\mathbb{Q}}))$$ compatible with the singular realisation. It extends to non-effective motives.
All motives have singular and de Rham cohomology and hence periods. We formalise as in [@period-buch Section 11.2].
\[defn:VV\] Let ${({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$ be the category of tuples $(V_{\mathbb{K}},V_{\mathbb{Q}},\phi)$ where $V_{\mathbb{K}}$ is a finite dimensional ${\mathbb{K}}$-vector space, $V_{\mathbb{Q}}$ a finite dimensional ${\mathbb{Q}}$-vector space and $\phi:V_{\mathbb{K}}{\otimes}_{\mathbb{K}}{\mathbb{C}}\to V_{\mathrm{sing}}{\otimes}_{\mathbb{Q}}{\mathbb{C}}$ an isomorphism. Morphisms are given by pairs of linear maps such that the obvious diagram commutes.
Given $V\in {({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$, we define the set of periods of $V$ as $${\mathcal{P}}(V)={\mathrm{Im}}(V_{\mathbb{K}}\times V_{\mathbb{Q}}^\vee\to {\mathbb{C}}),\hspace{2ex} (\omega,\sigma)\mapsto \sigma_{\mathbb{C}}(\phi(\omega))$$ and the space of periods ${\mathcal{P}}\langle H\rangle$ as the additive group generated by it. If ${\mathcal{C}}\subset {({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$ is a subcategory, we put $${\mathcal{P}}({\mathcal{C}})=\bigcup_{V\in{({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}}{\mathcal{P}}(V).$$
1. If ${\mathcal{C}}$ is additive, then ${\mathcal{P}}({\mathcal{C}})$ is a ${\mathbb{K}}$-vector space. In particular, ${\mathcal{P}}\langle V\rangle$ agrees with ${\mathcal{P}}(\langle V\rangle)$ where $\langle V\rangle$ is the full abelian subcategory of ${({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$ closed under subquotients generated by $V$.
2. If ${\mathcal{C}}$ is an additive tensor category, then ${\mathcal{P}}({\mathcal{C}})$ is a ${\mathbb{K}}$-algebra.
By the universal property of Nori motives, there is a faithful exact functor $${\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})\to {({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}.$$ This defines periods of Nori motives, and by composition with $M\mapsto \bigoplus H^i_{\mathrm{Nori}}(M)$ also periods of geometric motives.
\[defn:mot\_period\]A complex number is called an *(effective) motivic period* if is the period of an (effective) motive.
Every Nori motive is a subquotient of the image of a geometric motive, hence is does not matter whether we think of geometric or Nori motives in the above definition.
\[prop:all\_mot\]The set of effective motivic period agrees with ${\mathcal{P}}^{\mathrm{eff}}$. The set of all motivic periods agrees with ${\mathcal{P}}$. In particular, both are independent of the choice of ${\mathbb{K}}$.
As a consequence, we see that both ${\mathcal{P}}^{\mathrm{eff}}$ and ${\mathcal{P}}$ are ${{\overline{{\mathbb{Q}}}}}$-algebras. Of course this is also easy to deduce directly from any of the other descriptions.
There is a big advantage of the motivic description: any type of structural result on motives has consequences for periods. One such are the bounds on the spaces of multiple zeta-values by Deligne-Goncharov. Another example are the transcendence results for $1$-motives as discussed below.
In particular, the motivic point of view allows us (or rather Nori) to give a structural reinterpretation of the period conjecture.
Comparison
----------
We conclude by putting all information together:
The following sets of complex numbers agree:
1. the set of periods in the sense of Definition \[defn:first\];
2. the set of Kontsevich-Zagier periods, see Definition \[defn:KZ\];
3. the set of cohomological periods, see Definition \[defn:coh\_period\];
4. the set of all motivic periods, see Definition \[defn:mot\_period\].
Moreover, they are independent of the field of definition ${\mathbb{Q}}\subset{\mathbb{K}}\subset{{\overline{{\mathbb{Q}}}}}$ and form a countable ${{\overline{{\mathbb{Q}}}}}$-algebra.
Formal periods and the period conjecture
========================================
In reverse order to the historic development, we start with Kontsevich’s formulation and discuss the Grothendieck conjecture afterwards.
Kontsevich’s period conjecture
------------------------------
We take the point of view of cohomological periods. At this point we restrict to ${\mathbb{K}}={{\overline{{\mathbb{Q}}}}}$ for simplicity. There are obvious relations between periods:
- (bilinearity) Let $\omega_1,\omega_2\in H^i_{\mathrm{dR}}(X,Y)$, $\alpha_1,\alpha_2\in{{\overline{{\mathbb{Q}}}}}$, $\sigma_1,\sigma_2\in H_i^{\mathrm{sing}}(X,Y;{\mathbb{Q}})$, $a_1,a_2\in{\mathbb{Q}}$. Then $$\int_{\sigma_1+\sigma_2}(\alpha_1\omega_1+\alpha_2\omega_2)=\sum_{i,j=1}^2\alpha_j\int_{\sigma_i}\omega_j.$$
- (functoriality) For $f:(X,Y)\to (X',Z)$, $\omega'\in H^i_{\mathrm{dR}}(X',Y')$, $\sigma\in H_i^{\mathrm{sing}}(X,Y;{\mathbb{Q}})$ we have $$\int_{f_*\sigma}\omega'=\int_\sigma f^*\omega'.$$
- (boundaries) For $Z\subset Y\subset X$, $\partial:H^i_{\mathrm{dR}}(Y,Z)\to H^{i+1}(X,Y)$, $\delta:H_{i+1}^{\mathrm{sing}}(X,Y;{\mathbb{Q}})\to H_i^{\mathrm{sing}}(Y,Z;{\mathbb{Q}})$, $\omega\in H^i_{\mathrm{dR}}(Y,Z)$ and $\sigma\in H_{i+1}^{\mathrm{sing}}(X,Y;{\mathbb{Q}})$ we have $$\int_{\delta\sigma}\omega=\int_\sigma\partial\omega.$$
\[defn:perf\]The space of *formal effective periods* ${\tilde{{\mathcal{P}}}}^{\mathrm{eff}}$ is defined as the ${\mathbb{Q}}$-vector space generated by symbols $$(X,Y,i,\omega,\sigma)$$ with $X$ an algebraic variety over ${{\overline{{\mathbb{Q}}}}}$, $Y\subset X$ a closed subvariety, $i\in{\mathbb{N}}_0$, $\omega\in H^i_{\mathrm{dR}}(X,Y)$, $\sigma\in H_i^{\mathrm{sing}}(X,Y;{\mathbb{Q}})$ modulo the space of relations generated by bilinearity, functoriality and boundary maps as above.
It is turned into a ${\mathbb{Q}}$-algebra with the product using the external product induced by the Künneth decomposition $$H^*(X,Y){\otimes}H^*(X',Y'){\cong}H^*(X\times X',Y\times X'\cup Y'\times X).$$
Let $2\tilde{\pi}i$ be the class of the symbol $({{\mathbb{G}}_m},\{1\},dz/z,S^1)$. The space of *formal periods* ${\tilde{{\mathcal{P}}}}$ is defined as the localisation $${\tilde{{\mathcal{P}}}}={\tilde{{\mathcal{P}}}}^{\mathrm{eff}}[\tilde{\pi}^{-1}].$$
1. ${\tilde{{\mathcal{P}}}}^{\mathrm{eff}}$ and ${\tilde{{\mathcal{P}}}}$ are even ${{\overline{{\mathbb{Q}}}}}$-algebras. We define the scalar multiplication by $\alpha(X,Y,i,\omega,\sigma)=(X,Y,i,\alpha\omega,\sigma)$ for $\alpha\in{{\overline{{\mathbb{Q}}}}}$.
2. We do not know if it suffices to work with $X$ smooth, $D$ a divisor with normal crossings as Kontsevich does in [@kontsevich]. As explained in [@period-buch Remark 13.1.8] these symbols generate the algebra, but is is not clear if they also give all relations. Indeed, Kontsevich only imposes relations in an even more special case.
3. It is not at all obvious that the product is well-defined. This fact follows from Nori’s theory, see [@period-buch Lemma 14.1.3].
There is a natural evaluation map $${\tilde{{\mathcal{P}}}}^{\mathrm{eff}}\to{\mathbb{C}},\quad (X,Y,i,\omega,\sigma)\mapsto \int_\sigma\omega.$$ By definition its image is the set of effective cohomological periods, hence equal to ${\mathcal{P}}^{\mathrm{eff}}$.
\[conj:kont\] the evaluation map $${\tilde{{\mathcal{P}}}}\to{\mathcal{P}}$$ is injective. In other words, all ${{\overline{{\mathbb{Q}}}}}$-linear relations between periods are induced by the above trivial relations.
Categorical and motivic description {#ssec:form_cat}
-----------------------------------
The above description is “minimalistic” in using a small set of generators and relations. The downside is that it is somewhat arbitrary. We now explain alternative descriptions. Let again ${\mathbb{K}}\subset{{\overline{{\mathbb{Q}}}}}$ be a subfield. Recall that we have fixed an embedding ${{\overline{{\mathbb{Q}}}}}\subset{\mathbb{C}}$.
Let ${\mathcal{C}}$ be an additive category, $T:{\mathcal{C}}\to{({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$ an additive functor. We define the *space of formal periods of ${\mathcal{C}}$* as the ${\mathbb{Q}}$-vector space ${\tilde{{\mathcal{P}}}}({\mathcal{C}})$ generated by symbols $(M,\sigma,\omega)$ with $M\in{\mathcal{C}}$, $\sigma\in T(M)_{\mathbb{Q}}^\vee$, $\omega\in T(M)_{\mathbb{K}}$ with relations
- bilinearity in $\sigma$ and $\omega$,
- functoriality in $M$.
We are going to apply this mainly to subcategories of ${\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}},{\mathbb{Q}})$ or ${{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}},{\mathbb{Q}})$.
\[thm:indep\] $${\tilde{{\mathcal{P}}}}({{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}},{\mathbb{Q}}))={\tilde{{\mathcal{P}}}}({\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}},{\mathbb{Q}}))={\tilde{{\mathcal{P}}}},$$ independent of ${\mathbb{K}}$.
This is only implicit in [@period-buch]. It is a consequence of Galois theory of periods as discussed in the next section. We defer the proof.
In particular, the validity of the period conjecture does not depend on ${\mathbb{K}}\subset {{\overline{{\mathbb{Q}}}}}$. This is the reason that it was safe to consider only ${\mathbb{K}}={{\overline{{\mathbb{Q}}}}}$ in Definition \[defn:perf\].
The evaluation map ${({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}\to{\mathbb{C}}$ induces linear maps $${\tilde{{\mathcal{P}}}}({\mathcal{C}})\to{\mathbb{C}}.$$ We denote the image by ${\mathcal{P}}({\mathcal{C}})$.
We say that the period conjecture holds for ${\mathcal{C}}$ if the map ${\tilde{{\mathcal{P}}}}({\mathcal{C}})\to{\mathcal{P}}({\mathcal{C}})$ is injective.
In the case ${\mathcal{C}}={\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}},{\mathbb{Q}})$, we get back Kontsevich’s conjecture \[conj:kont\]. We are going to study the relation between the period conjecture for ${\mathcal{C}}$ and subcategories of ${\mathcal{C}}$ below using a Galois theory for formal periods.
Galois theory of (formal) periods
=================================
Torsors and semi-torsors
------------------------
\[defn:semi\]Let $k$ be a field.
1. A *torsor* in the category of affine $k$-schemes is an affine scheme $X$ with an operation of an affine $k$-group $G$ $$G\times X\to X$$ such that there is an affine $S$ over $k$ and a point $x\in X(S)$ such that the morphism $$G_S\to X_S,\quad g\mapsto gx$$ induced by the operation is an isomorphism.
2. An *semi-torsor* over $k$ is a $k$-vector space $V$ together with a coalgebra $A$ and a right co-operation $$V\to V{\otimes}A$$ such there is a a commutative $k$-algebra $K$ and a $k$-linear map $V\to K$ such that the map $$V_K\to A_K$$ induced by the comodule structure is an isomorphism.
\[rem:stor\]
1. The isomorphism makes $X$ a torsor in the fpqc-topology because $k\to K$ is faithfully flat. See also [@period-buch Section 1.7] for an extended discussion of different notions of torsors.
2. \[it:stor\_tor\] The second notion is newly introduced here. A torsor gives rise to a semi-torsor by taking global sections. A semi-torsor is a torsor, if, in addition, $A$ is a Hopf algebra, $V$ a $k$-algebra and the structure map is a ring homomorphism. Hence a semi-torsor only has half of the information of a torsor. Note that an operation $G\times X\to X$ that gives rise to a semi-torsor, is already a torsor because an algebra homomorphism which is a vector space isomorphism is also an algebra isomorphism.
3. If $B$ is a finite dimensional $k$-algebra, $N$ a left-$B$-module that is fpqc-locally free of rank $1$ (i.e., here is a commutative $k$-algebra $K$ such that $K{\otimes}_kN$ is free of rank $1$), then $N^\vee:={\mathrm{Hom}}_k(N,k)$ is a semi-torsor over the coalgebra $B^\vee$.
4. Once there is one $x$ satisfying the torsor condition, all elements of $X(S)$ for all $S$ will satisfy the condition. This is not the case for semi-torsors.
Galois theory revisited {#ssec:galois}
-----------------------
As promised in the introduction, we want to formulate Galois theory in torsor language.
Let $\Gamma={\mathrm{Gal}}({{\overline{{\mathbb{Q}}}}}/{\mathbb{Q}})$. It is profinite, hence proalgebraic where we view abstract finite groups as algebraic groups over ${\mathbb{Q}}$. Indeed, we identify a finite group $G$ with the algebraic group given by ${\mathrm{Spec}}( {\mathbb{Q}}[G]^\vee)$. Let $X_0={\mathrm{Spec}}({{\overline{{\mathbb{Q}}}}})$. We view it as a pro-algebraic variety over ${\mathbb{Q}}$. There is a natural operation $$\Gamma\times X_0\to X_0.$$ We describe it on finite level. Let $K/{\mathbb{Q}}$ be Galois. Then $${\mathrm{Gal}}(K/{\mathbb{Q}})\times {\mathrm{Spec}}(K)\to {\mathrm{Spec}}( K)$$ is given by the ring homomorphism $${\mathbb{Q}}[{\mathrm{Gal}}(K/{\mathbb{Q}})]^\vee{\otimes}K\leftarrow K$$ adjoint to $$K\leftarrow {\mathbb{Q}}[{\mathrm{Gal}}(K/{\mathbb{Q}})]{\otimes}K$$ induced by the defining operation of the abstract group $G(K/{\mathbb{Q}})$ on $K$.
The space $X_0$ is a torsor under $\Gamma$. More precisely, the choice ${\mathrm{id}}\in X_0({{\overline{{\mathbb{Q}}}}})$ induces an isomorphism $$\Gamma_{{\overline{{\mathbb{Q}}}}}\to (X_0)_{{\overline{{\mathbb{Q}}}}}.$$
We check the claim on the isomorphism on finite level. Let $K/{\mathbb{Q}}$ be a finite Galois extension. The projection $X_0\to {\mathrm{Spec}}(K)$ amounts to fixing an embedding $K\subset{{\overline{{\mathbb{Q}}}}}$.
We want to study the base change of $${\mathrm{Gal}}(K/{\mathbb{Q}})\times {\mathrm{Spec}}(K)\to{\mathrm{Spec}}(K)$$ to ${{\overline{{\mathbb{Q}}}}}$. On rings, this is $$\label{eq1} {{\overline{{\mathbb{Q}}}}}[{\mathrm{Gal}}(K/{\mathbb{Q}})]^\vee{\otimes}_{{\overline{{\mathbb{Q}}}}}(K{\otimes}{{\overline{{\mathbb{Q}}}}})\leftarrow K{\otimes}{{\overline{{\mathbb{Q}}}}}.$$ On the other hand, consider the map $$K{\otimes}{{\overline{{\mathbb{Q}}}}}\to \mathrm{Map}({\mathrm{Hom}}(K,{{\overline{{\mathbb{Q}}}}}),{{\overline{{\mathbb{Q}}}}}){\cong}{{\overline{{\mathbb{Q}}}}}[{\mathrm{Hom}}(K,{{\overline{{\mathbb{Q}}}}})]^\vee,\quad \lambda{\otimes}a\mapsto (\sigma\mapsto a\sigma(\lambda)).$$ It is Galois equivariant. As $K/{\mathbb{Q}}$ is separable, it is an isomorphism. Hence our map (\[eq1\]) can be written as $$\label{eq2}
{{\overline{{\mathbb{Q}}}}}[{\mathrm{Gal}}(K/{\mathbb{Q}})]^\vee{\otimes}{{\overline{{\mathbb{Q}}}}}[{\mathrm{Hom}}(K,{{\overline{{\mathbb{Q}}}}})]^\vee\leftarrow {{\overline{{\mathbb{Q}}}}}[{\mathrm{Hom}}(K,{{\overline{{\mathbb{Q}}}}})]^\vee.$$ It is obtained by duality from the right operation $${\mathrm{Gal}}(K/{\mathbb{Q}})\times{\mathrm{Hom}}(K/{{\overline{{\mathbb{Q}}}}})\to{\mathrm{Hom}}(K/{{\overline{{\mathbb{Q}}}}}).$$ By Galois theory, this operation is simply transitive. By evaluating on the fixed inclusion $K\subset {{\overline{{\mathbb{Q}}}}}$, we get a bijection $${\mathrm{Gal}}(K/{\mathbb{Q}})\to{\mathrm{Hom}}(K,{{\overline{{\mathbb{Q}}}}})$$ translating to the isomorphism $${{\overline{{\mathbb{Q}}}}}[{\mathrm{Gal}}(K/{\mathbb{Q}}]^\vee\leftarrow K{\otimes}{{\overline{{\mathbb{Q}}}}}$$ that we needed to show.
Nori’s semi-torsor
------------------
We return to the situation of Section \[ssec:form\_cat\]. We want to explain how formal periods are a semi-torsor or even torsor.
Let ${\mathbb{K}}\subset {\mathbb{C}}$, ${\mathcal{C}}$ an additive category, $$T:{\mathcal{C}}\to {({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}},\quad M\mapsto (T(M)_{\mathbb{K}}, T(M)_{\mathbb{Q}},\phi_M)$$ an additive functor. From now on we write $$T_{\mathrm{dR}}(M):= T(M)_{\mathbb{K}}, \hspace{2ex}T_{\mathrm{sing}}(M):=T(M)_{\mathbb{Q}}.$$ Both $T_{\mathrm{dR}}$ and $T_{\mathrm{sing}}$ are additive functors to ${\mathbb{K}}{\mathrm{-Vect}}$ and ${\mathbb{Q}}{\mathrm{-Vect}}$, respectively. In the Tannakian spirit we call them *fibre functors*.
We first give an alternative description of the formal period space. For a fixed $M$, the ${\mathbb{Q}}$-vector space generated by the symbols $(M,\sigma,\omega)$ with the relation of bilinearity is nothing but $$\begin{gathered}
T_{\mathrm{dR}}(M){\otimes}_{\mathbb{Q}}T_{\mathrm{sing}}(M)^\vee{\cong}T_{\mathrm{dR}}(M){\otimes}_{\mathbb{K}}T_{\mathrm{sing}}^\vee(M)_{\mathbb{K}}\\
{\cong}{\mathrm{Hom}}_{\mathbb{K}}(T_{\mathrm{dR}}(M),T_{\mathrm{sing}}(M)_{\mathbb{K}})^\vee.\end{gathered}$$ Hence we have an alternative description of the space of formal periods $${\tilde{{\mathcal{P}}}}({\mathcal{C}})=\left(\bigoplus_{M\in{\mathcal{C}}} {\mathrm{Hom}}_{{\overline{{\mathbb{Q}}}}}(T_{\mathrm{dR}}(M),T_{\mathrm{sing}}(M))^\vee\right)/\text{functoriality}.$$ Note that the ${\mathbb{Q}}$-sub vector space of relations induced by functoriality is even a ${\mathbb{K}}$-vector space. This makes clear that ${\tilde{{\mathcal{P}}}}({\mathcal{C}})$ is a ${\mathbb{K}}$-vector space.
In the next step, we introduce the coalgebra (or even affine ${\mathbb{K}}$-group) under which the formal period space is a semi-torsor (or even torsor). Actually, its definition is parallel to the definition of the formal period space, but with one fibre functor instead of two.
For $M\in{\mathcal{C}}$ put $${\mathcal{A}}(M)={\mathrm{End}}_{\mathbb{Q}}(T_{\mathrm{sing}}(M))^\vee$$ and $${\mathcal{A}}({\mathcal{C}})=\left(\bigoplus_{M\in{\mathcal{C}}}{\mathcal{A}}(M)\right)/\text{functoriality}.$$
Note that ${\mathcal{A}}(M)$ and ${\mathcal{A}}({\mathcal{C}})$ are ${\mathbb{Q}}$-coalgebras.
This object is at the very heart of Nori’s work. It is an explicit description of the coalgebra that Nori attaches to the additive functor $T_{\mathrm{sing}}:{\mathcal{C}}\to {\mathbb{Q}}{\mathrm{-Vect}}$.
We extend scalars to ${\mathbb{K}}$: $${\mathcal{A}}({\mathcal{C}})_{\mathbb{K}}=\left(\bigoplus_{M\in{\mathcal{C}}}{\mathrm{End}}_{\mathbb{K}}(T_{\mathrm{sing}}(M)_{\mathbb{K}})\right)^\vee/\text{functoriality}.$$
The left-module structure $$\begin{gathered}
{\mathrm{End}}_{\mathbb{K}}(T_{\mathrm{sing}}(M)_{\mathbb{K}},T_{\mathrm{sing}}(M)_{\mathbb{K}})\times {\mathrm{Hom}}_{\mathbb{K}}(T_{\mathrm{dR}}(M),T_{\mathrm{sing}}(M)_{\mathbb{K}})\\
\longrightarrow {\mathrm{Hom}}_{\mathbb{K}}(T_{\mathrm{dR}}(M),T_{\mathrm{sing}}(M)_{\mathbb{K}})\end{gathered}$$ induces a right-comodule structure $${\mathrm{Hom}}_{\mathbb{K}}(T_{\mathrm{dR}}(M),T_{\mathrm{sing}}(M)_{\mathbb{K}})^\vee\to {\mathcal{A}}(M)_{\mathbb{K}}{\otimes}{\mathrm{Hom}}_{\mathbb{K}}(T_{\mathrm{dR}}(M),T_{\mathrm{sing}}(M)_{\mathbb{K}})^\vee$$ and hence $${\tilde{{\mathcal{P}}}}({\mathcal{C}})\to{\mathcal{A}}({\mathcal{C}})_{\mathbb{K}}{\otimes}_{\mathbb{K}}{\tilde{{\mathcal{P}}}}({\mathcal{C}}){\cong}{\mathcal{A}}({\mathcal{C}}){\otimes}_{\mathbb{Q}}{\tilde{{\mathcal{P}}}}({\mathcal{C}}).$$ Recall that $T$ takes values in ${({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$, hence we have a distinguished isomorphism of functors $\phi_T:T_{\mathrm{dR}}(M)_{\mathbb{C}}\to T_{\mathrm{sing}}(M)_{\mathbb{C}}$. It is an element of ${\mathrm{Hom}}_{\mathbb{C}}(T_{\mathrm{dR}}(M)_{\mathbb{C}},T_{\mathrm{sing}}(M)_{\mathbb{C}})$ or equivalently a linear map $${\mathrm{Hom}}_{\mathbb{K}}(T_{\mathrm{dR}}(M),T_{\mathrm{sing}}(M)_{\mathbb{K}})^\vee \to{\mathbb{C}}.$$ Hence, this is precisely the data that we need for a semi-torsor. Indeed, because $\phi_T$ is an isomorphism, it induces an isomorphism $${\tilde{{\mathcal{P}}}}({\mathcal{C}})_{\mathbb{C}}\to {\mathcal{A}}({\mathcal{C}})_{\mathbb{C}}.$$ We have shown:
The space ${\tilde{{\mathcal{P}}}}({\mathcal{C}})$ is a ${\mathbb{K}}$-semi-torsor over ${\mathcal{A}}({\mathcal{C}})_{\mathbb{K}}$ in the sense of Definition \[defn:semi\].
Let ${C({\mathcal{C}},T_{\mathrm{sing}})}$ be the category of ${\mathcal{A}}({\mathcal{C}})$-comodules finite dimensional over ${\mathbb{Q}}$.
This is nothing but Nori’s diagram category, see [@period-buch Section 7.1.2].
There is a natural factorisation $$T_{\mathrm{sing}}:{\mathcal{C}}\to{C({\mathcal{C}},T_{\mathrm{sing}})}\xrightarrow{\tilde{T}_{\mathrm{sing}}}{\mathbb{Q}}{\mathrm{-Vect}}$$ with ${C({\mathcal{C}},T_{\mathrm{sing}})}$ abelian and ${\mathbb{Q}}$-linear, the functor $\tilde{T}$ faithful and exact, and the category is universal with this property.
\[rem:univ\]In particular, ${\mathcal{C}}$ is equivalent to ${C({\mathcal{C}},T_{\mathrm{sing}})}$ if ${\mathcal{C}}$ is itself abelian and $T_{\mathrm{sing}}$ faithful and exact.
\[prop:make\_ab\]We have $${\tilde{{\mathcal{P}}}}({\mathcal{C}}){\cong}{\tilde{{\mathcal{P}}}}({C({\mathcal{C}},T_{\mathrm{sing}})}).$$
We use the semi-torsor structure to reduce the claim to the comparison of the diagram algebra. It remains to check that the natural map $${\mathcal{A}}({\mathcal{C}})\to{\mathcal{A}}({C({\mathcal{C}},T_{\mathrm{sing}})})$$ is an isomorphism. This is only implicit in [@period-buch Section 7.3]. Here is the argument: By construction, ${\mathcal{A}}({\mathcal{C}})=\lim_i A_i$ for coalgebras finite dimensional over ${\mathbb{Q}}$ of the form $A_i=E_i^*$ for a ${\mathbb{Q}}$-algebra $E_i$. We also have ${C({\mathcal{C}},T_{\mathrm{sing}})}=\bigcup_i E_i{\mathrm{-Mod}}$, the category of finitely generated $E_i$-modules. Hence it suffices to show that $$E_i{\cong}{\mathcal{A}}(E_i{\mathrm{-Mod}})^\vee.$$ The category $E_i{\mathrm{-Mod}}$ has the projective generator $E_i$. By [@period-buch Lemma 7.3.14], we have ${\mathcal{A}}(E_i{\mathrm{-Mod}})^\vee{\cong}{\mathrm{End}}(T_{\mathrm{sing}}|_{E_i})$. The latter means all ${\mathbb{Q}}$-linear maps $E_i\to E_i$ commuting with all $E_i$-morphisms $E_i\to E_i$. We have ${\mathrm{End}}_{E_i{\mathrm{-Mod}}}(E_i)=E_i^\circ$ (the opposite algebra of $E_i$) and hence ${\mathrm{End}}(T_{\mathrm{sing}}|_{E_i})=E_i$ as claimed.
The Tannakian case
------------------
Assume now that, in addition, ${\mathcal{C}}$ is a rigid tensor category and $T$ a faithfully exact tensor functor. Using $T_{\mathrm{sing}}$ as fibre functor, this makes ${\mathcal{C}}$ a *Tannakian* category. By Tannaka duality, it is equivalent to the category of finite dimensional representations of a pro-algebraic group $G({\mathcal{C}})$. There is a second fibre functor given by $T_{\mathrm{dR}}$. Again by Tannaka theory, the comparison of the two fibre functors defines a pro-algebriac affine scheme $X({\mathcal{C}})$ over ${\mathbb{K}}$ such that $$X(S)=\left\{\Phi:T_{\mathrm{sing}}(\cdot)_S\to T_{\mathrm{dR}}(\cdot)_S| \text{isom. of tensor functors}\right\}.$$ It is a torsor under $G({\mathcal{C}})$ $$G({\mathcal{C}})\times X({\mathcal{C}})\to X({\mathcal{C}}).$$ When passing to the underlying semi-torsor, these objects are identical to the ones considered before.
There are natural isomorphisms ${\mathcal{O}}(X){\cong}{\tilde{{\mathcal{P}}}}({\mathcal{C}})$ and ${\mathcal{O}}(G({\mathcal{C}}))={\mathcal{A}}({\mathcal{C}})$.
The case of the group is [@period-buch Theorem 7.1.21]. The same arguments also applies to $X({\mathcal{C}})$, see also [@period-buch Remark 8.2.11].
The case of motives {#ssec:motive_torsor}
-------------------
A particularly interesting case ist ${\mathcal{C}}={\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})$. Put $$\begin{aligned}
G_{\mathrm{mot}}({\mathbb{K}})&:={\mathrm{Spec}}\left({\mathcal{A}}({\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}}))\right),\\
X({\mathbb{K}})&:={\mathrm{Spec}}\left({\tilde{{\mathcal{P}}}}({\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}}))\right).\end{aligned}$$ Then $$G_{\mathrm{mot}}({\mathbb{K}})_{\mathbb{K}}\times_{\mathbb{K}}X({\mathbb{K}})\to X({\mathbb{K}})$$ is a torsor. We use this to give the proof of Theorem \[thm:indep\] that was left open.
Fix ${\mathbb{K}}\subset{{\overline{{\mathbb{Q}}}}}$. Let ${\mathcal{A}}_{\mathrm{Nori}}({\mathbb{K}})$ be the coalgebra defined using the same data as in Definition \[defn:perf\], but with $H^i_{\mathrm{dR}}$ replaced by $H^i_{\mathrm{sing}}$ and with base field ${\mathbb{K}}$ instead of ${{\overline{{\mathbb{Q}}}}}$.
The considerations of the present section also apply to this case and show that ${\mathcal{A}}_{\mathrm{Nori}}({\mathbb{K}})$ is nothing but Nori’s diagram coalgebra for the diagram of pairs, see [@period-buch Definition 9.1.1]. By definition, see [@period-buch Definition 9.1.3], the category ${\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})$ is the category of representations of $G_{\mathrm{mot}}({\mathbb{K}})={\mathrm{Spec}}\left({\mathcal{A}}_{\mathrm{Nori}}({\mathbb{K}})\right)$. Hence ${\mathcal{A}}_{\mathrm{Nori}}({\mathbb{K}}){\cong}{\mathcal{A}}({\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}}))$.
By [@period-buch Corollary 10.1.7] we also have ${\mathcal{A}}_{\mathrm{Nori}}({\mathbb{K}}){\cong}{\mathcal{A}}({{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}}))$. By Galois theory of periods, i.e., the torsor property, this also implies $${\tilde{{\mathcal{P}}}}({\mathbb{K}}){\cong}{\tilde{{\mathcal{P}}}}({\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})){\cong}{\tilde{{\mathcal{P}}}}({{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})).$$
We now have to compare different ${\mathbb{K}}$’s. Let $K/k$ be an algebraic extension of subfields of ${{\overline{{\mathbb{Q}}}}}$. The comparison $${\tilde{{\mathcal{P}}}}(K){\cong}{\tilde{{\mathcal{P}}}}(k)$$ is claimed in [@period-buch Proposition 3.1.11]. Unfortunately, the argument is not complete. It only shows that the map induced by base change $${\mathcal{P}}(k)\to{\tilde{{\mathcal{P}}}}(K)$$ is surjective. We are going to complete the argument now. It suffices to consider the case $K/k$ finite and Galois. In this case, there is a natural short exact sequence $$0\to G_{\mathrm{mot}}(K)\to G_{\mathrm{mot}}(k)\to{\mathrm{Gal}}(K/k)\to 0$$ by [@period-buch Theorem 9.1.6]. Moreover, we have torsor structures $$\begin{gathered}
G_{\mathrm{mot}}(k)_K\times_K X(k)_K\to X(k)_K,\\
G_{\mathrm{mot}}(K)_K\times_K X(K)\to X(K).\end{gathered}$$ We have ${{\overline{{\mathbb{Q}}}}}\subset{\tilde{{\mathcal{P}}}}(K)$ as formal periods of zero-dimensional varieties. Hence the structure morphism $X(k)\to {\mathrm{Spec}}(k)$ factors via ${\mathrm{Spec}}(K)$. We write $X'(k)$ for $X(k)$ viewed as $K$-scheme. Then $$X(k)_K=X(k)\times_kK{\cong}X'(k)\times_K({\mathrm{Spec}}(K)\times_k{\mathrm{Spec}}(K)){\cong}\coprod_{\sigma\in{\mathrm{Gal}}(K/k)}X'(k).$$ Comparing with the structure of $G_{\mathrm{mot}}(k)$, we deduce the torsor $$G_{\mathrm{mot}}(K)_K\times_KX'(k)\to X'(k).$$ Hence the natural map $X'(k)\to X(K)$ is an isomorphism.
Consequences for the period conjecture
======================================
Abstract considerations
-----------------------
As before, let ${\mathcal{C}}$ be an additive category and $T:{\mathcal{C}}\to {({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$ an additive functor and ${C({\mathcal{C}},T_{\mathrm{sing}})}$ the diagram category.
The period conjecture for ${\mathcal{C}}$ is equivalent to the period conjecture for ${C({\mathcal{C}},T_{\mathrm{sing}})}$.
The formal period algebras agree by Proposition \[prop:make\_ab\].
Hence we only need to consider abelian categories and faithful exact functors $T$ when analysing the period conjecture.
\[prop:inj\]Let ${\mathcal{C}}$ be an abelian, ${\mathbb{Q}}$-linear category. Let ${\mathcal{C}}'\subset {\mathcal{C}}$ be an abelian subcategory such that the inclusion is exact. The following are equivalent:
1. \[it:1\]the category ${\mathcal{C}}'$ is closed under subquotients in ${\mathcal{C}}$;
2. \[it:2\]the map ${\tilde{{\mathcal{P}}}}({\mathcal{C}}')\to{\tilde{{\mathcal{P}}}}({\mathcal{C}})$ is injective.
Injectivity can be tested after base change to ${\mathbb{C}}$, hence the second assertion of equivalent to the injectivity of ${\mathcal{A}}({\mathcal{C}}')\to{\mathcal{A}}({\mathcal{C}})$. The implication from (\[it:1\]) to (\[it:2\]) is [@period-buch Proposition 7.5.9].
For the converse, consider $K=\ker({\mathcal{A}}({\mathcal{C}}')\to{\mathcal{A}}({\mathcal{C}})$. It is an ${\mathcal{A}}({\mathcal{C}})$-comodule, but not a ${\mathcal{A}}({\mathcal{C}}')$-comodule. By construction ${\mathcal{A}}({\mathcal{C}}')$ is a direct limit of coalgebras finite over ${\mathbb{Q}}$. Hence $K$ is a direct limit of ${\mathcal{A}}({\mathcal{C}})$-comodules of finite dimension over ${\mathbb{Q}}$. The cannot all be ${\mathcal{A}}({\mathcal{C}}')$-comodules, hence we have found objects of ${\mathcal{C}}$ that are not in ${\mathcal{C}}'$.
\[cor:faithful\]Suppose ${\mathcal{C}}$ is abelian and $T$ faithful and exact. If the period conjecture holds for ${\mathcal{C}}$, then $T:{\mathcal{C}}\to {({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$ is fully faithful with image closed under subquotients.
We view ${\mathcal{C}}$ as a subcategory of ${({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$ via $T$. Let $\tilde{{\mathcal{C}}}$ be its closure under subquotients. Note that the periods of ${\mathcal{C}}$ agree with the periods of $\tilde{{\mathcal{C}}}$. By assumption, the composition $${\tilde{{\mathcal{P}}}}({\mathcal{C}})\to {\tilde{{\mathcal{P}}}}(\tilde{{\mathcal{C}}})\to{\mathcal{P}}({\mathcal{C}})$$ is injective. Hence the first map injective. By Proposition \[prop:inj\] this implies that ${\mathcal{C}}$ is closed under subquotients in ${({{\overline{{\mathbb{Q}}}}},{\mathbb{Q}}){\mathrm{-Vect}}}$. Let $X,Y\in{\mathcal{C}}$ and $f:X\to Y$ a morphism in ${({{\overline{{\mathbb{Q}}}}},{\mathbb{Q}}){\mathrm{-Vect}}}$. Then its graph $\Gamma\subset X\times Y$ is a subobject in ${({{\overline{{\mathbb{Q}}}}},{\mathbb{Q}}){\mathrm{-Vect}}}$, hence in ${\mathcal{C}}$. This implies that $f$ is in ${\mathcal{C}}$. Hence ${\mathcal{C}}$ is a full subcategory.
\[cor:full\]Let ${\mathcal{C}}'\subset{\mathcal{C}}$ be a full abelian subcategory closed under subquotients. If the period conjecture holds for ${\mathcal{C}}$, then it holds for ${\mathcal{C}}'$.
We have ${\tilde{{\mathcal{P}}}}({\mathcal{C}}')\subset{\tilde{{\mathcal{P}}}}({\mathcal{C}})\to{\mathbb{C}}$. If the second map is injective, then so is the composition.
The case of motives {#the-case-of-motives}
-------------------
We now specialise to ${\mathcal{C}}={\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}},{\mathbb{Q}})$.
Let $M\in {\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}},{\mathbb{Q}})$. We denote by $\langle M\rangle$ the full abelian category of ${\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}},{\mathbb{Q}})$ closed under subquotients generated by $M$.
\[prop:equiv1\]The following are equivalent:
1. The period conjecture holds for ${\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}},{\mathbb{Q}})$.
2. The period conjecture holds for $\langle M\rangle $ for all $M\in{\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}},{\mathbb{Q}})$.
Note that ${\mathcal{P}}(\langle M\rangle)={\mathcal{P}}\langle M\rangle$ in the notation of Definition \[defn:VV\].
We have ${\tilde{{\mathcal{P}}}}(\langle M\rangle)\subset{\tilde{{\mathcal{P}}}}({\mathcal{MM}_{\mathrm{Nori}}})$ because the subcategory is full and closed under subquotients. Hence $${\tilde{{\mathcal{P}}}}={\tilde{{\mathcal{P}}}}({\mathcal{MM}_{\mathrm{Nori}}})=\bigcup_M{\tilde{{\mathcal{P}}}}(\langle M\rangle).$$ The map ${\tilde{{\mathcal{P}}}}\to{\mathbb{C}}$ is injective if and only if this is true for all ${\tilde{{\mathcal{P}}}}(\langle M\rangle)$.
Grothendieck’s period conjecture
--------------------------------
In the previous section, we did not make use of the tensor product on motives. Indeed, by Section \[ssec:motive\_torsor\] we have the torsor $$G_{\mathrm{mot}}({{\overline{{\mathbb{Q}}}}})_{{\overline{{\mathbb{Q}}}}}\times_{{\overline{{\mathbb{Q}}}}}X({{\overline{{\mathbb{Q}}}}})\to X({{\overline{{\mathbb{Q}}}}}).$$
\[galois\]If the period conjecture holds, then ${\mathcal{P}}$ is a $G_{\mathrm{mot}}({\mathbb{K}})_{\mathbb{K}}$-torsor.
Let $M\in{\mathcal{MM}_{\mathrm{Nori}}}({{\overline{{\mathbb{Q}}}}})$. We put $\langle M\rangle^{{\otimes}, \vee}\subset {\mathcal{MM}_{\mathrm{Nori}}}({{\overline{{\mathbb{Q}}}}})$ the full abelian rigid tensor subcategory closed under subquotients generated by $M$. We denote $G(M)$ its Tannaka dual.
\[prop:equiv2\]The following are equivalent:
1. the period conjecture holds for ${\mathcal{MM}_{\mathrm{Nori}}}({{\overline{{\mathbb{Q}}}}})$;
2. the period conjecture holds for $\langle M\rangle^{{\otimes},\vee}$ for all $M\in{\mathcal{MM}_{\mathrm{Nori}}}({{\overline{{\mathbb{Q}}}}})$.
Same proof as for Proposition \[prop:equiv1\]
The scheme $X(M):={\mathrm{Spec}}({\tilde{{\mathcal{P}}}}\langle M\rangle^{{\otimes},\vee})$ is a $G(M)$-torsor. Hence it inherits all properties of the algebraic group $G(M)$ that can be tested after a faithfully flat base change. In particular it is smooth.
Let $M\in{\mathcal{MM}_{\mathrm{Nori}}}$. Then $X(M)$ is connected and $$\dim G(M)={\mathrm{trdeg}}{\mathbb{Q}}( {\mathcal{P}}(M)).$$
In the case of pure motives, this is the formulation of André in [@andre2 Chapitre 23].
The following are equivalent:
1. the period conjecture holds for $\langle M\rangle^{{\otimes},\vee}$;
2. the point $\mathrm{ev}\in X(M)({\mathbb{C}})$ is generic and $X(M)$ is connected;
3. Grothendieck’s period conjecture holds for $M$.
See the proof of equivalence in [@period-buch]. The crucial input is that $X(M)$ is smooth. Hence being connected makes it integral, so that there is only one generic point. Moreover, $X(M)$ and $G(M)$ have the same dimension.
Cycles and the period conjecture
================================
We now discuss the relation between conjectures on fullness of cycle class maps and the period conjecture.
Let ${\mathcal{C}}\subset {\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})$ be a full abelian category closed under subquotients. If the period conjecture holds for ${\mathcal{C}}$, then $H:{\mathcal{C}}\to {({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$ is fully faithful with image closed under subquotients.
This is a special case of Corollary \[cor:full\].
In the Tannakian case this is already formulated in [@period-buch Proposition 13.2.8]. Indeed, it seems well-known.
By definition $H$ is faithful, hence the interesting part about this is fullness. This point is taken up in a version of the period conjecture formulated by Bost and Charles in [@bost-charles] and generalised by Andreatta, Barbieri-Viale and Bertapelle in [@ABB]. We formulate the latter version. Recall from [@harrer] the realisation functor $${{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})\to D^b({\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})).$$ By composition with ${\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})\to {({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$ this gives us a functor $$H^0:{{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})\to {({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}.$$
We say that the fullness conjecture holds for ${\mathcal{C}}\subset {{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})$ if $$H^0:{\mathcal{C}}\to{({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$$ is full.
\[conj:ABB\] Let ${\mathcal{C}}\subset {{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})$ be a full additive subcategory. Then the fullness conjecture holds for ${\mathcal{C}}$.
1. Andreatta, Barbieri-Viale and Bertapelle also consider refined integral information. We do not go into this here.
2. This terminology deviates from [@ABB] where the term period conjecture is used.
Arapura established in [@arapura] (see also [@period-buch Theorem 10.2.7]) that the category of pure Nori motives is equivalent to Andrés’s abelian category ${\mathrm{AM}}$ of motives via motivated cycles.
If the period conjecture holds for pure Nori motives and the Künneth and Lefschetz standard conjectures are true, then the fullness conjecture holds for Chow motives.
The functor from Chow motives to Grothendieck motives is full by definition. The Künneth and Lefschetz standard conjectures together imply that the category of Grothendieck motives is abelian and agrees with ${\mathrm{AM}}$, hence with the category of pure Nori motives. By the period conjecture, the functor to ${({{\overline{{\mathbb{Q}}}}},{\mathbb{Q}}){\mathrm{-Vect}}}$ is full. Together this proves fullness of the composition.
A similar result actually holds for the full category of geometric motives. Its assumptions are very strong, but are indeed expected to be true.
\[prop:full\_gm\]We assume
1. the period conjecture holds for all Nori motives;
2. the category ${{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})$ has a $t$-structure compatible with the singular realisation;
3. the singular realisation is conservative;
4. the category ${{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})$ is of cohomological dimension $1$.
Then the fullness conjecture holds for ${{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})$.
The period conjecture gives again fullness of ${\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})\to{({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$. It remains to check surjectivity of $$\label{eq:surj}\tag{*} {\mathrm{Hom}}_{{{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})}(M_1,M_2)\to {\mathrm{Hom}}_{{\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})}(H^0_{\mathrm{Nori}}(M_1),H^0_{\mathrm{Nori}}(M_2))$$ for all $M_1,M_2\in{{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})$.
Let ${\mathcal{MM}}$ be the heart of the $t$-structure. By conservativity, $H^0$ is faithful. By the universal property of Nori motives, this implies ${\mathcal{MM}}{\cong}{\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})$. As the cohomological dimension is $1$, this implies ${{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}}){\cong}D^b({\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}}))$. It also implies that every object of ${{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})$ is (uncanonically) isomorphic to the direct sum of its cohomology objects. Hence it suffices to consider $M_1=N_1[i_1]$, $M_2=N_2[i_2]$ for $N_1,N_2\in{\mathcal{MM}}$, $i_1,i_2\in{\mathbb{Z}}$. We have $H^0(M_k)=0$ unless $0=i_1,i_2$. Hence surjectivity in (\[eq:surj\]) is trivial for $i_1\neq 0$ or $i_2\neq 0$. In the remaining case $i_1=i_2=0$, we have equality in equation (\[eq:surj\]), so again, the map is surjective.
The converse is not true, the fullness conjecture does *not* imply the period conjecture. An explicit example is given in [@huber-wuestholz Remark 5.6]. See also the case of mixed Tate motives, Proposition \[prop:mt\].
On the other hand, the condition on the cohomological dimension is necessary if we want to conclude from fully faithfulness on the level of abelian categories to the triangulated category. Note that ${({\mathbb{K}},{\mathbb{Q}}){\mathrm{-Vect}}}$ has cohomological dimension $1$.
Examples
========
Artin motives
-------------
The category of Artin motives over ${\mathbb{K}}$ can be described as the full subcategory of ${\mathcal{MM}_{\mathrm{Nori}}}({\mathbb{K}})$ generated by motives of $0$-dimensional varieties. The Tannakian dual of this category is nothing but ${\mathrm{Gal}}(\bar{{\mathbb{K}}}/{\mathbb{K}})$. The space of periods is ${{\overline{{\mathbb{Q}}}}}$. The torsor structure is the one described in Section \[ssec:galois\]. For more details see also [@period-buch Section 13.3].
The period conjecture and the fullness conjecture hold true for Artin motives.
Mixed Tate motives
------------------
Inside the triangulated category ${{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathrm{Spec}}({\mathbb{Q}}),{\mathbb{Q}})$ let ${\mathsf{DMT}}$ the full thick rigid tensor triangulated subcategory generated by the Tate motive ${\mathbb{Q}}(1)$. Equivalently, it is the full thick triangulated subcategory generated by the set of ${\mathbb{Q}}(i)$ for $i\in{\mathbb{Z}}$. By deep results of Borel it carries a $t$-structure, so that we get a well-defined *abelian* category ${\mathcal{MT}}$ of mixed Tate motives over ${\mathbb{Q}}$, see [@LeMTM]. Deligne and Gochanarov find a smaller abelian tensor subcategory ${{\mathcal{MT}}^f}$ of *unramified* mixed Tate motives, see [@deligne-goncharov]. Its distinguishing feature is that $${\mathrm{Ext}}^1_{{{\mathcal{MT}}^f}}({\mathbb{Q}}(0),{\mathbb{Q}}(1))={\mathbb{Z}}^*{\otimes}_{\mathbb{Z}}{\mathbb{Q}}=0\subsetneq {\mathrm{Ext}}^1_{{\mathcal{MT}}}({\mathbb{Q}}(0),{\mathbb{Q}}(1))={\mathbb{Q}}^*{\otimes}_{\mathbb{Z}}{\mathbb{Q}}.$$ By [@deligne-goncharov], the periods of ${{\mathcal{MT}}^f}$ are precisely the famous multiple zeta values. The Tannaka dual of ${{\mathcal{MT}}^f}$ is very well-understood. Using Galois theory of periods this information translates into information on ${\tilde{{\mathcal{P}}}}({{\mathcal{MT}}^f})$. The surjectivity of ${\tilde{{\mathcal{P}}}}({{\mathcal{MT}}^f})\to {\mathcal{P}}({{\mathcal{MT}}^f})$ then yields upper bounds on the dimensions of spaces of multiple zeta values. For more details see [@deligne-goncharov] or [@period-buch Chapter 15].
The period conjecture and the fullness conjecture are open for ${{\mathcal{MT}}^f}$.
Indeed, the period conjecture for ${{\mathcal{MT}}^f}$ implies statements like the algebraic independence of the values of Riemann $\zeta$-function $\zeta(n)$ for $n\geq 3$ odd that are wide open. The fullness conjecture is a lot weaker.
\[prop:mt\]The fullness conjecture holds for ${\mathsf{DMT}}$ if and only if $\zeta(n)$ is irrational for $n\geq 3$ odd.
Assume the irrationality condition. By [@deligne-goncharov], the category ${\mathcal{MT}}$ has cohomological dimension $1$ and $D^b({\mathcal{MT}})$ is equivalent to ${\mathsf{DMT}}$. Hence we can use the same reasoning as in the proof of Proposition \[prop:full\_gm\]. It suffices to show that ${\mathcal{MT}}\to {({\mathbb{Q}},{\mathbb{Q}}){\mathrm{-Vect}}}$ is fully faithful. We address this issue with the same argument as in the proof of [@deligne-goncharov Proposition 2.14] treating the functor from ${{\mathcal{MT}}^f}$ to mixed realisations. It remains to check injectivity of $${\mathrm{Ext}}^1_{{\mathcal{MT}}}({\mathbb{Q}},{\mathbb{Q}}(n))\to {\mathrm{Ext}}^1_{{({\mathbb{Q}},{\mathbb{Q}}){\mathrm{-Vect}}}}({\mathbb{Q}},{\mathbb{Q}}(n))$$ for all $n\in{\mathbb{Z}}$. Here ${\mathbb{Q}}(n)$ denotes the object of rank $1$ given by ${\mathbb{Q}}(n)_{\mathrm{dR}}={\mathbb{Q}}, {\mathbb{Q}}(n)_{\mathrm{sing}}=(2\pi i)^n{\mathbb{Q}}$ and the natural comparison isomorphism.
By Borel’s work, the left hand side is known to vanish for $n\leq 0$ and $n\geq 2$ even, hence it suffices to consider $n=2m-1$ for $m\geq 1$. By [@huber_1604 Proposition 4.3.2 and Proposition 4.2.3] $${\mathrm{Ext}}^1_{{({\mathbb{Q}},{\mathbb{Q}}){\mathrm{-Vect}}}}={\mathrm{coker}}\left( {\mathbb{Q}}(n)_{\mathrm{dR}}\oplus {\mathbb{Q}}(n)_{\mathrm{sing}}\to {\mathbb{C}}\right)={\mathbb{C}}/\langle 1,(2\pi i)^n\rangle_{\mathbb{Q}}.$$ This is a quotient of $$\label{eq:ext} {\mathrm{Ext}}^1_{{\mathrm{MHS}}_{\mathbb{Q}}}({\mathbb{Q}},{\mathbb{Q}}(n)){\cong}{\mathbb{C}}/(2\pi i)^n{\mathbb{Q}}.$$ We first consider $n=m=1$, the Kummer case. The left hand side is ${\mathbb{Q}}^*{\otimes}{\mathbb{Q}}$ and the map is the Dirichlet regulator $u\mapsto \log|u|$. The prime numbers give a basis of ${\mathbb{Q}}^*{\otimes}{\mathbb{Q}}$. By Baker’s theorem, their logarithms are ${{\overline{{\mathbb{Q}}}}}$-linearly independent. They are also real whereas $2\pi i$ is imaginary. Hence the regulator map (\[eq:ext\]) is injective.
Now let $m\geq 2$. From the well-known explicit computation of the Deligne regulator, we know that the image of a generator of ${\mathrm{Ext}}^1_{{\mathcal{MT}}}({\mathbb{Q}},{\mathbb{Q}}(2m-1))$ is given by $\zeta(2m-1)\in {\mathbb{C}}/\langle 1,(2\pi i)^{2m-1}\rangle$. It is real hence ${\mathbb{Q}}$-linearly independent of $(2\pi i)^{2m-1}\in i{\mathbb{R}}$. If it is irrational, then it is ${\mathbb{Q}}$-linearly independent of $1$ as well.
Now assume conversely that the fullness conjecture holds for ${\mathsf{DMT}}$. This implies that ${\mathcal{MT}}\to {({\mathbb{Q}},{\mathbb{Q}}){\mathrm{-Vect}}}$ is fully faithful. We claim that (\[eq:ext\]) is injective. Take an element on the right, i.e., a short exact sequence $$0\to {\mathbb{Q}}(n)\to E\to {\mathbb{Q}}(0)\to 0$$ in ${\mathcal{MT}}$ whose image in ${({\mathbb{Q}},{\mathbb{Q}}){\mathrm{-Vect}}}$ is split. This means that there is a splitting morphism ${\mathbb{Q}}(0)\to E$ in ${({\mathbb{Q}},{\mathbb{Q}}){\mathrm{-Vect}}}$. By fullness it comes from a morphism in ${\mathcal{MT}}$. This shows injectivity of (\[eq:ext\]). By the explicit computation this implies that $\zeta(2m-1)$ is ${\mathbb{Q}}$-linearly independent of $1$, i.e., irrational.
We see that the fullness statement is much weaker that than the period conjecture. There are even partial results due to Apéry and Zudilin.
The case of $1$-motives
-----------------------
The situation for $1$-motives is quite similar to the case of $0$-motives: there is an abelian category ${1\mathrm{-Mot}}_{\mathbb{K}}$ of iso-$1$-motives over ${\mathbb{K}}$. It was originally defined by Deligne, see [@hodge3]. Its derived category is equivalent the subcategory of ${{\mathsf{DM}}_{{\mathrm{gm}}}}({\mathbb{K}})$ generated by motives of varieties of dimension at most $1$. This result is due to Orgogozo, see [@orgogozo], see also Barbieri-Viale and Kahn [@BVK]. There is a significant difference to the $0$-dimensional case, though: ${1\mathrm{-Mot}}_{\mathbb{K}}$ is not closed under tensor products, so it does not make sense to speak about torsors. However, our notion of a semi-torsor is built to cover this case: ${\tilde{{\mathcal{P}}}}({1\mathrm{-Mot}}_{\mathbb{K}})$ is semi-torsor under the coalgebra ${\mathcal{A}}({1\mathrm{-Mot}}_{\mathbb{K}})_{\mathbb{K}}$.
The main result of [@huber-wuestholz] settles the period conjecture:
\[thm:1-mot\] The evalution map ${\tilde{{\mathcal{P}}}}({1\mathrm{-Mot}}_{{\overline{{\mathbb{Q}}}}})\to{\tilde{{\mathcal{P}}}}({1\mathrm{-Mot}}_{{\overline{{\mathbb{Q}}}}})$ is injective, i.e., the period conjecture holds for ${1\mathrm{-Mot}}_{{\overline{{\mathbb{Q}}}}}$.
This is first and foremost a result in transcendence theory. With a few extra arguments, it implies:
Let $X$ be an algebraic curve over ${{\overline{{\mathbb{Q}}}}}$, $\omega\in\Omega^1_{{{\overline{{\mathbb{Q}}}}}(X)}$ an algebraic differential form, $\sigma=\sum_{i=1}^na_i\gamma_i$ a singular chain avoiding the poles of $\omega$ and with $\partial\sigma$ a divisor on $X({{\overline{{\mathbb{Q}}}}})$. Then the following are equivalent:
1. the number $\int_\sigma\omega$ is algebraic;
2. we have $\omega= df +\phi$ with $\int_\phi\phi=0$.
This contains the previously known results on transcendence of $\pi$ or values of $\log$ in algebraic numbers as well as periods of differential forms over closed paths. The really new case concerns differential forms of the third kind (so with non-vanishing residues) and non-closed paths. This settles questions of Schneider that had been open since the 1950s. Here is a very explicit example:
Let $E/{{\overline{{\mathbb{Q}}}}}$ be an elliptic curve. Recall the Weierstrass $\wp$-, $\zeta$- and $\sigma$-function for $E$. Let $u\in{\mathbb{C}}$ such that $\wp(u)\in{{\overline{{\mathbb{Q}}}}}$ and $\exp_E(u)$ is non-torsion in $E({{\overline{{\mathbb{Q}}}}})$. Then $$u\zeta(u)-2\log(\sigma(u))$$ is transcendental.
In the spirit of Baker’s theorem, Theorem \[thm:1-mot\] also allows us to give an explicit formula for $\dim_{{\overline{{\mathbb{Q}}}}}{\mathcal{P}}\langle M\rangle$ for all $1$-motives $M$. It is not easy to state, so we simply refer to [@huber-wuestholz Chapter 6].
On the other hand, Theorem \[thm:1-mot\] has consequences for the various conjectures discussed before. In [@ayoub-barbieri], Ayoub and Barbieri-Viale show that ${1\mathrm{-Mot}}_k$ is a full subcategory closed under subquotients in ${\mathcal{MM}_{\mathrm{Nori}}}^{\mathrm{eff}}(k)$. They also give descpription of ${\mathcal{A}}({1\mathrm{-Mot}}_k)$ in terms of generators and relations. Together with Theorem \[thm:1-mot\] this implies Kontsevich’s period conjecture for curves.
In detail: Following [@huber-wuestholz Definition 4.6] we call a period number of *curve type* if it is a period of a motive of the form $H^1(C,D)$ with $C$ a smooth affine curve over ${{\overline{{\mathbb{Q}}}}}$ and $D\subset C$ a finite set of ${{\overline{{\mathbb{Q}}}}}$-valued points.
All relations between periods of curve type are induced by bilinearity and functoriality of pairs $(C,D)\to (C',D')$ with $C, C'$ smooth affine curves and $D\subset C$, $D'\subset C'$ finite sets of points.
In a different direction:
The functor ${1\mathrm{-Mot}}_{{\overline{{\mathbb{Q}}}}}\to {({{\overline{{\mathbb{Q}}}}},{\mathbb{Q}}){\mathrm{-Vect}}}$ is fully faithful.
This is Corollary \[cor:full\]. An independent proof was given by Andreatta, Barbieri-Viale and Bertapelle.
On the other hand, Orgogozo [@orgogozo] and Barbieri-Viale, Kahn [@BVK] showed that the derived category of ${1\mathrm{-Mot}}_k$ is a full subcategory of ${{\mathsf{DM}}_{{\mathrm{gm}}}}(k)$.
The fullness conjecture holds for $d_{\leq 1}{{\mathsf{DM}}_{{\mathrm{gm}}}}({{\overline{{\mathbb{Q}}}}})$.
By [@orgogozo Proposition 3.2.4], the category $d_{\leq 1}{{\mathsf{DM}}_{{\mathrm{gm}}}}({{\overline{{\mathbb{Q}}}}}){\cong}D^b({1\mathrm{-Mot}}_{{\overline{{\mathbb{Q}}}}})$ has cohomological dimension $1$. We can now use the same argument as in the proof of Proposition \[prop:full\_gm\], unconditionally.
The natural functors ${1\mathrm{-Mot}}_{{\overline{{\mathbb{Q}}}}}\to {\mathcal{MM}_{\mathrm{Nori}}}({{\overline{{\mathbb{Q}}}}})$ (non-effective Nori motives) and ${1\mathrm{-Mot}}_{{\overline{{\mathbb{Q}}}}}\to{\mathrm{MHS}}_{{\overline{{\mathbb{Q}}}}}$ (mixed Hodge structures over ${{\overline{{\mathbb{Q}}}}}$) are full.
Their composition with the functor to ${({{\overline{{\mathbb{Q}}}}},{\mathbb{Q}}){\mathrm{-Vect}}}$ is full.
A much stronger version of the above was recently shown by André by comparing the Tannakian dual of the tensor category generated by $1$-motives to the Mumford Tate group.
Let $k\subset {\mathbb{C}}$ be algebraically closed. Let ${1\mathrm{-Mot}}_k^{\otimes}\subset {\mathcal{MM}_{\mathrm{Nori}}}(k)$ be the full Tannakian subcategory closed under subquotients generated by ${1\mathrm{-Mot}}_k$. Then the functor $${1\mathrm{-Mot}}_k^{\otimes}\to {\mathrm{MHS}}$$ is fully faithful.
In the case $k={{\overline{{\mathbb{Q}}}}}$, this means that the ${{\overline{{\mathbb{Q}}}}}$-structure can actually be recovered from the Hodge- and weight filtration – and conversely. His result also gives fullness of ${1\mathrm{-Mot}}_k\to {\mathcal{MM}_{\mathrm{Nori}}}(k)$ for all algebraically closed $k\subset {\mathbb{C}}$ (actually he has to show this on the way).
|
---
abstract: 'We report on a striking departure from the canonical step sequence of quantized conductance in a ballistic, quasi-one-dimensional metallic channel. Ideally, in such a structure, each sub-band population contributes its Landauer conductance quantum independently of the rest. In a picture based exclusively on coherent single-carrier transmission, unitary back-scattering can lower a conductance step below ideal, but it is absolutely impossible for it to enhance the ideal Landauer conductance of a sub-band. Precisely such an anomalous and robust nonlinear enhancement has already been observed over the whole density range between sub-band thresholds (de Picciotto R [*et al*]{} 2004 Phys. Rev. Lett. [**92**]{} 036805 and 2008 J. Phys. Condens. Matter [**20**]{} 164204). We show theoretically that the anomalous enhancement of ideal Landauer conductance is the hallmark of carrier transitions coupling the discrete sub-bands.'
address:
- '$^1$ School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia.'
- '$^2$ Department of Theoretical Physics, RSPE, The Australian National University, Canberra, ACT 2601, Australia.'
author:
- 'Frederick Green$^1$ and Mukunda P. Das$^2$'
title: 'Anomalous Conductance Quantization in the Inter-band Gap of a One-dimensional Channel'
---
= 10000
Introduction
=============
Understanding the conductance in quantum-confined metallic channels is a central aspect of electrical transport in meso- and nanoscopic structures. Quantization of the conductance in ballistic quantum-well channels is the unique property of their one-dimensional (1D), waveguide-like nature.
The standard model of 1D quantized conductance [@FG; @Davies; @Yaro] adopts the viewpoint of Landauer and associates [@RL; @RLYI] in which a highly constricted channel, interposed between macroscopic ohmic contacts, is conceived as a simple barrier potential modifying the propagation of free single-electron quantum states. Strong lateral confinement of these states by the device’s quantum-well structure leads to their segregation into discrete levels (sub-bands), separated by energy gaps whose magnitude may run from tens of meV in III-V heterojunctions to several eV in carbon nanotubes.
This picture of conductance as quantum transmission accounts elegantly for the well-documented resolution of the ohmic conductance in 1D structures into a sequence of integral plateaux. As a function of increasing carrier density in the device, each successive step in the conductance extends, unaltered, throughout the energy-gap region separating the discrete sub-bands. A new plateau appears as soon as the chemical potential and thus the carrier population cross the gap to access the next higher sub-band.
That the single-particle quantum transmission picture does not address all experimental observations is known [@Neder; @GTD; @dP1; @dP2]. The reasons that it does not are also known [@wims]. First, quantum transmission theory is restricted to weak-field linear response; second, it does not account for the resistive dissipation that is inevitable in every ohmic structure; third, the approach has no way to address scattering processes other than purely coherent, purely elastic, back-scattering. These and other fundamental drawbacks of the transmission approach have been critiqued in detail elsewhere [@DG; @DG2; @DG3].
One example of a crucial inelastic physical process is intra-band scattering by phonon emission, responsible for the above-mentioned dissipation. Another example is inter-band transitions, which are inherently inelastic two-body processes with substantial energy exchange between discrete bands. They lie beyond the scope of a single-carrier description.
The physics of discrete transitions between distinct conduction sub-bands lies at the heart of this paper. The experimental findings that animate our work are those of de Picciotto, Pfeiffer, Baldwin and West [@dP1; @dP2].
-6cm
{width="11truecm"}
[[**FIG. 1**]{} [Quantized differential conductance measured in a quasi-one-dimensional multi-sub-band ballistic channel, after de Picciotto [*et al*]{}, reference [@dP2]. Conductance is plotted in Landauer units $G_0=77.48\mu$S, as a function of a control-gate voltage modulating the total carrier density within the channel. Beyond the labile structures (grey box) at threshold of the ground-state band lies a series of very flat extended plateaux observed at different values of source-drain bias voltage inducing the current. The system is taken beyond linear response, as seen in the bias dependence of the conductance. Most remarkable is the raising of $G$ above the absolute upper limit $G_0$ imposed by unitarity. This shows that effects beyond simple quantum-coherent transmission dominate the transport physics. ]{}]{} 0.25cm
Let us explain the importance of the remarkable non-linear, non-Landauer 1D conductance plateaux documented by de Picciotto and colleagues. Figure 1 reproduces the core results of their references [@dP1] and [@dP2]. It shows a series of differential-conductance traces (normalized to the Landauer quantum $G_0=77.48\mu$S) for a nearly ideal ballistic quantum wire, taken at fixed source-drain driving voltage and plotted as functions of gate voltage controlling the channel chemical potential and so its carrier density. The greyed region contains a complex of highly mutable shoulder structures evident at the ground-state threshold, popularly termed the “0.7 anomaly”, which is not of interest here.
We focus, by contrast, upon the conductance plateaux extending over the larger part of the inter-band region up to the threshold of the first excited sub-band. Unlike the “0.7” features [@DG4], they are robustly regular with a highly systematic dependence on the source-drain driving voltage.
- The anomalous steps are extremely flat and extend, with carrier density, from the early-onset “0.7” feature sequence right up to the threshold of the next higher sub-band. Qualitatively and quantitatively, they are wholly distinct from the relatively ephemeral structures close to first threshold.
- The anomalous conductance steps are voltage dependent. They are beyond any linear-response description.
- These plateaux cover the entire region where the chemical potential of the carriers lies in the gap between ground- and first-excited-state bands. According to quantum-transmission theory, such a structure cannot be higher than the ideal limit $G_0$.
- With increasing source-drain voltage, the enhanced steps increase in size from the expected weak-field baseline, exceeding appreciably the absolute maximum set by $G_0$. We stress the [*impossibility*]{} of such a scenario within quantum transmission, which would otherwise see its unitarity (conservation of probability) wrecked.
The devices studied in references [@dP1] and [@dP2] are of unprecedented quality, perhaps the closest-to-perfect ballistic wires so far fabricated. We remark on the care taken by the cited authors to isolate the physics at work within their samples. It is a little surprising, then, that (to our knowledge) no authors since the original team [@dP2] have commented on the startling violation of accepted predictions by the quantized conductance data of figure 1.
Reference [@dP1] contains a prescient comment on the role of inter-mode coupling within the test structures; that is, that there should be some exchange of energy and possibly carriers between sub-bands, setting up a mutual dynamical feedback. To date, theoretical support for that hypothesis has not been at hand. The goal is to provide it.
In the following we present a brief description of our quantum kinetic analysis of the anomalous steps in 1D conductance. Our microscopically conserving model is based on the quantum Boltzmann formalism [@wims] extended to inter-band transitions. After this short account we survey our numerical results, comparing and contrasting them with the basic data of de Picciotto and colleagues. Particular features of the results shed light on the relevant physics. The paper ends with a summary and foreshadows novel theoretical possibilities that could be tested in a renewed series of experiments in similar high-quality structures.
Problem and Solution
====================
The problem is to try to replicate and thus unpack the physics of anomalous non-linear enhancement of the Landauer conductance. We posit a uniform one-dimensional ballistic channel.
Since, even in principle, the active 1D device region cannot be divorced from the large source and drain boundary leads, its operative length is no longer exclusively determined by its physical dimensions or the bulk mean free paths of its originating substrate. Rather, its length is dictated by the longest carrier mean free path (MFP) for the channel [*as embedded in its non-ideal bounding leads*]{}. Adopting the estimate suggested in reference [@dP1], we take an operational channel length $L = 2 {\mu}$m. Our uniform-channel results, however, do not depend on the absolute MFPs assumed for this ballistic scenario.
In essence we are describing carrier behaviour averaged over an abstract ensemble of such wires, seamlessly connected in series, each with maximum mean free path $L$. At cryogenic temperatures one expects the inelastic mean free path to set the longer scale, fixing $L$. On the other hand, the observed weak-field conductance falls short of the ideal Landauer quantum and we set the complementary elastic intra-band MFP somewhat below $L$. An elastic MFP of $0.769L$ matches the mean weak-field conductance after reference [@dP1].
Transport equations with inter-band coupling
--------------------------------------------
Our transport equations define the behaviour of two sub-band populations, separated by their energy gap $E_g$. Given two steady-state distributions $f_{kj}$ for the lower band ($j=1$) and the next higher ($j=2$) as functions of band momentum $k$, the equations are of modified Boltzmann-Drude form:
$$\begin{aligned}
{q{\cal E}\over \hbar} {\partial f_{k1}\over \partial k}
=&&
- R_{\rm in1}(f_{k1} - {\overline f}_{k1}) - R_{\rm el1} f^o_{k1}
\cr
&& {~~~ }
- R_{01}
{\Bigl( e^{-E_g/k_{\rm B}T} f_{k1} (1 - {\overline f}_{k2})
- {\overline f}_{k2} (1 - f_{k1}) \Bigr)},
\cr
%\cr
{q{\cal E}\over \hbar} {\partial f_{k2}\over \partial k}
=&&
- R_{\rm in2} (f_{k2} - {\overline f}_{k2}) - R_{\rm el2} f^o_{k2}
\cr
&& {~~~ }
- R_{02}
{\Bigl( f_{k2} (1 - {\overline f}_{k1}) -
e^{-E_g/k_{\rm B}T} ~{\overline f}_{k1} (1 - f_{k2}) \Bigr)}.
\label{nl01}\end{aligned}$$
The nature of the reference distributions ${\overline f}_{kj}$ is explained below. Other notation is as follows. On the left-hand side of this pair of steady-state equations, ${\cal E}$ is the uniform field exerted on the carriers by the applied source-drain voltage. On the right-hand sides the parameters $R_{{\rm in}j}$ and $R_{{\rm el}j}$ are, respectively, the intra-band inelastic and elastic scattering rates assigned to each band while $R_{0j}$ correspondingly is the inter-band transition rate for the coupling between the two populations. The elastic collision term scales with the odd part of the distributions since elastic scattering can only reverse the momentum direction with no change in energy: $f^o_{kj} = (f_{kj} - f_{-kj})/2$.
The final parameter is the Boltzmann factor $e^{-E_g/k_{\rm B}T}$ associated with promoting a carrier from lower to upper sub-band across their gap separation $E_g$. In the subsequent discussion we will express all energies in thermal units $k_{\rm B}T$, and momenta in thermal units $k_{\rm th} \equiv \sqrt{2k_{\rm B}T/\hbar^2}$.
We now discuss the meaning and role of the effective equilibrium functions
$$\begin{aligned}
&&
{~~~ ~~~ ~ }
{\overline f}_{k1}(\mu)
\equiv
1/(1 + \exp(k^2 - \mu)) ~~{\rm and}
\cr
\cr
&&{\overline f}_{k2}(\mu - E_g)
\equiv
1/(1 + \exp(k^2 + E_g - \mu))
\label{nl01.1}\end{aligned}$$
with momenta and energies in thermal units. Transition events redistribute the electron population between the two sub-bands of the channel. The change in their respective densities is presumed to depress the effective chemical potential of the lower band: $\mu$ is renormalized to $\mu\!-\!\zeta_1$ while the augmented population in the upper band, located above the lower by the band gap $E_g$, follows the rise in its effective chemical potential: $\mu\!-\!E_g$ goes to $\mu\!+\!\zeta_2\!-\!E_g$. Thus both quantities $\zeta_1$ and $\zeta_2$ should be non-negative.
For any choice of the pair of renormalized chemical potentials, the coupled equations (\[nl01\]) are solved systematically with the expressions for ${\overline f}_{k1}(\mu\!-\!\zeta_1)$ and ${\overline f}_{k2}(\mu\!+\!\zeta_2\!-\!E_g)$, equations (\[nl01.1\]), as input. Details of the relevant Green-function algorithm are left to a longer account.
Microscopic conservation
------------------------
Consider the equilibrium state at zero field, for which $\zeta_1 = 0 = \zeta_2$. The effective equilibria are now absolute and furthermore $f_{kj} = {\overline f}_{kj}$. The distinct terms on either side of the transport equations (\[nl01\]) all vanish individually. Detailed balance is satisfied.
What happens at finite field? The intra- and inter-band inelastic collision terms on the right-hand sides of equations (\[nl01\]) are not guaranteed to vanish identically when integrated separately, although the left-hand-side expressions always do so. As a consequence conservation must be imposed explicitly upon the solution to the joint response. [@DG; @Greene; @Mermin] We have a type of generation-recombination problem; it follows that conservation cannot apply to each sub-band individually, but only jointly.
The bi-linear coupling between distributions in the inter-band transition terms means that the full problem cannot be solved even semi-analytically save in the trivial case of independent sub-bands (zero transitions, $R_{01} = R_{02} = 0$). Conservation then appears as a mandatory relation linking the pair of potential shifts $\zeta_j$ so the system’s total density is invariant (the spin factor appears explicitly in the integrals, which are rendered in dimensionless units):
$$\begin{aligned}
&&
2\int dk {\Bigl( {\overline f}_{k1}(\mu) +
{\overline f}_{k2}(\mu - E_g) \Bigr)}
\equiv
n(\mu)
\cr
%\cr
&& {~~~ }
\equiv
2\int dk {\Bigl( f_{k1}(\mu - \zeta_1; \zeta_2) +
f_{k2}(\mu + \zeta_2 - E_g; \zeta_1) \Bigr)}.
\label{nl02}\end{aligned}$$
Since there are two undetermined quantities to solve, a second relation is needed. The new physical information to be adduced must be independent of anything contained in the transport equations themselves.
For the second, constitutive relation we take the Helmholtz free-energy density for the non-equilibrium carrier distribution and remove from it the formal energy of assembly for the system, mediated by the chemical potential. Under the action of the external driving field, the change in this net energy measures the internal dynamical rearrangement of the sub-band distributions induced by the field alone. This is manifested in the renormalization of the bands’ chemical potentials $\mu_1\!\equiv\!\mu\!-\!\zeta_1 ~{\rm and}~
\mu_2\!\equiv\!\mu\!+\!\zeta_2\!-\!E_g$, as well as the form of the distributions $f_1(\mu_1)$ and $f_2(\mu_2)$.
Extending the standard thermodynamic expression [@TD] for the Helmholtz free energy in each band, we write its difference with the energy of assembly as
$$\begin{aligned}
&&F[f_j(\mu_j)]
\equiv
2\int dk (k^2\!-\!\mu_j) f_{kj}
\cr
&& {~~~ ~~~ ~~~ ~~~ ~~~ }
+ 2\!\! \int dk
{\Bigl( f_{kj}\ln f_{kj}\!+\!(\!1-\!f_{kj})\ln(1\!-\!f_{kj}) \Bigr)}
\label{nl03}\end{aligned}$$
recognizing the leading right-hand integral in equation (\[nl03\]) as the total internal energy, less the assembly energy. The second right-hand integral is the Uehling-Uhlenbeck entropy entering into the H-theorem for fermions [@UU]. At global equilibrium, minimizing $F$ (understood as a functional of the distribution $f_j$ and subject to the latter’s variation) leads to the familiar Fermi-Dirac form for $f_j$.
We recapitulate.
- A given gate voltage fixes the global chemical potential and total density within the channel.
- The source-drain field, acting independently of this, excites the carriers so a portion from the lower band is promoted to the upper band.
- The density decrease in the lower band is determined by the decrement $\zeta_1$ in the value of the common chemical potential $\mu$, while the increase in the upper band is determined by the increment $\zeta_2$ in $\mu$.
- Any loss from the lower band must match the gain in the upper one. Thus the two shifts in chemical potential are coupled by the conservation relation (\[nl02\]).
- To close the self-consistent solution for the $\zeta_j$, we look for any change in the free energy of the system as defined in equation (\[nl03\]), induced solely by application of the driving field.
The behaviour of $F$ above, purely as a function of chemical potential, is quite different from its behaviour purely as a functional of $f$. In the equilibrium state it is readily seen that
$${dF\over d\mu}[{\overline f}(\mu)] = -2\int dk {\overline f}_k(\mu)$$
so $F$ has no lower bound with increasing sub-band density. Since the form of the $f_j$ is strictly prescribed by solving the kinetic equations, the behaviour of $F$ is strictly a function of $\mu_1$ and $\mu_2$. The total net energy of the system, based on equation (\[nl03\]), must then exhibit maxima in the space $(\zeta_1,\zeta_2)$.
As our working principle we look for the maximum in the total net energy summed over both bands, as a function of the $\zeta_j$. The physical picture is analogous to a tap continuously feeding fluid to a finite container, which finally overflows. In a similar way the driven system will accumulate as much of the inflowing excess energy as it can, up to the point that increasing resistive dissipation matches the inflow and precludes any further internal buildup.
Next we construct the difference of the sum of net energies over the non-equilibrium distributions, indexed by $\zeta_1$ and $\zeta_2$, and its analogous non-equilibrium sum with $\zeta_2$ set to zero. This difference vanishes for $\zeta_2 = 0$. Otherwise, according to circumstances, it may exhibit a nontrivial maximum in parameter space as $\zeta_2$ is increased systematically. The computed quantity is
$$\Delta F(\mu; \zeta_1, \zeta_2)
= \sum_{j=1,2} {\Bigl(
F[f_j(\mu_j)] - {\Bigl. F[f_j(\mu_j)] \Bigr|}_{\zeta_2=0} \Bigr)}.
\label{nl03.1}$$
Wherever we find the local maximum of $\Delta F$ on the locus of constant density $n(\mu)$ there is a unique self-consistent pair $(\zeta_1, \zeta_2)$. This fixes the desired physical solution for the interacting system. If no nontrivial solution exists, the maximum defaults to the $\zeta_1$-axis (that is, $\zeta_2=0$) and produces the standard Landauer conductance.
0.25cm
{width="10truecm"}
[[**FIG. 2**]{} [Landscape of self-consistent solutions as functions of $\zeta_1$ and $\zeta_2$ (in thermal units) at 1mV source-drain voltage over a channel 2$\mu$m long, at temperature 4K. The dotted horizontal line at $\zeta_2 \approx 2.6$, tracks the maximum in the bias-induced excess free energy $\Delta F(\mu; \zeta_1, \zeta_2)$, equation (\[nl03.1\]), over the latter’s density map. At global chemical potential $\mu$ the contours of constant channel density $n(\mu)$, equation (\[nl02\]), rise vertically from the $\zeta_1$-axis to intersect the maximum $\Delta F$ at the points of self-consistency (dots). For each fixed density the solution determines the system’s non-linear conductance as a function of $\mu$. Since the physical chemical potential is $\mu = \mu_0 - \zeta'_1$ at the point of departure $\zeta'_1$ on the $\zeta_1$-axis (with $\zeta'_1 = 0$ at origin), the self-consistent values $(\zeta_1, \zeta_2)$ referred to $\mu_0$ are offset to yield the physical renormalizations $(\zeta_1-\zeta_1', \zeta_2+\zeta_1')$ referred to the physical $\mu$. ]{}]{} 0.25cm
Figure 2 illustrates the typical landscape of non-equilibrium net free energy and total electron density in the space of the renormalized chemical potentials, encompassing the family of self-consistent solutions for a range of densities down from a given equilibrium starting value $n(\mu_0)$. The contour of constant density that matches the equilibrium value $n(\mu_0 - \zeta'_1)$, say, is superimposed on the contours of constant excess energy $\Delta F$. For $\zeta_2 = 0$ the upper band is practically empty so each contour of constant density departs orthogonally from the $\zeta_1$-axis. The landscape is mapped as $\zeta_2$ increases and the density in the upper band goes from near-empty to degenerate.
Implementation
--------------
Equation (\[nl03.1\]) is a measure of the non-equilibrium excess energy built up in the system when the driving field induces redistribution of the populations between sub-bands. Computing the self-consistent solution requires us to connect the intra-band MFPs, together of course with the inter-band transition probability, to the rates $R_{{\rm in}j}, R_{{\rm el}j}$ and $R_{0j}$ that parametrize the collision terms in the coupled equations (\[nl01\]).
It is assumed that the mean free paths are common to each sub-band, though this need not be so more generally. Let $\lambda_{\rm in}$ be the inelastic MFP and $\lambda_{\rm el}$ be the elastic MFP. (Recall that, by hypothesis, the operational channel length is given by $L = \lambda_{\rm in}$.) Any equilibrium distribution has its associated characteristic velocity
$${\overline v}(\mu) \equiv - {v_{\rm th}\over {\overline f}_0(\mu)}
\int^{\infty}_0 k dk {\partial {\overline f}_k\over \partial k}(\mu)
= v_{\rm th} (1 + e^{-\mu}) \int^{\infty}_0 dk {\overline f}_k(\mu).$$
We have scaled out the thermal velocity $v_{\rm th} = \hbar k_{\rm th}/m^*$ where $m^*$ is the electron effective mass. In the low-density classical limit, this becomes essentially $v_{\rm th}$ while in the high-density degenerate limit it is the Fermi velocity $v_{\rm th}\sqrt{\mu}$. The quantity ${\overline v}$ thus sets the typical velocity scale. Accordingly we define the rates in dimensionless units from the respective characteristic velocities:
$$\begin{aligned}
R_{{\rm in}j}
\equiv&&
{L\over v_{\rm th}}{{\overline v}(\mu_j)\over \lambda_{\rm in}};
~~~~~~~~~
R_{{\rm el}j}
\equiv
{L\over v_{\rm th}}{{\overline v}(\mu_j)\over \lambda_{\rm el}}
\label{nl04}\end{aligned}$$
The transition rates represent a different physical mechanism and are treated differently from the intra-band ones, as a single dimensionless parameter
$$R_{0j} \equiv L/\lambda_0
\label{nl05}$$
scaling inversely with a nominal “transition MFP” $\lambda_0$ which, however, is qualitatively distinct from intra-band MFPs. Its value is an experimental unknown. Moreover $\lambda_0$ is likely to depend strongly on device geometry and electrostatics [@dP1]. For this work we set it an order of magnitude larger than the operational length $L$.
The current response summed over both parabolic sub-bands is given by
$$\begin{aligned}
I(\mu, V_{\rm sd})
&\equiv&
qk_{\rm th}v_{\rm th}
\int k{dk\over \pi} \sum_{j=1,2} f_{kj}(\mu_j)
\cr
%\cr
&=&
V_{\rm sd} {q^2\over \pi\hbar}
\int kdk \sum_{j=1,2}
%
{\left(
{\hbar k_{\rm th}v_{\rm th}\over q{\cal E} L}
f^o_{kj}(\mu_j)
%
\right)}
\label{nl05.1}\end{aligned}$$
where the source-drain voltage is $V_{\rm sd} = {\cal E}L$, and we note that the even distributions do not contribute. From equation (\[nl05.1\]) all the transport properties are derived.
Results
=======
We come now to the consequences for the quantized conductance. In figure 3 below, for a total channel current $I(\mu, V_{\rm sd})$ at a series of fixed applied $V_{\rm sd}$, we show the computed chord conductance $G = I/V_{\rm sd}$ for a device conforming to the specifications of reference [@dP1]. The dynamical scattering parameters are those of the preceding section, namely: operational length $L \!=\! \lambda_{\rm in} \!=\! 2\mu$m; $\lambda_{\rm el} \!=\! 0.769L \!=\! 1.538\mu$m; $\lambda_0 \!=\! 10L \!=\! 20\mu$m.
Figure 4 of reference [@dP2] gives some evidence of thermal broadening of conductance at the sub-band thresholds presumably from localized Joule heating. We compute our curves at the nominal temperature 4K. The energy-gap value $E_g = 15$meV and the effective mass for GaAs are used. Finally, on the horizontal axis of our figure 3 we have mapped the global chemical potential $\mu$ to values of a corresponding gate-control voltage, using the parameters provided by reference [@dP1].
Our figure 3 should be compared directly with figure 1 as taken from reference [@dP2], figure 3(a). Both in the real data of figure 1 and in our calculation, the action of a substantial source-drain voltage driving the current through the channel leads to a series of elevated conductance plateaux which (a) are inherently non-linear in origin,
\(b) are extremely flat and robust,
\(c) anomalously exceed the Landauer upper bound on $G$ and thus
\(d) violate the unitary limit of linear-response transmission theory.
The striking confluence of behaviours between experiment and theory speaks for itself.
-6cm
{width="11truecm"}
0.25cm [[**FIG. 3**]{} [Anomalous enhancement of conductance $G$ calculated for a ballistic device equivalent to that of figure 1 (figure 3(a) of reference [@dP2]) at a nominal temperature 4K. Axis scales are as for that figure; $G$ is plotted (units of $G_0 = q^2/\pi\hbar$) versus gate voltage $V_g$ sweeping the channel density across the energy gap from the bottom of the ground-state sub-band to the threshold of the first excited-state sub-band. Bottom curve: in weak-field response the quantized conductance matches that for standard linear response. Higher curves: as the driving voltage $V_{\rm sd}$ increases, the conductance acquires a non-linear enhancement. The action of inter-band transitions dynamically redistributes carrier density between sub-bands. This is responsible for the strong enhancement of the step in $G$, beyond the unitarity limit posited by quantum-transmission theories of conductance. ]{}]{} 0.25cm
In figure 3(a) of reference [@dP2] (and figure 1 reproducing it in this paper) the plots show ${\partial I/\partial V_{\rm sd}}$: the rate of change of current with driving field, plotted as the density increases. It is easily seen that when that slope is essentially flat over a broad range of gate voltage (thus density) as in figure 1, the simple conductance $I/V_{\rm sd}$, as in figure 3, must track it closely and vice versa.
Before examining further characteristics of our theoretical conductance we discuss differences between the present implementation and the experiment. Our calculation here exhibits greater sensitivity with respect to $V_{\rm sd}$ than the experiment so that, while in figure 3 the step increase of $G$ resulting from $V_{\rm sd} = 1$mV coincides with that in figure 1, its height at 3mV is 1.43$G_0$ while its counterpart in figure 1 is 1.16$G_0$. This overestimate might be accounted for in part if the inelastic MFP $\lambda_{\rm in}$ suffered shortening via optical-phonon emission at higher driving voltages (the optical-phonon energy in GaAs is 35meV, not hugely larger than $E_g$ at 15meV). Use of energy-dependent mean free paths within equations (\[nl01\]), rather than fixed ones, is an obvious aspect for exploration. Increased local Joule heating with increased driving field may also suppress the plateaux, as figure 4 suggests.
In figure 4 we show the properties of the enhanced conductance taken at a typical mid-gap density where $G$ is steady. Device specifications are the same as for figure 3, with an additional choice of temperature, 8K as well as 4K. Our calculated $G$ exhibits an onset at finite field and asymptotic saturation at high fields. The threshold voltage value at onset depends on temperature; at low temperature the threshold value of $V_{\rm sd}$ tends to zero and at high temperatures it rises in rough proportion to $T$.
-6cm
{width="11truecm"}
0.25cm [[**FIG.4**]{} [Threshold and saturation behaviour of calculated conductance versus source-drain voltage, at temperatures $T = 4$K and 8K. Lower dotted line is the first Landauer level in this model, upper line is the second level. The threshold driving field for onset of the enhancement scales approximately with $T$. The saturation asymptote at high fields is independent of temperature.]{}]{} 0.25cm
The phenomenon above may partly explain why measurements prior to de Picciotto [*et al*]{} [@dP1; @dP2] have not recorded the anomalous enhancement. Predominantly, experiments in quantized conductance have been carried out either at weak fields below threshold, or at higher temperatures, or on noisy devices, or in any combination of the above. Any enhancement of $G$ under such conditions would tend to be washed out.
Saturation of $G$ in figure 4 sets in at driving fields considerably higher than those employed in reference [@dP1] and in our figure 3. The upper bound of $G$ is close to 1.6$G_0$, well short of the second occupied level at 1.77$G_0$ in the weak-field limit. This suggests that there is a field- and temperature-independent maximum transfer of carriers to the upper band, beyond which the feedback of transitions returning carriers to the lower band precludes any increase. Saturation behaviour may provide a further experimental window on the dynamics of the inter-band transition.
Summary and Implications
========================
De Picciotto [*et al*]{} [@dP1; @dP2] in the first instance addressed their experiments to the topic (still unresolved) of the “0.7 anomaly”. Yet the same data harbours a message that is perhaps more seminal to the understanding of ballistic transport at low temperatures; namely, the quite surprising violation of the unitary limit for quantized conductance.
That violation is illusory; the apparent paradox vanishes when a more appropriate kinetic-theoretical argument is brought to bear, going beyond the limits of single-particle transmission theory. Carrier transitions between well separated sub-bands are generation-recombination processes, viewed microscopically. This means that their quantitative description must address the direct creation and destruction of actual occupancies in such discretely separate energy bands. Unitarity still applies but, playing out as it must on the much larger stage of multi-particle dynamics, it cannot be accommodated by purely single-particle prescriptions.
Put succinctly, inelasticity and thermodynamic irreversibility rule the physics. It is crucial to build these into the theory [*explicitly*]{}. That is not feasible within the restrictive confines of reversible single-particle Hamiltonian dynamics.
De Picciotto and co-authors have presaged a role for inter-band transitions in the dynamics of their structures [@dP1]. Our motive here has been to advance a theory of such transitions in terms of textbook quantum kinetics.
Certainly the boundary conditions for this problem do require special care in interpretation to be given to the ballistic nature of mesoscopic one-dimensional conductors. Nevertheless it is unavoidable to confront a transport problem where both elastic and inelastic scattering processes act with equal physical status, as in all normal metallic transport.
Our calculation strongly reinforces the conjecture [@dP1] that inter-band transitions in a 1D ballistic device do indeed produce anomalous enhancement of the quantized conductance, within the density regime between a sub-band threshold and its next-higher neighbour’s. This enhancement can indeed readily exceed the presumed unitarity limit set by $G_0$, mandated as the absolute stepwise upper bound for 1D conductance.
The straightforward reason for that phenomenon rests with the physics of creation-annihilation across a band gap and is not beholden to single-particle conservation band-by-band, as it were in isolation. Only global conservation, subsuming the bands within one interacting system, applies. The central mechanism is non-linear feedback between the coupled sub-band populations.
Finally, beyond explaining theoretically the quizzical enhancement of ballistic conductance reported in references [@dP1] and [@dP2], our results on temperature behaviour from figure 4 offer a basis to predict thermal characteristics for transition-induced changes in conductance in clean quantum wires. Furthermore, there is a case for probing similar effects in sufficiently clean carbon nanotubes, whose energy scale and robustness at high fields far outstrip any device based on GaAs heterojunction technology. All of this would call for novel experiments.
References {#references .unnumbered}
==========
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Ferry D K and Goodnick S M 2009 [*Transport in Nanostructures*]{} 2nd ed (Cambridge: Cambridge University Press)
Davies J 1998 [*Physics of Low-Dimensional Semiconductors: An Introduction*]{} (Cambridge: Cambridge University Press)
Nazarov Y V and Blanter Y M 2009 [*Quantum Transport: Introduction to Nanoscience*]{} (Cambridge: Cambridge University Press)
Landauer R D 1996 J. Math. Phys [**37**]{}
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Neder I, Heiblum M, Levinson Y, Mahalu D and Umansky V 2006 Phys. Rev. Lett. [**96**]{} 016804
Green F, Thakur J and Das M P 2004 Phys. Rev. Lett. [**92**]{} 156804
de Picciotto R, Pfeiffer L N, Baldwin K W and West K W 2004 Phys. Rev. Lett. [**92**]{}, 036805
de Picciotto R, Baldwin K W, Pfeiffer L N and West K W 2008 J. Phys.: Condens. Matter [**20**]{} 164204
Magnus W and Schoenmaker W 2002 [*Quantum Transport in Submicron Devices*]{} (Berlin: Springer)
Das M P and Green F 2012 J. Phys.: Condens. Matter [**24**]{} 183201
Das M P and Green F 2003 J. Phys.: Condens. Matter [**15**]{} L687; 2009 J. Phys.: Condens. Matter [**21**]{} 101001
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Das M P and Green F 2017 Adv. Nat. Sci.: Nanosci. Nanotechnol. [**8**]{} 023001
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Mermin N D 1970 Phys. Rev. B [**1**]{} 2363
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Uehling E A and Uhlenbeck G E 1933 Phys. Rev. [**43**]{} 552
|
1.0cm
**Nonexistence of time-reversibility in**
**statistical physics**
C. Y. Chen
Department of Physics, Beijing University of Aeronautics
and Astronautics, Beijing, 100083, P.R.China
[[**Abstract:**]{} Contrary to the customary thought prevailing for long, the time reversibility associated with beam-to-beam collisions does not really exist. Related facts and consequences are presented. The discussion, though involving simple mathematics and physics only, is well-related to the foundation of statistical theory. ]{}
Introduction
============
More than one hundred years ago, the debate concerning time reversibility arose in a confusing way: Boltzmann derived his kinetic equation from the time reversibility of mechanics while the equation itself was of time irreversibility[@reif]. Even today, though a long time has passed and countless papers in the literature have revealed a variety of aspects of the issue, paradoxical things still bother some of us[@chen02].
Here, it will be shown that the real problem of Boltzmann’s theory is related not to the time irreversibility assumed by it, but to the time reversibility assumed by it. To make the topic more intriguing and more profound, the investigation will manifest that any attempt to formulate the ‘true distribution function’ will fail in the ultimate sense.
Particle-to-particle and beam-to-beam collisions
================================================
Before entering the detailed discussion, it is essential to establish distinction between particle-to-particle collisions and beam-to-beam collisions.
A particle-to-particle collision involves two individual particles. The time reversal symmetry of it has been fully elucidated in classical mechanics and it says that if the collision expressed by $({\bf v}_1,{\bf v}_2) \rightarrow ({\bf
v}'_1, {\bf v}'_2)$ is physically possible, where ${\bf v}_1$, ${\bf v}_2$ are respectively the velocities of the two particles before the collision and ${\bf v}'_1$, ${\bf v}'_2$ after the collision, then the inverse collision expressed by $(-{\bf v}'_1,
-{\bf v}'_2) \rightarrow (-{\bf v}_1,-{\bf v}_2)$ is also physically possible. This kind of time reversibility is not truly relevant to the subject herein and we shall not discuss it too much in this paper.
Beam-to-beam collisions, involving particle densities or distribution functions, are of great significance to statistical mechanics. For instance, Boltzmann’s theory treats $f({\bf
v})d{\bf v}$ as a beam and derives its formulation on the premise that certain types of time-reversibility are there.
However, we happen to realize that no time-reversibility of any form can be defined in the context of Boltzmann’s theory. This conclusion is surprising, seems very imprudent and directly contradicts what has been embedded in our mind. In view of such strong resistance, it is felt that a very clear and very detailed discussion should be given. In this section the subject will be studied intuitively and in the next two sections mathematical investigations will be presented.
(100,76) (24,23)(-1,2)[2]{}[(2,1)[22]{}]{} (24,52)(-1,-2)[2]{}[(2,-1)[22]{}]{} (52,41)[(1,1)[18]{}]{} (53,40)[(3,2)[20]{}]{} (51,42)[(2,3)[14]{}]{} (52,34)[(1,-1)[18]{}]{} (53,35)[(3,-2)[20]{}]{} (51,33)[(2,-3)[14]{}]{} (50,2)[(0,8)\[c\][**(a)**]{}]{}
(145,39.6)(1,2)[2]{}[(-2,1)[22]{}]{} (145,36.4)(1,-2)[2]{}[(-2,-1)[22]{}]{} (171,59.5)[(-1,-1)[18]{}]{} (174,53.5)[(-3,-2)[20]{}]{} (166,63.5)[(-2,-3)[14]{}]{} (171,16)[(-1,1)[18]{}]{} (174,22)[(-3,2)[20]{}]{} (166,12)[(-2,3)[14]{}]{} (150,2)[(0,8)\[c\][**(b)**]{}]{}
0.1cm
[Figure 1: A candidate for time reversibility of beam-to-beam collision: (a) the original collisions; and (b) the inverse collisions. ]{}
0.2cm
Take a look at Fig. 1. Fig. 1a shows a process in which two beams with two definite velocities collide and the produced particles diverge in space. Fig. 1b illustrates the inverse process, in which converging beams collide and the produced particles form two definite beams. In no need of detailed discussion, we surely know that the first process makes sense in statistical mechanics while the second one does not.
No time-reversibility in terms of cross sections
================================================
In the textbook treatment, the time reversibility concerning beam-to-beam collisions is expressed in terms of cross sections[@reif]: $$\label{equality} \sigma({\bf v}_1,{\bf v}_2
\rightarrow {\bf v}_1^\prime, {\bf v}_2^\prime) = \sigma ({\bf
v}_1^\prime, {\bf v}_2^\prime \rightarrow {\bf v}_1, {\bf v}_2)
,$$ where $\sigma({\bf v}_1,{\bf v}_2 \rightarrow {\bf
v}_1^\prime, {\bf v}_2^\prime )$ is defined in such a way that $$\label{sigma1}
N= \sigma({\bf v}_1,{\bf v}_2 \rightarrow {\bf v}_1^\prime, {\bf
v}_2^\prime ) d{\bf v}_1^\prime d{\bf v}_2^\prime$$ represents the number of type-1 particles emerging, after collisions, between ${\bf v}_1^\prime$ and ${\bf v}_1^\prime+d
{\bf v}_1^\prime$ per unit incident flux and unit time, while the type-2 particle emerges between ${\bf v}_2^\prime$ and ${\bf
v}_2^\prime+d {\bf v}_2^\prime$.
(200,143)
(-120,85)[(3,-1)[84.5]{}]{} (-120,85)[(1,1)[27.5]{}]{} (-120,85)[(1,0)[56.5]{}]{} (-91.5,113.5)(1.5,-1.5)[36]{} (-58.2,85.3)[(1,-1)[28.2]{}]{} (-58.2,85.3) (-106,111)[(35,8)\[l\][${\bf v}_1$]{}]{} (-58,53)[(35,8)\[l\][${\bf v}_2$]{}]{} (-90,77)[(35,8)\[c\][${\bf c}$]{}]{} (-55,68)[(35,8)\[c\][${\bf u}$]{}]{}
(-80,28)[(0,8)\[c\][**(a)**]{}]{}
(98,28)[(0,8)\[c\][**(b)**]{}]{} (46,85)[(1,0)[55.5]{}]{} (102.0,85)(-1.0,-2.0)[19]{} (102.0,85)(1.0,2.0)[19]{} (108,85) (108,85)[(-1,-2)[18.0]{}]{} (139,102)[(35,8)\[l\][$({\bf v}_1^\prime)$]{}]{} (45.5,85)[(5,1)[92]{}]{} (136.5,103.4)[(4,1)[1]{}]{}
(46.5,85)[(2,1)[72.5]{}]{} (121,121)[(35,8)\[l\][${\bf v}'_1$]{}]{}
(46,85)[(1,1)[28.5]{}]{} (57,114)[(35,8)\[l\][$({\bf v}_1^\prime)$]{}]{}
(46.5,85)[(1,-1)[36]{}]{}
(66,47)[(35,8)\[l\][${\bf v}_2^\prime$]{}]{} (69,76)[(35,8)\[c\][${\bf c}'$]{}]{} (88,64)[(35,8)\[c\][${\bf u}'$]{}]{}
(133,50)[(35,8)\[l\][$S$]{}]{} (143.00, 85.00) (142.78, 89.18) (142.13, 93.32) (141.04, 97.36) (139.54, 101.27) (137.64, 105.00) (135.36, 108.51) (132.73, 111.77) (129.77, 114.73) (126.51, 117.36) (123.00, 119.64) (119.27, 121.54) (115.36, 123.04) (111.32, 124.13) (107.18, 124.78) (103.00, 125.00) (98.82, 124.78) (94.68, 124.13) (90.64, 123.04) (86.73, 121.54) (83.00, 119.64) (79.49, 117.36) (76.23, 114.73) (73.27, 111.77) (70.64, 108.51) (68.36, 105.00) (66.46, 101.27) (64.96, 97.36) (63.87, 93.32) (63.22, 89.18) (63.00, 85.00) (63.22, 80.82) (63.87, 76.68) (64.96, 72.64) (66.46, 68.73) (68.36, 65.00) (70.64, 61.49) (73.27, 58.23) (76.23, 55.27) (79.49, 52.64) (83.00, 50.36) (86.73, 48.46) (90.64, 46.96) (94.68, 45.87) (98.82, 45.22) (103.00, 45.00) (107.18, 45.22) (111.32, 45.87) (115.36, 46.96) (119.27, 48.46) (123.00, 50.36) (126.51, 52.64) (129.77, 55.27) (132.73, 58.23) (135.36, 61.49) (137.64, 65.00) (139.54, 68.73) (141.04, 72.64) (142.13, 76.68) (142.78, 80.82)
-1.7cm
-0.5cm Unfortunately, the cross section in (\[equality\]) and (\[sigma1\]) is ill-defined. Notice that the energy and momentum conservation laws state that (assuming every particle to have the same mass for simplicity) $$\label{conservation}
{\bf c}={\bf c}' \quad{\rm and}\quad |{\bf u}|=|{\bf u}'|\equiv u,$$ where $2{\bf c}={\bf v}_1+{\bf v}_2$, $2{\bf c}'={\bf v}'_1+ {\bf
v}'_2$, $2{\bf u} = {\bf v}_2-{\bf v}_1$ and $2{\bf u}'={\bf
v}'_2- {\bf v}'_1$. Fig. 2a shows how ${\bf v}_1$ and ${\bf v}_2$ determine ${\bf c}$ and ${\bf u}$, while Fig. 2b shows how $\bf c$ and $u\equiv |{\bf u}|$ form four constraint constants on ${\bf
v}'_1$ and ${\bf v}'_2$. Notably, ${\bf v}'_1$, as well as ${\bf
v}'_2$, falls on the spherical shell $S$ of radius $u$ in the velocity space, which will be called the accessible shell. With these constraints in mind, two problems associated with the definition (\[sigma1\]) will surface by themselves. The first is that, after $d{\bf v}'_1$ is specified, specifying $d{\bf v}'_2$ in the definition is a work overdone (since ${\bf v}'_1$ and ${\bf
v}'_2$ are not independent of each other). The second is that the cross section should be defined in reference to surface elements rather than to volume elements.
To see the second problem aforementioned more vividly, let’s consider $d{\bf v}'_1$ shown in Fig. 3a, which is cube-shaped with equal sides $l$. If we let $\rho$ denote the area density of particles on the accessible shell caused by unit flux of type-1 particles, the number of type-1 particles found in $d{\bf v}'_1$ can be expressed as $N\approx \rho l^2$. Then, the cross section defined by (\[sigma1\]) is equal to, with $d{\bf v}_2^\prime$ omitted, $$\label{v-to-s}
\sigma=\frac N{d{\bf v}_1^\prime}=\frac{\rho l^2} {l^3}=
\frac{\rho}{l},$$ which depends on $l$ and tends to infinity as the cube becomes smaller and smaller. Nevertheless, if $d{\bf v}_1^\prime$ is chosen to be a slim box in Fig. 3b and the box becomes slimmer and slimmer, then $\sigma$ tends to zero; if $d{\bf v}_1^\prime$ is chosen to be a short box in Fig. 3c and the box becomes shorter and shorter, $\sigma$ tends to infinity again. These representative examples inform us that the cross section defined by (\[sigma1\]), and thus the time reversibility expressed by (\[equality\]), does not mean anything.
(300,138)
(85,87)(90,0)[3]{}[(8,8)\[c\][$S$]{}]{} (57,42)[(8,8)\[c\][**(a)**]{}]{} (147,42)[(8,8)\[c\][**(b)**]{}]{} (237,42)[(8,8)\[c\][**(c)**]{}]{}
(46,105)[(30,8)\[c\][$d{\bf v}_1^\prime$]{}]{} (136,118)[(30,8)\[c\][$d{\bf v}_1^\prime$]{}]{} (226,105)[(30,8)\[c\][$d{\bf v}_1^\prime$]{}]{}
(145,75)[(10,40)]{} (220,90)[(40,10)]{} (55,90)[(10,10)]{} (95.00,60.00)(90,0)[3]{} (95.00,60.00)(90,0)[3]{} (94.89,62.82)(90,0)[3]{} (94.55,65.61)(90,0)[3]{} (93.98,68.38)(90,0)[3]{} (93.20,71.08)(90,0)[3]{} (92.20,73.72)(90,0)[3]{} (90.99,76.27)(90,0)[3]{} (89.58,78.71)(90,0)[3]{} (87.98,81.03)(90,0)[3]{} (86.20,83.21)(90,0)[3]{} (84.25,85.24)(90,0)[3]{} (82.14,87.11)(90,0)[3]{} (79.88,88.80)(90,0)[3]{} (77.50,90.31)(90,0)[3]{} (75.00,91.62)(90,0)[3]{} (72.41,92.73)(90,0)[3]{} (69.74,93.62)(90,0)[3]{} (67.00,94.29)(90,0)[3]{} (64.22,94.74)(90,0)[3]{} (61.41,94.97)(90,0)[3]{} (58.59,94.97)(90,0)[3]{} (55.78,94.74)(90,0)[3]{} (53.00,94.29)(90,0)[3]{} (50.26,93.62)(90,0)[3]{} (47.59,92.73)(90,0)[3]{} (45.00,91.62)(90,0)[3]{} (42.50,90.31)(90,0)[3]{} (40.12,88.80)(90,0)[3]{} (37.86,87.11)(90,0)[3]{} (35.75,85.24)(90,0)[3]{} (33.80,83.21)(90,0)[3]{} (32.02,81.03)(90,0)[3]{} (30.42,78.71)(90,0)[3]{} (29.01,76.27)(90,0)[3]{} (27.80,73.72)(90,0)[3]{} (26.80,71.08)(90,0)[3]{} (26.02,68.38)(90,0)[3]{} (25.45,65.61)(90,0)[3]{} (25.11,62.82)(90,0)[3]{}
-2.0cm
-0.7cm As a matter of fact, the above problem can be examined more briefly. For any definite velocities ${\bf v}_1$ and ${\bf v}_2$, the six components of ${\bf v}'_1$ and ${\bf v}'_2$ are under four constraint equations imposed by the energy-momentum conservation laws. This literally means that the free space of ${\bf v}'_1$ and ${\bf v}'_2$ is just two-dimensional and there are no enough degrees of freedom allowing us to define a cross section in reference to six-dimensional volume elements.
No time-reversibility in terms of velocity volumes
==================================================
Another form of time reversibility is simultaneously employed in such textbooks[@reif]: $$\label{timerev2} d{\bf
v}_1 d{\bf v}_2=d{\bf v}'_1 d{\bf v}'_2.$$ It should be noted that there is a conceptual conflict between (\[equality\]) and (\[timerev2\]). In connection with (\[equality\]), when incident particles have two definite velocities the velocities of scattered particles are allowed to distribute over almost the entire velocity space; whereas, in connection with (\[timerev2\]), an infinitesimal velocity range of incident particles strictly corresponds to another infinitesimal velocity range of scattered particles. Nevertheless, for purposes of this paper, we shall leave this conflict alone.
The mathematical proof of (\[timerev2\]) goes as follows. First, $$\label{j1}
d{\bf v}_1 d{\bf v}_2=\|J\|d{\bf c}d{\bf u}\quad {\rm with} \quad
\|J\|=\left\| \frac{\partial ({\bf v}_1,{\bf v}_2)} {\partial
({\bf c},{\bf u})} \right\|.$$ Then, $$\label{j2}
d{\bf v}'_1 d{\bf v}'_2=\|J'\|d{\bf c}'d{\bf u}'\quad {\rm with}
\quad \|J'\|=\left\| \frac{\partial ({\bf v}'_1,{\bf v}'_2)}
{\partial ({\bf c}',{\bf u}')} \right\|.$$ In view of that $$\label{j3}
\|J\|=\|J'\|,\quad {\bf c}\equiv {{\bf c}'} \quad d{\bf u}=d{\bf
u}',$$ we obtain (\[timerev2\]).
Unfortunately again, the formulation given above also involves errors. All equations in (\[j1\]), (\[j2\]) and (\[j3\]) hold except $d{\bf u}=d{\bf u}'$. Referring to Fig. 2, we find that when ${\bf u}$ is a definite vector, ${\bf u}'$ distributes over a spherical shell, pointing in any direction. This means that if $d{\bf u}=u^2dud\Omega_{\bf u}$ is an infinitely thin line-shaped volume element (say, $d\Omega_{\bf u}$ is infinitesimal while $du$ finite), the corresponding volume element $d{\bf u}'$ will be a spherical shell with finite thickness. It is then obvious that the two elements are not equal in volume.
The issue can be analyzed more economically in terms of variable transformation. If we identify $v_{1x},v_{1y},v_{1z}$ and $v_{2x},v_{2y},v_{2z}$ as six variables and identify $v'_{1x},v'_{1y},v'_{1z}$ and $v'_{2x},v'_{2y},v'_{2z}$ as six new variables, then we have $$\label{jacobian} d{\bf
v}_1 d{\bf v}_2=\|\hat J\| d{\bf v}'_1 d{\bf v}'_2.$$ What has been proven by equations (\[j1\]), (\[j2\]) and (\[j3\]) is nothing but $\|\hat J\|=1$. However, in order for (\[jacobian\]) to make sense, there must exist six independent equations connecting those variables. In our case, we have four equations only. That is to say, the ‘variable transformation’ is incomplete and expression (\[jacobian\]) is not truly legitimate.
It is instructive to look at the issues discussed in the last and this sections in a unifying way. If there were no single constraint equation, defining the cross section could make sense; if there were six independent constraint equations, defining the Jacobian would be meaningful. Since there are four and only four constraint equations, neither the cross section nor the Jacobian can be defined.
Formulation of beam-to-beam collisions
======================================
It is now rather clear that the concepts and methodologies of Boltzmann’s theory are in need of reconsideration.
For instance, one of the major steps in deriving Boltzmann’s equation is to identify $f({\bf v}'_1)d{\bf v}'_1$ and $f({\bf
v}'_2)d{\bf v}'_2$ as two definite beams and then to determine how many beam-1 particles will emerge between ${\bf v}_1$ and ${\bf
v}_1+d{\bf v}_1$ due to collisions of the two beams. As has been shown, this context leads to nothing but an absurd result: the number of such emerging particles actually depends on the size and shape of $d{\bf v}_1$, varying drastically from zero to infinity.
In what follows, we shall propose a new context to do the job. Surprisingly, the formulation will reveal some of deep-rooted properties of statistical mechanics.
To involve less details, we adopt the following assumptions: (i) The zeroth-order, collisionless, distribution function of the gas is completely known. (ii) Each particles, though belonging to the same species, is still distinguishable (which is possible in terms of classical mechanics). (iii) No particle collides twice or more. (General treatments can be accomplished along this line[@chen99].)
Referring to Fig. 4, we consider that two typical beams, denoted by $ f^{(0)}_1({\bf v}'_1) d{\bf v}'_1$ and $f^{(0)}_2({\bf v}'_2)
d{\bf v}'_2$, collide with each other, and suppose that a particle detector has been placed somewhere in the region. Let $\Delta S$ be the entry area of the detector and $\Delta N_1$ be the number of the beam-1 particles entering the detector within the velocity range $\Delta v_1 v_1^2 \Delta \Omega_1$ during $d t$. Since any beam of the system can be regarded as the first beam, or the second beam, aforementioned, the total distribution function due to collisions is, at the detector entry, $$\label{deltaN} f^{(1)}_1(t,{\bf r}, {\bf v}_1,\Delta
S,\Delta v_1,\Delta \Omega_1 )\approx \frac {\sum_1\sum_2\Delta
N_1}{(\Delta S v_1 d t)(\Delta v_1 v_1^2 \Delta
\Omega_1)},$$ in which ${\bf r}$ is the representative position of $\Delta S$ and ${\bf v}_1$ is the representative velocity of $\Delta v_1 v_1^2 \Delta \Omega_1$. According to the customary thought, when $\Delta S$, $\Delta v_1$ and $\Delta
\Omega_1$ shrink to zero simultaneously, expression (\[deltaN\]) stands for the ‘true distribution function’ there; for reasons to be clear a bit later, we shall, in the following formulation, assume that $\Delta S$ and $\Delta v_1$ are infinitely small while $\Delta \Omega_1$ is kept fixed and finite (though rather small). The spatial region $-\Delta\Omega_1$, that has been shaded in Fig. 4 and ‘opposite’ to the velocity solid-angle range $\Delta
\Omega_1$ in (\[deltaN\]), will be called the effective cone. It is intuitively obvious that the particles that collide somewhere in the effective cone and move, after the collision, toward the detector entry along their free trajectories will contribute to $\Delta N_1$. (Even if such particles are allowed to collide again, some of them will still arrive at the detector freely, which means the concept of effective cone holds its significance rather generally.)
(200,98) (85,45)(0,4)[2]{}[(3,1)[15]{}]{} (85,70)(0,4)[3]{}[(3,-1)[15]{}]{}
(110,40)(12,0)[2]{}[(1,0)[8]{}]{} (110,40)(20,0)[2]{}[(0,-1)[12]{}]{}
(120,40)(-0.3,1.2)[34]{} (120,40)(0.3,1.2)[34]{}
(119,48)(1.5,0.4)[3]{} (118,52)(1.5,0.4)[4]{} (117,56)(1.5,0.4)[5]{} (116,60)(1.5,0.4)[7]{} (115,64)(1.5,0.4)[9]{} (114,68)(1.5,0.4)[10]{} (113,72)(1.5,0.4)[11]{}
(132,30)[(20,8)\[l\][Detector]{}]{} (110,80)[(20,8)\[c\][$\small -\Delta\Omega_1$]{}]{} (63,42)[(20,8)\[c\][$f^{(0)}_1d{\bf v}'_1 $]{}]{} (63,75)[(20,8)\[c\][$f^{(0)}_2d{\bf v}'_2 $]{}]{}
[ -0.3cm Figure 4: Two beams collide and a particle detector is placed in the region.]{}
Observing the colliding beams in the center-of-mass reference frame, we find that the number of collisions in a volume element $d{\bf r}'$, which is located inside the effective cone $-\Delta
\Omega$, can be represented by, as in Boltzmann’s theory, $$\label{twobeams} [d{\bf r}'f^{(0)}_1({\bf v}'_1)d{\bf v}'_1]
[f^{(0)}_2({\bf v}'_2)d{\bf v}'_2] [2u\sigma_c({\bf u}',{\bf u}) d
\Omega_c d t ],$$ where $\Omega_c$ is the solid angle between ${\bf u}'$ and ${\bf u}$, and $\sigma_c({\bf u}',{\bf u})$ is the cross section in the center-of-mass frame. By integrating (\[twobeams\]) over the effective cone and taking account of all the particles that are registered by the detector, the right side of (\[deltaN\]) becomes $$\label{entry} \int_{-\Delta \Omega_1}d{\bf r}'
\int_{\Delta
v_1\Delta \Omega_1} d\Omega_c \int d{\bf v}'_1\int d{\bf v}'_2
\frac{2u\sigma_c({\bf u}',{\bf u}) f^{(0)}_1({\bf v}'_1)
f^{(0)}_2({\bf v}'_2)} {(|{\bf r}-{\bf r}'|^2 \Delta \Omega_0
v_1)\dot (v_1^2\Delta v_1 \Delta \Omega_1 )},$$ where $\Delta \Omega_0$ is the solid-angle range formed by a representative point in $d{\bf r}'$ (as the apex) and the detector entry area $\Delta S$ (as the base). In view of that $\Delta S$ is truly small and $\Delta\Omega_1$ is fixed and finite by our assumption, we know that $\Delta\Omega_0 \ll
\Delta\Omega_1$ and every particle starting its free journey from the effective cone and entering the detector can be treated as one emerging within $\Delta \Omega_1$. Then, we can rewrite expression (\[entry\]) as, with help of the variable transformation from $({\bf v}'_1,{\bf v}'_2)$ to $({\bf c}',{\bf u}')$, $$\label{entry1} \int_{-\Delta \Omega_1}d{\bf
r}'\int_{\Delta v_1\Delta \Omega_0} d\Omega_c \int d{\bf c}' \int
d{\bf u}'\|\tilde J\| \frac{2u\sigma_c({\bf u}',{\bf u})
f^{(0)}_1({\bf c}'-{\bf u}') f^{(0)}_2({\bf c}'+{\bf u}')} {(|{\bf
r}-{\bf r}'|^2 \Delta \Omega_0 v_1)\dot (v_1^2\Delta v_1 \Delta
\Omega_1 )},$$ in which $\|\tilde J\|\equiv \partial({\bf v}'_1,
{\bf v}'_2) / \partial({\bf c}',{\bf u}' )$ and the subindex $\Delta v_1\Delta \Omega_1$ has been replaced by $\Delta v_1\Delta
\Omega_0$. In view of the energy-momentum conservation laws, we arrive at $$\label{entry2}
\begin{array}{l}\displaystyle \int_{-\Delta \Omega_1}d{\bf
r}' \int d{\bf c}' \int_{4\pi} d\Omega_{{\bf u}'}\int_{\Delta
v_1\Delta \Omega_0}d\Omega_c\int du \vspace{3pt}\\
\displaystyle \hspace{4cm}\cdot \|\tilde J\| \frac{2u\sigma_c({\bf
u}',{\bf u}) f^{(0)}_1({\bf c}'-{\bf u}') f^{(0)}_2({\bf c}'+{\bf
u}')} {(|{\bf r}-{\bf r}'|^2 \Delta \Omega_0 v_1)\dot (v_1^2\Delta
v_1 \Delta \Omega_1 )}, \end{array}$$
(200,117) (155,93)[(35,8)\[l\][${\bf c}$]{}]{} (165,45)[(35,8)\[l\][$\bf u$]{}]{} (220,90)[(-1,-1)[60]{}]{} (220,90)(-1.9,-2.3)[29]{} (70,90)[(1,0)[150]{}]{} (217.5,90.5)[(1,0)[1]{}]{} (70,90)(3.0,-2.3)[34]{} (70,90)(3.0,-1.7)[37]{} (170,14.7)(1.6,2.4)[6]{} (154,42.4)(-1.6,-2.4)[5]{} (72,63)[(35,8)\[l\][$\Delta\Omega_0$]{}]{} (207,65)[(35,8)\[l\][$d\Omega_c$]{}]{} (140,14)[(35,8)\[l\][$\Delta v_1$]{}]{}
(143.22,53.87) (144.80,50.69) (146.52,47.57) (148.36,44.53) (150.32,41.57) (152.41,38.70) (154.62,35.91) (156.94,33.22) (159.38,30.63) (161.91,28.15) (164.56,25.77) (167.29,23.50) (170.13,21.35) (173.04,19.32) (176.04,17.42) (179.12,15.64) (182.27,14.00) (185.49,12.48)
[ -0.54cm Figure 5: The relation between the velocity element $v_1^2\Delta v_1 \Delta\Omega_0 $ and the velocity element $u^2 du
d\Omega_c $.]{}
By examining the situation shown in Fig. 5, which is drawn in the velocity space, the following relation can be found out: $$\int_{\Delta v_1 \Delta \Omega_0}u^2 du d\Omega_c
\cdots \approx v_1^2\Delta v_1 \Delta\Omega_0\cdots
.$$ Therefore, the distribution function due to collisions is equal to $$\label{final}
\begin{array}{l}\displaystyle f^{(1)}_1(t,{\bf
r}, v_1, \Delta \Omega_1)=\frac{1}{v_1 \Delta \Omega_1}
\displaystyle \int_{-\Delta \Omega_1}d{\bf r}' \int d{\bf c}'
\int_{4\pi} d\Omega_{{\bf u}'}\vspace{3pt}\\\hspace{4.5cm}
\displaystyle \cdot\frac{2\|\tilde J\|\sigma_c({\bf u}',{\bf u})
f^{(0)}_1({\bf c}'-{\bf u}') f^{(0)}_2({\bf c}'+{\bf u}')} {u|{\bf
r}-{\bf r}'|^2},
\end{array}$$ in which $\bf u$ is determined by ${\bf u}={\bf c}-{\bf v}_1$ (${\bf v}_1$ is in principle along the direction of ${\bf r}-{\bf
r}'$) and ${\bf u}'$ by $u$ and $\Omega_{{\bf u}'}$. It should be noted that $f^{(1)}_1$ in (\[final\]) differs from that in (\[deltaN\]) in the sense that $\Delta S$ and $\Delta v_1$ cease to be arguments and $v_1$ takes the place of ${\bf v}_1$.
If the zeroth-order distribution functions depend on time and space, the replacement $$f^{(0)}_1({\bf c}'-{\bf u}')f^{(0)}_2({\bf c}'+
{\bf u}')\rightarrow
f^{(0)}_1(t',{\bf r}',{\bf c}'-{\bf u}')f^{(0)}_2(t',{\bf r}',{\bf
c}'+{\bf u}')$$ needs to be taken, where $t'=t-|{\bf
r}-{\bf r}'|/v_1$ stands for the time delay.
Since $\Delta\Omega_1$ has been set finite, expression (\[final\]) is nothing but the distribution function averaged over $\Delta\Omega_1$, which is not good enough according to the standard theory. At first glance, if we let $\Delta\Omega_1$ become smaller and smaller, expression (\[final\]) will finally represent the true distribution function there. However, the discussion after (\[entry\]) has shown that if $\Delta \Omega_1$ shrinks to zero the definition of the effective cone will collapse and thus the entire formalism will no longer be valid.
A careful inspection tells us that, if we assumed $\Delta S$ to be finite and $\Delta \Omega_1$ to be infinitesimal at the starting point, a different average distribution function, averaged over $\Delta S$, would be obtained. The fact that no distribution function can be determined if $\Delta \Omega_1$ and $\Delta S$ simultaneously take on their infinitesimal values suggests that even in classical statistical mechanics there also exists an uncertainty principle. The connection between this uncertainty principle and the uncertainty principle in quantum mechanics remains to be seen.
In view of that the integration in (\[final\]) is carried out over an effective region defined by free trajectories of particles, this methodology has been named as a path-integral approach[@chen99]. As (\[final\]) has partially shown, anything taking place in the effective region can make direct impact along ‘free’ paths and any macroscopic structures, continuous or not, will help shape microscopic structures elsewhere along ‘free’ paths. Concepts like the aforementioned, though appearing to be foreign from the viewpoint of differential approach, are in harmony with what takes place in realistic gases.
Summary
=======
It has been shown that, when we concern ourselves with beam-to-beam collisions, part of the system’s information has, knowingly or not, been disregarded, and the information loss is characterized by the fact that we have four and only four constraint equations (rather than six). Since statistical mechanics deals with beam-to-beam collisions, the time-reversibility related to each individual collision becomes irrelevant from the very beginning and will not reemerge at any later stage.
By proposing a new context in which what can be really measured in an experiment is of central interest, beam-to-beam collisions have been reformulated. The new formulation reveals that only distribution functions averaged over certain finite ranges make good physical sense.
Apparently, this paper raises many difficult questions related to the very foundation of statistical physics. Reference papers can be found in the regular and e-print literature[@chen02; @chen99; @chen05].
[**Acknowledgments:**]{} The author is very grateful to professor Oliver Penrose, who reminded me of possibility of defining the cross section of delta-function type. Such definition and its possible consequences are discussed in Appendix A. This paper is supported by School of Science, BUAA, PRC.
Appendix A. An alternative cross section {#appendix-a.-an-alternative-cross-section .unnumbered}
========================================
As revealed in Sect. 3, the original cross section $\sigma({\bf
v}_1,{\bf v}_2 \to {\bf v}'_1,{\bf v}'_2) $ defined by (\[sigma1\]) cannot be interpreted as a function of the usual kind due to the existence of the energy-momentum conservation laws. However, one may still wish to define cross sections in which the time reversibility related to a collision of two particles plays a certain role. Interestingly, this can be done and doing so will help us to find out what kind of problems Boltzmann’s equation really has.
If we adopt the definition of (\[sigma1\]) and, at the same time, take the energy-momentum conservation laws into account, we are led to a cross section of delta-function type $$\label{app1}
\begin{array}{l}\sigma({\bf v}_1,{\bf v}_2 \to {\bf v}_1',{\bf
v}_2') =\displaystyle\eta({\bf v}_1,{\bf v}_2 \to {\bf v}_1',{\bf
v}_2')\cdot \vspace{3pt}\\\hspace{20pt} \delta\left(
\sqrt{(v'_1)^2 +(v'_2)^2}- \sqrt{(v_1)^2 + (v_2)^2}\right)\cdot
\delta^3 \left[({\bf v}'_1 + {\bf v}'_2)-({\bf v}_1 + {\bf v}_2)
\right]. \end{array}$$ To work out the physical meaning of $\eta$, we integrate the above expression over a finite but small volume element $\Delta {\bf v}_1'\Delta {\bf v}_2'$ $$\label{app2}\begin{array}{l}\displaystyle
\int_{\Delta {\bf v}_1'\Delta {\bf v}_2'}\eta({\bf v}_1,{\bf v}_2
\to {\bf v}_1',{\bf v}_2')\cdot\vspace{3pt}\\ \delta\left(
\sqrt{(v'_1)^2 +(v'_2)^2}- \sqrt{(v_1)^2 + (v_2)^2}\right)\cdot
\delta^3 \left[({\bf v}'_1 + {\bf v}'_2)-({\bf v}_1 + {\bf v}_2)
\right] d{\bf v}_1'd{\bf v}_2'.
\end{array}$$ As shown in Sect. 3, these delta-functions define an accessible shell, denoted by $S$ there, and the integration becomes $$\label{eta-s}
\eta({\bf v}_1,{\bf v}_2 \to {\bf v}_1',{\bf v}_2') \Delta S,$$ where $\Delta S$ is enclosed by $\Delta {\bf v}_1'\Delta {\bf
v}_2'$ (actually by one of $\Delta {\bf v}_1'$ and $\Delta {\bf
v}_2'$ since the two are not independent of each other). By comparing this with the cross section defined in the center-of-mass system $\sigma_c({\bf u},{\bf u}') d\Omega $, it is clear that $$\label{eta}
\eta({\bf v}_1,{\bf v}_2 \to {\bf v}_1',{\bf v}_2') =\sigma_c({\bf
u},{\bf u}')/u^2 .$$ The above formalism has illustrated that the introduced delta-functions make good sense as long as they are integrated over an adequate volume (finite or infinitely large). Similarly, $$\label{app11}
\begin{array}{l}\sigma({\bf v}_1',{\bf v}_2' \to {\bf v}_1,{\bf
v}_2) =\displaystyle \eta({\bf v}_1',{\bf v}_2' \to {\bf v}_1,{\bf
v}_2)\cdot \vspace{3pt}\\ \hspace{20pt}\delta\left( \sqrt{(v_1)^2
+(v_2)^2}- \sqrt{(v_1')^2 + (v_2')^2}\right)\cdot \delta^3
\left[({\bf v}_1 + {\bf v}_2)-({\bf v}_1' +{\bf v}_2')
\right].\end{array}$$ With help of (\[eta\]), we see that $$\eta({\bf v}_1,{\bf v}_2 \to {\bf v}'_1,{\bf v}'_2) = \eta({\bf
v}'_1,{\bf v}'_2 \to {\bf v}_1,{\bf v}_2).$$ This reflects the fact that all collisions, including the original collision and the inverse collision, are of head-on type in the center-of-mass system.
It is now in order to find out whether or not the newly defined cross section is relevant to Boltzmann’s equation. We first examine the particles leaving $d{\bf r}d{\bf v}_1$ during $dt$ because of collisions. Following the textbook treatment, the number of collisions is represented by $$[2uf({\bf v}_1)d{\bf v}_1][f({\bf v}_2)d{\bf
r}d{\bf v}_2] \sigma({\bf v}_1,{\bf v}_2 \to {\bf v}'_1,{\bf
v}'_2)dt.$$ Integrating it over ${\bf v}_1'$, ${\bf v}_2'$ and ${\bf v}_2$ yields, by virtue of (\[app1\]), (\[eta-s\]) and (\[eta\]), $$dtd{\bf r}d{\bf v}_1\int 2uf({\bf
v}_1)f({\bf v}_2)\sigma_c({\bf u},{\bf u}')d{\bf v}_2 d\Omega.$$ Dividing it by $dtd{\bf r}d{\bf v}_1$, we obtain the collision number per unit time and unit phase volume $$\int 2uf({\bf v}_1)f({\bf v}_2)\sigma_c({\bf u},{\bf u}')d{\bf
v}_2 d\Omega.$$ If we adopt the assumption that the above number is identical to the number of particles leaving the unit phase volume per unit time because of collisions (though a different conclusion is offered in Ref. 2), we find that the above derivation is entirely consistent with that in the standard approach.
Then, we examine the particles entering $d{\bf r}\Delta {\bf v}_1$ during $dt$ because of collisions ($\Delta{\bf v}_1$ has been set finite for a reason that will be clarified). To make our discussion a bit simpler, it is assumed that there are only two incident beams $$\label{twobeams}
f({\bf v}_1') \Delta{\bf v}_1' \quad {\rm and}\quad f({\bf v}_2')
\Delta{\bf v}_2'.$$ Again, following the standard approach, we know that the collision number caused by the two beams within $d{\bf r}$ during $dt$ is $$[2uf({\bf v}_1')\Delta{\bf v}_1']\cdot[f({\bf v}_2')d{\bf
r}\Delta{\bf v}_2']\cdot \sigma({\bf v}_1',{\bf v}_2' \to {\bf
v}_1,{\bf v}_2)dt.$$ At this point, a sharp question arises. Can we identify these particles as those entering $d{\bf r}\Delta {\bf v}_1$ during $dt$? The answer is apparently a negative one. As one thing, only a small fraction of the scattering particles will enter $\Delta{\bf v}_1$, and the following integration needs to be done: $$\label{enter1}
\int_{\Delta {\bf v}_1\Delta {\bf v}_2} 2uf({\bf v}_1')\Delta{\bf
v}_1'\cdot f({\bf v}_2')d{\bf r}\Delta{\bf v}_2'\cdot \sigma({\bf
v}_1',{\bf v}_2' \to {\bf v}_1,{\bf v}_2) dtd{\bf v}_1d{\bf v}_2,$$ in which $\Delta {\bf v}_1$ has to be finite, since the integrand contains delta-functions, while $\Delta {\bf v}_2$ can be infinitely large. If we are interested in knowing the number of the entering particles per unit phase volume and unit time, we are supposed to evaluate $$\label{limit}
\lim\limits_{dtd{\bf r}\Delta{\bf v}_1\to 0} \frac{\Delta
N}{dtd{\bf r}\Delta{\bf v}_1},$$ where $\Delta N$ is nothing but expression (\[enter1\]). Unfortunately, there are problems. As one thing, it has just been pointed out that $\Delta{\bf v}_1$ has to be finite. As another, expression (\[v-to-s\]) and the accompanying discussion have shown that the limit expressed by (\[limit\]) does not exist.
The above discussion, verifiable with numerical computation, suggests that (i) The methodology of determining particles entering a phase volume element must be very different from that of determining particles leaving a phase volume element, namely there is no symmetry. (ii) Taking limits $dt\to 0$, $d{\bf r}\to
0$ and $d{\bf v}_1\to 0$, simultaneously or not, can create unexpected problems. (iii) In order to formulate the collisional dynamics, new concepts and new methodologies are in need.
[99]{}
F. Reif, [*Fundamentals of Statistical and Thermal Physics*]{}, (McGraw-Hill Book Company, 1965, 1987, 1988 in English and German); L. E. Reichl [*A Modern Course in Statistical Physics*]{}, 2nd ed., (John Wiley and Sons, New York, 1998). C. Y. Chen, Il Nuovo Cimento B [**V117B**]{}, 177 (2002), which reveals some problems of Boltzmann’s equation; C. Y. Chen, Journal of Physics A, 6589, (2002), which discusses something that can actually challenge Liouville’s theorem in quantum statistical physics. C. Y. Chen, [*Perturbation Methods and Statistical Theories*]{}, in English, (International Academic Publishers, Beijing, 1999). C. Y. Chen, cond-mat/0412396; physics/0312043, 0311120, 0305006, 0010015, 0006033, 0006009, 9908062; quant-ph/0009023, 0009015, 9911064, 9907058.
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---
abstract: 'Strongly multiplicative linear secret sharing schemes (LSSS) have been a powerful tool for constructing secure multi-party computation protocols. However, it remains open [*whether or not there exist efficient constructions of strongly multiplicative LSSS from general LSSS*]{}. In this paper, we propose the new concept of a [*$3$-multiplicative LSSS*]{}, and establish its relationship with strongly multiplicative LSSS. More precisely, we show that any 3-multiplicative LSSS is a strongly multiplicative LSSS, but the converse is not true; and that any strongly multiplicative LSSS can be efficiently converted into a 3-multiplicative LSSS. Furthermore, we apply 3-multiplicative LSSS to the computation of unbounded fan-in multiplication, which reduces its round complexity to four (from five of the previous protocol based on strongly multiplicative LSSS). We also give two constructions of 3-multiplicative LSSS from Reed-Muller codes and algebraic geometric codes. We believe that the construction and verification of 3-multiplicative LSSS are easier than those of strongly multiplicative LSSS. This presents a step forward in settling the open problem of efficient constructions of strongly multiplicative LSSS from general LSSS.'
author:
- Zhifang Zhang
- Mulan Liu
- Yeow Meng Chee
- San Ling
- Huaxiong Wang
title: 'Strongly Multiplicative and 3-Multiplicative Linear Secret Sharing Schemes '
---
[monotone span program, secure multi-party computation, strongly multiplicative linear secret sharing scheme]{}
Introduction
============
Secure multi-party computation (MPC) [@Yao:Protocols; @Goldreich:How] is a cryptographic primitive that enables $n$ players to jointly compute an agreed function of their private inputs in a secure way, guaranteeing the correctness of the outputs as well as the privacy of the players’ inputs, even when some players are malicious. It has become a fundamental tool in cryptography and distributed computation. Linear secret sharing schemes (LSSS) play an important role in building MPC protocols. Cramer [*et al.*]{} [@Cramer:General] developed a generic method of constructing MPC protocols from LSSS. Assuming that the function to be computed is represented as an arithmetic circuit over a finite field, their protocol ensures that each player share his private input through an LSSS, and then evaluates the circuit gate by gate. The main idea of their protocol is to keep the intermediate results secretly shared among the players with the underlying LSSS. Due to the nature of linearity, secure additions (and linear operations) can be easily achieved. For instance, if player $P_i$ holds the share $x_{1i}$ for input $x_1$ and $x_{2i}$ for input $x_2$, he can locally compute $x_{1i}+x_{2i}$ which is actually $P_i$’s share for $x_1+x_2$. Unfortunately, the above homomorphic property does not hold for multiplication. In order to securely compute multiplications, Cramer [*et al.*]{} [@Cramer:General] introduced the concept of [*multiplicative*]{} LSSS, where the product $x_1x_2$ can be computed as a linear combination of the local products of shares, that is, $x_1x_2=\sum_{i=1}^na_ix_{1i}x_{2i}$ for some constants $a_i, 1\leq i\leq n$. Since $x_{1i}x_{2i}$ can be locally computed by $P_i$, the product can then be securely computed through a linear combination. Furthermore, in order to resist against an active adversary, they defined [*strongly*]{} multiplicative LSSS, where $x_1x_2$ can be computed as a linear combination of the local products of shares by all players excluding any corrupted subset. Therefore, multiplicativity becomes an important property in constructing secure MPC protocols. For example, using strongly multiplicative LSSS, we can construct an error-free MPC protocol secure against an active adversary in the information-theoretic model [@Cramer:General]. Cramer [*et al.*]{} [@Cramer:OnCodes] also gave an efficient reconstruction algorithm for strongly multiplicative LSSS that recovers the secret even when the shares submitted by the corrupted players contain errors. This implicit “built-in” verifiability makes strongly multiplicative LSSS an attractive building block for MPC protocols.
Due to their important role as the building blocks in MPC protocols, efficient constructions of multiplicative LSSS and strongly multiplicative LSSS have been studied by several authors in recent years. Cramer [*et al.*]{} [@Cramer:General] developed a generic method of constructing a multiplicative LSSS from any given LSSS with a double expansion of the shares. Nikov [*et al.*]{} [@Nikov:multiplicativeLSSS] studied how to securely compute multiplications in a dual LSSS, without blowing up the shares. For some specific access structures there exist very efficient multiplicative LSSS. Shamir’s threshold secret sharing scheme is a well-known example of an ideal (strongly) multiplicative LSSS. Besides, self-dual codes give rise to ideal multiplicative LSSS [@Cramer:OnCodes], and Liu [*et al.*]{} [@Liu:IEEE] provided a further class of ideal multiplicative LSSS for graph access structures. We note that for strongly multiplicative LSSS, the known general construction is of exponential complexity. Käsper [*et al.*]{} [@KNN:SMHTSS] gave some efficient constructions for specific access structures (hierarchical threshold structures). It remains open whether there exists an efficient transformation from a general LSSS to a strongly multiplicative one.
On the other hand, although in a multiplicative LSSS, multiplication can be converted into a linear combination of inputs from the players, each player has to [*reshare*]{} the product of his shares, that is, for $1\leq i\leq n$, $P_i$ needs to reshare the product $x_{1i}x_{2i}$ to securely compute the linear combination $\sum_{i=1}^na_ix_{1i}x_{2i}$. This resharing process involves costly interactions among the players. For example, if the players are to securely compute multiple multiplications, $\prod_{i=1}^lx_i$, the simple sequential multiplication requires interaction of round complexity proportional to $l$. Using the technique developed by Bar-Ilan and Beaver [@BB:ConstantRd], Cramer [*et al.*]{} [@CKP:LinearAlgbra] recently showed that the round complexity can be significantly reduced to a constant of five for unbounded fan-in multiplications. However, the method does not seem efficient when $l$ is small. For example, considering $x_1x_2$ and $x_1x_2x_3$, extra rounds of interactions seem unavoidable for computing $x_1x_2x_3$ if we apply the method of Cramer [*et al.*]{} [@CKP:LinearAlgbra].
Our Contribution
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In this paper, we propose the concept of 3-multiplicative LSSS. Roughly speaking, a 3-multiplicative LSSS is a generalization of multiplicative LSSS, where the product $x_1x_2x_3$ is a linear combination of the local products of shares. As one would expect, a 3-multiplicative LSSS achieves better round complexity for the computation of $\prod_{i=1}^lx_i$ compared to a multiplicative LSSS, if $l \geq 3$. Indeed, it is easy to see that computing the product $\prod_{i=1}^9x_i$ requires two rounds of interaction for a 3-multiplicative LSSS but four rounds for a multiplicative LSSS. We also extend the concept of a 3-multiplicative LSSS to the more general $\lambda$-multiplicative LSSS, for all integers $\lambda \geq 3$, and show that $\lambda$-multiplicative LSSS reduce the round complexity by a factor of $\frac{1}{\log\lambda}$ from multiplicative LSSS. In particular, 3-multiplicative LSSS reduce the constant round complexity of computing the unbounded fan-in multiplication from five to four, thus improving a result of Cramer [*et al.*]{} [@CKP:LinearAlgbra].
More importantly, we show that 3-multiplicative LSSS are closely related to strongly multiplicative LSSS. The latter is known to be a powerful tool for constructing secure MPC protocols against active adversaries. More precisely, we show the following:
(i) 3-multiplicative LSSS are also strongly multiplicative;
(ii) there exists an efficient algorithm that transforms a strongly multiplicative LSSS into a 3-multiplicative LSSS;
(iii) an example of a strongly multiplicative LSSS that is not 3-multiplicative.
Our results contribute to the study of MPC in the following three aspects:
- The 3-multiplicative LSSS outperform strongly multiplicative LSSS with respect to round complexity in the construction of secure MPC protocols.
- The 3-multiplicative LSSS are easier to construct than strongly multiplicative LSSS. First, the existence of an efficient transformation from a strongly multiplicative LSSS to a 3-multiplicative LSSS implies that efficiently constructing 3-multiplicative LSSS is not a harder problem. Second, verification of a strongly multiplicative LSSS requires checking the linear combinations for all possibilities of adversary sets, while the verification of a 3-multiplicative LSSS requires only one checking. We give two constructions of LSSS based on Reed-Muller codes and algebraic geometric codes that can be easily verified for 3-multiplicativity, but it does not seem easy to give direct proofs of their strong multiplicativity.
- This work provides two possible directions toward solving the open problem of determining the existence of efficient constructions for strongly multiplicative LSSS. On the negative side, if we can prove that in the information-theoretic model and with polynomial size message exchanged, computing $x_1x_2x_3$ inevitably needs more rounds of interactions than computing $x_1x_2$, then we can give a negative answer to this open problem. On the positive side, if we can find an efficient construction for 3-multiplicative LSSS, which also results in strongly multiplicative LSSS, then we will have an affirmative answer to this open problem.
Organization
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Section \[sec:pre\] gives notations, definition of multiplicative LSSS, and general constructions for strongly multiplicative LSSS. Section \[sec:3-m\] defines 3-multiplicative LSSS. Section \[sec:3-strong\] shows the relationship between 3-multiplicative LSSS and strongly multiplicative LSSS. Section \[sec:constructions\] gives two constructions of 3-multiplicative LSSS from error-correcting codes, and Section \[sec:implication\] discusses the implications of 3-multiplicative LSSS in MPC. Section \[sec:conclusion\] concludes the paper.
Preliminaries {#sec:pre}
=============
Throughout this paper, let $P=\{P_1,\ldots,P_n\}$ denote the set of $n$ players and let $\mathcal{K}$ be a finite field. In a secret sharing scheme, the collection of all subsets of players that are authorized to recover the secret is called its [*access structure*]{}, and is denoted $AS$. An access structure possesses the monotone ascending property: if $A'\in AS$, then for all $A\subseteq P$ with $A\supseteq A'$, we also have $A\in AS$. Similarly, the collection of subsets of players that are possibly corrupted is called the [*adversary structure*]{}, and is denoted $\mathcal{A}$. An adversary structure possesses the monotone descending property: if $A'\in
\mathcal{A}$, then for all $A\subseteq P$ with $A\subseteq A'$, we also have $A\in \mathcal{A}$. Owing to these monotone properties, it is often sufficient to consider the [*minimum access structure*]{} $AS_{min}$ and the [*maximum adversary structure*]{} $\mathcal{A}_{max}$ defined as follows: $$\begin{aligned}
AS_{min} &=\{A\in AS\mid\forall B\subseteq P, \text{ we have } B\subsetneq A\Rightarrow B\not\in
AS\},\\
\mathcal{A}_{max} &=\{A\in \mathcal{A}\mid
\forall B\subseteq P, \text{ we have } B\supsetneq A\Rightarrow B\not\in\mathcal{A}\}.\end{aligned}$$ In this paper, we consider the [*complete*]{} situation, that is, $\mathcal{A}=2^P-AS$. Moreover, an adversary structure $\mathcal{A}$ is called $Q^2$ (respectively, $Q^3$) if any two (respectively, three) sets in $\mathcal{A}$ cannot cover the entire player set $P$. For simplicity, when an adversary structure $\mathcal{A}$ is $Q^2$ (respectively, $Q^3$) we also say the corresponding access structure $AS=2^P-\mathcal{A}$ is $Q^2$ (respectively, $Q^3$).
Linear Secret Sharing Schemes and Monotone Span Programs
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Suppose $S$ is the secret-domain, $R$ is the set of random inputs, and $S_i$ is the share-domain of $P_i$, where $1\leq i\leq n$. Let [S]{} and [R]{} denote random variables taking values in $S$ and $R$, respectively. Then $\Pi:S\times R\rightarrow S_1\times\cdots\times S_n$ is called a [*secret sharing scheme*]{} (SSS) with respect to the access structure $AS$, if the following two conditions are satisfied:
1. for all $A\in AS$, $H({\sf S}\mid \Pi({\sf S},{\sf R})|_A)=0$;
2. for all $B\not\in AS$, $H({\sf S}\mid \Pi({\sf S},{\sf R})|_B)=H({\sf S})$,
where $H(\cdot)$ is the entropy function. Furthermore, the secret sharing scheme $\Pi$ is called [*linear*]{} if we have $S=\mathcal{K}$, $R=\mathcal{K}^{l-1}$, and $S_i=\mathcal{K}^{d_i}$ for some positive integers $l$ and $d_i$, $1\leq i\leq n$, and the reconstruction of the secret can be performed by taking a linear combination of shares from the authorized players. The quantity $d =
\sum_{i=1}^nd_i$ is called the [*size*]{} of the LSSS.
Karchmer and Wigderson [@Karchmer:On] introduced monotone span programs (MSP) as a linear model for computing monotone Boolean functions. We denote an MSP by $\mathcal{M}(\mathcal{K},M,\psi,\vec{v})$, where $M$ is a $d\times
l$ matrix over $\mathcal{K}$, $\psi:\{1,\ldots,d\}\rightarrow\{P_1,\ldots,P_n\}$ is a surjective labeling map, and $\vec{v}\in\mathcal{K}^l$ is a nonzero vector. We call $d$ the [*size*]{} of the MSP and $\vec{v}$ the [*target vector*]{}. A monotone Boolean function $f:\{0,1\}^n\rightarrow
\{0,1\}$ satisfies $f(\vec{\delta}')\geq f(\vec{\delta})$ for any $\vec{\delta}'\geq\vec{\delta}$, where $\vec{\delta}=(\delta_1,\ldots,\delta_n)$, $\vec{\delta}'=(\delta'_1,\ldots,\delta'_n)\in \{0,1\}^n$, and $\vec{\delta}'\geq\vec{\delta}$ means $\delta'_i\geq\delta_i$ for $1\leq i\leq n$. We say that an MSP $\mathcal{M}(\mathcal{K},M,\psi,\vec{v})$ [*computes the monotone Boolean function $f$*]{} if $\vec{v}\in span\{M_A\}$ if and only if $f(\vec{\delta_A})=1$, where $A$ is a set of players, $M_A$ denotes the matrix constricted to the rows labeled by players in $A$, $span\{M_A\}$ denotes the linear space spanned by the row vectors of $M_A$, and $\vec{\delta_A}$ is the characteristic vector of $A$.
\[thLSSS&MSP\] Suppose $AS$ is an access structure over $P$ and $f_{AS}$ is the characteristic function of $AS$, that is, $f_{AS}(\vec{\delta})=1$ if and only if $\vec{\delta}=\vec{\delta}_A$ for some $A\in AS$. Then there exists an LSSS of size $d$ that realizes $AS$ if and only if there exists an MSP of size $d$ that computes $f_{AS}$.
Since an MSP computes the same Boolean function under linear transformations, we can always assume that the target vector is $\vec{e}_1=(1,0,\ldots,0)$. From an MSP $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$ that computes $f_{AS}$, we can derive an LSSS realizing $AS$ as follows: to share a secret $s\in\mathcal{K}$, the dealer randomly selects $\vec{\rho}\in\mathcal{K}^{l-1}$, computes $M(s,\vec{\rho})^\tau$ and sends $M_{P_i}(s,\vec{\rho})^\tau$ to $P_i$ as his share, where $1\leq i\leq n$ and $\tau$ denotes the transpose. The following property of MSP is useful in the proofs of our results.
\[prop1\] Let $\mathcal{M}(\mathcal{K},M,\psi,\vec{e_1})$ be an MSP that computes a monotone Boolean function $f$. Then for all $A\subseteq P$, $\vec{e}_1\not\in span\{M_A\}$ if and only if there exists $\vec{\rho}\in\mathcal{K}^{l-1}$ such that $M_A(1,\vec{\rho})^\tau=\vec{0}^\tau$.
Multiplicative Linear Secret Sharing Schemes
--------------------------------------------
From Theorem \[thLSSS&MSP\], an LSSS can be identified with its corresponding MSP in the following way. Let $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$ be an LSSS realizing the access structure $AS$. Given two vectors $\vec{x}=(x_1,\ldots,x_d)$, $\vec{y}=(y_1,\ldots,y_d)\in\mathcal{K}^d$, we define $\vec{x}\diamond \vec{y}$ to be the vector containing all entries of the form $x_i\cdot y_j$ with $\psi(i)=\psi(j)$. More precisely, let $$\begin{aligned}
\vec{x}&=(x_{11},\ldots,x_{1d_1},\ldots,x_{n1},\ldots,x_{nd_n}),\\
\vec{y}&=(y_{11},\ldots,y_{1d_1},\ldots,y_{n1},\ldots,y_{nd_n}),\end{aligned}$$ where $\sum_{i=1}^nd_i=d$, and $(x_{i1},\ldots,x_{id_i})$, $(y_{i1},\ldots,y_{id_i})$ are the entries distributed to $P_i$ according to $\psi$. Then $\vec{x}\diamond \vec{y}$ is the vector composed of the $\sum_{i=1}^nd_i^{\;2}$ entries $x_{ij}y_{ik}$, where $1\leq j,k\leq d_i, 1\leq i\leq n$. For consistency, we write the entries of $\vec{x}\diamond \vec{y}$ in some fixed order. We also define $(\vec{x}\diamond
\vec{y})^\tau=\vec{x}^\tau\diamond\vec{y}^\tau$.
\[defMLSS\] Let $\mathcal{M}(\mathcal{K},M,\psi,\vec{e_1})$ be an LSSS realizing the access structure $AS$ over $P$. Then $\mathcal{M}$ is called [*multiplicative*]{} if there exists a recombination vector $\vec{z}\in\mathcal{K}^{\sum_{i=1}^nd_i^2}$, such that for all $s,s'\in\mathcal{K}$ and $\vec{\rho},\vec{\rho}'\in\mathcal{K}^{l-1}$, we have $$ss'=\vec{z}(M(s,\vec{\rho})^\tau\diamond M(s',\vec{\rho}')^\tau).$$ Moreover, $\mathcal{M}$ is [*strongly multiplicative*]{} if for all $A\in \mathcal{A}=2^P-AS$, $\mathcal{M}_{\overline{A}}$ is multiplicative, where $\mathcal{M}_{\overline{A}}$ denotes the MSP $\mathcal{M}$ constricted to the subset $\overline{A}=P-A$.
\[propQ3\] Let $AS$ be an access structure over $P$. Then there exists a multiplicative (respectively, strongly multiplicative) LSSS realizing $AS$ if and only if $AS$ is $Q^2$ (respectively, $Q^3$).
General Constructions of Strongly Multiplicative LSSS {#subsec:2.3}
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For all $Q^2$ access structure $AS$, Cramer [*et al.*]{} [@Cramer:General] gave an efficient construction to build a multiplicative LSSS from a general LSSS realizing the same $AS$. It remains [*open*]{} if we can [*efficiently*]{} construct a strongly multiplicative LSSS from an LSSS. However, there are general constructions with exponential complexity, as described below.
Since Shamir’s threshold secret sharing scheme is strongly multiplicative for all $Q^3$ threshold access structure, a proper composition of Shamir’s threshold secret sharing schemes results in a general construction for strongly multiplicative LSSS [@Cramer:General]. Here, we give another general construction based on multiplicative LSSS.
Let $AS$ be any $Q^3$ access structure and $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$ be an LSSS realizing $AS$. For all $A\in \mathcal{A}=2^P-AS$, it is easy to see that $\mathcal{M}_{\overline{A}}$ realizes the restricted access structure $AS_{\overline{A}}=\{B\subseteq \overline{A}\mid B\in
AS\}$. The access structure $AS_{\overline{A}}$ is $Q^2$ over $\overline{A}$ because $AS$ is $Q^3$ over $\overline{A}\cup A$. Thus, we can transform $\mathcal{M}_{\overline{A}}$ into a multiplicative LSSS following the general construction of Cramer [*et al.*]{} [@Cramer:General] to obtain a strongly multiplicative LSSS realizing $AS$. The example in Section \[subsec:example\] gives an illustration of this method.
We note that both constructions above give LSSS of exponential sizes, and hence are not [*efficient*]{} in general.
3-Multiplicative and $\lambda$-Multiplicative LSSS {#sec:3-m}
==================================================
In this section, we give an equivalent definition for (strongly) multiplicative LSSS. We then define 3-multiplicative LSSS and give a necessary and sufficient condition for its existence. The notion of 3-multiplicativity is also extended to $\lambda$-multiplicativity for all integer $\lambda>1$. Finally, we present a generic (but inefficient) construction of $\lambda$-multiplicative LSSS.
Under the same notations used in Section 2.2, it is straightforward to see that we have an induced labeling map $\psi':\{1,\ldots,\sum_{i=1}^nd_i^2\}\rightarrow\{P_1,\ldots,P_n\}$ on the entries of $\vec{x}\diamond\vec{y}$, distributing the entry $x_{ij}y_{ik}$ to $P_i$, since both $x_{ij}$ and $y_{ik}$ are labeled by $P_i$ under $\psi$. For an MSP $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$, denote $M=(M_1,\ldots,M_l)$, where $M_i\in\mathcal{K}^d$ is the $i$-th column vector of $M$, $1\leq i\leq l$. We construct a new matrix $M_{\diamond}$ as follows: $$M_{\diamond}=(M_1\diamond M_1,\ldots,M_1\diamond M_l,M_2\diamond M_1,\ldots,M_2\diamond M_l,\ldots,M_l\diamond M_1,\ldots,M_l\diamond M_l).$$ For consistency, we also denote $M_{\diamond}$ as $M\diamond M$. Obviously, $M_{\diamond}$ is a matrix over $\mathcal{K}$ with $\sum_{i=1}^nd_i^2$ rows and $l^2$ columns. For any two vectors $\vec{u},\vec{v}\in \mathcal{K}^l$, it is easy to verify that $$(M\vec{u}^\tau)\diamond(M\vec{v}^\tau)=M_{\diamond}(\vec{u}\otimes\vec{v})^\tau,$$ where $\vec{u}\otimes\vec{v}$ denotes the tensor product with its entries written in a proper order. Define the induced labeling map $\psi'$ on the rows of $M_{\diamond}$. We have the following proposition.
\[prop2\] Let $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$ be an LSSS realizing the access structure $AS$, and let $M_{\diamond}$ be with the labeling map $\psi'$. Then $\mathcal{M}$ is multiplicative if and only if $\vec{e}_1\in span\{M_{\diamond}\}$, where $\vec{e}_1 = (1, 0, \dots, 0)$. Moreover, $\mathcal{M}$ is strongly multiplicative if and only if $\vec{e}_1\in
span\{(M_{\diamond})_{\overline{A}}\}$ for all $A\in\mathcal{A}=2^P-AS$.
By Definition \[defMLSS\], $\mathcal{M}$ is multiplicative if and only if $ss'=\vec{z}(M(s,\vec{\rho})^\tau\diamond
M(s',\vec{\rho}')^\tau)$ for all $s,s'\in\mathcal{K}$ and $\vec{\rho},\vec{\rho}'\in\mathcal{K}^{l-1}$. Obviously, $$\label{eq-product}
M(s,\vec{\rho})^\tau\diamond
M(s',\vec{\rho}')^\tau=M_{\diamond}((s,\vec{\rho})\otimes(s',\vec{\rho}'))^\tau=M_{\diamond}(ss',\vec{\rho}'')^\tau,$$ where $(ss',\vec{\rho}'')=(s,\vec{\rho})\otimes(s',\vec{\rho}')$. On the other hand, $ss'=\vec{e}_1(ss',\vec{\rho}'')^\tau$. Thus $\mathcal{M}$ is multiplicative if and only if $$\label{eq11}
(\vec{e}_1-\vec{z}M_{\diamond})(ss',\vec{\rho}'')^\tau=0.$$ Because of the arbitrariness of $s,s',\vec{\rho}$ and $\vec{\rho}'$, equality (\[eq11\]) holds if and only if $\vec{e}_1-\vec{z}M_{\diamond}=\vec{0}$. Thus $\vec{e}_1\in
span\{M_{\diamond}\}$. The latter part of the proposition can be proved similarly. $\Box$
Now we are ready to give the definition of 3-multiplicative LSSS. We extend the diamond product “$\diamond$" and define $\vec{x}\diamond\vec{y}\diamond\vec{z}$ to be the vector containing all entries of the form $x_i y_jz_k$ with $\psi(i)=\psi(j)=\psi(k)$, where the entries of $\vec{x}\diamond\vec{y}\diamond\vec{z}$ are written in some fixed order.
\[def3M\] Let $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$ be an LSSS realizing the access structure $AS$. Then $\mathcal{M}$ is called [*$3$-multiplicative*]{} if there exists a recombination vector $\vec{z}\in\mathcal{K}^{\sum_{i=1}^nd_i^3}$ such that for all $s_1,s_2,s_3\in\mathcal{K}$ and $\vec{\rho}_1,\vec{\rho}_2,\vec{\rho}_3\in\mathcal{K}^{l-1}$, we have $$s_1s_2s_3=\vec{z}(M(s_1,\vec{\rho}_1)^\tau\diamond M(s_2,\vec{\rho}_2)^\tau\diamond M(s_3,\vec{\rho}_3)^\tau).$$
We can derive an equivalent definition for 3-multiplicative LSSS, similar to Proposition \[prop2\]: $\mathcal{M}$ is 3-multiplicative if and only if $\vec{e}_1\in span\{(M\diamond
M\diamond M)\}$. The following proposition gives a necessary and sufficient condition for the existence of 3-multiplicative LSSS.
For all access structures $AS$, there exists a $3$-multiplicative LSSS realizing $AS$ if and only if $AS$ is $Q^3$.
Suppose $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$ is a 3-multiplicative LSSS realizing $AS$, and suppose to the contrary, that $AS$ is not $Q^3$, so there exist $A_1,A_2,A_3\in\mathcal{A}=2^P-AS$ such that $A_1\cup A_2\cup
A_3=P$. By Proposition \[prop1\], there exists $\vec{\rho}_i\in\mathcal{K}^{l-1}$ such that $M_{A_i}(1,\vec{\rho}_i)^\tau=\vec{0}^\tau$ for $1\leq i\leq 3$. Since $A_1\cup A_2\cup A_3=P$, we have $M(1,\vec{\rho}_1)^\tau\diamond
M(1,\vec{\rho}_2)^\tau\diamond M(1,\vec{\rho}_3)^\tau=\vec{0}^\tau$, which contradicts Definition \[def3M\].
On the other hand, a general construction for building a 3-multiplicative LSSS from a strongly multiplicative LSSS is given in the next section, thus sufficiency is guaranteed by Proposition \[propQ3\]. $\Box$
A trivial example of 3-multiplicative LSSS is Shamir’s threshold secret sharing scheme that realizes any $Q^3$ threshold access structure. Using an identical argument for the case of strongly multiplicative LSSS, we have a general construction for 3-multiplicative LSSS based on Shamir’s threshold secret sharing schemes, with exponential complexity.
For any $\lambda$ vectors $\vec{x}_i=(x_{i1},\ldots,x_{id})\in\mathcal{K}^d,1\leq i\leq \lambda$, we define $\diamond_{i=1}^{\lambda}\vec{x}_i$ to be the $\sum_{i=1}^n{d_i^{\lambda}}$-dimensional vector which contains entries of the form $\prod_{i=1}^{\lambda}x_{ij_i}$ with $\psi(j_1)=\cdots=\psi(j_{\lambda})$.
Let $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$ be an LSSS realizing the access structure $AS$, and let $\lambda>1$ be an integer. Then $\mathcal{M}$ is [*$\lambda$-multiplicative*]{} if there exists a recombination vector $\vec{z}$ such that for all $s_1,\ldots,s_{\lambda}\in\mathcal{K}$ and $\vec{\rho}_1,\ldots,\vec{\rho}_{\lambda}\in\mathcal{K}^{l-1}$, we have $$\prod_{i=1}^{\lambda}s_i=\vec{z}(\diamond_{i=1}^{\lambda}M(s_i,\vec{\rho}_i)^\tau).$$ Moreover, $\mathcal{M}$ is [*strongly $\lambda$-multiplicative*]{} if for all $A\not\in AS$, the constricted LSSS $\mathcal{M}_{\overline{A}}$ is $\lambda$-multiplicative.
Again, we can define a new matrix by taking the diamond product of $\lambda$ copies of $M$. This gives an equivalence to (strongly) $\lambda$-multiplicative LSSS. Also, since Shamir’s threshold secret sharing scheme is trivially $\lambda$-multiplicative and strongly $\lambda$-multiplicative, a proper composition of Shamir’s threshold secret sharing schemes results in a general construction for both $\lambda$-multiplicative LSSS and strongly $\lambda$-multiplicative LSSS. Let $Q^\lambda$ be a straightforward extension of $Q^2$ and $Q^3$, that is, an access structure $AS$ is $Q^\lambda$ if the player set $P$ cannot be covered by $\lambda$ sets in $\mathcal{A}=2^P-AS$. The following corollary is easy to prove.
Let $AS$ be an access structure over $P$. Then there exists a $\lambda$-multiplicative (respectively, strongly $\lambda$-multiplicative) LSSS realizing $AS$ if and only if $AS$ is $Q^{\lambda}$ (respectively, $Q^{\lambda+1}$).
Since a $\lambda$-multiplicative LSSS transforms the products of $\lambda$ entries into a linear combination of the local products of shares, it can be used to simplify the secure computation of sequential multiplications. In particular, when compared to using only the multiplicative property (which corresponds to the case when $\lambda=2$), a $\lambda$-multiplicative LSSS can lead to reduced round complexity by a factor of $\frac{1}{\log\lambda}$ in certain cases.
We also point out that $Q^\lambda$ is not a necessary condition for secure computation. Instead, the necessary condition is $Q^2$ for the passive adversary model, or $Q^3$ for the active adversary model [@Cramer:General]. The condition $Q^\lambda$ is just a necessary condition for the existence of $\lambda$-multiplicative LSSS which can be used to simplify computation. In practice, many threshold adversary structures satisfy the $Q^\lambda$ condition for some appropriate integer $\lambda$, and the widely used Shamir’s threshold secret sharing scheme is already $\lambda$-multiplicative. By using this $\lambda$-multiplicativity, we can get more efficient MPC protocols. However, since the special case $\lambda=3$ shows a close relationship with strongly multiplicative LSSS, a fundamental tool in MPC, this paper focuses on 3-multiplicative LSSS.
Strong Multiplicativity and 3-Multiplicativity {#sec:3-strong}
==============================================
In this section, we show that strong multiplicativity and 3-multiplicativity are closely related. On the one hand, given a strongly multiplicative LSSS, there is an [*efficient*]{} transformation that converts it to a 3-multiplicative LSSS. On the other hand, we show that any 3-multiplicative LSSS is a strongly multiplicative LSSS, but the converse is not true. It should be noted that strong multiplicativity, as defined, has a combinatorial nature. The definition of 3-multiplicativity is essentially algebraic, which is typically easier to verify.
From Strong Multiplicativity to 3-Multiplicativity {#subsec:s->3}
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We show a general method to efficiently build a 3-multiplicative LSSS from a strongly multiplicative LSSS, for all $Q^3$ access structures. As an extension, the proposed method can also be used to efficiently build a $(\lambda+1)$-multiplicative LSSS from a strongly $\lambda$-multiplicative LSSS.
\[prop5\] Let $AS$ be a $Q^3$ access structure and $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$ be a strongly multiplicative LSSS realizing $AS$. Suppose that $\mathcal{M}$ has size $d$ and $|\psi^{-1}(P_i)|=d_i$, for $1\leq i\leq n$. Then there exists a $3$-multiplicative LSSS for $AS$ of size $O(d^2)$.
We give a constructive proof. Let $M_{\diamond}$ be the matrix defined in Section \[sec:3-m\], and $\psi'$ be the induced labeling map on the rows of $M_{\diamond}$. Then we have an LSSS $\mathcal{M}_{\diamond}(\mathcal{K},M_{\diamond},\psi',\vec{e}_1)$ that realizes an access structure $AS_{\diamond}$. Because $\mathcal{M}$ is strongly multiplicative, by Proposition \[prop2\] we have $\vec{e}_1\in span\{(M_{\diamond})_{\overline{A}}\}$ for all $A\not\in AS$. Therefore $\overline{A}\in AS_{\diamond}$ and it follows that $AS^*\subseteq AS_{\diamond}$, where $AS^*$ denotes the dual access structure of $AS$, defined by $AS^*=\{A\subseteq P\mid P-A\not\in AS\}$.
The equality (\[eq-product\]) in the proof of Proposition \[prop2\] shows that the diamond product of two share vectors equals sharing the product of the two secrets by the MSP $\mathcal{M}_{\diamond}(\mathcal{K},M_{\diamond},\psi',\vec{e}_1)$, that is, $$(M(s_1,\vec{\rho}'_1)^\tau)\diamond(M(s_2,\vec{\rho}'_2)^\tau)=M_{\diamond}(s_1s_2,\vec{\rho})^\tau, ~\mbox{~for some $\vec{\rho}'_1, \vec{\rho}'_2, \vec{\rho} \in {\cal K}^{l-1}$}.$$ Thus, using a method similar to Nikov [*et al.*]{} [@Nikov:multiplicativeLSSS], we can get the product $(s_1s_2)\cdot s_3$ by sharing $s_3$ through the dual MSP of $\mathcal{M}_{\diamond}$, denoted by $(\mathcal{M}_{\diamond})^*$. Furthermore, since $(\mathcal{M}_{\diamond})^*$ realizes the dual access structure $(AS_\diamond)^*$ and $(AS_\diamond)^*\subseteq
(AS^*)^*=AS$, we can build a 3-multiplicative LSSS by the union of $\mathcal{M}$ and $(\mathcal{M}_{\diamond})^*$, which realizes the access structure $AS \cup (AS_\diamond)^*=AS$. Now following the same method of Cramer [*et al.*]{} and Fehr [@Cramer:General; @Fehr:DualMSP], we prove the required result via the construction below.
Compute the column vector $\vec{v}_0$ as a solution to the equation $(M_{\diamond})^\tau\vec{v}=\vec{e_1}^\tau$ for $\vec{v}$, and compute $\vec{v}_1,\ldots,\vec{v}_k$ as a basis of the solution space to $(M_{\diamond})^\tau\vec{v}=\vec{0}^\tau$. Note that $(M_{\diamond})^\tau\vec{v}=\vec{e_1}^\tau$ is solvable because $\vec{e}_1\in span\{(M_{\diamond})_{\overline{A}}\}$ for all $A\not\in AS$, while $(M_{\diamond})^\tau\vec{v}=\vec{0}^\tau$ may only have the trivial solution $\vec{v}=\vec{0}$ and $k=0$. Let $$M'=\left(\begin{array}{cccccc}m_{11}&\cdots&m_{1l}& & &
\\\vdots&\ddots&\vdots& & & \\m_{d1}&\cdots&m_{dl}& & &
\\\vec{v_0}& &
&\vec{v_1}&\cdots&\vec{v_k}\end{array}\right),$$ where $\left(\begin{array}{ccc}m_{11}&\cdots&m_{1l}\\\vdots&\ddots&\vdots\\m_{d1}&\cdots&m_{dl}\end{array}\right)=M$ and the blanks in $M'$ denote zeros. Define a labeling map $\psi''$ on the rows of $M'$ which labels the first $d$ rows of $M'$ according to $\psi$ and the other $\sum_{i=1}^nd_i^2$ rows according to $\psi'$.
As mentioned above, $\mathcal{M}'(\mathcal{K},M',\psi'',\vec{e}_1)$ obviously realizes the access structure $AS$. We now verify its 3-multiplicativity.
Let $N=(\vec{v}_0, \vec{v}_1,\ldots,\vec{v}_k)$, a matrix over $\mathcal{K}$ with $\sum_{i=1}^nd_i^2$ rows and $k+1$ columns. For $s_i\in\mathcal{K}$ and $\vec{\rho}_i=(\vec{\rho}'_i,\vec{\rho}''_i)\in
\mathcal{K}^{l-1}\times \mathcal{K}^k$, $1\leq i\leq 3$, denote $M'(s_i,\vec{\rho}_i)^\tau=(\vec{u}_i,\vec{w}_i)^\tau$, where $\vec{u}_i^{\;\tau}=M(s_i,\vec{\rho}'_i)^\tau$ and $\vec{w}_i^{\;\tau}=N(s_i,\vec{\rho}''_i)^\tau$. We have $$\vec{u}_1^{\;\tau}\diamond\;
\vec{u}_2^{\;\tau}=(M(s_1,\vec{\rho}'_1)^\tau)\diamond(M(s_2,\vec{\rho}'_2)^\tau)=M_{\diamond}(s_1s_2,\vec{\rho})^\tau,$$ where $(s_1s_2,\vec{\rho})=(s_1,\vec{\rho}'_1)\otimes(s_2,\vec{\rho}'_2)$. Then, $$\begin{aligned}
(\vec{u}_1\diamond\vec{u}_2)\cdot\vec{w}_3^{\;\tau} &=(s_1s_2,\vec{\rho})(M_{\diamond})^{\tau}\cdot N\left(\begin{array}{c}s_3\\ \\ {\vec{\rho}_3''}^{\;\tau}\end{array}\right) \\
&=(s_1s_2,\vec{\rho})
\left(\begin{array}{cccc}1&0&\cdots&0\\0&0&\cdots&0\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&0\end{array}\right)\left(\begin{array}{c}s_3\\ \\ {\vec{\rho}_3''}^{\;\tau}\end{array}\right)\\
&=s_1s_2s_3.\end{aligned}$$
It is easy to see that $(\vec{u}_1\diamond\vec{u}_2)\cdot\vec{w}_3^{\;\tau}$ is a linear combination of the entries from $(\vec{u}_1\diamond\vec{u}_2)\diamond \vec{w}_3$, and so is a linear combination of the entries from $M'(s_1,\vec{\rho}_1)^\tau\diamond
M'(s_2,\vec{\rho}_2)^\tau\diamond M'(s_3,\vec{\rho}_3)^\tau$.
Hence $\mathcal{M}'$ is a 3-multiplicative LSSS for $AS$. Obviously, the size of $\mathcal{M}'$ is $O(d^2)$, since $d+\sum_{i=1}^nd_i^2 <
d^2 + d.$ $\Box$
If we replace the matrix $M_{\diamond}$ above by the diamond product of $\lambda$ copies of $M$, using an identical argument, the construction from Theorem \[prop5\] gives rise to a $(\lambda+1)$-multiplicative LSSS from a strongly $\lambda$-multiplicative LSSS.
\[coro2\] Let $AS$ be a $Q^{\lambda+1}$ access structure and $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$ be a strongly $\lambda$-multiplicative LSSS realizing $AS$. Suppose the size of $\mathcal{M}$ is $d$ and $|\psi^{-1}(P_i)|=d_i$, for $1\leq i\leq n$. Then there exists a $(\lambda+1)$-multiplicative LSSS for $AS$ of size $O(d^{\lambda})$.
From 3-Multiplicativity to Strong Multiplicativity
--------------------------------------------------
\[th5\] Any $3$-multiplicative LSSS is strongly multiplicative.
Let $\mathcal{M}(\mathcal{K},M,\psi,\vec{e}_1)$ be a 3-multiplicative LSSS realizing the access structure $AS$ over $P$. For all $A\in\mathcal{A}=2^P-AS$, by Proposition \[prop1\], we can choose a fixed vector $\vec{\rho}''\in\mathcal{K}^{l-1}$ such that $M_{A}(1,\vec{\rho}'')^\tau=\vec{0}^\tau$. There exists a recombination vector $\vec{z}\in\mathcal{K}^{\sum_{i=1}^nd_i^3}$ such that for all $s,s'\in\mathcal{K}$ and $\vec{\rho},\vec{\rho}'\in\mathcal{K}^{l-1}$, we have $$ss'=\vec{z}(M(s,\vec{\rho})^\tau\diamond M(s',\vec{\rho}')^\tau\diamond M(1,\vec{\rho}'')^\tau).$$ Since $M_{A}(1,\vec{\rho}'')^\tau=\vec{0}^\tau$, and $M_{\overline{A}}(1,\vec{\rho}'')^\tau$ is a constant vector for fixed $\vec{\rho}''$, the vector $\vec{z}'\in\mathcal{K}^{\sum_{P_i\not\in A}d_i^2}$ that satisfies $$\vec{z}(M(s,\vec{\rho})^\tau\diamond M(s',\vec{\rho}')^\tau\diamond M(1,\vec{\rho}'')^\tau)=\vec{z}'(M_{\overline{A}}(s,\vec{\rho})^\tau\diamond M_{\overline{A}}(s',\vec{\rho}')^\tau)$$ can be easily determined. Thus $ss'=\vec{z}'(M_{\overline{A}}(s,\vec{\rho})^\tau\diamond
M_{\overline{A}}(s',\vec{\rho}')^\tau)$. Hence, $\mathcal{M}$ is strongly multiplicative. $\Box$
Although 3-multiplicative LSSS is a subclass of strongly multiplicative LSSS, one of the advantages of 3-multiplicativity is that its verification admits a simpler process. For 3-multiplicativity, we need only to check that $\vec{e}_1\in span\{(M\diamond M\diamond M)\}$, while strong multiplicativity requires the verification of $\vec{e}_1\in span\{(M\diamond M)_{\overline{A}}\}$ for [*all*]{} $A\not\in AS$.
Using a similar argument, the following results for $(\lambda+1)$-multiplicativity can be proved:
(i) A $(\lambda+1)$-multiplicative LSSS is a strongly $\lambda$-multiplicative LSSS.
(ii) A $\lambda$-multiplicative LSSS is a $\lambda'$-multiplicative LSSS, where $1<\lambda'<\lambda$.
An Example of a Strongly Multiplicative LSSS that is Not 3-Multiplicative {#subsec:example}
-------------------------------------------------------------------------
We give an example of a strongly multiplicative LSSS that is not 3-multiplicative. It follows that 3-multiplicative LSSS are strictly contained in the class of strongly multiplicative LSSS. The construction process is as follows. Start with an LSSS that realizes a $Q^3$ access structure but is not strongly multiplicative. We then apply the general construction given in Section \[subsec:2.3\] to convert it into a strongly multiplicative LSSS. The resulting LSSS is however not 3-multiplicative.
Let $P=\{P_1,P_2,P_3,P_4,P_5,P_6\}$ be the set of players. Consider the access structure $AS$ over $P$ defined by $$AS_{min}=\{(1,2),(3,4),(5,6),(1,5),(1,6),(2,6),(2,5),(3,6),(4,5)\},$$ where we use subscript to denote the corresponding player. For example, $(1,2)$ denotes the subset $\{P_1,P_2\}$. It is easy to verify that the corresponding adversary structure is $$\mathcal{A}_{max}=\{(1,3),(1,4),(2,3),(2,4),(3,5),(4,6)\},$$ and that $AS$ is a $Q^3$ access structure.
Let $\mathcal{K}=\mathbb{F}_2$. Define the matrix $M$ over $\mathbb{F}_2$ with the labeling map $\psi$ such that $$\begin{aligned}
M_{P_1}=\left(\begin{array}{ccccc}1&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\end{array}\right),\;
M_{P_2}=\left(\begin{array}{ccccc}0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\end{array}\right),\;
M_{P_3}=\left(\begin{array}{ccccc}1&1&0&0&0\\0&0&0&0&1\end{array}\right),
\\
M_{P_4}=\left(\begin{array}{ccccc}0&1&0&0&0\\0&0&0&1&0\end{array}\right),\;
M_{P_5}=\left(\begin{array}{ccccc}1&1&1&0&0\\1&0&0&1&0\end{array}\right),\;
M_{P_6}=\left(\begin{array}{ccccc}0&1&1&0&0\\1&0&0&0&1\end{array}\right).\end{aligned}$$ It can be verified that the LSSS $\mathcal{M}(\mathbb{F}_2,M,\psi,\vec{e}_1)$ realizes the access structure $AS$. Moreover, for all $A\in\mathcal{A}-\{(1,3),(1,4)\}$, the constricted LSSS $\mathcal{M}_{\overline{A}}$ is multiplicative. Thus in order to get a strongly multiplicative LSSS, we just need to expand $\mathcal{M}$ with multiplicativity when constricted to both $\{P_2,P_4,P_5,P_6\}$ and $\{P_2,P_3,P_5,P_6\}$.
Firstly, consider the LSSS $\mathcal{M}$ constricted to $P'=\{P_2,P_4,P_5,P_6\}$. Obviously, $\mathcal{M}_{P'}$ realizes the access structure $AS'_{min}=\{(5,6),(2,6),(2,5),(4,5)\}$, which is $Q^2$ over $P'$. By the method of Cramer [*et al.*]{} [@Cramer:General], we can transform $\mathcal{M}_{P'}$ into the multiplicative LSSS $\mathcal{M}'_{P'}(\mathbb{F}_2,M',\psi',\vec{e}_1)$ defined as follows: $$\begin{aligned}
M'_{P_2}=\left(\begin{array}{ccccccccc}0&0&1&0&0& & & & \\0&0&0&1&0& & & & \\0&0&0&0&1& & & & \\ & & & & &0&1&1&1\\ & & & & &1&1&0&0\\ & & & & &0&0&0&1\end{array}\right),~~~~
M'_{P_4}=\left(\begin{array}{ccccccccc}0&1&0&0&0& & & &
\\0&0&0&1&0& & & & \\ & & & & &0&1&1&1\\ & & & &
&1&0&0&0\end{array}\right),\\
M'_{P_5}=\left(\begin{array}{ccccccccc}1&1&1&0&0& & & & \\1&0&0&1&0&
& & & \\1& & & & &0&1&0&1\\0& & & &
&0&1&0&0\end{array}\right),~~~~
M'_{P_6}=\left(\begin{array}{ccccccccc}0&1&1&0&0& & & &
\\1&0&0&0&1& & & & \\1& & & & &0&0&1&0\\0& & & &
&0&0&0&1\end{array}\right),\end{aligned}$$ where the blanks in the matrices denote zeros.
For consistency, we define $$\begin{aligned}
M'_{P_1} &=(M_{P_1}\;O_{3\times4}),\\
M'_{P_3} &=(M_{P_3}\;O_{2\times4}),\end{aligned}$$ where $O_{m\times n}$ denotes the $m\times n$ matrix of all zeros. It can be verified that for the subset $P''=\{P_2,P_3,P_5,P_6\}$, the constricted LSSS $\mathcal{M}'_{P''}$ is indeed multiplicative. Therefore, $\mathcal{M}'(\mathbb{F}_2,M',\psi',\vec{e}_1)$ is a strongly multiplicative LSSS realizing the access structure $AS$. Furthermore, it can be verified that $\mathcal{M}'$ is not 3-multiplicative (the verification involves checking a $443\times 729$ matrix using Matlab).
The scheme $\mathcal{M}(\mathbb{F}_2,M,\psi,\vec{v}_1)$ given above is the first example of an LSSS which realizes a $Q^3$ access structure but is not strongly multiplicative.
Constructions for $3$-multiplicative LSSS {#sec:constructions}
=========================================
It is tempting to find efficient constructions for 3-multiplicative LSSS. In general, it is a hard problem to construct LSSS with polynomial size for any specified access structure, and it seems to be an even harder problem to construct polynomial size 3-multiplicative LSSS with general $Q^3$ access structures. We mention two constructions for 3-multiplicative LSSS. These constructions are generally inefficient, which can result in schemes with exponential sizes. The two constructions are:
1. The Cramer-Damg[å]{}rd-Maurer construction based on Shamir’s threshold secret sharing scheme [@Cramer:General].
2. The construction given in Subsection \[subsec:s->3\] based on strongly multiplicative LSSS.
There exist, however, some efficient LSSS with specific access structures that are multiplicative or 3-multiplicative. For instance, Shamir’s $t$ out of $n$ threshold secret sharing schemes are multiplicative if $n
\geq 2t+1$, and 3-multiplicative if $n \geq 3t+1$.
On the other hand, secret sharing schemes from error-correcting codes give good multiplicative properties. It is well known that a secret sharing scheme from a linear error-correcting code is an LSSS. We know that such an LSSS is multiplicative provided the underlying code is a self dual code [@Cramer:OnCodes]. The LSSS from a Reed-Solomon code is $\lambda$-multiplicative if the corresponding access structure is $Q^{\lambda}$. In this section, we show the multiplicativity of two other classes of secret sharing schemes from error-correcting codes:
(i) schemes from Reed-Muller codes are $\lambda$-multiplicative LSSS; and
(ii) schemes from algebraic geometric codes are $\lambda$-multiplicative ramp LSSS.
A Construction from Reed-Muller Codes
-------------------------------------
Let $\vec{v}_0,\vec{v}_1,\ldots,\vec{v}_{2^m-1}$ be all the points in the space $\mathbb{F}_2^{\;m}$. The binary Reed-Muller code $\mathcal{R}(r,m)$ is defined as follows: $$\mathcal{R}(r,m)=\{(f(\vec{v}_0),f(\vec{v}_1),\ldots,f(\vec{v}_{2^m-1}))\mid f\in\mathbb{F}_2[x_1,\ldots,x_m],\;\deg f\leq
r\}.$$
Take $f({\vec{v}_0})$ as the secret, and $f({\vec{v}_i})$ as the share distributed to player $P_i$, $1\leq i\leq 2^m-1$. Then $\mathcal{R}(r,m)$ gives rise to an LSSS for the set of players $\{P_1,\ldots,P_n\}$, with the secret-domain being $\mathbb{F}_2$, where $n=2^m-1$. For any three codewords $$\vec{c}_i=(s_i,s_{i1},\ldots,s_{in})=(f_i(\vec{v}_0),f_i(\vec{v}_1),\ldots,f_i(\vec{v}_n))\in\mathcal{R}(r,m),~~1\leq i\leq 3,$$ it is easy to see that $$\begin{aligned}
\vec{c}_1\diamond\vec{c}_2\diamond\vec{c}_3 &=(s_1s_2s_3,s_{11}s_{21}s_{31},\ldots,s_{1n}s_{2n}s_{3n}) \\
&=(g(\vec{v}_0),g(\vec{v}_1),\ldots,g(\vec{v}_n))\in\mathcal{R}(3r,m),\end{aligned}$$ where $g=f_1f_2f_3\in\mathbb{F}_2[x_1,\ldots,x_m]$ and $\deg g\leq 3r$. From basic results on Reed-Muller codes [@Lint:Code], we know that $\mathcal{R}(3r,m)$ has dual code $\mathcal{R}(m-3r-1,m)$ when $m>3r$, and the dual code $\mathcal{R}(m-3r-1,m)$ trivially contains the codeword $(1,1,\ldots,1)$. It follows that $s_1s_2s_3=\sum_{j=1}^ns_{1j}s_{2j}s_{3j}$, which shows that the LSSS from $\mathcal{R}(r,m)$ is 3-multiplicative when $m>3r$. Certainly, this LSSS is strongly multiplicative. In general, we have the following result:
\[thm:RM-code\] The LSSS constructed above from $\mathcal{R}(r,m)$ is $\lambda$-multiplicative, provided $m >
\lambda r$.
A Construction from Algebraic Geometric Codes
---------------------------------------------
Chen and Cramer [@CC:AGRamp] constructed secret sharing schemes from algebraic geometric (AG) codes. These schemes are [*quasi-threshold*]{} (or [*ramp*]{}) schemes, which means that any $t$ out of $n$ players can recover the secret, and any fewer than $t'$ players have no information about the secret, where $t'\leq t \leq n$. In this section, we show that ramp schemes from some algebraic geometric codes are $\lambda$-multiplicative.
Let $\chi$ be an absolutely irreducible, projective, and nonsingular curve defined over $\mathbb{F}_q$ with genus $g$, and let $D=\{v_0,v_1,\ldots,v_n\}$ be the set of $\mathbb{F}_q$-rational points on $\chi$. Let $G$ be an $\mathbb{F}_q$-rational divisor with degree $m$ satisfying $supp(G) \cap D=\emptyset$ and $2g-2<m<n+1$. Let $\overline{\mathbb{F}}_q$ denote the algebraic closure of $\mathbb{F}_q$, let $\overline{\mathbb{F}}_q(\chi)$ denote the function field of the curve $\chi$, and let $\Omega(\chi)$ denote all the differentials on $\chi$. Define the linear spaces: $$\begin{aligned}
\mathcal{L}(G) &=\{f\in\overline{\mathbb{F}}_q(\chi)\mid (f)+G\geq 0\},\\
\Omega(G) &=\{\omega\in\Omega(\chi)\mid (\omega)\geq G\}.\end{aligned}$$ Then the functional AG code $C_\mathcal{L}(D,G)$ and residual AG code $C_{\Omega}(D,G)$ are respectively defined as follows: $$\begin{aligned}
C_\mathcal{L}(D,G) &=\{(f(v_0),f(v_1),\ldots,f(v_n))\mid f\in\mathcal{L}(G)\}\subseteq \mathbb{F}_q^{\;n+1}, \\
C_{\Omega}(D,G) &=\{(Res_{v_0}(\eta),Res_{v_1}(\eta),\ldots,Res_{v_n}(\eta))\mid\eta\in\Omega(G-D)\}\subseteq\mathbb{F}_q^{\;n+1},\end{aligned}$$ where $Res_{v_i}(\eta)$ denotes the residue of $\eta$ at $v_i$.
As above, $C_{\Omega}(D,G)$ induces an LSSS for the set of players $\{P_1,\ldots,P_n\}$, where for every codeword $(f(v_0),f(v_1),\ldots,f(v_n))\in
C_{\Omega}(D,G)=C_{\mathcal{L}}(D,D-G+(\eta))$, $f(v_0)$ is the secret and $f(v_i)$ is $P_i$’s share, $1\leq i\leq n$. For any $\lambda$ codewords $$\begin{aligned}
\vec{c}_i &=(s_i,s_{i1},\ldots,s_{in}) \\
&=(f_i(v_0),f_i(v_1),\ldots,f_i(v_n))\in C_{\mathcal{L}}(D,D-G+(\eta)),~~~1\leq
i\leq \lambda,\end{aligned}$$ it is easy to see that $$\diamond_{i=1}^\lambda\vec{c}_i=\left(\prod_{i=1}^\lambda s_i,\prod_{i=1}^\lambda s_{i1},\ldots,\prod_{i=1}^\lambda s_{in}\right)\in C_\mathcal{L}(D,\lambda(D-G+(\eta))).$$
If $2g-2<\deg(\lambda(D-G+(\eta)))<n$, then $C_\mathcal{L}(D,\lambda(D-G+(\eta)))$ has the dual code $C_{\Omega}(D,\lambda(D-G+(\eta)))=C_{\mathcal{L}}(D,\lambda
G-(\lambda-1)(D+(\eta)))$. When $\deg(\lambda
G-(\lambda-1)(D+(\eta)))\geq 2g$, $C_{\Omega}(D,\lambda(D-G+(\eta)))$ has a codeword with a nonzero first coordinate, implying $\prod_{i=1}^\lambda
s_i=\sum_{j=1}^na_j\prod_{i=1}^\lambda s_{ij}$ for some constants $a_j\in\mathbb{F}_q$. Thus, the LSSS induced by the AG code $C_{\Omega}(D,G)$ is $\lambda$-multiplicative. It is easy to see that if $\deg G = m \geq\frac{(\lambda-1)(n-1)}{\lambda}+2g$ then we have $2g-2<\deg(\lambda(D-G+(\eta)))<n$ and $\deg(\lambda
G-(\lambda-1)(D+(\eta)))\geq 2g$. Therefore, we have the following theorem.
Let $\chi$ be an absolutely irreducible, projective, and nonsingular curve defined over $\mathbb{F}_q$ with genus $g$, let $D=\{v_0,v_1,\ldots,v_n\}$ be the set of $\mathbb{F}_q$-rational points on $\chi$. Let $G$ be an $\mathbb{F}_q$-rational divisor with degree $m$ satisfying $supp(G) \cap D=\emptyset$ and $2g-2<m<n+1$. Then the LSSS induced by the AG code $C_{\Omega}(D,G)$ is $\lambda$-multiplicative, provided $m\geq\frac{(\lambda-1)(n-1)}{\lambda}+2g$.
Implications of the Multiplicativity of LSSS {#sec:implication}
============================================
The property of 3-multiplicativity implies strong multiplicativity, and so is sufficient for building MPC protocols against active adversaries. The conditions for 3-multiplicativity are easy to verify, while verification for strong multiplicativity involves checking an exponential number of equations (each subset in the adversary structure corresponds to an equation).
With 3-multiplicative LSSS, or more generally $\lambda$-multiplicative LSSS, we can simplify local computation for each player and reduce the round complexity in MPC protocols. For example, using the technique of Bar-Ilan and Beaver [@BB:ConstantRd], we can compute $\prod_{i=1}^lx_i,\;
x_i\in\mathbb{F}_q$, in a constant number of rounds, independent of $l$. For simplicity, we consider passive adversaries in the information-theoretic model. Suppose for $1\leq i\leq l$, the shares of $x_i$, denoted by $[x_i]$, have already been distributed among the players. To compute $\prod_{i=1}^lx_i,\; x_i\in\mathbb{F}_q$, we follow the process of Cramer [*et al.*]{} [@CKP:LinearAlgbra]:
(1) Generate $[b_0\in_R\mathbb{F}_q^{\;*}],[b_1\in_R\mathbb{F}_q^{\;*}],\ldots,[b_l\in_R\mathbb{F}_q^{\;*}]$ and $[b_0^{-1}],[b_1^{-1}],\ldots,[b_l^{-1}]$, where $b_i\in_R\mathbb{F}_q^{\;*}$ means that $b_i$ is a random element in $\mathbb{F}_q^{\;*}$.
(2) For $1\leq i\leq l$, each player computes $[b_{i-1}x_ib_i^{-1}]$ from $[b_{i-1}],[b_i^{-1}]$ and $[x_i]$.
(3) Recover $d_i=b_{i-1}x_ib_i^{-1}$ from $[b_{i-1}x_ib_i^{-1}]$ for $1\leq i\leq l$, and compute $d=\prod_{i=1}^{l}d_i$.
(4) Compute $[db_0^{-1}b_l]$ from $[b_0^{-1}],[b_l]$ and $d$.
It is easy to see that $db_0^{-1}b_l=\prod_{i=1}^lx_i$. Using a strongly multiplicative LSSS, the above process takes five rounds of interactions as two rounds are required in Step (2). However, if we use a 3-multiplicative LSSS instead, then only one round is needed for Step (2). Thus, 3-multiplicative LSSS reduce the round complexity of computing unbounded fan-in multiplication from five to four. This in turn simplifies the computation of many problems, such as polynomial evaluation and solving linear systems of equations.
In general, the relationship between $\lambda$-multiplicative LSSS and strongly $\lambda$-multiplicative LSSS can be described as follows: $$\cdots\subseteq SMLSSS_{\lambda+1}\subsetneq MLSSS_{\lambda+1}\subseteq SMLSSS_{\lambda}\subsetneq MLSSS_{\lambda}\subseteq\cdots,$$ where $MLSSS_{\lambda}$ (respectively, $SMLSSS_{\lambda}$) denotes the class of $\lambda$-multiplicative (respectively, strongly $\lambda$-multiplicative) LSSS. It is easy to see that $SMLSSS_{\lambda}\subsetneq
MLSSS_{\lambda}$ because they exist under the conditions $Q^{\lambda+1}$ and $Q^\lambda$, respectively. Since $SMLSSS_{\lambda}$ and $MLSSS_{\lambda+1}$ both exist under the same necessary and sufficient condition of $Q^{\lambda+1}$, it is not straightforward to see whether $MLSSS_{\lambda+1}$ is strictly contained in $SMLSSS_{\lambda}$. For $\lambda=2$, we already know that $MLSSS_3\subsetneq SMLSSS_2$ (Section \[subsec:example\]). It would be interesting to find out if this is also true for $\lambda >
2$. We have also given an efficient transformation from $SMLSSS_{\lambda}$ to $MLSSS_{\lambda+1}$. It remains open whether an efficient transformation from $MLSSS_{\lambda}$ to $SMLSSS_{\lambda}$ exists when the access structure is $Q^{\lambda+1}$. When $\lambda=2$, this is a well-known open problem [@Cramer:General].
Conclusions {#sec:conclusion}
===========
In this paper, we propose the new concept of 3-multiplicative LSSS, which form a subclass of strongly multiplicative LSSS. The 3-multiplicative LSSS are easier to construct compared to strongly multiplicative LSSS. They can also simplify the computation and reduce the round complexity in secure multiparty computation protocols. We believe that 3-multiplicative LSSS are a more appropriate primitive as building blocks for secure multiparty computations, and deserve further investigation. We stress that finding efficient constructions of 3-multiplicative LSSS for general access structures remains an important open problem.
Acknowledgement {#acknowledgement .unnumbered}
===============
The work of M. Liu and Z. Zhang is supported in part by the 973 project of China (No. 2004CB318000). Part of the work was done while Z. Zhang was visiting Nanyang Technological University supported by the Singapore Ministry of Education under Research Grant T206B2204.
The work of Y. M. Chee, S. Ling, and H. Wang is supported in part by the Singapore National Research Foundation under Research Grant NRF-CRP2-2007-03.
In addition, the work of Y. M. Chee is also supported in part by the Nanyang Technological University under Research Grant M58110040, and the work of H. Wang is also supported in part by the Australian Research Council under ARC Discovery Project DP0665035.
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---
abstract: 'The total mass ${\,M_\mathrm{GCS}}$ in the globular cluster (GC) system of a galaxy is empirically a near-constant fraction of the total mass $M_h \equiv M_{bary} + M_{dark}$ of the galaxy, across a range of $10^5$ in galaxy mass. This trend is radically unlike the strongly nonlinear behavior of total stellar mass $M_{\star}$ versus $M_h$. We discuss extensions of this trend to two more extreme situations: (a) entire clusters of galaxies, and (b) the Ultra-Diffuse Galaxies (UDGs) recently discovered in Coma and elsewhere. Our calibration of the ratio $\eta_M = {\,M_\mathrm{GCS}}/ M_h$ from normal galaxies, accounting for new revisions in the adopted mass-to-light ratio for GCs, now gives $\eta_M = 2.9 \times 10^{-5}$ as the mean absolute mass fraction. We find that the same ratio appears valid for galaxy clusters and UDGs. Estimates of $\eta_M$ in the four clusters we examine tend to be slightly higher than for individual galaxies, but more data and better constraints on the mean GC mass in such systems are needed to determine if this difference is significant. We use the constancy of $\eta_M$ to estimate total masses for several individual cases; for example, the total mass of the Milky Way is calculated to be $M_h = 1.1 \times 10^{12} M_{\odot}$. Physical explanations for the uniformity of $\eta_M$ are still descriptive, but point to a picture in which massive, dense star clusters in their formation stages were relatively immune to the feedback that more strongly influenced lower-density regions where most stars form.'
author:
- 'William E. Harris, John P. Blakeslee, and Gretchen L. H. Harris'
bibliography:
- 'gc.bib'
title: 'Galactic Dark Matter Halos and Globular Cluster Populations. III: Extension to Extreme Environments'
---
Introduction
============
Two decades ago, @blakeslee_etal1997 used observations of $\sim20$ brightest cluster galaxies (BCGs) to propose that the number of globular clusters (GCs) in giant galaxies is directly proportional to the total mass $M_h$ of the host galaxy, which in turn is dominated by the dark matter (DM) halo. This suggestion was followed up with increasing amounts of observational evidence in several papers, including @mclaughlin1999 [@blakeslee1999; @spitler_forbes2009; @georgiev_etal2010; @kruijssen2015], @hudson_etal2014 (hereafter Paper I), @harris_etal2015 (hereafter Paper II), and @forbes_etal2016b, among others (see Paper II for a more complete review of the literature). This remarkable trend may be a strong signal that the formation of GCs, most of which happened in the redshift range $8 \gtrsim z \gtrsim 2$, was relatively resistant to the feedback processes that hampered field-star formation [e.g. @kravtsov_gnedin2005; @moore_etal2006; @li_etal2016]. If that is the case, then the total mass inside GCs at their time of formation may have been closely proportional to the original gas mass present in the dark-matter halo of their parent galaxy [Papers I, II, and @kruijssen2015], very unlike the total stellar mass $M_{\star}$.
@blakeslee_etal1997 introduced the *number* ratio $\eta_N \equiv N_{\mathrm{GC}}/M_h$ (where $N_{GC}$ is the total number of GCs in a galaxy) and showed that it was approximately constant for their sample of BCGs, at least when measured within a fixed physical radius. Following @spitler_forbes2009, @georgiev_etal2010, and our Papers I and II in this series, here we discuss the link between GC populations and galaxy total mass in terms of the dimensionless *mass* ratio $\eta_M \equiv {\,M_\mathrm{GCS}}/ M_h$, where ${\,M_\mathrm{GCS}}$ is the total mass of all the GCs combined and $M_h$ is the total galaxy mass including its dark halo plus baryonic components. In some sense, $\eta_M$ represents an absolute efficiency of GC formation, after accounting for subsequent dynamical evolution up to the present day [cf. @katz_ricotti2014; @kruijssen2015; @forbes_etal2016b]. Direct estimates of $\eta_M$ for several hundred galaxies indicate that this ratio is indeed nearly uniform (Papers I and II), far more so than the more well known ratio $M_{\star}/M_h$. A second-order dependence on galaxy type has been found in the sense that S/Irr galaxies have a $\sim 30$% lower $\eta_M$ than do E/S0 galaxies (Paper II).
So far, the case that $\eta_M \simeq const$ has been built on the observed GC populations in ‘normal’ galaxies covering the range from dwarfs to giants. However, opportunities have recently arisen to extend tests of its uniformity in two new directions. The first is by using the GC populations in entire clusters of galaxies, including their GCs in the Intracluster Medium (ICM). The second is from the newly discovered ultra-diffuse galaxies (UDGs) and the GCs within them. It is also worth noting that in both these cases, the masses of the systems concerned have been measured with different methods than by the gravitational-lensing approach used to build the calibration of $\eta_M$ in our Papers I and II, which makes the test of the hypothesis even more interesting.
The plan of this paper is as follows. In Section 2, we describe improved methodology for determining ${\,M_\mathrm{GCS}}$, followed by a recalibration of the mass ratio $\eta_M$. In Section 3 the discussion is extended to include four clusters of galaxies in which measurements now exist for both ${\,M_\mathrm{GCS}}$ and $M_h$, while in Section 4 the relation is extended in quite a different direction to selected UDGs. In Section 5, the key ratio $\eta_M$ is applied to several interesting sample cases with accompanying predictions for their GCS sizes. We finish with brief comments about the implications of the constancy of $\eta_M$ for understanding the conditions of formation for GCs.
Recalibration of the Mass Ratio: Method
=======================================
The total GCS mass within a galaxy or cluster of galaxies is calculated as $$M_{GCS} \, = \, \int (\frac{M}{L}) L \, n(L) \, dL
\label{eqn:mlv}$$ where $n(L)$ represents the number of GCs per unit luminosity $L$, and $M/L$ is their mass-to-light ratio. Here, we use $V-$band luminosities $L_V$ following most previous studies. The GC luminosity function (GCLF) is assumed to have a Gaussian distribution in number per unit magnitude. The parameters defining $n(L)$, namely the Gaussian turnover magnitude $\mu_0$ and intrinsic dispersion $\sigma$, are in turn weak functions of galaxy luminosity, as described in @villegas_etal2010 [@harris_etal2013; @harris_etal2014]. From the observations covering a large range of host galaxies, $\mu_0$ and $\sigma$ both exhibit shallow increases as galaxy luminosity increases.
In the present paper, unlike all previous discussions of this topic, we now assume that the mass-to-light ratio $M/L_V$ for individual GCs is also a function of GC mass, following empirical evidence from recent literature [e.g. @rejkuba_etal2007; @kruijssen2008; @kruijssen_mieske2009; @strader2011]. Of necessity, however, we assume that $M/L$ follows the same function of GC mass (or luminosity, which is the more easily observable quantity) within all galaxies.
In the Appendix below, we assemble recent observational data and define a convenient interpolation function for $M/L_V$ versus GC luminosity. Using Eq. \[eqn:mlv\], we can then readily derive a mean mass-to-light ratio averaged over all GCs in any given galaxy, defined as $\langle M/L_V \rangle(tot) = M_{GCS}/L_{GC}(tot)$. We can also define a mean GC mass as $\langle M_{GC} \rangle = M_{GCS} / N_{GC}$.[^1] For example, for a typical $L_{\star}$ galaxy at $M_V^T \simeq -21.4$ [the type of galaxy within which most GCs in the universe are found; see @harris2016], we obtain $\langle M/L_V \rangle \simeq 1.73$. The mean ranges from $\langle M/L_V \rangle \simeq 1.3$ for very small dwarfs ($M_V^T \sim -15$) up to $\langle M/L_V \rangle \simeq 2.1$ for very luminous supergiants ($M_V^T \sim -23$).
In Figure \[fig:mlav\] the trends of GC mean mass and global $\langle M/L_V \rangle$ are plotted versus galaxy luminosity $M_V^T$. Accurate numerical approximations to these results are given by the interpolation curves $${\rm log} \langle M_{GC} \rangle \, = \, 5.698 + 0.1294 M_V^T + 0.0054 (M_V^T)^2$$ and $${\rm log} \langle M/L_V \rangle \, = \, 1.041 + 0.117 M_V^T + 0.0037 (M_V^T)^2 \, .$$ The intrinsic galaxy-to-galaxy scatter in $\langle M_{GC} \rangle$ is $\pm 0.2$ dex, based on the observations from the Virgo and Fornax clusters [@villegas_etal2010].
We have used these calibrations to recalculate the values of ${\,M_\mathrm{GCS}}$ and $\eta_M$ in the complete sample of galaxies discussed in Papers I and II. In particular, for this paper we use these recalibrated values for the 175 ‘best’ galaxies of all types used in Paper II.[^2] In addition, to help tie down the high-mass end of the range of galaxies, we have updated the $N_{GC}$ values for five Brightest Cluster Galaxies (BCGs) including NGC 4874, 4889, 6166, UGC 9799, and UGC 10143 from the recent data of Harris et al. (2016, ApJ in press).
The results are shown in Figure \[fig:eta\]. First, the upper panel shows the *number* per unit mass $\eta_N$ versus $M_h$ (following the notation convention of @georgiev_etal2010). An unweighted least-squares fit to the E/S0 galaxies gives the simple linear relation $${\rm log} \eta_N \, = \, (-8.56 \pm 0.34) - (0.11 \pm 0.03) {\rm log} (M_{h}/M_{\odot}) \, .$$ Thus the total number of GCs in galaxies, observed at redshift 0, scales approximately as $N_{GC} \sim M_{h}^{0.9}$. The rms scatter around this relation is $\pm 0.26$ dex.
Second, the lower panel shows the trend for the *mass* ratio $\eta_M$. Because the *mean mass* of the GCs increases progressively with galaxy mass, the shallow decrease of $\eta_N$ with $M_h$ nearly cancels, leaving the mass ratio $\eta_M$ approximately constant over the entire run of galaxies. The weighted mean is $\langle \log\eta_M \rangle = -4.54$, or $\langle\eta_M\rangle = 2.9\times10^{-5}$, with a residual rms scatter $\pm 0.28$ dex.
{width="45.00000%"}
\[fig:mlav\]
In Fig. \[fig:eta\], the lower panel ($\eta_M$) and the upper panel ($\eta_N$) are obviously closely linked, but they are not entirely equivalent. As described in Paper I, $M_h$ is deduced from the $K-$band luminosity $M_K$ as calibrated through gravitational lensing. On the other hand, ${\,M_\mathrm{GCS}}$ is derived from $N_{GC}$ and the visual luminosity $M_V^T$ through Eq. (1).
Five points near $M_h \sim 10^{13} M_{\odot}$ sit anomalously high above the mean relations. These five are NGC 4636 and the BCGs NGC 3258, 3268, 3311, 5193. The cluster populations for these systems are well determined and very unlikely to be overestimated by the factor of $\sim 5$ that would be needed to bring them back to the mean lines. The alternate possibility is that their halo masses may have been underestimated, which in turn means that the $K-$band luminosities of these very large galaxies [from 2MASS; see @harris_etal2013] would have to be underestimated [see @schombert_smith2012; @scott_etal2013 for more extensive comments in this direction].
{width="52.00000%"}
\[fig:eta\]
Clusters of Galaxies
====================
Useful estimates of the total GC populations within entire clusters of galaxies have now been made for four rich clusters: Virgo, Coma, Abell 1689, and Abell 2744 (sources listed below). This material has in each case taken advantage of wide-field surveys or unusually deep sets of images taken with HST. The quoted $N_{GC}$ values include both the GCs associated directly with the individual member galaxies, and the intracluster globular clusters (IGCs) that are now known to be present in rich clusters. It is likely that the IGCs represent GCs stripped from many different systems during the extensive history of galaxy/galaxy merging and harassment that takes place within such environments [@peng_etal2011]. Simulations [e.g. @purcell_etal2007] show that $L_{\star}-$type galaxies contribute the most to the Intracluster light. For such galaxies the mean globular cluster mass is $\langle M_{GC} \rangle \simeq 2.6 \times 10^5 M_{\odot}$ (Fig. \[fig:mlav\]), and the GCLF mean and dispersion are $\mu_0(M_V) = -7.4$, $\sigma = 1.2$ mag [@harris_etal2013]. However, the total GC population in the *entire* cluster of galaxies consists of the IGCs plus the individual galaxies in roughly similar amounts. For the giant ellipticals and BCGs that dominate the individual galaxies, the GCLFs are broader with $\sigma \simeq 1.4$ and $\langle M_{GC} \rangle$ is higher. For the present purpose, we therefore adopt averages $\sigma \simeq 1.3$ mag and $\langle M_{GC} \rangle \simeq 2.8 \times 10^5 M_{\odot}$ for an entire galaxy cluster. Knowing $N_{GC}$, the total mass $M_{GCS}$ in all the GCs within the cluster can then be directly estimated, albeit with more uncertainty than for a single galaxy.
*Virgo:* @durrell_etal2014, from the Next Generation Virgo Cluster Survey, calculate that the entire cluster contains $(37700 \pm 7500)$ GCs brighter than $g_0 = 24.0$, which is 0.2 mag fainter than the GCLF turnover point. Adopting the fiducial value $\sigma = 1.3$ mag as explained above, we then obtain $N_{GC} = (67000 \pm 13000)$, which then gives ${\,M_\mathrm{GCS}}= (1.88 \pm 0.37) \times 10^{10} M_{\odot}$. Durrell et al. also quote a total Virgo mass $5.5 \times 10^{14} M_{\odot}$, leading to $\eta_M = (3.4 \pm 0.7) \times 10^{-5}$. This $\eta_M$ estimate is 17% higher than the value quoted by @durrell_etal2014, but the increase results entirely from the difference in assumed ${\langle M_\mathrm{GC}\rangle}$. Just as for the Coma survey data (discussed below), in this case the limiting magnitude of the survey is close to the GCLF turnover (peak) magnitude, so that approximately half the total GC population is directly observed, and the estimated $N_{GC}$ count is virtually independent of the assumed GCLF dispersion $\sigma$.
*Coma:* @peng_etal2011 find that within a projected radius of $r \simeq 520$ kpc, the cluster contains $71000\pm 5000$ GCs (here we combine their estimates of internal and systematic uncertainties, and renormalize the total to $\sigma = 1.3$ mag instead of their value of 1.37 mag), and thus ${\,M_\mathrm{GCS}}= (1.99 \pm 0.14) \times 10^{10} M_{\odot}$. This survey radius is well within the Coma virial radius, which is $2.5 - 3.0 $ Mpc [e.g. @hughes1989; @colless_dunn1996; @rines_etal2003; @lokas_mamon2003; @kubo_etal2007; @gavazzi_etal2009; @okabe_etal2014; @falco_etal2014]. From these sources, which use methods including X-ray light, galaxy dynamics, weak lensing, and galaxy sheets to derive the Coma mass profile, the mass within the GC survey radius of 520 kpc is $M \simeq (4 \pm 1) \times 10^{14} M_{\odot}$. The resulting mass ratio *within this radius* is then $\eta_M = (4.98 \pm 1.25) \times 10^{-5}$.
*A1689:* @alamo-martinez_etal2013 used unusually deep HST imaging of the core region of this massive cluster, which is at redshift $z = 0.183$, to find a large and extended population of GCs. In contrast to Virgo and Coma, the limiting magnitude of their photometry was 2.3 mag brighter than the expected GCLF turnover and thus only the brightest 4.0% of the GC population was measurable. From an analysis of the known statistical and systematic uncertainties, they deduced $N_\mathrm{GC} = 162{,}850 \,(+75{,}450, -51{,}300)$ GCs within a projected 400-kpc radius of the central cD-type galaxy after using a GCLF-extrapolated fit to the observed number of GCs brighter than the photometric limit. However, that number assumed $\sigma = 1.4$ mag; converting to our fiducial $\sigma = 1.3$ mag, the total becomes $N_\mathrm{GC} = 209{,}600$ ($+$97,100, $-$66,000) within $r<400$ kpc. From gravitational lensing, the total mass within the same radius is $M_h = 6.4 \times 10^{14} M_{\odot}$. The resulting mass ratio is then $\eta_M(r < 400 \mathrm{kpc}) = 9.17 (+4.25, -2.90) \times 10^{-5}$. We can attempt to define something closer to a global value out to larger radius by extrapolating the best-fit $r^{1/4}$-law derived to the full GC distribution in A1689 (their fit for the case without masking the regions around the galaxies), which suggests that the estimated number of GCs out to 1 Mpc is larger by a factor of 1.48. From the multi-probe mass profile of @umetsu_etal2015 [using $h=0.7$], the total mass within the same radius is $M_h = 1.3 \times 10^{15} M_{\odot}$. We therefore estimate, for the whole cluster, $\eta_M = 6.7 (+3.1, -2.1) \times 10^{-5}$ within $r<1$ Mpc. The total mass of A1689 out to the virial radius exceeds $2\times 10^{15} M_{\odot}$, and thus the global value of $\eta_M$ might be still lower. In any case, given the uncertainties, the value of $\eta_M$ in A1689 appears consistent with the standard range calculated above.
*A2744:* @lee_jang2016b have used HST imaging from the HST Frontier Field to find the brightest GCs and UCDs in this very rich cluster, which is at redshift $z = 0.308$. Intracluster Light (ICL) is also detectably present at least in the inner part of the cluster [@montes_trujillo2014]. From M. G. Lee (private communication), the raw number of observed GCs brighter than $F814W = 29.0$, after photometric completeness correction, field subtraction and removal of UCDs, is $(664 \pm 41)$ (Poisson uncertainty). This limiting magnitude is $\simeq 3.8$ mag brighter than the GCLF turnover point, or 2.93$\sigma$ short of the turnover. Here, we have adopted a $0.2-$mag brightening of the GCLF turnover luminosity to account for passive evolution for its redshift of $z = 0.3$; note that @alamo-martinez_etal2013 determined a net brightening in $F814W$ of 0.12 mag in A1689 ($z = 0.18$), and for small $z$ the correction scales nearly linearly. Extrapolating from the observed total, then $N_{GC} \simeq 390,000$.
Given that only $\sim\,$0.2% of the inferred total was actually detected, the dominant sources of error in this case are not the internal count statistics, but instead the uncertainties in the adopted GCLF turnover and dispersion, as well as the basic assumption that the GCLF follows a strictly Gaussian shape all the way to luminosities far above the turnover magnitude. These uncertainties are unimportant for situations like Virgo and Coma where the raw observations reach to limiting magnitudes approaching the turnover magnitude $\mu_0$, but they become critically important where only the bright tip of the GCLF is observed. The turnover $\mu_0$ exhibits an intrinsic galaxy-to-galaxy scatter at the level of $\pm 0.2$ mag, while the dispersion $\sigma$ has an intrinsic scatter of $\pm 0.05-0.1$ mag [@jordan_etal2006; @villegas_etal2010; @rejkuba2012]. For the baseline values of $\sigma(\mathrm{GCLF}) = 1.3$ mag and a limiting magnitude 2.93$\sigma$ brighter than the turnover, the fraction $N(obs)/N(tot)$ equals 0.0017 $(+0.0010,-0.00066)$ due to a $\pm0.2-$mag uncertainty in the turnover luminosity, and an additional $(+0.00166,-0.00094)$ due to a $\pm0.1-$mag uncertainty in $\sigma$. Treating the uncertainties as independent, we obtain $N_{GC} = 3.9 (+7.2,-2.1) \times 10^5$. If we further assume the GC counts extend out to 600 kpc (the limit that can be probed within the ACS field of view at this redshift) and follow a similar profile to that of A1689, then the correction to a radius of 1 Mpc is a factor of 1.21, [**yielding $N_{GC} = 4.7 (+8.6,-2.5) \times 10^5$.** ]{}
The best-estimate mass of A2744 is also difficult to pin down, since the cluster has complex internal dynamics with two major subclusterings at quite different mean velocities, indicative of merging in progress [@boschin_etal2006; @owers_etal2011; @jauzac_etal2016]. For the dominant central “a” subcluster @boschin_etal2006 determine $M_{vir}(<2.4 Mpc) = 2.2 (+0.7,-0.6) \times 10^{15} M_{\odot}$. With this value for the mass, we conclude $\eta_M = 6.0 (+11,-3.2) \times 10^{-5}$ for A2744.
The underlying assumption about the Gaussian GCLF shape is harder to quantify. The strongest empirical test available is the measurement of GCLFs in seven BCG galaxies from @harris_etal2014, which extend from the turnover point to an upper limit almost 5 magnitudes brighter (equivalent to almost $4 \sigma$), thanks to the extremely large numbers of GCs per galaxy. The results indicate that the Gaussian assumption fits the data remarkably well to that level. However, these tests all apply to *single* galaxies, whereas the IGCs are a composite population of GCs stripped from galaxies of all types. This composite GCLF will in general not have a simple Gaussian shape even if all the progenitor galaxies had individually normal GCLFs [see @gebhardt_beers1991]. Unfortunately, no direct measurements of the GCLF for a pure IGC population are yet available. For the present time, we simply adopt the parameters for galaxy clusters as described above and recognize that further observations and analysis could change the results for A2744 quite significantly. While uncertainties are also sizable for A1689, the fraction of the GC population directly observed there is 20 to 40 times greater than in A2744; thus, the results for A1689 are robust by comparison.
It is evident from the preceding discussion that estimates of $\eta_M$ on the scale of entire clusters of galaxies put us in a much more challenging regime of uncertainties than is the case for single galaxies. In particular, better results will be possible if $\eta_M$ as a function of $r$ can be more accurately established. Ideally the global value should be estimated out to the virial radius, but so far this is the case only for Virgo, the cluster for which the estimated $\eta_M$ is closest to the mean for individual galaxies. Nevertheless, for all four galaxy clusters, the weighted average result is $\eta_M = (3.9 \pm 0.6) \times 10^{-5}$, slightly larger than the mean value of $2.9 \times 10^{-5}$ characterizing the individual galaxies. It is difficult to say at this point whether or not the mean difference is significant. One possible evolutionary difference between the GCs within galaxies and the IGCs is that the IGCs are subjected to much lower rates of dynamical destruction than the GCs deep within the potential wells of individual galaxies, and so may have kept a higher fraction of their initial mass.
Ultra-Diffuse Galaxies
======================
UDGs have recently been found within the Virgo, Fornax, and Coma clusters in large numbers [e.g. @mihos_etal2015; @vandokkum_etal2015; @koda_etal2015; @munoz_etal2015]. Deep imaging has revealed GC populations around two of the Coma UDGs, Dragonfly 44 [@vandokkum_etal2016] and Dragonfly 17 [@peng_lim2016], as well as VCC1287 in Virgo [@beasley_etal2016]. All three of these systems have anomalously high specific frequencies $S_N \sim N_{GC}/L_V$, but the interest for this discussion is their *mass* ratio. @beasley_etal2016 and @vandokkum_etal2016 have already commented that $\eta_M$ for the two UDGs they studied falls close to the standard level applicable for more normal galaxies.
For the Virgo dwarf VCC1287, @beasley_etal2016 find a GC population $N_{GC} = 22 \pm 8$, which would translate to ${\,M_\mathrm{GCS}}= (2.2 \pm 0.8) \times 10^6 M_{\odot}$ if we adopt a mean GC mass of $1.0 \times 10^5 M_{\odot}$ appropriate for a moderate dwarf galaxy. Fortunately, $\langle M_{GC} \rangle$ does not vary strongly across the dwarf luminosity range (see Fig. \[fig:mlav\]). From a combination of the velocity dispersion of the GCs themselves, and comparison with EAGLE simulations, they estimate a virial mass for the galaxy of $M_{200} = (8 \pm 4) \times 10^{10} M_{\odot}$, giving an approximate mass ratio $\eta_M = (2.75 \pm 1.70) \times 10^{-5}$.
For Dragonfly 44, @vandokkum_etal2016 find a relatively rich GC population $N_{GC} \simeq 94 \pm 25$ and, again, a specific frequency $S_N$ higher by an order of magnitude than normal galaxies with similar $L_V$. With $\langle M_{GC} \rangle = 1.1 \times 10^5 M_{\odot}$ for the galaxy luminosity $M_V^T = -16.1$, then ${\,M_\mathrm{GCS}}= (1.0 \pm 0.3) \times 10^7 M_{\odot}$. Their measurement of the velocity dispersion of the stellar light gives $M(<r_{1/2}) = 7.1 \times 10^9 M_{\odot}$. The total (virial) mass $M_h$ is likely to be at least an order of magnitude higher, but scaling from the comparable results for VCC1287 above [see @beasley_etal2016], we obtain a *very* rough estimate $M_h \sim 2 \times 10^{11} M_{\odot}$. In turn, the final result is $\eta_M \sim 5.5 \times 10^{-5}$, again very uncertain.
Discussion and Conclusions
==========================
From Fig. \[fig:eta\], we find that the near-constancy of the mass ratio $\eta_M$ over 5 orders of magnitude in $M_h$ remains valid. The addition of two quite different environments to the mix, UDGs and entire galaxy clusters, has not changed the essential result. It is worth noting that the $M_h$ estimates for the galaxy clusters and UDGs relied on internal dynamics rather than the weak-lensing calibration used for the normal galaxies (except A1689, for which the mass is based on a combination of weak and strong lensing). The new calibration of the mean mass ratio $\eta_M = (2.9 \pm 0.2) \times 10^{-5}$ is significantly lower than in previous papers (see Paper II for comparisons); the difference is due mainly to our lower adopted GC mass-to-light ratio and its nonlinear dependence on GC mass. An additional residual outcome is that the slight nonlinearity in the trend of $\eta_M$ (see Papers I and II) is now reduced.
Galaxy Mass Estimation
----------------------
In a direct practical sense, $\eta_M$ is usable as a handy way to estimate the total mass of a galaxy (dark plus baryonic) to better than a factor of 2. This direction is the emphasis taken notably by @spitler_forbes2009 [@vandokkum_etal2016; @peng_lim2016] and @beasley_etal2016 and was also used in Paper I to apply to M31 and the Milky Way. As these authors emphasize, it remains surprising that a method apparently unconnected with internal satellite dynamics, lensing, or other fairly direct measures of a galaxy’s gravitational field can yield such an accurate result. The prescription for determining $M_{h}$ is:
1. Count the total GC population, $N_{GC}$. Given the uncertainties discussed in Section 3 above and in @harris_etal2013, it is important to have raw photometric data that reach or approach the GCLF turnover point, which helps avoid uncomfortably large extrapolations from observations that resolve only the bright tip of the GCLF.
2. Use the galaxy luminosity $M_V^T$ to define the appropriate mean GC mass $\langle M_{GC} \rangle$ (from Fig. \[fig:mlav\] and Eq. 2). The total mass in the galaxy’s GC system follows as ${\,M_\mathrm{GCS}}= N_{GC} \langle M_{GC} \rangle$.
3. Divide ${\,M_\mathrm{GCS}}$ by $\eta_M$ to obtain $M_{h}$.
More generally, this method for estimating galaxy mass is workable only for systems near enough that the GC population can be resolved and counted. The practical ‘reach’ of the procedure is $d \lesssim 200 - 300$ Mpc through deep HST imaging [e.g. @harris_etal2014], though with the use of SBF techniques, the effective limit may be extended somewhat further; cf. @blakeslee_etal1997 [@blakeslee1999; @marin-franch_aparicio2002].
Sample Cases
------------
We provide some sample specific examples of the procedure:
*Milky Way:* @harris1996 (2010 edition) lists 157 GCs in the Milky Way, but many of these are extremely faint or sparse objects that would be undetectable in almost all other galaxies. To treat this system in the same way as other galaxies, we note that the GCLF turnover is at $M_V \simeq -7.3$ and that there are 72 clusters brighter than that point. Doubling this, we then adopt $N_{GC} = 144$ in the sense that it is defined here. At $M_V^T(MW) \simeq -21.0$ [@licquia_etal2015; @bland-hawthorn_gerhard2016], close to an $L_{\star}$ galaxy, the mean GC mass is then $\langle M_{GC} \rangle = 2.3 \times 10^5 M_{\odot}$, from which ${\,M_\mathrm{GCS}}= 3.3 \times 10^7 M_{\odot}$. Finally then, $M_{h} \simeq 1.1 \times 10^{12} M_{\odot}$, which is well within the mix of estimates obtained from a wide variety of methods involving satellite dynamics and Local Group timing arguments [see, e.g. @wang_etal2015; @eadie_harris2016 for compilations and comparisons].
*Dragonfly 17:* For this UDG in Coma, @peng_lim2016 estimate a total population $N_{GC} = 28 \pm 14$ based on a well sampled GCLF and radial distribution. For $M_V^T = -15.1$, we have $\langle M_{GC} \rangle = 0.94 \times 10^5 M_{\odot}$ and then predict a total mass $M_{h} = (9.1 \pm 4.5) \times 10^{10} M_{\odot}$ for this galaxy. @peng_lim2016 predicted $M_{h} = (9.3 \pm 4.7) \times 10^{10} M_{\odot}$, although the agreement with our estimate is something of a coincidence: they used a higher value of $\eta$ from an older calibration, but also a higher mean GC mass, and these two differences cancelled out.
*M87:* This centrally dominant Virgo giant is the classic “high specific frequency” elliptical. From the GCS catalog [@harris_etal2013], $N_{GC} = 13000 \pm 800$ and the luminosity is $M_V^T = -22.61$, giving a mean cluster mass $\langle M_{GC}\rangle = 3.4 \times 10^5 M_{\odot}$. Thus ${\,M_\mathrm{GCS}}= (4.4 \pm 0.3) \times 10^9 M_{\odot}$, leading to the prediction $M_{h} = 1.5 \times 10^{14} M_{\odot}$. For comparison, @oldham_auger2016 give $M_{vir} = 7.4 (+12.1, -4.1) \times 10^{13}M_{\odot}$ from the kinematics of the GC system and satellites. A plausible upper limit from @mclaughlin1999b is $M_{DM}(r_{200}) = (4.2 \pm 0.5) \times 10^{14} M_{\odot}$, obtained from the GC dynamics and the X-ray gas in the Virgo potential well, while @durrell_etal2014 quote $5.5 \times 10^{14} M_{\odot}$ for the entire Virgo cluster. For BCGs it is difficult to isolate the galaxy’s dark-matter halo from that of the entire galaxy cluster, but given that M87 holds $\simeq 20$% of the Virgo GC population, these estimates of $M_{h}$ seem mutually consistent.
*Fornax Cluster:* Fornax is the next nearest rich cluster of galaxies after Virgo, with the cD giant NGC 1399 near its center. GC populations in the individual galaxies have been studied as part of the HST/ACS Fornax Cluster Survey [@villegas_etal2010; @jordan_etal2015], and the NGC 1399 system particularly has been well studied photometrically and spectroscopically [e.g. @dirsch_etal2003; @schuberth_etal2010]. The Fornax cluster as a whole is not as rich as Virgo or the others discussed here, but an ICM is present at least in the form of substructured X-ray gas [@paolillo_etal2002]. Global mass measurements from galaxy dynamics in and around Fornax [@drinkwater_etal2001; @nasonova_etal2011] find $M$($<$1.4 Mpc)$ = (7 \pm 2) \times 10^{13} M_{\odot}$, rising to perhaps as high as $3 \times 10^{14} M_{\odot}$ within 3 Mpc. For NGC 1399 itself, @schuberth_etal2010 find $M_{vir}$ approaching $10^{13} M_{\odot}$. If we use the mean value $\eta_M = 3.9 \times 10^{-5}$ determined above for the other galaxy clusters, and we also use $M_h = 7 \times 10^{13} M_{\odot}$ within a radius of 1.4 Mpc, then we predict that the entire Fornax cluster should contain at least $N_{GC} \sim 10,000$ clusters. NGC 1399 alone has $N_{GC} = (6450 \pm 700)$ [@dirsch_etal2003] and the other smaller galaxies together are likely to contribute at least 6000 more [@jordan_etal2015]. Thus the prediction and the actual known total agree to within the internal scatter of $\eta_M$, which would indicate that the GC population within Fornax is already accounted for and that few IGCs remain to be found. It will be intriguing to see if wide-field surveys [see, e.g. @dabrusco_etal2016] will reveal a significant IGC population.
*Fornax dSph:* The dwarf spheroidal satellite of the Milky Way, Fornax, is an especially interesting case. It has 5 GCs of its own and a total luminosity of just $M_V^T = -13.4$, placing it among the very faintest galaxies known to contain GCs. Since it lacks a measured $K-$band luminosity it does not appear in the GCS catalog with a lensing-calibrated $M_h$. The existence of this handful of GCs presents something of an interpretive puzzle for understanding their survival over a Hubble time, with several discussions of dynamical modelling in the recent literature [e.g. @cole_etal2012; @strigari_etal2006; @penarrubia_etal2008; @martinez2015]. The 5 Fornax clusters add up to a total mass ${\,M_\mathrm{GCS}}= 3.8 \times 10^5 M_{\odot}$ assuming $(M/L_V) = 1.3$ [@mackey_gilmore2003]. The predicted halo mass of the galaxy should then be $M_h \simeq 1.3 \times 10^{10} M_{\odot}$ if we adopt our normal $\eta_M$, which would give Fornax a global mass-to-light ratio $M_h/L_V(tot) \gtrsim 600$ (assuming that it has kept its entire halo to the present day against tidal stripping from the Milky Way). By contrast, dynamical modelling of Fornax with various assumed dark-matter profiles tends to infer virial masses near $M_h \sim 10^9 M_{\odot}$ [@penarrubia_etal2008; @cole_etal2012; @angus_diaferio2009]. In that case, the derived mass ratio would then be $\eta_M \simeq 3.8 \times 10^{-4}$, an order of magnitude larger than our standard value. Fornax is plotted in Fig. \[fig:eta\] with this higher value; although it is not located outrageously far from the scatter of points defined by the larger galaxies, it does leave a continuing problem for interpretation. With $M_h = 10^9 M_{\odot}$ and our baseline value $\eta_M \sim 3 \times 10^{-5}$, the normal prediction would be that Fornax should contain only one GC.
The Fornax dSph case hints that the argument for a constant $\eta$ across the range of galaxies may begin to break down at the very lowest luminosities. For a standard $\eta_M \simeq 3 \times 10^{-5}$, the boundary below which dwarf galaxies should be too small to have any remaining GCs is near $M_h \lesssim 3 \times 10^9 M_{\odot}$. In this very low-mass regime, other physical factors should also come into play determining the number of surviving GCs within the galaxy, particularly massive gas loss during the earliest star-forming period in such tiny potential wells.
Cluster Formation Conditions
----------------------------
In the longer term, we suggest that the more important implications for the near-constancy of $\eta_M$ are for helping understand the formation conditions of the dense, massive star clusters that evolved into the present-day GCs (see Papers I and II). These clusters should have formed within very massive host Giant Molecular Clouds (GMCs) under conditions of unusually high pressure and turbulence, analogous to the Young Massive Clusters (YMCs) seen today in starburst dwarfs, galactic nuclei, and merging systems [@harris_pudritz1994; @elmegreen2012; @kruijssen2015].
Under such conditions, star formation within these dense protoclusters will be shielded from *external* feedback such as active galactic nuclei, UV and stellar winds from more dilute field star formation, and cosmic reionization [e.g. @kravtsov_gnedin2005; @li_gnedin2014; @howard_etal2016]. In strong contrast, these forms of feedback were much more damaging to the majority of star formation happening in lower-density, lower-pressure local environments. Though such a picture needs more quantitative modelling, it is consistent with the idea (Paper II) that ${\,M_\mathrm{GCS}}$ *at high redshift* may be nearly proportional to the total initial gas mass – at least, much more so than the total stellar mass $M_{\star}$.
An alternate route [@kruijssen2015] is that the empirical result $\eta_M \sim const$ can be viewed in some sense as a coincidence if it is written as $$\eta_M \, = \, \frac{{\,M_\mathrm{GCS}}}{M_h} \, = \, \frac{{\,M_\mathrm{GCS}}}{M_{\star}} \cdot \frac{M_{\star}}{M_h} \, .$$ The two ratios on the right-hand side show well known opposite trends with $M_h$ that happen to cancel out when multiplied together. The stellar-to-halo mass ratio (SHMR) ($M_{\star}/M_h$) reaches a peak efficiency near $M_h \simeq 10^{12} M_{\odot}$ and falls off to both higher and lower mass by more than an order of magnitude [@leauthaud_etal2012; @moster_etal2013; @behroozi_etal2010; @behroozi_etal2013; @hudson_etal2015]. By contrast, the GCS number per unit $M_{\star}$ (that is, the specific frequency $S_N$ or its mass-weighted version $T_N$) reaches a minimum near $M_h \sim 10^{12} M_{\odot}$ and rises by an order of magnitude on both sides. Both these trends are extremely nonlinear, and we suggest that it is difficult to see how their mutual cancellation can be so exact over such a wide range of galaxy mass if it is only a coincidence.
The common factor between the two mass ratios ($T_N$, SHMR) is the galaxy stellar mass $M_{\star}$. As in Paper II, we suggest that the result can be seen as the outcome of a *single* physical process (the role of galaxy-scale feedback on $M_{\star}$). If instead we essentially ignore $M_{\star}$, then ${\,M_\mathrm{GCS}}$ is closer to representing the total initial gass mass, and thus $M_h$. We recognize, however, that this argument is as yet unsatisfactorily descriptive and will require full-scale numerical simulations that track GC formation within hierarchical galaxy assembly over the full range of redshifts $8 \gtrsim z \gtrsim 1$ [@kravtsov_gnedin2005; @li_etal2016; @griffen_etal2010; @tonini2013].
Conclusions
-----------
In this paper we have revisited the empirical relation between the total mass ${\,M_\mathrm{GCS}}$ of the globular clusters in a galaxy, and that galaxy’s total mass $M_h$. Our findings are the following:
1. A recalibration of the mass ratio $\eta_M \equiv ({\,M_\mathrm{GCS}}/ M_h)$, now including the systematic trend of GC mass-to-light ratio with GC mass, yields $\langle \eta_M \rangle = 2.9 \times 10^{-5}$ with a $\pm 0.28-$dex scatter between individual galaxies.
2. Evaluation of $\eta_M$ for four *clusters of galaxies* (Virgo, Coma, A1689, A2744) including all GCs in both the cluster galaxies and the IGM, shows that very much the same mass ratio applies for entire clusters as for individual galaxies. For these four clusters $\langle \eta_M \rangle = (3.9 \pm 0.6) \times 10^{-5}$.
3. Two of the recently discovered Ultra-Diffuse Galaxies in Virgo and Coma can also now be included in the relation. Within the (large) measurement uncertainties, both such galaxies fall within the normal value of $\eta_M$ at the low-mass end of the galaxy range. By contrast, the Fornax dSph in the Local Group may be a genuine extreme outlier, containing perhaps 5 times more clusters than expected.
4. The near-constant mass ratio between GC systems and their host galaxy masses is strikingly different from the highly nonlinear behavior of total stellar mass $M_{\star}$ versus $M_h$. We favor the interpretation that GC formation – in essence, star formation under conditions of extremely dense gas in proto-GCs embedded in turn within giant molecular clouds – was nearly immune to the violent external feedback that hampered most field star formation.
Acknowledgements {#acknowledgements .unnumbered}
================
WEH acknowledges financial support from NSERC (Natural Sciences and Engineering Research Council of Canada). We are grateful to Myung Gyoon Lee for helpfully transmitting their observed numbers of globular clusters in Abell 2744. JPB thanks K. Alamo-Martínez for helpful discussions about Abell 1689.
Mass-to-Light Ratios for Globular Clusters
==========================================
A key ingredient in the calculation of ${\,M_\mathrm{GCS}}$, the total mass in the globular cluster system of a given galaxy, is the assumed mass-to-light ratio $M/L_V$ for GCs. Along with many other GC studies in recent years, we simply used a constant $M/L_V = 2$ in Papers I and II. However, much recent data and modelling for internal dynamical studies of GCs supports (a) a lower mean value, (b) a significant cluster-to-cluster scatter probably because of differing dynamical histories, and (c) a systematic trend for $M/L_V$ to increase with GC mass (or luminosity) particularly for masses above $10^6 M_{\odot}$.
Figure \[fig:ml3\] shows estimates of $M/L_V$ for Milky Way GCs, determined from measurements of GC internal velocity dispersion combined with dynamical modelling. Data from two widely used previous compilations [@mandushev_etal1991; @mclaughlin_vandermarel2005] are shown in the upper two panels of the Figure. Since then, dynamical studies have been carried out on numerous individual GCs based on both radial-velocity and proper-motion data, that are built on larger and more precise samples than in the earlier eras. Results from these post-2005 studies are listed in Table \[tab:ml\] and plotted in the bottom panel of Fig. \[fig:ml3\]. (Note that many clusters appear more than once because each individual study is plotted. However, the values listed in Table \[tab:ml\] present the weighted mean $M/L_V$, in Solar units, for each cluster.) The cluster luminosities $M_V^T$ are from the catalog of @harris1996 (2010 edition). For the clusters listed in Table \[tab:ml\], the weighted mean value for 36 GCs excluding NGC 5139 and NGC 6535 is $\langle M/L_V \rangle = (1.3 \pm 0.08) M_{\odot}/L_{V{\odot}}$ with a cluster-to-cluster rms scatter of $\pm 0.45$. For comparison, the values in the upper panel of Fig. \[fig:ml3\] have a mean $\langle M/L_V \rangle = (1.49 \pm 0.11) M_{\odot}/L_{V{\odot}}$ with an rms scatter of $\pm 0.58$, while in the middle panel the mean is $\langle M/L_V \rangle = (1.53 \pm 0.11) M_{\odot}/L_{V{\odot}}$ with an rms scatter of $\pm 0.64$.
[llccl]{}\
& & & &\
\
NGC104 & 1.32& (0.03,0.03) & $-9.42$ & 6,15\
NGC288 & 1.53& (0.17,0.17) &$-6.75$ & 6,10,15\
NGC362 & 1.10& (0.10,0.10) &$-8.43$ & 6,15\
NGC2419 & 1.55& (0.10,0.10) &$-9.42$ & 1,15\
NGC2808 & 2.24& (0.19,0.19) &$-9.39$ & 6,9\
NGC3201 & 1.91& (0.17,0.17) &$-7.45$ & 15\
NGC4147 & 1.47& (0.54,0.54) &$-6.17$ & 6\
NGC4590 & 1.40& (0.43,0.43) &$-7.38$ & 6\
NGC5024 & 1.38& (0.16,0.16) &$-8.72$ & 6,10\
NGC5053 & 1.30& (0.26,0.26) &$-6.76$ & 6\
NGC5139 & 2.45& (0.04,0.04) &$-10.26$ & 12,13,15\
NGC5272 & 1.32& (0.14,0.14) &$-8.88$ & 4,6\
NGC5466 & 0.72& (0.23,0.23) &$-6.98$ & 6\
NGC5904 & 1.36& (0.21,0.21) &$-8.81$ & 6\
NGC6121 & 1.32& (0.14,0.14) &$-7.19$ & 6,15\
NGC6205 & 2.10& (0.27,0.17) &$-8.55$ & 4\
NGC6218 & 1.12& (0.10,0.10) &$-7.31$ & 6,10,15\
NGC6254 & 1.61& (0.19,0.19) &$-7.48$ & 15\
NGC6341 & 1.69& (0.07,0.07) &$-8.21$ & 4,6,15\
NGC6388 & 1.45& (0.13,0.13) &$-9.41$ & 8,14\
NGC6397 & 1.9 & (0.10,0.10) &$-6.64$ & 5\
NGC6402 & 1.82& (0.35,0.35) &$-9.10$ & 6\
NGC6440 & 1.23& (0.27,0.27) &$-8.75$ & 14\
NGC6441 & 1.07& (0.19,0.19) &$-9.63$ & 6,14\
NGC6528 & 1.12& (0.39,0.42) &$-6.57$ & 14\
NGC6535 &11.06& (2.68,2.12) &$-4.75$ & 14\
NGC6553 & 1.02& (0.31,0.36) &$-7.77$ & 14\
NGC6656 & 1.30& (0.13,0.13) &$-8.51$ & 6,15\
NGC6715 & 1.52& (0.45,0.45) &$-9.98$ & 6\
NGC6752 & 2.65& (0.19,0.19) &$-7.73$ & 6,10\
NGC6809 & 0.81& (0.05,0.05) &$-7.57$ & 3,6,10,15\
NGC6838 & 1.36& (0.39,0.39) &$-5.61$ & 6\
NGC6934 & 1.52& (0.49,0.49) &$-7.45$ & 6\
NGC7078 & 1.14& (0.05,0.05) &$-9.19$ & 6,11,15\
NGC7089 & 1.66& (0.38,0.38) &$-9.03$ & 6\
NGC7099 & 1.84& (0.19,0.19) &$-7.45$ & 6,10\
Pal 5 & 1.60& (0.85,0.59) &$-5.17$ & 7\
Pal 13 & 2.4 & (5.0,2.4 ) &$-3.76$ & 2\
\
*Sources:* (1) @bellazzini_etal2012; (2) @bradford_etal2011; (3) @diakogiannis_etal2014; (4) @kamann_etal2014; (5) @kamann_etal2016; (6) @kimmig_etal2015; (7) @kupper_etal2015; (8) @lutzgendorf_etal2011; (9) @lutzgendorf_etal2012; (10) @sollima_etal2012; (11) @vandenbosch_etal2006; (12) @vandeven_etal2006; (13) @watkins_etal2013; (14) @zaritsky_etal2014; (15) @zocchi_etal2012.
![Mass-to-light ratios for Milky Way globular clusters plotted versus GC luminosity, from three different observational eras. Top panel: @mandushev_etal1991. Middle panel: @mclaughlin_vandermarel2005. Bottom panel: Dynamically based measurements from a variety of individual studies (see text). The low-luminosity clusters NGC 6535 and Pal 13 do not appear on the plot; see Table \[tab:ml\].](ml3.pdf){width="50.00000%"}
\[fig:ml3\]
There is also now much new observational material for GC mass measurements in other nearby galaxies. For other galaxies, the spatial structures of the GCs are unresolved or only partially resolved, and the measurements most often consist of a luminosity-weighted average of the internal velocity dispersion of each cluster, converted to mass via some appropriate form of the virial theorem or mass profile model. Data from several galaxies are displayed in Figure \[fig:mltot\], including M31 [@meylan_etal2001; @strader2011], M33 [@larsen_etal2002], NGC 5128 [@martini_ho2004; @rejkuba_etal2007; @taylor_etal2010], and M87 [@hasegan_etal2005].
The $M/L_V$ results from these different studies occupy similar ranges as in the Milky Way, but perhaps not surprisingly the scatter is much larger for these fainter targets. Some puzzling issues remain, for example in NGC 5128 for which the @taylor_etal2010 values are roughly 50% larger than those from @rejkuba_etal2007 and @martini_ho2004 for 14 objects in common, though again with considerable scatter. The source of this discrepancy is unclear. Possible trends of $M/L$ with GC metallicity as deduced from the M31 sample are discussed by @strader2011 [@shanahan_gieles2015; @zonoozi_etal2016] and are also not yet clear.
By contrast, there is general agreement that $M/L$ should increase systematically with GC mass [@kruijssen2008; @kruijssen_mieske2009; @rejkuba_etal2007; @strader2011], since the high-mass clusters have relaxation times large enough that the preferential loss of low-mass stars has not yet taken place to the same degree as for lower-mass clusters. The mean $M/L_V$ is expected to increase from $\sim 1 - 2$ at $M < 10^6 M_{\odot}$ progressively up to the level $\simeq 5 - 6$ for the mass range $10^7 - 10^8 M_{\odot}$ characterizing UCDs (Ultra-Compact Dwarfs) and dwarf E galaxies [e.g. @baumgardt_mieske2008; @mieske_etal2008]. For convenience of later calculation, we define a simple interpolation curve for $M/L$ as a function of cluster luminosity, $$\frac{M}{L_V} \, = \, 1.3 + \frac{4.5}{1 + e^{2.0 (M_V^T + 10.7)}} \, .$$ This function gives a roughly linear increase of $M/L$ from $M_V^T \simeq -10$ up to $-12$, saturating at the level of $\sim 5 - 6$ appropriate for UCDs and dE’s. The ‘baseline’ at $M/L_V = 1.3$ is chosen to match the observed data for Milky Way clusters. At very low luminosity, the mass-to-light ratio should increase again because low-mass and highly dynamically evolved clusters should become relatively more dominated by binary stars and stellar remnants (cf. the datapoint for Palomar 13 as an example). However, clusters at the low-mass end are also relatively few in number, and contribute little mass per cluster to the system in any case, so the particular $M/L$ value adopted for them has negligible effects on the total mass of the system ${\,M_\mathrm{GCS}}$.
![Dynamically measured mass-to-light ratios $M/L_V$ in Solar units, for globular clusters in several galaxies. *Solid dots:* Milky Way; *blue triangles:* M31 GCs with \[Fe/H\] $< -1.0$; *open green circles:* NGC 5128 GCs; *magenta dots:* M33 GCs; and *solid squares:* M87 GCs and M31-G1. Literature sources are given in the text. Larger symbol sizes correspond to smaller internal measurement uncertainties. The solid line at $M/L_V = 1.3$ is the weighted mean value for the Milky Way GCs excluding NGC 5139 ($\omega$ Cen) and NGC 6535, while the dashed line is the interpolation function stated in the text.](mltot.pdf){width="70.00000%"}
\[fig:mltot\]
chicagotrue
\[lastpage\]
[^1]: By convention, to maintain a consistent calculation procedure, the GCLF is assumed to be Gaussian and $N_{GC}$ is defined for our purposes as twice the number of GCs brighter than the GCLF turnover point; see @harris_etal2013.
[^2]: This ‘best’ sample consists of all the galaxies for which the raw GC photometry was of high enough quality to separate the conventional red and blue GC subpopulations. In most cases this criterion also means that the limiting magnitude of the photometry was faint enough to be near the GCLF turnover, making the calculation of the total population $N_{GC}$ relatively secure.
|
---
abstract: 'We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure $\mu$ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Lie algebra” of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the $L^{2}\left(\nu\right)$-closure of holomorphic polynomials by their values on the Cameron-Martin subgroup.'
address:
- |
Department of Mathematics, 0112\
University of California, San Diego\
La Jolla, CA 92093-0112
- |
Department of Mathematics\
University of Connecticut\
Storrs, CT 06269, U.S.A.
author:
- 'Bruce K. Driver[$^{\dagger }$]{}'
- 'Maria Gordina[$^{*}$]{}'
date: ' *File:*'
title: 'Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups'
---
¶
\#1
[^1]
[^2]
Introduction\[s.h1\]
====================
The aim of this paper is to study spaces of holomorphic functions on an infinite-dimensional Heisenberg like group based on a complex abstract Wiener space. In particular, we prove Taylor, skeleton, and holomorphic chaos isomorphism theorems. The tools we use come from properties of heat kernel measures on such groups which have been constructed and studied in [@DG07b]. We will state the main results of our paper and then conclude this introduction with a brief discussion of how our results relate to the existing literature.
Statements of the main results\[s.h1.1\]
----------------------------------------
### The Heisenberg like groups and heat kernel measures\[s.h1.1.1\]
The basic input to our theory is a complex abstract Wiener space, $\left( W,H,\mu\right)$, as in Notation \[n.h2.4\] which is equipped with a continuous skew-symmetric bi-linear form $\omega:W\times W\rightarrow \mathbf{C}$ as in Notation \[n.h3.1\]. Here and throughout this paper, $\mathbf{C}$ is a finite dimensional complex inner product space. The space, $G:=W\times\mathbf{C}$, becomes an infinite-dimensional Heisenberg like group when equipped with the following multiplication rule $$\left( w_{1},c_{1}\right) \cdot\left( w_{2},c_{2}\right) =\left(
w_{1}+w_{2},c_{1}+c_{2}+\frac{1}{2}\omega\left( w_{1},w_{2}\right) \right)
. \label{e.h1.1}$$ A typical example of such a group is the Heisenberg group of a symplectic vector space, but in our setting we have an additional structure of an abstract Wiener space to carry out the heat kernel measure analysis.
The group $G$ contains the *Cameron–Martin* group, $G_{CM}:=H\times\mathbf{C}$, as a subgroup. The Lie algebras of $G$ and $G_{CM}$ will be denoted by $\mathfrak{g}$ and $\mathfrak{g}_{CM}$ respectively which, as sets, may be identified with $G$ and $G_{CM}$ respectively — see Definition \[d.h3.2\], Notation \[n.h3.3\], and Proposition \[p.h3.5\] for more details.
Let $b\left( t\right) =\left( B\left( t\right), B_{0}\left(
t\right) \right)$ be a Brownian motion on $\mathfrak{g}$ associated to the natural Hilbertian structure on $\mathfrak{g}_{CM}$ as described in Eq. (\[e.h4.1\]). The Brownian motion $\left\{
g\left( t\right) \right\}_{t\geq0}$ on $G$ is then the solution to the stochastic differential equation, $$dg\left( t\right)=g\left( t\right) \circ db\left( t\right)
\text{ with
}g\left( 0\right)=\mathbf{e}=\left( 0, 0\right). \label{e.h1.2}$$ The explicit solution to Eq. (\[e.h1.2\]) may be found in Eq. (\[e.h4.2\]). For each $T>0$ we let $\nu_{T}:=\operatorname*{Law}\left( g\left(
T\right) \right)$ be the *heat kernel* measure on $G$ at time $T$ as explained in Definitions \[d.h4.1\] and \[d.h4.2\]. Analogous to the abstract Wiener space setting, $\nu_{T}$ is left (right) quasi-invariant by an element, $h\in G$, iff $h\in G_{CM}$, while $\nu_{T}\left( G_{CM}\right)=0$, see Theorem \[t.h4.5\], Proposition \[p.h4.6\], and [@DG07b Proposition 6.3].
In addition to the above infinite-dimensional structures we will need corresponding finite dimensional approximations. These approximations will be indexed by $\operatorname*{Proj}\left(
W\right)$ which we now define.
\[n.h1.1\]Let $\operatorname*{Proj}\left( W\right)$denote the collection of finite rank continuous linear maps, $P:W\rightarrow
H$, such that $P|_{H}$ is an orthogonal projection. (Explicitly, $P$ must be as in Eq. below.) Further, let $G_{P}:=PW\times\mathbf{C}$ (a subgroup of $G_{CM})$ and $\pi_{P}:G\rightarrow G_{P}$ be the projection map defined by $\pi_{P}\left( w, c\right):=\left( Pw, c\right)$.
To each $P\in\operatorname*{Proj}\left( W\right)$, $G_{P}$ is a finite dimensional Lie group. The Brownian motions and heat kernel measures, $\left\{ \nu_{t}^{P}\right\}_{t>0}$, on $G_{P}$ are constructed similarly to those on $G$–see Definition \[d.h4.10\]. We will use $\left\{ \left( G_{P}, \nu_{T}^{P}\right)
\right\}_{P\in\operatorname*{Proj}\left( W\right) }$ as finite dimensional approximations to $\left( G, \nu_{T}\right)$.
### The Taylor isomorphism theorem\[s.h1.1.2\]
The Taylor map, $\mathcal{T}_{T}$, is a unitary map relating the square integrable holomorphic functions on $G_{CM}$ with the collection of their derivatives at $\mathbf{e\in}G_{CM}$. Before we can state this theorem we need to introduce the two Hilbert spaces involved.
In what follows, $\mathcal{H}\left( G_{CM}\right)$ and $\mathcal{H}\left( G\right)$ will denote the space of holomorphic functions on $G_{CM}$ and $G$ respectively. (See Section \[s.h5\] for the properties of these function spaces which are used throughout this paper.) We also let $\mathbf{T:=T} \left(
\mathfrak{g}_{CM}\right)$ be the algebraic tensor algebra over $\mathfrak{g}_{CM}$, $\mathbf{T}^{\prime}$ be its algebraic dual, $J$ be the two-sided ideal in $\mathbf{T}$ generated by $$\{h\otimes k-k\otimes h-[h,k]:h,k\in\mathfrak{g}_{CM}\}, \label{e.h1.3}$$ and $J^{0}=\{\alpha\in\mathbf{T}^{\prime}:\alpha\left( J\right)
=0\}$ be the backwards annihilator of $J$–see Notation \[n.h6.1\]. Given $f\in\mathcal{H}\left( G\right)$ we let $\alpha:=\mathcal{T}f$ denote the element of $J^{0}$ defined by $\left\langle \alpha, 1\right\rangle =f\left( \mathbf{e}\right)$ and $$\left\langle \alpha, h_{1}\otimes\dots\otimes h_{n}\right\rangle
:=\left( \tilde{h}_{1}\dots\tilde{h}_{n}f\right) \left(
\mathbf{e}\right)$$ where $h_{i}\in\mathfrak{g}_{CM}$ and $\tilde{h}_{i}$ is the left invariant vector field on $G_{CM}$ agreeing with $h_{i}$ at $\mathbf{e}$–see Proposition \[p.h3.5\] and Definition \[d.h6.2\]. We call $\mathcal{T}$ the Taylor map since $\mathcal{T}f\in J^{0}\left( \mathfrak{g}_{CM}\right)$ encodes all of the derivatives of $f$ at $\mathbf{e}$.
\[$L^{2}$–holomorphic functions on $G_{CM}$\]\[d.h1.2\] For $T>0$, let $$\begin{aligned}
\left\Vert f\right\Vert_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }
&
=\sup_{P\in\operatorname*{Proj}\left( W\right) }\left\Vert f|_{G_{P}}\right\Vert_{L^{2}\left( G_{P},\nu_{T}^{P}\right) }\text{ for all }f\in\mathcal{H}\left( G_{CM}\right),\text{ and }\label{e.h1.4}\\
\mathcal{H}_{T}^{2}\left( G_{CM}\right) & :=\left\{ f\in\mathcal{H}\left( G_{CM}\right) :\left\Vert
f\right\Vert_{\mathcal{H}_{T}^{2}\left(
G_{CM}\right) }<\infty\right\}. \label{e.h1.5}$$
In Corollary \[c.h6.6\] below, we will see that $\mathcal{H}_{T}^{2}\left( G_{CM}\right)$ is not empty and in fact contains the space of holomorphic cylinder polynomials $\left(
\mathcal{P}_{CM}\right)$on $G_{CM}$ described in Eq. (\[e.h1.7\]) below. Despite the fact that $\nu_{T}\left( G_{CM}\right)=0$, $\mathcal{H}_{T}^{2}\left( G_{CM}\right)$ should roughly be thought of as the $\nu_{T}$–square integrable holomorphic functions on $G_{CM}$.
\[Non-commutative Fock space\]\[d.h1.3\]Let $T>0$ and $$\left\Vert \alpha\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right)
}^{2}:=\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\sum_{h_{1},\dots,h_{n}\in
S}\left\vert \left\langle \alpha,h_{1}\otimes\dots\otimes
h_{n}\right\rangle
\right\vert ^{2}\text{ for all }\alpha\in J^{0}\left( \mathfrak{g}_{CM}\right),$$ where $S$ is any orthonormal basis for $\mathfrak{g}_{CM}$. The *non-commutative Fock space* is defined as $$J_{T}^{0}\left( \mathfrak{g}_{CM}\right) :=\left\{ \alpha\in
J^{0}\left( \mathfrak{g}_{CM}\right) :\left\Vert \alpha\right\Vert
_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }^{2}<\infty\right\}.$$
It is easy to see that $\left\Vert \cdot\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) }$ is a Hilbertian norm on $J_{T}^{0}\left(
\mathfrak{g}_{CM}\right)$–see Definition \[d.h6.4\] and Eq. (\[e.h6.8\]). For a detailed introduction to such Fock spaces we refer to [@GrossMall].
\[r.h1.4\] When $\omega=0$, $G\left( \omega\right)$ is commutative and the Fock space, $J_{T}^{0}\left(
\mathfrak{g}_{CM}\right)$, becomes the standard commutative bosonic Fock space of symmetric tensors over $\mathfrak{g}_{CM}^{\ast}$.
The following theorem is proved in Section \[s.h6\]–see Theorem \[t.h6.10\].
\[The Taylor isomorphism\]\[t.h1.5\]For all $T>0$, $\mathcal{T}\left( \mathcal{H}_{T}^{2}\left( G_{CM}\right) \right)
\subset J_{T}^{0}\left( \mathfrak{g}_{CM}\right)$ and the linear map, $$\mathcal{T}_{T}:=\mathcal{T}|_{\mathcal{H}_{T}^{2}\left( G_{CM}\right)
}:\mathcal{H}_{T}^{2}\left( G_{CM}\right) \rightarrow J_{T}^{0}\left(
\mathfrak{g}_{CM}\right), \label{e.h1.6}$$ is unitary.
Associated to this theorem is an analogue of Bargmann’s pointwise bounds which appear in Theorem \[t.h6.11\] below.
### The skeleton isomorphism theorem\[s.h1.1.3\]
Similarly to how it has been done on a complex abstract Wiener space by H. Sugita in [@Sugita1994a; @Sugita1994b], the quasi-invariance of the heat kernel measure $\nu_{T}$ allows us to define the skeleton map from $L^{p}\left( G, \nu_{T}\right)$ to a space of functions on the Cameron-Martin subgroup $G_{CM}$, a set of $\nu_{T}$-measure $0$.
\[d.h1.6\]A **holomorphic cylinder polynomial on** $G$ is a holomorphic cylinder function (see Definition \[d.h4.3\]) of the form, $f=F\circ\pi_{P}:G\rightarrow\mathbb{C}$, where $P\in\operatorname*{Proj}\left( W\right)$ and $F:PW\times\mathbf{C}\mathbb{\rightarrow C}$ is a holomorphic polynomial. The space of holomorphic cylinder polynomials will be denoted by $\mathcal{P}$.
The Gaussian heat kernel bounds in Theorem \[t.h4.11\] easily imply that $\mathcal{P}\subset
L^{p}\left( \nu_{T}\right)$ for all $p<\infty$–see Corollary \[c.h5.10\].
\[Holomorphic $L^{p}$–functions\]\[d.h1.7\] For $T>0$ and $1\leqslant p<\infty$, let $\mathcal{H}_{T}^{p}\left( G\right)$ denote the $L^{p}\left( \nu_{T}\right)$ – closure of $\mathcal{P}\subset L^{p}\left( \nu_{T}\right)$.
From Corollary \[c.h4.8\] below, if $T>0$, $p\in(1,\infty]$, $f\in L^{p}\left( G,\nu_{T}\right)$, and $h\in G_{CM}$, then $\int_{G}\left\vert f\left( h\cdot g\right) \right\vert
d\nu_{T}\left( g\right) <\infty$. Thus, if $f\in\mathcal{H}_{T}^{2}\left( G\right)$ we may define the *skeleton map* (see Definition \[d.h4.7\]) by$$\left( S_{T}f\right) \left( h\right) :=\int_{G}f\left( h\cdot
g\right) d\nu_{T}\left( g\right).$$ It is shown in Theorem \[t.h5.12\] that $S_{T}\left( \mathcal{H}_{T}^{2}\left( G\right) \right) \subset\mathcal{H}_{T}^{2}\left(
G_{CM}\right)$ for all $T>0$.
\[The skeleton isomorphism\]\[t.h1.8\]For each $T>0$, the skeleton map, $S_{T}:\mathcal{H}_{T}^{2}\left( G\right) \rightarrow\mathcal{H}_{T}^{2}\left( G_{CM}\right)$, is unitary.
Following Sugita’s results [@Sugita1994a; @Sugita1994b] in the case of an abstract Wiener space, we call $S_{T}|_{\mathcal{H}_{T}^{2}\left( G \right)}$ the skeleton map since it characterizes $f\in\mathcal{H}_{T}^{2}\left( G\right)$ by its values, $S_{T}f$, on $G_{CM}$. Sugita would refer to $G_{CM}$ as the skeleton of $G\left(
\omega\right)$ owing to the fact that $\nu_{T}\left( G_{CM}\right)
=0$ as we show in Proposition \[p.h4.6\].
Theorem \[t.h1.8\] is proved in Section \[s.h8\] and relies on two key density results from Section \[s.h7\]. The first is Lemma \[l.h7.3\] (an infinite-dimensional version of [@D-G-SC07b Lemma 3.5]) which states that the finite rank tensors (see Definition \[d.h7.2\]) are dense inside of $J_{T}^{0}\left(
\mathfrak{g}_{CM}\right)$. The second is Theorem \[t.h7.1\] which states that $$\mathcal{P}_{CM}:=\left\{ p|_{G_{CM}}:p\in\mathcal{P}\right\} \label{e.h1.7}$$ is a dense subspace of $\mathcal{H}_{T}^{2}\left( G_{CM}\right)$. Matt Cecil [@Cecil2008] has modified the arguments presented in Section \[s.h7\] to cover the situation of path groups over graded nilpotent Lie groups. Cecil’s arguments are necessarily much more involved because his Lie groups have nilpotency of arbitrary step.
### The holomorphic chaos expansion\[s.h1.1.4\]
So far we have produced (for each $T>0$) two unitary isomorphisms, the skeleton map $S_{T}$ and the Taylor isomorphism $\mathcal{T}_{T}$, $$\mathcal{H}_{T}^{2}\left( G\right) \overset{S_{T}}{\underset{\cong
}{\longrightarrow}}\mathcal{H}_{T}^{2}\left( G_{CM}\right)
\overset
{\mathcal{T}_{T}}{\underset{\cong}{\longrightarrow}}J_{T}^{0}\left(
\mathfrak{g}_{CM}\right).$$ The next theorem gives an explicit formula for $\left(
\mathcal{T}_{T}\circ S_{T}\right)^{-1}:J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) \rightarrow \mathcal{H}_{T}^{2}\left(
G\right)$.
\[The holomorphic chaos expansion\]\[t.h1.9\] If $f\in\mathcal{H}_{T}^{2}\left( G\right)$ and $\alpha_{f}:=\mathcal{T}_{T}S_{T}f$, then $$f\left( g\left( T\right) \right) =\sum_{n=0}^{\infty}\left\langle
\alpha_{f},\int_{0\leq s_{1}\leq s_{2}\leq\dots\leq s_{n}\leq T}db\left(
s_{1}\right) \otimes\dots\otimes db\left( s_{n}\right) \right\rangle
\label{e.h1.8}$$ where $b\left( t\right)$ and $g\left( t\right)$ are related as in Eq. (\[e.h1.2\]) or equivalently as in Eq. (\[e.h4.2\]).
This result is proved in Section \[s.h9\] and in particular, see Theorem \[t.h9.10\]. The precise meaning of the right hand side of Eq. (\[e.h1.8\]) is also described there.
Discussion\[s.h1.2\]
--------------------
As we noticed in Remark \[r.h1.4\] when the form $\omega\equiv0$ the Fock space $J_{T}^{0}\left( \mathfrak{g}_{CM}\right)$ is the standard commutative bosonic Fock space [@Fock1928]. In this case the Taylor map is one of three isomorphisms between different representations of a Fock space, one other being the Segal-Bargmann transform. The history of the latter is described in [@GrossMall] beginning with works of V. Bargmann [@Bargmann61] and I. Segal in [@Segal1960a]. For other relevant results see [@HallSengupta1998; @Driver1998c].
To put our results into perspective, recall that the classical Segal-Bargmann space is the Hilbert space of holomorphic functions on $\mathbb{C}^{n}$ that are square-integrable with respect to the Gaussian measure $d\mu_{n}(z)=\pi^{-n}e^{-|z|^{2}}dz$, where $dz$ is the $2n$–dimensional Lebesgue measure. One of the features of functions in the Segal-Bargmann space is that they satisfy the pointwise bounds $\left\vert f(z)\right\vert \leqslant\Vert
f\Vert_{L^{2}(\mu_{n})}\exp(|z|^{2}/2)$ (compare with Theorem \[t.h6.11\]). As it is described in [@GrossMall], if $\mathbb{C}^{n}$ is replaced by an infinite-dimensional complex Hilbert space $H$, one of the first difficulties is to find a suitable version of the Gaussian measure. It can be achieved, but only on a certain extension $W$ of $H$, which leads one to consider the complex abstract Wiener space setting. From H. Sugita’s [@Sugita1994a; @Sugita1994b] work on holomorphic functions over a complex abstract Wiener space, it is known that the pointwise bounds control only the values of the holomorphic functions on $H$. This difficulty explains, in part, the need to consider two function spaces: one is of holomorphic functions on $H$ (or $G_{CM}$ in our case) versus the square-integrable (weakly) holomorphic functions on $W$ (or $G$ in our case).
The Taylor map has also been studied in other non-commutative infinite-dimensional settings. M. Gordina [@Gordina2000a; @Gordina2000b; @Gordina2002] considered the Taylor isomorphism in the context of Hilbert-Schmidt groups, while M. Cecil [@Cecil2008] considered the Taylor isomorphism for path groups over stratified Lie groups. The nilpotentcy of the Heisenberg like groups studied in this paper allow us to give a more complete description of the square integrable holomorphic function spaces than was possible in [@Gordina2000a; @Gordina2000b; @Gordina2002] for the Hilbert-Schmidt groups.
Complex analysis in infinite dimensions in a somewhat different setting has been studied by L. Lempert (e.g.[@Lempert2004]), and for more results on Gaussian-like measures on infinite-dimensional curved spaces see papers by D. Pickrell (e.g.[@Pickrell1987; @Pickrell2000]). For another view of different representations of Fock space, one can look at results in the field of white noise, as presented in the book by N. Obata [@ObataBook]. The map between an $L^{2}$-space and a space of symmetric tensors sometimes is called the Segal isomorphism as in [@Kondratiev1980; @KLPSW1996]. For more background on this and related topics see [@HKPSbook].
Complex abstract Wiener spaces\[s.h2\]
======================================
Suppose that $W$ is a complex separable Banach space and $\mathcal{B}_{W}$ is the Borel $\sigma$–algebra on $W$. Let $W_{\operatorname{Re}}$ denote $W$ thought of as a real Banach space. For $\lambda\in\mathbb{C}$, let $M_{\lambda}:W\rightarrow W$ be the operation of multiplication by $\lambda$.
\[d.h2.1\] A measure $\mu$ on $(W,\mathcal{B}_{W})$ is called a (mean zero, non-degenerate) **Gaussian measure** provided that its characteristic functional is given by $$\hat{\mu}(u):=\int_{W}e^{iu\left( w\right) }d\mu\left( w\right)
=e^{-\frac{1}{2}q(u,u)}\text{ for all }u\in W_{\operatorname{Re}}^{\ast},
\label{e.h2.1}$$ where $q=q_{\mu}:W_{\operatorname{Re}}^{\ast}\times W_{\operatorname{Re}}^{\ast}\rightarrow\mathbb{R}$ is an inner product on $W_{\operatorname{Re}}^{\ast}$. If in addition, $\mu$ is invariant under multiplication by $i$, that is, $\mu\circ M_{i}^{-1}=\mu$, we say that $\mu$ is a **complex Gaussian measure** on $W$.
\[r.h2.2\] Suppose $W=\mathbb{C}^{d}$ and let us write $w\in W$ as $w=x+iy$ with $x,y\in\mathbb{R}^{d}$. Then the most general Gaussian measure on $W$ is of the form $$d\mu\left( w\right) =\frac{1}{Z}\exp\left( -\frac{1}{2}Q\left[
\begin{array}
[c]{c}x\\
y
\end{array}
\right] \cdot\left[
\begin{array}
[c]{c}x\\
y
\end{array}
\right] \right) dx \, dy$$ where $Q$ is a real positive definite $2d \times 2d$ matrix and $Z$ is a normalization constant. The matrix $Q$ may be written in $2\times2$ block form as $$Q=\left[
\begin{array}
[c]{cc}A & B\\
B^{\operatorname{tr}} & C
\end{array}
\right].$$ A simple exercise shows $\mu=\mu\circ M_{i}^{-1}$ iff $B=0$ and $A=C$. Thus the general complex Gaussian measure on $\mathbb{C}^{d}$ is of the form $$\begin{aligned}
d\mu\left( w\right) & =\frac{1}{Z}\exp\left( -\frac{1}{2}\left( Ax\cdot
x+Ay\cdot y\right) \right) dx\, dy\\
& =\frac{1}{Z}\exp\left( -\frac{1}{2}Aw\cdot\bar{w}\right) dx\,
dy,\end{aligned}$$ where $A$ is a real positive definite matrix.
Given a complex Gaussian measure $\mu$ as in Definition \[d.h2.1\], let $$\left\Vert w\right\Vert_{H}:=\sup\limits_{u\in
W_{\operatorname{Re}}^{\ast }\setminus\left\{ 0\right\}
}\frac{|u(w)|}{\sqrt{q(u,u)}}\text{ for all
}w\in W, \label{e.h2.2}$$ and define the **Cameron-Martin subspace**, $H\subset W$, by $$H=\left\{ h\in W:\left\Vert h\right\Vert_{H}<\infty\right\}.
\label{e.h2.3}$$ The following theorem summarizes some of the standard properties of the triple $\left( W, H, \mu\right)$.
\[t.h2.3\]Let $\left( W, H, \mu\right)$ be as above, where $\mu$ is a complex Gaussian measure on $\left( W,
\mathcal{B}_{W}\right)$. Then
1. $H$ is a dense complex subspace of $W$.
2. There exists a unique inner product, $\left\langle \cdot,\cdot
\right\rangle_{H}$, on $H$ such that $\left\Vert h\right\Vert_{H}^{2}=\left\langle h,h\right\rangle $ for all $h\in H$. Moreover, with this inner product $H$ is a complete separable complex Hilbert space.
3. There exists $C<\infty$ such that$$\left\Vert h\right\Vert_{W}\leqslant C\left\Vert h\right\Vert
_{H}\text{ for
any }h\in H. \label{e.h2.4}$$
4. If $\left\{ e_{j}\right\}_{j=1}^{\infty}$ is an orthonormal basis for $H$ and $u, v\in W_{\operatorname{Re}}^{\ast}$, then$$q\left( u, v\right) =\left\langle u, v\right\rangle_{H_{\operatorname{Re}}^{\ast}}=\sum_{j=1}^{\infty}\left[ u\left( e_{j}\right) v\left(
e_{j}\right) + u\left( ie_{j}\right) v\left( ie_{j}\right)
\right].
\label{e.h2.5}$$
5. $\mu\circ M_{\lambda}^{-1}=\mu$ for all $\lambda\in\mathbb{C}$ with $\left\vert \lambda\right\vert =1$.
We will begin with the proof of item 5. From Eq. (\[e.h2.1\]), the invariance of $\mu$ under multiplication by $i$ ( $\mu\circ
M_{i}^{-1}=\mu$ ) is equivalent to assuming that $q\left( u\circ
M_{i},u\circ M_{i}\right) =q\left( u, u\right)$ for all $u\in
W_{\operatorname{Re}}^{\ast}$. By polarization, we may further conclude that $$q\left( u\circ M_{i}, v\circ M_{i}\right)=q\left( u, v\right)
\text{ for
all }u, v\in W_{\operatorname{Re}}^{\ast}. \label{e.h2.6}$$ Taking $v=u\circ M_{i}$ in this identity then shows that $q\left(
u\circ M_{i}, -u\right) =q\left( u, u\circ M_{i}\right)$ and hence that $$q\left( u,u\circ M_{i}\right) =0\text{ for any }u\in W_{\operatorname{Re}}^{\ast}. \label{e.h2.7}$$ Therefore if $\lambda=a+ib$ with $a, b\in\mathbb{R}$, we see that $$\begin{aligned}
q\left( u\circ M_{\lambda},u\circ M_{\lambda}\right) & =q\left(
au+bu\circ M_{i}, au+bu\circ M_{i}\right) \nonumber\\
& =\left( a^{2}+b^{2}\right) q\left( u, u\right) =\left\vert
\lambda\right\vert ^{2}q\left( u, u\right), \label{e.h2.8}$$ from which it follows that $q\left( u\circ M_{\lambda}, u\circ
M_{\lambda }\right) =q\left( u, u\right) $ for all $u\in
W_{\operatorname{Re}}^{\ast}$ and $\left\vert \lambda\right\vert
=1$. Coupling this observation with Eq. (\[e.h2.1\]) implies $\mu\circ M_{\lambda}^{-1}=\mu$ for all $\left\vert
\lambda\right\vert =1$. If $\left\vert \lambda\right\vert =1$, from Eqs. (\[e.h2.2\]) and (\[e.h2.8\]), it follows that$$\begin{aligned}
\left\Vert \lambda w\right\Vert_{H} & =\sup_{u\in W_{\operatorname{Re}}^{\ast}\setminus\left\{ 0\right\} }\frac{\left\vert u\left( \lambda
w\right) \right\vert }{\sqrt{q\left( u,u\right) }}=\sup_{u\in
W_{\operatorname{Re}}^{\ast}\setminus\left\{ 0\right\} }\frac{\left\vert
u\circ M_{\lambda}\left( w\right) \right\vert }{\sqrt{q\left( u\circ
M_{\lambda},u\circ M_{\lambda}\right) }}\\
& =\sup_{u\in W_{\operatorname{Re}}^{\ast}\setminus\left\{ 0\right\} }\frac{\left\vert u\left( w\right) \right\vert }{\sqrt{q\left(
u,u\right) }}=\left\Vert w\right\Vert_{H}\text{ for all }w\in W.\end{aligned}$$ In particular, if $\left\Vert h\right\Vert_{H}<\infty$ and $\left\vert \lambda\right\vert =1$, then $\left\Vert \lambda
h\right\Vert_{H}=\left\Vert h\right\Vert_{H}<\infty$ and hence $\lambda H\subset H$ which shows that $H$ is a complex subspace of $W$. From [@DG07b Theorem 2.3] summarizing some well-known properties of Gaussian measures, we know that item 3. holds, $H$ is a dense subspace of $W_{\operatorname{Re}}$, and there exists a unique real Hilbertian inner product, $\left\langle
\cdot,\cdot\right\rangle_{H_{\operatorname{Re}}}$, on $H$ such that $\left\Vert h\right\Vert_{H}^{2}=\left\langle h, h\right\rangle
_{H_{\operatorname{Re}}}$ for all $h\in H$. Polarizing the identity $\left\Vert \lambda h\right\Vert_{H}=\left\Vert h\right\Vert_{H}$ implies $\left\langle \lambda h,\lambda k\right\rangle_{H_{\operatorname{Re}}}=\left\langle h, k\right\rangle_{H_{\operatorname{Re}}}$ for all $h,k\in H$. Taking $\lambda=i$ and $k=-ih$ then shows $\left\langle
ih,h\right\rangle_{\operatorname{Re}}=\left\langle
h,-ih\right\rangle_{\operatorname{Re}}$, and hence that $\left\langle ih,h\right\rangle_{\operatorname{Re}}=0$ for all $h\in
H$. Using this information it is a simple matter to check that $$\left\langle h, k\right\rangle_{H}:=\left\langle h, k\right\rangle
_{H_{\operatorname{Re}}}+i\left\langle h, ik\right\rangle
_{H_{\operatorname{Re}}}\text{ for all }h, k\in H, \label{e.h2.9}$$ is the unique complex inner product on $H$ such that $\operatorname{Re}\left\langle \cdot, \cdot\right\rangle_{H}=\left\langle \cdot,\cdot
\right\rangle_{H_{\operatorname{Re}}}$. [^3]
So it only remains to prove Eq. (\[e.h2.5\]). For a proof of the first equality in Eq. (\[e.h2.5\]), see [@DG07b Theorem 2.3]. To prove the second equality in this equation, it suffices to observe that $\left\{ e_{j},ie_{j}\right\}_{j=1}^{\infty}$ is an orthonormal basis for $\left( H_{\operatorname{Re}},\left\langle
\cdot,\cdot\right\rangle
_{H_{\operatorname{Re}}}\right)$ and therefore,$$\left\langle u, v\right\rangle_{H_{\operatorname{Re}}^{\ast}}=\sum
_{j=1}^{\infty}\left[ u\left( e_{j}\right) v\left( e_{j}\right)
+u\left( ie_{j}\right) v\left( ie_{j}\right) \right] \text{ for
any }u, v\in H_{\operatorname{Re}}^{\ast}.$$
\[n.h2.4\]The triple, $\left( W, H, \mu\right)$, appearing in Theorem \[t.h2.3\] will be called a **complex abstract Wiener space**. (Notice that there is redundancy in this notation since $\mu$ is determined by $H$, and $H$ is determined by $\mu$.)
\[l.h2.5\]Suppose that $u, v\in W_{\operatorname{Re}}^{\ast}$ and $a, b\in\mathbb{C}$, then $$\int_{W}e^{au+bv}d\mu=\exp\left( \frac{1}{2}\left( a^{2}q\left(
u,u\right) +b^{2}q\left( v, v\right) +2abq\left( u, v\right)
\right) \right).
\label{e.h2.10}$$
Equation (\[e.h2.10\]) is easily verified when both $a$ and $b$ are real. This suffices to complete the proof, since both sides of Eq. (\[e.h2.10\]) are analytic functions of $a,b\in\mathbb{C}$.
\[l.h2.6\]Let $\left( W,H,\mu\right) $ be a complex abstract Wiener space, then for any $\varphi\in W^{\ast}$, we have $$\int_{W}e^{\varphi\left( w\right) }d\mu\left( w\right) =1=\int
_{W}e^{\overline{\varphi\left( w\right) }}d\mu\left( w\right),
\label{e.h2.11}$$$$\int_{W}\left\vert \operatorname{Re}\varphi\left( w\right) \right\vert
^{2}d\mu\left( w\right) =\int_{W}\left\vert \operatorname{Im}\varphi\left(
w\right) \right\vert ^{2}d\mu\left( w\right) =\left\Vert \varphi\right\Vert
_{H^{\ast}}^{2}, \label{e.h2.12}$$ and$$\int_{W}\left\vert \varphi\left( w\right) \right\vert ^{2}d\mu\left(
w\right) =2\left\Vert \varphi\right\Vert_{H^{\ast}}^{2}. \label{e.h2.13}$$ More generally, if $\mathbf{C}$ is another complex Hilbert space and $\varphi\in L\left( W,\mathbf{C}\right)$, then $$\int_{W}\left\Vert \varphi\left( w\right) \right\Vert_{\mathbf{C}}^{2}d\mu\left( w\right) =2\left\Vert \varphi\right\Vert_{H^{\ast}\otimes\mathbf{C}}^{2}. \label{e.h2.14}$$
If $u=\operatorname{Re}\varphi$, then $\varphi\left( w\right)
=u\left( w\right) -iu\left( iw\right)$. Therefore by Eqs. (\[e.h2.6\]), (\[e.h2.7\]), and (\[e.h2.10\]),$$\begin{aligned}
\int_{W}e^{\varphi}d\mu & =\int_{W}e^{u-iu\circ M_{i}}d\mu\\
& =\exp\left( \frac{1}{2}\left( q\left( u,u\right) -q\left( u\circ
M_{i},u\circ M_{i}\right) -2iq\left( u,u\circ M_{i}\right) \right)
\right) =1.\end{aligned}$$ Taking the complex conjugation of this identity shows $\int_{W}e^{\overline {\varphi\left( w\right) }}d\mu\left(
w\right) =1$. Also using Lemma \[l.h2.5\], we have$$\begin{aligned}
\int_{W}\left\vert \operatorname{Re}\varphi\left( w\right) \right\vert
^{2}d\mu\left( w\right) & =q\left( u, u\right) \text{ and}\\
\int_{W}\left\vert \operatorname{Im}\varphi\left( w\right)
\right\vert ^{2}d\mu\left( w\right) & =\int_{W}\left\vert
u\left( iw\right) \right\vert ^{2}d\mu\left( w\right) =q\left(
u\circ M_{i}, u\circ M_{i}\right) =q\left( u, u\right).\end{aligned}$$ To evaluate $q\left( u,u\right)$, let $\left\{ e_{k}\right\}
_{k=1}^{\infty}$ be an orthonormal basis for $H$ so that $\left\{
e_{k},ie_{k}\right\}_{k=1}^{\infty}$ is an orthonormal basis for $\left( H_{\operatorname{Re}},\operatorname{Re}\left\langle
\cdot,\cdot\right\rangle_{H}\right)$. Then by Eq. (\[e.h2.5\]), $$q\left( u, u\right) =\sum_{k=1}^{\infty}\left[ \left\vert u\left(
e_{k}\right) \right\vert ^{2}+\left\vert u\left( ie_{k}\right)
\right\vert ^{2}\right] =\sum_{k=1}^{\infty}\left\vert
\varphi\left( e_{k}\right) \right\vert ^{2}=\left\Vert
\varphi\right\Vert_{H^{\ast}}^{2}.$$ To prove Eq. (\[e.h2.14\]), apply [@DG07b Eq. (2.13)] to find $$\begin{aligned}
\int_{W}\left\Vert \varphi\left( w\right) \right\Vert_{\mathbf{C}}^{2}d\mu\left( w\right) & =\sum_{k=1}^{\infty}\left[ \left\Vert
\varphi\left( e_{k}\right) \right\Vert_{\mathbf{C}}^{2}+\left\Vert
\varphi\left( ie_{k}\right) \right\Vert_{\mathbf{C}}^{2}\right] \\
& =2\sum_{k=1}^{\infty}\left\Vert \varphi\left( e_{k}\right) \right\Vert
_{\mathbf{C}}^{2}=2\left\Vert \varphi\right\Vert_{H^{\ast}\otimes\mathbf{C}}^{2}.\end{aligned}$$
\[Heat kernel interpretation of Lemma \[l.h2.6\]\]\[r.h2.7\]The measure $\mu$ formally satisfies $$\int_{W}f\left( w\right) d\mu\left( w\right) =\left( e^{\frac{1}{2}\Delta_{H_{\operatorname{Re}}}}f\right) \left( 0\right),$$ where $\Delta_{H_{\operatorname{Re}}}=\sum_{j=1}^{\infty}\partial_{e_{j}}^{2}$ and $\left\{ e_{j}\right\}_{j=1}^{\infty}$ is an orthonormal basis for $H_{\operatorname{Re}}$. If $f$ is holomorphic or anti-holomorphic, then $f$ is harmonic and therefore $$\int_{W}f\left( w\right) d\mu\left( w\right) =\left( e^{\frac{1}{2}\Delta_{H_{\operatorname{Re}}}}f\right) \left( 0\right) =f\left( 0\right)
.$$ Applying this identity to $f\left( w\right) =e^{\varphi\left( w\right) }$ or $f\left( w\right) =\overline{e^{\varphi\left( w\right) }}$ with $\varphi\in W^{\ast}$ gives Eq. . If $u\in W_{\operatorname{Re}}^{\ast}$, we have $$\begin{aligned}
\int_{W}u^{2}\left( w\right) d\mu\left( w\right) & =\left( e^{\frac
{1}{2}\Delta_{H_{\operatorname{Re}}}}u^{2}\right) \left( 0\right)
=\sum_{n=0}^{\infty}\frac{1}{2^{n}n!}\left( \Delta_{H_{\operatorname{Re}}}^{n}u^{2}\right) \left( 0\right) \\
& =\frac{1}{2}\left( \Delta_{H_{\operatorname{Re}}}u^{2}\right) \left(
0\right) =\frac{1}{2}\sum_{j=1}^{\infty}\left( \partial_{e_{j}}^{2}u^{2}\right) \left( 0\right) \\
& =\sum_{j=1}^{\infty}u\left( e_{j}\right)^{2}=\left\Vert
u\right\Vert _{H_{\operatorname{Re}}}^{2}.\end{aligned}$$ Eqs. and now follow easily from this identity.
The structure of the projections\[s.h2.1\]
------------------------------------------
Let $i:H\rightarrow W$ be the inclusion map and $i^{\ast}:W^{\ast}\rightarrow H^{\ast}$ be its transpose, i.e. $i^{\ast}\ell:=\ell\circ i$ for all $\ell\in W^{\ast}$. Also let $$H_{\ast}:=\left\{ h\in H:\left\langle \cdot,h\right\rangle_{H}\in\operatorname*{Ran}\left( i^{\ast}\right) \subset H^{\ast}\right\}
\label{e.h2.15}$$ or in other words, $h\in H$ is in $H_{\ast}$ iff $\left\langle \cdot
, h\right\rangle_{H}\in H^{\ast}$ extends to a continuous linear functional on $W$. (We will continue to denote the continuous extension of $\left\langle \cdot,h\right\rangle_{H}$ to $W$ by $\left\langle \cdot,h\right\rangle_{H}.)$ Because $H$ is a dense subspace of $W$, $i^{\ast}$ is injective, and because $i$ is injective, $i^{\ast}$ has a dense range. Since $h\in
H\rightarrow\left\langle \cdot,h\right\rangle_{H}\in H^{\ast}$ is a conjugate linear isometric isomorphism, it follows from the above comments that $H_{\ast}\ni h\rightarrow\left\langle
\cdot,h\right\rangle_{H}\in W^{\ast}$ is a conjugate linear isomorphism too, and that $H_{\ast}$ is a dense subspace of $H$.
\[l.h2.8\] There is a one to one correspondence between $\operatorname*{Proj}\left( W\right)$ (see Notation \[n.h1.1\]) and the collection of finite rank orthogonal projections, $P$, on $H$ such that $PH\subset H_{\ast}$.
If $P\in\operatorname*{Proj}\left( W\right)$ and $u\in PW\subset
H$, then, because $P|_{H}$ is an orthogonal projection, we have$$\left\langle Ph,u\right\rangle_{H}=\left\langle h,Pu\right\rangle
_{H}=\left\langle h,u\right\rangle_{H}\text{ for all }h\in H. \label{e.h2.16}$$ Since $P:W\rightarrow H$ is continuous, it follows that $u\in
H_{\ast}$, i.e. $PW\subset H_{\ast}$.
Conversely, suppose that $P:H\rightarrow H$ is a finite rank orthogonal projection such that $PH\subset H_{\ast}$. Let $\left\{
e_{j}\right\}_{j=1}^{n}$ be an orthonormal basis for $PH$ and $\ell_{j}\in W^{\ast}$ such that $\ell_{j}|_{H}=\left\langle \cdot,
e_{j}\right\rangle_{H}$. Then we may extend $P$ uniquely to a continuous operator from $W$ to $H$ (still denoted by $P)$ by letting$$Pw:=\sum_{j=1}^{n}\ell_{j}\left( w\right) e_{j}=\sum_{j=1}^{n}\left\langle
w,e_{j}\right\rangle_{H}e_{j}\text{ for all }w\in W. \label{e.h2.17}$$ From [@DG07b Eq. 3.43], there exists $C=C\left( P\right) <\infty$ such that$$\left\Vert Pw\right\Vert_{H}\leqslant C\left\Vert w\right\Vert
_{W}\text{ for
all }w\in W. \label{e.h2.18}$$
Complex Heisenberg like groups\[s.h3\]
======================================
In this section we review the infinite-dimensional Heisenberg like groups and Lie algebras which were introduced in [@DG07b Section 3].
\[n.h3.1\]Let $\left( W, H, \mu\right)$ be a complex abstract Wiener space, $\mathbf{C}$ be a complex finite dimensional inner product space, and $\omega: W\times W\rightarrow\mathbf{C}$ be a continuous skew symmetric bilinear quadratic form on $W$. Further, let $$\left\Vert \omega\right\Vert_{0}:=\sup\left\{ \left\Vert
\omega\left( w_{1},w_{2}\right) \right\Vert
_{\mathbf{C}}:w_{1},w_{2}\in W\text{ with }\left\Vert
w_{1}\right\Vert_{W}=\left\Vert w_{2}\right\Vert_{W}=1\right\}
\label{e.h3.1}$$ be the uniform norm on $\omega$ which is finite by the assumed continuity of $\omega$.
\[d.h3.2\]Let $\mathfrak{g}$ denote $W\times\mathbf{C}$ when thought of as a Lie algebra with the Lie bracket operation given by $$\left[ \left( A,a\right),\left( B,b\right) \right] :=\left(
0,\omega\left( A,B\right) \right). \label{e.h3.2}$$ Let $G=G\left( \omega\right)$ denote $W\times\mathbf{C}$ when thought of as a group with the multiplication law given by $$g_{1}g_{2}=g_{1}+g_{2}+\frac{1}{2}\left[ g_{1},g_{2}\right] \text{ for any
}g_{1},g_{2}\in G \label{e.h3.3}$$ or equivalently by Eq. (\[e.h1.1\]).
It is easily verified that $\mathfrak{g}$ is a Lie algebra and $G$ is a group. The identity of $G$ is the zero element, $\mathbf{e:}=\left( 0,0\right)$.
\[n.h3.3\]Let $\mathfrak{g}_{CM}$ denote $H\times\mathbf{C}$ when viewed as a Lie subalgebra of $\mathfrak{g}$ and $G_{CM}$ denote $H\times\mathbf{C}$ when viewed as a subgroup of $G=G\left(
\omega\right)$. We will refer to $\mathfrak{g}_{CM}$ ($G_{CM})$ as the **Cameron–Martin subalgebra (subgroup)** of $\mathfrak{g}$ $\left( G\right)$. (For explicit examples of such $\left(
W,H,\mathbf{C}, \omega\right)$, see [@DG07b].)
We equip $G=\mathfrak{g}=W\times\mathbf{C}$ with the Banach space norm $$\left\Vert \left( w, c\right) \right\Vert
_{\mathfrak{g}}:=\left\Vert
w\right\Vert_{W}+\left\Vert c\right\Vert_{\mathbf{C}} \label{e.h3.4}$$ and $G_{CM}=\mathfrak{g}_{CM}=H\times\mathbf{C\ }$with the Hilbert space inner product,$$\left\langle \left( A, a\right), \left( B, b\right) \right\rangle
_{\mathfrak{g}_{CM}}:=\left\langle A, B\right\rangle
_{H}+\left\langle
a, b\right\rangle_{\mathbf{C}}. \label{e.h3.5}$$ The associate Hilbertian norm is given by $$\left\Vert \left( A,\delta\right) \right\Vert_{\mathfrak{g}_{CM}}:=\sqrt{\left\Vert A\right\Vert_{H}^{2}+\left\Vert \delta\right\Vert
_{\mathbf{C}}^{2}}. \label{e.h3.6}$$ As was shown in [@DG07b Lemma 3.3], these Banach space topologies on $W\times\mathbf{C}$ and $H\times\mathbf{C}$ make $G$ and $G_{CM}$ into topological groups.
\[Linear differentials\]\[n.h3.4\] Suppose $f:G\rightarrow\mathbb{C}$, is a Frechét smooth function. For $g\in G$ and $h, k\in\mathfrak{g}$ let $$f^{\prime}\left( g\right) h:=\partial_{h}f\left( g\right) =\frac{d}{dt}\Big|_{0}f\left( g+th\right)$$ and$$f^{\prime\prime}\left( g\right) \left( h\otimes k\right)
:=\partial_{h}\partial_{k}f\left( g\right).$$ (Here and in the sequel a prime on a symbol will be used to denote its derivative or differential.)
As $G$ itself is a vector space, the tangent space, $T_{g}G$, to $G$ at $g$ is naturally isomorphic to $G$. Indeed, if $v, g\in G$, then we may define a tangent vector $v_{g}\in T_{g}G$ by $v_{g}f=f^{\prime}\left( g\right)v$ for all Frechét smooth functions $f:G\rightarrow\mathbb{C}$. We will identify $\mathfrak{g}$ with $T_{\mathbf{e}}G$ and $\mathfrak{g}_{CM}$ with $T_{\mathbf{e}}G_{CM}$. Recall that as sets $\mathfrak{g}=G$ and $\mathfrak{g}_{CM}=G_{CM}$. For $g\in G$, let $l_{g}:G\rightarrow G$ be the left translation by $g$. For $h\in\mathfrak{g}$, let $\tilde{h}$ be the **left invariant vector field** on $G$ such that $\tilde{h}\left( g\right) =h$ when $g=\mathbf{e}$. More precisely, if $\sigma\left( t\right) \in G$ is any smooth curve such that $\sigma\left( 0\right) =\mathbf{e}$ and $\dot{\sigma }\left( 0\right) =h$ (e.g. $\sigma\left( t\right)
=th)$, then $$\tilde{h}\left( g\right)=,
l_{g\ast}h:=\frac{d}{dt}\left|_{0}\right.g\cdot
\sigma\left( t\right). \label{e.h3.7}$$ As usual, we view $\tilde{h}$ as a first order differential operator acting on smooth functions, $f:G\rightarrow\mathbb{C}$, by $$\left( \tilde{h}f\right) \left( g\right) =\frac{d}{dt}\Big|_{0}f\left(
g\cdot\sigma\left( t\right) \right). \label{e.h3.8}$$ The proof of the following easy proposition may be found in [@DG07b Proposition 3.7].
\[p.h3.5\]Let $f:G\rightarrow\mathbb{C}$ be a smooth function, $h=(A,a)\in\mathfrak{g}$ and $g=\left( w, c\right) \in G$. Then$$\widetilde{h}\left( g\right) :=, l_{g\ast}h=\left( A, a+\frac{1}{2}\omega\left( w, A\right) \right) \text{ for any }g=\left( w,
c\right) \in
G \label{e.h3.9}$$ and, in particular, $$\widetilde{\left( A, a\right) }f\left( g\right) =f^{\prime}\left(
g\right) \left( A, a+\frac{1}{2}\omega\left( w, A\right) \right).
\label{e.h3.10}$$ If $h,k\in\mathfrak{g}$, then$$\left( \tilde{h}\tilde{k}f-\tilde{k}\tilde{h}f\right) =\widetilde{\left[
h,k\right] }f. \label{e.h3.11}$$ The one parameter group in $G$, $e^{th}$, determined by $h=\left(
A, a\right) \in\mathfrak{g}$, is given by $e^{th}=th=t\left( A,
\delta\right)$.
Brownian motion and heat kernel measures\[s.h4\]
================================================
This section will closely follow [@DG07b Section 4] except for the introduction of a certain factor of $1/2$ into the formalism which will simplify later formulas. Let $\left\{ b\left( t\right)
=\left( B\left( t\right), B_{0}\left( t\right) \right) \right\}
_{t\geqslant0}$ be a Brownian motion on $\mathfrak{g}=W\times\mathbf{C}$ with the variance determined by $$\mathbb{E}\left[ \operatorname{Re}\left\langle b\left( s\right) ,
h\right\rangle
_{\mathfrak{g}_{CM}}\cdot\operatorname{Re}\left\langle
b\left( t\right), k\right\rangle_{\mathfrak{g}_{CM}}\right] =\frac{1}{2}\operatorname{Re}\left\langle h, k\right\rangle
_{\mathfrak{g}_{CM}}s\wedge
t \label{e.h4.1}$$ for all $s, t\in\lbrack0,\infty)$, $h=\left( A, a\right)$, and $k\mathbf{:=}\left( C, c\right)$, where $A,C\in H_{\ast}$ and $a,c\in\mathbf{C}$. (Recall the definition of $H_{\ast}$ from Eq. (\[e.h2.15\]).)
\[d.h4.1\] The associated **Brownian motion** on $G$ starting at $\mathbf{e}=\left( 0, 0\right) \in G$ is defined to be the process $$g\left( t\right) =\left( B\left( t\right), B_{0}\left( t\right)
+\frac{1}{2}\int_{0}^{t}\omega\left( B\left( \tau\right), dB\left(
\tau\right) \right) \right). \label{e.h4.2}$$ More generally, if $h\in G$, we let $g_{h}\left( t\right) :=h\cdot
g\left( t\right)$, the Brownian motion on $G$ starting at $h$.
\[d.h4.2\] Let $\mathcal{B}_{G}$ be the Borel $\sigma$–algebra on $G$ and for any $T>0$, let $\nu_{T}:\mathcal{B}_{G}\rightarrow\left[ 0, 1\right]$ be the distribution of $g\left( T\right)$. We will call $\nu_{T}$ the **heat kernel measure** on $G$.
To be more explicit, the measure $\nu_{T}$ is the unique measure on $\left( G,\mathcal{B}_{G}\right)$ such that $$\nu_{T}\left( f\right):=\int_{G}fd\nu_{T}=\mathbb{E}\left[ f\left(
g\left( T\right) \right) \right] \label{e.h4.3}$$ for all bounded measurable functions $f:G\rightarrow\mathbb{C}$. Our next goal is to describe the generator of the process $\left\{
g_{h}\left( t\right) \right\}_{t\geqslant0}$.
\[d.h4.3\] A function $f:G\rightarrow\mathbb{C}$ is said to be a **cylinder function** if it may be written as $f=F\circ\pi_{P}$ for some $P\in\operatorname*{Proj}\left( W\right)$ and some function $F:G_{P} \rightarrow\mathbb{C}$, where $G_{P}$ is defined as in Notation \[n.h1.1\]. We say that $f$ is a **holomorphic** (**smooth**) **** cylinder function if $F:G_{P}\rightarrow \mathbb{C}$ is holomorphic (smooth). We will denote the space of holomorphic (analytic) cylinder functions by $\mathcal{A}$.
\[Generator of $g_{h}$\]\[p.h4.4\]If $f:G\rightarrow\mathbb{C}$ is a smooth cylinder function, let$$Lf:=\sum_{j=1}^{\infty}\left[ \widetilde{\left( e_{j},0\right) }^{2}+\widetilde{\left( ie_{j},0\right) }^{2}\right]
f+\sum_{j=1}^{d}\left[ \widetilde{\left( 0, f_{j}\right)
}^{2}+\widetilde{\left( 0, if_{j}\right)
}^{2}\right] f, \label{e.h4.4}$$ where $\left\{ e_{j}\right\}_{j=1}^{\infty}$ and $\left\{
f_{j}\right\}_{j=1}^{d}$ are complex orthonormal bases for $\left(
H, \left\langle \cdot, \cdot\right\rangle_{H}\right)$ and $\left(
\mathbf{C},\left\langle \cdot,\cdot\right\rangle
_{\mathbf{C}}\right)$ respectively. Then $Lf$ is well defined, i.e. the sums in Eq. (\[e.h4.4\]) are convergent and independent of the choice of bases. Moreover, for all $h\in G$, $\frac{1}{4}L$ is the generator for $\left\{ g_{h}\left( t\right) \right\}_{t\geqslant
0}$. More precisely, $$M_{t}^{f}:=f\left( g_{h}\left( t\right) \right) -\frac{1}{4}\int_{0}^{t}Lf\left( g_{h}\left( \tau\right) \right) d\tau\label{e.h4.5}$$ is a local martingale for any smooth cylinder function, $f:G\rightarrow \mathbb{C}$.
After bearing in mind the factor of $1/2$ used in defining the Brownian motion $b\left( t \right)$ in Eq. (\[e.h4.1\]), this proposition becomes a direct consequence of Proposition 3.29 and Theorem 4.4 of [@DG07b]. Indeed, the Brownian motions in this paper are equal in distribution to the Brownian motions used in [@DG07b] after making the time change, $t\rightarrow t/2$. It is this time change that is responsible for the $1/4$ factor (rather than $1/2)$ in Eq. (\[e.h4.5\]).
Heat kernel quasi-invariance properties\[s.h4.1\]
-------------------------------------------------
In this subsection we are going to recall one of the key theorems from [@DG07b]. We first need a little more notation.
Let $C_{CM}^{1}$ denote the collection of $C^{1}$-paths, $g:\left[
0,1\right] \rightarrow G_{CM}$. The length of $g$ is defined as $$\ell_{G_{CM}}\left( g\right) =\int_{0}^{1}\left\Vert
\,l_{g^{-1}\left( s\right) \ast}g^{\prime}\left( s\right)
\right\Vert_{\mathfrak{g}_{CM}}ds.
\label{e.h4.6}$$ As usual, the Riemannian distance between $x,y\in G_{CM}$ is defined as$$d_{G_{CM}}(x,y)=\inf\left\{ \ell_{G_{CM}}\left( g\right) :g\in C_{CM}^{1}~\ni~g\left( 0\right) =x\text{ and }g\left( 1\right)
=y\right\}.
\label{e.h4.7}$$ Let us also recall the definition of $k\left( \omega\right)$ from [@DG07b Eq. 7.6]; $$\begin{aligned}
k\left( \omega\right) & =-\frac{1}{2}\sup_{\left\Vert
A\right\Vert_{H_{\operatorname{Re}}}=1}\left\Vert \omega\left(
\cdot, A\right)
\right\Vert_{H_{\operatorname{Re}}^{\ast}\otimes\mathbf{C}_{\operatorname{Re}}}^{2}\nonumber\\
& =-\sup_{\left\Vert A\right\Vert_{H}=1}\left\Vert \omega\left(
\cdot, A\right) \right\Vert
_{H^{\ast}\otimes\mathbf{C}}^{2}\geqslant
-\left\Vert \omega\right\Vert_{H^{\ast}\otimes H^{\ast}\otimes C}^{2}>-\infty, \label{e.h4.8}$$ wherein we have used [@DG07b Lemma 3.17] in the second equality. It is known by Fernique’s or Skhorohod’s theorem that $\left\Vert \omega\right\Vert
_{2}^{2}=\left\Vert \omega\right\Vert_{H^{\ast}\otimes H^{\ast}\otimes C}^{2}<\infty$, see [@DG07b Proposition 3.14] for details.
\[t.h4.5\]For all $h\in G_{CM}$ and $T>0$, the measures, $\nu_{T}\circ $$l_{h}^{-1}$ and $\nu_{T}\circ r_{h}^{-1}$, are absolutely continuous relative to $\nu_{T}$. Let $Z_{h}^{l}:=\frac{d\left( \nu_{T}\circ\, l_{h}^{-1}\right) }{d\nu_{T}}$ and $Z_{h}^{r}:=\frac{d\left( \nu_{T}\circ r_{h}^{-1}\right)
}{d\nu_{T}}$ be the respective Randon-Nikodym derivatives, $k\left( \omega\right)$ is given in Eq. (\[e.h4.8\]), and$$c\left( t\right) :=\frac{t}{e^{t}-1} \text{ for any }t\in\mathbb{R}$$ with the convention that $c\left( 0\right) =1$. Then for all $1\leqslant p<\infty$, $Z_{h}^{l}$ and $Z_{h}^{r}\ $ are both in $L^{p}\left( \nu_{T}\right)$ and satisfy the estimate $$\left\Vert Z_{h}^{\ast}\right\Vert_{L^{p}\left( \nu_{T}\right) }\leqslant\exp\left( \frac{c\left( k\left( \omega\right)
T/2\right) \left( p-1\right) }{T}d_{G_{CM}}^{2}\left( \mathbf{e},
h\right) \right),
\label{e.h4.9}$$ where $\ast=l$ or $\ast=r$.
This is [@DG07b Theorem 8.1] (also see [@DG07b Corollary 7.3]) with the modification that $T$ should be replaced by $T/2$. This is again due to the fact that the Brownian motions in this paper are equal in distribution to those in [@DG07b] after making the time change, $t\rightarrow t/2$.
It might be enlightening to note here that we call $G_{CM}$ the Cameron-Martin subgroup not only because it is constructed from the Cameron-Martin subspace, $H$, but also because it has properties similar to $H$. In particular, the following statement holds.
\[p.h4.6\]The heat kernel measure does not charge $G_{CM}$, i.e. $\nu_{T}\left( G_{CM}\right) =0$.
Note that for a bounded measurable function $f: W\times C
\to\mathbb{C}$ that depends only on the the first component in $W\times C$, that is, $f \left( w, c\right) =f\left( w\right)$ we have
$$\int_{G} f\left( w\right) d\nu_{T}\left( w, c \right) =
\mathbb{E} \left[ f \left( B \left( T\right) \right) \right]
=\int_{W}f\left( w\right) d\mu_{T}\left( w\right).$$ Note that for the projection $\pi: W \times C \to W$, $\pi\left( w, c
\right) =w$ we have $\pi_{\ast}\nu_{T}=\mu_{T}$ and therefore $$\nu_{T}\left( G_{CM} \right) =\nu_{T}\left( \pi^{-1}\left( H
\right) \right) =\pi_{\ast}\nu_{T}\left( H \right) =\mu_{T}\left(
H \right) =0.$$
For later purposes, we would like to introduce the heat operator, $S_{T}:=e^{TL/4}$, acting on $L^{p}\left( G, \nu_{T}\right)$. To motivate our definition, suppose $f:G\rightarrow\mathbb{C}$ is a smooth cylinder function and suppose we can make sense of $u\left(
t,y\right) =\left( e^{\left( T-t\right) L/4}f\right) \left(
y\right)$. Then working formally, by Itô’s formula, Eq. (\[e.h4.5\]), and the left invariance of $L$, we expect $u\left(
t, hg\left( t\right) \right)$ to be a martingale for $0\leqslant
t\leqslant T$ and in particular, $$\mathbb{E}\left[ f\left( hg\left( T\right) \right) \right]
=\mathbb{E}\left[ u\left( T, hg\left( T\right) \right) \right]
=\mathbb{E}\left[ u\left( 0, hg\left( 0\right) \right) \right]
=\left(
e^{TL/4}f\right) \left( h\right). \label{e.h4.10}$$
\[d.h4.7\]For $T>0$, $p\in(1,\infty]$, and $f\in L^{p}\left(
G,\nu_{T}\right)$, let $S_{T}f:G_{CM}\rightarrow\mathbb{C}$ be defined by $$\left( S_{T}f\right) \left( h\right) =\int_{G}f\left( h\cdot g\right)
d\nu_{T}\left( g\right) =\mathbb{E}\left[ f\left( hg\left( T\right)
\right) \right]. \label{e.h4.11}$$
The following result is a simple corollary of Theorem \[t.h4.5\] and Hölder’s inequality along with the observation that $p^{\prime}-1=\left( p-1\right)^{-1}$, where $p^{\prime}$ is the conjugate exponent to $p\in(1,\infty]$.
\[c.h4.8\]If $p>1$, $T>0$, $f\in L^{p}\left( G,\nu_{T}\right)
$, $h\in G_{CM}$, and $$Z_{h}^{l}\in L^{\infty-}\left( \nu_{T}\right) :=\cap_{1\leqslant q<\infty
}L^{q}\left( \nu_{T}\right) \label{e.h4.12}$$ is as in Theorem \[t.h4.5\], then $S_{T}f$ is well defined and may be computed as $$\left( S_{T}f\right) \left( h\right) =
\int_{G}f\left( g\right) Z_{h}^{l}\left( g\right) d\nu_{T}\left( g\right). \label{e.h4.13}$$ Moreover, we have the following pointwise Gaussian bounds $$\left\vert \left( S_{T}f\right) \left( h\right) \right\vert
\leqslant \left\Vert f\right\Vert_{L^{p}\left( \nu_{T}\right)
}\exp\left( \frac{c\left( k\left( \omega\right) T/2\right)
}{T\left( p-1\right)
}d_{G_{CM}}^{2}\left( \mathbf{e}, h\right) \right). \label{e.h4.14}$$
We will see later that when $f$ is holomorphic and $p=2,$ the above estimate in Eq. (\[e.h4.14\]) may be improved to$$\left\vert \left( S_{T}f\right) \left( h\right) \right\vert
\leqslant \left\Vert f\right\Vert_{L^{2}\left( \nu_{T}\right)
}\exp\left( \frac {1}{2T}d_{G_{CM}}^{2}\left( \mathbf{e}, h\right)
\right) \text{ for any } h
\in G_{CM}. \label{e.h4.15}$$ This bound is a variant of Bargmann’s pointwise bounds (see [@Bargmann61 Eq. (1.7)] and [@Driver1997c Eq. (5.4)]).
\[l.h4.9\]Let $T>0$ and suppose that $f:G\rightarrow\mathbb{C}$ is a continuous and in $L^{p}\left( \nu_{T}\right) $ for some $p>1$. Then $S_{T}f:G_{CM}\rightarrow\mathbb{C}$ is continuous.
For $q\in\left( 1, p\right)$ and $h\in G_{CM}$ we have by Hölder’s inequality and Theorem \[t.h4.5\] that$$\begin{aligned}
\mathbb{E}\left\vert f\left( hg\left( T\right) \right)
\right\vert ^{q} & =v_{T}\left( \left\vert f\right\vert
^{q}Z_{h}^{l}\right) \leqslant \left\Vert f\right\Vert_{L^{p}\left(
\nu_{T}\right) }^{q/p}\cdot\left\Vert Z_{h_{n}}^{l}\right\Vert
_{L^{\frac{p}{p-q}}\left( \nu_{T}\right)
}\nonumber\\
& \leqslant\left\Vert f\right\Vert_{L^{p}\left( \nu_{T}\right) }^{q/p}\exp\left( \frac{c\left( k\left( \omega\right) T/2\right) q}{T\left(
p-q\right) }d_{G_{CM}}^{2}\left( \mathbf{e},h\right) \right)
\label{e.h4.16}$$ Hence if $\left\{ h_{n}\right\}_{n=1}^{\infty}\subset G_{CM}$ is a sequence converging to $h\in G_{CM}$, it follows that $$\sup_{n}\mathbb{E}\left\vert f\left( h_{n}g\left( T\right) \right)
\right\vert ^{q}\leqslant\left\Vert f\right\Vert_{L^{p}\left(
\nu_{T}\right) }^{q/p}\exp\left( \frac{c\left( k\left(
\omega\right) T/2\right) q}{T\left( p-q\right)
}\sup_{n}d_{G_{CM}}^{2}\left( \mathbf{e}, h_{n}\right) \right)
<\infty\label{e.h4.17},$$ which implies that $\left\{ f\left( h_{n}g\left( T\right) \right)
\right\}_{n=1}^{\infty}$ is uniformly integrable. Since by continuity of $f$, $\lim_{n\rightarrow\infty}f\left( h_{n}g\left(
T\right) \right) =f\left( hg\left( T\right) \right)$, we may pass to the limit under the expectation to find $$\lim_{n\rightarrow\infty}S_{T}f\left( h_{n}\right)
=\lim_{n\rightarrow \infty}\mathbb{E}f\left( h_{n}g\left( T\right)
\right) =\mathbb{E}\left[ f\left( hg\left( T\right) \right)
\right] =S_{T}f\left( h\right).$$
Finite dimensional approximations\[s.h4.2\]
-------------------------------------------
\[d.h4.10\]For each $P\in\operatorname*{Proj}\left( W\right) $, let $g_{P}\left( t\right)$ denote the $G_{P}$–valued Brownian motion defined by $$g_{P}\left( t\right) =\left( PB\left( t\right),B_{0}\left(
t\right) +\frac{1}{2}\int_{0}^{t}\omega\left( PB\left( \tau\right)
,dPB\left(
\tau\right) \right) \right). \label{e.h4.18}$$ Also, for any $t>0$, let $\nu_{t}^{P}:=\operatorname*{Law}\left(
g_{P}\left( t\right) \right)$ be the corresponding heat kernel measure on $G_{P}$.
The following Theorem is a restatement of [@DG07b Theorem 4.16].
\[Integrated heat kernel bounds\]\[t.h4.11\]Suppose that $\rho
^{2}:G\rightarrow\lbrack0,\infty)$ be defined as $$\rho^{2}\left( w,c\right) :=\left\Vert
w\right\Vert_{W}^{2}+\left\Vert
c\right\Vert_{\mathbf{C}}. \label{e.h4.19}$$ Then there exists a $\delta>0$ such that for all $\varepsilon\in\left( 0, \delta\right)$ and $T>0$ $$\sup_{P\in\operatorname*{Proj}\left( W\right) }\mathbb{E}\left[
e^{\frac{\varepsilon}{T}\rho^{2}\left( g_{P}\left( T\right) \right)
}\right] <\infty\text{ and }\int_{G}e^{\frac{\varepsilon}{T}\rho^{2}\left(
g\right) }d\nu_{T}\left( g\right) <\infty. \label{e.h4.20}$$
\[p.h4.12\]Let $P_{n}\in\operatorname*{Proj}\left( W\right)$ such that $P_{n}|_{H}\uparrow I_{H}$ on $H$ and let $g_{n}\left( T\right) :=g_{P_{n}}\left( T\right)$. Further suppose that $\delta>0$ is as in Theorem \[t.h4.11\], $p\in\lbrack1,\infty)$, and $f:G\rightarrow\mathbb{C}$ is a continuous function such that $$\left\vert f\left( g\right) \right\vert \leqslant Ce^{\varepsilon\rho
^{2}\left( g\right) /\left( pT\right) }\text{ for all }g\in G
\label{e.h4.21}$$ for some $\varepsilon\in\left( 0,\delta\right)$. Then $f\in
L^{p}\left(
\nu_{T}\right)$ and for all $h\in G$ we have$$\lim_{n\rightarrow\infty}\mathbb{E}\left\vert f\left( hg\left( T\right)
\right) -f\left( hg_{n}\left( T\right) \right) \right\vert ^{p}=0,
\label{e.h4.22}$$ and$$\lim_{n\rightarrow\infty}\mathbb{E}\left\vert f\left( g\left( T\right)
h\right) -f\left( g_{n}\left( T\right) h\right) \right\vert ^{p}=0.
\label{e.h4.23}$$
If $q\in\left( p,\infty\right)$ is sufficiently close to $p$ so that $qp^{-1}\varepsilon<\delta$, then$$\sup_{n}\mathbb{E}\left\vert f\left( g_{n}\left( T\right) \right)
\right\vert ^{q}\leqslant C^{q}\sup_{n}\mathbb{E}\left[ e^{p^{-1}q\varepsilon\rho^{2}\left( g\right) /T}\right]$$ which is finite by Theorem \[t.h4.11\]. This shows that $\left\{
\left\vert f\left( g_{n}\left( T\right) \right) \right\vert
^{p}\right\}_{n=1}^{\infty}$ is uniformly integrable. As a consequence of [@DG07b Lemma 4.7] and the continuity of $f$, we also know that $f\left( g_{n}\left( T\right) \right) \rightarrow
f\left( g\left( T\right) \right)$ in probability as $n\rightarrow\infty$. Thus we have shown Eqs. (\[e.h4.22\]) and (\[e.h4.23\]) hold when $h=\mathbf{e}=0$. Now suppose that $g=\left( w,c\right)$ and $h=\left( A,a\right)$ are in $G$. Then for all $\alpha>0,$$$\begin{aligned}
\rho^{2}\left( gh\right) & =\left\Vert
w+A\right\Vert_{W}^{2}+\left\Vert
a+c+\frac{1}{2}\omega\left( w,A\right) \right\Vert_{\mathbf{C}}\nonumber\\
& \leqslant\left\Vert w\right\Vert_{W}^{2}+\left\Vert A\right\Vert_{W}^{2}+2\left\Vert A\right\Vert_{W}\left\Vert w\right\Vert
_{W}+\left\Vert
a\right\Vert_{\mathbf{C}}+\left\Vert c\right\Vert_{\mathbf{C}}+\frac{1}{2}\left\Vert \omega\left( w,A\right) \right\Vert_{\mathbf{C}}\nonumber\\
& \leqslant\rho^{2}\left( g\right) +\rho^{2}\left( h\right) +C\left\Vert
A\right\Vert_{W}\left\Vert w\right\Vert_{W}\label{e.h4.24}\\
& \leqslant\rho^{2}\left( g\right) +\rho^{2}\left( h\right) +\frac{C}{2}\left[ \alpha^{-1}\left\Vert
A\right\Vert_{W}^{2}+\alpha\left\Vert
w\right\Vert_{W}^{2}\right] \nonumber\\
& \leqslant\left( 1+\frac{C\alpha}{2}\right) \rho^{2}\left(
g\right) +\left( 1+\frac{C}{2\alpha}\right) \rho^{2}\left(
h\right),\nonumber\end{aligned}$$ where $C:=\left( 2+\frac{1}{2}\left\Vert \omega\right\Vert
_{0}\right)$. As Eq. (\[e.h4.24\]) is invariant under interchanging $g$ and $h$ the same bound also hold for $\rho^{2}\left( hg\right)$. By choosing $\alpha>0$ sufficiently small so that $\left( 1+\frac{C\alpha}{2}\right) \varepsilon
<\delta$, we see that $g\rightarrow f\left( gh\right)$ and $g\rightarrow f\left( hg\right)$ satisfy the same type of bound as in Eq. (\[e.h4.21\]) for $\,g\rightarrow f\left( g\right)$. Therefore, by the first paragraph, we have now verified Eqs. (\[e.h4.22\]) and (\[e.h4.23\]) hold for any $h\in G$.
Holomorphic functions on $G$ and $G_{CM}$\[s.h5\]
=================================================
We will begin with a short summary of the results about holomorphic functions on Banach spaces that will be needed in this paper.
Holomorphic functions on Banach spaces\[s.h5.1\]
------------------------------------------------
Let $X$ and $Y$ be two complex Banach space and for $a\in X$ and $\delta>0$ let $$B_{X}\left( a,\delta\right) :=\left\{ x\in X:\left\Vert x-a\right\Vert
_{X}<\delta\right\}$$ be the open ball in $X$ with center $a$ and radius $\delta$.
\[[Hille and Phillips [@HP74 Definition 3.17.2, p. 112.]]{}\]\[d.h5.1\]Let $\mathcal{D}$ be an open subset of $X$. A function $u:\mathcal{D}\rightarrow Y\ $is said to be **holomorphic (or analytic)** if the following two conditions hold.
1. $u$ is locally bounded, namely for all $a\in\mathcal{D}$ there exists an $r_{a}>0$ such that $$M_{a}:=\sup\left\{ \left\Vert u\left( x\right)
\right\Vert_{Y}:x\in B_{X}\left( a,r_{a}\right) \right\} <\infty.$$
2. The function $u$ is complex Gâteaux differentiable on $\mathcal{D}$, i.e. for each $a\in\mathcal{D}$ and $h\in X$, the function $\lambda\rightarrow u\left( a+\lambda h\right)$ is complex differentiable at $\lambda =0\in\mathbb{C}$.
(Holomorphic and analytic will be considered to be synonymous terms for the purposes of this paper.)
The next theorem gathers together a number of basic properties of holomorphic functions which may be found in [@HP74]. (Also see [@Herve89].) One of the key ingredients to all of these results is Hartog’s theorem, see [@HP74 Theorem 3.15.1].
\[t.h5.2\]If $u:\mathcal{D}\rightarrow Y$ is holomorphic, then there exists a function $u^{\prime}:\mathcal{D}\rightarrow\operatorname{Hom}\left(
X,Y\right)$, the space of bounded **complex** linear operators from $X$ to $Y$, satisfying
1. If $a\in\mathcal{D}$, $x\in B_{X}\left( a,r_{a}/2\right)$, and $h\in
B_{X}\left( 0,r_{a}/2\right)$, then $$\left\Vert u\left( x+h\right) -u\left( x\right) -u^{\prime}\left(
x\right) h\right\Vert_{Y}\leqslant\frac{4M_{a}}{r_{a}\left( r_{a}-2\left\Vert h\right\Vert_{X}\right) }\left\Vert h\right\Vert
_{X}^{2}.
\label{e.h5.1}$$ In particular, $u$ is continuous and Frechét differentiable on $\mathcal{D}$.
2. The function $u^{\prime}:\mathcal{D}\rightarrow\operatorname{Hom}\left(
X,Y\right)$ is holomorphic.
\[r.h5.3\]By applying Theorem \[t.h5.2\] repeatedly, it follows that any holomorphic function, $u:\mathcal{D}\rightarrow Y$ is Frechét differentiable to all orders and each of the Frechét differentials are again holomorphic functions on $\mathcal{D}$.
By [@HP74 Theorem 26.3.2 on p. 766.], for each $a\in\mathcal{D}$ there is a linear operator, $u^{\prime}\left(
a\right): X\rightarrow Y$ such that $du\left( a+\lambda h\right)
/d\lambda|_{\lambda=0}=u^{\prime}\left( a\right) h$. The Cauchy estimate in Theorem 3.16.3 (with $n=1$) of [@HP74] implies that if $a\in\mathcal{D}$, $x\in B_{X}\left( a,r_{a}/2\right)$ and $h\in
B_{X}\left( 0,r_{a}/2\right)$ (so that $x+h\in B_{X}\left( a,
r_{a}\right) )$, then $\left\Vert u^{\prime}\left( x\right)
h\right\Vert_{Y}\leqslant M_{a}$. It follows from this estimate that$$\sup\left\{ \left\Vert u^{\prime}\left( x\right) \right\Vert
_{\operatorname{Hom}\left( X,Y\right) }:x\in B_{X}\left( a,r_{a}/2\right)
\right\} \leqslant2M_{a}/r_{a}. \label{e.h5.2}$$ and hence that $u^{\prime}:\mathcal{D}\rightarrow\operatorname{Hom}\left(
X,Y\right)$ is a locally bounded function. The estimate in Eq. (\[e.h5.1\]) appears in the proof of the Theorem 3.17.1 in [@HP74] which completes the proof of item 1.
To prove item 2. we must show $u^{\prime}$ is Gâteaux differentiable on $\mathcal{D}$. We will in fact show more, namely, that $u^{\prime}$ is Frechét differentiable on $\mathcal{D}$. Given $h\in X$, let $F_{h}:\mathcal{D}\rightarrow Y$ be defined by $F_{h}\left( x\right) :=u^{\prime}\left( x\right) h$. According to [@HP74 Theorem 26.3.6], $F_{h}$ is holomorphic on $\mathcal{D}$ as well. Moreover, if $a\in\mathcal{D}$ and $x\in
B\left( a,r_{a}/2\right)$ we have by Eq. (\[e.h5.2\]) that $$\left\Vert F_{h}\left( x\right)
\right\Vert_{Y}\leqslant2M_{a}\left\Vert h\right\Vert_{X}/r_{a}.$$ So applying the estimate in Eq. (\[e.h5.1\]) to $F_{h}$, we learn that$$\left\Vert F_{h}\left( x+k\right) -F_{h}\left( x\right)
-F_{h}^{\prime }\left( x\right)
k\right\Vert_{Y}\leqslant\frac{4\left( 2M_{a}\left\Vert
h\right\Vert_{X}/r_{a}\right) }{\frac{r_{a}}{2}\left( \frac{r_{a}}{2}-2\left\Vert k\right\Vert_{X}\right) }\cdot\left\Vert
k\right\Vert
_{X}^{2} \label{e.h5.3}$$ for $x\in B\left( a,r_{a}/4\right)$ and $\left\Vert k\right\Vert_{X}<r_{a}/4$, where $$F_{h}^{\prime}\left( x\right)
k=\frac{d}{d\lambda}\left|_{0}F_{h}\left( x+\lambda k\right)\right.
=\frac{d}{d\lambda}|_{0}u^{\prime}\left( x+\lambda k\right)
h=:\left( \delta^{2}u\right) \left( x;h,k\right).$$ Again by [@HP74 Theorem 26.3.6], for each fixed $x\in\mathcal{D}$, $\left( \delta^{2}u\right) \left(
x;h,k\right) $ is a continuous symmetric bilinear form in $\left(
h,k\right) \in X\times X$. Taking the supremum of Eq. (\[e.h5.3\]) over those $h\in X$ with $\left\Vert h\right\Vert_{X}=1$, we may conclude that$$\begin{aligned}
\big\Vert u^{\prime}\left( x+k\right) -u^{\prime}\left( x\right) &
-\delta^{2}u\left( x;\cdot,k\right) \big\Vert_{\operatorname{Hom}\left(
X,Y\right) }\\
& =\sup_{\left\Vert h\right\Vert_{X}=1}\left\Vert F_{h}\left(
x+k\right)
-F_{h}\left( x\right) -F_{h}^{\prime}\left( x\right) k\right\Vert_{Y}\\
& \leqslant\frac{4\left( 2M_{a}/r_{a}\right)
}{\frac{r_{a}}{2}\left( \frac{r_{a}}{2}-2\left\Vert
k\right\Vert_{X}\right) } \left\Vert k\right\Vert_{X}^{2}.\end{aligned}$$ This estimate shows $u^{\prime}$ is Frechét differentiable with $u^{\prime\prime}\left( x\right) \in\operatorname{Hom}\left(
X,\operatorname{Hom}\left( X,Y\right) \right)$ being given by $u^{\prime\prime}\left( x\right) k=\left( \delta^{2}u\right)
\left( x;\cdot,k\right) \in\operatorname{Hom}\left( X,Y\right)$ for all $k\in X$ and $x\in\mathcal{D}$.
Holomorphic functions on $G$ and $G_{CM}$\[s.h5.2\]
---------------------------------------------------
For the purposes of this section, let $G_{0}=G$ and $\mathfrak{g}_{0}=\mathfrak{g}$ or $G_{0}=G_{CM}$ and $\mathfrak{g}_{0}=\mathfrak{g}_{CM}$. Also for $g,h\in\mathfrak{g}$, let (as usual) $ad_{g}h:=\left[ g,h\right]$.
\[l.h5.4\]For each $g\in G_{0}$, $l_{g}:G_{0}\rightarrow G_{0}$ is holomorphic in the $\left\Vert \cdot\right\Vert_{\mathfrak{g}_{0}}$–topology. Moreover, a function $u:G_{0}\rightarrow\mathbb{C}$ defined in a neighborhood of $g\in G_{0}$ is Gâteaux (Frechét) differentiable at $g$ iff $u\circ$$l_{g}$ is Gâteaux (Frechét) differentiable at $0$. In addition, if $u$ is Frechét differentiable at $g$, then$$\left( u\circ\,l_{g}\right)^{\prime}\left( 0\right)
h=u^{\prime}\left(
g\right) \left( h+\frac{1}{2}\left[ g,h\right] \right). \label{e.h5.4}$$ (See [@GrossMall Theorem 5.7] for an analogous result in the context of path groups.)
Since $$\,l_{g}\left( h\right) =gh=g+h+\frac{1}{2}\left[ g,h\right] =g+\left(
Id_{\mathfrak{g}_{0}}+\frac{1}{2}ad_{g}\right) h,$$ it is easy to see that $l_{g}$ is holomorphic and $l_{g}^{\prime}$ is the constant function equal to $Id_{\mathfrak{g}_{0}}+\frac{1}{2}ad_{g}\in\operatorname*{End}\left( \mathfrak{g}_{0}\right)$. Using $ad_{g}^{2}=0$ or the fact that $l_{g}^{-1}=l_{g^{-1}}$, we see that $l_{g}^{\prime}$ is invertible and that $$l_{g}^{\prime}{}^{-1}=\left( Id_{\mathfrak{g}_{0}}+\frac{1}{2}ad_{g}\right)
^{-1}=Id_{\mathfrak{g}_{0}}-\frac{1}{2}ad_{g}.$$ These observations along with the chain rule imply the Frechét differentiability statements of the lemma and the identity in Eq. (\[e.h5.4\]).
If $u$ is Gâteaux differentiable at $g$, $h\in\mathfrak{g}_{0}$, and $k:=h+\frac{1}{2}\left[ g,h\right]$, then $$\frac{d}{d\lambda}|_{0}u\circ\,l_{g}\left( \lambda h\right) =\frac
{d}{d\lambda}|_{0}u\left( g\cdot\left( \lambda h\right) \right) =\frac
{d}{d\lambda}|_{0}u\left( g+\lambda k\right)$$ and the existence of $\frac{d}{d\lambda}|_{0}u\left( g+\lambda
k\right)$ implies the existence of $\frac{d}{d\lambda}|_{0}u\circ$$l_{g}\left( \lambda
h\right)$. Conversely, if $u\circ$$l_{g}$ is Gâteaux differentiable at $0$, $h\in\mathfrak{g}_{0}$, and $$k:=h-\frac{1}{2}\left[ g,h\right] =\left( Id_{\mathfrak{g}_{0}}+\frac{1}{2}ad_{g}\right)^{-1}h,$$ then $$\,l_{g}\left( \lambda k\right) =g+\lambda\left( Id_{\mathfrak{g}_{0}}+\frac{1}{2}ad_{g}\right) k=g+\lambda h.$$ So the existence of $\frac{d}{d\lambda}|_{0}\left( u\circ\,l_{g}\right)
\left( \lambda k\right)$ implies the existence of $\frac{d}{d\lambda}|_{0}u\left( g+\lambda h\right)$.
\[c.h5.5\]A function $u:G_{0}\rightarrow\mathbb{C}$ is holomorphic iff it is locally bounded and $h\rightarrow u\left(
ge^{h}\right) =u\left( g\cdot h\right)$ is Gâteaux (Frechét) differentiable at $0$ for all $g\in G_{0}$. Moreover, if $u$ is holomorphic and $h\in\mathfrak{g}_{0}$, then $$\left( \tilde{h}u\right) \left( g\right) =\frac{d}{d\lambda}|_{0}u\left(
ge^{\lambda h}\right) =u^{\prime}\left( g\right) \left( h+\left[
g,h\right] \right)$$ is holomorphic as well.
\[n.h5.6\]The space of globally defined holomorphic functions on $G$ and $G_{CM}$ will be denoted by $\mathcal{H}\left( G\right)$ and $\mathcal{H}\left( G_{CM}\right)$ respectively.
Notice that the space $\mathcal{A}$ of holomorphic cylinder functions as described in Definition \[d.h4.3\] is contained in $\mathcal{H}\left( G\right)$. Also observe that a simple induction argument using Corollary \[c.h5.5\] allows us to conclude that $\tilde{h}_{1}\dots\tilde{h}_{n}u\in\mathcal{H}\left( G_{0}\right)$ for all $u\in\mathcal{H}\left(
G_{0}\right)$ and $h_{1},\dots,h_{n}\in\mathfrak{g}_{0}$.
\[p.h5.7\]If $f\in\mathcal{H}\left( G\right)$ and $h\in\mathfrak{g}$, then $\widetilde{ih}f=i\tilde{h}f$, $\widetilde{ih}\bar{f}=-i\tilde{h}\bar {f}$, $$\begin{aligned}
\left[
\left( \widetilde{ih}\right)^{2}+\tilde{h}^{2}\right] f &
=0,\text{ and}\label{e.h5.5}\\
\left( \tilde{h}^{2}+\widetilde{ih}^{2}\right) \left\vert f\right\vert ^{2}
& =4\left\vert \tilde{h}f\right\vert ^{2}. \label{e.h5.6}$$
The first assertions are directly related to the definition of $f$ being holomorphic. Using the identity $\widetilde{ih}f=i\tilde{h}f$ twice implies Eq. (\[e.h5.5\]). Eq.(\[e.h5.6\]) is a consequence of summing the following two identities $$\tilde{h}^{2}\left\vert f\right\vert ^{2}=\tilde{h}\left( f\cdot\bar
{f}\right) =\tilde{h}^{2}f\cdot\bar{f}+f\cdot\tilde{h}^{2}\bar{f}+2\tilde
{h}f\cdot\tilde{h}\bar{f}$$ and$$\begin{aligned}
\widetilde{ih}^{2}\left\vert f\right\vert ^{2} & =\widetilde{ih}\left(
f\cdot\bar{f}\right) =\widetilde{ih}^{2}f\cdot\bar{f}+f\cdot\widetilde
{ih}^{2}\bar{f}+2\widetilde{ih}f\cdot\widetilde{ih}\bar{f}\\
& =-\widetilde{h}^{2}f\cdot\bar{f}-f\cdot\widetilde{h}^{2}\bar{f}+2\widetilde{h}f\cdot\widetilde{h}\bar{f},\end{aligned}$$ and using $\widetilde{h}\bar{f}=\overline{\widetilde{h}f}$.
\[c.h5.8\]Let $L$ be as in Proposition \[p.h4.4\]. Suppose that $f:G\rightarrow\mathbb{C}$ is a holomorphic cylinder function (i.e. $f\in\mathcal{A})$, then $Lf=0$ and$$L\left\vert f\right\vert ^{2}=\sum_{h\in\Gamma}\left\vert \tilde
{h}f\right\vert ^{2}, \label{e.h5.7}$$ where $\Gamma$ is an orthonormal basis for $\mathfrak{g}_{CM}$ of the form $$\Gamma=\Gamma_{e}\cup\Gamma_{f}=\left\{ \left( e_{j},0\right)
\right\}_{j=1}^{\infty}\cup\left\{ \left( 0,f_{j}\right)
\right\}_{j=1}^{d}
\label{e.h5.8}$$ with $\left\{ e_{j}\right\}_{j=1}^{\infty}$ and $\left\{
f_{j}\right\}_{j=1}^{d}$ being complex orthonormal bases for $H$ and $\mathbf{C}$ respectively.
These assertions follow directly form Eqs. (\[e.h4.4\]), (\[e.h5.5\]), and (\[e.h5.6\]).
Formally, if $f:G\rightarrow\mathbb{C}$ is a holomorphic function, then $e^{TL/4}f=f$ and therefore we should expect $S_{T}f=f|_{G_{CM}}$ where $S_{T}$ is defined in Definition \[d.h4.7\]. Theorem \[t.h5.9\] below is a precise version of this heuristic.
\[t.h5.9\]Suppose $p\in\left( 1,\infty\right)$ and $f:G\rightarrow
\mathbb{C}$ is a continuous function such that $f|_{G_{CM}}\in\mathcal{H}\left( G_{CM}\right)$ and there exists $P_{n}\in\operatorname*{Proj}\left( W\right)$ such that $P_{n}|_{H}\uparrow I_{H}$, then $$\left\Vert f\right\Vert_{L^{p}\left( \nu_{T}\right) }\leqslant\sup
_{n}\left\Vert f\right\Vert_{L^{p}\left(
G_{P_{n}},\nu_{T}^{P_{n}}\right)
}. \label{e.h5.9}$$ If we further assume that$$\sup_{n}\left\Vert f\right\Vert_{L^{p}\left( G_{P_{n}},\nu_{T}^{P_{n}}\right) }<\infty, \label{e.h5.10}$$ then $f\in L^{p}\left( \nu_{T}\right)$, $S_{T}f=f|_{G_{CM}}$, and $f$ satisfies the Gaussian bounds $$\left\vert f\left( h\right) \right\vert \leqslant\left\Vert f\right\Vert
_{L^{p}\left( \nu_{T}\right) }\exp\left( \frac{c\left( k\left(
\omega\right) T/2\right) }{T\left( p-1\right) }d_{G_{CM}}^{2}\left(
\mathbf{e},h\right) \right) \text{ for any } h\in G_{CM}. \label{e.h5.11}$$
According to [@DG07b Lemma 4.7], by passing to a subsequence if necessary, we may assume that $g_{P_{n}}\left( T\right) \rightarrow
g\left( T\right)$ almost surely. Hence an application of Fatou’s lemma implies Eq. (\[e.h5.9\]). In particular, if we assume Eq. (\[e.h5.10\]) holds, then $f\in L^{p}\left( \nu_{T}\right)$ and so $S_{T}f$ is well defined.
Now suppose that $P\in\operatorname*{Proj}\left( W\right)$ and $h\in G_{P}$. Working exactly as in the proof of Lemma \[l.h4.9\], we find for any $q\in\left( 1,p\right)$ that$$\mathbb{E}\left\vert f\left( hg_{P}\left( T\right) \right)
\right\vert ^{q}\leqslant\left\Vert f\right\Vert_{L^{p}\left(
G_{P},\nu_{T}^{P}\right) }^{q/p}\exp\left( \frac{c\left(
k_{P}\left( \omega\right) T/2\right) q}{T\left( q-p\right)
}d_{G_{P}}^{2}\left( \mathbf{e},h\right) \right),
\label{e.h5.12}$$ where $d_{G_{P}}\left( \cdot,\cdot\right)$ is the Riemannian distance on $G_{P}$ and (see [@DG07b Eq. (\[e.h5.13\])]),$$k_{P}\left( \omega\right) :=-\frac{1}{2}\sup\left\{ \left\Vert
\omega\left( \cdot,A\right) \right\Vert_{\left( PH\right)^{\ast}\otimes\mathbf{C}}^{2}:A\in PH,\ \left\Vert A\right\Vert
_{PH}=1\right\}.
\label{e.h5.13}$$ Observe that $k_{P}\left( \omega\right) \geqslant k\left(
\omega\right)$ and therefore, as $c$ is a decreasing function, $c\left( k\left( \omega\right) \right) \geqslant c\left(
k_{P}\left( \omega\right) \right)$. Let $m\in\mathbb{N}$ be given and $h\in G_{P_{m}}$. Then for $n\geqslant m$ we have from Eq. (\[e.h5.12\]) that$$\begin{aligned}
\mathbb{E}\left\vert f\left( hg_{P_{n}}\left( T\right) \right)
\right\vert ^{q} & \leqslant\left\Vert f\right\Vert_{L^{p}\left(
G_{P_{n}},\nu_{T}^{P_{n}}\right) }^{q/p}\exp\left( \frac{c\left(
k_{P_{n}}\left( \omega\right) T/2\right) q}{T\left( q-p\right)
}d_{G_{P_{n}}}^{2}\left(
\mathbf{e},h\right) \right) \\
& \leqslant\left\Vert f\right\Vert_{L^{p}\left( G_{P_{n}},\nu_{T}^{P_{n}}\right) }^{q/p}\exp\left( \frac{c\left( k\left( \omega\right)
T/2\right) q}{T\left( q-p\right) }d_{G_{P_{m}}}^{2}\left( \mathbf{e},h\right) \right)\end{aligned}$$ wherein in the last inequality we have used $c\left( k\left(
\omega\right) \right) \geqslant c\left( k_{P}\left( \omega\right)
\right)$ and the fact that $d_{G_{P_{n}}}^{2}\left(
\mathbf{e},h\right)$ is decreasing in $n\geqslant m$. Hence it follows that $\sup_{n\geqslant m}\mathbb{E}\left\vert f\left(
hg_{P_{n}}\left( T\right) \right) \right\vert ^{q}<\infty$ and thus that $\left\{ f\left( hg_{P_{n}}\left( T\right) \right)
\right\}_{n\geqslant m}$ is uniformly integrable. Therefore, $$S_{T}f\left( h\right) =\mathbb{E}f\left( hg\left( T\right) \right)
=\lim_{n\rightarrow\infty}\mathbb{E}f\left( hg_{P_{n}}\left( T\right)
\right) =\lim_{n\rightarrow\infty}\int_{G_{P_{n}}}f\left( hx\right)
d\nu_{T}^{P_{n}}\left( x\right). \label{e.h5.14}$$
On the other hand by [@DG07b Lemma 4.8] (with $T$ replaced by $T/2$ because of our normalization in Eq. (\[e.h4.1\])), $\nu_{T}^{P_{n}}$ is the heat kernel measure on $G_{P_{n}}$ based at $\mathbf{e}\in G_{P_{n}}$, i.e. $\nu_{T}^{P_{n}}\left( dx\right)
=p_{T/2}^{P_{n}}\left( e, x\right) dx$, where $dx$ is the Riemannian volume measure (equal to a Haar measure) on $G_{P_{n}}$ and $p_{T}^{P_{n}}\left( x,y\right)$ is the heat kernel on $G_{P_{n}}$. Since $f|_{G_{P_{n}}}$ is holomorphic, the previous observations allow us to apply [@DG07a Proposition 1.8] to conclude that $$\int_{G_{P_{n}}}f\left( hx\right) d\nu_{T}^{P_{n}}\left( x\right)
=f\left( \mathbf{e}\right) \text{ for all }n\geqslant m. \label{e.h5.15}$$ As $m\in\mathbb{N}$ was arbitrary, combining Eqs. (\[e.h5.14\]) and (\[e.h5.15\]) implies that $S_{T}f\left( h\right) =f\left(
h\right)$ for all $h\in G_{0}:=\cup_{m\in\mathbb{N}}G_{P_{m}}$. Recall from Lemma \[l.h4.9\] that $S_{T}f:G_{CM}\rightarrow\mathbb{C}$ is continuous and from the proof of [@DG07b Theorem 8.1] that $G_{0}$ is a dense subgroup of $G_{CM}$. Therefore we may conclude that in fact $S_{T}f\left(
h\right) =f\left( h\right)$ for all $h\in G_{CM}$. The Gaussian bound now follows immediately from Corollary \[c.h4.8\].
\[c.h5.10\]Suppose that $\delta>0$ is as in Theorem \[t.h4.11\] and $f:G\rightarrow\mathbb{C}$ is a continuous function such that $f|_{G_{CM}}$ is holomorphic and $\left\vert
f\right\vert \leqslant Ce^{\varepsilon\rho ^{2}/\left( pT\right) }$ for some $\varepsilon\in\lbrack0,\delta)$. Then $f\in L^{p}\left(
\nu_{T}\right)$, $S_{T}f=f$, and the Gaussian bounds in Eq. (\[e.h5.11\]) hold.
By Theorem \[t.h4.11\], the given function $f$ verifies Eq. (\[e.h5.10\]) for any choice of $\left\{ P_{n}\right\}
_{n=1}^{\infty}\subset \operatorname*{Proj}\left( W\right)$ with $P_{n}|_{H}\uparrow P$ strongly as $n\uparrow\infty$. Hence Theorem \[t.h5.9\] is applicable.
As a simple consequence of Corollary \[c.h5.10\], we know that $\mathcal{P}\subset L^{p}\left( \nu_{T}\right)$ (see Definition \[d.h1.6\]) and that $\left( S_{T}p\right) \left( h\right)
=p\left( h\right)$ for all $h\in G_{CM}$ and $p\in\mathcal{P}$.
\[n.h5.11\]For $T>0$ and $1\leqslant p<\infty$, let $\mathcal{A}_{T}^{p}$ and $\mathcal{H}_{T}^{p}\left( G\right)$ denote the $L^{p}\left( \nu_{T}\right)$ – closure of $\mathcal{A}\cap L^{p}\left( \nu_{T}\right)$ and $\mathcal{P}$, where $\mathcal{A}$ and $\mathcal{P}$ denote the holomorphic cylinder functions (see Definition \[d.h4.3\]) and holomorphic cylinder polynomials on $G$ respectively.
\[t.h5.12\]For all $T>0$ and $p\in\left( 1,\infty\right)$, we have $S_{T}\left( \mathcal{H}_{T}^{p}\left( G\right) \right)
\subset \mathcal{H}\left( G_{CM}\right)$.
Let $f\in\mathcal{H}_{T}^{p}\left( G\right)$ and $p_{n}\in\mathcal{P}$ such that $\lim_{n\rightarrow\infty}\left\Vert
f-p_{n}\right\Vert_{L^{p}\left( \nu_{T}\right) }=0$. If $h\in
G_{CM}$, then by Corollary \[c.h4.8\] $$\begin{aligned}
\left\vert S_{T}f\left( h\right) -p_{n}\left( h\right) \right\vert &
=\left\vert S_{T}\left( f-p_{n}\right) \left( h\right) \right\vert \\
& \leqslant\left\Vert f-p_{n}\right\Vert_{L^{p}\left( \nu_{T}\right) }\exp\left( \frac{c\left( k\left( \omega\right) T/2\right)
}{T\left( p-1\right) }d_{G_{CM}}^{2}\left( \mathbf{e},h\right)
\right).\end{aligned}$$ This shows that $S_{T}f$ is the limit of $p_{n}|_{G_{CM}}\in\mathcal{H}\left( G_{CM}\right)$ with the limit being uniform over any bounded subset of $h$’s contained in $G_{CM}$. This is sufficient to show that $S_{T}f\in
\mathcal{H}\left( G_{CM}\right)$ via an application of [@HP74 Theorem 3.18.1].
\[r.h5.13\] It seems reasonable to conjecture that $\mathcal{A}_{T}^{2}=\mathcal{H}_{T}^{2}\left( G\right)$, nevertheless we do not know if these two spaces are equal! We also do not know if $S_{T}f=f$ for every $f\in\mathcal{A}\cap L^{2}\left(
\nu_{T}\right)$. However, Theorem \[t.h5.9\] does show that $S_{T}f=f$ for all $f\in\mathcal{A}\cap
_{P\in\operatorname*{Proj}\left( W\right) }L^{p}\left(
\nu_{T}^{P}\right)$ with $L^{p}\left( \nu_{T}^{P}\right)$–norms of $f$ being bounded.
The Taylor isomorphism theorem\[s.h6\]
======================================
The main purpose of this section is to prove the Taylor isomorphism Theorem \[t.h1.5\] (or Theorem \[t.h6.10\]). We begin with the formal development of the algebraic setup. In what follows below for a vector space $V$ we will denote the algebraic dual to $V$ by $V^{\prime}$. If $V$ happens to be a normed space, we will let $V^{\ast}$ denote the topological dual of $V$.
A non-commutative Fock space\[s.h6.1\]
--------------------------------------
\[n.h6.1\]For $n\in\mathbb{N}$ let $\mathfrak{g}_{CM}^{\otimes
n}$ denote the $n$–fold algebraic tensor product of $\mathfrak{\
g}_{CM}$ with itself, and by convention let $\mathfrak{g}_{CM}^{\otimes0}:=\mathbb{C}$. Also let$$\mathbf{T:=T}\left( \mathfrak{g}_{CM}\right) =\mathbb{C}\oplus
\mathfrak{g}_{CM}\oplus\mathfrak{g}_{CM}^{\otimes2}\oplus\mathfrak{g}_{CM}^{\otimes3}\oplus\dots$$ be the algebraic tensor algebra over $\mathfrak{g}_{CM}$, $\mathbf{T}^{\prime }$ be its algebraic dual, and $J$ be the two-sided ideal in $\mathbf{T}$ generated by the elements in Eq. (\[e.h1.3\]). The backwards annihilator of $J$ is$$J^{0}=\{\alpha\in\mathbf{T}^{\prime}:\alpha\left( J\right) =0\}.
\label{e.h6.1}$$ For any $\alpha\in\mathbf{T}^{\prime}$ and $n\in\mathbb{N\cup}\left\{
0\right\}$, we let $\alpha_{n}:=\alpha|_{\mathfrak{g}_{CM}^{\otimes n}}\in\left( \mathfrak{g}_{CM}^{\otimes n}\right)^{\prime}$.
After the next definition we will be able to give numerous examples of elements in $J^{0}$.
\[Left differentials\]\[d.h6.2\]For $f\in\mathcal{H}\left(
G_{CM}\right) $, $n\in\mathbb{N\cup}\left\{ 0\right\}$, and $g\in
G_{CM}$, define $\hat{f}_{n}\left( g\right) :=D^{n}f\left(
g\right) \in\left( \mathfrak{g}_{CM}^{\otimes n}\right)^{\prime}$ by $$\begin{aligned}
& \left( D^{0}f\right) \left( g\right) =f\left( g\right) \text{
and}\nonumber\\
& \left\langle D^{n}f\left( g\right), h_{1}\otimes\dots\otimes
h_{n}\right\rangle =\left( \tilde{h}_{1}\dots\tilde{h}_{n}f\right)
\left(
g\right) \label{e.h6.2}$$ for all and $h_{1},...,h_{n}\in\mathfrak{g}_{0}$, where $\tilde{h}f$ is given as in Eq. (\[e.h3.8\]) or Eq. (\[e.h3.10\]). We will write $Df$ for $D^{1}f$ and $\hat{f}\left( g\right) $ to be the element of $\mathbf{T}\left( \mathfrak{g}_{CM}\right)^{\prime}$ determined by$$\left\langle \hat{f}\left( g\right),\beta\right\rangle
=\left\langle \hat{f}_{n}\left( g\right), \beta\right\rangle \text{
for all }\beta
\in\mathfrak{g}_{CM}^{\otimes n}\text{ and }n\in\mathbb{N}_{0}. \label{e.h6.3}$$
\[ex.h6.3\]As a consequence of Eq. (\[e.h3.11\]), $\hat{f}\left( g\right) \in J^{0}$ for all $f\in\mathcal{H}\left(
G_{CM}\right)$ and $g\in G_{CM}$.
In order to put norms on $J^{0}$, let us equip $\mathfrak{g}_{CM}^{\otimes n}$ with the usual inner product determined by $$\left\langle h_{1}\otimes\dots\otimes h_{n},
k_{1}\otimes\dots\otimes
k_{n}\right\rangle_{\mathfrak{g}_{CM}^{\otimes n}}=\prod_{j=1}^{n}\left\langle h_{j}, k_{j}\right\rangle_{\mathfrak{g}_{CM}}\text{ for
any
}h_{i},k_{j}\in\mathfrak{g}_{CM}. \label{e.h6.4}$$ For $n=0$ we let $\left\langle z,w\right\rangle
_{\mathfrak{g}_{CM}^{\otimes 0}}:=z\bar{w}$ for all $z,w\in\mathfrak{g}_{CM}^{\otimes0}=\mathbb{C}$. The inner product $\left\langle \cdot,\cdot\right\rangle_{\mathfrak{g}_{CM}^{\otimes n}}$ induces a dual inner product on $\left( \mathfrak{g}_{CM}^{\otimes n}\right)^{\ast}$ which we will denote by $\left\langle
\cdot,\cdot\right\rangle_{n}$. The associated norm on $\left( \mathfrak{g}_{CM}^{\otimes n}\right)^{\ast}$ will be denoted by $\left\Vert
\cdot\right\Vert_{n}$. We extend $\left\Vert \cdot\right\Vert_{n}$ to all of $\left( \mathfrak{g}_{CM}^{\otimes n}\right)^{\prime}$ by setting $\left\Vert \beta\right\Vert_{n}=\infty$ if $\beta\in\left( \mathfrak{g}_{CM}^{\otimes n}\right)^{\prime}\setminus\left(
\mathfrak{g}_{CM}^{\otimes
n}\right)^{\ast}$. If $\Gamma$ is any orthonormal basis for $\mathfrak{g}_{CM}$, then $\left\Vert \beta\right\Vert_{n}$ may be computed using $$\left\Vert \beta\right\Vert_{\mathfrak{g}_{CM}^{\otimes
n}}^{2}:=\sum_{h_{1},\dots,h_{n}\in\Gamma}\left\vert \left\langle
\beta,h_{1}\otimes
\dots\otimes h_{n}\right\rangle \right\vert ^{2}. \label{e.h6.5}$$
\[Non-commutative Fock space\]\[d.h6.4\]Given $T>0$ and $\alpha\in
J^{0}\left( \mathfrak{g}_{CM}\right)$, let $$\left\Vert \alpha\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right)
}^{2}:=\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\left\Vert
\alpha_{n}\right\Vert
_{n}^{2}. \label{e.h6.6}$$ Further let$$J_{T}^{0}\left( \mathfrak{g}_{CM}\right) :=\left\{ \alpha\in
J^{0}\left( \mathfrak{g}_{CM}\right) :\left\Vert
\alpha\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) }^{2}<\infty\right\}. \label{e.h6.7}$$
The space, $J_{T}^{0}\left( \mathfrak{g}_{CM}\right)$, is then a Hilbert space when equipped with the inner product $$\left\langle \alpha,\beta\right\rangle_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }=\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\left\langle \alpha
_{n},\beta_{n}\right\rangle_{n}\text{ for any }\alpha,\beta\in J_{T}^{0}\left( \mathfrak{g}_{CM}\right). \label{e.h6.8}$$
The Taylor isomorphism\[s.h6.2\]
--------------------------------
\[l.h6.5\]Let $f\in\mathcal{H}\left( G_{CM}\right) $ and $T>0$ and suppose that $\left\{ P_{n}\right\}_{n=1}^{\infty}\subset
\operatorname*{Proj}\left( W\right) $ is a sequence such that $P_{n}|_{\mathfrak{g}_{CM}}\uparrow I_{\mathfrak{g}_{CM}}$ as $n\rightarrow\infty$. Then$$\lim_{n\rightarrow\infty}\left\Vert \hat{f}\left( \mathbf{e}\right)
\right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{P_{n}}\right)
}=\left\Vert \hat{f}\left( \mathbf{e}\right) \right\Vert
_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) }=\left\Vert f\right\Vert_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }=\lim_{n\rightarrow\infty}\left\Vert f\right\Vert
_{L^{2}\left( G_{P_{n}},\nu_{T}^{P_{n}}\right) }, \label{e.h6.9}$$ where $\left\Vert \cdot\right\Vert_{\mathcal{H}_{T}^{2}\left(
G_{CM}\right) }$ is defined in Eq. (\[e.h1.4\]).
By Theorem 5.1 of [@Driver1997c], for all $P\in\operatorname*{Proj}\left( W\right)$, $$\left\Vert f\right\Vert_{L^{2}\left( G_{P},\nu_{T}^{P}\right)
}=\left\Vert \hat{f}\left( \mathbf{e}\right) \right\Vert
_{J_{T}^{0}\left(
\mathfrak{g}_{P}\right) }, \label{e.h6.10}$$ where $$\left\Vert \hat{f}\left( \mathbf{e}\right)
\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{P}\right) }^{2}=\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\sum_{\left\{ h_{j}\right\}_{j=1}^{n}\subset\Gamma_{P}}\left\vert
\left\langle \hat{f}\left( \mathbf{e}\right)
,h_{1}\otimes\dots\otimes
h_{n}\right\rangle \right\vert ^{2} \label{e.h6.11}$$ and $\Gamma_{P}$ is an orthonormal basis for $\mathfrak{g}_{P}$. In particular, it follows that $$\left\Vert f\right\Vert_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }=\sup_{P\in\operatorname*{Proj}\left( W\right) }\left\Vert
\hat{f}\left( \mathbf{e}\right) \right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{P}\right) }
\label{e.h6.12}$$ and hence we must now show $$\sup_{P\in\operatorname*{Proj}\left( W\right) }\left\Vert
\hat{f}\left( \mathbf{e}\right) \right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{P}\right) }=\left\Vert \hat{f}\left( \mathbf{e}\right)
\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) }. \label{e.h6.13}$$ If $\Gamma$ is an orthonormal basis for $\mathfrak{g}_{CM}$ containing $\Gamma_{P}$, it follows that $$\left\Vert \hat{f}\left( \mathbf{e}\right)
\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{P}\right) }^{2}=\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\sum_{\left\{ h_{j}\right\}_{j=1}^{n}\subset\Gamma}\left\vert
\left\langle \hat{f}\left( \mathbf{e}\right)
,h_{1}\otimes\dots\otimes h_{n}\right\rangle \right\vert
^{2}=\left\Vert \hat{f}\left( \mathbf{e}\right) \right\Vert
_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }^{2},$$ which shows that $\sup_{P\in\operatorname*{Proj}\left( W\right)
}\left\Vert \hat{f}\left( \mathbf{e}\right)
\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{P}\right) }\leqslant\left\Vert \hat{f}\left( \mathbf{e}\right) \right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }$. We may choose orthonormal bases, $\Gamma_{P_{n}}$, for $\mathfrak{g}_{P_{n}}$ such that $\Gamma_{P_{n}}\uparrow\Gamma$ as $n\uparrow\infty$. Then it is easy to show that $$\begin{aligned}
\lim_{n\rightarrow\infty}\left\Vert f\right\Vert_{L^{2}\left( G_{P_{n}},\nu_{T}^{P_{n}}\right) } & =\lim_{n\rightarrow\infty}\left\Vert \hat
{f}\left( \mathbf{e}\right) \right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{P_{n}}\right) }\\
&
=\lim_{n\rightarrow\infty}\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\sum_{\left\{
h_{j}\right\}_{j=1}^{n}\subset\Gamma_{P_{n}}}\left\vert \left\langle
\hat {f}\left( \mathbf{e}\right),h_{1}\otimes\dots\otimes
h_{n}\right\rangle
\right\vert ^{2}\\
& =\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\sum_{\left\{ h_{j}\right\}
_{j=1}^{n}\subset\Gamma}\left\vert \left\langle \hat{f}\left( \mathbf{e}\right),h_{1}\otimes\dots\otimes h_{n}\right\rangle \right\vert
^{2}=\left\Vert \hat{f}\left( \mathbf{e}\right) \right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }$$ from which it follows that $\sup_{P\in\operatorname*{Proj}\left(
W\right) }\left\Vert \hat{f}\left( \mathbf{e}\right)
\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{P}\right) }\geqslant\left\Vert \hat{f}\left( \mathbf{e}\right) \right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }$.
For the next corollary, recall that $\mathcal{P}$ and $\mathcal{P}_{CM}$ denote the spaces of holomorphic cylinder polynomials on $G$ and $G_{CM}$ respectively, see Definition \[d.h1.6\] and Eq. (\[e.h1.7\]).
\[c.h6.6\]If $f:G\rightarrow\mathbb{C}$ is a continuous function satisfying the bounds in Proposition \[p.h4.12\] with $p=2$, then $f|_{G_{CM}}\in\mathcal{H}_{T}^{2}\left( G_{CM}\right)$ and $\hat{f}\left( \mathbf{e}\right) \in J_{T}^{0}\left(
\mathfrak{g}_{CM}\right)$. In particular, for all $T>0$, $\mathcal{P}_{CM}\subset\mathcal{H}_{T}^{2}\left(
G_{CM}\right)$ and for any $p\in\mathcal{P}$, $\hat{p}\left( \mathbf{e}\right) \in J_{T}^{0}\left( \mathfrak{g}_{CM}\right)$. This shows that $\mathcal{H}_{T}^{2}\left( G_{CM}\right)$ and $J_{T}^{0}\left(
\mathfrak{g}_{CM}\right)$ are non-trivial spaces.
\[d.h6.7\]For each $T>0$, the **Taylor map** is the linear map, $\mathcal{T}_{T}:\mathcal{H}_{T}^{2}\left( G_{CM}\right)
\rightarrow
J_{T}^{0}\left( \mathfrak{g}_{CM}\right)$, defined by $\mathcal{T}_{T}f:=\hat{f}\left( \mathbf{e}\right)$.
\[c.h6.8\]The Taylor map, $\mathcal{T}_{T}:\mathcal{H}_{T}^{2}\left( G_{CM}\right) \rightarrow
J_{T}^{0}\left( \mathfrak{g}_{CM}\right)$, is injective. Moreover, the function $\left\Vert \cdot\right\Vert_{\mathcal{H}_{T}^{2}\left( G_{CM}\right)}$ is a norm on $\mathcal{H}_{T}^{2}\left(
G_{CM}\right)$ which is induced by the inner product on $\mathcal{H}_{T}^{2}\left( G_{CM}\right)$ defined by$$\left\langle u, v\right\rangle_{\mathcal{H}_{T}^{2}\left(
G_{CM}\right) }:=\left\langle \hat{u}\left( \mathbf{e}\right)
,\hat{v}\left( \mathbf{e}\right) \right\rangle_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) }\text{ for any }u,
v\in\mathcal{H}_{T}^{2}\left( G_{CM}\right).
\label{e.h6.14}$$
If $\hat{f}\left( \mathbf{e}\right) =0$, then $\left\Vert
f\right\Vert_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }=0$ which then implies that $f|_{G_{P}}\equiv0$ for all $P\in\operatorname*{Proj}\left( W\right)$. As $f:G_{CM}\rightarrow\mathbb{C}$ is continuous and $\cup_{P\in
\operatorname*{Proj}\left( W\right) }G_{P}$ is dense in $G_{CM}$ (see the end of the proof of Theorem \[t.h5.9\]), it follows that $f\equiv0$. Hence we have shown $\mathcal{T}_{T}$ injective. Since $\left\Vert \cdot\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) }$ is a Hilbert norm and, by Lemma \[l.h6.9\], $\left\Vert f\right\Vert_{\mathcal{H}_{T}^{2}\left(
G_{CM}\right) }=\left\Vert \mathcal{T}_{T}f\right\Vert
_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right)}$, it follows that $\left\Vert \cdot\right\Vert
_{\mathcal{H}_{T}^{2}\left( G_{CM}\right)}$ is the norm on $\mathcal{H}_{T}^{2}\left( G_{CM}\right) $ induced by the inner product defined in Eq. (\[e.h6.14\]).
Our next goal is to show that the Taylor map, $\mathcal{T}_{T}$, is surjective. The following lemma motivates the construction of the inverse of the Taylor map.
\[l.h6.9\]For every $f\in\mathcal{H}\left( G_{CM}\right),$$$f\left( g\right) =\sum_{n=0}^{\infty}\frac{1}{n!}\left\langle \hat{f}_{n}\left( \mathbf{e}\right), g^{\otimes n}\right\rangle \text{ for
any
}g\in G_{CM}, \label{e.h6.15}$$ where the above sum is absolutely convergent. By convention, $g^{\otimes 0}=1\in\mathbb{C}$. (For a more general version of this Lemma, see Proposition 5.1 in [@Driver1995a].)
The function $u\left( z\right) :=f\left( zg\right) $ is a holomorphic function of $z\in\mathbb{C}$. Therefore, $$f\left( g\right) =u\left( 1\right) =\sum_{n=0}^{\infty}\frac{1}{n!}u^{\left( n\right) }\left( 0\right)$$ and the above sum is absolutely convergent. In fact, one easily sees that for all $R>0$ there exists $C\left( R\right) <\infty$ such that $\frac{1}{n!}\left\vert u^{\left( n\right) }\left( 0\right) \right\vert
\leqslant C\left( R\right) R^{-n}$ for all $n\in\mathbb{N}$. The proof is now completed upon observing $$\begin{aligned}
u^{\left( n\right) }\left( 0\right) & =\left(
\frac{d}{dt}\right)^{n}u\left( t\right)\left|_{t=0}\right.=\left(
\frac{d}{dt}\right)^{n}f\left(
tg\right)\left|_{t=0}\right. \\
& =\left( \frac{d}{dt}\right)^{n}f\left( e^{tg}\right)
\left|_{t=0}\right.=\left(
\tilde{g}^{n}f\right) \left( \mathbf{e}\right) =\left\langle \hat{f}_{n}\left( \mathbf{e}\right), g^{\otimes n}\right\rangle .\end{aligned}$$
The next theorem is a more precise version of Theorem \[t.h1.5\].
\[Taylor isomorphism theorem\]\[t.h6.10\]For all $T>0$, the space $\mathcal{H}_{T}^{2}\left( G_{CM}\right) $ equipped with the inner product $\left\langle \cdot,\cdot\right\rangle
_{\mathcal{H}_{T}^{2}\left(
G_{CM}\right) }$ is a Hilbert space, $\mathcal{T}\left( \mathcal{H}_{T}^{2}\left( G_{CM}\right) \right) \subset J_{T}^{0}\left( \mathfrak{g}_{CM}\right)$, and $\mathcal{T}_{T}:=\mathcal{T}|_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }:\mathcal{H}_{T}^{2}\left( G_{CM}\right)
\rightarrow J_{T}^{0}\left( \mathfrak{g}_{CM}\right) $ is a unitary transformation.
Given Corollary \[c.h6.8\], it only remains to prove $\mathcal{T}_{T}$ is surjective. So let $\alpha\in J_{T}^{0}\left(
\mathfrak{g}_{CM}\right)$. By Lemma \[l.h6.9\], if $f=\mathcal{T}_{T}^{-1}\alpha$ exists it must be given by $$f\left( g\right) :=\sum_{n=0}^{\infty}\frac{1}{n!}\left\langle \alpha
_{n},g^{\otimes n}\right\rangle \text{ for any }g \in G_{CM}. \label{e.h6.16}$$ We now have to check that the sum is convergent, the resulting function $f$ is in $\mathcal{H}\left( G_{CM}\right)$, and $\hat{f}\left( \mathbf{e}\right) =\alpha$. Once this is done, we may apply Lemma \[l.h6.5\] to conclude that $\left\Vert f\right\Vert
_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }=\left\Vert
\alpha\right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right)
}<\infty$ and hence we will have shown that $f\in\mathcal{H}_{T}^{2}\left( G_{CM}\right) $ and $\mathcal{T}_{T}f=\alpha$. For each $n\in\mathbb{N}\cup\left\{ 0\right\}$, the function $u_{n}\left( g\right) :=\frac{1}{n!}\left\langle
\alpha_{n},g^{\otimes n}\right\rangle $ is a continuous complex $n$–linear form in $g\in G_{CM}$ and therefore holomorphic. Since $\left\vert \left\langle \alpha_{n},g^{\otimes n}\right\rangle
\right\vert \leqslant\left\Vert \alpha_{n}\right\Vert_{n}\left\Vert
g\right\Vert_{\mathfrak{g}_{CM}}^{n}$, then for $R>0$ $$\sup\left\{ \left\vert u_{n}\left( g\right) \right\vert
:\left\Vert g\right\Vert_{\mathfrak{g}_{CM}}\leqslant R\right\}
\leqslant\left\Vert \alpha_{n}\right\Vert_{n}R^{n}.$$ Therefore it follows that$$\begin{aligned}
\sum_{n=0}^{\infty}\sup\left\{ \left\vert u_{n}\left( g\right)
\right\vert :\left\Vert g\right\Vert_{\mathfrak{g}_{CM}}\leqslant
R\right\} & \leqslant\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\left\Vert
\alpha_{n}\right\Vert
_{n}\frac{R^{n}}{T^{n}}\nonumber\\
& \leqslant\sqrt{\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\left\Vert
\alpha
_{n}\right\Vert_{n}^{2}}\sqrt{\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\left(
\frac{R^{n}}{T^{n}}\right)^{2}}\nonumber\\
& =\left\Vert \alpha\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right)
} e^{R^{2}/\left( 2T\right) }<\infty. \label{e.h6.17}$$ This shows $f\left( g\right) =\lim_{N\rightarrow\infty}\sum_{n=0}^{N}u_{n}\left( g\right) $ with the limit being uniform over $g$ in bounded subsets of $\mathfrak{g}_{CM}$. Hence, the sum in Eq. (\[e.h6.16\]) is convergent and (see [@HP74 Theorem 3.18.1]) the resulting function, $f$, is in $\mathcal{H}\left(
G_{CM}\right)$. Since$$f\left( zh\right) =\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\left\langle
\alpha_{n},h^{\otimes n}\right\rangle \text{ for any }z\in\mathbb{C}\text{ and
}h\in\mathfrak{g}_{CM},$$ it follows that $$\left\langle \alpha_{n},h^{\otimes n}\right\rangle =\left( \frac{d}{dz}\right)^{n}f\left( zh\right) |_{z=0}=\left(
\frac{d}{dt}\right) ^{n}f\left( e^{th}\right) |_{t=0}=\left\langle
\hat{f}_{n}\left( \mathbf{e}\right),h^{\otimes n}\right\rangle .$$ This is true for all $n$ and $h\in\mathfrak{g}_{CM}$, so we may use the argument following Eq. (6.13) in [@Driver1995a] (or see the proof of Theorem 2.5 in [@D-G-SC07b]) to show $\hat{f}\left(
\mathbf{e}\right) =\alpha$.
As a consequence of Eq. (\[e.h6.17\]) we see that if $f\in\mathcal{H}_{T}^{2}\left( G_{CM}\right) $ then$$\left\vert f\left( g\right) \right\vert \leqslant\left\Vert
f\right\Vert _{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }
e^{\left\Vert g\right\Vert _{\mathfrak{g}_{CM}}^{2}/\left( 2T\right)
}\text{ for any }g\in G_{CM}.
\label{e.h6.18}$$ The next theorem, which is an analogue of Bargmann’s pointwise bounds (see [@Bargmann61 Eq. (1.7)] and [@Driver1997c Eq. (5.4)]), improves upon the estimate in Eq. (\[e.h6.18\]).
\[Pointwise bounds\]\[t.h6.11\]If $f\in\mathcal{H}_{T}^{2}\left(
G_{CM}\right)$ and $g\in G_{CM}$, then for all $g\in G_{CM},$$$\left\vert f\left( g\right) \right\vert \leqslant\left\Vert
f\right\Vert _{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }
e^{d_{CM}^{2}\left(
\mathbf{e}, g\right) /\left( 2T\right)}, \label{e.h6.19}$$ where $d_{CM}^{2}\left( \cdot,\cdot\right)$ is the distance function on $G_{CM}$ defined in Eq. (\[e.h4.7\]).
Let $P_{n}\in\operatorname*{Proj}\left( W\right) $ be chosen so that $P_{n}|_{\mathfrak{g}_{CM}}\uparrow I_{\mathfrak{g}_{CM}}$ as $n\rightarrow \infty$ and recall that $G_{0}:=\cup_{n=1}^{\infty}G_{P_{n}}$ is a dense subgroup of $G_{CM}$ as explained in the proof of Theorem \[t.h5.9\]. Let $g\in G_{P_{m}}$ for some $m\in\mathbb{N}$ and let $\sigma:\left[
0,1\right] \rightarrow G_{CM}$ be a $C^{1}$–curve such that $\sigma\left( 0\right) =\mathbf{e}$ and $\sigma\left( 1\right)
=g$. Then for $n\geqslant m$, $\sigma_{n}\left( t\right)
:=\pi_{P_{n}}\left( \sigma\left( t\right) \right) $ is a $C^{1}$ curve in $G_{n}$ such that $\sigma_{n}\left( 0\right) =\mathbf{e}$ and $\sigma_{n}\left( 1\right) =g$. Therefore by [@Driver1997c Eq. (5.4)], we have$$\left\vert f\left( g\right) \right\vert \leqslant\left\Vert f|_{G_{P_{n}}}\right\Vert_{L^{2}\left( G_{P_{n}},\nu_{T}^{P_{n}}\right) }\cdot
e^{d_{G_{P_{n}}}^{2}\left( \mathbf{e},g\right) /\left( 2T\right)
}\leqslant\left\Vert f\right\Vert_{\mathcal{H}_{T}^{2}\left(
G_{CM}\right) }\cdot e^{\ell_{G_{CM}}^{2}\left( \sigma_{n}\right)
/\left( 2T\right) },
\label{e.h6.20}$$ where $\ell_{G_{CM}}\left( \sigma_{n}\right) $ is the length of $\sigma_{n}$ as in Eq. (\[e.h4.6\]). In the proof [@DG07b Theorem 8.1], it was shown that $\lim_{n\rightarrow\infty}\ell_{G_{CM}}\left( \sigma_{n}\right)
=\ell_{G_{CM}}\left( \sigma\right)$. Hence we may pass to the limit in Eq. (\[e.h6.20\]) to find, $\left\vert f\left( g\right)
\right\vert \leqslant\left\Vert f\right\Vert
_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }\cdot
e^{\ell_{G_{CM}}^{2}\left( \sigma\right) /\left( 2T\right) }$. Optimizing this last inequality over all $\sigma$ joining $\mathbf{e}$ to $g$ then shows that Eq. (\[e.h6.19\]) holds for all $g\in G_{0}$. This suffices to prove Eq. (\[e.h6.19\]) as both sides of this inequality are continuous in $g\in G_{CM}$ and $G_{0}$ is dense in $G_{CM}$.
Density theorems\[s.h7\]
========================
The following density result is the main theorem of this section and is crucial to the next section. Techniques similar to those used in this section have appeared in Cecil [@Cecil2008] to prove an analogous result for path groups over stratified Lie groups.
\[Density theorem\]\[t.h7.1\] For all $T>0$, $\mathcal{P}_{CM}$ defined by Eq. (\[e.h1.7\]) is a dense subspace of $\mathcal{H}_{T}^{2}\left( G_{CM}\right)$.
This theorem is a consequence of Corollary \[c.h7.4\] and Proposition \[p.h7.12\] below.
The remainder of this section will be devoted to proving the results used in the proof of the theorem. We will start by constructing some auxiliary dense subspaces of $J_{T}^{0}\left(
\mathfrak{g}_{CM}\right)$ and $\mathcal{H}_{T}^{2}\left(
G_{CM}\right)$.
Finite rank subspaces\[s.h7.1\]
-------------------------------
\[d.h7.2\]A tensor, $\alpha\in J^{0}\left(
\mathfrak{g}_{CM}\right)$, is said to have **finite rank** if $\alpha_{n}=0$ for all but finitely many $n\in\mathbb{N}$.
The next lemma is essentially a special case of [@D-G-SC07b Lemma 3.5].
\[Finite Rank Density Lemma\]\[l.h7.3\]The finite rank tensors in $J_{T}^{0}\left( \mathfrak{g}_{CM}\right) $ are dense in $J_{T}^{0}\left( \mathfrak{g}_{CM}\right)$.
For $\theta\in\mathbb{R}$, let $\varphi_{\theta}:\mathfrak{g}_{CM}\rightarrow\mathfrak{g}_{CM}$ be defined by $$\varphi_{\theta}\left( A, a\right) =\left( e^{i\theta}A,
e^{i2\theta }a\right).$$ Since $$\begin{aligned}
\left[ \varphi_{\theta}\left( A, a\right), \varphi_{\theta}\left(
B, b\right) \right] & =\left[ \left( e^{i\theta}A,
e^{i2\theta}a\right)
, \left( e^{i\theta}B, e^{i2\theta}b\right) \right] \\
& =\left( 0, \omega\left( e^{i\theta}A, e^{i\theta}B\right)
\right) =\left( 0, e^{i2\theta}\omega\left( A, B\right) \right)
=\varphi_{\theta }\left[ \left( A, a\right), \left( B, b\right)
\right]\end{aligned}$$ we see that $\varphi_{\theta}$ is a Lie algebra homomorphism.
Now let $\Phi_{\theta}:\mathbf{T}\left( \mathfrak{g}_{CM}\right)
\rightarrow\mathbf{T}\left( \mathfrak{g}_{CM}\right)$ be defined by $\Phi_{\theta}1=1$ and $$\Phi_{\theta}\left( h_{1}\otimes\dots\otimes h_{n}\right) =\varphi_{\theta
}h_{1}\otimes\dots\otimes\varphi_{\theta}h_{n}\text{ for all }h_{i}\in\mathfrak{g}_{CM}\text{ and }n\in\mathbb{N}.$$ If we write $\xi\wedge\eta$ for $\xi\otimes\eta-\eta\otimes\xi$, then$$\begin{aligned}
\Phi_{\theta}(\xi\wedge\eta-[\xi,\eta]) & =(\varphi_{e^{i\theta}}\xi
)\wedge(\varphi_{e^{i\theta}}\eta)-\varphi_{e^{i\theta}}[\xi,\eta]\\
& =(\varphi_{e^{i\theta}}\xi)\wedge(\varphi_{e^{i\theta}}\eta)-[\varphi
_{e^{i\theta}}\xi,\varphi_{e^{i\theta}}\eta].\end{aligned}$$ From this it follows that $\Phi_{\theta}\left( J\right) \subset J$ and therefore if $\alpha\in J^{0}\left( \mathfrak{g}_{CM}\right)$, then $\alpha\circ\Phi_{\theta}\in J^{0}\left(
\mathfrak{g}_{CM}\right)$. Letting $\Gamma$ be an orthonormal basis as in Eq. (\[e.h5.8\]), we have $\varphi_{\theta}h=e^{i2\theta}h$ or $\varphi_{\theta}h=e^{i\theta}h$ for all $h\in\Gamma$. Therefore it follows that $$\begin{aligned}
\left\vert \langle\alpha\circ\Phi_{\theta},k_{1}\otimes k_{2}\otimes
\dots\otimes k_{n}\rangle\right\vert ^{2} & =\left\vert \langle
\alpha,\varphi_{\theta}k_{1}\otimes\varphi_{\theta}k_{2}\otimes\dots
\otimes\varphi_{\theta}k_{n}\rangle\right\vert ^{2}\\
& =\left\vert \langle\alpha, k_{1}\otimes k_{2}\otimes\dots\otimes
k_{n}\rangle\right\vert ^{2}$$ and hence that $$\begin{aligned}
\left\Vert \alpha\circ\Phi_{\theta}\right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) }^{2} & =\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\sum_{k_{1},k_{2},\dots, k_{n}\in\Gamma}\left\vert
\langle\alpha\circ \Phi_{\theta}, k_{1}\otimes
k_{2}\otimes\dots\otimes k_{n}\rangle\right\vert
^{2} \\
& =\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\sum_{k_{1},k_{2},\dots,
k_{n}\in\Gamma
}\left\vert \langle\alpha, k_{1}\otimes k_{2}\otimes\dots\otimes k_{n}\rangle\right\vert ^{2}=\left\Vert
\alpha\right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }^{2}.\end{aligned}$$ So the map $\alpha\in J_{T}^{0}\left( \mathfrak{g}_{CM}\right)
\rightarrow\alpha\circ\Phi_{\theta}\in J_{T}^{0}\left( \mathfrak{g}_{CM}\right) $ is unitary. Moreover, since$$\left\vert \langle\alpha,\varphi_{\theta}k_{1}\otimes\varphi_{\theta}k_{2}\otimes\dots\otimes\varphi_{\theta}k_{n}\rangle-\langle\alpha
,k_{1}\otimes k_{2}\otimes\dots\otimes k_{n}\rangle\right\vert ^{2}\leqslant2\left\vert \langle\alpha, k_{1}\otimes
k_{2}\otimes\dots\otimes
k_{n}\rangle\right\vert ^{2}$$ we may apply the dominated convergence theorem to conclude $$\begin{aligned}
& \lim_{\theta\rightarrow0}\left\Vert \alpha\circ\Phi_{\theta}-\alpha
\right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }^{2}\\
& =\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\sum_{k_{1}, k_{2}, \dots,
k_{n}\in\Gamma
}\lim_{\theta\rightarrow0}\left\vert \langle\alpha, \varphi_{\theta}k_{1}\otimes\varphi_{\theta}k_{2}\otimes\dots\otimes\varphi_{\theta}k_{n}\rangle-\langle\alpha, k_{1}\otimes k_{2}\otimes\dots\otimes k_{n}\rangle\right\vert ^{2}\\
& =0,\end{aligned}$$ so that $\alpha\rightarrow\alpha\circ\Phi_{\theta}$ is continuous. (Notice that $\Phi_{\theta}\circ\Phi_{\alpha}=\Phi_{\theta+\alpha}$, so it suffices to check continuity at $\theta=0$.)
Let $$F_{n}(\theta)=\frac{1}{2\pi n}\ \sum_{k=0}^{n-1}\ \sum_{\ell=-k}^{k}e^{i\ell\theta}=\frac{1}{2\pi n}\ \frac{\sin^{2}(k\theta/2)}{\sin^{2}(\theta/2)}$$ denote Fejer’s kernel [@T p. 143]. Then $\int_{-\pi}^{\pi}F_{n}(\theta)d\theta=1$ for all $n$ and $$\lim_{n\rightarrow\infty}\int_{-\pi}^{\pi}F_{n}(\theta)u(\theta)d\theta
=u\left( 0\right) \text{ for all }u\in C\left( [-\pi, \pi],\mathbb{C}\right).$$ We now let $$\alpha\left( n\right) :=\int_{-\pi}^{\pi}\alpha\circ\Phi_{\theta}F_{n}\left( \theta\right) d\theta.$$ Then $$\begin{aligned}
\limsup_{n\rightarrow\infty}\left\Vert \alpha-\alpha\left( n\right)
\right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }^{2} &
\leqslant\limsup_{n\rightarrow\infty}\left\Vert
\int_{-\pi}^{\pi}\left[ \alpha-\alpha\circ\Phi_{\theta}\right]
F_{n}\left( \theta\right)
d\theta\right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }\\
& \leqslant\limsup_{n\rightarrow\infty}\int_{-\pi}^{\pi}\left\Vert
\alpha-\alpha\circ\Phi_{\theta}\right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }F_{n}\left( \theta\right) d\theta=0.\end{aligned}$$
Moreover if $\beta:=k_{1},\dots, k_{m}\in\mathfrak{g}_{CM}$ with $m>n$, then there exits $\beta_{l}\in\mathfrak{g}_{CM}^{\otimes m}$ such that$$\Phi_{\theta}\beta=\sum_{l=m}^{2m}e^{il\theta}\beta_{l}.$$ From this it follows that $$\left\langle \alpha\left( n\right), \beta\right\rangle
:=\int_{-\pi}^{\pi }\left\langle
\alpha,\Phi_{\theta}\beta\right\rangle F_{n}\left(
\theta\right) d\theta=\sum_{l=m}^{2m}\left\langle \alpha,\beta_{l}\right\rangle \int_{-\pi}^{\pi}e^{il\theta}F_{n}\left( \theta\right)
d\theta=0$$ from which it follows that $\alpha\left( n\right)_{m}\equiv0$ for all $m>n$. Thus $\alpha\left( n\right) $ is a finite rank tensor for all $n\in\mathbb{N}$ and $\limsup_{n\rightarrow\infty}\left\Vert
\alpha
-\alpha\left( n\right) \right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right)}^{2}=0$.
\[c.h7.4\]The vector space,$$\mathcal{H}_{T,\text{fin}}^{2}\left( G_{CM}\right) :=\left\{ u\in
\mathcal{H}_{T}^{2}\left( G_{CM}\right) :\hat{u}\left( \mathbf{e}\right)
\in J_{T}^{0}\left( \mathfrak{g}_{CM}\right) \text{ has a finite
rank}\right\} \label{e.h7.1}$$ is a dense subspace of $\mathcal{H}_{T}^{2}\left( G_{CM}\right)$.
This follows directly from Lemma \[l.h7.3\] and the Taylor isomorphism Theorem \[t.h6.10\].
Polynomial approximations\[s.h7.2\]
-----------------------------------
To prove Theorem \[t.h7.1\], it suffices to show that every element $u\in\mathcal{H}_{T, \text{fin}}^{2}\left( G_{CM}\right) $ may be well approximated by an element from $\mathcal{H}_{T}^{2}\left( G\right)$. In order to do this, let $\left\{ e_{j}:j=1,2,\right\} \subset H_{\ast}$ be an orthonormal basis for $H$ and for $N\in\mathbb{N}$, define $P_{N}\in\operatorname*{Proj}\left( W\right)$ as in Eq. (\[e.h2.17\]), i.e.$$P_{N}\left( w\right) =\sum_{j=1}^{N}\left\langle w,e_{j}\right\rangle
_{H}e_{j}\text{ for all }w\in W. \label{e.h7.2}$$ Let us further define $\pi_{N}:=\pi_{P_{N}}$ and $$u_{N}:=u\circ\pi_{N}\text{ for all }N\in\mathbb{N}. \label{e.h7.3}$$ We are going to prove Theorem \[t.h7.1\] by showing $u_{N}\in\mathcal{P}$ and $u_{N}\rightarrow u$ in $\mathcal{H}_{T}^{2}\left( G_{CM}\right)$.
\[r.h7.5\]A complicating factor in showing $u_{N}|_{G_{CM}}\rightarrow u$ in $\mathcal{H}_{T}^{2}\left( G_{CM}\right) $ is the fact that for general $\omega$ and $P\in\operatorname*{Proj}\left( W\right)$, $\pi_{P}:G\rightarrow G_{P}\subset G_{CM}$ is **not** a group homomorphism. In fact we have, $$\pi_{P}\left[ \left( w, c\right) \cdot\left( w^{\prime},
c^{\prime}\right) \right] -\pi_{P}\left( w, c\right)
\cdot\pi_{P}\left( w^{\prime}, c^{\prime
}\right) =\Gamma_{P}\left( w, w^{\prime}\right) \label{e.h7.4}$$ where$$\Gamma_{P}\left( w, w^{\prime}\right) =\frac{1}{2}\left(
0,\omega\left( w, w^{\prime}\right) -\omega\left(
Pw,Pw^{\prime}\right) \right)
\label{e.h7.5}$$ So unless $\omega$ is supportedon the range of $P$, $\pi_{P}$ is not a group homomorphism. Since, $\left( w,b\right) +\left( 0,c\right) =\left( w,b\right)
\cdot\left( 0, c\right)$ for all $w\in W$ and $b, c\in\mathbf{C}$, we may also write equation \[e.h7.4\] as $$\pi_{P}\left[ \left( w, c\right) \cdot\left( w^{\prime},
c^{\prime}\right) \right] =\pi_{P}\left( w, c\right)
\cdot\pi_{P}\left( w^{\prime},c^{\prime
}\right) \cdot\Gamma_{P}\left( w, w^{\prime}\right). \label{e.h7.6}$$
\[l.h7.6\]To each $k:=\left( A, a\right) \in\mathfrak{g}_{CM}$, $g=\left( w, c\right) \in G$, and $P\in\operatorname*{Proj}\left(
W\right)$, let $$k^{P}\left( g\right) =k^{P}\left( w, c\right)
:=\pi_{P}k+\Gamma_{P}\left(
w,A\right) \in\mathfrak{g}_{P} \label{e.h7.7}$$ where $\Gamma_{P}$ is defined in Eq. (\[e.h7.5\]) above. If $u:G_{CM}\rightarrow\mathbb{C}$ is a holomorphic function and $g\in G$, then$$\left( \tilde{k}\left( u\circ\pi_{P}\right) \right) \left(
g\right) =\left\langle Du\left( \pi_{P}\left( g\right) \right),
k^{P}\left(
g\right) \right\rangle \label{e.h7.8}$$ or equivalently put,$$\left\langle \widehat{u\circ\pi_{P}}\left( g\right),k\right\rangle
=\left\langle D\left( u\circ\pi_{P}\right) \left( g\right)
,k\right\rangle =\left\langle Du\left( \pi_{P}\left( g\right)
\right), k^{P}\left(
g\right) \right\rangle . \label{e.h7.9}$$
By direct computation,$$\begin{aligned}
\left( \tilde{k}\left( u\circ\pi_{P}\right) \right) \left( g\right) &
=\frac{d}{dt}\Big|_{0}u\left( \pi_{P}\left( g\cdot e^{tk}\right) \right) \\
& =\frac{d}{dt}\Big|_{0}\left\langle Du\left( \pi_{P}\left(
g\right) \right), \left[ \pi_{P}\left( g\right)
\right]^{-1}\cdot\pi_{P}\left( g\cdot e^{tk}\right) \right\rangle\end{aligned}$$ where by Eq. (\[e.h7.6\]), $$\begin{aligned}
\frac{d}{dt}\Big|_{0}\left( \left[ \pi_{P}\left( g\right) \right]
^{-1}\cdot\pi_{P}\left( g\cdot e^{tk}\right) \right) & =\frac{d}{dt}\Big|_{0}\left( P\left( tA\right), a+\frac{1}{2}\omega\left(
w, tA\right) -\omega\left( Pw, tPA\right) \right) \\
& =\left( PA, a+\frac{1}{2}\omega\left( w, A\right)
-\omega\left(
Pw, PA\right) \right) \\
& =\pi_{P}k+\Gamma_{P}\left( w, A\right).\end{aligned}$$
\[n.h7.7\]Given $P\in\operatorname*{Proj}\left( W\right)$ and $k_{j}=\left( A_{j}, c_{j}\right) \in\mathfrak{g}_{CM}$, let $K_{j}:=k_{j}^{P}: G_{CM}\rightarrow\mathfrak{g}_{CM}$ and $\kappa_{n}:G_{CM}\rightarrow\oplus_{j=1}^{n}\mathfrak{g}_{CM}^{\otimes j}$ be defined by $$\begin{aligned}
\kappa_{n} & =\left( \tilde{k}_{n}+K_{n}\otimes\right) \left( \tilde
{k}_{n-1}+K_{n-1}\otimes\right) \dots\left( \tilde{k}_{1}+K_{1}\otimes\right) 1\nonumber\\
& =\left( \tilde{k}_{n}+K_{n}\otimes\right) \left( \tilde{k}_{n-1}+K_{n-1}\otimes\right) \dots\left( \tilde{k}_{2}+K_{2}\otimes\right) K_{1}.
\label{e.h7.10}$$ In these expressions, $K_{j}\otimes$ denotes operation of left tensor multiplication by $K_{j}$.
\[ex.h7.8\]The functions $\kappa_{n}$ are determined recursively by $\kappa_{1}=K_{1}$ and then $$\kappa_{n}=\left( K_{n}\otimes+\tilde{k}_{n}\right) \kappa_{n-1}=K_{n}\otimes\kappa_{n-1}+\tilde{k}_{n}\kappa_{n-1}\text{ for all }n\geqslant2. \label{e.h7.11}$$ The first four $\kappa_{n}$ are easily seen to be given by, $\kappa_{1}=K_{1}$, $$\kappa_{2}=K_{2}\otimes
K_{1}+\tilde{k}_{2}K_{1}=K_{2}\otimes K_{1}+\Gamma_{P}\left(
A_{2},A_{1}\right),$$$$\begin{aligned}
\kappa_{3} & =\left( K_{3}\otimes+\tilde{k}_{3}\right) \left(
K_{2}\otimes K_{1}+\Gamma_{P}\left( A_{2},A_{1}\right) \right) \\
& =K_{3}\otimes K_{2}\otimes K_{1}+K_{3}\otimes\Gamma_{P}\left( A_{2},A_{1}\right) +\Gamma_{P}\left( A_{3},A_{2}\right) \otimes K_{1}+K_{2}\otimes\Gamma_{P}\left( A_{3},A_{1}\right),\end{aligned}$$ and$$\begin{aligned}
\kappa_{4} & =K_{4}\otimes K_{3}\otimes K_{2}\otimes K_{1}\\
& +\left(
\begin{array}
[c]{c}K_{4}\otimes K_{3}\otimes\Gamma_{P}\left( A_{2}, A_{1}\right) +K_{4}\otimes\Gamma_{P}\left( A_{3}, A_{2}\right) \otimes
K_{1}+K_{4}\otimes
K_{2}\otimes\Gamma_{P}\left( A_{3}, A_{1}\right) \\
+\Gamma_{P}\left( A_{4}, A_{3}\right) \otimes K_{2}\otimes K_{1}+K_{3}\otimes\Gamma_{P}\left( A_{4}, A_{2}\right) \otimes
K_{1}+K_{3}\otimes K_{2}\otimes\Gamma_{P}\left( A_{4}, A_{1}\right)
\end{array}
\right) \\
& +\Gamma_{P}\left( A_{4}, A_{3}\right) \otimes\Gamma_{P}\left(
A_{2}, A_{1}\right) +\Gamma_{P}\left( A_{3}, A_{2}\right)
\otimes\Gamma_{P}\left( A_{4}, A_{1}\right) +\Gamma_{P}\left(
A_{4}, A_{2}\right) \otimes\Gamma_{P}\left( A_{3}, A_{1}\right).\end{aligned}$$ At the end we will only use $\kappa_{n}$ evaluated at $\mathbf{e}\in
G_{CM}$. Evaluating the above expressions at $\mathbf{e}$ amounts to replacing $K_{j}$ by $\pi_{P}k_{j}$ in all of the previous formulas.
\[p.h7.9\]If $u\in\mathcal{H}\left( G_{CM}\right)$, then, with the setup in Notation \[n.h7.7\], we have$$\left\langle \widehat{u\circ\pi_{P}},k_{n}\otimes\dots\otimes k_{1}\right\rangle =\left\langle \hat{u}\circ\pi_{P},\kappa_{n}\right\rangle \text{
for any }n\in\mathbb{N}, \label{e.h7.12}$$ where both sides of this equation are holomorphic functions on $G_{CM}$.
The proof is by induction with the case $n=1$ already completed via Equation (\[e.h7.9\]). To proceed with the induction argument, suppose that Eq. (\[e.h7.12\]) holds for some $n\in\mathbb{N}$. Then by induction and the product rule $$\begin{aligned}
\left\langle \widehat{u\circ\pi_{P}}, k_{n+1}\otimes
k_{n}\otimes\dots\otimes
k_{1}\right\rangle & =\tilde{k}_{n+1}\left\langle \widehat{u\circ\pi_{P}},k_{n+1}\otimes k_{n}\otimes\dots\otimes k_{1}\right\rangle \nonumber\\
& =\tilde{k}_{n+1}\left\langle \hat{u}\circ\pi_{P},\kappa_{n}\right\rangle
\nonumber\\
& =\left\langle
\hat{u}\circ\pi_{P},\tilde{k}_{n+1}\kappa_{n}\right\rangle
+\left\langle \tilde{k}_{n+1}\left[
\hat{u}\circ\pi_{P}\right],\kappa
_{n}\right\rangle . \label{e.h7.13}$$ To evaluate $\tilde{k}_{n+1}\left[ \hat{u}\circ\pi_{P}\right] $ let $v\in\mathbf{T}\left( \mathfrak{g}_{CM}\right) $ and let $\tilde{v}$ denote the corresponding left invariant differential operator on $G_{CM}$. Then $$\begin{aligned}
\left\langle \tilde{k}_{n+1}\left[ \hat{u}\circ\pi_{P}\right],
v\right\rangle \left( g\right) & =\left(
\tilde{k}_{n+1}\left\langle \left[ \hat{u}\circ\pi_{P}\right],
v\right\rangle \right) \left( g\right)
\nonumber\\
& =\left( \tilde{k}_{n+1}\left[ \left( \tilde{v}u\right) \circ\pi
_{P}\right] \right) \left( g\right) \nonumber\\
& =\left\langle D\left( \tilde{v}u\right) \left( \pi_{P}\left( g\right)
\right),k_{n+1}^{P}\left( g\right) \right\rangle \nonumber\\
& =\left( \widetilde{k_{n+1}^{P}\left( g\right) }\tilde{v}u\right)
\left( \pi_{P}\left( g\right) \right) \nonumber\\
& =\left\langle \hat{u}\left( \pi_{P}\left( g\right) \right),k_{n+1}^{P}\left( g\right) \otimes v\right\rangle . \label{e.h7.14}$$ Combining Eqs. (\[e.h7.13\]) and (\[e.h7.14\]) shows,$$\begin{aligned}
\left\langle \widehat{u\circ\pi_{P}},k_{n+1}\otimes k_{n}\otimes\dots\otimes
k_{1}\right\rangle & =\left\langle \hat{u}\circ\pi_{P},\tilde{k}_{n+1}\kappa_{n}\right\rangle +\left\langle \hat{u}\circ\pi_{P},k_{n+1}^{P}\otimes\kappa_{n}\right\rangle \\
& =\left\langle \hat{u}\circ\pi_{P},\tilde{k}_{n+1}\kappa_{n}+k_{n+1}^{P}\otimes\kappa_{n}\right\rangle =\left\langle \hat{u}\circ\pi_{P},\kappa_{n+1}\right\rangle\end{aligned}$$ wherein we have used Eq. (\[e.h7.11\]) for the last equality.
The induction proof of the following lemma will be left to the reader with Example \[ex.h7.8\] as a guide.
\[l.h7.10\]Let $k_{j}=\left( A_{j},c_{j}\right)
\in\mathfrak{g}_{CM}$ for $1\leqslant j\leqslant n$, $\left\lfloor
\frac{n}{2}\right\rfloor =n/2$ if $n$ is even and $\left(
n-1\right) /2$ if $n$ is odd, and $\kappa_{n}$ be as in Eq. (\[e.h7.10\]). Then$$\kappa_{n}\left( \mathbf{e}\right)
=\pi_{P}k_{n}\otimes\dots\otimes\pi
_{P}k_{2}\otimes\pi_{P}k_{1}+R\left( P:k_{n}, \dots, k_{1}\right),
\label{e.h7.15}$$ where $$R\left( P:k_{n},,\dots,k_{1}\right) =\sum_{j=1}^{\left\lfloor \frac{n}{2}\right\rfloor }R_{j}\left( P:k_{n},,\dots,k_{1}\right) \label{e.h7.16}$$ with $R_{j}\left( P:k_{1},\dots,k_{n}\right) \in\mathfrak{g}_{CM}^{\otimes\left( n-j\right) }$. Each remainder term, $R_{j}\left(
P:k_{1},\dots,k_{n}\right)$, is a linear combination (with coefficients coming from $\left\{ \pm1,0\right\} )$ of homogenous tensors which are permutations of the indices and order of the terms in the tensor product of the form$$\Gamma_{P}\left( A_{1},A_{2}\right) \otimes\dots\otimes\Gamma_{P}\left(
A_{2j-1},A_{2j}\right) \otimes k_{2j+1}\otimes\dots\otimes k_{n}.
\label{e.h7.17}$$
\[p.h7.11\]Let $P_{N}\in\operatorname*{Proj}\left( W\right) $ and $\pi_{N}:=\pi_{P_{N}}$ be as in Notation \[n.h1.1\] and suppose that $u\in\mathcal{H}\left( G_{CM}\right) $ satisfies $\left\Vert \hat{u}_{n}\left( \mathbf{e}\right) \right\Vert_{n}<\infty$ for all $n$. Then $$\lim_{N\rightarrow\infty}\left\Vert \hat{u}_{n}\left( \mathbf{e}\right)
-\left[ \widehat{u\circ\pi_{N}}\left( \mathbf{e}\right) \right]
_{n}\right\Vert_{n}=0\text{ for }n=0,1,2,\dots.. \label{e.h7.18}$$
To simplify notation, let $\alpha_{n}:=\hat{u}_{n}\left(
\mathbf{e}\right) $ and $\alpha_{n}\left( N\right) :=\left[
\widehat{u\circ\pi_{N}}\left( \mathbf{e}\right) \right]_{n}$. Let $\Gamma$ be an orthonormal basis for $\mathfrak{g}_{CM}$ of the form in Eq. (\[e.h5.8\]) and let $\mathbf{k}:=\left( k_{1},k_{2},\dots,k_{n}\right) \in\Gamma^{n}$. Then$$\left\langle \alpha-\alpha\left( N\right),k_{1}\otimes\dots\otimes
k_{n}\right\rangle =\left\langle \alpha,k_{1}\otimes\dots\otimes
k_{n}-\pi _{N}k_{1}\otimes\dots\otimes\pi_{N}k_{n}\right\rangle
+\left\langle \alpha,R\left( P_{N}:\mathbf{k}\right) \right\rangle$$ where $R\left( P_{N}:\mathbf{k}\right) $ is as in Lemma \[l.h7.10\]. Therefore, $\left\Vert \alpha_{n}-\alpha_{n}\left( N\right) \right\Vert
_{n}\leqslant C_{N}+D_{N}$ where$$\begin{aligned}
C_{N} & :=\sqrt{\sum_{\mathbf{k}\in\Gamma^{n}}\left\vert \left\langle
\alpha,R\left( P_{N}:\mathbf{k}\right) \right\rangle \right\vert ^{2}}\text{
and}\\
D_{N} & :=\sqrt{\sum_{\mathbf{k}\in\Gamma^{n}}\left\vert \left\langle
\alpha_{n},k_{1}\otimes\dots\otimes k_{n}-\pi_{N}k_{1}\otimes\dots\otimes
\pi_{N}k_{n}\right\rangle \right\vert ^{2}}.\end{aligned}$$ We will complete the proof by showing that, $\lim_{N\rightarrow\infty}C_{N}=0=\lim_{N\rightarrow\infty}D_{N}$. To estimate $C_{N}$, use Lemma \[l.h7.10\] and the triangle inequality for $\ell_{2}\left(
\Gamma^{n}\right) $ to find, $$C_{N}=\sqrt{\sum_{\mathbf{k}\in\Gamma^{n}}\left\vert \sum_{j=1}^{\left\lfloor
\frac{n}{2}\right\rfloor }\left\langle \alpha,R_{j}\left( P_{N}:\mathbf{k}\right) \right\rangle \right\vert ^{2}}\leqslant\sum
_{j=1}^{\left\lfloor \frac{n}{2}\right\rfloor }\sqrt{\sum_{\mathbf{k}\in
\Gamma^{n}}\left\vert \left\langle \alpha,R_{j}\left( P_{N}:\mathbf{k}\right) \right\rangle \right\vert ^{2}}.$$ But $\sum_{\mathbf{k}\in\Gamma^{n}}\left\vert \left\langle \alpha,R_{j}\left(
P_{N}:\mathbf{k}\right) \right\rangle \right\vert ^{2}$ is bounded by a sum of terms (the number of these terms depends only on $j$ and $n$ and **not** $N)$ of which a typical term (see Eq. (\[e.h7.17\])) is;$$\sum_{\mathbf{k}\in\Gamma^{n}}\left\vert \left\langle \alpha_{n-j},\Gamma_{P_{N}}\left( A_{1},A_{2}\right) \otimes\dots\otimes\Gamma_{P_{N}}\left( A_{2j-1},A_{2j}\right) \otimes k_{2j+1}\otimes\dots\otimes
k_{n}\right\rangle \right\vert ^{2}. \label{e.h7.19}$$ The sum in Eq. (\[e.h7.19\]) may be estimated by,$$\left\Vert
\alpha_{n-j}\right\Vert_{n-j}\sum_{l_{1},\dots,l_{2j}=1}^{\infty
}\left\Vert \Gamma_{P_{N}}\left( e_{l_{1}},e_{l_{2}}\right)
\right\Vert
_{\mathfrak{g}_{CM}}^{2}\dots\left\Vert \Gamma_{P_{N}}\left( e_{l_{2j-1}},e_{l_{2j}}\right) \right\Vert_{\mathfrak{g}_{CM}}^{2}=\left\Vert
\alpha_{n-j}\right\Vert_{n-j}^{2}\varepsilon_{N}^{j},$$ where$$\begin{aligned}
\varepsilon_{N} & =\frac{1}{4}\sum_{k,l=1}^{\infty}\left\Vert \omega\left(
e_{k},e_{l}\right) -\omega\left( P_{N}e_{k},P_{N}e_{l}\right) \right\Vert
_{\mathbf{C}}^{2}\\
& =\frac{1}{4}\sum_{\max\left( k,l\right) >N}^{\infty}\left\Vert
\omega\left( e_{k},e_{l}\right) -\omega\left( P_{N}e_{k},P_{N}e_{l}\right)
\right\Vert_{\mathbf{C}}^{2}\\
& \leqslant\frac{1}{2}\sum_{\max\left( k,l\right)
>N}^{\infty}\left\Vert \omega\left( e_{k},e_{l}\right)
\right\Vert_{\mathbf{C}}^{2}\rightarrow 0\text{ and
}N\rightarrow\infty.\end{aligned}$$ Thus we have shown $\lim_{N\rightarrow\infty}C_{N}=0$
For $N\in\mathbb{N}$, let $\Gamma_{N}=\left\{ \left(
0,f_{j}\right)
\right\}_{j=1}^{d}\cup\left\{ \left( e_{j},0\right) \right\}_{j=1}^{N}$. Since $k_{1}\otimes\dots\otimes
k_{n}=\pi_{N}k_{1}\otimes\dots \otimes\pi_{N}k_{n}$ if $\mathbf{k}:=\left( k_{1},k_{2},\dots,k_{n}\right)
\in\Gamma_{N}^{n}$, it follows that$$\begin{aligned}
D_{N}^{2} & =\sum_{\mathbf{k}\in\Gamma^{n}\setminus\Gamma_{N}^{n}}\left\vert
\left\langle \alpha_{n},k_{1}\otimes\dots\otimes k_{n}-\pi_{N}k_{1}\otimes\dots\otimes\pi_{N}k_{n}\right\rangle \right\vert ^{2}\nonumber\\
& \leqslant2\sum_{\mathbf{k}\in\Gamma^{n}\setminus\Gamma_{N}^{n}}\left\vert
\left\langle \alpha_{n},k_{1}\otimes\dots\otimes k_{n}\right\rangle
\right\vert ^{2}. \label{e.h7.20}$$ Because $$\sum_{\mathbf{k}\in\Gamma^{n}}\left\vert \left\langle \alpha_{n},k_{1}\otimes\dots\otimes k_{n}\right\rangle \right\vert ^{2}=\left\Vert
\alpha _{n}\right\Vert_{n}^{2}<\infty$$ and $\Gamma_{N}^{n}\uparrow\Gamma_{N}$ as $N\uparrow\infty$, the sum in Eq. (\[e.h7.20\]) tends to zero as $N\rightarrow\infty$. Thus $\lim_{N\rightarrow\infty}D_{N}=0$ and the proof is complete.
\[p.h7.12\]If $u\in\mathcal{H}_{T,\text{fin}}^{2}\left(
G_{CM}\right) $ and $u_{N}:=u\circ\pi_{N}$ as in Eq. (\[e.h7.3\]), then $u_{N}\in \mathcal{P}$ and $u_{N}|_{G_{CM}}\rightarrow u$ in $\mathcal{H}_{T}^{2}\left(
G_{CM}\right)$.
Suppose $m\in\mathbb{N}$ is chosen so that $\hat{u}_{n}\left( \mathbf{e}\right) =0$ if $n>m$. According to Proposition \[p.h7.9\], $$\left\langle \hat{u}_{N}\left(
\mathbf{e}\right),k_{n}\otimes\dots\otimes k_{1}\right\rangle
=\left\langle \hat{u}\left( \mathbf{e}\right),\kappa _{n}\left(
\mathbf{e}\right) \right\rangle$$ where $\kappa_{n}\left( \mathbf{e}\right)
\in\bigoplus_{j=1}^{\left\lfloor \frac{n}{2}\right\rfloor
}\mathfrak{g}_{CM}^{\otimes\left( n-j\right) }$. From this it follows that $\left\langle \hat{u}_{N}\left( \mathbf{e}\right)
,k_{n}\otimes\dots\otimes k_{1}\right\rangle =0$ if $n\geqslant2m+2$. Therefore, $u_{N}$ restricted to $P_{N}H\times\mathbf{C}$ is a holomorphic polynomial and since $u_{N}=u_{N}|_{P_{N}H\times\mathbf{C}}\circ\pi_{N}$, it follows that $u_{N}\in\mathcal{P}$. Moreover, $$\lim_{N\rightarrow\infty}\left\Vert \hat{u}\left( \mathbf{e}\right)
-\hat {u}_{N}\left( \mathbf{e}\right) \right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) }^{2}=\lim_{N\rightarrow\infty}\sum_{n=0}^{2m+2}\frac{T^{n}}{n!}\left\Vert \hat{u}_{n}\left(
\mathbf{e}\right) -\left[ \hat{u}_{N}\left( \mathbf{e}\right)
\right]_{n}\right\Vert_{n}^{2}=0,$$ wherein we have used Proposition \[p.h7.11\] to conclude $\lim
_{N\rightarrow\infty}\left\Vert \hat{u}_{n}\left( \mathbf{e}\right)
-\left[ \hat{u}_{N}\left( \mathbf{e}\right) \right]
_{n}\right\Vert_{n}=0$ for all $n$. It then follows by the Taylor isomorphism Theorem \[t.h6.10\] that $\lim_{N\rightarrow\infty}\left\Vert u-u_{N}\right\Vert_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }=0$.
The skeleton isomorphism\[s.h8\]
================================
This section is devoted to the proof of the skeleton Theorem \[t.h1.8\]. Let us begin by gathering together a couple of results that we have already proved.
\[p.h8.1\]If $f:G\rightarrow\mathbb{C}$ is a continuous function such that $f|_{G_{CM}}$ is holomorphic, then$$\left\Vert f\right\Vert_{L^{2}\left( \nu_{T}\right)
}\leqslant\left\Vert f|_{G_{CM}}\right\Vert
_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }=\left\Vert
\hat{f}\left( \mathbf{e}\right) \right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) }. \label{e.h8.1}$$ If $\left\Vert f|_{G_{CM}}\right\Vert_{\mathcal{H}_{T}^{2}\left(
G_{CM}\right) }^{2}<\infty$, then $S_{T}f=f$ and $f$ satisfies the Gaussian pointwise bounds in Eq. (\[e.h6.19\]). (See Corollary \[c.h8.3\] for a more sophisticated version of this proposition.)
See Theorems \[t.h5.9\] and \[t.h6.11\].
\[l.h8.2\]Let $f:G\rightarrow\mathbb{C}$ be a continuous function such that $f|_{G_{CM}}$ is holomorphic and let $\delta>0$ be as in Theorem \[t.h4.11\]. If there exists an $\varepsilon\in\left( 0,\delta\right) $ such that $\left\vert f\left( \cdot\right) \right\vert \leqslant
Ce^{\varepsilon\rho^{2}\left( \cdot\right) /\left( 2T\right) }$ on $G$, then$$\left\Vert f\right\Vert_{L^{2}\left( \nu_{T}\right) }=\left\Vert
f\right\Vert_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }=\left\Vert
\hat
{f}\left( \mathbf{e}\right) \right\Vert_{J_{T}^{0}\left( \mathfrak{g}_{CM}\right) }<\infty. \label{e.h8.2}$$ (It will be shown in Corollary \[c.h8.4\] that $f$ is actually in $\mathcal{H}_{T}^{2}\left( G\right)$.) In particular, Eq. (\[e.h8.2\]) holds for all $f\in\mathcal{P}$.
Let $\left\{ P_{n}\right\}
_{n=1}^{\infty}\subset\operatorname*{Proj}\left( W\right) $ be a sequence such that $P_{n}|_{\mathfrak{g}_{CM}}\uparrow
I_{\mathfrak{g}_{CM}}$ as $n\rightarrow\infty$. Then, by Lemma \[l.h6.5\] and Proposition \[p.h4.12\] with $h=0$, $$\infty>\left\Vert f\right\Vert_{L^{2}\left( \nu_{T}\right) }=\lim
_{n\rightarrow\infty}\left\Vert f\right\Vert_{L^{2}\left(
G_{P_{n}}\nu_{T}^{P_{n}}\right) }=\left\Vert f\right\Vert
_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }=\left\Vert
\hat{f}\left( \mathbf{e}\right) \right\Vert_{J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) }.$$
We are now ready to complete the proof of the Skeleton isomorphism Theorem \[t.h1.8\].
Proof of Theorem \[t.h1.8\]\[s.h8.1\]
-------------------------------------
By Corollary \[c.h5.10\], $S_{T}f=f|_{G_{CM}}$ for all $f\in\mathcal{P}$ and hence by Lemma \[l.h8.2\], $\left\Vert S_{T}f\right\Vert_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }=\left\Vert f\right\Vert
_{L^{2}\left( \nu_{T}\right) }$. It therefore follows that $S_{T}|_{\mathcal{P}}$ extends uniquely to an isometry, $\bar{S}_{T}$, from $\mathcal{H}_{T}^{2}\left( G\right) $ to $\mathcal{H}_{T}^{2}\left( G_{CM}\right)$ such that $\bar
{S}_{T}\left( \mathcal{P}\right) =\mathcal{P}_{CM}$. Since $\bar{S}_{T}$ is isometric and $\mathcal{P}_{CM}$ is dense in $\mathcal{H}_{T}^{2}\left( G_{CM}\right)$, it follows that $\bar{S}_{T}$ is surjective, i.e. $\bar
{S}_{T}:\mathcal{H}_{T}^{2}\left( G\right) \rightarrow\mathcal{H}_{T}^{2}\left( G_{CM}\right)$ is a unitary map. To finish the proof we only need to show $S_{T}f=\bar{S}_{T}f$ for all $f\in\mathcal{H}_{T}^{2}\left( G\right)$. Let $p_{n}\in\mathcal{P}$ such that $p_{n}\rightarrow f$ in $L^{2}\left( \nu_{T}\right)$. Then $p_{n}=S_{T}p_{n}\rightarrow\bar{S}_{T}f$ in $\mathcal{H}_{T}^{2}\left( G_{CM}\right) $ and hence by the Gaussian pointwise bounds in Eq. (\[e.h6.19\]), $\bar{S}_{T}f\left( g\right) =\lim_{n\rightarrow\infty}p_{n}\left(
g\right) $ for all $g\in G_{CM}$. Similarly, using the Gaussian bounds in Corollary \[c.h4.8\], it follows that $$\begin{aligned}
\left\vert S_{T}f\left( g\right) -p_{n}\left( g\right) \right\vert &
=\left\vert S_{T}\left( f-p_{n}\right) \left( g\right) \right\vert
\nonumber\\
& \leqslant\left\Vert f-p_{n}\right\Vert_{L^{2}\left( \nu_{T}\right) }\exp\left( \frac{c\left( k\left( \omega\right) T/2\right) }{T}d_{G_{CM}}^{2}\left( \mathbf{e},g\right) \right) \label{e.h8.3}$$ and hence we also have, $S_{T}f\left( g\right)
=\lim_{n\rightarrow\infty }p_{n}\left( g\right)$ for all $g\in
G_{CM}$. Therefore, $S_{T}f=\bar {S}_{T}f$ as was to be shown.
\[c.h8.3\]If $f:G\rightarrow\mathbb{C}$ is a continuous function such that $f|_{G_{CM}}\in\mathcal{H}_{T}^{2}\left( G_{CM}\right) $, then $f\in\mathcal{H}_{T}^{2}\left( G\right)$, $S_{T}f=f|_{G_{CM}}$, and $\left\Vert f\right\Vert_{L^{2}\left(
\nu_{T}\right) }=\left\Vert f\right\Vert
_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }$.
By Proposition \[p.h8.1\] we already know that $S_{T}f=f|_{G_{CM}}$. By Theorem \[t.h1.8\], there exists $u\in\mathcal{H}_{T}^{2}\left( G\right) $ such that $f|_{G_{CM}}=S_{T}u$. Let $p_{n}\in\mathcal{P}$ be chosen so that $p_{n}\rightarrow u$ in $L^{2}\left( \nu_{T}\right) $ and hence $p_{n}|_{G_{CM}}=S_{T}p_{n}\rightarrow S_{T}u=S_{T}f$ in $\mathcal{H}_{T}^{2}\left( G_{CM}\right) $ as $n\rightarrow\infty$. Hence it follows from Proposition \[p.h8.1\] that$$\left\Vert f-p_{n}\right\Vert_{L^{2}\left( \nu_{T}\right)
}\leqslant
\left\Vert \left( f-p_{n}\right) |_{G_{CM}}\right\Vert_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) }=\left\Vert S_{T}\left( f-p_{n}\right)
\right\Vert_{\mathcal{H}_{T}^{2}\left( G_{CM}\right) },$$ and therefore, $\lim_{n\rightarrow\infty}\left\Vert
f-p_{n}\right\Vert_{L^{2}\left( \nu_{T}\right) }=0$, i.e. $p_{n}\rightarrow f$ in $L^{2}\left( \nu_{T}\right)$. Since $p_{n}\rightarrow u$ in $L^{2}\left(
\nu_{T}\right) $ as well, we may conclude that $f=u\in\mathcal{H}_{T}^{2}\left( G\right)$.
\[c.h8.4\]Suppose that $f:G\rightarrow\mathbb{C}$ is a continuous function such that $\left\vert f\right\vert \leqslant Ce^{\varepsilon\rho^{2}/\left(
2T\right) }$ and $f|_{G_{CM}}$ is holomorphic, then $f\in\mathcal{H}_{T}^{2}\left( G\right) $ and $S_{T}f=f$.
This is a consequence of Lemma \[l.h8.2\] and Corollary \[c.h8.3\].
The holomorphic chaos expansion\[s.h9\]
=======================================
This section is devoted to the proof of the holomorphic chaos expansion Theorem \[t.h1.9\] (or equivalently Theorem \[t.h9.10\]). Before going to the proof we will develop the machinery necessary in order to properly define the right side of Eq. (\[e.h1.8\]).
Generalities about multiple Itô integrals\[s.h9.1\]
---------------------------------------------------
Let $\left( \mathbb{H},\mathbb{W}\right) $ be a complex abstract Wiener space. Analogous to the notation used in Subsection \[s.h6.1\] we will denote the norm on $\mathbb{H}^{\ast\otimes n}$ by $\Vert\cdot\Vert_{n}$.
\[n.h9.1\]For $\alpha\in\mathbb{H}^{\ast\otimes n}$ and $P\in
\operatorname*{Proj}\left( \mathbb{W}\right)$, let $\alpha_{P}:=\alpha\circ P^{\otimes n}\in\mathbb{H}^{\ast\otimes
n}$.
\[p.h9.2\]Let $n\in\mathbb{N}$ and $\alpha\in\mathbb{H}^{\ast\otimes n}$ and $P_{k}\in\operatorname*{Proj}\left( \mathbb{W}\right) $ with $P_{k}|_{\mathbb{H}}\uparrow I|_{\mathbb{H}}$. Then $\alpha_{P_{k}}\rightarrow\alpha$ in $\mathbb{H}^{\ast\otimes n}$.
Let $\Lambda:=\cup_{k}\Lambda_{k}$ be an orthonormal basis for $\mathbb{H}$ where $\Lambda_{k}$ is chosen to be an orthonormal basis for $\operatorname*{Ran}\left( P_{k}\right) $ such that $\Lambda_{k}\subset\Lambda_{k+1}$ for all $k$. Since $P_{k}u=u$ or $P_{k}u=0$ for all $u\in\Lambda$ and $k\in\mathbb{N}$, we have$$\left\vert \left\langle \alpha,u_{1}\otimes\dots\otimes u_{n}-P_{k}u_{1}\otimes\dots\otimes P_{k}u_{n}\right\rangle \right\vert ^{2}\leq\left\vert \left\langle \alpha,u_{1}\otimes\dots\otimes u_{n}\right\rangle
\right\vert ^{2}$$ where$$\sum_{u_{1},\dots,u_{n}\in\Lambda}\left\vert \left\langle \alpha,u_{1}\otimes\dots\otimes u_{n}\right\rangle \right\vert ^{2}=\left\Vert
\alpha\right\Vert_{n}^{2}<\infty.$$ An application of the dominated convergence theorem then implies, $$\begin{aligned}
\lim_{k\rightarrow\infty}\left\Vert
\alpha-\alpha_{P_{k}}\right\Vert_{n}^{2} &
=\lim_{k\rightarrow\infty}\sum_{u_{1},\dots,u_{n}\in\Lambda}\left\vert
\left\langle \alpha,u_{1}\otimes\dots\otimes u_{n}-P_{k}u_{1}\otimes
\dots\otimes P_{k}u_{n}\right\rangle \right\vert ^{2}\\
& =\sum_{u_{1},\dots,u_{n}\in\Lambda}\lim_{k\rightarrow\infty}\left\vert
\left\langle \alpha,u_{1}\otimes\dots\otimes u_{n}-P_{k}u_{1}\otimes
\dots\otimes P_{k}u_{n}\right\rangle \right\vert ^{2}=0.\end{aligned}$$
\[l.h9.3\]Suppose that $\left\{ b\left( t\right)
\right\}_{t\geq0}$ is a $\mathbb{W}$–valued Brownian motion normalized by$$\mathbb{E}\left[ \ell_{1}\left( b\left( t\right) \right)
\ell_{2}\left( b\left( s\right) \right) \right]
=\frac{1}{2}s\wedge t\left( \ell
_{1},\ell_{2}\right)_{\mathbb{H}_{\operatorname{Re}}^{\ast}}\text{
for all
}\ell_{1},\ell_{2}\in\mathbb{W}_{\operatorname{Re}}^{\ast}. \label{e.h9.1}$$ If $P\in\operatorname*{Proj}\left( \mathbb{W}\right)$, $T>0$, and $\left\{ f_{s}\right\}_{s\geq0}$ is a $\left( P\mathbb{H}\right)
^{\ast}$–valued continuous adapted process, such that $\mathbb{E}\int_{0}^{T}\left\vert f_{s}\right\vert_{\left(
P\mathbb{H}\right)^{\ast}}^{2}ds<\infty$, then $$\mathbb{E}\left\vert \int_{0}^{T}\left\langle f_{s},d\left( Pb\right)
\left( s\right) \right\rangle \right\vert ^{2}=\int_{0}^{T}\mathbb{E}\left\vert f_{s}\right\vert_{\left( P\mathbb{H}\right)
^{\ast}}^{2}ds.
\label{e.h9.2}$$
Let $\left\{ e_{j}\right\}_{j=1}^{d}$ be an orthonormal basis for $P\mathbb{H}$ and write $$Pb\left( s\right) =\sum_{j=1}^{d}\left[ X_{j}\left( s\right) e_{j}+Y_{j}\left( s\right) ie_{j}\right]$$ where $X_{j}\left( s\right) =\operatorname{Re}\left( Pb\left(
s\right) ,e_{j}\right) $ and $Y_{j}\left( s\right)
=\operatorname{Im}\left( Pb\left( s\right),e_{j}\right)$. From the normalization in Eq. (\[e.h9.1\]) it follows that $\left\{
\sqrt{2}X_{j},\sqrt{2}Y_{j}\right\}_{j=1}^{d}$ is a sequence of independent standard Brownian motions, and therefore the quadratic covariations of these processes are given by:$$dX_{j}dY_{k}=0\text{ and }dX_{j}dX_{k}=dY_{j}dY_{k}=\frac{1}{2}\delta
_{jk}dt\text{ for all }j,k=1,\dots,d. \label{e.h9.3}$$ Using Eq. (\[e.h9.3\]) along with the identity,$$\left\langle f_{s},d\left( Pb\right) \left( s\right) \right\rangle
=\sum_{j=1}^{d}\left[ \left\langle f_{s},e_{j}\right\rangle dX_{j}\left(
s\right) +\left\langle f_{s},ie_{j}\right\rangle dY_{j}\left( s\right)
\right], \label{e.h9.4}$$ it follows by the basic isometry property of the stochastic integral that$$\begin{aligned}
\mathbb{E}\left\vert \int_{0}^{T}\left\langle f_{s},d\left( Pb\right)
\left( s\right) \right\rangle \right\vert ^{2} & =\frac{1}{2}\sum
_{j=1}^{d}\mathbb{E}\left[ \int_{0}^{T}\left\vert \left\langle f_{s},e_{j}\right\rangle \right\vert ^{2}ds+\int_{0}^{T}\left\vert \left\langle
f_{s},ie_{j}\right\rangle \right\vert ^{2}ds\right] \\
& =\mathbb{E}\int_{0}^{T}\sum_{j=1}^{d}\left\vert \left\langle f_{s},e_{j}\right\rangle \right\vert
^{2}ds=\int_{0}^{T}\mathbb{E}\left\vert f_{s}\right\vert_{\left(
P\mathbb{H}\right)^{\ast}}^{2}ds.\end{aligned}$$
\[d.h9.4\]For $P\in\operatorname*{Proj}\left( \mathbb{W}\right)$, $n\in\mathbb{N}$, and $T>0$, let$$M_{n}^{P}\left( T\right) :=\int_{0\leq s_{1}\leq
s_{2}\leq\dots\leq s_{n}\leq T}dPb\left( s_{1}\right) \otimes
dPb\left( s_{2}\right) \otimes\dots\otimes dPb\left( s_{n}\right).$$ Alternatively put, $M_{0}^{P}\left( T\right) \equiv1$ and $M_{n}^{P}\left( t\right) \in\left( P\mathbb{H}\right)^{\otimes
n}$ is defined inductively by$$M_{n}^{P}\left( t\right) :=\int_{0}^{t}M_{n-1}^{P}\left( s\right) \otimes
dPb\left( s\right) \text{ for all }t\geq0. \label{e.h9.5}$$
\[c.h9.5\]Suppose that $T>0$, $\alpha\in\mathbb{H}^{\ast\otimes
n}$, and $P\in\operatorname*{Proj}\left( \mathbb{W}\right)$, then $\left\langle \alpha,M_{n}^{P}\left( T\right) \right\rangle $ is a square integrable random variable and$$\mathbb{E}\left\vert \left\langle \alpha,M_{n}^{P}\left( T\right)
\right\rangle \right\vert ^{2}=\frac{T^{n}}{n!}\left\Vert \alpha
_{P}\right\Vert_{n}^{2}.$$
The proof is easily carried out by induction with the case $n=1$ following directly from Lemma \[l.h9.3\]. Similarly from Lemma \[l.h9.3\], Eq. (\[e.h9.5\]), and induction we have$$\begin{aligned}
\mathbb{E}\left\vert \tilde{\alpha}_{P}\right\vert ^{2} & =\mathbb{E}\left\vert \int_{0}^{T}\left\langle \alpha,M_{n-1}^{P}\left( s\right)
\otimes dPb\left( s\right) \right\rangle \right\vert ^{2}\\
& =\int_{0}^{T}\sum_{j=1}^{d}\mathbb{E}\left\vert \left\langle \alpha
,M_{n-1}^{P}\left( s\right) \otimes e_{j}\right\rangle \right\vert ^{2}ds\\
& =\sum_{j=1}^{d}\int_{0}^{T}\frac{s^{n-1}}{\left( n-1\right) !}\left\Vert
\left\langle \alpha,\left( \cdot\right) \otimes e_{j}\right\rangle
\right\Vert_{n-1}^{2}ds=\frac{T^{n}}{n!}\left\Vert \alpha\right\Vert_{n}^{2}.\end{aligned}$$
\[n.h9.6\]We now fix $T>0$ and for $P\in\operatorname*{Proj}\left(
\mathbb{W}\right)$, let $\tilde{\alpha}_{P}=\left\langle \alpha,M_{n}^{P}\left( T\right) \right\rangle $, i.e.$$\tilde{\alpha}_{P}=\left\langle \alpha,\int_{0\leq s_{1}\leq s_{2}\leq
\dots\leq s_{n}\leq T}dPb\left( s_{1}\right) \otimes dPb\left(
s_{2}\right) \otimes\dots\otimes dPb\left( s_{n}\right) \right\rangle .$$
\[l.h9.7\]If $P,Q\in\operatorname*{Proj}\left( \mathbb{W}\right)$, then$$\left\Vert \tilde{\alpha}_{P}-\tilde{\alpha}_{Q}\right\Vert_{L^{2}}^{2}:=\mathbb{E}\left\vert
\tilde{\alpha}_{P}-\tilde{\alpha}_{Q}\right\vert
^{2}=\frac{T^{n}}{n!}\left\Vert
\alpha_{P}-\alpha_{Q}\right\Vert_{n}^{2}.$$
Let $R\in\operatorname*{Proj}\left( \mathbb{W}\right) $ be the orthogonal projection onto $\operatorname*{Ran}\left( P\right) +\operatorname*{Ran}\left( Q\right)$. We then have $\left( \alpha_{P}\right)
_{R}=\alpha_{P}$ and $\left( \alpha_{Q}\right)_{R}=\alpha_{Q}$ and therefore, by Corollary \[c.h9.5\],$$\begin{aligned}
\mathbb{E}\left\vert
\tilde{\alpha}_{P}-\tilde{\alpha}_{Q}\right\vert ^{2} &
=\mathbb{E}\left\vert \left(
\alpha_{P}\right)_{R}^{\symbol{126}}-\left( \alpha_{Q}\right)
_{R}^{\symbol{126}}\right\vert ^{2}=\mathbb{E}\left\vert
\left( \alpha_{P}-\alpha_{P}\right)_{R}^{\symbol{126}}\right\vert ^{2}\\
& =\frac{T^{n}}{n!}\left\Vert \left( \alpha_{P}-\alpha_{P}\right)
_{R}\right\Vert_{n}^{2}=\frac{T^{n}}{n!}\left\Vert \alpha_{P}-\alpha
_{P}\right\Vert_{n}^{2}.\end{aligned}$$
\[p.h9.8\]Let $\alpha\in\mathbb{H}^{\ast\otimes n}$ and $P_{k}\in\operatorname*{Proj}\left( \mathbb{W}\right) $ with $P_{k}|_{\mathbb{H}}\uparrow I|_{\mathbb{H}}$, then $\left\{
\tilde{\alpha}_{P_{k}}\right\}_{k=1}^{\infty}$ is an $L^{2}$–convergent series. We denote the limit by $\tilde{\alpha}$. This limit is independent of the choice of orthogonal projections used in constructing $\tilde{\alpha}$.
For $k,l\in\mathbb{N}$, by Lemma \[l.h9.7\], $$\left\Vert \tilde{\alpha}_{P_{l}}-\tilde{\alpha}_{P_{k}}\right\Vert_{L^{2}}=\left\Vert
\alpha_{P_{l}}-\alpha_{P_{k}}\right\Vert_{n}\rightarrow0\text{ as
}l,k\rightarrow\infty,$$ because, as we have already seen, $\alpha_{P_{l}}\rightarrow\alpha$ in $\mathbb{H}^{\ast\otimes n}$. Therefore $\tilde{\alpha}:=L^{2}$– $\lim_{k\rightarrow\infty}\tilde{\alpha}_{P_{k}}$ exists.
Now suppose that $Q_{l}\in\operatorname*{Proj}\left(
\mathbb{W}\right) $ also increases to $I|_{\mathbb{H}}$. By Lemma \[l.h9.7\] and the fact that both $\alpha_{P_{l}}$ and $\alpha_{Q_{l}}$ converge to $\alpha$ in $\mathbb{H}^{\ast\otimes
n}$, we have $$\left\Vert \tilde{\alpha}_{P_{l}}-\tilde{\alpha}_{Q_{l}}\right\Vert_{L^{2}}=\left\Vert \alpha_{P_{l}}-\alpha_{Q_{l}}\right\Vert_{\mathbb{H}^{\ast\otimes n}}\rightarrow0\text{ as }l\rightarrow\infty.$$
By polarization of the identity, $\left\Vert \tilde{\alpha}\right\Vert
_{L^{2}}^{2}=T^{n}\left\Vert \alpha\right\Vert_{n}^{2}/n!$, it follows that$$\left(
\tilde{\alpha},\tilde{\beta}\right)_{L^{2}}=\frac{T^{n}}{n!}\left(
\alpha,\beta\right)_{\mathbb{H}^{\ast\otimes n}}\text{ for all
}\alpha ,\beta\in\mathbb{H}^{\ast\otimes n}.$$ Moreover, if $\alpha\in\mathbb{H}^{\ast\otimes n}$ and $\beta\in
\mathbb{H}^{\ast\otimes m}$ with $m\neq n$, by the orthogonality of the finite dimensional approximations, $\tilde{\alpha}_{P_{l}}$ and $\tilde{\beta}_{P_{l}}$, we have that $\left( \tilde{\alpha},\tilde{\beta}\right)_{L^{2}}=0$.
\[Itô’s isometry\]\[c.h9.9\]Suppose that $\alpha=\left\{ \alpha
_{n}\right\}_{n=0}^{\infty}\in\bigoplus\limits_{n=0}^{\infty}\frac{T^{n}}{n!}\mathbb{H}^{\ast\otimes n}$, i.e. $\alpha_{n}\in\mathbb{H}^{\ast\otimes n}$ for all $n$ such that $$\left\Vert \alpha\right\Vert_{T}^{2}=\sum_{n=0}^{\infty}\frac{T^{n}}{n!}\left\Vert \alpha_{n}\right\Vert_{n}^{2}<\infty.$$ Then $\tilde{\alpha}:=\sum_{n=0}^{\infty}\tilde{\alpha}_{n}$ is $L^{2}\left(
\mathbf{P}\right) $–convergent and the map,$$\bigoplus\limits_{n=0}^{\infty}\frac{T^{n}}{n!}\mathbb{H}^{\ast\otimes n}\ni\alpha\mapsto\tilde{\alpha}\in L^{2}\left( \mathbf{P}\right),$$ is an isometry, where $\mathbf{P}$ is the probability measure used in describing the law of $\left\{ b\left( t\right) \right\}
_{t\geq0}$.
The stochastic Taylor map\[s.h9.2\]
-----------------------------------
Let $b\left( t\right) =\left( B\left( t\right),B_{0}\left(
t\right) \right) \in\mathfrak{g}$ and $g\left( t\right) \in G$ be the Brownian motions introduced at the start of Section \[s.h4\]. We are going to use the results of the previous subsection with $\mathbb{H}=\mathfrak{g}_{CM}$, $\mathbb{W}=\mathfrak{g,}$ and $b\left( t\right) =\left( B\left( t\right) ,B_{0}\left( t\right)
\right)$. Let $f\in\mathcal{H}_{T}^{2}\left( G\right) $ and $\alpha_{f}:=\mathcal{T}_{T}S_{T}f\in J_{T}^{0}\left(
\mathfrak{g}_{CM}\right) $. The following theorem is a (precise) restatement of Theorem \[t.h1.9\].
\[t.h9.10\]For any $f\in\mathcal{H}_{T}^{2}\left( G\right) $ $$f\left( g\left( T\right) \right) =\tilde{\alpha}_{f}, \label{e.h9.6}$$ where $\tilde{\alpha}_{f}$ was introduced in Corollary \[c.h9.9\]. (The right hand side of Eq. (\[e.h1.8\]) is to be interpreted as $\tilde{\alpha
}_{f}.$)
First suppose that $f$ is a holomorphic polynomial and $P\in
\operatorname*{Proj}\left( W\right) $ so that $\pi_{P}\in
\operatorname*{Proj}\left( \mathfrak{g}\right)$. Then by Itô’s formula,$$f\left( g_{P}\left( T\right) \right) =f\left( \mathbf{e}\right)
+\int_{0}^{T}\left\langle Df\left( g_{P}\left( t\right)
\right),d\pi _{P}b\left( t\right) \right\rangle .$$ Iterating this equation as in the proof of [Driver1995a]{}, if $N\in\mathbb{N}$ is sufficiently large, then$$\begin{aligned}
f\left( g_{P}\left( T\right) \right) & =f\left(
\mathbf{e}\right) +\sum_{n=1}^{N}\int_{0\leq s_{1}\leq
s_{2}\leq\dots\leq s_{n}\leq T}\left\langle D^{n}f\left(
\mathbf{e}\right),d\pi_{P}b\left( s_{1}\right) \otimes\dots\otimes
d\pi_{P}b\left( s_{n}\right) \right\rangle
\\
& =f\left( \mathbf{e}\right) +\sum_{n=1}^{N}\left[ D^{n}f\left(
\mathbf{e}\right) \right]_{\pi_{P}}^{\symbol{126}}.\end{aligned}$$ We now replace $P$ by $P_{k}\in\operatorname*{Proj}\left( W\right) $ with $P_{k}\uparrow I$ in this identity. Using Propositions \[p.h4.12\] and \[p.h9.8\], we may now pass to the limit as $k\rightarrow\infty$ in order to conclude, $$f\left( g\left( T\right) \right) =f\left( \mathbf{e}\right) +\sum
_{n=1}^{N}\left[ D^{n}f\left( \mathbf{e}\right) \right]^{\symbol{126}}=\tilde{\alpha}_{f}. \label{e.h9.7}$$
Now suppose that $f\in\mathcal{H}_{T}^{2}\left( G\right)$. By Theorem \[t.h7.1\] we can find a sequence of holomorphic polynomials $\left\{ f_{n}\right\}
_{n=1}^{\infty}\subset\mathcal{P}$ such that $$\mathbb{E}\left\vert f\left( g\left( T\right) \right) -f_{n}\left(
g\left( T\right) \right) \right\vert ^{2}=\left\Vert f-f_{n}\right\Vert
_{L^{2}\left( \nu_{T}\right) }^{2}\rightarrow0\text{ as }n\rightarrow
\infty.$$ The isometry property of the Taylor and skeleton maps (Theorem \[t.h6.10\] and Corollary \[c.h8.3\]), shows that $\alpha_{f_{n}}\rightarrow\alpha_{f}$ in $J_{T}^{0}$ and therefore by Corollary \[c.h9.9\] $\tilde{\alpha}_{f_{n}}\rightarrow\tilde{\alpha}_{f}$ as $n\rightarrow\infty$. Hence we may pass to the limit in Eq. (\[e.h9.7\]) applied to the sequence $f_{n}\left( g\left( T\right) \right) =\tilde{\alpha}_{f_{n}}$, to complete the proof of Eq. (\[e.h9.6\]).
Future directions and questions\[s.h10\]
========================================
In this last section we wish to speculate on a number of ways that the results in this paper might be generalized.
1. It should be possible to remove the restriction on $\mathbf{C}$ being finite dimensional, i.e. we expect much of what have done in this paper to go through when $\mathbf{C}$ is replaced by a separable Hilbert space. In doing so one would have to modify the finite dimensional approximations used in our construction to truncate $\mathbf{C}$ as well.
2. We also expect that the level of non-commutativity of $G$ may be increased. To be more precise, under suitable hypothesis it should be possible to handle more general graded nilpotent Lie groups.
3. Open questions:
1. as we noted in Remark \[r.h5.13\] we do not know if $\mathcal{A}_{T}^{p}=\mathcal{H}_{T}^{p}\left( G\right)$. It might be easier to try to answer this question for $p=2$.
2. give an intrinsic characterization of $\mathcal{H}_{T}^{2}\left( G\right)$ as in Shigekawa [@Shigekawa1991] in terms of functions in $L^{2}\left(
\nu_{T}\right)$ solving a weak form of the Cauchy–Riemann equations.
We are grateful to Professor Malliavin whose question during a workshop at the Hausdorff Institute (Bonn, Germany) led us to include a section on a holomorphic chaos expansion.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[10]{}
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[^1]: This research was supported in part by NSF Grant DMS-0504608 and the Miller Institute at the University of California, at Berkeley.
[^2]: Research was supported in part by NSF Grant DMS-0706784 and the Humboldt Foundation Research Fellowship.
[^3]: Here is the proof which should probably be omitted.
Using $\left\langle
k,ih\right\rangle_{H_{\operatorname{Re}}}=\left\langle ik,
i^{2}h\right\rangle_{H_{\operatorname{Re}}}=-\left\langle h,
ik\right\rangle_{H_{\operatorname{Re}}}$ we see that $$\begin{aligned}
\left\langle ih, k\right\rangle_{H} & =\left\langle ih,
k\right\rangle_{H_{\operatorname{Re}}}+i\left\langle ih,
ik\right\rangle
_{H_{\operatorname{Re}}}=\left\langle ih, k\right\rangle_{H_{\operatorname{Re}}}+i\left\langle h, k\right\rangle_{H_{\operatorname{Re}}}\\
& =i\left\langle h,
k\right\rangle_{H_{\operatorname{Re}}}-\left\langle h,
ik\right\rangle_{H_{\operatorname{Re}}}=i\left\langle h,
k\right\rangle\end{aligned}$$ and$$\begin{aligned}
\left\langle k, h\right\rangle_{H} & =\left\langle k,
h\right\rangle_{H_{\operatorname{Re}}}+i\left\langle k,
ih\right\rangle
_{H_{\operatorname{Re}}}\\
& =\left\langle h, k\right\rangle
_{H_{\operatorname{Re}}}-i\left\langle h, ik\right\rangle
_{H_{\operatorname{Re}}}=\overline{\left\langle h, k\right\rangle
}_{H}.\end{aligned}$$
|
---
abstract: 'The growing demand for structured knowledge has led to great interest in relation extraction, especially in cases with limited supervision. However, existing distance supervision approaches only extract relations expressed in single sentences. In general, cross-sentence relation extraction is under-explored, even in the supervised-learning setting. In this paper, we propose the first approach for applying distant supervision to cross-sentence relation extraction. At the core of our approach is a graph representation that can incorporate both standard dependencies and discourse relations, thus providing a unifying way to model relations within and across sentences. We extract features from multiple paths in this graph, increasing accuracy and robustness when confronted with linguistic variation and analysis error. Experiments on an important extraction task for precision medicine show that our approach can learn an accurate cross-sentence extractor, using only a small existing knowledge base and unlabeled text from biomedical research articles. Compared to the existing distant supervision paradigm, our approach extracted twice as many relations at similar precision, thus demonstrating the prevalence of cross-sentence relations and the promise of our approach.'
author:
- Chris Quirk
- |
Hoifung Poon\
Microsoft Research\
One Microsoft Way\
Redmond, WA 98052\
[{chrisq,hoifung}@microsoft.com]{}\
bibliography:
- 'refs.bib'
title: Distant Supervision for Relation Extraction beyond the Sentence Boundary
---
|
---
abstract: 'In this paper, we derive a new representation for the Anger–Weber function, employing the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A **439** (1992) 373–396). As a consequence of this representation, we deduce a number of properties of the large order asymptotic expansion of the Anger–Weber function, including explicit and realistic error bounds, asymptotic approximations for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.'
address: 'Central European University, Department of Mathematics and its Applications, H-1051 Budapest, Nádor utca 9, Hungary'
author:
- Gergő Nemes
title: |
The resurgence properties of\
the large order asymptotics of\
the Anger–Weber function II
---
Introduction {#section1}
============
In the first part of this series of papers [@Nemes2], we proved new resurgence-type representations for the remainder term of the asymptotic expansion of the Anger–Weber function $\mathbf{A}_{ - \nu } \left( \lambda \nu \right)$ with complex $\nu$ and $\lambda \geq 1$. These resurgence formulas have different forms according to whether $\lambda >1$ or $\lambda=1$. The main goal of this paper is to derive a similar representation for the Anger–Weber function $\mathbf{A}_{\nu} \left( \lambda \nu\right)$ with complex $\nu$ and $\lambda > 0$. Our derivation is based on the reformulation of the method of steepest descents by Howls [@Howls]. Using this representation, we obtain a number of properties of the large order asymptotic expansion of the Anger–Weber function, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.
Our first theorem describes the resurgence properties of the asymptotic expansion of $\mathbf{A}_{\nu} \left(\lambda \nu \right)$ for $\lambda > 0$. The notations follow the ones given in [@NIST p. 298]. Throughout this paper, empty sums are taken to be zero.
\[thm1\] Let $\lambda>0$ be a fixed positive real number, and let $N$ be a non-negative integer. Then we have $$\label{eq11}
\mathbf{A}_\nu \left( {\lambda \nu } \right) = \frac{1}{\pi}\sum\limits_{n = 0}^{N - 1} {\frac{{\left( {2n} \right)!a_n \left( {\lambda } \right)}}{\nu ^{2n + 1}}} + R_N \left( {\nu ,\lambda } \right)$$ for $\left|\arg \nu\right| < \frac{\pi}{2}$, with $$\label{eq9}
a_n \left( \lambda \right) = \frac{1}{{\left( {2n} \right)!}}\left[ {\frac{d^{2n}}{dt^{2n}}\left( {\frac{t}{{\lambda \sinh t + t}}} \right)^{2n + 1} } \right]_{t = 0} = \frac{{\left( { - 1} \right)^n }}{{\left( {2n} \right)!}}\int_0^{ + \infty } {t^{2n} e^{ - \pi t} iH_{it}^{\left( 1 \right)} \left( {\lambda it} \right)dt}$$ and $$\label{eq10}
R_N \left( {\nu ,\lambda } \right) = \frac{{\left( { - 1} \right)^N }}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} e^{ - \pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( \lambda it \right)dt} .$$
In a previous paper [@Nemes], we proved similar representations for the Hankel function $H_\nu^{\left(1\right)}\left(\lambda \nu\right)$ with $\lambda \geq 1$. In particular, for any non-negative integer $N$ and fixed $0 < \beta <\frac{\pi}{2}$, we have $$\label{eq30}
H_\nu ^{\left( 1 \right)} \left( {\nu \sec \beta } \right) = \frac{{e^{i\nu \left( {\tan \beta - \beta } \right) - \frac{\pi}{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\left( {\sum\limits_{m = 0}^{M - 1} {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}} + R_M^{\left(H\right)} \left( {\nu ,\beta } \right)} \right)$$ and $$\label{eq31}
H_\nu ^{\left( 1 \right)} \left( \nu \right) = - \frac{2}{{3\pi }}\sum\limits_{n = 0}^{N - 1} {d_{2n} e^{\frac{{2\left( {2n + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2n + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}{{\nu ^{\frac{{2n + 1}}{3}} }}} + R_N^{\left( H \right)} \left( \nu \right),$$ for $-\frac{\pi}{2} < \arg \nu < \frac{3\pi}{2}$, with $$\begin{gathered}
\label{eq32}
\begin{split}
U_m \left( {i\cot \beta } \right) & = \left( { - 1} \right)^m \frac{{\left( {i\cot \beta } \right)^m }}{{2^m m!}}\left[ {\frac{{d^{2m} }}{{dt^{2m} }}\left( {\frac{1}{2}\frac{{t^2 }}{{i\cot \beta \left( {t - \sinh t} \right) + \cosh t - 1}}} \right)^{m + \frac{1}{2}} } \right]_{t = 0}\\
& = \frac{i^m}{2\left( {2\pi \cot \beta } \right)^{\frac{1}{2}}}\int_0^{ + \infty } {t^{m - \frac{1}{2}} e^{ -t \left( {\tan \beta - \beta } \right)} \left( {1 + e^{ - 2\pi t} } \right)i H_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt}
\end{split}\end{gathered}$$ and $$\label{eq33}
d_{2n} = \frac{1}{{\left( {2n} \right)!}}\left[ {\frac{{d^{2n} }}{{dt^{2n} }}\left( {\frac{{t^3 }}{{\sinh t - t}}} \right)^{\frac{{2n + 1}}{3}} } \right]_{t = 0} = \frac{{\left( { - 1} \right)^n }}{{\Gamma \left( {\frac{{2n + 1}}{3}} \right)}}\int_0^{ + \infty } {t^{\frac{{2n - 2}}{3}} e^{ - 2\pi t} i H_{it}^{\left( 1 \right)} \left( {it} \right)dt} .$$ The remainder terms $R_M^{\left(H\right)} \left( {\nu ,\beta } \right)$ and $R_N^{\left( H \right)} \left( \nu \right)$ can be expressed as $$\label{eq34}
R_M^{\left(H\right)} \left( {\nu ,\beta } \right) = \frac{1}{{2\left( {2\pi \cot \beta } \right)^{\frac{1}{2}} \left(i\nu\right)^M }}\int_0^{ + \infty } {\frac{{t^{M - \frac{1}{2}} e^{ - t\left( {\tan \beta - \beta } \right)} }}{{1 + it/\nu }}\left( {1 + e^{ - 2\pi t} } \right)i H_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt}$$ and $$\label{eq35}
R_N^{\left( H \right)} \left( \nu \right) = \frac{{\left( { - 1} \right)^N }}{{3\pi \nu ^{\frac{{2N + 1}}{3}} }}\int_0^{ + \infty } {t^{\frac{{2N - 2}}{3}} e^{ - 2\pi t} \left( {\frac{{e^{\frac{{\left( {2N + 1} \right)\pi i}}{3}} }}{{1 + \left( {t/\nu } \right)^{\frac{2}{3}} e^{\frac{{2\pi i}}{3}} }} + \frac{1}{{1 + \left( {t/\nu } \right)^{\frac{2}{3}} }}} \right)H_{it}^{\left( 1 \right)} \left( {it} \right)dt} .$$ These representations of the Hankel function will play an essential role in later sections of this paper. It is important to note that for the case $0 < \lambda < 1$, no simple explicit expression is known for the remainder of the asymptotic series of the Hankel function $H_\nu^{\left(1\right)}\left(\lambda \nu\right)$.
If $\mathbf{J}_{\nu}\left(z\right)$ denotes the Anger function, then $\sin \left( {\pi \nu } \right)\mathbf{A}_{\nu} \left( \lambda \nu \right) =\mathbf{J}_{\nu} \left(\lambda\nu\right)- J_{\nu} \left(\lambda\nu\right)$ (see [@NIST p. 296]). From these and the continuation formulas for the Bessel and Hankel functions (see [@NIST p. 222 and p. 226]), we find $$\begin{aligned}
\sin \left( {\pi \nu } \right)\mathbf{A}_{\nu} \left( {\lambda \nu e^{2\pi im} } \right) & = \mathbf{J}_{\nu} \left( {\lambda \nu e^{2\pi im} } \right) - J_{\nu } \left( {\lambda \nu e^{2\pi im} } \right)
\\ & = \sin \left( {\pi \nu } \right)\mathbf{A}_\nu \left( {\lambda \nu } \right) + \left( {1 - e^{2\pi im\nu } } \right)J_\nu \left( {\lambda \nu } \right)
\\ & = \sin \left( {\pi \nu } \right)\mathbf{A}_\nu \left( {\lambda \nu } \right) - ie^{\pi im\nu } \sin \left( {\pi m\nu } \right)\left( {H_\nu ^{\left( 1 \right)} \left( {\lambda \nu } \right) + H_\nu ^{\left( 2 \right)} \left( {\lambda \nu } \right)} \right)\end{aligned}$$ for every integer $m$. From this expression and the resurgence formulas , and , we can derive analogous representations in sectors of the form $$\left( {2m - \frac{1}{2}} \right)\pi < \arg \nu < \left( {2m + \frac{1}{2}} \right)\pi ,\; m \in \mathbb{Z},$$ as long as $\lambda \geq 1$. Similarly, applying the continuation formulas $$\begin{gathered}
\label{eq50}
\begin{split}
& - \sin \left( {\pi \nu } \right)\mathbf{A}_{-\nu} \left( {\lambda \nu e^{\left( {2m + 1} \right)\pi i} } \right) = \mathbf{J}_{-\nu} \left( {\lambda \nu e^{\left( {2m + 1} \right)\pi i} } \right) - J_{-\nu} \left( {\lambda \nu e^{\left( {2m + 1} \right)\pi i} } \right)\\
& = \sin \left( {\pi \nu } \right)\mathbf{A}_\nu \left( {\lambda \nu } \right) + J_\nu \left( {\lambda \nu } \right) - e^{ - \left( {2m + 1} \right)\pi i\nu } J_{ - \nu } \left( {\lambda \nu } \right)\\
& = \sin \left( {\pi \nu } \right) \mathbf{A}_\nu \left( {\lambda \nu } \right) + ie^{ - \pi i m \nu} \sin \left( {\pi m\nu } \right)H_\nu ^{\left( 1 \right)} \left( {\lambda \nu } \right) + ie^{ - \left( {m + 1} \right)\pi i\nu } \sin \left( {\pi \left( {m + 1} \right)\nu } \right)H_\nu ^{\left( 2 \right)} \left( {\lambda \nu } \right)
\end{split}\end{gathered}$$ and the representations , and , we can obtain resurgence formulas in any sector of the form $$\left( {2m + \frac{1}{2}} \right)\pi < \arg \nu < \left( {2m + \frac{3}{2}} \right)\pi ,\; m \in \mathbb{Z},$$ provided that $\lambda \geq 1$. The lines $\arg \nu = \left( {2m \pm \frac{1}{2}} \right)\pi$ are the Stokes lines for the function $\mathbf{A}_{\nu} \left(\lambda \nu\right)$.
When $\nu$ is an integer, the limiting values have to be taken in these continuation formulas.
If we neglect the remainder term and extend the sum to $N = \infty$ in Theorem \[thm1\], we recover the known asymptotic series of the Anger–Weber function. Some other formulas for the coefficients $a_n\left(\lambda\right)$ can be found in [@Nemes2 Appendix A].
In the following two theorems, we give exponentially improved asymptotic expansions for the function $\mathbf{A}_{\nu } \left(\lambda \nu\right)$ when $\lambda>1$ and $\lambda=1$, respectively. Since there is no simple resurgence formula for the Hankel function $H_\nu^{\left(1\right)}\left(\lambda \nu\right)$ when $0<\lambda <1$, at least with our method, we can not prove exponentially improved expansions for the function $\mathbf{A}_{\nu } \left(\lambda \nu\right)$ in the range $0<\lambda <1$. We express our expansions in terms of the Terminant function $\widehat T_p\left(w\right)$ whose definition and basic properties are given in Section \[section5\]. In these theorems, $R_N \left( {\nu ,\lambda } \right)$ is defined by and it is extended to the sector $\left|\arg \nu\right| \leq \frac{3\pi}{2}$ via analytic continuation. In Theorem \[thm2\], we employ the substitution $1 < \lambda = \sec \beta$ with a suitable $0 < \beta <\frac{\pi}{2}$. Throughout this paper, we write $\mathcal{O}_{M,\rho }$ to indicate the dependence of the implied constant on the parameters $M$ and $\rho$.
\[thm2\] Suppose that $\left|\arg \nu\right| \leq \frac{3\pi}{2}$, $\left|\nu\right|$ is large and $N = \frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta +\pi} \right) + \rho$ is a positive integer with $\rho$ being bounded. Then $$\begin{gathered}
R_N \left( {\nu ,\sec \beta } \right) = i\frac{{e^{i\nu \left( {\tan \beta - \beta +\pi} \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^{M - 1} {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\widehat T_{2N - m + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta +\pi} \right)} \right)}\\ - i\frac{{e^{ - i\nu \left( {\tan \beta - \beta +\pi} \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^{M - 1} {\frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta +\pi} \right)} \right)} + R_{N,M} \left( {\nu ,\sec \beta } \right)\end{gathered}$$ with $M$ being an arbitrary fixed non-negative integer, and $$R_{N,M} \left( {\nu ,\sec \beta } \right) = \mathcal{O}_{M,\rho } \left( {\frac{{e^{ - \left| \nu \right|\left( {\tan \beta - \beta +\pi} \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right)$$ for $\left|\arg \nu\right| \leq \frac{\pi}{2}$; $$R_{N,M} \left( {\nu ,\sec \beta } \right) = \mathcal{O}_{M,\rho } \left( {\frac{{e^{ \mp \Im \left( \nu \right)\left( {\tan \beta - \beta +\pi} \right)} }}{{\left( {\frac{1}{2}\left| \nu \right|\pi \tan \beta } \right)^{\frac{1}{2}} }}\frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{\left| \nu \right|^M }}} \right)$$ for $\frac{\pi}{2} \leq \pm \arg \nu \leq \frac{3\pi}{2}$.
\[thm3\] Suppose that $\left|\arg \nu\right| \leq \frac{3\pi}{2}$, $\left|\nu\right|$ is large and $N = \frac{1}{2}\pi \left| \nu \right| + \rho$ is a positive integer with $\rho$ being bounded. Then $$\begin{gathered}
R_N \left( {\nu ,1} \right) = - ie^{\pi i\nu } \frac{2}{{3\pi }}\sum\limits_{m = 0}^{M - 1} {d_{2m} e^{\frac{{2\left( {2m + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}\widehat T_{2N - \frac{{2m - 2}}{3}} \left( {\pi i\nu } \right)}
\\ - ie^{ - \pi i\nu } \frac{2}{{3\pi }}\sum\limits_{m = 0}^{M - 1} {d_{2m} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}\widehat T_{2N - \frac{{2m - 2}}{3}} \left( { - \pi i\nu } \right)} + R_{N,M} \left( {\nu ,1} \right).\end{gathered}$$ with $M$ being an arbitrary fixed non-negative integer, and $$R_{N,M} \left( {\nu ,1} \right) = \mathcal{O}_{M,\rho} \left( {e^{ - \pi \left| \nu \right|} \left| {d_{2M} } \right|\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 1}}{3}} }}} \right) \; \text{ if } \; M \equiv 0,2 \mod 3,$$ $$R_{N,M} \left( {\nu ,1} \right) = \mathcal{O}_{M,\rho} \left( {e^{ - \pi \left| \nu \right|} \left| {d_{2M + 2} } \right|\frac{{\Gamma \left( {\frac{{2M + 3}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 3}}{3}} }}} \right) \; \text{ if } \; M \equiv 1 \mod 3$$ for $\left|\arg \nu\right| \leq \frac{\pi}{2}$; $$R_{N,M} \left( {\nu ,1} \right) = \mathcal{O}_{M,\rho } \left( {e^{ \mp \pi \Im \left( \nu \right)} \left| {d_{2M} } \right|\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 1}}{3}} }}} \right) \; \text{ if } \; M \equiv 0,2 \mod 3,$$ $$R_{N,M} \left( {\nu ,1} \right) = \mathcal{O}_{M,\rho } \left( {e^{ \mp \pi \Im \left( \nu \right)} \left| {d_{2M + 2} } \right|\frac{{\Gamma \left( {\frac{{2M + 3}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 3}}{3}} }}} \right) \; \text{ if } \; M \equiv 1 \mod 3$$ for $\frac{\pi}{2} \leq \pm \arg \nu \leq \frac{3\pi}{2}$.
The rest of the paper is organized as follows. In Section \[section2\], we prove the resurgence formula stated in Theorem \[thm1\]. In Section \[section3\], we give explicit and realistic error bounds for the asymptotic expansions of $\mathbf{A}_{\nu}\left(\lambda \nu \right)$ using the results of Section \[section2\]. In Section \[section4\], asymptotic approximations for $a_n \left( \lambda \right)$ as $n \to +\infty$ are given. In Section \[section5\], we prove the exponentially improved expansions presented in Theorems \[thm2\] and \[thm3\], and provide a detailed discussion of the Stokes phenomenon related to the expansions of $\mathbf{A}_{\nu}\left( \lambda \nu \right)$, $\lambda \geq 1$. The paper concludes with a short discussion in Section \[section6\].
Proof of the resurgence formula {#section2}
===============================
Our analysis is based on the integral definition of the Anger–Weber function $$\mathbf{A}_{\nu } \left( z \right) = \frac{1}{\pi }\int_0^{ + \infty } {e^{-\nu t - z\sinh t} dt} \quad \left| {\arg z} \right| < \frac{\pi }{2}.$$ If $z = \lambda \nu$, where $\lambda$ is a positive constant, then $$\label{eq1}
\mathbf{A}_\nu \left( {\lambda \nu} \right) = \frac{1}{\pi }\int_0^{ + \infty } {e^{ - \nu \left( {\lambda\sinh t + t} \right)} dt} \quad \left| {\arg \nu } \right| < \frac{\pi }{2}.$$ The saddle points of the integrand are the roots of the equation $\lambda \cosh t = -1$. Hence, the saddle points are given by $t_{\pm}^{\left( k \right)} = \pm \mathrm{sech}^{ - 1} \lambda + \left(2k+1\right)\pi i$ where $k$ is an arbitrary integer. We denote by $\mathscr{C}_{\pm}^{\left( k \right)}\left(\theta\right)$ the portion of the steepest paths that pass through the saddle point $t_{\pm}^{\left( k \right)}$. Here, and subsequently, we write $\theta = \arg \nu$. As for the path of integration $\mathscr{P}\left(\theta \right)$ in , we take that connected component of $$\left\{ {t \in \mathbb{C}:\arg \left[ {e^{i\theta } \left( {\lambda \sinh t + t} \right)} \right] = 0} \right\} \cup \left\{0\right\},$$ which contains the origin. We remark that $\mathscr{P}\left(0\right)$ is the positive real axis.
First, we suppose that $\lambda > 1$ and take $\lambda = \sec \beta$ with a suitable $0<\beta<\frac{\pi}{2}$. With this notation, $t_{\pm}^{\left( k \right)} = \pm i \beta + \left(2k+1\right)\pi i$. For simplicity, we assume that $\theta = 0$. In due course, we shall appeal to an analytic continuation argument to extend our results to complex $\nu$. Let $f\left( {t,\beta } \right) = \sec \beta \sinh t + t$. If $$\label{eq2}
\tau = f\left( {t,\beta } \right),$$ then $\tau$ is real on the curve $\mathscr{P}\left(0\right)$, and, as $t$ travels along this curve from $0$ to $+\infty $, $\tau$ increases from $0$ to $+\infty$. Therefore, corresponding to each positive value of $\tau$, there is a value of $t$, say $t\left(\tau\right)$, satisfying with $t\left(\tau\right)>0$. In terms of $\tau$, we have $$\mathbf{A}_\nu \left( {\nu \sec \beta } \right) = \frac{1}{\pi }\int_0^{ + \infty } {e^{ - \nu \tau } \frac{{dt\left( \tau \right)}}{{d\tau }}d\tau } = \frac{1}{\pi }\int_0^{ + \infty } {e^{ - \nu \tau } \frac{1}{{\sec \beta \cosh t\left( \tau \right) + 1}}d\tau } .$$ Following Howls, we express the function involving $t\left(\tau\right)$ as a contour integral using the residue theorem, to find $$\mathbf{A}_\nu \left( {\nu \sec \beta } \right) = \frac{1}{\pi }\int_0^{ + \infty } {e^{ - \nu \tau } \frac{1}{{2\pi i}}\oint_\Gamma {\frac{{f^{ - 1} \left( {u,\beta } \right)}}{{1 - \tau ^2 f^{ - 2} \left( {u,\beta } \right)}}du} d\tau }$$ where the contour $\Gamma$ encircles the path $\mathscr{P}\left(0\right)$ in the positive direction and does not enclose any of the saddle points $t_ \pm ^{\left( k \right)}$ (see Figure \[fig1\]). Now, we employ the well-known expression for non-negative integer $N$ $$\label{eq3}
\frac{1}{1 - z} = \sum\limits_{n = 0}^{N-1} {z^n} + \frac{z^N}{1 - z},\; z \neq 1,$$ to expand the function under the contour integral in powers of $\tau ^2 f^{ - 2} \left( {u,\beta } \right)$. The result is $$\mathbf{A}_\nu \left( {\nu \sec \beta } \right) = \frac{1}{\pi}\sum\limits_{n = 0}^{N - 1}{ \int_0^{ + \infty } {\tau ^{2n} e^{ - \nu \tau } \frac{1}{{2\pi i}}\oint_\Gamma {\frac{{du}}{{f^{2n + 1} \left( {u,\beta } \right)}}} d\tau }} + R_N \left( {\nu ,\sec\beta } \right),$$ where $$\label{eq4}
R_N \left( {\nu ,\sec\beta } \right) = \frac{1}{\pi }\int_0^{ + \infty } {\tau ^{2N} e^{ - \nu \tau } \frac{1}{{2\pi i}}\oint_\Gamma {\frac{{f^{ - 2N - 1} \left( {u,\beta } \right)}}{{1 - \tau ^2 f^{ - 2} \left( {u,\beta } \right)}}du} d\tau } .$$ The path $\Gamma$ in the sum can be shrunk into a small circle around $0$, and we arrive at $$\label{eq5}
\mathbf{A}_\nu \left( {\nu \sec \beta } \right) = \frac{1}{\pi }\sum\limits_{n = 0}^{N - 1} {\frac{{\left( {2n} \right)!a_n \left( {\sec \beta } \right)}}{{\nu ^{2n + 1} }}} + R_N \left( {\nu ,\sec\beta } \right),$$ where $$a_n \left( {\sec \beta } \right) = \frac{1}{{2\pi i}}\oint_{\left( {0^ + } \right)} {\frac{{du}}{{f^{2n + 1} \left( {u,\beta } \right)}}} = \frac{1}{{\left( {2n} \right)!}}\left[ {\frac{{d^{2n} }}{{dt^{2n} }}\left( {\frac{t}{{\sec \beta \sinh t + t}}} \right)^{2n + 1} } \right]_{t = 0} .$$
Performing the change of variable $\nu \tau = s$ in yields $$\label{eq6}
R_N \left( {\nu ,\sec\beta } \right) = \frac{1}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {s^{2N} e^{ - s} \frac{1}{{2\pi i}}\oint_\Gamma {\frac{{f^{ - 2N - 1} \left( {u,\beta } \right)}}{{1 - \left( {s/\nu } \right)^2 f^{ - 2} \left( {u,\beta } \right)}}du} ds}$$ This representation of $R_N \left( {\nu ,\sec\beta } \right)$ and the formula can be continued analytically if we choose $\Gamma = \Gamma\left(\theta\right)$ to be an infinite contour that surrounds the path $\mathscr{P}\left(\theta\right)$ in the anti-clockwise direction and that does not encircle any of the saddle points $t_ \pm ^{\left( k \right)}$. This continuation argument works until the path $\mathscr{P}\left(\theta\right)$ runs into a saddle point. In the terminology of Howls, such saddle points are called adjacent to the endpoint $0$. As $$\left|\arg \left( {f\left(0 ,\beta\right) - f\left( {t_ \pm ^{\left( k \right)} ,\beta } \right)} \right)\right| = \frac{\pi}{2}$$ for any saddle point $t_\pm ^{\left( k \right)}$, we infer that is valid as long as $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ with a contour $\Gamma\left(\theta\right)$ specified above. When $\theta = -\frac{\pi}{2}$, the path $\mathscr{P}\left(\theta\right)$ connects to the saddle point $t_{-} ^{\left( 0 \right)} = - i\beta+\pi i$. Similarly, when $\theta = \frac{\pi}{2}$, the path $\mathscr{P}\left(\theta\right)$ connects to the saddle point $t_{+} ^{\left( -1 \right)} = i\beta-\pi i$. These are the adjacent saddles. The set $$\Delta = \left\{ {u \in \mathscr{P}\left( \theta \right) : - \frac{\pi}{2} < \theta < \frac{\pi}{2}} \right\}$$ forms a domain in the complex plane whose boundary contains portions of steepest descent paths through the adjacent saddles (see Figure \[fig2\]). These paths are $\mathscr{C}_{-}^{\left( 0 \right)} \left( { -\frac{\pi }{2}} \right)$ and $\mathscr{C}_{+} ^{\left( -1 \right)} \left( {\frac{\pi }{2}} \right)$, and they are called the adjacent contours to the endpoint $0$. The function under the contour integral in is an analytic function of $u$ in the domain $\Delta$, therefore we can deform $\Gamma$ over the adjacent contours. We thus find that for $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ and $N \geq 0$, may be written $$\begin{gathered}
\label{eq7}
\begin{split}
R_N \left( {\nu ,\sec\beta } \right) = \; & \frac{1}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {s^{2N} e^{ - s} \frac{1}{{2\pi i}}\int_{\mathscr{C}_{-}^{\left( 0 \right)} \left( { -\frac{\pi }{2}} \right)} {\frac{{f^{ - 2N - 1} \left( {u,\beta } \right)}}{{1 - \left( {s/\nu } \right)^2 f^{ - 2} \left( {u,\beta } \right)}}du} ds} \\ & + \frac{1}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {s^{2N} e^{ - s} \frac{1}{{2\pi i}}\int_{\mathscr{C}_{+} ^{\left( -1 \right)} \left( {\frac{\pi }{2}} \right)} {\frac{{f^{ - 2N - 1} \left( {u,\beta } \right)}}{{1 - \left( {s/\nu } \right)^2 f^{ - 2} \left( {u,\beta } \right)}}du} ds} .
\end{split}\end{gathered}$$
Now we make the changes of variable $$s = t\frac{{\left| {f\left( { - i\beta + \pi i,\beta } \right) - f\left( {0,\beta } \right)} \right|}}{{f\left( { - i\beta + \pi i,\beta } \right) - f\left( {0,\beta } \right)}}f\left( {u,\beta } \right) = - i t f\left( {u,\beta } \right)$$ in the first, and $$s = t\frac{{\left| {f\left( {i\beta - \pi i,\beta } \right) - f\left( {0,\beta } \right)} \right|}}{{f\left( {i\beta - \pi i,\beta } \right) - f\left( {0,\beta } \right)}}f\left( {u,\beta } \right) = i t f\left( {u,\beta } \right)$$ in the second double integral. Clearly, by the definition of the adjacent contours, $t$ is positive. The quantities $f\left( { - i\beta + \pi i,\beta } \right) - f\left( {0,\beta } \right) = i\left( {\tan \beta - \beta +\pi} \right)$ and $f\left( {i\beta - \pi i,\beta } \right) - f\left( {0,\beta } \right) = -i\left( {\tan \beta - \beta +\pi } \right)$ were essentially called the “singulants" by Dingle [@Dingle p. 147]. With these changes of variable, the representation for $R_N \left( {\nu ,\sec\beta } \right)$ becomes $$\label{eq8}
R_N \left( {\nu ,\sec\beta } \right) = \frac{{\left( { - 1} \right)^N }}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} }}{{1 + \left( {t/\nu } \right)^2 }}\left( {\frac{1}{{2\pi }}\int_{\mathscr{C}_{+} ^{\left( -1 \right)} \left( {\frac{\pi }{2}} \right)} {e^{ - itf\left( {u,\beta } \right)} du} - \frac{1}{{2\pi }}\int_{\mathscr{C}_{-}^{\left( 0 \right)} \left( { -\frac{\pi }{2}} \right)} {e^{itf\left( {u,\beta } \right)} du} } \right)ds} ,$$ for $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ and $N \geq 0$. Finally, we shift the contour $\mathscr{C}_{+} ^{\left( -1 \right)} \left( {\frac{\pi }{2}} \right)$ upward by $\pi$ and the contour $\mathscr{C}_{-}^{\left( 0 \right)} \left( { -\frac{\pi }{2}} \right)$ downward by $\pi$. Let us denote these new paths by $\widetilde{\mathscr{C}}_{+} ^{\left( -1 \right)} \left( {\frac{\pi }{2}} \right)$ and $\widetilde{\mathscr{C}}_{-}^{\left( 0 \right)} \left( { -\frac{\pi }{2}} \right)$, respectively. We therefore find that the contour integrals in can be expressed in terms of the Hankel functions since $$\frac{1}{{2\pi }}\int_{\mathscr{C}_{+} ^{\left( -1 \right)} \left( {\frac{\pi }{2}} \right)} {e^{ - itf\left( {u,\beta } \right)} du} = \frac{{e^{ - \pi t} }}{2}i\frac{1}{{\pi i}}\int_{\widetilde{\mathscr{C}}_{+} ^{\left( -1 \right)} \left( {\frac{\pi }{2}} \right)} {e^{it\left( {\sec \beta \sinh u - u} \right)} du} = \frac{{e^{ - \pi t} }}{2}iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right),$$ and $$\begin{aligned}
- \frac{1}{{2\pi }}\int_{\mathscr{C}_{-}^{\left( 0 \right)} \left( { -\frac{\pi }{2}} \right)} {e^{itf\left( {u,\beta } \right)} du} = - \frac{{e^{ - \pi t} }}{2}i\frac{1}{{\pi i}}\int_{\widetilde{\mathscr{C}}_{-}^{\left( 0 \right)} \left( { -\frac{\pi }{2}} \right)} {e^{ - it\left( {\sec \beta \sinh u - u} \right)} du} & = - \frac{{e^{ - \pi t} }}{2}iH_{ - it}^{\left( 2 \right)} \left( { - it\sec \beta } \right) \\ & = \frac{{e^{ - \pi t} }}{2}iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right).\end{aligned}$$ Substituting these into gives $$R_N \left( {\nu ,\sec\beta } \right) = \frac{{\left( { - 1} \right)^N }}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} e^{ - \pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right)dt} ,$$ for $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ and $N \geq 0$. Thus, we have proved and for $\lambda>1$.
Now, we extend our results to every $\lambda>0$. For fixed $\nu$, $\Re\left(\nu\right)>0$, we can extend $R_N \left( {\nu ,\lambda } \right)$ to an analytic function of $\lambda>0$ using . From the known behaviours $$iH_{it}^{\left( 1 \right)} \left( {\lambda it} \right) \sim - \frac{2}{\pi }\log t \; \text{ as } \; t \to 0+$$ and $$iH_{it}^{\left( 1 \right)} \left( {\lambda it} \right) = o\left( 1 \right) \; \text{ as } \; t \to +\infty,$$ it is seen that the right-hand side of is a well-defined analytic function of $0<\lambda \leq 1$, for every fixed $\nu$ with $\Re\left(\nu\right)>0$. Whence, by analytic continuation the equality holds for every $\lambda>0$ and $\nu$ with $\Re\left(\nu\right)>0$.
The first formula in has been proved for $\lambda>1$, however, by analytic continuation, it holds for every $\lambda>0$. To prove the second representation in , we apply for the right-hand side of $$a_n \left( \lambda \right) = \pi \frac{{\nu ^{2n + 1} }}{{\left( {2n} \right)!}}\left( {R_n \left( {\nu ,\lambda } \right) - R_{n + 1} \left( {\nu ,\lambda } \right)} \right).$$
Error bounds {#section3}
============
In this section, we derive explicit and realistic error bounds for the large order asymptotic series of the Anger–Weber function. The proofs are based on the resurgence formula given in Theorem \[thm1\].
We comment on the relation between Meijer’s work [@Meijer] on the asymptotic expansion of $\mathbf{A}_{\nu} \left( \lambda \nu \right)$, $\lambda > 1$ and ours. Some of the estimates in [@Meijer] coincide with ours and are valid in wider sectors of the complex $\nu$-plane. However, it should be noted that those bounds become less effective outside the sectors of validity of the representation due to the Stokes phenomenon. For those sectors we recommend the use of the continuation formulas given in Section \[section1\].
To estimate the remainder terms, we shall use the elementary result that $$\label{eq12}
\frac{1}{{\left| {1 - re^{i\varphi } } \right|}} \le \begin{cases} \left|\csc \varphi \right| & \; \text{ if } \; 0 < \left|\varphi \text{ mod } 2\pi\right| <\frac{\pi}{2} \\ 1 & \; \text{ if } \; \frac{\pi}{2} \leq \left|\varphi \text{ mod } 2\pi\right| \leq \pi \end{cases}$$ holds for any $r>0$. We will also need the fact that $$\label{eq13}
iH_{it}^{\left(1\right)} \left( {\lambda it} \right) \ge 0$$ for any $t>0$ and $\lambda \geq 1$ (see [@Nemes]).
Case (i): $\lambda \geq 1$
--------------------------
We observe that from and it follows that $$\left| {a_n \left( \lambda \right)} \right| = \frac{1}{{\left( {2n} \right)!}}\int_0^{ + \infty } {t^{2n} e^{ - \pi t} iH_{it}^{\left( 1 \right)} \left( {\lambda it} \right)dt} .$$ Using this formula, together with the representation and the estimate , we obtain the error bound $$\label{eq14}
\left| {R_N \left( {\nu ,\lambda } \right)} \right| \le \frac{1}{\pi }\frac{{\left( {2N} \right)!\left| {a_N \left( \lambda \right)} \right|}}{{\left| \nu \right|^{2N + 1} }} \begin{cases} \left|\csc\left(2\theta\right)\right| & \; \text{ if } \; \frac{\pi}{4} < \left|\theta\right| <\frac{\pi}{2} \\ 1 & \; \text{ if } \; \left|\theta\right| \leq \frac{\pi}{4}. \end{cases}$$ When $\nu$ is real and positive, we can obtain more precise estimates. Indeed, as $0 < \frac{1}{{1 + \left( {t/\nu } \right)^2 }} < 1 $ for $t,\nu>0$, from and we find $$R_N \left( {\nu ,\lambda } \right) = \frac{1}{\pi }\frac{{\left( {2N} \right)!a_N \left( \lambda \right)}}{{\nu ^{2N + 1} }}\Theta ,$$ where $0 < \Theta < 1$ is an appropriate number depending on $\nu,\lambda$ and $N$. In particular, when $N=0$, we have $$0 < \mathbf{A}_\nu \left( {\lambda \nu } \right) < \frac{1}{{\pi \nu \left( {1 + \lambda } \right)}} \; \text{ for } \; \nu >0.$$ Therefore, the leading order asymptotic approximation for $\mathbf{A}_{\nu} \left( \lambda \nu \right)$ is always in error by excess, for $\lambda \geq 1$ and for all positive values of $\nu$.
The error bound becomes singular as $\theta \to \pm \frac{\pi}{2}$, and therefore unrealistic near the Stokes lines. A better bound for $R_N \left( {\nu ,\lambda } \right)$ near these lines can be derived as follows. Let $0 < \varphi < \frac{\pi }{2}$ be an acute angle that may depend on $N$. Suppose that $\frac{\pi}{4} +\varphi < \theta \le \frac{\pi}{2}$. An analytic continuation of the representation to this sector can be found by rotating the path of integration in by $\varphi$: $$R_N \left( {\nu ,\lambda } \right) = \frac{{\left( { - 1} \right)^N }}{{\pi \nu ^{2N + 1} }}\int_0^{ + \infty e^{i\varphi } } {\frac{{t^{2N} e^{ - \pi t} }}{{1 + \left( {t/\nu } \right)^2 }}iH_{it}^{\left( 1 \right)} \left( {\lambda it} \right)dt} .$$ Substituting $t = \frac{se^{i\varphi }}{\cos \varphi}$ and applying the estimation , we obtain $$\left| {R_N \left( {\nu ,\lambda } \right)} \right| \le \frac{{\csc \left( {2\left( {\theta - \varphi } \right)} \right)}}{{\pi \cos ^{2N + 1} \varphi \left| \nu \right|^{2N + 1} }}\int_0^{ + \infty } {s^{2N} e^{ - \pi s} \left| {H_{\frac{{ise^{i\varphi } }}{{\cos \varphi }}}^{\left( 1 \right)} \left( {\lambda \frac{{ise^{i\varphi } }}{{\cos \varphi }}} \right)} \right|ds} .$$ In [@Nemes], it was shown that $$\label{eq60}
\left| {H_{\frac{{ise^{i\varphi } }}{{\cos \varphi }}}^{\left( 1 \right)} \left( {\lambda \frac{{ise^{i\varphi } }}{{\cos \varphi }}} \right)} \right| \le \frac{1}{{\sqrt {\cos \varphi } }}\left| {H_{is}^{\left( 1 \right)} \left( {\lambda is} \right)} \right| = \frac{1}{{\sqrt {\cos \varphi } }}iH_{is}^{\left( 1 \right)} \left( {\lambda is} \right)$$ for any $s>0$, $\lambda\geq 1$ and $0 < \varphi < \frac{\pi }{2}$. It follows that $$\label{eq15}
\left| {R_N \left( {\nu ,\lambda } \right)} \right| \le \frac{{\csc \left( {2\left( {\theta - \varphi } \right)} \right)}}{{\pi \cos ^{2N + \frac{3}{2}} \varphi \left| \nu \right|^{2N + 1} }}\int_0^{ + \infty } {s^{2N} e^{ - \pi s} iH_{is}^{\left( 1 \right)} \left( {\lambda is} \right)ds} = \frac{{\csc \left( {2\left( {\theta - \varphi } \right)} \right)}}{{\cos ^{2N + \frac{3}{2}} \varphi }}\frac{1}{\pi }\frac{{\left( {2N} \right)!\left| {a_N \left( \lambda \right)} \right|}}{{\left| \nu \right|^{2N + 1} }}.$$ The angle $\varphi = \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)$ minimizes the function $\csc \left( {2\left( {\frac{\pi }{2} - \varphi } \right)} \right)\cos ^{ - 2N - \frac{3}{2}} \varphi$, and $$\begin{aligned}
\frac{{\csc \left( {2\left( {\theta - \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)} \right)} \right)}}{{\cos ^{2N + \frac{3}{2}} \left( {\arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)} \right)}} & \le \frac{{\csc \left( {2\left( {\frac{\pi }{2} - \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)} \right)} \right)}}{{\cos ^{2N + \frac{3}{2}} \left( {\arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)} \right)}} \\ &= \frac{1}{{\sqrt 2 }}\left( 1+\frac{2}{4N + 5} \right)^{N + \frac{7}{4}} \sqrt {N + \frac{5}{4}} \le \sqrt {\frac{e}{2}\left( {N + \frac{3}{2}} \right)}\end{aligned}$$ for all $\frac{\pi }{4} + \varphi = \frac{\pi }{4} + \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right) < \theta \le \frac{\pi }{2}$ with $N \geq 0$. Applying this in yields the upper bound $$\label{eq16}
\left| {R_N \left( {\nu ,\lambda } \right)} \right| \le \sqrt {\frac{e}{2}\left( {N + \frac{3}{2}} \right)} \frac{1}{\pi }\frac{{\left( {2N} \right)!\left| {a_N \left( \lambda \right)} \right|}}{{\left| \nu \right|^{2N + 1} }},$$ which is valid for $\frac{\pi }{4} + \varphi = \frac{\pi }{4} + \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right) < \theta \le \frac{\pi }{2}$ with $N \geq 0$. Since $\left| {R_N \left( {\bar \nu ,\lambda } \right)} \right| = \left| {\overline {R_N \left( {\nu ,\lambda } \right)} } \right| = \left| {R_N \left( {\nu ,\lambda } \right)} \right|$, this bound also holds when $-\frac{\pi}{2} \leq \theta < -\frac{\pi }{4} - \arctan \left( {\left( {\frac{{4N + 5}}{2}} \right)^{ - \frac{1}{2}} } \right)$. In the ranges $\frac{\pi }{4} < \left| \theta \right| \leq \frac{\pi }{4} + \arctan \left( {\frac{{\sqrt 2 }}{3}} \right)$ it holds that $\left| {\csc \left( {2\theta } \right)} \right| \le \sqrt {\frac{e}{2}\left( {1 + \frac{3}{2}} \right)}$, whence the estimate is valid in the wider sectors $\frac{\pi }{4} < \left| \theta \right| \le \frac{\pi }{2}$ as long as $N\geq 1$.
Case (ii): $0 < \lambda < 1$
----------------------------
In this case, we cannot prove error bounds involving the first omitted term, since $iH_{it}^{\left(1\right)} \left( {\lambda it} \right)$ has an oscillatory behaviour when $0 < \lambda < 1$ and $t>0$. Nevertheless, we define $\widetilde a_n \left( \lambda \right)$ via the integral $$\widetilde a_n \left( \lambda \right) = \frac{1}{{\left( {2n} \right)!}}\int_0^{ + \infty } {t^{2n} e^{ - \pi t} \left| {H_{it}^{\left( 1 \right)} \left( {\lambda it} \right)} \right|dt};$$ and by the representation and the inequality , we deduce the error bound $$\left| {R_N \left( {\nu ,\lambda } \right)} \right| \le \frac{1}{\pi }\frac{{\left( {2N} \right)! \widetilde a_N \left( \lambda \right)}}{{\left| \nu \right|^{2N + 1} }} \begin{cases} \left|\csc\left(2\theta\right)\right| & \; \text{ if } \; \frac{\pi}{4} < \left|\theta\right| <\frac{\pi}{2} \\ 1 & \; \text{ if } \; \left|\theta\right| \leq \frac{\pi}{4}. \end{cases}$$ Simple estimates for the quantities $\widetilde a_n \left(\lambda\right)$ may perhaps be derived from the connection formula with the modified Bessel function of the third kind of purely imaginary order $$H_{it}^{\left( 1 \right)} \left(\lambda it \right) = \frac{2}{\pi i}e^{\frac{\pi}{2}t} K_{it} \left( \lambda t \right),$$ and the known bounds for this latter function (see, e.g., Booker et al. [@Booker]).
Since for $0<\lambda <1$ we do not have an inequality like , it seems hard to obtain any usable error bound which is appropriate when $\arg \nu$ is close to $\pm \frac{\pi}{2}$.
Asymptotics for the late coefficients {#section4}
=====================================
In this section, we investigate the asymptotic nature of the coefficients $a_n\left(\lambda\right)$ as $n \to +\infty$ with $\lambda$ being fixed. For our purposes, the most appropriate representation of these coefficients is the second integral formula in . Although the representation is valid for all $\lambda>0$, we shall find that the asymptotic form of $a_n\left(\lambda\right)$ is significantly different according to whether $\lambda>1$, $\lambda=1$ or $0<\lambda<1$.
Case (i): $\lambda>1$
---------------------
For this case, we take $\lambda = \sec \beta$ with a suitable $0 < \beta <\frac{\pi}{2}$. From , it follows that for any $t>0$ and $0 < \beta <\frac{\pi}{2}$, it holds that $$\label{eq17}
iH_{it}^{\left( 1 \right)} \left( {it\sec \beta } \right) = \frac{{e^{ - t\left( {\tan \beta - \beta } \right)} }}{{\left( {\frac{1}{2}t\pi \tan \beta } \right)^{\frac{1}{2}} }}\left( {\sum\limits_{m = 0}^{M - 1} {\frac{{i^m U_m \left( {i\cot \beta } \right)}}{{t^m }}} + R_M^{\left( H \right)} \left( {it,\beta } \right)} \right).$$ In [@Nemes], it was proved that the remainder $R_M^{\left( H \right)} \left( {it,\beta } \right)$ satisfies $$\label{eq18}
\left| {R_M^{\left( H \right)} \left( {it,\beta } \right)} \right| \le \frac{{\left| {U_M \left( {i\cot \beta } \right)} \right|}}{{t^M }} .$$ Substituting the formula into leads us to the expansion $$\begin{gathered}
\label{eq19}
\begin{split}
\left( {2n} \right)!a_n \left( {\sec \beta } \right) = \; & \left( {\frac{{2\cot \beta }}{{\pi \left( {\tan \beta - \beta + \pi } \right)}}} \right)^{\frac{1}{2}} \frac{{\left( { - 1} \right)^n \Gamma \left( {2n + \frac{1}{2}} \right)}}{{\left( {\tan \beta - \beta + \pi } \right)^{2n} }} \\ & \times \left( {\sum\limits_{m = 0}^{M - 1} {\left( {i\left( {\tan \beta - \beta + \pi } \right)} \right)^m U_m \left( {i\cot \beta } \right)\frac{{\Gamma \left( {2n - m + \frac{1}{2}} \right)}}{{\Gamma \left( {2n + \frac{1}{2}} \right)}}} + A_M \left( {n,\beta } \right)} \right),
\end{split}\end{gathered}$$ for any fixed $0 \le M \le 2n$, provided that $n\geq 1$. The remainder term $A_M \left( {n,\beta } \right)$ is given by the integral formula $$A_M \left( {n,\beta } \right) = \frac{{\left( {\tan \beta - \beta + \pi } \right)^{2n + \frac{1}{2}} }}{{\Gamma \left( {2n + \frac{1}{2}} \right)}}\int_0^{ + \infty } {t^{2n - \frac{1}{2}} e^{ - \left( {\tan \beta - \beta + \pi } \right)t} R_M^{\left( H \right)} \left( {it,\beta } \right)dt} .$$ To bound this error term, we apply the estimate to find $$\label{eq20}
\left| {A_M \left( {n,\beta } \right)} \right| \le \left( {\tan \beta - \beta + \pi } \right)^M \left| {U_M \left( {i\cot \beta } \right)} \right|\frac{{\Gamma \left( {2n - M + \frac{1}{2}} \right)}}{{\Gamma \left( {2n + \frac{1}{2}} \right)}}.$$ Expansions of type are called inverse factorial series in the literature. Numerically, their character is similar to the character of asymptotic power series, because the consecutive Gamma functions decrease asymptotically by a factor $2n$.
From the asymptotic behaviour of the coefficients $U_m \left( {i\cot \beta } \right)$ (see [@Nemes]), we infer that for large $n$, the least value of the bound occurs when $$M \approx \frac{\tan \beta - \beta }{3\left( {\tan \beta - \beta } \right) + \pi}\left(4n+1\right) .$$ Whence, the smaller $\beta$ is the larger $n$ has to be to get a reasonable approximation from .
Numerical examples illustrating the efficacy of the expansion , truncated optimally, are given in Table \[table1\].
\[c\][ l r @c@ l]{} &\
\[-1ex\] values of $\beta$ and $M$ & $\beta=\frac{\pi}{6}$, $M=4$ & &\
\[1ex\] exact numerical value of $a_{50}\left(\sec\beta\right)$ & $0.2004926124399177097019512509947129$ & $\times$ & $10^{-51}$\
\[1ex\] approximation to $a_{50}\left(\sec\beta\right)$ & $0.1997204566354320191164985775448290$ & $\times$ & $10^{-51}$\
\[1ex\] error & $0.7721558044856905854526734498839$ & $\times$ & $10^{-54}$\
\[1ex\] error bound using & $0.16182537012652011778281419657176$ & $\times$ & $10^{-53}$\
\[1ex\] &\
\[-1ex\] values of $\beta$ and $M$ & $\beta=\frac{\pi}{3}$, $M=27$ & &\
\[1ex\] exact numerical value of $a_{50}\left(\sec\beta\right)$ & $0.1619316740481494064448396260188866$ & $\times$ & $10^{-59}$\
\[1ex\] approximation to $a_{50}\left(\sec\beta\right)$ & $0.1619316740481497277978573226174596$ & $\times$ & $10^{-59}$\
\[1ex\] error & $-0.3213530176965985730$ & $\times$ & $10^{-74}$\
\[1ex\] error bound using & $0.6473043619300051742$ & $\times$ & $10^{-74}$\
\[1ex\] &\
\[-1ex\] values of $\beta$ and $M$ & $\beta=\frac{5\pi}{12}$, $M=47$ & &\
\[1ex\] exact numerical value of $a_{50}\left(\sec\beta\right)$ & $0.4989354184460076118014557886550703$ & $\times$ & $10^{-76}$\
\[1ex\] approximation to $a_{50}\left(\sec\beta\right)$ & $0.4989354184460076118014557886641359$ & $\times$ & $10^{-76}$\
\[1ex\] error & $-0.90656$ & $\times$ & $10^{-105}$\
\[1ex\] error bound using & $0.181989$ & $\times$ & $10^{-104}$\
\[-1ex\] &\
Case (ii): $\lambda=1$
----------------------
Using , we can write $$\label{eq24}
iH_{it}^{\left( 1 \right)} \left( {it} \right) = \frac{2}{{3\pi }}\sum\limits_{m = 0}^{M - 1} {\left( { - 1} \right)^m d_{2m} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{t^{\frac{{2m + 1}}{3}} }}} + iR_M^{\left( H \right)} \left( {it} \right)$$ for any $t>0$. It was shown in [@Nemes] that $$\label{eq25}
\left| {R_M^{\left( H \right)} \left( {it} \right)} \right| \le \frac{2}{3\pi}\left| {d_{2M} } \right|\frac{\sqrt 3 }{2}\frac{{\Gamma \left( {\frac{2M + 1}{3}} \right)}}{{t^{\frac{2M + 1}{3}} }} + \frac{2}{3\pi}\left| {d_{2M + 2} } \right|\frac{\sqrt 3}{2}\frac{{\Gamma \left( {\frac{2M + 3}{3}} \right)}}{{t^{\frac{2M + 3}{3}} }},$$ $$\label{eq26}
\left| {R_M^{\left( H \right)} \left( {it} \right)} \right| \le \frac{2}{3\pi}\left| {d_{2M + 2} } \right|\frac{\sqrt 3}{2}\frac{{\Gamma \left( {\frac{2M + 3}{3}} \right)}}{{t^{\frac{2M + 3}{3}} }},$$ $$\label{eq27}
\left| {R_M^{\left( H \right)} \left( {it} \right)} \right| \le \frac{2}{3\pi}\left| {d_{2M} } \right|\frac{\sqrt 3}{2}\frac{{\Gamma \left( {\frac{2M + 1}{3}} \right)}}{{t^{\frac{2M + 1}{3}} }}$$ according to whether $M \equiv 0 \mod 3$, $M \equiv 1 \mod 3$ or $M \equiv 2 \mod 3$, respectively. Substituting the expression into yields the expansion $$\begin{gathered}
\label{eq21}
\begin{split}
\left( {2n} \right)!a_n \left( 1 \right) = \; & \left( { - 1} \right)^n \frac{2\Gamma \left( {2n + \frac{2}{3}} \right)}{3\pi ^{2n + \frac{5}{3}} }\\ & \times \left( {\sum\limits_{m = 0}^{M - 1} {\left( { - 1} \right)^m \pi ^{\frac{{2m}}{3}} d_{2m} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\Gamma \left( {\frac{{2m + 1}}{3}} \right)\frac{{\Gamma \left( {\frac{{6n - 2m + 2}}{3}} \right)}}{{\Gamma \left( {2n + \frac{2}{3}} \right)}}} + A_M \left( n \right)} \right),
\end{split}\end{gathered}$$ for any fixed $0 \leq M \leq 3n - 1$, provided that $n \geq 1$. The remainder term $A_M\left( n \right)$ is given by the formula $$A_M \left( n \right) = \frac{{3\pi ^{2n + \frac{5}{3}} }}{{2\Gamma \left( {2n + \frac{2}{3}} \right)}}\int_0^{ + \infty } {t^{2n} e^{ - \pi t} iR_M^{\left( H \right)} \left( {it} \right)dt} .$$ Bounds for this error term follow from the estimates – since $$\left| {A_M \left( n \right)} \right| \le \pi ^{\frac{{2M}}{3}} \left| {d_{2M} } \right|\frac{{\sqrt 3 }}{2}\Gamma \left( {\frac{{2M + 1}}{3}} \right)\frac{{\Gamma \left( {\frac{{6n - 2M + 2}}{3}} \right)}}{{\Gamma \left( {2n + \frac{2}{3}} \right)}} + \pi ^{\frac{{2M + 2}}{3}} \left| {d_{2M + 2} } \right|\frac{{\sqrt 3 }}{2}\Gamma \left( {\frac{{2M + 3}}{3}} \right)\frac{{\Gamma \left( {\frac{{6n - 2M}}{3}} \right)}}{{\Gamma \left( {2n + \frac{2}{3}} \right)}},$$ $$\label{eq22}
\left| {A_M \left( n \right)} \right| \le \pi ^{\frac{{2M + 2}}{3}} \left| {d_{2M + 2} } \right|\frac{{\sqrt 3 }}{2}\Gamma \left( {\frac{{2M + 3}}{3}} \right)\frac{{\Gamma \left( {\frac{{6n - 2M}}{3}} \right)}}{{\Gamma \left( {2n + \frac{2}{3}} \right)}},$$ $$\label{eq23}
\left| {A_M \left( n \right)} \right| \le \pi ^{\frac{{2M}}{3}} \left| {d_{2M} } \right|\frac{{\sqrt 3 }}{2}\Gamma \left( {\frac{{2M + 1}}{3}} \right)\frac{{\Gamma \left( {\frac{{6n - 2M + 2}}{3}} \right)}}{{\Gamma \left( {2n + \frac{2}{3}} \right)}}$$ according to whether $M\equiv 0 \mod 3$, $M\equiv 1 \mod 3$ or $M\equiv 2 \mod 3$, respectively.
From the asymptotic behaviour of the coefficients $d_{2m}$ (see [@Nemes]), for large $n$, the least values of these bounds occur when $M \approx 2n$. With this choice of $M$, the error bounds are $\mathcal{O}\left( {n^{ - \frac{1}{2}} 9^{-n} } \right)$. This is the best accuracy we can achieve using the expansion . Numerical examples for various $n$ are provided in Table \[table2\].
\[c\][ l r @c@ l]{} &\
\[-1ex\] values of $n$ and $M$ & $n=5$, $M=10$ & &\
\[1ex\] exact numerical value of $a_n\left(1\right)$ & $-0.2039315629047481261022927689594356$ & $\times$ & $10^{-5}$\
\[1ex\] approximation to $a_n\left(1\right)$ & $-0.2039317236866484733447636037370858$ & $\times$ & $10^{-5}$\
\[1ex\] error & $0.1607819003472424708347776502$ & $\times$ & $10^{-11}$\
\[1ex\] error bound using & $0.5218454726884724646870658288$ & $\times$ & $10^{-11}$\
\[1ex\] &\
\[-1ex\] values of $n$ and $M$ & $n=10$, $M=20$ & &\
\[1ex\] exact numerical value of $a_n\left(1\right)$ & $0.1740499192613222665759959822566006$ & $\times$ & $10^{-10}$\
\[1ex\] approximation to $a_n\left(1\right)$ & $0.1740499192631695872689300620308834$ & $\times$ & $10^{-10}$\
\[1ex\] error & $-0.18473206929340797742828$ & $\times$ & $10^{-21}$\
\[1ex\] error bound using & $0.52455141471539645254342$ & $\times$ & $10^{-21}$\
\[1ex\] &\
\[-1ex\] values of $n$ and $M$ & $n=25$, $M=50$ & &\
\[1ex\] exact numerical value of $a_n\left(1\right)$ & $-0.1567780710784896492198553870128892$ & $\times$ & $10^{-25}$\
\[1ex\] approximation to $a_n\left(1\right)$ & $-0.1567780710784896492198553919627602$ & $\times$ & $10^{-25}$\
\[1ex\] error & $0.49498710$ & $\times$ & $10^{-51}$\
\[1ex\] error bound using & $0.145150293$ & $\times$ & $10^{-50}$\
\[-1ex\] &\
Case (iii): $0<\lambda<1$
-------------------------
For this case, we take $\lambda = \mathop{\text{sech}} \alpha$ with a suitable $\alpha>0$. It is known that $$\label{eq28}
iH_{it}^{\left( 1 \right)} \left( {it \mathop{\text{sech}} \alpha } \right) = 2 \Re \left( {\frac{{e^{it\left( {\alpha - \tanh \alpha } \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\pi t\tanh \alpha } \right)^{\frac{1}{2}} }}\left( {\sum\limits_{m = 0}^{M - 1} {\frac{{i^m U_m \left( {\coth \alpha } \right)}}{{t^m }}} + R_M^{\left( H \right)} \left( {it,\alpha } \right)} \right)} \right),$$ where $R_M^{\left( H \right)} \left( {it,\alpha } \right) = \mathcal{O}_{M,\alpha } \left( {t^{ - M} } \right)$ as $t \to +\infty$. Here $U_m \left( {\coth \alpha } \right) = \left[U_m \left( x \right)\right]_{x = \coth \alpha }$ with $U_m\left(x\right)$ being a polynomial in $x$ of degree $3m$. As far as we know, there is no simple closed expression nor a realistic estimate for the remainder term $R_M^{\left( H \right)} \left( {it,\alpha } \right)$. Nevertheless, we assume that $$\int_0^{ + \infty } {t^{2n - \frac{1}{2}} e^{ - \pi t} \left| {R_M^{\left( H \right)} \left( {it,\alpha } \right)} \right|dt} < +\infty$$ and substitute the expansion into to obtain $$\begin{gathered}
\label{eq29}
\begin{split}
\left( {2n} \right)!a_n \left( {\mathop{\text{sech}} \alpha } \right) = \;& \Re \left( {\left( {\frac{{2\coth \alpha }}{{\pi \left( {\alpha - \tanh \alpha + \pi i} \right)}}} \right)^{\frac{1}{2}} \frac{2 \Gamma \left( {2n + \frac{1}{2}} \right)}{{\left( {\alpha - \tanh \alpha + \pi i} \right)^{2n} }} }\right. \\ & \times \left.{\left( {\sum\limits_{m = 0}^{M - 1} {\left( {\alpha - \tanh \alpha + \pi i} \right)^m U_m \left( {\coth \alpha } \right)\frac{\Gamma \left( {2n - m + \frac{1}{2}} \right)}{\Gamma \left( {2n + \frac{1}{2}} \right)}} + A_M \left( {n,\alpha } \right)} \right)} \right),
\end{split}\end{gathered}$$ for any fixed $0 \le M \le 2n$, provided that $n\geq 1$. The remainder term $A_M \left( {n,\alpha } \right)$ is given by the integral formula $$A_M \left( {n,\alpha } \right) = \left( { - 1} \right)^n e^{ - \frac{\pi }{4}i} \frac{{\left( {\alpha - \tanh \alpha + \pi i} \right)^{2n + \frac{1}{2}} }}{{\Gamma \left( {2n + \frac{1}{2}} \right)}}\int_0^{ + \infty } {t^{2n - \frac{1}{2}} e^{i\left( {\alpha - \tanh \alpha + \pi i} \right)t} R_M^{\left( H \right)} \left( {it,\alpha } \right)dt} .$$ To achieve the best accuracy using the expansion , we need to determine the index of the least term of the expansion. This can be done if we know the large $m$ behaviour of the coefficients $U_m \left(\coth \alpha \right)$. Such an asymptotic formula was derived by Dingle [@Dingle p. 168], using formal, non-rigorous methods. At leading order, his formula can be written as $$\left| {U_m \left( {\coth \alpha } \right)} \right| \sim \frac{{\Gamma \left( m \right)}}{{2\pi \left( {2\left( {\alpha - \tanh \alpha } \right)} \right)^m }}.$$ Numerical calculations indicate that this approximation is correct, and assuming so, the optimal truncation occurs at $$M \approx \frac{{\alpha - \tanh \alpha }}{{2\left( {\alpha - \tanh \alpha } \right) + \sqrt {\left( {\alpha - \tanh \alpha } \right)^2 + \pi ^2 } }}\left( {4n + 1} \right).$$ Therefore, the smaller $\alpha$ is the larger $n$ has to be to get a reasonable approximation from .
Numerical examples illustrating the efficacy of the formula , truncated optimally, are given in Table \[table3\].
\[c\][ l r @c@ l]{} &\
\[-1ex\] values of $\alpha$ and $M$ & $\alpha=\frac{1}{2}$, $M=3$ & &\
\[1ex\] exact numerical value of $a_{50}\left(\mathop{\text{sech}} \alpha\right)$ & $0.2315627683882018769175712540082165$ & $\times$ & $10^{-50}$\
\[1ex\] approximation to $a_{50}\left(\mathop{\text{sech}} \alpha\right)$ & $0.2303064844873166640986637287015961$ & $\times$ & $10^{-50}$\
\[1ex\] error & $0.12562839008852128189075253066203$ & $\times$ & $10^{-52}$\
\[1ex\] &\
\[-1ex\] values of $\alpha$ and $M$ & $\alpha=1$, $M=14$ & &\
\[1ex\] exact numerical value of $a_{50}\left(\mathop{\text{sech}} \alpha\right)$ & $0.1279482878426982457824386451759845$ & $\times$ & $10^{-50}$\
\[1ex\] approximation to $a_{50}\left(\mathop{\text{sech}} \alpha\right)$ & $0.1279482903067682761677730364825915$ & $\times$ & $10^{-50}$\
\[1ex\] error & $-0.24640700303853343913066070$ & $\times$ & $10^{-58}$\
\[1ex\] &\
\[-1ex\] values of $\alpha$ and $M$ & $\alpha=5$, $M=62$ & &\
\[1ex\] exact numerical value of $a_{50}\left(\mathop{\text{sech}} \alpha\right)$ & $-0.9536145099812834565097014294181624$ & $\times$ & $10^{-72}$\
\[1ex\] approximation to $a_{50}\left(\mathop{\text{sech}} \alpha\right)$ & $-0.9536145099812834565097014294180344$ & $\times$ & $10^{-72}$\
\[1ex\] error & $0.1280$ & $\times$ & $10^{-102}$\
\[-1ex\] &\
Exponentially improved asymptotic expansions {#section5}
============================================
We shall find it convenient to express our exponentially improved expansions in terms of the (scaled) Terminant function, which is defined by $$\widehat T_p \left( w \right) = \frac{{e^{\pi ip} w^{1 - p} e^{ - w} }}{{2\pi i}}\int_0^{ + \infty } {\frac{{t^{p - 1} e^{ - t} }}{w + t}dt} \; \text{ for } \; p>0 \; \text{ and } \; \left| \arg w \right| < \pi ,$$ and by analytic continuation elsewhere. Olver [@Olver4] showed that when $p \sim \left|w\right|$ and $w \to \infty$, we have $$\label{eq36}
ie^{ - \pi ip} \widehat T_p \left( w \right) = \begin{cases} \mathcal{O}\left( {e^{ - w - \left| w \right|} } \right) & \; \text{ if } \; \left| {\arg w} \right| \le \pi \\ \mathcal{O}\left(1\right) & \; \text{ if } \; - 3\pi < \arg w \le - \pi. \end{cases}$$ Concerning the smooth transition of the Stokes discontinuities, we will use the more precise asymptotic formulas $$\label{eq37}
\widehat T_p \left( w \right) = \frac{1}{2} + \frac{1}{2}\mathop{\text{erf}} \left( {c\left( \varphi \right)\sqrt {\frac{1}{2}\left| w \right|} } \right) + \mathcal{O}\left( {\frac{{e^{ - \frac{1}{2}\left| w \right|c^2 \left( \varphi \right)} }}{{\left| w \right|^{\frac{1}{2}} }}} \right)$$ for $-\pi +\delta \leq \arg w \leq 3 \pi -\delta$, $0 < \delta \le 2\pi$; and $$\label{eq38}
e^{ - 2\pi ip} \widehat T_p \left( w \right) = - \frac{1}{2} + \frac{1}{2}\mathop{\text{erf}} \left( { - \overline {c\left( { - \varphi } \right)} \sqrt {\frac{1}{2}\left| w \right|} } \right) + \mathcal{O}\left( {\frac{{e^{ - \frac{1}{2}\left| w \right|\overline {c^2 \left( { - \varphi } \right)} } }}{{\left| w \right|^{\frac{1}{2}} }}} \right)$$ for $- 3\pi + \delta \le \arg w \le \pi - \delta$, $0 < \delta \le 2\pi$. Here $\varphi = \arg w$ and erf denotes the Error function. The quantity $c\left( \varphi \right)$ is defined implicitly by the equation $$\frac{1}{2}c^2 \left( \varphi \right) = 1 + i\left( {\varphi - \pi } \right) - e^{i\left( {\varphi - \pi } \right)},$$ and corresponds to the branch of $c\left( \varphi \right)$ which has the following expansion in the neighbourhood of $\varphi = \pi$: $$\label{eq39}
c\left( \varphi \right) = \left( {\varphi - \pi } \right) + \frac{i}{6}\left( {\varphi - \pi } \right)^2 - \frac{1}{{36}}\left( {\varphi - \pi } \right)^3 - \frac{i}{{270}}\left( {\varphi - \pi } \right)^4 + \cdots .$$ For complete asymptotic expansions, see Olver [@Olver5]. We remark that Olver uses the different notation $F_p \left( w \right) = ie^{ - \pi ip} \widehat T_p \left( w \right)$ for the Terminant function and the other branch of the function $c\left( \varphi \right)$. For further properties of the Terminant function, see, for example, Paris and Kaminski [@Paris3 Chapter 6].
Proof of the exponentially improved expansions for $\mathbf{A}_{\nu}\left(\lambda \nu \right)$
----------------------------------------------------------------------------------------------
### Case (i): $\lambda>1$
The proof goes exactly the same way as the proof of Theorem 3 in the first paper of this series [@Nemes2]. One has to replace $R_N \left( {\nu ,\beta } \right)$, $R_{N,M} \left( {\nu ,\beta } \right)$ and $\tan \beta - \beta$ by $R_N \left( {\nu ,\sec \beta } \right)$, $R_{N,M} \left( {\nu ,\sec \beta } \right)$ and $\tan \beta - \beta + \pi$ in the corresponding formulas.
### Case (ii): $\lambda=1$
First, we suppose that $\left|\arg \nu\right| < \frac{\pi}{2}$. Our starting point is the representation , written in the form $$\label{eq40}
R_N \left( {\nu ,1 } \right) = \frac{{\left( { - 1} \right)^N }}{{2\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} e^{ - \pi t} }}{{1 - it/\nu }}iH_{it}^{\left( 1 \right)} \left( {it } \right)dt} + \frac{{\left( { - 1} \right)^N }}{{2\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} e^{ - \pi t} }}{{1 + it/\nu }}iH_{it}^{\left( 1 \right)} \left( {it } \right)dt} .$$ Let $0 \leq M <3N$ be a fixed integer such that $M \equiv 0 \mod 3$. We use to expand the function $H_{it}^{\left( 1 \right)} \left( {it } \right)$ under the integrals in , to obtain $$\begin{gathered}
\label{eq41}
\begin{split}
R_N \left( {\nu ,1} \right) = \; & \frac{2}{{3\pi }}\sum\limits_{m = 0}^{M - 1} {d_{2m} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}\frac{{\left( { - 1} \right)^{N + m} \nu ^{\frac{{2m - 2}}{3} - 2N} }}{{2\pi }}\int_0^{ + \infty } {\frac{{t^{2N - \frac{{2m - 2}}{3} - 1} e^{ - \pi t} }}{{1 - it/\nu }}dt} }\\
& + \frac{2}{{3\pi }}\sum\limits_{m = 0}^{M - 1} {d_{2m} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}\frac{{\left( { - 1} \right)^{N + m } \nu ^{\frac{{2m - 2}}{3} - 2N} }}{{2\pi }}\int_0^{ + \infty } {\frac{{t^{2N - \frac{{2m - 2}}{3} - 1} e^{ - \pi t} }}{{1 + it/\nu }}dt} } \\
& + R_{N,M} \left( {\nu ,1} \right),
\end{split}\end{gathered}$$ with $$\label{eq42}
R_{N,M} \left( {\nu ,1} \right) = \frac{{\left( { - 1} \right)^N }}{{2\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} e^{ - \pi t} }}{{1 - it/\nu }}iR_M^{\left( H \right)} \left( {it} \right)dt} + \frac{{\left( { - 1} \right)^N }}{{2\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} e^{ - \pi t} }}{{1 + it/\nu }}iR_M^{\left( H \right)} \left( {it} \right)dt} .$$ The integrals in can be identified in terms of the Terminant function since $$\frac{{\left( { - 1} \right)^{N + m} \nu ^{\frac{{2m - 2}}{3} - 2N} }}{{2\pi }}\int_0^{ + \infty } {\frac{{t^{2N - \frac{{2m - 2}}{3} - 1} e^{ - \pi t} }}{{1 - it/\nu }}dt} = -ie^{\pi i\nu } e^{\frac{{2\left( {2m + 1} \right)\pi i}}{3}} \widehat T_{2N - \frac{{2m - 2}}{3}} \left( {\pi i\nu } \right)$$ and $$\frac{{\left( { - 1} \right)^{N + m} \nu ^{\frac{{2m - 2}}{3} - 2N} }}{{2\pi }}\int_0^{ + \infty } {\frac{{t^{2N - \frac{{2m - 2}}{3} - 1} e^{ - \pi t} }}{{1 + it/\nu }}dt} = -ie^{ - \pi i\nu } \widehat T_{2N - \frac{{2m - 2}}{3}} \left( { - \pi i\nu } \right).$$ Hence, we have the following expansion $$\begin{gathered}
R_N \left( {\nu ,1} \right) = -ie^{\pi i\nu } \frac{2}{{3\pi }}\sum\limits_{m = 0}^{M - 1} {d_{2m} e^{\frac{{2\left( {2m + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}\widehat T_{2N - \frac{{2m - 2}}{3}} \left( {\pi i\nu } \right)}
\\ - ie^{ - \pi i\nu } \frac{2}{{3\pi }}\sum\limits_{m = 0}^{M - 1} {d_{2m} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}\widehat T_{2N - \frac{{2m - 2}}{3}} \left( { - \pi i\nu } \right)} + R_{N,M} \left( {\nu ,1} \right).\end{gathered}$$ Taking $\nu = r e^{i\theta}$, the representation takes the form $$\label{eq43}
R_{N,M} \left( {\nu ,1} \right) = \frac{{\left( { - 1} \right)^N }}{{2\pi \left( {e^{i\theta } } \right)^{2N + 1} }}\int_0^{ + \infty } {\frac{{\tau ^{2N} e^{ - \pi r\tau } }}{{1 - i\tau e^{ - i\theta } }}iR_M^{\left( H \right)} \left( {ir\tau } \right)d\tau } + \frac{{\left( { - 1} \right)^N }}{{2\pi \left( {e^{i\theta } } \right)^{2N + 1} }}\int_0^{ + \infty } {\frac{{\tau ^{2N} e^{ - \pi r\tau } }}{{1 + i\tau e^{ - i\theta } }}iR_M^{\left( H \right)} \left( {ir\tau } \right)d\tau } .$$ In [@Nemes Appendix B] it was shown that $$\frac{{1 - \left( {s/r\tau } \right)^{\frac{4}{3}} }}{{1 - \left( {s/r\tau } \right)^2 }} = \frac{{1 - \left( {s/r} \right)^{\frac{4}{3}} }}{{1 - \left( {s/r} \right)^2 }} + \left( {\tau - 1} \right)f\left( {r,\tau ,s} \right)$$ for positive $r$, $\tau$ and $s$, with some $f\left(r,\tau ,s\right)$ satisfying $\left|f\left(r,\tau ,s\right)\right| \leq 2$. Using the integral formula , $R_M^{\left( H \right)} \left( {ir\tau } \right)$ can be written as $$\begin{aligned}
R_M^{\left( H \right)} \left( {ir\tau } \right) = \; & \frac{1}{{\sqrt 3 \pi \left( {r\tau } \right)^{\frac{{2M + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{2M - 2}{3}} e^{ - 2\pi s} \frac{{1 - \left( {s/r\tau } \right)^{\frac{4}{3}} }}{{1 - \left( {s/r\tau } \right)^2 }}H_{is}^{\left( 1 \right)} \left( {is} \right)ds} \\
= \; & \frac{1}{{\sqrt 3 \pi \left( {r\tau } \right)^{\frac{{2M + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{{2M - 2}}{3}} e^{ - 2\pi s} \frac{{1 - \left( {s/r} \right)^{\frac{4}{3}} }}{{1 - \left( {s/r} \right)^2 }}H_{is}^{\left( 1 \right)} \left( {is} \right)ds} \\
& + \frac{{\tau - 1}}{{\sqrt 3 \pi \left( {r\tau } \right)^{\frac{{2M + 1}}{3}} }}\int_0^{ + \infty } {s^{\frac{{2M - 2}}{3}} e^{ - 2\pi s} f\left( {r,\tau ,s} \right)H_{is}^{\left( 1 \right)} \left( {is} \right)ds} .\end{aligned}$$ Noting that $$0< \frac{{1 - \left( {s/r} \right)^{\frac{4}{3}} }}{{1 - \left( {s/r} \right)^2 }} < 1$$ for any positive $r$ and $s$, substitution into yields the upper bound $$\begin{aligned}
\left| {R_{N,M} \left( {\nu ,1} \right)} \right| \le \; &\frac{{\left| {d_{2M} } \right|\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\sqrt 3 \pi \left| \nu \right|^{\frac{{2M + 1}}{3}} }}\left| {\frac{1}{{2\pi }}\int_0^{ + \infty } {\frac{{\tau ^{2N - \frac{{2M - 2}}{3} - 1} e^{ - \pi r\tau } }}{{1 - i\tau e^{ - i\theta } }}d\tau } } \right| \\ &+ \frac{{\left| {d_{2M} } \right|\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\sqrt 3 \pi ^2 \left| \nu \right|^{\frac{{2M + 1}}{3}} }}\int_0^{ + \infty } {\tau ^{2N - \frac{{2M - 2}}{3} - 1} e^{ - \pi r\tau } \left| {\frac{{\tau - 1}}{{\tau + ie^{i\theta } }}} \right|d\tau } \\
& + \frac{{\left| {d_{2M} } \right|\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\sqrt 3 \pi \left| \nu \right|^{\frac{{2M + 1}}{3}} }}\left| {\frac{1}{{2\pi }}\int_0^{ + \infty } {\frac{{\tau ^{2N - \frac{{2M - 2}}{3} - 1} e^{ - \pi r\tau } }}{{1 + i\tau e^{ - i\theta } }}d\tau } } \right| \\ & + \frac{{\left| {d_{2M} } \right|\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\sqrt 3 \pi ^2 \left| \nu \right|^{\frac{{2M + 1}}{3}} }}\int_0^{ + \infty } {\tau ^{2N - \frac{{2M - 2}}{3} - 1} e^{ - \pi r\tau } \left| {\frac{{\tau - 1}}{{\tau - ie^{i\theta } }}} \right|d\tau } .\end{aligned}$$ As $\left| \left(\tau - 1\right)/\left(\tau \pm i e^{i\theta}\right) \right| \le 1$, we find that $$\begin{aligned}
\left| {R_{N,M} \left( {\nu ,1} \right)} \right| \le \frac{{\left| {d_{2M} } \right|\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\sqrt 3 \pi \left| \nu \right|^{\frac{{2M + 1}}{3}} }}\left| {e^{\pi i\nu } \widehat T_{2N - \frac{{2M - 2}}{3}} \left( {\pi i\nu } \right)} \right| & + \frac{{\left| {d_{2M} } \right|\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\sqrt 3 \pi \left| \nu \right|^{\frac{{2M + 1}}{3}} }}\left| {e^{ - \pi i\nu } \widehat T_{2N - \frac{{2M - 2}}{3}} \left( { - \pi i\nu } \right)} \right|\\
& + \frac{{2\left| {d_{2M} } \right|\Gamma \left( {\frac{{2M + 1}}{3}} \right)\Gamma \left( {2N - \frac{{2M - 2}}{3}} \right)}}{{\sqrt 3 \pi ^2 \pi ^{2N - \frac{{2M - 2}}{3}} \left| \nu \right|^{2N + 1} }}.\end{aligned}$$ By continuity, this bound holds in the closed sector $\left|\arg \nu\right| \le \frac{\pi}{2}$. Assume that $N = \frac{1}{2}\pi \left|\nu\right|+\rho$ where $\rho$ is bounded. Employing Stirling’s formula, we find that $$\frac{{2\left| {d_{2M} } \right|\Gamma \left( {\frac{{2M + 1}}{3}} \right)\Gamma \left( {2N - \frac{{2M - 2}}{3}} \right)}}{{\sqrt 3 \pi ^2 \pi ^{2N - \frac{{2M - 2}}{3}} \left| \nu \right|^{2N + 1} }} = \mathcal{O}_{M,\rho } \left( {\frac{{e^{ - \pi \left| \nu \right|} }}{{\left| \nu \right|^{\frac{1}{2}} }}\left| {d_{2M} } \right|\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 1}}{3}} }}} \right)$$ as $\nu \to \infty$. Olver’s estimation shows that $$\left| {e^{ \pm \pi i\nu } \widehat T_{2N - \frac{{2M - 2}}{3}} \left( { \pm \pi i\nu } \right)} \right| = \mathcal{O}_{M,\rho } \left( {e^{ - \pi \left| \nu \right|} } \right)$$ for large $\nu$. Therefore, we have that $$\label{eq44}
R_{N,M} \left( {\nu ,1} \right) = \mathcal{O}_{M,\rho } \left( {e^{ - \pi \left| \nu \right|} \left| {d_{2M} } \right|\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 1}}{3}} }}} \right)$$ as $\nu \to \infty$ in the sector $\left|\arg \nu\right| \le \frac{\pi}{2}$.
Rotating the path of integration in and applying the residue theorem yields $$\begin{aligned}
R_{N,M} \left( {\nu ,1} \right) & = ie^{\pi i\nu } R_M^{\left( H \right)} \left( \nu \right) + \frac{{\left( { - 1} \right)^N }}{{2\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} e^{ - \pi t} }}{{1 - it/\nu }}iR_M^{\left( H \right)} \left( {it} \right)dt} + \frac{{\left( { - 1} \right)^N }}{{2\pi \nu ^{2N + 1} }}\int_0^{ + \infty } {\frac{{t^{2N} e^{ - \pi t} }}{{1 + it/\nu }}iR_M^{\left( H \right)} \left( {it} \right)dt}\\
& = ie^{\pi i\nu } R_M^{\left( H \right)} \left( \nu \right) - R_{N,M} \left( {\nu e^{ - \pi i} ,1} \right),\end{aligned}$$ when $\frac{\pi}{2} < \arg \nu < \frac{3\pi}{2}$. It follows that $$\left| {R_{N,M} \left( {\nu ,1} \right)} \right| \le e^{ - \pi \Im \left( \nu \right)} \left| {R_M^{\left( H \right)} \left( \nu \right)} \right| + \left| {R_{N,M} \left( {\nu e^{ - \pi i} ,1} \right)} \right|$$ in the closed sector $\frac{\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$, using continuity. It was proved in [@Nemes] that $R_M^{\left( H \right)} \left( \nu \right) = \mathcal{O}_M \left( {\left| {d_{2M} } \right|\Gamma \left( {\frac{{2M + 1}}{3}} \right)\left| \nu \right|^{ - \frac{{2M + 1}}{3}} } \right)$ as $\nu \to \infty$ in the closed sector $-\frac{\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$, whence, by , we deduce that $$\begin{gathered}
\label{eq58}
\begin{split}
R_{N,M} \left( {\nu ,1} \right) & = \mathcal{O}_M \left( {e^{ - \pi \Im \left( \nu \right)} \left| {d_{2M} } \right|\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 1}}{3}} }}} \right) + \mathcal{O}_{M,\rho } \left( {e^{ - \pi \left| \nu \right|} \left| {d_{2M} } \right|\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 1}}{3}} }}} \right)
\\& = \mathcal{O}_{M,\rho } \left( {e^{ - \pi \Im \left( \nu \right)} \left| {d_{2M} } \right|\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 1}}{3}} }}} \right)
\end{split}\end{gathered}$$ as $\nu \to \infty$ in the sector $\frac{\pi}{2} \leq \arg \nu \leq \frac{3\pi}{2}$.
The reflection principle gives the relation $$R_{N,M} \left( {\nu ,1} \right) = \overline {R_{N,M} \left( {\bar \nu ,1} \right)} = - ie^{ - \pi i\nu } \overline {R_M^{\left( H \right)} \left( {\bar \nu } \right)} - R_{N,M} \left( {\nu e^{\pi i} ,1} \right) = - ie^{ - \pi i\nu } R_M^{\left( H \right)} \left( {\nu e^{\pi i} } \right) - R_{N,M} \left( {\nu e^{\pi i} ,1} \right),$$ valid when $-\frac{3\pi}{2} < \arg \nu < -\frac{\pi}{2}$. Trivial estimation and a continuity argument show that $$\left| {R_{N,M} \left( {\nu ,1} \right)} \right| \le e^{\pi \Im \left( \nu \right)} \left| {R_M^{\left( H \right)} \left( {\nu e^{\pi i} } \right)} \right| + \left| {R_{N,M} \left( {\nu e^{\pi i} ,1} \right)} \right|$$ in the closed sector $-\frac{3\pi}{2} \leq \arg \nu \leq -\frac{\pi}{2}$. Since $R_M^{\left( H \right)} \left( \nu e^{\pi i} \right) = \mathcal{O}_M \left( {\left| {d_{2M} } \right|\Gamma \left( {\frac{{2M + 1}}{3}} \right)\left| \nu \right|^{ - \frac{{2M + 1}}{3}} } \right)$ as $\nu \to \infty$ in the range $-\frac{3\pi}{2} \leq \arg \nu \leq -\frac{\pi}{2}$, by , we find that $$\begin{gathered}
\label{eq59}
\begin{split}
R_{N,M} \left( {\nu ,1} \right) & = \mathcal{O}_M \left( {e^{\pi \Im \left( \nu \right)} \left| {d_{2M} } \right|\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 1}}{3}} }}} \right) + \mathcal{O}_{M,\rho } \left( {e^{ - \pi \left| \nu \right|} \left| {d_{2M} } \right|\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 1}}{3}} }}} \right)
\\& = \mathcal{O}_{M,\rho } \left( {e^{\pi \Im \left( \nu \right)} \left| {d_{2M} } \right|\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\left| \nu \right|^{\frac{{2M + 1}}{3}} }}} \right)
\end{split}\end{gathered}$$ as $\nu \to \infty$ with $-\frac{3\pi}{2} \leq \arg \nu \leq -\frac{\pi}{2}$.
If $M \equiv 1 \mod 3$ or $M \equiv 2 \mod 3$, we write the remainder $R_{N,M} \left( {\nu ,1} \right)$ in the form $$\begin{aligned}
R_{N,M} \left( {\nu ,1} \right) = & - ie^{\pi i\nu } \frac{2}{{3\pi }}d_{2M + 2} e^{\frac{\pi }{3}i} \frac{{\sqrt 3 }}{2}\frac{{\Gamma \left( {\frac{{2M + 3}}{3}} \right)}}{{\nu ^{\frac{{2M + 3}}{3}} }}\widehat T_{2N - \frac{{2M}}{3}} \left( {\pi i\nu } \right)\\
& + ie^{ - \pi i\nu } \frac{2}{{3\pi }}d_{2M + 2} \frac{{\sqrt 3 }}{2}\frac{{\Gamma \left( {\frac{{2M + 3}}{3}} \right)}}{{\nu ^{\frac{{2M + 3}}{3}} }}\widehat T_{2N - \frac{{2M}}{3}} \left( { - \pi i\nu } \right) + R_{N,M + 2} \left( {\nu ,1} \right)\end{aligned}$$ and $$\begin{aligned}
R_{N,M} \left( {\nu ,1} \right) = & - ie^{\pi i\nu } \frac{2}{{3\pi }}d_{2M} e^{\frac{\pi }{3}i} \frac{{\sqrt 3 }}{2}\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\nu ^{\frac{{2M + 1}}{3}} }}\widehat T_{2N - \frac{{2M - 2}}{3}} \left( {\pi i\nu } \right)\\
& + ie^{ - \pi i\nu } \frac{2}{{3\pi }}d_{2M} \frac{{\sqrt 3 }}{2}\frac{{\Gamma \left( {\frac{{2M + 1}}{3}} \right)}}{{\nu ^{\frac{{2M + 1}}{3}} }}\widehat T_{2N - \frac{{2M - 2}}{3}} \left( { - \pi i\nu } \right) + R_{N,M + 1} \left( {\nu ,1} \right),\end{aligned}$$ respectively. Applying the connection formula $\widehat T_p \left( w \right) = e^{2\pi ip} \left( {\widehat T_p \left( {we^{2\pi i} } \right) - 1} \right)$ together with Olver’s result and the bounds , , we have established, the estimates for the cases $M \equiv 1 \mod 3$ and $M \equiv 2 \mod 3$ follow.
Stokes phenomenon and Berry’s transition
----------------------------------------
### Case (i): $\lambda>1$
As usual, let $\lambda = \sec \beta$ with some $0 <\beta <\frac{\pi}{2}$. We study the Stokes phenomenon related to the asymptotic expansion of $\mathbf{A}_{\nu} \left(\nu \sec \beta\right)$ occurring when $\arg \nu$ passes through the values $\pm \frac{\pi}{2}$. In the range $\left|\arg \nu\right|<\frac{\pi}{2}$, the asymptotic expansion $$\label{eq47}
\mathbf{A}_{\nu} \left( {\nu \sec \beta } \right) \sim \frac{1}{\pi }\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!a_n \left(\sec \beta\right)}}{{\nu ^{2n + 1} }}}$$ holds as $\nu \to \infty$. From we have $$\mathbf{A}_\nu \left( {\nu \sec \beta } \right) = i e^{\pi i\nu } H_\nu ^{\left( 1 \right)} \left( {\nu \sec \beta } \right) - \mathbf{A}_{-\nu} \left( {\nu e^{ - \pi i} \sec \beta } \right)$$ when $\frac{\pi}{2} < \arg \nu < \frac{3\pi}{2}$, and $$\mathbf{A}_\nu \left( {\nu \sec \beta } \right) = - ie^{ - \pi i\nu } H_\nu ^{\left( 2 \right)} \left( {\nu \sec \beta } \right) - \mathbf{A}_{-\nu} \left( {\nu e^{\pi i} \sec \beta } \right)$$ for $-\frac{3\pi}{2} < \arg \nu < -\frac{\pi}{2}$. For the right-hand sides, we can apply the asymptotic expansions of the Hankel functions and the Anger–Weber function to deduce that $$\label{eq54}
\mathbf{A}_{\nu } \left( {\nu \sec \beta } \right) \sim i\frac{{e^{i\nu \left( {\tan \beta - \beta +\pi} \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^\infty {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}} + \frac{1}{\pi}\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!a_n \left( {\sec \beta } \right)}}{{\nu ^{2n + 1} }}}$$ as $\nu \to \infty$ in the sector $\frac{\pi}{2} < \arg \nu < \frac{3\pi}{2}$, and $$\label{eq55}
\mathbf{A}_{\nu } \left( {\nu \sec \beta } \right) \sim - i\frac{{e^{ - i\nu \left( {\tan \beta - \beta +\pi} \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^\infty {\frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}} + \frac{1}{\pi}\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!a_n \left( {\sec \beta } \right)}}{{\nu ^{2n + 1} }}}$$ as $\nu \to \infty$ in the sector $-\frac{3\pi}{2} < \arg \nu < -\frac{\pi}{2}$. Therefore, as the line $\arg \nu = \frac{\pi}{2}$ is crossed, the additional series $$\label{eq48}
i\frac{{e^{i\nu \left( {\tan \beta - \beta +\pi} \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^\infty {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{\nu ^m}}$$ appears in the asymptotic expansion of $\mathbf{A}_{\nu} \left( {\nu \sec \beta } \right)$ beside the original one . Similarly, as we pass through the line $\arg \nu = -\frac{\pi}{2}$, the series $$\label{eq49}
- i\frac{{e^{ - i\nu \left( {\tan \beta - \beta +\pi} \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0}^\infty {\frac{{U_m \left( {i\cot \beta } \right)}}{\nu ^m }}$$ appears in the asymptotic expansion of $\mathbf{A}_{\nu} \left( {\nu \sec \beta } \right)$ beside the original series . We have encountered a Stokes phenomenon with Stokes lines $\arg \nu = \pm\frac{\pi}{2}$.
In his important paper [@Berry2], Berry gave a new interpretation of the Stokes phenomenon; he proved that assuming optimal truncation, the transition between compound asymptotic expansions is of Error function type, thus yielding a smooth and rapid transition as a Stokes line is crossed.
Using the exponentially improved expansion given in Theorem \[thm2\], we show that the asymptotic expansion of $\mathbf{A}_{\nu} \left( {\nu \sec \beta } \right)$ exhibits the Berry transition between the two asymptotic series across the Stokes lines $\arg \nu = \pm\frac{\pi}{2}$. More precisely, we shall find that the first few terms of the series in and “emerge" in a rapid and smooth way as $\arg \nu$ passes through $\frac{\pi}{2}$ and $-\frac{\pi}{2}$, respectively.
From Theorem \[thm2\], we conclude that if $N \approx \frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta +\pi} \right)$, then for large $\nu$, $ \left|\arg \nu\right| < \pi$, we have $$\begin{aligned}
\mathbf{A}_{\nu} \left( {\nu \sec \beta } \right) \approx \; & \frac{1}{\pi }\sum\limits_{n = 0}^{N - 1} {\frac{{\left( {2n} \right)!a_n \left( {\sec \beta } \right)}}{\nu ^{2n + 1}}} \\ & + i\frac{{e^{i\nu \left( {\tan \beta - \beta +\pi} \right) - \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0} {\left( { - 1} \right)^m \frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\widehat T_{2N - m + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta +\pi} \right)} \right)}
\\ & - i\frac{{e^{ - i\nu \left( {\tan \beta - \beta +\pi} \right) + \frac{\pi }{4}i} }}{{\left( {\frac{1}{2}\nu \pi \tan \beta } \right)^{\frac{1}{2}} }}\sum\limits_{m = 0} {\frac{{U_m \left( {i\cot \beta } \right)}}{{\nu ^m }}\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta +\pi} \right)} \right)} ,\end{aligned}$$ where $\sum\nolimits_{m = 0}$ means that the sum is restricted to the leading terms of the series.
In the upper half-plane, the terms involving $\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta +\pi} \right)} \right)$ are exponentially small, the dominant contribution comes from the terms involving $\widehat T_{2N - m + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta +\pi} \right)} \right)$. Under the above assumption on $N$, from and , the Terminant functions have the asymptotic behaviour $$\widehat T_{2N - m + \frac{1}{2}} \left( {i\nu \left( {\tan \beta - \beta +\pi} \right)} \right) \sim \frac{1}{2} + \frac{1}{2} \mathop{\text{erf}} \left( {\left( {\theta - \frac{\pi }{2}} \right)\sqrt {\frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta +\pi} \right)} } \right)$$ provided that $\arg \nu = \theta$ is close to $\frac{\pi}{2}$, $\nu$ is large and $m$ is small in comparison with $N$. Therefore, when $\theta < \frac{\pi}{2}$, the Terminant functions are exponentially small; for $\theta = \frac{\pi }{2}$, they are asymptotically $\frac{1}{2}$ up to an exponentially small error; and when $\theta > \frac{\pi}{2}$, the Terminant functions are asymptotic to $1$ with an exponentially small error. Thus, the transition across the Stokes line $\arg \nu = \frac{\pi}{2}$ is effected rapidly and smoothly. Similarly, in the lower half-plane, the dominant contribution is controlled by the terms involving $\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta +\pi} \right)} \right)$. From and , we have $$\widehat T_{2N - m + \frac{1}{2}} \left( { - i\nu \left( {\tan \beta - \beta +\pi} \right)} \right) \sim \frac{1}{2} - \frac{1}{2} \mathop{\text{erf}} \left( {\left( {\theta + \frac{\pi }{2}} \right)\sqrt {\frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta +\pi} \right)} } \right)$$ under the assumptions that $\arg \nu = \theta$ is close to $-\frac{\pi}{2}$, $\nu$ is large and $m$ is small in comparison with $N \approx \frac{1}{2}\left| \nu \right|\left( {\tan \beta - \beta +\pi} \right)$. Thus, when $\theta > - \frac{\pi}{2}$, the Terminant functions are exponentially small; for $\theta = -\frac{\pi}{2}$, they are asymptotic to $\frac{1}{2}$ with an exponentially small error; and when $\theta < - \frac{\pi}{2}$, the Terminant functions are asymptotically $1$ up to an exponentially small error. Therefore, the transition through the Stokes line $\arg \nu = -\frac{\pi}{2}$ is carried out rapidly and smoothly.
We remark that from the expansions and , it follows that is an asymptotic expansion of $\mathbf{A}_{\nu} \left( {\nu \sec \beta } \right)$ in the wider sector $\left|\arg \nu\right| \leq \pi -\delta < \pi$, with any fixed $0 < \delta \leq \pi$.
### Case (ii): $\lambda=1$
The analysis of the Stokes phenomenon for the asymptotic expansion of $\mathbf{A}_{\nu} \left( {\nu} \right)$ is similar to the case $\lambda > 1$. In the range $\left|\arg \nu\right| <\frac{\pi}{2}$, the asymptotic expansion $$\label{eq51}
\mathbf{A}_{\nu} \left(\nu\right) \sim \frac{1}{\pi }\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!a_n \left(1\right)}}{{\nu ^{2n + 1} }}}$$ holds as $\nu \to \infty$. Employing the continuation formulas stated in Section \[section1\], we find that $$\mathbf{A}_\nu \left( {\nu } \right) = i e^{\pi i\nu } H_\nu ^{\left( 1 \right)} \left( {\nu } \right) - \mathbf{A}_{-\nu} \left( {\nu e^{ - \pi i}} \right)$$ and $$\mathbf{A}_\nu \left( {\nu } \right) = - ie^{ - \pi i\nu } H_\nu ^{\left( 2 \right)} \left( {\nu } \right) - \mathbf{A}_{-\nu} \left( {\nu e^{\pi i} } \right).$$ For the right-hand sides, we can apply the asymptotic expansions of the Hankel functions and the Anger–Weber function to deduce that $$\label{eq56}
\mathbf{A}_{\nu} \left(\nu\right) \sim - ie^{\pi i \nu} \frac{2}{{3\pi }}\sum\limits_{m = 0}^\infty {d_{2m} e^{\frac{{2\left( {2m + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}} + \frac{1}{\pi }\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!a_n \left( 1 \right)}}{{\nu ^{2n + 1} }}}$$ as $\nu \to \infty$ in the sector $\frac{\pi}{2}<\arg \nu < \frac{3\pi}{2}$, and $$\label{eq57}
\mathbf{A}_{\nu} \left(\nu\right) \sim ie^{ - \pi i \nu} \frac{2}{{3\pi }}\sum\limits_{m = 0}^\infty {d_{2m} e^{ - \frac{{2\left( {2m + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}} + \frac{1}{\pi }\sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!a_n \left( 1 \right)}}{{\nu ^{2n + 1} }}}$$ as $\nu \to \infty$ in the sector $-\frac{3\pi}{2}<\arg \nu < -\frac{\pi}{2}$. Therefore, as the line $\arg \nu = \frac{\pi}{2}$ is crossed, the additional series $$\label{eq52}
- ie^{\pi i \nu} \frac{2}{{3\pi }}\sum\limits_{m = 0}^\infty {d_{2m} e^{\frac{{2\left( {2m + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}}$$ appears in the asymptotic expansion of $\mathbf{A}_{\nu} \left(\nu\right)$ beside the original one . Similarly, as we pass through the line $\arg \nu = -\frac{\pi}{2}$, the series $$\label{eq53}
ie^{ - \pi i \nu} \frac{2}{{3\pi }}\sum\limits_{m = 0}^\infty {d_{2m} e^{ - \frac{{2\left( {2m + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}}$$ appears in the asymptotic expansion of $\mathbf{A}_{\nu} \left(\nu\right)$ beside the original series . We have encountered a Stokes phenomenon with Stokes lines $\arg \nu = \pm\frac{\pi}{2}$. With the aid of the exponentially improved expansion given in Theorem \[thm3\], we shall find that the asymptotic series of $\mathbf{A}_{\nu} \left( \nu \right)$ shows the Berry transition property: the two series in and “emerge” in a rapid and smooth way as the Stokes lines $\arg \nu = \frac{\pi}{2}$ and $\arg \nu = -\frac{\pi}{2}$ are crossed.
From Theorem \[thm3\], we infer that if $N \approx \frac{1}{2}\pi \left|\nu\right|$, then for large $\nu$, $\left|\arg \nu\right|<\pi$, we have $$\begin{gathered}
\mathbf{A}_\nu \left( \nu \right) \approx \frac{1}{\pi }\sum\limits_{n = 0}^{N - 1} {\frac{{\left( {2n} \right)!a_n \left( 1 \right)}}{{\nu ^{2n + 1} }}} - ie^{\pi i\nu } \frac{2}{{3\pi }}\sum\limits_{m = 0} {d_{2m} e^{\frac{{2\left( {2m + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}\widehat T_{2N - \frac{{2m - 2}}{3}} \left( {\pi i\nu } \right)} \\ - ie^{ - \pi i\nu } \frac{2}{{3\pi }}\sum\limits_{m = 0} {d_{2m} e^{ - \frac{{2\left( {2m + 1} \right)\pi i}}{3}} \sin \left( {\frac{{\left( {2m + 1} \right)\pi }}{3}} \right)\frac{{\Gamma \left( {\frac{{2m + 1}}{3}} \right)}}{{\nu ^{\frac{{2m + 1}}{3}} }}e^{\frac{{2\left( {2m - 2} \right)\pi i}}{3}} \widehat T_{2N - \frac{{2m - 2}}{3}} \left( { - \pi i\nu } \right)} ,\end{gathered}$$ where, as before, $\sum\nolimits_{m = 0}$ means that the sum is restricted to the leading terms of the series.
In the upper half-plane, the main contribution comes from the terms involving $\widehat T_{2N - \frac{{2m - 2}}{3}} \left( {\pi i\nu } \right)$. Under the above assumption on $N$, from and , the Terminant functions have the asymptotic behaviour $$\widehat T_{2N - \frac{{2m - 2}}{3}} \left( {\pi i\nu } \right) \sim \frac{1}{2} + \frac{1}{2}\mathop{\text{erf}} \left( {\left( {\theta - \frac{\pi }{2}} \right)\sqrt {\frac{1}{2}\pi \left| \nu \right|} } \right),$$ provided that $\arg \nu = \theta$ is close to $\frac{\pi}{2}$, $\nu$ is large and $m$ is small in comparison with $N$. Therefore, when $\theta < \frac{\pi}{2}$, the Terminant functions are exponentially small; for $\theta = \frac{\pi }{2}$, they are asymptotically $\frac{1}{2}$ up to an exponentially small error; and when $\theta > \frac{\pi}{2}$, the Terminant functions are asymptotic to $1$ with an exponentially small error. Thus, the transition across the Stokes line $\arg \nu = \frac{\pi}{2}$ is effected rapidly and smoothly. Similarly, in the lower half-plane, the dominant contribution is controlled by the terms containing $\widehat T_{2N - \frac{{2m - 2}}{3}} \left( { - \pi i\nu } \right)$. From and , we have $$e^{\frac{{2\left( {2m - 2} \right)\pi i}}{3}} \widehat T_{2N - \frac{{2m - 2}}{3}} \left( { - \pi i\nu } \right) \sim - \frac{1}{2} + \frac{1}{2}\mathop{\text{erf}}\left( {\left( {\theta + \frac{\pi }{2}} \right)\sqrt {\frac{1}{2}\pi \left| \nu \right|} } \right),$$ under the assumptions that $\arg \nu = \theta$ is close to $-\frac{\pi}{2}$, $\nu$ is large and $m$ is small in comparison with $N \approx \frac{1}{2}\pi \left|\nu\right|$. Thus, when $\theta > - \frac{\pi}{2}$, the normalized Terminant functions are exponentially small; for $\theta = -\frac{\pi}{2}$, they are asymptotic to $-\frac{1}{2}$ with an exponentially small error; and when $\theta < - \frac{\pi}{2}$, the normalized Terminant functions are asymptotically $-1$ up to an exponentially small error. Therefore, the transition through the Stokes line $\arg \nu = -\frac{\pi}{2}$ is carried out rapidly and smoothly.
We note that from the expansions and , it follows that is an asymptotic series of $\mathbf{A}_{\nu} \left( \nu \right)$ in the wider range $\left|\arg \nu\right| \leq \pi -\delta < \pi$, with any fixed $0 < \delta \leq \pi$.
Discussion {#section6}
==========
In this paper, we have discussed in detail the large order and argument asymptotics of the Anger–Weber function $\mathbf{A}_{\nu}\left(\lambda \nu\right)$ when $\lambda > 0$, using Howls’ method. The resurgence properties and the exponentially improved versions of the large $\nu$ asymptotics of the associated Anger function $\mathbf{J}_\nu \left(\lambda \nu\right)$ and Weber function $\mathbf{E}_\nu \left(\lambda \nu\right)$ can be obtained from the relations $$\mathbf{J}_\nu \left( {\lambda \nu } \right) = J_\nu \left( {\lambda \nu } \right) + \sin \left( {\pi \nu } \right)\mathbf{A}_\nu \left( {\lambda \nu } \right),$$ $$\mathbf{E}_\nu \left( {\lambda \nu } \right) = - Y_\nu \left( {\lambda \nu } \right) - \cos \left( {\pi \nu } \right)\mathbf{A}_\nu \left( {\lambda \nu } \right) - \mathbf{A}_{ - \nu } \left( {\lambda \nu } \right),$$ our previous results on the Bessel functions [@Nemes] and the results of the present series of papers on the Anger–Weber function. Note that the resulting resurgence formulas have different forms according to whether $\lambda=1$ or $\lambda>1$. From these new representations, error bounds for the asymptotic expansions of the Anger and Weber functions can be derived which, in the case $\lambda=1$, may be compared with the ones given earlier by Olver [@Olver2].
Acknowledgement {#acknowledgement .unnumbered}
===============
I would like to thank the anonymous referee for his/her constructive and helpful comments and suggestions on the manuscript.
[10]{}
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F. W. J. Olver, Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral, *SIAM J. Math. Anal.* **22** (1991), pp. 1460–1474.
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F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), *NIST Handbook of Mathematical Functions*, Cambridge University Press, New York, 2010.
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---
author:
- 'Simon Caron-Huot,'
- 'Anh-Khoi Trinh'
bibliography:
- 'references.bib'
title: |
All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity:\
Hidden Ten-Dimensional Conformal Symmetry
---
Introduction
============
Among strongly coupled quantum systems, a distinguished class are dual to weakly coupled gravitational theories and are therefore efficiently tackled by the gauge-gravity correspondence. Models in this class are characterized by a large-$N$ expansion and a sparse spectrum of local operators: all single-trace primary operators with small scaling dimension have spin two or less [@Heemskerk:2009pn]. While the gravity side of the correspondence typically remains more straightforward to use, new analytic bootstrap techniques are becoming applicable and should be vigorously pursued being potentially more general. They appear especially advantageous for observables like four-point correlation functions, which can now be studied to unprecedented precision.
In this paper, we study the four-point function of general half-BPS states in maximally supersymmetric super Yang-Mills theory, or $\mathcal{N}=4$ SYM for short, dual to type IIB superstring theory on AdS${}_5\times$S${}_5$ geometry. Through the AdS/CFT correspondence, half-BPS operators are dual to $S_5$ spherical harmonics (Kaluza-Klein modes) of the graviton, and we will consider the general correlator of four such spherical harmonics. This provides an essential ingredient for studying the duality at loop level, where all modes can run inside the loop, and also provides an explicit model for how local physics in an internal manifold gets encoded from the CFT perspective.
Four-point correlators have been much studied since the early days of the AdS/CFT correspondence. Initially, techniques were developed for computing tree-level Witten diagrams in position space, which led to the first complete results for the correlators of four dilatons or stress tensor multiplets [@DHoker:1999kzh; @Arutyunov:2000py]. In the case of the SYM model, part of these correlators are fixed by non-renormalization theorem and agree with calculations at weak ‘t Hooft coupling [@Lee:1998bxa; @Eden:2000bk], but there is in addition an unprotected part which contains interesting dynamical information. For example, a suitable limit reveals the flat space S-matrix of the underlying ten-dimensional superstring theory [@Gary:2009ae; @Penedones:2010ue; @Maldacena:2015iua]. This program was gradually extended to higher spherical harmonics [@Arutyunov:2002fh; @Arutyunov:2003ae; @Berdichevsky:2007xd; @Uruchurtu:2008kp; @Uruchurtu:2011wh].
Meanwhile, by analyzing the operator product expansion (OPE), it was discovered how to exploit the large gap property of holographic theories, that is the fact that all non-protected single-traces get lifted from the spectrum. This led to various conjectures for correlators [@Nirschl:2004pa; @Dolan:2006ec]. However, it took almost a decade for a conjecture to appear in the general case [@Rastelli:2016nze], thanks to a remarkably simple pattern which was observed in Mellin-space [@Penedones:2010ue; @Fitzpatrick:2011ia]. The corresponding position space calculation of tree-level Witten diagrams was only very recently completed after a longstanding effort [@Arutyunov:2017dti; @Arutyunov:2018tvn], which confirmed the conjecture in all considered examples.
In this paper we will revisit correlators of half-BPS operators from the perspective of the analytic bootstrap, which offers a constructive method to build correlators from their light-cone singularities [@Fitzpatrick:2012yx; @Komargodski:2012ek; @Alday:2015ewa]. This can be formulated nonperturbatively as a Kramers-Kronig type dispersion relation, known as the Lorentzian inversion formula, which explicitly reconstructs OPE data from a suitably defined absorptive part [@Caron-Huot:2017vep; @Simmons-Duffin:2017nub; @Kravchuk:2018htv]. For holographic theories at tree-level, the absorptive part is a finite sum of conformal blocks, thus reducing the computation in a given theory to essentially a group-theoretic exercise (see also [@Alday:2017gde; @Liu:2018jhs]). The result is unique up to AdS contact interactions, whose size can be bounded by considering the Regge limit [@Alday:2016htq; @Caron-Huot:2017vep]. These are further restricted by supersymmetry and in fact absent in $\mathcal{N}=4$ SYM as we will see. The method, therefore, rigorously and unambiguously determines the correlators. Interestingly, in analogy with the unitarity method for one-loop S-matrices [@Bern:1994cg], “squaring” this tree-level four-point data gives sufficient information to go to one-loop [@Aharony:2016dwx], as has now been successfully carried out for low spherical harmonics [@Aprile:2017bgs; @Alday:2017xua; @Alday:2017vkk].
Study of $\mathcal{N}=4$ SYM specifically is further motivated by the possibility of finding unexpected structures. In the planar limit, this theory is controlled by a rich and remarkable integrable system, making it a rather unique nontrivial four-dimensional quantum field theory in which the spectrum can be computed exactly at finite ‘t Hooft coupling, see [@Beisert:2010jr; @Gromov:2014caa]. It is important to determine what, if anything, of this structure persists beyond the planar limit. Correlation functions at strong coupling offer a clean, natural environment in which to discuss this question. Furthermore, we expect the correlators in this limit to provide important cross-checks and guidance as integrability-based methods are now being developed to tackle the $1/N$ expansion [@Bargheer:2017nne; @Ben-Israel:2018ckc], as well as novel AdS-based techniques [@Cardona:2017tsw; @Yuan:2017vgp; @Yuan:2018qva]. Hints of a hidden symmetry in correlators have been uncovered recently [@Aprile:2018efk], who studied the anomalous dimension matrix governing the mixing between double-trace operators built from different S${}_5$ spherical harmonics, starting from the conjectured Mellin-space formula of [@Rastelli:2016nze]. Amazingly, the eigenvalues of the anomalous dimension matrix are simple rational numbers, for which the authors of ref. [@Aprile:2018efk] conjectured a general formula.
This paper is organized into two parts. First in section \[Sec:Generalities\] we review the three ideas behind our calculations: the Operator Product Expansion, the supersymmetric Ward identities satisfied by $\mathcal{N}=4$ SYM correlators, and the Lorentzian inversion formula. As a warm-up, we then apply these formulas at order $1/N_c^0$ in section \[Sec:1/c0\]. Including the identity in two cross-channels yields double-trace data in the direct channel. While conceptually straightforward, the supersymmetric OPE differs from the usual generalized free field result in simple but important ways. Section \[Sec:Tree-level\] contains our main computations: the tree-level correlator (order $1/N_c^2$) at strong ‘t Hooft coupling. The full result is determined by just the singular part arising from half-BPS single-trace operators. While their couplings are well-known and could be obtained from non-renormalization theorems [@Lee:1998bxa], we show that crossing symmetry in fact suffices to bootstrap them from scratch. This computation yields double trace mixing matrices, and we confirm the conjecture of [@Aprile:2018efk] in many examples.
In a second part of this paper, section \[sec:10D\], we explore the remarkable structure found in this result. We will attribute the simplicity of the eigenvalues to an unexpected SO(10,2) symmetry, representing an effective conformal invariance of ten-dimensional supergravity at tree-level. This symmetry will be used to unify all spherical harmonics into a single ten-dimensional object. Proof of the symmetry at the level of four-point correlators amounts to showing that this generating function correctly predicts the singular part of each correlator; we check this for a large number of examples. We also show how the symmetry readily implies the conjectured Mellin-space formula of [@Rastelli:2016nze], and use it to obtain the leading-logarithmic term at each order in $1/N$. Implications are discussed in the conclusion.
Generalities {#Sec:Generalities}
============
At any coupling, $\mathcal{N}=4$ SYM contains a special class of operators, which are half-BPS and thus annihilated by the maximal possible number of supercharges. They transform as Lorentz scalars and traceless symmetric tensors with respect to the SO(6)${}_R\simeq$ SU(4)${}_R$ global symmetry (Dynkin label $\big[0,p,0\big]$). In the Lagrangian description of $\mathcal{N}=4$ SYM, they are described as symmetrical traceless polynomials in the 6 adjoint scalar fields $\phi^a$ of the theory. Denoting $x^\mu$ the spacetime coordinates and introducing null six-vectors $y^a$ to parametrize the SO(6)${}_R$ dependence, they can be written as \[def Op\] \^p(x,y) + O(1/N\^2) , which will be normalized in this paper so that ł\^p(x\_1,y\_1)\^p(x\_2,y\_2)= ( )\^p where $x_{ij}^2=(x_i-x_j)^2$, $y_{ij}^2=y_i\cdot y_j$. Due to the BPS condition, the scaling dimension of $\OO^p$ is exactly $p$, where $p\geq 2$. At strong ‘t Hooft coupling, they admit a dual description as Kaluza-Klein modes of the graviton on AdS${}_5\times$ S${}_5$. According to the gravity analysis, all other operators in this limit are either multi-trace composites built of products of the $\OO^p$, or are heavier than the string scale, $\Delta{\mathrel{\hbox{\rlap{\lower.55ex \hbox{$\sim$}} \kern-.3em \raise.4ex \hbox{$>$}}}}\lambda^{1/4}$.
Four-point correlators depend on AdS${}_5$ and S${}_5$ cross-ratios: $$\begin{aligned}
u= \frac{x_{12}^{2}x_{34}^2}{x_{13}^2x_{24}^2} = z \bar{z}, \qquad &v = \frac{x_{23}^2x_{14}^2}{x_{13}^2x_{24}^2} = (1-z)(1-\bar{z}), \label{cross-ratios u,v} \\
\sigma= \frac{y_{12}^2 y_{34}^2}{y_{13}^2 y_{24}^2} = \alpha \bar{\alpha}, \qquad& \tau = \frac{y_{23}^2 y_{14}^2}{y_{13}^2 y_{24}^2} = (1-\alpha)(1-\bar{\alpha}), \label{cross-ratios sigma tau}\end{aligned}$$ up to an overall prefactor conventionally written as follows [@Dolan:2003hv]: $$\begin{aligned}
\big\l \OO^{p_1}(x_1,y_1) \cdots \OO^{p_4}(x_4,y_4) \big\r
\equiv&
\left(\frac{y_{12}^2}{x_{12}^2}\right)^{\frac{p_1+p_2}{2}}
\left(\frac{y_{34}^2}{x_{34}^2}\right)^{\frac{p_3+p_4}{2}}
\left(\frac{x_{14}^2 y_{24}^2}{x_{24}^2 y_{14}^2}\right)^{\frac{p_{2}-p_1}{2}}
\left(\frac{x_{14}^2 y_{13}^2}{x_{13}^2 y_{14}^2}\right)^{\frac{p_{3}-p_4}{2}}
\nl
&\times \mathcal{G}_{\{p_i\} }(z,\zb,\a,\ab)\,.
\label{Eq:OriginalCorrelator}\end{aligned}$$
We will be studying the correlator in the regime appropriate to the gauge-gravity duality, where the ’t Hooft coupling $\lambda = g_{YM}^2 N$ is large but finite, and work order by order in the gravity loop expansion $1/c$ where $c=\frac{N^2-1}{4}$: $$\mathcal{G}_{\{p_i\} } = \mathcal{G}_{\{p_i\}}^{(0)} + \frac{1}{c}\mathcal{G}_{\{p_i\}}^{(1)} + \frac{1}{c^2}\mathcal{G}_{\{p_i\}}^{(2)} + ...
\label{Eq:CorrelatorExpansion}$$
The Operator Product Expansion (OPE) offers a series expansion of the correlator around kinematic limits. In its most straightforward form, where we do not try to exploit supersymmetry, it involves standard four-dimensional conformal blocks, times S${}_5$ spherical harmonics for each $R$-symmetry representation which can be exchanged. The conformal blocks for a given spin and dimension are written as $$\begin{aligned}
G_{\ell,\Delta}^{r,s} (z,\bar{z}) &= \frac{z \bar{z}}{\bar{z}-z} \left[ k_{\frac{\Delta-\ell-2}{2}}^{r,s}(z) k_{\frac{\Delta+\ell}{2}}^{r,s}(\bar{z}) - k_{\frac{\Delta+\ell}{2}}^{r,s}(z) k_{\frac{\Delta-\ell-2}{2}}^{r,s}(\bar{z}) \right], \label{Eq:G-block}
\\
k_h^{r,s}(z) &= z^h \; _{2}F_1 \left(h+ \frac{r}{2}, h + \frac{s}{2}; 2h, z \right),\end{aligned}$$ with $r=p_{21}\equiv p_2-p_1$, $s=p_{34}$. Since SO(6)${}_R$ and the SO(4,2) Lie algebra are analytic continuations of each other, the S${}_5$ spherical harmonics admit identical expressions up to reversal of some quantum numbers: Z\_[m,n]{}\^[r,s]{} (,) = (-1)\^[m]{} G\_[m,-n]{}\^[-r,-s]{}(,), where the labels $m,n$ are related to the R-symmetry Dynkin labels by $\big[m, n-m, m\big]$. In the literature these are often written as Jacobi polynomials, see [@Bissi:2015qoa]. The OPE decomposition of the correlator (\[Eq:OriginalCorrelator\]), in the $(12)$ or $s$-channel, is then \_[{p\_i} ]{}(z,,,) = \_[,,m,n]{} \_[{p\_i}]{}(,,m,n) G\_[,]{}\^[p\_[21]{},p\_[34]{}]{}(z,)Z\_[m,n]{}\^[p\_[21]{},p\_[34]{}]{}(,). \[naive OPE\] Here the spin $\ell\geq 0$ is an arbitrary nonnegative integer and $\Delta$ runs over all operators with the given spin and $R$-symmetry. The summations over $m,n$ however are finite, ranging over the R-symmetry representations which can appear in the tensor product of each of the pairs $(1,2)$ and $(3,4)$. Using the general formula for the tensor product of SU(4) representations (see [@Dolan:2002zh; @Bissi:2015qoa]): $$\big[0,p,0 \big] \times \big[0,q,0 \big] = \sum_{m=0}^{p} \sum_{s=0}^{p-m} \big[ m, q-p +2s, m\big],
\label{Eq:TensorProductStates}$$ where we have assumed $p \leq q$, we get the summation range in eq. (\[naive OPE\]): 0m{ p\_i}, (|p\_[12]{}|, |p\_[34]{}|)+m n ( p\_1+p\_2, p\_3+p\_4)-m, where the difference between $n$ and its lower/upper bound is restricted to be an even integer.
The OPE (\[naive OPE\]) accounts for all the bosonic symmetries of the correlator. It is still rather redundant because it does not exploit supersymmetry. A natural refinement would be to use superconformal blocks instead, but here we will follow a simpler route which is applicable thanks to the half-BPS nature of our external operators.
Superconformal Ward identities
------------------------------
A half-BPS supermultiplet is annihilated by half of the 32 supercharges of the theory. Since the remaining charges split into pairs of raising/lowering operators acting within the multiplet, only 1/4 of the supercharges actually act nontrivially on a given bosonic primary $\OO^p(x,y)$. This is significant because for four external operators, these 1/4 are generically linearly independent and span the full algebra. Thus the correlators of superconformal descendents are fully determined from those of the primaries [@Chicherin:2014uca; @Bissi:2015qoa].
Linear independence however fails when the $x$ and $y$ cross-ratios are aligned in a specific way; this leads to a Ward identity satisfied by the bosonic correlator [@Nirschl:2004pa; @Dolan:2004mu; @Bissi:2015qoa]: \_z ((z,,,)|\_[=z]{}) =0. \[ward identity\] That is, the $z$ dependence of the correlator disappears upon setting $\a=z$.
Since the dependence of the correlator on $\a,\ab$ is purely rational (the left-hand-side of eq. (\[Eq:OriginalCorrelator\]) being polynomial in the $y_{ij}^2$), the Ward identities can be solved by factoring out powers of $z-\a$, together with its conjugates under the $(z\leftrightarrow \zb)$ and $(a\leftrightarrow\ab)$ symmetries. The most general solution, consistent with these symmetries, is [@Dolan:2004iy; @Bissi:2015qoa]: $$\begin{aligned}
\mathcal{G}_{\{p_i\} }(z, \zb, \a, \ab) &= k \chi(z, \a) \chi(\zb, \ab) + \frac{(z-\a)(z-\ab)(\zb-\a) (\zb - \ab) }{(\a - \ab)(z - \zb)} \nonumber \\
&\times \left(- \frac{ \chi(\zb,\ab) f(z,\a)}{\a z (\zb-\ab)} + \frac{\chi(\zb,\a) f(z,\ab)}{\ab z (\zb-\a)} + \frac{\chi(z,\ab) f(\zb,\a)}{\a \zb (z-\ab)} - \frac{\chi(z,\a) f(\zb,\ab)}{\ab \zb (z-\a)} \right) \nonumber \\
&+ \frac{(z-\a) (z - \ab) (\zb- \a) (\zb - \ab)}{(z \zb)^2 (\a \ab)^2} \HH_{\{p_i\}}(z,\zb, \a, \ab), \label{G ansatz}\end{aligned}$$ where $\chi$ is a fixed function satisfying $\chi(z,z)=1$, given shortly. Note that all the functions above depend on $\{p_i\}$, which we omitted in the above for readability. In practice, starting from a correlator which fulfills the Ward identity (\[ward identity\]), $k_{\{p_i\} }$, which we will call the *unit* contribution, is obtained simply by setting $z=\a$, $\zb=\ab$. The *chiral correlator* $f_{\{p_i\} }$ is obtained by taking only one such limit and subtracting the unit: \[kf from G\]
k\_[{p\_i} ]{} &= \_[{p\_i} ]{}(z, , z, ),\
f\_[{p\_i} ]{}(,) &= ( \_[{p\_i} ]{}(z, , z, ) - k\_[{p\_i}]{}\_[{p\_i} ]{}(,) ).
Finally, the *reduced correlator* $\HH_{\{p_i\} }$ can be extracted from $\GG_{\{p_i\} }$ by subtracting everything else that comes before it in eq. (\[G ansatz\]).
There is a rather unique, convenient choice for the function $\chi(z,\ab)$, which ensures that the superconformal Casimir equation commutes with the preceding decomposition [@Bissi:2015qoa]: $$\chi_{\{p_i\}}(z,\a) = \left( \frac{z}{\alpha} \right)^{\max(|p_{21}|,|p_{34}|)/2} \left( \frac{1-a}{1-z} \right)^{\max(p_{21}+p_{34},0)/2}.
\label{Eq:ChiWard}$$ The Casimir operator then annihilates the $k$ contribution, in particular. In fact there are four possible solutions to this constraint, obtained by analytic continuation in the $p_i$’s: the above solution is singled out by the fact that it does not introduce spurious negative exponents at $z\to 0$ and $\a\to 1$. The same solution was used in [@Bissi:2015qoa] (who restricted to a specific ordering of the $p_i$’s).
We now review the implications of the Casimir equations, following [@Bissi:2015qoa]. Its action on the *chiral correlator* takes on a separated form, whose general solution involves products of the hypergeometric functions in eq. (\[Eq:G-block\]), thus giving the OPE: $$f_{\{p_i\}}(z,\a) = \sum_{j=0}^\infty \sum_m b_{\{p_i\}}(j,m) k_{1+m/2+j}^{p_{21},p_{34}}(z) k_{-m/2}^{-p_{21},-p_{34}}(\a).
\label{Eq:fGeneric}$$ The sum over $m$ is finite since $f$ is a polynomial in $1/\a$, whose degree determines the range: (|p\_[12]{}|,|p\_[34]{}|) m (p\_1+p\_2,p\_3+p\_4) -2, where in addition $m$ should differ from its lower bound by an even integer. Single-valuedness of the correlator (\[Eq:OriginalCorrelator\]) forces $j$ to be an integer. It must be nonnegative due to the unitarity bound, since the superconformal Casimir eigenvalue $(m+j)(m+j+1) - m(m+1)$ must be nonnegative.
The Casimir equation for the *reduced correlator* $\HH$ also takes separated form, and its solution is similar to the naive OPE (\[naive OPE\]) but with the dimension shifted by $4$ in accordance to the $z\zb$ denominator in the prefactor multiplying $H$: $$\HH_{\{p_i\}}(z,\zb,\a,\ab)=\sum_{\ell,\Delta,m,n} a_{\{p_i\}}(\ell,\Delta,m,n)
G_{\ell,\Delta+4}^{p_{21},p_{34}} (z,\zb) Z_{m,n}^{p_{21},p_{34}}(\a,\ab).
\label{Eq:Hpart}$$ The representations appearing in $H$ can be worked out from the $\a$, $\ab$ dependence of the coefficient of $\HH$ in eq. (\[G ansatz\]), and are only those which appear in the tensor products of $[0,p_i-2,0]$; this gives the range 0m{ p\_i-2}, (|p\_[12]{}|, |p\_[34]{}|)+m n [min]{}(p\_1+p\_2, p\_3+p\_4)-m-4. \[small range\] While formally similar to the bosonic expansion enjoyed by $\mathcal{G}$ in eq. (\[naive OPE\]), we see that the OPE for the susy-decomposed correlators $f$ and $\HH$ contains much fewer terms. For example, for the simplest case of the 2222 correlator, eq. (\[naive OPE\]) contains six distinct R-symmetry representations whereas there is a single one in eq. (\[Eq:Hpart\]), involving only $Z_{0,0}^{(0,0)}(\a,\ab)=1$. The full $\a$ dependence is then entirely produced by the chiral correlator $f(z,\a)$ and the various factors in (\[G ansatz\]).
Lorentzian inversion formula {#Sec:InversionIntegral}
----------------------------
To determine the OPE data $b^{\{p_i\}}_{j,m}$ and $a^{\{p_i\}}_{\ell,\Delta,m,n}$ entering eqs. (\[Eq:fGeneric\]) and eqs. (\[Eq:Hpart\]), our main tool will be a recently derived inversion integral which plucks out the OPE coefficients from the correlator. The key point is that this formula does not require to know the full correlator, but just its “double discontinuity” which is much easier to compute.
Let us first deal with the global symmetry. Using that the $Z_{m,n}(\a,\ab)$ represent S${}_5$ spherical harmonics, which are mutually orthogonal, it is straightforward to write down an integral which projects the reduced correlator onto any desired representation. As noted, the $Z_{m,n}$’s are secretly Jacobi polynomials, with argument $\cos\theta=1-\frac{2}{\a}$, which suggests that the natural integration range is $\cos\theta\in [-1,1]$ or $\a,\ab\in [1,\infty)$. One can then show that the following orthogonality relation holds: \_[{p\_i}]{}(z,,m,n) = (8\^4 \^[r,s]{}\_[2+]{}\^[r,s]{}\_[1+]{})\^[-1]{} \_1\^ ( )\^2 Z\^[-r,-s]{}\_[m,n]{}(,) \_[{p\_i}]{}(z,,,) \[Eq:Z projection\] where $r=p_{21}$, $s=p_{34}$, and \^[r,s]{}\_h = = , r\_h\^[r,s]{} = . \[kappa tilde\] This diagonalizes the S${}_5$ part of the OPE (\[Eq:Hpart\]), leaving only sums over dimension and spin: \[rfct\] \_[{p\_i}]{}(z,,m,n) = \_[,]{} a\_[{p\_i}]{}(,,m,n) G\_[,+4]{}\^[(p\_[21]{},p\_[34]{})]{} (z,). \[OPE Hmn\] The Lorentzian inversion formula inverts this $z,\zb$ OPE: it extracts the coefficient of a given $G_{\ell,\Delta}$ block by integrating against $z,\zb$. Writing its left-hand-side as $c_{m,n}(\ell,\Delta-4)$ to compensate for the argument of the block being $\Delta+4$, the formula from [@Caron-Huot:2017vep] is written as a sum of $t$-channel and $u$-channel contributions:
c\_[{p\_i}]{}(,-4,m,n) &= 14\^[p\_[21]{},p\_[34]{}]{}\_[+2]{} \_0\^1 ( )\^2 G\_[-3,+3]{}\^[-p\_[21]{},-p\_[34]{}]{}(z,)\
&+(-1)\^[+m]{} ( p\_1p\_2) \[Eq:cInversion\]
with $p_{ij}=p_i-p_j$.[^1] The function $c_{\{p_i\}}(\ell,\Delta,m,n)$ serves as a generating function of the OPE data, which, for a fixed integer $\ell$, are encoded in its $\Delta$-plane poles: $$\lim_{\Delta\to \Delta_k} c_{\{p_i\}}(\ell,\Delta,m,n) = \displaystyle \frac{a_{\{p_i\}}(\ell,\Delta_k,m,n)}{\Delta_k - \Delta},
\label{naive c poles}$$ where $k$ labels the the different exchanged operators.
The double discontinuity in the integrand is a natural Lorentzian object defined as the expectation value of a double commutator $\frac12\l 0| [\OO_4,\OO_1][\OO_2,\OO_3]|0\r$ in a region where the two invariants $x_{14}^2$ and $x_{23}^2$ are timelike [@Caron-Huot:2017vep]. Since $(1-\zb) \propto x_{14}^2x_{23}^2$ switches sign when either invariant crosses the light-cone, the different operator orderings can be reached by analytically continuing around $\zb=1$ with $z$ fixed. Specifically, working in the region $0<z,\zb<1$ and denoting the usual Euclidean-branch correlator simply as $\mathcal{G}$, the double commutator is equal to: = () (z,|[z]{}) - e\^[i]{} \^(z, |[z]{}) - e\^[-i]{}\^(z, |[z]{}), \[Eq:doubleDisc\] where $\alpha= \frac{p_{21}+p_{34}}{2}$.
It is important to stress that eq. (\[Eq:cInversion\]), contrary to eq. (\[Eq:Z projection\]), is not a straightforward consequence of an orthogonality condition applied termwise to the OPE. In fact, the [dDisc]{} operation naively annihilates each term in eq. (\[OPE Hmn\]). In reality it is nonzero because the OPE sum diverges for $\zb>1$ which generally creates a branch cut or pole at $\zb=1$. The formula is thus more analogous to a Kramers-Kronig dispersion relation which reconstructs a function from its discontinuities, or the Froissart-Gribov formula in the scattering amplitude context.
The OPE data in one channel is thus obtained from that in two cross-channels, which can be used to efficiently compute the two $H$ functions on the right of eq. (\[Eq:cInversion\]). This reorganization of crossing symmetry is advantageous in large-$N$ theories, because the $\dDisc$ operation kills cross-channel operators with double-trace dimensions, see [@Caron-Huot:2017vep]. Only single-traces contribute in the cross-channels (up to order $1/c^2$, that is one-loop order in supergravity). We will see this in action in the next sections.
Using the explicit form (\[Eq:G-block\]) for the conformal blocks, the formula (\[Eq:cInversion\]) can be written in a more useful factorized form as
c\_[{p\_i}]{}(,,m,n) &= \_0\^1 k\_[1-h]{}\^[-p\_[21]{},-p\_[34]{}]{}(z)k\_\^[-p\_[21]{},-p\_[34]{}]{}()\
& + (-1)\^[+m]{} ( p\_1p\_2), \[Eq:cInversion 2\]
where $h=1+\frac{\Delta-\ell}{2}$ and $\hb=2+\frac{\Delta+\ell}{2}$. In practice, the OPE data is encoded in poles with respect to $h$ which comes from the $z\rightarrow 0$ limit of the $z$ integration. The $\bar{z}$ integral is dual to $\bar{h}$. Analytic results for the integrals we will need are given in appendix \[Appendix: Inversion\]. Some words about convergence. In an abstract (unitary) CFT, convergence of the integral for $\ell>1$ (along the principal series $\Delta=d/2+i\nu$) was proven in [@Caron-Huot:2017vep] using boundedness of $\mathcal{G}$ from the Regge limit $z,\zb\to 0$ (more precisely, boundedness of each term in eq. (\[Eq:doubleDisc\])). For the present supersymmetric correlator, tracing through the factors in eq. (\[G ansatz\]), boundedness of $\mathcal{G}$ actually implies that $H/(z\zb)^2$ is bounded. Supersymmetry thus buys us four units of spin: convergence is ensured for $\ell >-3$, that is, all spins.[^2]
Order by order in the $1/c$ expansion, the situation can be less favorable than nonperturbatively, reflecting the worsening ultraviolet behavior of supergravity with increasing loop order. At order $1/c$, convergence for $\ell>2$ was nonetheless proven in [@Caron-Huot:2017vep] from a stress tensor sum rule, which thus ensures convergence for $\ell>-2$ in our supersymmetric case. Therefore, at order $1/c$, the integral (\[Eq:cInversion 2\]) is guaranteed to recover *all* OPE data in $\mathcal{N}=4$ super Yang-Mills. The coefficients $c(\ell,\Delta,m,n)$ contain information about both protected *and* unprotected contributions, which are effectively treated on the same footing.
Finally, for the chiral correlator $f$, the OPE (\[Eq:fGeneric\]) can be similarly inverted using the one-dimensional inversion integral from ref. [@Simmons-Duffin:2017nub]. This formula deals with the case where the sum runs over integer $h$, which is precisely the case for the chiral correlator![^3] Using orthogonality on a simple contour integral around $\a=0$ to deal with the $R$-symmetry part, f\_[{p\_i}]{}(z,m) = k\_[1+m/2]{}\^[p\_[21]{},p\_[34]{}]{}() f\_[{p\_i}]{}(z,), the inversion integral then gives the coefficient as b\_[{p\_i}]{}(j,m) = \_0\^1 k\_[1+m/2+j]{}\^[-p\_[21]{},-p\_[34]{}]{}() . \[f Inversion\] In this way all protected and unprotected data for $f$ and $H$ in eqs. (\[Eq:fGeneric\]) and are recovered from the double-discontinuity.
We will be particularly interested in double-trace operators $[\mathcal{O}_p \mathcal{P}_q]_{k,\ell}$, which in the large-$N$ limit should appear as poles near twist $\Delta-\ell= p+q+2k+\gamma$, that is $h=1+k+\frac{p+q}{2}+\frac{\gamma}{2}$, where $\gamma$ is a small anomalous dimension. Actually, many nearly-degenerate operators generally contribute, so summing over them, eq. becomes $$c_{\{p_i\}}(h,\hb=h+\ell+1,m,n) = \left\langle \frac{a_{\{p_i\}}(\ell,\Delta,m,n)}{2(1+k+\frac12(p+q)+\frac{1}{2}\gamma - h)} \right\rangle,
\label{Eq:cGamma}$$ where the brackets imply that we sum over all superconformal primary operators with spin $\ell$ and approximate twist $p+q+2k$, and $R$-symmetry representation $[m,n-m,m]$. One can use this equation to organize the OPE data as a $1/c$ expansion, with $c=\frac{N^2-1}{4}$. Indeed, one can consider the $1/c$ expansion of the scaling dimension and OPE coefficients as $$\begin{aligned}
\Delta_{n,\ell} &= \Delta + \frac{1}{c} \gamma_{n,\ell}^{(1)} + \frac{1}{c^2} \gamma_{n,\ell}^{(2)} +... \label{Eq:massExpansion} \\
a_{n,\ell} &= a_{n,\ell}^{(0)} + \frac{1}{c} a_{n,\ell}^{(1)} + \frac{1}{c^2} a_{n,\ell}^{(2)} + ...
\label{Eq:aExpansion}\end{aligned}$$ In particular, expanding in $1/c$, leading anomalous dimensions $\gamma_{n,\ell}^{(1)}$ appear as double poles in the tree-level data $c_{\{p_i\}}^{(1)}$.
Below we will first recover the OPE data for $f$ and $H$ at order $1/c^0$ (disconnected correlator), where only the identity contributes in the cross-channel, which we will resolve into protected and unprotected double-traces in the $s$-channel. We will then proceed to order $1/c$, where we will only need to insert a finite number of single-trace half-BPS blocks in the cross-channels.
Other decompositions in the literature
--------------------------------------
Before proceeding, let us briefly contrast our organization scheme, based on the Ward identity (\[ward identity\]) and Casimir equation, to other similar but distinct formulas used in the literature. A common way of writing the OPE is to separate “short” (protected) and “long” (unprotected) superconformal blocks, schematically: (z,,,) = c\^[short]{} g\^[short]{}(z,,,) + c\^[long]{} g\^[long]{}(z,,,). In terms of the supersymmetry decomposition (\[G ansatz\]), the short blocks contribute to all of $k,f,H$, whereas long blocks contribute only to $H$ (as the single term in eq. (\[Eq:Hpart\])). Another popular scheme exploits nonrenormalization theorems to separate out contributions which do not depend on the coupling: (z,,,) = \^[free]{}(z,,,) + \^[interacting]{}\_[{p\_i}]{}(z,, , ) where $\GG^{\rm free}$ can be computed from the limit of weak ‘t Hooft coupling $\lambda\to 0$ and $\HH^{\rm interacting}(z,\zb,\a,\ab)$ only receives contribution from long blocks (of both the strong and weakly interacting theories). Since the $g^{\rm long}$ only contribute to the $H$ part, these schemes both isolate the nontrivial corrections into $H$-like functions, which differ only by the amount of protected physics subtracted from it.
The decomposition used in this paper is distinct since we do not subtract anything from $H$, neither its short block contribution nor its free theory limit. Rather we directly obtain the functions $f$ and $H$ at strong coupling, by computing their double-discontinuity and using the inversion integrals (\[Eq:cInversion 2\]) and (\[f Inversion\]) to find the expansion coefficients (\[Eq:fGeneric\]) and (\[Eq:Hpart\]). This method eschews the use of nonrenormalization theorems and detailed knowledge of superconformal blocks.
Disconnected correlator: leading order $1/c^{0}$ {#Sec:1/c0}
================================================
As a first step toward anomalous dimensions of double-trace operators, here we detail the disconnected correlator OPE coefficients $\langle a^{(0)} \rangle_{pqqp}$, which set the overall normalization. The disconnected correlator itself is simply (see fig. \[Fig:disconnected correlator\]) \_[pqqp]{}\^[(0)]{} =\_[p,q]{} + ()\^ . \[G disconnected\] The OPE decomposition of such correlators is well known [@Fitzpatrick:2011dm]. However, we want to first apply the supersymmetry decomposition (\[G ansatz\]). While we find that the unit part is given by a simple general formula: $k^{(0)}_{pqqp} = 1+2\delta_{pq}$, the chiral and reduced correlators are generally given by more lengthy expressions. To give a few examples:
f\^[(0)]{}\_[2222]{} &= + +z + , &H\^[(0)]{}\_[2222]{} &= u\^2 +,\
f\^[(0)]{}\_[2332]{}&= (a+z-2az), &H\^[(0)]{}\_[2332]{}&= .
Generally, starting from $p,q\geq 3$, neither $f$ nor $H$ are simple sums and the expressions quickly grow lengthier with increasing $p$ and $q$. Nonetheless, the corresponding OPE data admits a simple uniform expression, as we now describe.
OPE coefficients {#Sec:a0}
----------------
The OPE coefficients $\langle a^{(0)}(h,\hb,m,n)\rangle_{pqqp}$ for a given correlator depend on $h,\bar{h}$, the scaling dimensions $p,q$ and the R-symmetry representation labels $m,n$ of the exchanged operator. They are obtained by applying the R-symmetry projection (\[Eq:Z projection\]) and inversion integral (\[Eq:cInversion 2\]) to the double-discontinuity of $H^{(0)}$. (We will discuss the chiral part $f^{(0)}$ below.) Since there are no branch cut at $\zb=1$, the double-discontinuity here simply picks out the polar terms. For instance, in the above examples, switching to the variables $x=\frac{z}{1-z}$ and $y=\frac{1-\zb}{\zb}$, we find for the poles in $y$: H\_[2222]{}\^[(0)]{} &=& Z\_[0,0]{}\^[0,0]{}(,),\
H\_[2332]{}\^[(0)]{} &=& Z\_[0,1]{}\^[1,1]{}(,). To give a less trivial example:
H\_[3333]{}\^[(0)]{} &= Z\_[0,2]{}\^[0,0]{}(,) + Z\_[1,1]{}\^[0,0]{}(,) \
& + Z\_[0,0]{}\^[0,0]{}(,) .
We notice that all expressions are antisymmetric under the interchange $x\leftrightarrow y^{-1}$. This stems from the manifest antisymmetry in $(z,\zb)$ of the left-hand-side together with the fact that the expressions are simple polynomials in $x$ and $y^{-1}$ (after factoring the same power of $z$ and $\zb$ that appeared in the integral (\[Eq:cInversion 2\])). The integral can be immediately performed using the elementary integrals (\[Eq:zbarIntegral\]) and (\[Eq:zIntegral\]):
\[c2222 zero\] c\^[(0)]{}\_[2222]{}(h,,0,0)&=& r\_h\^[0,0]{}r\_\^[0,0]{} (h) (1+(-1)\^) ((-1)-h(h-1)),\
\[c2332 zero\] c\^[(0)]{}\_[2332]{}(h,,0,1) &=& r\_h\^[1,1]{}r\_\^[1,1]{} (h) 12((-1)-h(h-1))(h-12)(-12),
whereas in the $3333$ case the three non-vanishing R-symmetry structures give three coefficients: $c^{(0)}_{3333}(h,\hb,0,2)$, $c^{(0)}_{3333}(h,\hb,1,1)$ and $c^{(0)}_{3333}(h,\hb,0,0)$.
Let us briefly interpret such results in terms of double-trace and protected operators. Double-traces have twist $p+q+2k$ with $k=0,1,2\ldots$, which correspond to $h=\frac{p+q+2}{2}+k$ and therefore $h\geq 3$ in the $2222$ case. However, we see from the formula (\[c2222 zero\]) that $c_{2222}$ has poles (from the cotangent) at each integer $h$. The poles at $h\leq 0$ can ignored (due to the terms discarded in doing the integral (\[Eq:zIntegral\])), but the poles at $h=1$ and $h=2$ would seem puzzling: these naively correspond to twists $0$ and $2$, which shouldn’t be there. Similarly, for $2332$, the formula exhibits two poles below the $h=\frac72$ double-trace threshold: at $h=\frac32$ and $\frac52$. As we will see shortly, the resolution is that these poles originate from short (and so-called semi-short) multiplets, which can contribute to the reduced correlator $H$ with the two apparent twists: h= h= . \[protected h\]
We are now ready to state the OPE coefficients in the general case. Due to the general form of the integrals (\[Eq:zbarIntegral\])-(\[Eq:zIntegral\]), the result is always the $r$ factors times a polynomial in $h,\hb$. This polynomial is easily determined using the following observations:
- It is odd under interchange of $h$ and $\hb$ (due to the $x\leftrightarrow y^{-1}$ symmetry just mentioned).
- It is symmetrical under $\hb\to 1-\hb$ (up to an overall minus sign when $p+q$ is odd).
- It vanishes for all values of $h$ below the double-trace threshold $h=\frac{p+q+2}{2}$, *except* at the protected locations allowed by eq. (\[protected h\]).
It is easy to show that the result is always simply the lowest-order polynomial with these properties, up to a $h,\hb$-independent constant. It can be written in general as: $$\begin{aligned}
\langle a^{(0)}_{pqqp} \rangle(h,\hb,m,n) &=
\left(1+\delta_{p,q}(-1)^{\ell+m}\right)
\frac{r_{h}^{q-p,q-p}\,r_{\hb}^{q-p,q-p}}{r_{1+\frac{n-m}{2}}^{q-p,q-p}r_{2+\frac{m+n}{2}}^{q-p,q-p}} \nonumber
\\
&
\times
\frac{\Gamma(h+ \frac{p+q}{2}) \Gamma(\bar{h} + \frac{p+q}{2}) }{\Gamma(h-\frac{p-q}{2}) \Gamma(\bar{h}-\frac{p+q}{2})}
\frac{\hb(\hb-1)-h(h-1)}{\Delta_{m,n}^{(8)}(h,\bar{h})}
\nonumber \\
& \times \frac{(3+m+n)(1+n-m)(m+1)(n+2)}{\Gamma(\frac{p+q+4+m+n}{2}) \Gamma(\frac{p+q+2+n-m}{2}) \Gamma(\frac{p+q-(n-m)}{2}) \Gamma(\frac{p+q-(2+m+n)}{2})},
\label{Eq:a0pqqp}\end{aligned}$$ where $$\begin{aligned}
\Delta_{m,n}^{(8)}(h,\bar{h}) &= \left(h- \frac{n-m+2}{2} \right) \left(h+ \frac{n-m}{2} \right) \left( h - \frac{m+n+4}{2} \right) \left( h + \frac{m+n+2}{2} \right) \nonumber \\
&\times \left(\bar{h}- \frac{n-m+2}{2} \right) \left(\bar{h}+ \frac{n-m}{2} \right) \left(\bar{h} - \frac{m+n+4}{2} \right) \left(\bar{h} + \frac{m+n+2}{2} \right).
\label{Eq:Delta8}\end{aligned}$$ Notice that the factor $\Delta^{(8)}$ simply provides poles at the locations (\[protected h\]) and their orbits under $h\mapsto 1-h$ and $h\mapsto \hb$.
For the chiral correlator $f$, which depends only on $z,\alpha$, we similarly apply the one-dimensional inversion formula in eq. (\[f Inversion\]). We find a structurally similar result (loosely, one drops all the factors which depend either on $h$ or $m+n$, and substitutes $\frac{n-m}{2}\mapsto \frac{m}{2}$): $$\begin{aligned}
\langle b^{(0)}_{pqqp} \rangle (j,m)
&= \Big(1+\delta_{p,q}(-1)^{j}\Big) \frac{r_{\bar{h}}^{q-p,q-p}}{r_{1+m/2}^{q-p,q-p}}
\times \frac{\Gamma(\bar{h}+\frac{p+q}{2})}{\Gamma(\bar{h} - \frac{p+q}{2})}\frac{(-1)^{(p+q-m-2)/2}}{(\hb-m/2-1) (\hb+m/2)}\Bigg|_{\hb=1+m/2+j}
\nonumber \\
&
\times \frac{m+1}{\Gamma(\frac{p+q+m}{2}+1)\Gamma(\frac{p+q-m}{2})}.
\label{Eq:a0chiral}\end{aligned}$$ The last line of both eqs. (\[Eq:a0pqqp\]) and (\[Eq:a0chiral\]) contains a $h,\hb$-independent normalization factor, which in all cases is determined by the following nice property: the coefficient becomes unity when setting $h=1+\frac{n-m}{2}$ and $\hb=2+\frac{m+n}{2}$ (or $j=0$ in the chiral case). This is the expected half-BPS contribution to the correlator as we now explain.
Reorganizing into superconformal blocks
---------------------------------------
Although this will not be needed below, it is an interesting cross-check that the above results can be reorganized into superconformal blocks.
A general expectation is that only operators with double-trace dimensions contribute to the disconnected correlator (with the lone exception of the identity operator). However, looking at the result for $a^{(0)}_{pqqp}$ in eq. (\[Eq:a0pqqp\]), we found contributions to $H$ below the double-trace threshold $h\geq \frac{p+q+2}{2}$, exemplified by eq. (\[protected h\]). Looking more closely into them, we find that they always appear with identical coefficients as contributions to the chiral part $f$ in eq. (\[Eq:a0chiral\]). These together organize into superconformal blocks with double-trace dimensions.
Multiplet Dynkin labels Dimension $\Delta$ and spin $\ell$
-------------------------------------- ----------------------- -------------------------------------
Half-BPS $\mathcal{B}_{0,n}$ $\big[0,n,0\big]$ $\Delta=q$, $\ell=0$
Quarter-BPS $\mathcal{B}_{m,n}$ $\big[m,n{-}m,m\big]$ $\Delta = m+n$, $\ell=0$, $m\geq 1$
Semi-short $\mathcal{C}_{\ell,m,n}$ $\big[m,n{-}m,m\big]$ $\Delta=m+n+2+\ell$
Long $\mathcal{A}_{\ell,\Delta,m,n}$ $\big[m,n{-}m,m\big]$ $\Delta > m+n+2+\ell$
: Supermultiplets which can appear in the four point function of half-BPS operators.[]{data-label="Tab:Multiplets"}
An effective way to group these contributions is to use the superconformal (quadratic) Casimir invariant. The different types of supermultiplets that can contribute to the four-point function are listed in table \[Tab:Multiplets\] (see for e.g. [@Bissi:2015qoa]). The corresponding Casimir eigenvalues are $$\begin{aligned}
\label{C2 short}
\mathcal{C}_2 &\equiv \tfrac12\left(\Delta(\Delta+4)+\ell(\ell+2)-m(m+2)-n(n+4)\right)
\\
&= \left\{\begin{array}{l@{\hspace{10mm}}l}
0, & \mbox{Half-BPS } \mathcal{B}_{0,n}\,,\\
m(n+1), &\mbox{Quarter-BPS } \mathcal{B}_{m,n}\,,\\
(\ell+m+2)(\ell+n+3), & \mbox{Semi-short } \mathcal{C}_{\ell,m,n}\,. \end{array}\right.\end{aligned}$$ As mentioned, the superconformal Casimir commutes with the supersymmetry decomposition (\[G ansatz\]) and is diagonalized by the OPE expansions (\[Eq:fGeneric\]) and . For convenience let us recall the eigenvalues: $$\begin{aligned}
\label{C2 susy}
\mathcal{C}_2 & \left\{ \begin{array}{l} k \\
f_{j,m}(z,\a)\equiv k_{1+m/2+j}(z)k_{-m/2}(\a) \\
H_{\ell,\Delta,m,n}(z,\zb,a,\ab)\equiv G_{\Delta+4,\ell}(z,\zb)Z_{m,n}(\a,\ab)
\end{array}\right.
\quad \nonumber \\
=
& \ \left\{\begin{array}{l} 0 \\ j(m+j+1) \\
\tfrac12\left(\Delta(\Delta+4)+\ell(\ell+2)-m(m+2)-n(n+4)\right)\end{array}\right.\end{aligned}$$ Equating the eigenvalues (\[C2 short\]) and (\[C2 susy\]) will give a simple way of understanding the $k,f,H$-decomposition of superconformal blocks. Let us first discuss the short half-BPS blocks $\mathcal{B}_{0,\Delta}$. These are the only blocks for which the Casimir eigenvalue vanishes, and therefore the only blocks for which $k\neq 0$. However, for each $R$-symmetry representation, both the chiral and reduced correlator can also contain a term with vanishing Casimir eigenvalues: the functions $f_{0,m}(z,\a)$ and $H_{m,n,m,n}(z,\zb,a,\ab)$. Looking at the disconnected OPE data $a_{m,n,m,n}^{(0)}$ and $b_{0,m}^{(0)}$ in eqs. (\[Eq:a0pqqp\]) and (\[Eq:a0chiral\]), we find that these come with unit coefficient for all the allowed $R$-symmetry representations. On the other hand, only one half-BPS block, with double-trace dimension $\Delta=p+q$, is expected physically to contribute to the disconnected correlator. Consistency thus requires this half-BPS block to be equal to the sum over all the vanishing-Casimir contributions to $f^{(0)}$ and $H^{(0)}$, and the non-identity contribution to $k$: \[B multiplet\] \_[0,]{}\^[r,s]{} = {
[l]{} k=1, f(z,)= \_\^[-2]{} k\_[1+]{}\^[r,s]{}(z) k\_[-]{}\^[-r,-s]{}() ,\
H(z,|[z]{},, |)= \_[i=(|r|,|s|)]{}\^[-4]{}\_[j=0]{}\^[(-i)/2]{} G\_[j,i+j+4]{}\^[r,s]{}(z,|[z]{}) Z\_[j,i+j]{}\^[r,s]{}(,).
. Since this holds for any $\Delta=p+q$, this must be equal to the half-BPS block for any $\Delta$.
For half-BPS blocks, a simple alternative is to use formulas from [@Dolan:2001tt] (see also appendix A of [@Basso:2017khq]) which express the blocks as a finite sum of bosonic blocks, according to the finite list of primaries contained within the supermultiplet. The formulas there are given for identical external operators but are easy to generalize, since the list of primaries is the same and the coefficients can be fixed from the Ward identity in eq. . Performing this exercise we obtain $$\begin{aligned}
\mathcal{B}_{0,\Delta}^{r,s}&= G_{0,\Delta}^{r,s} Z_{0, \Delta}^{r,s} \nonumber \\
&+ \zeta(\Delta) G_{1,\Delta+1}^{r,s} Z_{1,\Delta-1}^{r,s} + \zeta(\Delta) \zeta(\Delta+2) G_{2,\Delta+2}^{r,s} Z_{0,\Delta-2}^{r,s} \nonumber \\
&+ \zeta(\Delta-2) \zeta(\Delta) G_{0,\Delta+2}^{r,s} Z_{2,\Delta-2}^{r,s} + \zeta(\Delta-2)\zeta(\Delta)\zeta(\Delta+2) G_{1,\Delta+3}^{r,s} Z_{1,\Delta-3}^{r,s} \nonumber \\
&+ \zeta(\Delta-2) \zeta^2(\Delta) \zeta(\Delta+2) G_{0,\Delta+4}^{r,s} Z_{0,\Delta-4}^{r,s},
\label{Eq:HalfBPS}\end{aligned}$$ where $\zeta(\Delta) = \frac{(\Delta^2 - r^2)(\Delta^2-s^2)}{2^4 \Delta^2 (\Delta^2 -1)}$. The block in the first line is always present and just has the quantum numbers of $\mathcal{B}_{0,\Delta}$. The remaining lines account for superconformal descendants. Because of the coefficient $\zeta(\Delta)$, the second line should only be kept for $\Delta \geq 2 + \min(|r|,|s|)$, while the last two lines only contribute for $\Delta \geq 4+ \min(|r|,|s|)$. (For another approach to superconformal blocks, see [@Aprile:2017xsp].) Although eq. was derived using the disconnected correlator (where $r=s$), we find that this expression is precisely equal to eq. (\[Eq:HalfBPS\]) for the large set of values of $\Delta$ and $r\neq s$ that we have verified. We believe that this equality is an exact mathematical identity.
The analog of eq. (\[B multiplet\]) for semi-short and quarter-BPS blocks is discussed in appendix \[Appendix susy blocks\]. The fact that the data obtained from the inversion integral organizes into a super-OPE with only double trace dimensions provides a very non-trivial consistency check.
$\Delta^{(8)}$: A remarkable eighth-order differential operator {#Sec:useful operator}
---------------------------------------------------------------
The combination $\Delta^{(8)}$ in eq. (\[Eq:Delta8\]) will play an important role below. It depends only on Casimir invariants like $h(h-1)$ and can in fact be viewed as a eight-order differential operator: \^[(8)]{} H\_[{p\_i}]{} ( \_z -\_)( \_z -\_)( \_ -\_)( \_ -\_) H\_[{p\_i}]{}, \[delta8 diff op\] where \_x x\^2\_x(1-x)\_x -12(r+s)x\^2\_x -14 rs x \[Casimir\] with $r=p_{21}$, $s=p_{34}$. Acting on the product $G_{\ell,\Delta+4}(z,\zb)Z_{m,n}(\a,\ab)$ this has precisely the eigenvalue in eq. (\[Eq:Delta8\]). Its presence in the denominator of eq. (\[Eq:a0pqqp\]) suggests that $\Delta^{(8)}H^{(0)}$ is a simple function; indeed by looking at individual cases we find that: \^[(8)]{} H\^[(0)]{}\_[pqqp]{} = ( + \_[p,q]{})C(p)C(q) \[Delta8 H0\]. This is precisely the form of a disconnected correlator of complex scalar primaries with dimensions $p_i+2$ and $R$-symmetry $[0,p_i{-}2,0]$, up to a normalization $C(p)=p^2(p^2-1)$, explaining the form of eq. (\[Eq:a0pqqp\]).
The operator $\Delta^{(8)}$ has two important properties. First, it annihilates all the protected multiplets, as listed in table \[Tab:Multiplets\]. Second, $\Delta^{(8)} H_{\{p_i\}}$ generally transforms under $z\mapsto z/(z-1)$ crossing like the correlator of scalar primaries with the mentioned quantum numbers, which are precisely those of the superconformal descendant $\mathcal{L}^{p_i-2}\equiv Q^4\mathcal{O}^{p_i}$. (The $\mathcal{L}^{p_i-2}$ are primaries with respect to bosonic isometries.) We thus identify $\Delta^{(8)} H_{\{p_i\}}$ as the correlator of the product $\mathcal{L}^{p_1-2}\mathcal{L}^{p_2-2}$ with its complex conjugate: indeed this correlator has to annihilate protected multiplets since the first pair is proportional to $Q^8$. This generalizes the $p=2$ case studied in [@Drummond:2006by], where $\mathcal{L}^{0}$ was the chiral Lagrangian density.
The correlators with different $p$’s will be combined into a single ten-dimensional dilaton amplitude in section \[sec:10D\].
Tree-level correlator at strong coupling: order $1/c^{1}$ {#Sec:Tree-level}
=========================================================
At order $1/c$ in the limit of strong ‘t Hooft coupling, the correlator can be computed in principle in terms of tree-level Witten diagrams in IIB supergravity on AdS${}_5\times$S${}_5$ [@Witten:1998qj]. The present formalism offers a powerful alternative which relies only on CFT ideas. The only input from supergravity is the assumption that all the single-trace operators are the half-BPS ones. The double-discontinuity of the correlator comes exclusively from their exchange in the cross-channels, which can be computed easily. The inversion integral then reconstructs the full tree-level correlator, in the form of its OPE data, as depicted in fig. \[Fig:crossing symmetry\].
The double-discontinuity is computed explicitly by inserting in the cross-channel the half-BPS blocks given in eq. (\[Eq:HalfBPS\]) or equivalently eq. (\[B multiplet\]), together with the crossing relation $$G_{p_1p_2p_3p_4}(z,\bar{z},\alpha, \bar{\alpha}) = \left( \frac{z \bar{z}}{\alpha \bar{\alpha}} \right)^{\frac{p_1+p_2}{2}} \left( \frac{(1-\alpha)(1-\bar{\alpha})}{(1-z)(1-\bar{z})} \right)^{\frac{p_2+p_3}{2}} G_{p_3p_2p_1p_4}(1-z,1-\bar{z},1-\alpha,1-\bar{\alpha}).
\label{Eq:Crossing}$$ If using eq. (\[B multiplet\]), it is important to note that the different supersymmetry components $f,k,H$ in the decomposition (\[G ansatz\]) mix under crossing: to find $H$ in one channel one needs all of $f,k,H$ in the cross-channel.
Fixing three-point coefficients from crossing symmetry {#ssec:3points}
------------------------------------------------------
For the correlator of eq. (\[Eq:OriginalCorrelator\]), the possible half-BPS states exchanged in the t-channel are the representations of the form $\big[0,\Delta,0\big]$ in the intersection $(\big[0,p_1,0\big] \times \big[0,p_4,0\big]) \cap (\big[0,p_2,0\big] \times \big[0,p_3,0\big])$. From the tensor product formula in eq. (\[Eq:TensorProductStates\]) one finds the allowed range (|p\_1-p\_4|,|p\_2-p\_3|) (p\_1+p\_4,p\_2+p\_3), \[range half BPS\] where the allowed values of $\Delta$ differ from the endpoints by an even integer.
We will now demonstrate that, at order $1/c$, the three-point function of single-trace half-BPS operators is uniquely fixed by crossing symmetry, to take the well-known value [@Lee:1998bxa]: f\_[pqr]{} = , |p-q|+2 r p+q-2. \[f coeff\] The coefficients $f_{pqr}$ with $r=|p-q|$ or $r=p+q$ can be set to zero without loss of generality by a suitable choice of basis for the “single-trace” operators. One simply chooses them to be orthogonal to all multi-traces. This can always be done by including suitable multiple of the latter in eq. (\[def Op\]), see for example [@Aprile:2018efk]. (This assumes simply the existence of linearly independent half-BPS double-trace operators $[\mathcal{O}^p \mathcal{O}^q]$ for each $p,q\geq 2$, which follows from the $1/c^0$ analysis of the preceding section. One also needs them to be linearly independent from the single traces $\mathcal{O}^{p+q}$, which is physically expected at large $N$.)
Our proof of eq. (\[f coeff\]) uses only the chiral part of the correlator. We illustrate the method starting with the simplest correlator, $f_{2222}$. The only half-BPS block in the range (\[range half BPS\]) which contributes to its double-discontinuity is the stress-tensor multiplet $\Delta=2$, since the $\Delta=4$ block has double-trace dimensions and therefore no singularity. This block is given as \_[0,2]{}(z,,,) = 1+. Only this block and the identity contribute to the double discontinuity of $f$. We insert them into the crossing relation (\[Eq:Crossing\]), extract the $f$ part by setting $\alpha=z$ (see eq. (\[kf from G\])), and focus on the polar terms as $\zb\to 1$:
f\_[2222]{}(,) &=\
&= k\_0\^[(0,0)]{}() + k\_[-1]{}\^[(0,0)]{}(),
where on the second line $y=\frac{1-\zb}{\zb}$ and we have decomposed the $\ab$ dependence into eigenfunctions. Plugging this into the inversion integral (\[f Inversion\]) then gives the *full* OPE decomposition for the chiral part: \[OPE f2222\] f\_[2222]{}(,) = \_[j=0]{}\^(1+(-1)\^j) (
[l]{} ( + ) r\^[(0,0)]{}\_[j+1]{} f\_[j,0]{}(,)\
+ ( (j[+]{}1)(j[+]{}2)+)r\^[(0,0)]{}\_[j+2]{}f\_[j,2]{}(,)
), where $f_{j,m}$ is the function defined in eq. (\[C2 susy\]), and the $(-1)^j$ terms account for the $u$-channel contribution (obtained generally by permuting operators $1$ and $2$, which does nothing here). This OPE can be easily resummed, as discussed shortly. However, what is most important here is to project it into its half-BPS contributions, which is simply the $j=0$ term:
f\_[2222]{}|\_[1/2-BPS]{} &= (2+ ) f\_[0,0]{}(,) + (2+ ) f\_[0,1]{}(,)\
&= f\_[222]{}\^2 \_[0,2]{} + (2+) \_[0,4]{}, \[f2222 crossing\]
where we have used from eq. (\[B multiplet\]) that the $f$ part of $\mathcal{B}_{0,2} = f_{0,0}$ and $\mathcal{B}_{0,4} = f_{0,0}+f_{0,1}$.
The second line of eq. (\[f2222 crossing\]) is the main result of this exercise. It shows that, if one inserts the stress tensor block $\mathcal{B}_{0,2}$ in the cross-channels with coefficient $f_{222}^2$, the resulting double-discontinuity forces it to reappear in the $s$-channel with the same coefficient. Thus crossing can be satisfied for any value of $f_{222}$. For the present purposes, its value, $f_{222}=\sqrt{2/c}$ from eq. (\[f coeff\]), is simply a definition of the central charge $c$. The double-trace block $\mathcal{B}_{0,4}$ then has a fixed coefficient.
We now proceed similarly for the $2323$ correlator, and find that the double-discontinuity of the $t$- and $u$-channel OPE force its half-BPS part to be f\_[2323]{}|\_[1/2-BPS]{} = ( 13f\_[222]{}f\_[233]{} + f\_[233]{}\^2)\_[0,3]{} + (1+ 13 f\_[222]{}f\_[233]{} + 19 f\_[233]{}\^2)\_[0,5]{}+O(1/c\^2). \[f2323 crossing\] Focusing on the first term, which is the only single-trace block, the $s$-channel OPE is satisfied only if the coefficient of $\mathcal{B}_{0,3}$ is $f_{233}^2$. Thus crossing symmetry requires that f\_[233]{}(f\_[222]{} - f\_[233]{}) =O(1/c\^2), \[f233 eq\] which has a nontrivial solution, $f_{233}=\frac{3}{2}f_{222}$, in precise agreement with eq. (\[f coeff\]). The trivial solution $f_{233}=0$ could be ruled out using the stress tensor Ward identities: the $\mathcal{O}^2$ multiplet contains the stress tensor, whose coupling to any pair of identical scalars is nonzero and proportional to their dimension (consistent with the nontrivial solution). More generally, the $2p2p$ crossing relation requires that f\_[2pp]{}(f\_[222]{} - f\_[2pp]{} ) = O(1/c\^2), \[f2pp eq\] which has the nontrivial solution $f_{2pp} = p/2 f_{222}$ in agreement with both eq. (\[f coeff\]) and the stress tensor Ward identity. We find empirically that it seems possible to rule out the trivial solutions without relying on the Ward identities: for example, setting $f_{233}=0$ (but not the other ones) would eventually lead to a crossing equation with no solution for a higher correlator (namely $5566$).
We include $O(1/c^2)$ errors in the above, because double-trace operators, such as semi-short ones, generally contribute to the double-discontinuity at this order.
Further coefficients can be fixed by looking at more correlators. For example, the $3p3p$ correlators gives two equations each, from the coefficients of the two single-trace blocks $\mathcal{B}_{0,p-1}$ and $\mathcal{B}_{0,p+1}$. From these we can uniquely fix the $f_{3p\,p+1}$ and $f_{4pp}$ three-point functions for all $p$. We find that the solution is always unique (up to an overall sign ambiguity for each operator: $\mathcal{O}^p\mapsto \pm \mathcal{O}^p$, which we fix by taking all three-point functions to be positive).
From the $4p4p$ correlators one similarly fixes all $f_{4p\,p+2}$ and $f_{6pp}$ three-point functions, then from $5p5p$ with $p\geq 5$ one can fix $f_{5p\,p+1}$, $f_{5p\,p+3}$ and $f_{8pp}$, and so on. There appears to be a simple pattern: the $qpqp$ family with a given $q$, once families with lower $q$’s have been used, fixes all the $f_{qmn}$ coefficients in the range of eq. (\[f coeff\]) as well as $f_{2q{-}2\,pp}$.
We conclude that, under the assumption that the only single-trace operators are the half-BPS ones (as expected at strong ‘t Hooft coupling), crossing symmetry at order $1/c$ uniquely determines the three-point functions to have the values in eq. (\[f coeff\]) at this order. A full analysis of the crossing relations at finite $c$ would be very interesting but is beyond our scope. See [@Beem:2016wfs] for further results.
Chiral (protected) correlator: deducing the free correlator {#ssec:free}
-----------------------------------------------------------
Interestingly, from this solution for the chiral correlator obtained at strong coupling (this is how we justified having all single-traces operators being half-BPS), we can recover Feynman diagrams of the free theory!
Consider again the protected correlator $f_{2222}$, given by the OPE in eq. (\[OPE f2222\]). The infinite sum over $j$ can be computed easily if one puts in the expectation that the result is $1/(1-\zb)^2$ times a polynomial, which can be found using a small number of terms. We obtain: f\_[2222]{}(z,) = + + + 1c . \[f2222\] This does not look very suggestive but can be illuminated by comparing with the free theory. Due to nonrenormalization theorems [@Seiberg:1993vc; @Eden:2000bk; @Bissi:2015qoa], the protected quantity $f$ should be independent of the ‘t Hooft coupling and therefore computable from the *free theory* limit (small ‘t Hooft coupling) of the correlator.
In the free theory the correlator is a sum of Wick contractions which give products of $y_{ij}^2/x_{ij}^2$ factors. In terms of cross-ratios (see eq. (\[Eq:OriginalCorrelator\])), the free correlator thus depends only on two variables: $\mathcal{G}^{\rm free}_{pqrs}(u/\sigma, v/\tau)$. Given that $f(z,\a)$ is itself a two-variable function, it is actually possible to go the other direction! Indeed, we can solve eq. (\[kf from G\]) for the former in terms of $f$ and $k$: \^[free]{}\_[pqrs]{} = f\_[pqrs]{}(z,)|\_[z= , =]{} + k\_[pqrs]{} ()\^[12[max]{}(|p\_[12]{}|,|p\_[34]{}|)]{} ()\^[12[max]{}(p\_2+p\_3-p\_1-p\_4,0)]{}. \[Gfree from f\] The unit part $k$ can be computed generally by summing the coefficients of all $\mathcal{B}_{0,\Delta}$ blocks in the OPE (including the double-trace one), plus the identity block. For example, eq. (\[f2222 crossing\]) gives $k_{2222}=3+\frac{3}{c}$. Plugging the innocuous-looking eq. (\[f2222\]) into eq. (\[Gfree from f\]), we then get the free correlator: \_[2222]{}\^[free]{} = 1+ + + 1c ( + + ). This precisely matches with computing Wick contractions at weak coupling! Note that this computation leveraged non-renormalization theorems starting from the limit of *strong* ‘t Hooft coupling.
It is straightforward to repeat these stops for other correlators, for example $$\begin{aligned}
\mathcal{G}^{\rm free}_{22pp} &= 1+ \frac1c\left(\frac{p}2 \frac{u}{\sigma} + \frac{p}2\frac{u\tau}{\sigma v} + p\frac{u^2\tau}{\sigma^2v}\right) +O(1/c^2)
\qquad (p>2),
\\
\mathcal{G}^{\rm free}_{3333} &= 1+\frac{u^2}{\sigma^2} + \frac{u^2\tau^2}{\sigma^2v^2}
+ \frac{9}{4c} \left(\frac{u^2}{\sigma^2}+ \frac{u\tau}{\sigma v}+ 2\frac{u^2\tau}{\sigma^2v} + \frac{u^2\tau^2}{\sigma^2v^2}+ \frac{u^3\tau^2}{\sigma^3v^2} + \frac{u^3\tau}{\sigma^3v} \right) + O(1/c^2).\end{aligned}$$ We found that these formulas agree precisely with the result of summing Wick contractions. For $p\geq 4$ it is important to define external “single-trace” operators to be orthogonal to all double-trace ones, reflecting the choice of basis stated below eq. (\[f coeff\]). For the reader’s convenience, a general formula for the free correlator at order $1/c$ is recorded in eq. (\[Gfree Appendix\]).
Reduced correlator: Anomalous dimensions at order $1/c$
-------------------------------------------------------
Let us now turn to our main target, the correlator at order $1/c$ for external operators with general $R$-symmetry representations. This gives information on both the $1/c$ anomalous dimensions of double-trace operators and $1/c$ corrections to their OPE coefficients.
Our input again is the double-discontinuity written as a sum of half-BPS blocks with coefficients in eq. (\[f coeff\]), and crossing relation (\[Eq:Crossing\]). Explicitly, it is:
\_[{p\_i}]{}= ( )\^ ( )\^ \_[=(|p\_[23]{}|,|p\_[14]{}|)+2]{}\^[(p\_2+p\_3,p\_1+p\_4)-2]{} \_\^[p\_[23]{},p\_[14]{}]{}(z’,’,’,’) \[Eq:CorrelatordDisc\]
where $x'=(1-x)$. To get information on unprotected operators we first extract the $H$ part following the SUSY decomposition (\[G ansatz\]); in the context of the $2222$ correlator (discussed at length in [@Alday:2017vkk]), this gives, for reference, H\_[2222]{}\^[(1)]{} = where we have dropped all terms with no poles at $v=0$.[^4] We then plug into the Lorentzian inversion integral (\[Eq:cInversion 2\]).
Loosely speaking, the $\log u$ term informs us about anomalous dimensions, whereas the constant term corrects in addition the OPE coefficients. We begin with the former. Converting to $x=\frac{z}{1-z}$ and $y=\frac{1-\zb}{\zb}$ variables, the above gives H\_[2222]{}\^[(1)]{}|\_[u]{} = -2x\^3 and the inversion integral (\[Eq:cInversion 2\]) gives straightforwardly, using formulas (\[Eq:zbarIntegral\]) and (\[Eq:zIntegral\]) for the $x$ and $y$ powers, a\^[(0)]{}\^[(1)]{}(h,,0,0) \_[2222]{} = -2(h+1)(h)(h-1)(h-2) r\^[0,0]{}\_h r\^[0,0]{}\_Z\_[0,0]{}\^[0,0]{}. This is in agreement with [@Alday:2017vkk]. The bracket notation indicates that the result is generally the sum of multiple nearly-degenerate operators. The set of correlators really gives a matrix of anomalous dimensions matrix, whose eigenvalues are the actual anomalous dimensions.
The operators which can mix with each other have the same R-symmetry representation $[m,n-m,m]$ and twist $\tau=\Delta-\ell$. To obtain their matrix element in the basis of disconnected theory operators, we need to divide by the disconnected OPE data in eq. (\[Eq:a0pqqp\]), and since $ \langle a^{(0)} \rangle_{pqqp} = \langle a^{(0)} \rangle_{qppq}$, we have: $$\gamma^{(1)}_{pq,rs}\equiv \frac{\langle a^{(0)} \gamma^{(1)} \rangle_{pqrs} }{\sqrt{\langle a^{(0)} \rangle_{pqqp} \langle a^{(0)} \rangle_{rssr}}} = \frac{\langle a^{(0)} \gamma^{(1)} \rangle_{pqrs}^t + (-1)^{\ell+m} \langle a^{(0)} \gamma^{(1)} \rangle_{qprs}^t }{\sqrt{\langle a^{(0)} \rangle_{pqqp} \langle a^{(0)} \rangle_{rssr} }},
\label{Eq:anomDim}$$ where the $t$ and $u$ superscripts denote the t and u-channel exchanges respectively. For a given twist and R-symmetry representation, one can find the list of pairs $(p,q)$, with $p\leq q$, for which double-trace operators $[pq]_{\ell,\Delta,m,n}$ exist, and construct the matrix in eq. (\[Eq:anomDim\]). Its eigenvalues are then the *actual* OPE coefficients of the double-trace operators.
To perform the computation, we wish to organize our data in terms of twist $\tau$ and R-symetry representations labelled by $\big[m, n-m, m\big]$. One must also consider separately the cases where the spin $\ell$ is either even or odd. For a given twist and representation, one must find the possible scaling dimensions of the operators in the four-point function. One can simply consider the constraints on the three-point vertices. Given the possible $(pqrs)$ scaling dimensions and values of twist, R-symmetry Dynkin labels and spin, one can diagonalize the corresponding matrix.
We will change our notation to write the anomalous dimension matrix $\left( \gamma^{(1)} \right)_{\tau,m,n}^{\ell}$ labelled by the twist $\tau$, R-symmetry labels $m,n$ and spin $\ell= \{+,-\}$ corresponding to either even or odd spin. For example, let us consider the nontrivial representation $\big[0,2,0\big]$. The minimal twist associated with this representation is $\tau=6$. For odd spin, the only three-point vertex is given by external operators $(p,q)=(2,4)$, in which case it is straightforward to diagonalize. For even spin, one has $(2,4)$ and $(3,3)$. The corresponding matrix is $$\begin{pmatrix}
\dfrac{\langle a^{(0)} \gamma^{(1)} \rangle_{2442}^t + (-1)^{\ell} \langle a^{(0)} \gamma^{(1)} \rangle_{4242}^t }{\sqrt{\langle a^{(0)} \rangle_{2442} \langle a^{(0)} \rangle_{2442}} } & \dfrac{\langle a^{(0)} \gamma^{(1)} \rangle_{2433}^t + (-1)^{\ell} \langle a^{(0)} \gamma^{(1)} \rangle_{4233}^t}{\sqrt{\langle a^{(0)} \rangle_{2442} \langle a^{(0)} \rangle_{3333}} } \\
\dfrac{\langle a^{(0)} \gamma^{(1)} \rangle_{3342}^t+ (-1)^{\ell} \langle a^{(0)} \gamma^{(1)} \rangle_{3342}^t}{\sqrt{\langle a^{(0)} \rangle_{3333} \langle a^{(0)} \rangle_{2442}} } & \dfrac{\langle a^{(0)} \gamma^{(1)} \rangle_{3333}^t+ (-1)^{\ell} \langle a^{(0)} \gamma^{(1)} \rangle_{3333}^t}{\sqrt{\langle a^{(0)} \rangle_{3333} \langle a^{(0)} \rangle_{3333}} }
\end{pmatrix}.$$
Consider the $2442$ component. One finds that the only possible R-symmetry representation is $\big[0,2,0\big]$. The reduced correlator inversion integral yields $$\begin{aligned}
c(\ell,\Delta,0,2) &=
\int_0^1 \frac{dz}{z^2}(1-z)^2 k_{1-h}^{2,2}(z) z^{-1} \int_0^1 \frac{d\bar{z}}{\zb^2} (1-\zb)^2 \kappa(\hb) k_{\hb}^{2,2}(\zb) \zb^{-1} \nonumber \\
& \times \left[2 \left( \frac{z}{1-z} \right)^3 - 4 \left( \frac{z}{1-z} \right)^4 - 4 \left( \frac{z}{1-z} \right)^5 \log(z) \right]
\dDisc\left( \frac{\zb}{1-\zb} \right)^3 \end{aligned}$$ where we have omitted non-singular terms. This formula contains corrections to the OPE coefficients $a^{(1)}$ and anomalous dimensions $\gamma^{(1)}$. Focusing now on the latter, which are related to double-poles with respect to $h$, they can be obtained simply by integrating the coefficient of $\log(z)$, see eq. .
Using eqs. -, the above matrix with $h=1+\frac{\tau}{2}$ gives (\^[(1)]{})\^+\_[6,0,2]{}=
12+(-1) & 6\
6 & 6+(-1)
\[02 matrix\] which has the two eigenvalues: $$\lambda_{6,0,2}^+ = \left\{-\frac{60(\hb+2)}{(\hb-4)(\hb-1)(\hb)},\
-\frac{60(\hb-3)}{(\hb-1)(\hb)(\hb+3)}\right\}
= \Big\{\frac{-\Delta_{0,2}^{(8)}(4,\hb)}{(\hb-4)_6},\ \frac{-\Delta_{0,2}^{(8)}(4,\hb)}{(\hb-2)_6}\Big\}
$$ where $(...)_6$ is the Pochhammer symbol and $\Delta^{(8)}$ is the polynomial introduced in eq. (\[Eq:Delta8\]). This way of writing the result was suggested by a recent conjecture by [@Aprile:2018efk], discussed shortly. For $\tau=8$, again for the $\big[0,2,0\big]$ representation, the matrix is somewhat bigger: $$\begin{pmatrix}
(2442) & (2433) & (2435) & (2444) \\
(3342) & (3333) & (3335) & (3344) \\
(3542) & (3533) & (3553) & (3544) \\
(4442) & (4433) & (4453) & (4444)
\end{pmatrix}.$$ For odd spin, one removes the second and last rows and columns. We then find the eigenvalues, for even and odd spins:
\_[8,0,2]{}\^- &= -\_[0,2]{}\^[(8)]{}(5,) { , },\
\_[8,0,2]{}\^+ &= -\_[0,2]{}\^[(8)]{}(5,) { , , , }.
Note that in the even spin case, the second and third eigenvalues are degenerate.
It is amazing that the eigenvalues of nontrivial matrices give rational functions of $\hb$.
Reciprocity symmetry, $\hb\to 1-\hb$, is preserved in an interesting way. The matrix elements in eq. (\[02 matrix\]) are all invariants under $\hb\to 1-\hb$ (this is always true and trivially follows from the form of the integrals (\[Eq:zbarIntegral\])-(\[Eq:zIntegral\])). The individual eigenvalues are not invariant, but the *set* of eigenvalues is, as it should: reciprocity interchanges the first and last eigenvalue.
One can continue this process to obtain anomalous dimension eigenvalues in different R-symmetry representations and higher twist which would involve diagonalizing larger dimensional matrices. A conjecture for the outcome was formulated recently in [@Aprile:2018efk] (who obtained it by OPE-decomposing a Mellin-space formula for the general $pqrs$ correlators previously conjectured in [@Rastelli:2016nze]). These authors conjectured that the eigenvalues always take the form: $$\gamma^{(1)}_{pqrs} (\tau,\hb,\ell, m,n) = - \frac{1}{c} \frac{\Delta_{m,n}^{(8)}(1+\frac{\tau}{2},\hb)}{\left(\ell+m+1 + \nu \right)_6}
\label{Eq:Anomalous Dimension}$$ where $\ell= \hb - \frac{\tau}{2}-2$ is the angular momentum and $\nu$ is an integer shift that distinguishes different eigenvalues of the matrix.
Generically, diagonalizing the anomalous dimension matrix will yield degenerate states. This degeneracy can be worked out experimentally. We find that the allowed shifts are $$\begin{aligned}
\nu &= \displaystyle \frac{1-(-1)^{\ell+m}}{2}+2u, \quad 0\leq u\leq \tfrac{1}{2}(\tau-2m-5+ \delta) \label{multiplicity shift} \\
\delta &= \begin{cases}
(-1)^{\ell+m} & \text{if} \mod(n-m,2)=0 \\
0 & \text{otherwise}.
\end{cases}\end{aligned}$$ The multiplicity for a given shift is then $$\text{min}\left(u+1, \tfrac{1}{2} \left(n-m+1+ \delta \right), \tfrac{1}{2}\left( \tau-2m-3+\delta \right)-u \right).
\label{multiplicity}$$ The first case sets the degeneracy for the first few cases, which increases incrementally from 1 until it reaches its maximum multiplicity given by the second condition. If the twist is large enough, the multiplicity will plateau until the third condition becomes relevant, after which it decreases symmetrically. In all cases, the multiplicities add up to the dimension of the mixing matrix (which is equal to the number in the second condition, times $\frac{\tau-m-n-2}{2}$).
Consider for example the $\big[2,12,2\big]$ representation with twist up to $\tau=20$ and $\tau=50$. One can find the multiplicity of the degenerate states in fig. \[Fig:multiplicity\]. This matches with the finding in ref. [@Aprile:2018efk], where the allowed states were nicely organized into a $45^\circ$-rotated rectangle in the $(p,q)$ plane.
[0.45]{} ![Even and odd spins are in blue and orange respectively. On the left is the multiplicity for $\tau=20$ and on the right is the multiplicity for $\tau=50$ of the same R-symmetry representation. Given eq. , the maximal multiplicity for \[2, 12, 2\] is 7 and 6 for even and odd spin respectively.[]{data-label="Fig:multiplicity"}](multiplicity.png "fig:"){width="\textwidth"}
[0.45]{} ![Even and odd spins are in blue and orange respectively. On the left is the multiplicity for $\tau=20$ and on the right is the multiplicity for $\tau=50$ of the same R-symmetry representation. Given eq. , the maximal multiplicity for \[2, 12, 2\] is 7 and 6 for even and odd spin respectively.[]{data-label="Fig:multiplicity"}](multiplicity-large-twist "fig:"){width="\textwidth"}
From this analysis, we see that the Pochhammer term serves as a useful quantity to resolve the double trace mixing problem. This observation will be recycled in the next section.
An accidental 10-dimensional conformal symmetry {#sec:10D}
===============================================
In this section we propose an explanation for the remarkable conjecture (\[Eq:Anomalous Dimension\]): that it originates from the conformal flatness of the AdS${}_5\times $S${}_5$ geometry and an accidental SO(10,2) conformal symmetry of the supergravity four-point amplitude. This will lead to new conjectures which we will test.
A mysterious equality with 10-dimensional $S$-matrix partial waves
------------------------------------------------------------------
Our main inspiration will be the following empirical observation: the conjectured anomalous dimension is equal to a quantity computed in *flat* ten-dimensional space. We consider the $2\to 2$ scattering of identical complex axi-dilatons in IIB supergravity: A\_[10]{}\^[tree]{} = 8G\_N .\[Eq:10D-amplitude\] Below we’ll use that $8\pi G_N=\frac{\pi^5L^8}{c}$ with $c=\frac{N^2-1}{4}$ in the AdS${}_5\times $S${}_5$ context, to express this in terms of $c$ and the AdS radius $L$. We now consider the partial-wave decomposition of this amplitude; in general dimension this can be written using Gegenbauer polynomials (see for example [@Giddings:2007qq]) as: $$\label{S matrix partial waves}
i A_{10}(s, \cos\theta) =
\frac{2^{d} \pi^{\frac{d-2}{2}}}{s^{\frac{d-4}{2}}\Gamma\big(\frac{d-2}{2}\big)}
\displaystyle \sum_{\ell>0, \text{ even}} (\ell+1)_{d-4}(2\ell+d-3) A_\ell(s) \ \tilde{C}^{(\frac{d-3}{2})}_\ell(\cos\theta) ,$$ where $\tilde{C}^{(\lambda)}_\ell(x)= C^{(\lambda)}_\ell(x)/C^{(\lambda)}_\ell(1)$ is a Gegenbauer polynomial normalized to unity at 1, the scattering angle is $\cos(\theta)=1+\frac{2t}{s}$, and we should put $d=10$ in the above. The normalization factors are such that $A_\ell(s)=1$ for the disconnected part of the $S$-matrix of two identical particles. Retaining the disconnected part and computing the $A_\ell$ using the orthogonality relation[^5] we find: A\_(s) = 1 + i . \[Eq:10D-partial-wave\] This should be compared with the exponential of the anomalous dimension in eq. (\[Eq:Anomalous Dimension\]), which controls the phase of each conformal block in the bulk point limit (see for example [@Heemskerk:2009pn; @Maldacena:2015iua]): e\^[-i]{} = 1 + i + (1/c). \[Eq:4D-partial-wave\] The resemblance of the phases (\[Eq:10D-partial-wave\]) and (\[Eq:4D-partial-wave\]) is too striking to be an accident and this suggests a direct relation between the four-dimensional theory and the *flat* ten dimensional supergravity. We will not fully derive such a relation from first principles here, but we will try to guess what form it could take and deduce precise implications.
Statement of the conjecture
---------------------------
To explain how an observable of 10-dimensional supergravity on AdS${}_5\times$S${}_5$ might be entirely determined by the $S$-matrix of the same theory on *flat* space, we propose a scenario based on the following two observations:
- The AdS${}_5\times$S${}_5$ metric is conformally equivalent to flat space. To check this, write the 10-dimensional flat space metric using radial and angular variables for the 6 extra dimensions, with $d\Omega_5^2$ is the metric on $S_5$:
ds\^2\_[Minkowski]{} &= \_[=0]{}\^3 dx\^dx\_+ dr\^2 + r\^2 d\_5\^2\
&= r\^2 ( + d\_5\^2) ds\^2\_[[AdS]{}\_5\_5]{}.
- By a happy coincidence, the combination $G_N \delta^{16}(Q)$ is dimensionless. Divided by this “coupling”, the four-point tree amplitude (\[Eq:10D-amplitude\]) is in fact conformally invariant: K\_ =0. This can be checked using the conformal generators in momentum space, K\_= \_[i=1]{}\^4 ( 2 -p\_i\^ - ).
In effect, we will now play a game where we pretend that IIB supergravity is a conformal theory and work out the implications for four-point correlators. In particular, we should find an action of the ten-dimensional conformal group SO(10,2).
This SO(10,2) contains as a subgroup the SO(4,2)$\times$ SO(6) bosonic isometries of AdS${}_5\times$S${}_5$, so a natural action can be defined by building 12-vectors $w_i$: w\_i(X\_i,y\_i). Here $X_i$ is the embedding coordinate corresponding to the space-time point $x_i^\mu$, on which SO(4,2) acts, and $y^i$ is the null 6-vector introduced in eq. (\[cross-ratios sigma tau\]) and on which the SO(6) $R$-symmetry acts.
Motivated by the above, we conjecture that all tree-level four-point correlators arise from a single SO(10,2)-invariant object.
To state this more explicitly we consider the correlator of Lagrangian insertions $\mathcal{L}^{p_i-2}$ defined in section \[Sec:useful operator\] (which are susy descendent $Q^4 \mathcal{O}^{p_i}$ of scaling dimension $2+p_i$), since these can be easily identified with axi-dilatons in ten dimensions. Indeed, the simplest case, $p_i=0$, has scaling dimension 4 which is the correct value for a free field in ten dimensions. The operators with $p_i> 0$ are then identified to be its S${}_5$ spherical harmonics. To extract a given spherical harmonic we simply need to extract the component of a ten-dimensional expression with the correct homogeneity degree in the null vectors $y_i$: \#1[[\_[\#1]{}]{}]{} 0| \^(x\_1) \^(x\_2) \^(x\_3) \^(x\_4)|0 = (w\_1) (w\_2) |(w\_3) |(w\_4) \_[10]{} \_[y\_1\^y\_2\^y\_3\^y\_4\^]{}, \[LLLL from 10D\] with $\tp{i}=p_i-2$. The expectation value on the right is formal since we haven’t defined a 10D CFT (and won’t define one). It simply denotes a function of the four points $w_i$ which transforms like a 10D CFT correlator of four scalars with scaling dimension 4, that is: (w\_1) (w\_2) |(w\_3) |(w\_4) \_[10]{} \[10d four scalars\] where u\_[10]{} ,v\_[10]{} are the then-dimensional cross-ratios.
Let us now write the prediction in eq. (\[LLLL from 10D\]) in terms of cross-ratios, by dividing the left-hand-side by the factor in eq. (\[Eq:OriginalCorrelator\]) (with the $p_i$ shifted by $\pm 2$ for the $x$ and $y$ dependence, respectively). We can extract the component of correct homogeneity degree in $y$ by performing contour integrals in auxiliary variables $a_i$ introduced via the rescaling $y_{ij}^2\equiv y_i\cdot y_j \mapsto a_i a_j y_{ij}^2$. With a suitable rescaling of the $a$’s this gives a formula with cross-ratios only: $$\begin{aligned}
\tilde{H}_{p_1 p_2 p_3 p_4}(u,v,\sigma,\tau) &=
\oint \prod_{i=1}^4 \left[\frac{da_i\,a_i^{1-p_i}}{2\pi i} \right]\ \frac{(u/\sigma)^{\frac{p_1+p_2}{2}-2}}{(1- \frac{\sigma}{u} a_1 a_2 )^4 (1- a_3 a_4)^4}
\nonumber\\&\quad \times G_{10}
\left(u\frac{(1-\frac{\sigma}{u} a_1 a_2)(1- a_3 a_4)}{(1- a_1 a_3)(1-a_2 a_4)}, v\frac{(1-\frac{\tau}{v} a_2 a_3)(1 - a_1 a_4)}{(1 -a_1 a_3)(1- a_2 a_4)} \right)
\nonumber\\ &\equiv \mathcal{D}_{p_1p_2p_3p_4}G_{10}(u,v).\label{Generating Function 10-4D}\end{aligned}$$ The last line emphasizes the fact that the contour integral really just defines a differential operator. In the first few cases for example we find:
\_[2222]{} &= 1, \[D examples\]\
\_[2332]{} &= - \_v,\
\_[2233]{} &= 4 - u \_u,\
\_[3333]{} &= 16 -8 u\_u + (u\_u)\^2 +2 u\_u v\_v + (v\_v)\^2.
Relation (\[Generating Function 10-4D\]) is the main result of this section: it expresses all tree-level correlators as derivatives of a single one, $G_{10}=\tilde{H}_{2222}$, the stress-tensor multiplet correlator. This is depicted in fig. \[Fig:AdS flat space\].
It remains to precisely define the left-hand-side $\tilde{H}_{2222}$. The correct guess is of course to be ultimately justified by explicit computations, as done in the next subsection, but it may be helpful to handwave-motivate the result here. Since the spherical harmonics of the ten-dimensional dilaton most naturally map to correlators of $\mathcal{L}^{p_i}$, the most natural guess would be $\tilde{H}=\Delta^{(8)}H$ as defined in section \[Sec:useful operator\]. *However*, to reveal its hidden conformal symmetry, we had to divide the 10D tree amplitude by the dimensionless “coupling” $G_N\delta^{16}(Q)/16$, which eqs. (\[Eq:10D-partial-wave\]) and (\[Eq:4D-partial-wave\]) suggest to identify as a multiple of $\Delta^{(8)}/c$. Thus a better guess is to divide the order $1/c$ dilaton correlator by $\Delta^{(8)}$ and thus consider $H$ itself, which turns out to be a correct prediction. We have to bear in mind that this division is ill-defined however unless one cancels the protected contributions killed by $\Delta^{(8)}$; a simple work-around is to subtract the free correlator. The correct guess for the functions $\tilde{H}$ which satisfies eq. (\[Generating Function 10-4D\]), including a normalization factor, thus turns out to be: \^[(0)]{}\_[pqrs]{} \^[(8)]{} H\^[(0)]{}\_[pqrs]{}, \^[(1)]{}\_[pqrs]{} (H\^[(1)]{}\_[pqrs]{}-H\^[(1)free]{}\_[pqrs]{}). \[def Htilde\] If this sequence were to continue at loop level supergravity, a natural form of the next term would be $\tilde{H}^{(2)}_{pqrs}=\frac{1}{\Delta^{(8)}}(H^{(2)}_{pqrs}-H^{(2)\rm free}_{pqrs})$, but we leave this for further study.
Eqs. (\[Generating Function 10-4D\]) and (\[def Htilde\]), independent of their origins, yield precise and testable relations among tree-level correlators.
Checking the conjecture from the calculated correlators {#ssec:tests}
-------------------------------------------------------
Let us first check the relation for the disconnected correlator $\tilde{H}^{(0)}$. According to eq. (\[Delta8 H0\]) G\_[10]{}\^[(0)]{}(u,v)= \^[(8)]{} H\^[(0)]{}\_[2222]{} = C(2)\^2 ( u\^4+), \[Htilde 2222 free\] with $C(p)=p^2(p^2-1)$. The formula in eq. (\[Generating Function 10-4D\]) then predict the other correlators as
\^[(0)]{}\_[p\_1p\_2p\_3p\_4]{} &= \_[i=1]{}\^4 ( + )\
&= C(p\_1)C(p\_2) ( \_[p\_1,p\_3]{}\_[p\_2,p\_4]{}+ \_[p\_1,p\_4]{}\_[p\_2,p\_3]{} ),
where we have used the integral $\oint \frac{6 da\ a^{1-p}}{2\pi i(1-a)^4} = C(p)/p$. This is in precise agreement with eq. (\[Delta8 H0\]), a first sanity check. Essentially this just fixes the normalization factors $\sqrt{p_i}$ in eq. (\[def Htilde\]).
The real test now is the tree-level correlator $\tilde{H}^{(1)}$. The formula in eq. (\[Generating Function 10-4D\]) predicts the higher correlators as specific derivatives of the first instance [@Arutyunov:2000py; @Dolan:2006ec],[^6] G\_[10]{}\^[(1)]{}(u,v) = \^[(1)]{}\_[2222]{} = -u\^4|[D]{}\_[2,4,2,2]{}(u,v). \[H2222 Dbar\] Acting with $\mathcal{D}_{p_1 p_2 p_3 p_4}$ on this gives predictions for the correlators of spherical harmonics. It is instructive to compare with the literature. For some of the cases presented in eqs. , refs. [@Dolan:2004iy; @Dolan:2006ec; @Berdichevsky:2007xd; @Uruchurtu:2008kp; @Aprile:2017xsp] provided explicit results in terms of $\bar{D}$ functions. Using derivative relations to generate higher weight $\bar{D}$ functions (see e.g. ref. [@Arutyunov:2002fh]), we evaluated these results as functions of $z,\bar{z}$ variables and compared with $\tilde{H}^{(1)}_{p_1 p_2 p_3 p_4}$ as computed from the action of the differential operator $\mathcal{D}_{p_1 p_2 p_3 p_4}$ on eq. . We find agreement with the literature up to a normalization constant:
\_[2233]{}\^[(1)]{} &= -6 u\^5 |[D]{}\_[3,5,2,2]{}, \[H1 Comparison\]\
\_[3333]{}\^[(1)]{} &= -9 u\^5 ( (1++ ) |[D]{}\_[3,5,3,3]{} + ( |[D]{}\_[2,5,2,3]{} + |[D]{}\_[3,5,2,2]{} + |[D]{}\_[2,5,3,2]{} ) ) ,\
\_[4444]{}\^[(1)]{} &=-16 u\^6 ( (+ + + (1+\^2 + \^2) ) |[D]{}\_[4,6,4,4]{}\
& - ( |[D]{}\_[4,6,2,4]{} + |[D]{}\_[2,6,4,4]{} + |[D]{}\_[4,6,4,2]{} )\
& + 2 (|[D]{}\_[3,6,2,3]{} + |[D]{}\_[2,6,3,3]{} + |[D]{}\_[3,6,3,2]{} )\
& + (|[D]{}\_[2,6,2,4]{} + \^2 |[D]{}\_[4,6,2,2]{} + \^2 |[D]{}\_[2,6,4,2]{} )\
& + (|[D]{}\_[3,6,3,4]{} + \^2 |[D]{}\_[4,6,3,3]{} + \^2 |[D]{}\_[3,6,4,3]{} ) ) .
We stress that in those papers, the result came from explicit supergravity or Mellin-space calculations whereas our result came from the action of the differential operator $\mathcal{D}_{p_1 p_2 p_3 p_4}$.
With hindsight, the existence of an operator such as $\mathcal{D}_{p_1 p_2 p_3 p_4}$, which acts on $\tilde{H}_{2222}$ to give the other correlators, can be understood from the derivative relations between $\bar{D}$ functions. Note however this way of writing the correlator is not unique, which is made apparent e.g. by comparing the 4444 cases from [@Dolan:2006ec] and [@Aprile:2017xsp]. The fact that the operators $\mathcal{D}_{p_1 p_2 p_3 p_4}$ can be taken from a unique generating function (\[Generating Function 10-4D\]) is very surprising and had not been observed before. It is also completely invisible in the supergravity Feynman rules.
From the perspective of the analytic bootstrap, since the formula (\[Generating Function 10-4D\]) manifests crossing symmetry, to rigorously establish it it suffices to show that it correctly predicts the double-discontinuity in one channel, say $v\to 0$. This will be entirely generated by the $v\to 0$ poles of the seed: \_[2222]{} = - - + . Indeed, acting with the differential operators (\[D examples\]) on this seed we reproduce precisely the double discontinuity of the finite sum (\[Eq:CorrelatordDisc\]). We checked this explicitly for all correlators $p_1,p_2,p_3,p_4$ with $p_i\leq 10$, making us confident that a group-theoretic identity underwrites the formula (\[Generating Function 10-4D\]). It would be interesting to prove it analytically in the general case.
Deriving the Mellin-space formula by Rastelli and Zhou {#ssec:Rastelli-Zhou}
------------------------------------------------------
A frequently useful language for conformal correlators is based on Mellin-space, see refs. [@Penedones:2010ue; @Dolan:2011dv; @Gopakumar:2016cpb; @Gopakumar:2016wkt] for example. Correlators of $\mathcal{N}=4$ SYM at strong coupling for arbitrary external scalar operators were conjectured in ref. [@Rastelli:2016nze; @Rastelli:2017udc; @Aprile:2018efk] in Mellin-space. In this section we show that our proposed formula is in precise agreement with that conjecture.
The Mellin representation of the generating functional is an integral over Mandelstam variables such that $$\tilde{H}_{p_1 p_2 p_3 p_4} = \int \frac{ds dt}{(2\pi i)^2} \ \tilde{M}_{p_1 p_2 p_3 p_4} \label{mellin}$$ where the region of integration is detailed in ref. [@Rastelli:2017udc]. We will omit the use of the Mandelstam variable $u$ and express everything in terms of $s,t$ to avoid confusion with the cross-ratio $u$. The reduced correlator for 2222 in position space is given by eq. . According to the first of eq. (\[D examples\]) this function serves as the seed of our generating functional. We thus start with the 2222 correlator with the normalization of eq. , written in terms of $\bar{D}_{2,4,2,2}$ in its Mellin representation as written in the Appendix of ref. [@Rastelli:2017udc]: $$\tilde{M}_{2222} = - u^4 \frac{1}{4} u^{\frac{s}{2}-2} v^{\frac{t}{2}-2} \frac{\Gamma(2-\frac{s}{2})^2 \Gamma(2-\frac{t}{2})^2 \Gamma(\frac{s+t}{2})^2}{\left(1 - \frac{s}{2} \right)\left(1 - \frac{t}{2} \right)\left( \frac{s+t}{2}-1 \right)},$$ where $u,v$ are cross-ratio variables. To uplift to a 10-dimensional generating functional, we promote the cross-ratios to be 10-dimensional, normalize according to eq. and expand in terms of 6-dimensional embedding coordinates similar to eq. with the auxiliary contour integral variables $a_i$: $$\begin{aligned}
\tilde{M}_{p_1 p_2 p_3 p_4}&= -2
\oint \prod_{i=1}^4 \left[\frac{da_i\,a_i^{1-p_i}}{2\pi i} \right] \frac{\Gamma(2-\frac{s}{2})^2 \Gamma(2-\frac{t}{2})^2 \Gamma(\frac{s+t}{2})^2}{\left(-2 + s \right)\left(-2 + t \right)\left( -2+s+t \right)} \frac{(u/\sigma)^{\frac{p_1+p_2}{2}-2}}{(1-a_1 a_3)^4 (1-a_2 a_4)^4} \nonumber \\
& \times \left(u \frac{(1-\frac{\sigma}{u} a_1 a_2)(1- a_3 a_4)}{(1- a_1 a_3)(1-a_2 a_4)} \right)^{\frac{s}{2}-2} \left(v \frac{(1-\frac{\tau}{v} a_2 a_3)(1 - a_1 a_4)}{(1 -a_1 a_3)(1- a_2 a_4)}\right)^{\frac{t}{2}-2}. \label{Mellin Generating Functional}\end{aligned}$$
By construction, this satisfies $\tilde{M}_{2222}=M_{2222}$. The key feature of this equation is that each of the six $\Gamma(-\alpha)$ functions is naturally paired with one of the six factors of the form $(1-c a_i a_j)^{\alpha}$, which originate from the six distances $w_{ij}^2$. To extract a given spherical harmonics, we need to series-expand in powers of the $a$’s. Thanks to the pairing, this series can be organized into shifted $\Gamma$-functions using the identity $-\alpha\Gamma(-\alpha) = \Gamma(-\alpha+1)$: $$(1-x)^\alpha \Gamma(-\alpha) = \displaystyle \sum_{i=0}^{\infty} \frac{x^i}{i!} \Gamma(i-\alpha).
\label{shift-expansion of gammas}$$ Inserting this identity six times into eq. (\[Mellin Generating Functional\]), we find a sum of the form \_[i,j,k,l,m,n0]{} ()\^i ( )\^k (a\_1 a\_2)\^i (a\_3 a\_4)\^j (a\_2 a\_3)\^k (a\_1 a\_4)\^l (a\_1 a\_3)\^m (a\_2 a\_4)\^n. \[six shifts\] The four contour integrals then select out the terms which have the correct exponent for each $a$. Consider for example the 3333 case, where each $a$ must appear linearly: three terms contribute, $$\begin{aligned}
\tilde{M}_{3333}&= \frac{-2}{(-2+s)(-2+t)(-2+s+t)} \Big(u^{\frac{s}{2}-2} v^{\frac{t}{2}-2} \Gamma(3-\frac{s}{2})^2 \Gamma(2-\frac{t}{2})^2 \Gamma(\frac{s+t}{2})^2 \nonumber \\
& \quad + u^{\frac{s}{2}-1} v^{\frac{t}{2}-3} \frac{\tau}{\sigma} \Gamma(2-\frac{s}{2})^2 \Gamma(3-\frac{t}{2})^2 \Gamma(\frac{s+t}{2})^2 \nonumber \\
& \quad + u^{\frac{s}{2}-1} v^{\frac{t}{2}-2} \frac{1}{\sigma} \Gamma(2-\frac{s}{2})^2 \Gamma(2-\frac{t}{2})^2 \Gamma(1+\frac{s+t}{2})^2 \Big).\end{aligned}$$ If one shifts $s$ and $t$ for each term above such that there is an overall factor of $u^{\frac{s}{2}-2} v^{\frac{t}{2}-3}$, one obtains $$\begin{aligned}
\tilde{M}_{3333}&= -2u^{\frac{s}{2}-2} v^{\frac{t}{2}-3} \Gamma(3-\frac{s}{2})^2 \Gamma(3-\frac{t}{2})^2 \Gamma(\frac{s+t-2}{2})^2 \Big( \frac{1}{(-2+s)(-4+t)(-4+s+t)} \nonumber \\
& \quad + \frac{\tau}{\sigma} \frac{1}{(-4+s)(-2+t)(-4+s+t)} + \frac{1}{\sigma} \frac{1}{(-4+s)(-4+t)(-6+s+t)} \Big),\end{aligned}$$ which is precisely the form found in ref. [@Rastelli:2016nze][^7] . Notice how all the $\Gamma$-functions automatically line up after the shifts. This can be tracked to the identity in eq. (\[shift-expansion of gammas\]). In general, one should perform appropriate shifts for each term such that an overall factor of $u^{\frac{s}{2}-2} v^{\frac{t-(p_2+p_3)}{2}}$ can be retrieved.
Let us now derive the general Mellin amplitude from eq. . The Mellin amplitude in [@Rastelli:2016nze] was conjectured by solving a set of algebraic crossing-symmetry constraints. (These authors assumed, with no loss of generality, an ordering $p_1 \geq p_2 \geq p_3 \geq p_4$, and ref. [@Aprile:2018efk] also assumed $p_{43}\geq p_{21}\geq 0$, but note that here we will make no assumption on the orderings.) Using eq. , the contour integrals in eq. (\[Mellin Generating Functional\]) imposes four constraints on the six indices, such that after lining up the powers of $u$ and $v$ we obtain: $$\begin{aligned}
\tilde{M}_{p_1 p_2 p_3 p_4} &= -2 u^{\frac{s}{2}-2} v^{\frac{t-(p_2+p_3)}{2}} \Gamma(\frac{p_1+p_2-s}{2}) \Gamma(\frac{p_3+p_4-s}{2}) \Gamma(\frac{p_2+p_3-t}{2}) \nonumber \\
&\times \Gamma(\frac{p_1+p_4-t}{2}) \Gamma(\frac{s+t+4-p_1-p_3}{2}) \Gamma(\frac{s+t+4-p_2-p_4}{2}) \nonumber \\
&\times \displaystyle \sum_{i,k=0}^{\infty} \frac{1}{i! j! k! l! m! n!} \frac{ \sigma^{i+2-\frac{p_1+p_2}{2}} \tau^k}{(s-2-2s_s)(t-2-t_s)(s+t-2-2(s_s+t_s))}, \label{General Mellin Amplitude}\end{aligned}$$ where the variables above must obey the following constraints: $$\begin{aligned}
j&= \frac{1}{2}(2i -p_1 -p_2 + p_3 + p_4),& l&= \frac{1}{2}(2k+ p_1 -p_2 - p_3 + p_4), \\
m&= \frac{1}{2}(p_1 +p_2 + p_3 -p_4 - 2(2+i+k)), & n&= p_2 - 2 - i -k, \\
s_s&= \frac{p_1 + p_2}{2} - 2 -i ,&t_s&= \frac{p_2+p_3}{2} - 2 - k.\end{aligned}$$ The shifts in Mandelstam variables $s_s,t_s$ were constrained by shifting $s$ and $t$ to obtain the appropriate power of $\frac{p_1+p_2}{2}$ and $\frac{p_2+p_3}{2}$ respectively as explained previously. Because of the inverse factorials, the summand is only nonvanishing when $i,j,k,l,m,n\geq 0$. Denoting $i=\frac{p_1+p_2}{2}-2-p$, demanding $m,n\geq 0$ yields for example $k \leq p + \frac{1}{2} \text{min}( p_{21},p_{34})$. Similarly, one can work out the remaining constraints to find that (this is similar to the summation range for the free correlator in eq. with $k\mapsto q$): $$\begin{aligned}
p_{\rm min}&= \tfrac12\max(|p_{12}|,|p_{34}|), & p_{\rm max} &=\tfrac12 \min(p_1+p_2,p_3+p_4) - 2,
\\
k_{\rm min}&=\tfrac12\max(0,p_{21}+p_{34}),& k_{\rm max}&=p+\tfrac12\min(p_{21},p_{34}).\end{aligned}$$ This matches precisely the results by ref. [@Rastelli:2016nze], with some extra factors fixed in ref. [@Aprile:2018efk].[^8] This completes our derivation of the Mellin amplitude from a more fundamental 10-dimensional symmetry.
Ten-dimensional OPE and the leading logarithmic part to all loops
-----------------------------------------------------------------
The SO(10,2) symmetry suggests using ten-dimensional blocks to express the correlators: these should diagonalize the mixing problem. This will also explain the flat space relation (\[Eq:4D-partial-wave\]), which originally motivated our looking for a SO(10,2) symmetry.
We start with the free correlator, eq. (\[Htilde 2222 free\]), and perform its OPE decomposition into 10-dimensional conformal blocks. As may be seen from the generalized free field formulas from ref. [@Fitzpatrick:2011dm], an important simplification occur for fields that saturate the unitarity bound, such as the axi-dilaton: a *single* double-twist trajectory then contributes, \_[2222]{}\^[(0)]{}(z,) = \_[=0, [even]{}]{}\^\^[(0)]{}\_ G\^[(10)]{}\_[,8+]{}(z,) a\^[(0)]{}\_[(10)]{}() = (+1)\_6. \[10D free sum\] Our blocks are normalized so that $\lim_{z\ll \zb \ll 1}G^{(d)}_{\ell,\Delta}(z,\zb) = z^{\frac{\Delta-\ell}{2}}\zb^{\frac{\Delta+\ell}{2}}$, and we have verified the sum by series expanding the blocks in $z,\zb$ using ref. [@Hogervorst:2013sma]. In fact, because we are in even dimensions and on the unitarity bound, the required blocks admit very simple closed-form expressions as derivatives of hypergeometric functions [@Dolan:2011dv], given explicitly in eq. (\[10D block\]). (Note that since the dimension of the dilaton is protected, the putative 10D CFT cannot be unitary since otherwise this would be the exact result to all orders in $1/c$.)
To extract ten-dimensional anomalous dimension, we OPE-decompose the $\log u$ terms in the tree-level correlator $\tilde{H}_{2222}^{(1)}=-u^4 \bar{D}_{2,4,2,2}$: \_[2222]{}\^[(1)]{}(z,)|\_[u]{} &=& -u\^4\_[u]{}\_v(1+u\_u +v\_v) \[H2222 log\]\
&=& \_[=0, [even]{}]{}\^12\^[(0)]{}\_\^[(1)]{}\_ G\^[(10)]{}\_[,8+]{}(z,) \^[(1)]{}\_. \[gamma10 sum\] This precisely matches the S-matrix phase shift in eq. (\[Eq:10D-partial-wave\]). This agreement has a simple intuitive explanation: in the bulk point limit $z\to \zb$ (reached after an analytic continuation, see [@Heemskerk:2009pn; @Maldacena:2015iua] for further detail), the correlator must reproduce the flat space amplitude. On the other hand, in this limit, each block in the sum (\[gamma10 sum\]) reduces to a single Gegenbauer polynomial matching that in the partial wave expansion (\[S matrix partial waves\]), weighted by $e^{-i\pi \Delta}$. Because a single ten-dimensional block appears for each spin, its anomalous dimension at order $1/c$ must be identical to the flat space phase shift.
The fact that there is a single 10D block for each spin at the lowest order suggests these blocks “diagonalize” the double trace mixing problem. Let us illustrate this with the nontrivial R-symmetry representation $[0,2,0]$ at twist 6, which led to the 2$\times$2 mixing matrix in eq. (\[02 matrix\]). We start by considering the contribution of a single ten-dimensional block to the numerator of the anomalous dimension matrix in eq. (\[Eq:anomDim\]), using the generating function (\[Generating Function 10-4D\]):
\_[2424]{} & \_[2433]{}\
\_[3324]{}&\_[3333]{}
G\^[(10)]{}\_[,8+]{}(z,), \[2x2 matrix diff eq\] where we now project each term onto the $[0,2,0]$ representation using the orthogonality relation (\[Eq:Z projection\]). Using the explicit form of the block in eq. (\[10D block\]) and expanding at small $z$, we find that this matrix contains the following three *four-dimensional* blocks of twist 6 (plus an infinite tower of blocks of twist 8 and higher):[^9] $$\begin{aligned}
&\ell(\ell-1)G_{\ell,\ell+4+6}(z,\zb) \begin{pmatrix}1 & -3/\sqrt{2} \\ -3/\sqrt{2} & 9/2\end{pmatrix}\nonumber
-\frac{\ell(\ell+4)(\ell+7)}{2(\ell+3)}G_{\ell+1,\ell+4+7}(z,\zb)
\begin{pmatrix} 1&0\\0&0\end{pmatrix}\nonumber
\\
&+\frac{(\ell+7)(\ell+8)}{4(2\ell+7)(2\ell+9)} G_{\ell+2,\ell+4+8}(z,\zb) \begin{pmatrix} (\ell+5)^2&3/\sqrt{2}(\ell+4)(\ell+5)\\
3/\sqrt{2}(\ell+4)(\ell+5)&9/2 (\ell+4)^2\end{pmatrix} + O(z^6)\end{aligned}$$ This is not very illuminating yet, but upon shifting the value of $\ell$ in each term so as to extract the coefficient of $G_{\ell,\ell+10}$ for a given even $\ell$ in the sum of eq. (\[gamma10 sum\]), and dividing by the free theory data in the denominator of eq. (\[Eq:anomDim\]), this yields the mixing matrix in the form: = \^[(1)]{}\_
&\
&
+ \^[(1)]{}\_[+2]{}
&\
&
. It turns out that each of these matrices is a projector! The two matrices square to themselves, and are orthogonal to each other. The right-hand-side precisely reproduces the matrix in eq. (\[02 matrix\]), if one inserts the value of $\tilde\gamma^{(1)}$, however it is now evident that the eigenvalues are $\tilde\gamma^{(1)}_\ell$ and $\tilde\gamma^{(1)}_{\ell+2}$ regardless of the functional form of $\tilde\gamma^{(1)}$ used in eq. (\[gamma10 sum\]). That is, the derivative operators in eq. (\[2x2 matrix diff eq\]) have simply turned each ten-dimensional block into a projector matrix. This example shows that the double trace mixing matrices are far from random but have an entirely group-theoretical origin, which encodes the expansion of SO(10,2) blocks into those of its SO(4,2)$\times$SO(6) subgroup. This explains why the eigenvalues are rational numbers, and also why these numbers are specifically equal to S-matrix phase shifts.
We have verified in many more examples that the ten-dimensional blocks always turn into projectors onto eigenspaces, including in cases with nontrivial multiplicity. Presumably, a more thorough study of these matrices would be more easily carried out at the level of 3-point vertices rather than four-point correlators.
As proposed in ref. [@Aprile:2018efk], the solution to the double-trace mixing problem at order $1/c$ enables to compute the leading logarithmic terms at each loop order. Thanks to the ten-dimensional blocks this can be done for any correlator without actually computing any matrix. We simply add more powers of $\gamma$ to the ten-dimensional OPE in eq. (\[gamma10 sum\]) to find $\tilde{H}_{2222}$, and then obtain the others by taking derivatives: \^[(k)]{}\_[pqrs]{}(z,,,) |\_[\^[k]{} u]{} = \^[k-1]{} \_[pqrs]{} \_[(3)]{}h\^[(k)]{}(z). \[conjecture leading log\] This result is a product of very many differential operators. The third-order operator $\mathcal{D}_{(3)}$, defined in eq. (\[10D block\]), builds (extremal) ten-dimensional blocks from single-variable functions $h^{(k)}$. The operator $\mathcal{D}_{pqrs}$, from eq. (\[Generating Function 10-4D\]), then extracts various 4D correlators; for the stress tensor multiplet this operation is trivial, $\mathcal{D}_{2222}=1$. Finally, the power of $\Delta^{(8)}$, defined in eq. (\[delta8 diff op\]), accounts for the fact that it is only the ratio $\gamma/\Delta^{(8)}$ which is compatible with the ten-dimensional symmetry. The ordering of operations is important: in general $\Delta^{(8)}$ destroys the ten-dimensional structure and must act to the left of $\mathcal{D}_{pqrs}$.
Using the explicit form of the block in eq. (\[10D block A\]), together with the coefficients in eqs. (\[10D free sum\]) and eqs. (\[gamma10 sum\]), the single-variable seed is given explicitly as: h\^[(k)]{}(z)()\^k \_[=0, [even]{}]{}\^ z\^[+1]{} \_2F\_1(+1,+4,2+8,z). \[llog seed\] In the first few cases the sum is readily computed, h\^[(0)]{}(z) &=& 2880 ,\
h\^[(1)]{}(z) &=& 120 + 10\
h\^[(2)]{}(z) &=& -\
&& ++, with the three-loop case in eq. (\[three loops\]). Plugging $h^{(0)}(z)$ into eq. (\[conjecture leading log\]) (and multiplying both sides by $\Delta^{(8)}$) reproduces eq. (\[Delta8 H0\]). Plugging $h^{(1)}(z)$ we reproduce the logarithmic term given in eq. (\[H2222 log\]) and its extension to all $H_{pqrs}$. Finally, plugging $h^{(2)}(z)$ we reproduce the one-loop double-discontinuity of $H_{2222}$ computed in refs. [@Aprile:2017bgs; @Alday:2017xua; @Alday:2017vkk] and now predict the $\log^2u$ terms for all $H_{pqrs}$ correlators.[^10] Needless to say, since the one-loop expressions require taking 11 or more derivatives of the compact seed $h^{(2)}(z)$, these expressions are much lengthier!
Conclusion
==========
In this paper we revisited the problem of computing correlation functions in holographic conformal field theories, focusing on the important example of $\mathcal{N}=4$ super Yang-Mills. We used a conformal field theory approach which constructs OPE data from the singular part of correlators, the Lorentzian inversion formula [@Caron-Huot:2017vep]. Other approaches to this problem include the direct evaluation of supergravity Witten diagrams and the solution of crossing symmetry constraints in Mellin-space; a chief interest of the present method is that the singular part can be computed rather straightforwardly in holographic theories, and the approach extends seamlessly to one-loop level.
By inserting the identity operator in the cross-channels, we thus obtained the OPE data at order $1/N^0$ in section \[Sec:1/c0\]; inserting a finite set of protected single-trace operators in section \[Sec:Tree-level\] then yielded the full $1/N^2$ corrections, corresponding to tree-level supergravity. In particular, we obtained mixing matrices and anomalous dimensions of double trace operators, see eq. (\[Eq:Anomalous Dimension\]), confirming a recent conjecture [@Aprile:2018efk].
Perhaps our most important result is that an unexpected symmetry controls this mixing matrix. For one thing, the eigenvalues are rational, as previously observed. We further observed in section \[sec:10D\] that they coincide with ten-dimensional flat space partial waves. We have argued that this agreement can be explained by an accidental SO(10,2) conformal symmetry of the four-particle supergravity amplitude. This symmetry correctly predicts an infinite number of relation between the correlators of different spherical harmonics, in fact unifying them into a single object. The generating function in eqs. (\[Generating Function 10-4D\]) and (\[def Htilde\]) is the main result of this paper. It is equivalent to the Mellin-space formula conjectured in [@Rastelli:2016nze], as shown in subsection \[ssec:Rastelli-Zhou\]. The symmetry explanation provides further evidence for the validity of this conjecture. An analytic proof of the symmetry itself, as noted at the end of subsection \[ssec:tests\], is now reduced to essentially group-theoretical identities between the singular part of four and ten-dimensional conformal blocks, which we have verified in many cases.
The SO(10,2) symmetry neatly diagonalizes the mixing between double trace operators which otherwise carry the same four-dimensional quantum numbers. Using this, we obtained a compact formula, see eqs. (\[conjecture leading log\]) and (\[llog seed\]), which yields the leading-logarithmic part of any correlator at each loop order. (At tree-level, the generating function (\[Generating Function 10-4D\]) gives more than the leading-log part and correctly predicts finite parts as well.) This data should suffice in the future to determine the one-loop correlator, up to contact ambiguities related to a $R^4$ counter-term, analogous to [@Aprile:2017bgs; @Alday:2017vkk]. An interesting question is whether the contact ambiguities for different spherical harmonics can be correlated with each other: this is related to locality in the internal S${}_5$ manifold.
It is very surprising to find that a theory of gravity, which evidently has a dimensionful coupling, effectively behaves like a conformal theory. Since AdS${}_5\times$S${}_5$ is conformally flat, this relates observables on this geometry to observables in ten-dimensional *flat* space: in our case, anomalous dimensions to $S$-matrix phase shifts. An analog of this may be the relation between four-dimensional Einstein and conformal gravity, see [@Maldacena:2011mk]; conformal-like Ward identities satisfied by tree amplitudes of $n$ gravitons in any dimension were discovered in [@Loebbert:2018xce].
A most pressing question is to determine if such a conformal symmetry can somehow be formulated for further CFT observables in this or other theories - higher point correlation functions, higher loops, superstring or M-theory corrections - or if the symmetry is just an accident of tree-level four-point supergravity amplitudes. One also wonders if the supergravity Lagrangian could be re-organized to streamline calculations.
Work of SCH is supported by the National Science and Engineering Council of Canada and the Simons Collaboration on the Nonperturbative Bootstrap. The authors would like to thank Leonardo Rastelli and Xinan Zhou for useful discussions.
Basic inversion integrals {#Appendix: Inversion}
=========================
Let us record generic results of the inversion integral. It will be convenient to isolate powers of $\frac{\bar{z}}{1-\bar{z}}$ and $\frac{z}{1-z}$ since they integrate nicely. We now perform the $\bar{z}$ integral first as shown in equation 4.7 of [@Caron-Huot:2017vep]. First, we compute the double discontinuity using eq. to find that $$\dDisc \left[ \left( \frac{1-\bar{z}}{\bar{z}} \right)^\lambda \bar{z}^{-p_{34}/2} \right] = 2 \sin( \lambda \pi ) \sin( \left( \lambda+ p_{21}+p_{34}\right)\pi ) \left( \frac{1-\bar{z}}{\bar{z}} \right)^\lambda \bar{z}^{-p_{34}/2}.$$ To evaluate the $z$ integral, we first rewrite the hypergeometric function in its integral form (with dummy variable $v$) and use the change of variable $t=\frac{\bar{z}(1-v)}{1-\bar{z} v}$. Details of the computation can be found in [@Caron-Huot:2017vep]. The $\bar{z}$ integral yields $$\begin{aligned}
\int_0^1 \frac{d\bar{z}}{\bar{z}^2} (1-\bar{z})^{\frac{1}{2}(p_{21}+p_{34})} &\tilde{\kappa} \textstyle \left(\bar{h}/2 \right) \displaystyle k_{\bar{h}/2}^{p_{21},p_{34}}(\bar{z}) \dDisc\left[ \left( \frac{1-\bar{z}}{\bar{z}} \right)^{\lambda} \bar{z}^{-p_{34}/2} \right] \nonumber \\
& \qquad = \frac{r_{\bar{h}}^{p_{21},p_{34}}}{\Gamma(-\lambda) \Gamma(-\lambda-\frac{p_{21}+p_{34}}{2})} \frac{\Gamma(\bar{h}-\lambda-\frac{p_{34}}{2} - 1)}{\Gamma(\bar{h}+\lambda+\frac{p_{34}}{2}+1)},
\label{Eq:zbarIntegral}\end{aligned}$$ where $r_{h}^{r,s}$ is defined in eq. (\[rfct\]).
A similar result can be found for the $z$ integral. This integral differs from the previous one by the double discontinuity term which only acted on the $\bar{z}$ sector, and the extra gamma functions from $\tilde{\kappa}$. We therefore have $$\begin{aligned}
\int_0^1& \frac{dz}{z^2} (1-z)^{\frac{1}{2}(p_{21}+p_{34})} k_{1-h}^{p_{21},p_{34}}(z) \left[ \frac{z}{1-z} \right]^\lambda z^{-p_{34}/2} \nonumber \\
=& \frac{\Gamma (2-2 h) \Gamma (1-\lambda ) \Gamma \left(-h+\lambda -\frac{p_{34}}{2}\right) \Gamma \left(\frac{p_{21}}{2}+\frac{p_{34}}{2}-\lambda +1\right)}{\Gamma \left(-h+\frac{p_{21}}{2}+1\right) \Gamma \left(-h-\frac{p_{34}}{2}+1\right) \Gamma \left(-h+\frac{p_{34}}{2}-\lambda +2\right)} \nonumber \\
=& \pi \frac{\sin(\pi (h-\frac{p_{21}}{2})) \sin(\pi(h+\lambda - \frac{p_{34}}{2})) \sin(\pi(h+\frac{p_{34}}{2})) }{\sin(2\pi h) \sin(\pi \lambda) \sin(\pi(h-\lambda+\frac{p_{34}}{2})) \sin(\pi(\frac{p_{21}+p_{34}}{2}-\lambda)) } \nonumber \\
& \times \frac{r_{h}^{p_{21},p_{34}}}{\Gamma(\lambda) \Gamma(\lambda - \frac{p_{21}}{2} - \frac{p_{34}}{2})} \frac{\Gamma(h+\lambda - \frac{p_{34}}{2}-1)}{\Gamma(h-\lambda+\frac{p_{34}}{2}+1)} \nonumber \\
\simeq & \pi \cot(\pi(\lambda-\frac{p_{34}}{2}-h)) \frac{r_{h}^{p_{21},p_{34}}}{\Gamma(\lambda) \Gamma(\lambda - \frac{p_{21}}{2} - \frac{p_{34}}{2})} \frac{\Gamma(h+\lambda - \frac{p_{34}}{2}-1)}{\Gamma(h-\lambda+\frac{p_{34}}{2}+1)}
\label{Eq:zIntegral}\end{aligned}$$ where we have assumed $\lambda>0$ and dropped terms with no poles near positive integer $h$. Notice the similarity with setting $\lambda\mapsto\lambda$.
To study higher $1/c$ corrections, one would need to integrate $\left[ \frac{z}{1-z} \right]^\lambda \log(z)$. This will generically yield the following structure: $$\begin{aligned}
\int_0^1& \frac{dz}{z^2} (1-z)^{\frac{1}{2}(p_{21}+p_{34})} k_{1-h}^{p_{21},p_{34}}(z) \left[ \frac{z}{1-z} \right]^\lambda \log(z) z^{-p_{34}/2} \nonumber \\
\simeq & \pi \partial_h \left( \cot(\pi(\lambda-\frac{p_{34}}{2}-h)) \right) \frac{r_{h}^{p_{21},p_{34}}}{\Gamma(\lambda) \Gamma(\lambda - \frac{p_{21}}{2} - \frac{p_{34}}{2})} \frac{\Gamma(h+\lambda - \frac{p_{34}}{2}-1)}{\Gamma(h-\lambda+\frac{p_{34}}{2}+1)} + ... \nonumber \\
\simeq & \frac{1}{\sin^2(\pi h)} \frac{r_{h}^{p_{21},p_{34}}}{\Gamma(\lambda) \Gamma(\lambda - \frac{p_{21}}{2} - \frac{p_{34}}{2})} \frac{\Gamma(h+\lambda - \frac{p_{34}}{2}-1)}{\Gamma(h-\lambda+\frac{p_{34}}{2}+1)} + ...
\label{x log(z) integral}\end{aligned}$$ where the ellipsis correspond to terms with single poles only. At order $1/c$, the subleading poles would correspond to contributions to OPE coefficients $a^{(1)}$, see [@Alday:2017vkk] for example computations.
Superconformal blocks, and weak coupling correlator to order $1/c$ {#Appendix susy blocks}
==================================================================
Although these were not needed in the main text, for reference we record here expressions for superconformal blocks, as deduced from the disconnected OPE data of section similarly to the half-BPS case in eq. (\[B multiplet\]). For the semi-short multiplets $\mathcal{C}_{\ell,m,n}$, equating the Casimirs in eqs. (\[C2 short\]) and (\[C2 susy\]) we find (for generic integer $m,n$) a single matching chiral block: f\_[+m+2,n-m]{}(z,). For the $H$ part, on each of the two families (\[protected h\]) we find a one-parameter family of solutions, and looking at the coefficients $\langle a^{(0)} \rangle_{pqqp}$ we find that they all appear with the same coefficient up to sign. It is thus natural to group them together into a single block: \_[,m,n]{}\^[r,s]{} = {
[l]{} k=0, f(z,) = (-1)\^m k\_[3++]{}\^[r,s]{}(z) k\_[-]{}\^[-r,-s]{}(),\
H(z,,,) = \_[i=1]{}\^[m]{} (-1)\^[i-1]{} G\_[+i,6++m+n-i]{}\^[r,s]{}(z,|[z]{}) Z\_[m-i,n-i]{}\^[r,s]{}(,)\
+ \_[i=0]{}\^[i\_[max]{}]{} (-1)\^[m+i]{} G\_[2++m+i,4++n-i]{}\^[r,s]{}(z,|[z]{}) Z\_[i,n-m-i-2]{}\^[r,s]{}(,)
. with $i_{\rm max}=(n-m-2-\max(|r|,|s|))/2$ on the last line. Again, although deduced from the disconnected correlator, we find that this expression has the correct $z,\zb\to 0$ limit for all the values of $m,n,\ell,r,s$ that we have verified, and so we identify it to be a general expression for any semi-short block.
We also find that quarter-BPS blocks are simply a special case of the semi-short blocks, where one sets $\ell=-1$: $$\mathcal{B}_{m,n}^{r,s} = \mathcal{C}_{-1,m-1,n-1}^{r,s}\,. \label{quarter BPS}$$ The long block $\mathcal{A}_{\ell,\Delta,m,n}$ only has a $H$ contribution, written already on the last line of (\[C2 susy\]).
In terms of these superconformal blocks, the OPE decomposition of the disconnected correlator given in eqs. (\[Eq:a0pqqp\])-(\[Eq:a0chiral\]) turns out to involve exclusively double-traces:
\[super OPE disconnected\] \^[(0)]{}\_[pqqp]{} &= \_[p,q]{} + (1+\_[p,q]{})\_[0,p+q]{}\
& +\_[m=0]{}\^[[min]{}(p,q)-1]{}\_[=-1]{}\^\_[,m,p+q-2-m]{}\
&+ \_[t=0]{}\^\_[=0]{}\^\_[m,n]{} a\^[(0)]{}\_[pqqp]{}(,p+q+2t+,m,n)\_[,p+q+2t+,m,n]{}.
The range of $m,n$ is as in eq. (\[small range\]) and the coefficients are in eqs. (\[Eq:a0pqqp\]) and (\[Eq:a0chiral\]). The fact that the OPE data can be reorganized into such a super-OPE with only double trace dimensions is a non-trivial consistency check.
Let us briefly comment on the so-called multiplet merging phenomenon: there is an inherent ambiguity in extracting the coefficients of $\mathcal{C}$ blocks, because a pair of semi-short multiplets can be equivalent to a long one: \_[,m,n]{} + \_[-1,m+1,n+1]{} = \_[,+m+n+2,m,n]{} (0). For $\ell=0$, the second term is to be interpreted as a quarter-BPS block according to eq. (\[quarter BPS\]). However, given that all the $\mathcal{C}_{\ell,m,n}$-blocks in eq. (\[super OPE disconnected\]) have the maximum value of $m+n$ allowed by R-symmetry considerations, this identity could not be used to rewrite the above without introducing terms with negative coefficients. We thus conclude that the decomposition of the disconnected correlator into superconformal blocks is unique if we assume unitarity. More generally, the multiplet merging phenomenon does not affect the $a$ and $b$ coefficients defined in eqs. (\[Eq:a0pqqp\]) and (\[Eq:a0chiral\]), which are the fundamental objects in this paper.
Weak coupling limit
-------------------
Finally, let us record another way to look at the protected part of the OPE data. As discussed in subsection \[ssec:free\], due to nonrenormalization theorems and the structure of Wick contractions, from it one can uniquely determine the *weak* ‘t Hooft coupling limit. At order $1/c$, one needs to sum over Wick contractions which can be drawn on a sphere. There are three cases, as shown in fig. \[Fig:Gfree\]; the sum, explicitly, can be written as \^[free]{}\_[{p\_i}]{} &=& \_[p\_1,p\_2]{}\_[p\_3,p\_4]{} + ()\^\
&& + \_[p=p\_[min]{}]{}\^[p\_[max]{}]{}\
&& + O(1/c\^2), \[Gfree Appendix\] with the summation ranges determined so that all bundles are non-empty, $$\begin{aligned}
p_{\rm min}&=1+\tfrac12\max(|p_{12}|,|p_{34}|),& p_{\rm max}&=\tfrac12\min(p_1+p_2,p_3+p_4)-1, \label{Gfree p constraints}
\\
q_{\rm min}&=1+\tfrac12\max(0,p_{21}+p_{34}),& q_{\rm max}&=p+\tfrac12\min(p_{21},p_{34})-1. \label{Gfree q constraints}\end{aligned}$$ This formula holds when the single-trace operators are defined to be orthogonal to multi-traces, see discussion below eq. (\[f coeff\]).
Ten-dimensional conformal blocks at the unitarity bound
=======================================================
Some calculations in the text are greatly simplified using the explicit form of even-dimensional conformal blocks, in particular right at the unitarity bound. Following arguments in [@Dolan:2011dv] we find that these blocks satisfy a simple differential equation: G\^[(d=10)]{}\_[,8+]{}(z,)=0, which holds for any $\ell\geq 0$. By a simple generalization of eq. (5.10) of [@Dolan:2011dv], a general solution can be written as a differential operator acting on a single-variable function: G\^[(d=10)]{}\_[,8+]{}(z,) &=& +(z) \_[(3)]{} f\_j(z). \[10D block\] Comparing with the series expansion in powers of $z,\zb$ (see [@Hogervorst:2013sma]) we find that the block indeed takes this form, with the function specifically chosen as f\_j(z) = z\^[j+1]{}\_2F\_1(j+1,j+4,2j+8,z). \[10D block A\] Finally, for future reference, let us record the seed controlling the two-loop leading-log term, following eq. (\[llog seed\]): $$\begin{aligned}
h^{(3)} =& \frac{(2-z) (249714 - 249714 z + 217855 z^2)}{12441600 z^3}
+ \frac{(1 - z)(2903 - 2903 z + 1258 z^2)}{172800 z^4}\log(1-z)
\nonumber\\ &+\frac{(-2435 + 4870 z - 3670 z^2 + 1235 z^3 + 2 z^4)}{172800 z^4}{\rm Li}_2(z)
\nonumber\\& - \frac{(1-z)(227 + 877 z + 177 z^2 - 23 z^3 + 2 z^4)}{345600 z^5}\log^2 (1-z)
\nonumber\\& + \frac{(1 - z)^3 (1 - 12 z + 21 z^2)}{2880 z^5} \Big(2 g_3(z) + {\rm Li}_3(z)- \log(1-z){\rm Li}_2(z)\Big)
\nonumber\\& +\frac{(1 + 10 z + 10 z^2)}{2880 z^5} \Big(g_3(z) - {\rm Li}_3(z)\Big),
\label{three loops}\end{aligned}$$ where $g_3(z) = {\rm Li}_3(1-z)- \log(1-z){\rm Li}_2(1-z) -\frac12\log(z)\log^2(1-z)-\zeta_3 $.
[^1]: Comparison with [@Caron-Huot:2017vep] requires the identity: $\big[(1-z)(1-\zb)\big]^{r+s}
G_{\Delta-3,\ell+3}^{r,s}(z,\zb) = G_{\Delta-3,\ell+3}^{-r,-s}(z,\zb)
$.
[^2]: A simply analogy is the flat space supergravity $S$-matrix, where supermomentum conservation $\delta^{(16)}(Q)$ removes a factor $s^4$ in the Regge limit $s\to\infty$.
[^3]: We find that the result in [@Simmons-Duffin:2017nub] holds also in the case of non-identical external operators provided that the sum runs over half-integer $h$ when $p_{12}$ is odd.
[^4]: For unequal operators, one can generally drop all powers of $v$ higher than $\frac12 {\rm min}(0,p_1+p_4-p_2-p_3)$, as these correspond to double-trace dimensions. These vanish upon integration as can be seen from the zeros of the $1/\Gamma$ factors in eq. (\[Eq:zbarIntegral\]).
[^5]: Specifically, the relevant relation is: A\_(s) = 1 + \_0\^d\^7() \^[(7/2)]{}\_() A\_[10]{}(s, ).
[^6]: Explicitly, $\bar{D}_{2,4,2,2}(u,v)=\partial_u\partial_v(1+u\partial_u +v\partial_v)
\frac{2{\rm Li}_2(z)-2{\rm Li}_2(\zb)+ \log(z\zb)\log\frac{1-z}{1-\zb}}{z-\zb}$.
[^7]: Their convention of $\sigma,\tau$ differs from ours: in their paper, they have $$\sigma = \frac{y_{13}^2 y_{24}^2}{y_{12}^2 y_{34}^2}, \qquad \tau = \frac{y_{14}^2 y_{23}^2}{y_{12}^2 y_{34}^2},$$ which should be compared to our definitions given by eq. .
[^8]: The precise comparison with ref. [@Aprile:2018efk], accounting for eq. (\[def Htilde\]) and different overall normalizations should be: \^[(1)]{}\_[p\_1p\_2p\_3p\_4]{}=(H\_[p\_1p\_2p\_3p\_4]{}\^[(1)]{}-H\_[p\_1p\_2p\_3p\_4]{}\^[(1)[free]{}]{}) = (N\_c\^2 H\_[p\_1p\_2p\_3p\_4]{})|\_[their]{}. Comparing with ref. [@Aprile:2018efk] we thus find precise agreement for the $p_i$ dependence, up to perhaps an overall constant factor $(4\pi i)^2$ which we didn’t track down.
[^9]: There is a slight abuse of notation here as we omit the superscripts on the blocks $G_{\ell,\Delta}^{p_{21},p_{34}}$. In reality the blocks sit inside the matrix and have a different superscript for each position inside the matrix.
[^10]: Note that the $\log^2u$ gives the full double-discontinuity at one-loop for $H_{2222}$, but for other correlators one may also need polar terms. We have not tested whether they are controlled by the same generating function.
|
---
abstract: 'We calculate the electromagnetic form factor of the pion in quenched lattice QCD. The non-perturbatively improved Sheikoleslami-Wohlert lattice action is used together with the ${\mathcal O}(a)$ improved current. We calculate form factor for pion masses down to $m_\pi = 360\; MeV$. We compare the mean square radius for the pion extracted from our form factors to the value obtained from the ’Bethe Salpeter amplitude’. Using (quenched) chiral perturbation theory, we extrapolate our results towards the physical pion mass.'
author:
- 'J. van der Heide'
title: The pion form factor from first principles
---
[ address=[National Institute for Nuclear Physics and High-Energy Physics (NIKHEF), 1009 DB Amsterdam, The Netherlands]{}, email=[r86@nikhef.nl]{} ]{}
Introduction
============
The pion, being the lightest and simplest particle in the hadronic spectrum has been studied intensively in the past. Using a variety of effective and phenemenological models, the properties of the pion have been desribed with varying succes. However, these models make assumptions, for example, confinement is put in by hand in contrast to being the result of the underlying dynamics. Lattice QCD (LQCD) doesn’t have this drawback since it is solves QCD directly.
Using LQCD, global properties of the pion such as the mass and the decay width have been calculated to satisfying accuracy. The form factor, which directly reflects the internal structure, is clearly an important challenge. The first lattice results were obtained by Martinelli and Sachrajda [@Martinelli]. It was followed by a more detailed study by Draper [*et al.*]{} [@Draper], who showed that the form factor could be described by a simple monopole form as suggested by vector meson dominance [@Williams]. We extend [@us] these studies by adopting improved lattice techniques [@Luscher; @Sheikholeslami; @Sachrajda; @Luscher2; @Sommer], which means that we include extra operators in order to systematically eliminate all the $\mathcal{O}(a)$ discretisation errors. Some aspects of the pion structure have been obtained [@Chu; @Hecht; @Gupta; @Lacock; @Schmidt] using ’the Bethe-Salpeter method’. We also use this approach and compare its predictions to the results of our direct calculation of the pion form factor.
Finally we study a chiral extrapolation to reach the physical limit.
The method
==========
To extract the form factor we calculate the two- and three point functions of the pion, analogously to [@Draper]. The two point function is given by $$G_2 (t, {\bf p}) = \sum_{\bf x} \left<\phi({t, \bf x})\; \phi ^{\dagger}
(0,{\bf 0})\right> \; e^{i\; {\bf p}\cdot {\bf x}} \; ,$$ where $\phi^{\dagger}$ is an operator creating a state with the quantum numbers of the pion. By varying the interquark distance at the sink, $t_f$, we can improve the overlap with the physical pion and obtain information on the ’Bethe-Salpeter amplitudes’. The three point function is calculated as $$G_3 (t_f, t; {\bf p}_f , {\bf p}_i) = \sum_{{\bf x}_f, {\bf x}} \;
\left<\phi (x_f) \; j_4 (x) \; \phi^{\dagger} (0)\right> e^{-i\; {\bf
p}_f \cdot ({\bf x}_f - {\bf x}) \; - i \; {\bf p}_i \cdot {\bf x}}$$ with $j_4$ the fourth component of the current, inserted at time $t$. Since the local current $$j_{\mu}^{\>L} (x) = {\bar \psi (x) }\; \gamma_\mu \; \psi (x),$$ is not conserved on the lattice, one can construct the Noether current belonging to our action $$j_{\mu}^{\>C}=\kappa\left({\bar\psi(x)}(1-\gamma_\mu)U_\mu(x)\psi(x+{\hat
\mu})-{\bar\psi(x+{\hat\mu})}(1+\gamma_\mu)U^{\dagger}_\mu (x)\psi (x)
\right).$$ This current however, still has $\mathcal{O}(a)$ corrections for $Q^2\; > \; 0$. A conserved and improved current can be constructed [@Sachrajda; @Luscher2; @Sommer] $$j_{\mu}^{\>I} = Z_{V} \, \{ j_{\mu}^{\>L} (x) + a \> c_V \> \partial _\nu \,
T_{\mu \nu} \} \;,$$ with $$\begin{aligned}
T_{\mu \nu} & = & {\bar \psi}(x)\; i \; \sigma_{\mu \nu} \; \psi (x)
\ , \\
Z_V & = & Z_V^{\>0} \: (1+ a\,b_V\,m_q) \; .\nonumber\end{aligned}$$ The bare-quark mass is defined as $m_q=\frac{1}{2a}\left(1/\kappa-1/\kappa_{c}\right)$, where $\kappa_{c}$ is the kappa value in the chiral limit and $a$ denotes the lattice spacing. For our simulation, $\kappa_{c}=0.13525$ [@Bowler]. Comparison of the currents will give us information on the importance of improvement.
Details of the calculation
==========================
On a $24^3x32$ lattice, we generated $\mathcal{O}(100)$ quenched gluon configurations at $\beta=6.0$. Subsequently, we calculated non-perturbatively improved ($c_{SW}=1.796$ [@Luscher]) quark propagators for 5 different values of the hopping parameter, $\kappa$. These propagators were then combined to two- and three point functions for the pion with masses ranging from 360 to 970 MeV.[^1] For the improved current, we use the parameters $Z_V^{\>0}$, $b_V$ and $c_V$ as determined by Bhattacharya [*et al.*]{} [@Bhattacharya]. To extract the significant parameters from our numerical data, we use the following parametrisations. For the two point function, we have $$G_2 (t,{\bf p}) = \sum_{n=0}^1{\sqrt {Z_R^n({\bf p})\: Z_0^n({\bf p})}}
\left\{e^{-E^n_{\bf p}t}+e^{-E^n_{\bf p}(N_{\tau}-t)}\right\}\; ,$$ including the contribution of the ground state (n=0) and a first excited one (n=1). The $Z^n_R$ denote the matrix elements, $$Z^n_R({\bf p}) \equiv |\left<\Omega | \phi_R | n, {\bf p} \right>|^2 \;,$$ and are related to the ’Bethe-Salpeter amplitude’, $\Phi(R)=\sqrt {Z^\pi_R ({\bf 0}) \, / \, Z^\pi_0 ({\bf 0})}$.
$E^0_{\bf p}$ and $E^1_{\bf p}$ are the energies of ground and excited state, respectively; $R$ denotes the interquark distance. The three-point function is parametrised as $$\begin{aligned}
G_3(t_f,t;{\bf p}_f,{\bf p}_i) = F(Q^2) \; \sqrt {Z_R^0 ({\bf p}_f)\;
Z_0^0 ({\bf p}_i) } e^{- E^0_{{\bf p}_f }\; (t_f - t)\, - E^0_{{\bf
p}_i}\; t } \nonumber\\
+\left\{\sqrt {Z_R^1 ({\bf p}_f) Z_0^0 ({\bf p}_i)}\left<1,{\bf p}_f|
j_\mu (0) |0,{\bf p}_f \right>e^{-\, E^1_{{\bf p}_f} \, (t_f - t) -\,
E^0_{{\bf p}_i} \, t} \;+ (1 \leftrightarrow 0) \right\}\;.\end{aligned}$$ The production of pion pairs, ’wrap around effects’ being due to the propagation of states beyond $t_f$ and an elastic contribution from the excited state were ignored in our parametrisation after being estimated negligable.
All parameters in the two- and three-point functions - energies $E$, $Z$-factors and the form factor $F(Q^2)$ - were fit simultaneously to the data per configuration. For the three-point function, we let the current insertion time $t$ vary from $0$ to $t_f$. The value for the parameters and their error in these simultaneous fits are obtained through a single elimination jackknife procedure.
Results
=======
As a byproduct of our simulations we also obtain pion masses for the 5 different $\kappa$-values. They agree with the literature. We also checked the energie-momentum relation and up to the energies involved we found that a continuum relation provides the best description. These non-trivial tests indicate that our simulations are done correctly.
Using the different currents, we extract the form factor for our five $\kappa$-values. Comparison of the results for the Noether current and the improved current yields that the effect of improvement can be as large as 25$\%$ for the highest momenta considered. The improved results are shown in Fig.\[fig:ff\_VMD\].
![[*Form factors as a function of $Q^2$ for the five pion masses. Curves: monopole fits to the data.*]{}[]{data-label="fig:ff_VMD"}](FF_VMD_plot_mono.eps){height="80mm"}
The high accuracy for the datapoint at $Q^2=0$ is due to the fact that we satisfy the Ward-Takahashi identity to 1 ppm.
As in the previous study [@Draper] of the pion form factor we compare our results to a monopole form factor $$F(Q^2) = \{ 1 + \frac{Q^2}{M_\rho^2}\}^{-1} \;,
\label{eq:VMD}$$ which is suggested by the vector meson dominance ansatz. Fitting our data to this model, we extract a vector meson mass which is within 5$\%$ of the corresponding rho mass on the lattice [@Bowler]. From the behaviour of the form factor at low $Q^2$, we can extract the mean-square charge radius of the pion, $$\left. \frac{dF(Q^2)}{dQ^2}\right|_{Q^2=0}=-\frac{1}{6}\left<
r^2\right>_{FF} = -\frac{1}{M_{V}^2}
\label{eq:RMS_FF}$$ where in the last step we assume Eq. \[eq:VMD\] and use the fitted parameter $M_V$. The results are shown in Fig. \[fig:RMS\] as a function of the pion mass. Previously, the ’Bethe-Salpeter-amplitude’ $\Phi(R)$ has been used to obtain estimates of the charge radius, $$\left<r^2\right>_{BS} := \frac{1}{4}
\frac{
\int d^3 \vec r \; \vec r^{\,2} \; \Phi^2(|\vec r|)
}{
\int d^3 \vec r \; \Phi^2(|\vec r|)
}\; .
\label{eq:RMS_BSA}$$ The results based on this procedure are also shown in Fig. \[fig:RMS\]. As can be seen these values are much lower than the actual values obtained from the form factor. Moreover, the Bethe-Salpeter results are almost mass independent, in accordance with the observations of Refs. [@Chu; @Hecht; @Gupta; @Schmidt]. This is a known[@Gupta] deficit of the approach, which Fig. \[fig:RMS\] makes quantitative. We extrapolate our results obtained with Eq. \[eq:RMS\_FF\] using three different parametrisations. First, we try chiral perturbation theorie ($\chi$pt). At one-loop order the prediction for the radius [@GL] is $$\left< r^2\right>_{\chi {\rm PT}} ^{\rm one-loop}
= c_1 + c_2 \log m_\pi^2
\label{eq:cPT}$$ In quenched $\chi$pt the rms is constant to this order. There are however indications that a mass dependence appears at higher order [@Colangelo1]. For our masses we restrict ourselves to a term linear in $m_{\pi}^2$ [@Colangelo2]. Lastly, we have also used the VMD prescription and assume a linear dependence of $M_V$ on $m_{\pi}^2$. These three expectations are also plotted in Fig. \[fig:RMS\].
![\[fig:RMS\] [*Radius of the pion as obtained from Bethe-Salpeter amplitudes, Eq. \[eq:RMS\_BSA\], and from the form factor, Eq. \[eq:RMS\_FF\]. Also shown are three different parametrizations of $\left< r^2 \right>$ (see text).*]{} ](RMS_ex_mono.eps){height="80mm"}
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been done in collaboration with Justus Koch and Edwin Laermann.
[2]{} G. Martinelli and C.T. Sachrajda, Nucl. Phys. [**B306**]{} (1988) 865. T. Draper, R.M. Woloshyn, W. Wilcox and K. Liu, Nucl. Phys. [**B318**]{} (1989) 319. H.B. O’Connell, B.C. Pearce, A.W. Thomas and A.G. Williams, Prog. Part. Nucl. Phys. [**39**]{} (1997) 201. J. van der Heide [*et al.,*]{}, Phys. Lett. [**B566**]{} (2003) 131. M.-C. Chu, M. Lissia and J.W. Negele, Nucl. Phys. [**B360**]{} (1991) 31. M.W. Hecht and T.A. DeGrand, Phys. Rev. [**D46**]{} (1992) 2155. R. Gupta, D. Daniel and J. Grandy, Phys. Rev. [**D48**]{} (1994) 3330 P. Lacock [*et al.*]{}, Phys. Rev. [**D51**]{} (1995) 6403. E. Laermann and P. Schmidt, Eur. Phys. J [**C20**]{} (2001) 541. M. Lüscher [*et al.*]{}, Nucl. Phys. [**B491**]{} (1997) 323. B. Sheikholeslami and R. Wohlert, Nucl. Phys. [**B259**]{} (1985) 572. G. Martinelli, C.T. Sachrajda and A. Vladikas, Nucl. Phys. [**B358**]{} (1991) 212. M. Lüscher, S. Sint, R. Sommer and H. Wittig, Nucl. Phys. [**B491**]{} (1997) 344. M. Guagnelli and R. Sommer, Nucl. Phys. Proc. Suppl. [**63**]{} (1998) 886. UKQCD Collaboration (K.C. Bowler [*et al.*]{}), Phys. Rev. [**D62**]{} (2000) 054506. R.G. Edwards, U.M. Heller and T.R. Klassen, Nucl. Phys. [**B517**]{} (1998) 377. T. Bhattacharya, R. Gupta, W. Lee and S. Sharpe, Phys. Rev. [**63**]{} (2001) 074505. J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) [**158**]{} (1984) 142. G. Colangelo and E. Pallante, Nucl. Phys. [**B520**]{} (1998) 433. G. Colangelo, private communication.
[^1]: The lattice spacing $a = 0.105\; {\rm fm}$ is taken from [@Edwards].
|
---
abstract: |
Recently we have used the Carlitz exponential map to define a finitely generated submodule of the Carlitz module having the right properties to be a function field analogue of the group of units in a number field. Similarly, we constructed a finite module analogous to the class group of a number field.
In this short note more algebraic constructions of these “unit” and “class” modules are given and they are related to $\Ext$ modules in the category of shtukas.
author:
- Lenny Taelman
bibliography:
- '../master.bib'
title: The Carlitz shtuka
---
Introduction and statement of the main results
==============================================
Notation
--------
Let $k$ be a finite field of $q$ elements. Without mention to the contrary schemes are understood to be over $ \operatorname{Spec}k $ and tensor products over $ k $.
Let $ t $ be the standard coordinate on the projective line $\PP^1$ over $k$, let $ F=k(t) $ the function field of $ \PP^1/k $ and let $ A =k[t] $ the ring of functions regular away from the “point at infinity” $\infty \in \PP^1$.
Let $ X $ be a smooth projective geometrically connected curve over $ k $ and $ X \to \PP^1 $ a surjective map. Denote the function field of $X$ by $K$. Let $ Y \subset X $ be the inverse image of $ \operatorname{Spec}A = \PP^1 -\infty $.
The Carlitz module
------------------
\[defCarlitz\] The Carlitz module is the functor $$C_0\colon \{ \operatorname{Spec}A\text{-schemes}\} \to \{A\text{-modules}\}$$ which associates to a scheme $S$ over $ \operatorname{Spec}A $ the $ A $-module $C(S)$ given by $C(S)= \Gamma(S, \O_S) $ as a $k$-vector space, with $A$-module structure $$\phi\colon A \to \operatorname{End}_k \Gamma(S, \O_S) \colon
t \mapsto \left( c \mapsto tc + c^q \right).$$
The functor $ C_0 $ is in many ways an analogue of the multiplicative group $$\G_m\colon \{\operatorname{Spec}\Z \text{-schemes}\} \to \{\Z\text{-modules}\}\colon
S \mapsto \Gamma(S,\O_S)^\times.$$ Yet, in contrast with Dirichlet’s unit theorem we have the following negative result:
The $ A $-module $ C_0( Y ) $ is not finitely generated.
A construction using the Carlitz exponential
--------------------------------------------
In [@Taelman10] we have used the Carlitz exponential map to cut out a canonical finitely generated sub-$A$-module from $ C_0(Y) $. We recall and reformulate this construction.
A simple recursion shows that there is a unique power series $\exp x$ in $ F[[x]] $ which is of the form $$\exp x = x + e_1x^q + e_2x^{q^2} + \cdots$$ and which satisfies $$\label{expfuneq}
\exp (tx) = t\exp x + (\exp x)^q.$$ This power series is called the *Carlitz exponential*. It is entire and for every point $z$ of $X\backslash Y$ it defines an $A$-linear map $ \exp\colon K_z \to C_0( K_z ) $.
Note that $ U \mapsto C_0(U) $ defines a (Zariski) sheaf of $ A $-modules on $ Y $. We extend it to a sheaf $C$ on $ X$ as follows: $$C( U ) :=
\left\{ ( c, ( \gamma_z )_z )
\in C_0( U\cap Y ) \times \!\!\prod_{z\in U\backslash Y}\! K_z
\,\,\big|\,\, \forall z\, \exp \gamma_z = c \right\}.$$ One verifies easily that this indeed defines a sheaf on $ X $. The main result of [@Taelman10] can be restated as follows:
\[finiteness\] $~$
1. $ \H^0( X, C ) $ is a finitely generated $ A $-module;
2. $ \H^1( X, C ) $ is a finite $ A $-module.
In §\[comparison\] we will show how to deduce this result from [@Taelman10].
In particular, the image of the restriction map $ C(X) \to C(Y) = C_0(Y) $ is a canonical finitely generated submodule of $ C_0(Y) $, it is a Carlitz analogue of the group of units in a number field. Similarly, $ \H^1( X, C ) $ is a Carlitz analogue of the class group of a number field (this should of course be compared with the isomorphism $\H^1( \O_L, \G_m ) = \operatorname{Pic}\O_L$).
We *do* need to pass to the completed curve $X$ to get something interesting: By Poonen’s theorem $\H^0( Y, C_0 ) $ is not finitely generated, and since $ C_0 \cong \O_Y $ as sheaves of abelian groups we have that $ \H^1( Y, C_0 ) = 0 $.
Unfortunately the above definition of the sheaf $C$ is analytic in nature, and it would be desirable to have a purely algebraic description of $C$. The aim of this paper is to provide such a description, as well as a more “motivic” interpretation of it.
An algebraic description of the sheaf $ C $
-------------------------------------------
For an integer $n$, denote by $ \O_{\PP^1}(n\infty) $ the sheaf of functions on $ \PP^1 $ that have a pole of order at most $n$ at $\infty$ and by $ \O_X(n\infty) $ its pullback over $X\to \PP^1$.
\[thm1\] There is a short exact sequence of sheaves of $A$-modules on $ X $ $$\label{thm1seq}
0 \longrightarrow \O_X \otimes A \overset{\partial}{\longrightarrow}
\O_X(\infty) \otimes A \longrightarrow C \longrightarrow 0$$ where $$\partial\colon f\otimes a \mapsto f\otimes ta - (tf+f^q) \otimes a.$$
The proof of this theorem will be given in section §\[sec1\].
Interpretation in terms of shtukas
----------------------------------
The short exact sequence of Theorem \[thm1\] can be reinterpreted in terms of shtukas.
For any $k$-scheme $ S $ denote by $ S_A $ the base change of $ S $ to $ \operatorname{Spec}A $ and by $ \tau_A\colon S_A \to S_A $ the base change of the $q$-th power Frobenius endomorphism $\tau\colon S \to S $.
A (right) *$A$-shtuka* on a $k$-scheme $ S $ is a diagram $$\M = \left[
\M \overset{\sigma}{\longrightarrow} \M' \overset{j}{\longleftarrow} \tau_A^\ast \M
\right]$$ of quasi-coherent $ \O_{S_A} $-modules.
With the obvious notion of morphism, the shtukas on $ S $ form an $A$-linear abelian category. In particular, if $ \M_1 $ and $\M_2$ are two shtukas on $ S $ then the Yoneda extension groups $\Ext^i( \M_1, \M_2 ) $ are $ A $-modules.
We have a natural isomorphism of sheaves of $ \O_{X} $-modules $$\tau^\ast \O_{X} \overset{{}_\sim}{\longrightarrow} \O_{X},$$ and will identify source and target in what follows.
If $\F$ is a coherent sheaf of $\O_X$-modules and $M$ an $A$-module we denote by $\F \boxtimes M$ the coherent sheaf of $\O_{X\times \operatorname{Spec}(A)}$-modules $$\F \boxtimes M = \pr_1^\ast \F \otimes_{\O_{X\times \operatorname{Spec}A }} \pr_2^\ast \tilde{M}$$ where $\pr_1$ and $\pr_2$ denote the projections from $X\times \operatorname{Spec}{A}$ to $X$ and $\operatorname{Spec}A$ respectively.
The *unit shtuka* on $X$ is defined to be the shtuka $$\1 = \left[
\O_{X} \boxtimes A
\overset{1}{\longrightarrow}
\O_{X} \boxtimes A
\overset{1}{\longleftarrow}
\tau^\ast\O_{X} \boxtimes A
\right].$$
We define the *Carlitz shtuka* on $ X $ to be the sthuka $$\C = \left[
\O_{X} \boxtimes A
\overset{ \sigma }{\longrightarrow}
\O_{X}( \infty ) \boxtimes A
\overset{ 1 }{ \longleftarrow }
\tau^\ast\O_{X} \boxtimes A
\right]$$ with $$\sigma = 1\otimes t - t\otimes 1.$$
The following is essentially a formal consequence of Theorem \[thm1\]:
\[thm2\] There are natural isomorphisms $$\Ext^i( \1, \C ) \overset{{}_\sim}{\longrightarrow} \H^{i-1}( X, C )$$ for all $i$.
The proof will be given in §\[sec2\].
Remarks
-------
Our notion of shtuka is the same as the one in V. Lafforgue [@Lafforgue09]. It is similar to the one used by Drinfeld [@Drinfeld77E] and L. Lafforgue [@Lafforgue98], but of a more arithmetic nature. Rather than compactifying the “coefficients” $\operatorname{Spec}A$ to a complete curve, we compactify the “base” $\AA^1_k$ to $\PP^1_k$.
Shtukas are function field toy models for (conjectural) mixed motives. The Carlitz shtuka $\C$ is an analogue of the Tate motive $\Z(1)$ and Theorem \[thm2\] should be compared with the isomorphisms $$\Ext^1_X(\1,\Z(1))=\Gamma(X,\O_X^\times) \,\,\text{ and }\,\,
\Ext^2_X(\1,\Z(1))=\operatorname{Pic}X$$ from motivic cohomology, see for example [@Mazza06 p. 25].
In the ($\infty$-adic) “class number formula” proven in [@Taelman10b], the $A$-modules $\H^0(X,C)$ and $\H^1(X,C)$ play a role analogous to the groups of units and the class group in the classical class number formula. In the guise of $\Ext^1(\1,\C)$ and $\Ext^2(\1,\C)$ they play a similar role in V. Lafforgue’s result [@Lafforgue09] on ($v$-adic, $v\neq \infty$) special values.
For any $m$ there is a natural isomorphism $\tau^\ast\O_X(m\infty){\overset{{}_\sim}{\rightarrow}}\O_X(qm\infty)$. So in the definition of the Carlitz shtuka one could twist both line bundles with $\O_X(-n\infty)$ for some $n\geq 0$ to obtain $$\O_{X}(-n\infty) \boxtimes A
\overset{ \sigma }{\longrightarrow}
\O_{X}( (1-n)\infty ) \boxtimes A
\overset{ 1 }{ \longleftarrow }
\tau^\ast\O_{X}(-n\infty) \boxtimes A.$$ The same results with the same proofs hold for this shtuka. We have chosen $n=0$ in our definition somewhat arbitrarily, distinguishing it from the other choices only by its minimality.
We have treated in this note only a very special case. One should try to obtain similar results for higher rank Drinfeld modules over general Drinfeld rings $A$, and even for the abelian $t$-modules of Anderson [@Anderson86]. Unfortunately it seems that these generalizations are not without difficulty, and even for higher rank Drinfeld modules it is not clear to me what the precise statement should be.
Acknowledgements
----------------
This work has been inspired by work of Anderson and Thakur [@Anderson90], Woo [@Woo95], Papanikolas and Ramachandran [@Papanikolas03], and V. Lafforgue [@Lafforgue09]. The author is grateful to David Goss for his feedback and constant encouragement, and to the referee for several useful suggestions.
The author is supported by a VENI Grant from the Netherlands Organization for Scientific Research (NWO). Part of the research leading to this paper was carried out at the Ecole Polytechnique Fédérale de Lausanne.
The cohomology of the sheaf $C$ {#comparison}
===============================
In this section we show how the modules $\H^0(X,C)$ and $\H^1(X,C)$ compare with the modules studied in [@Taelman10]. We recall the main result of *loc. cit.* Consider the map $$\delta\colon
C_0(Y) \times \!\!\prod_{z\in X\backslash Y}\! K_z
\to
\prod_{z\in X\backslash Y}\! C_0(K_z)
\colon
(c, (\gamma_z)_z ) \mapsto ( c - \exp \gamma_z )_z.$$
\[oldthm\] $\ker \delta$ is a finitely generated $A$-module and $\operatorname{coker}\delta$ is a finite $A$-module.
We now show that $\H^0(X,C)$ and $\H^1(X,C)$ coincide with the modules $\ker \delta$ (“$\exp^{-1}C(R)$” in the notation of *loc. cit.*) and $\operatorname{coker}\delta$ (“$H_R$”) above, and hence that Proposition \[finiteness\] follows from Theorem \[oldthm\].
\[lemmaC\] There is an exact sequence of $ A $-modules $$0 \longrightarrow
C( X )
\longrightarrow
C_0(Y) \times \!\!\prod_{z\in X\backslash Y}\! K_z
\overset{\delta}{\longrightarrow}
\prod_{z\in X\backslash Y}\! C_0(K_z)
\longrightarrow
\H^1( X, C )
\longrightarrow 0.$$
Denote by $i\colon Y\to X$ and by $ i_z\colon \{z\} \to X$ the inclusions of $Y$ and the points $z$ in $ X $. Then the following sequence of sheafs on $ X $ is exact: $$\label{sheafshort}
0 \longrightarrow
C
\longrightarrow
i_\ast C_0 \times \!\!\prod_{z\in X\backslash Y}\! i_{z,\ast} K_z
\longrightarrow
\prod_{z\in X\backslash Y}\! i_{z,\ast} C_0(K_z)
\longrightarrow 0.$$ (Here the middle map is the difference of the natural map and the map induced by $\exp$.) Left exactness follows from the definition of $ C $. For right exactness, one uses the fact that for all $z\in X\backslash Y$ we have $C_0(K_z) = C_0(K) + \exp K_z $ (which follows, for example, from Corollary \[corexpinv\] below).
Note that $\H^1(X,i_\ast C_0)=\H^1(Y,C_0)=0$ so that the desired exact sequence is precisely the long exact sequence of cohomology obtained from taking global sections in (\[sheafshort\]).
Proof of Theorem \[thm1\] {#sec1}
=========================
Away from $\infty$
------------------
Let $ R $ be an $ A $-algebra. Denote by $\alpha$ the $A$-linear map $$\alpha\colon R\otimes A \to C_0(R)\colon r \otimes a \mapsto \phi(a)(r).$$
\[propaway\] The sequence of $ A $-modules $$\label{away}
0 \longrightarrow R\otimes A \overset{\partial}{\longrightarrow} R\otimes A
\overset{\alpha}{\longrightarrow} C_0(R) \longrightarrow 0$$ is exact.
Straightforward.
In particular this provides the desired short exact sequence of sheaves (\[thm1seq\]) on the affine curve $Y\subset X$. In the following paragraphs we will extend it to the whole of $ X $.
Inversion of the exponential map
--------------------------------
Let $z\in X\backslash Y$ and let $|\cdot|$ be an absolute value on $K_z$ inducing the $z$-adic topology, so in particular $|t|>1$.
For all $ x \in K_z $ with $ |x| < |t|^{q/(q-1)} $ we have $ |\exp x-x| < |x| $.
Write $ \exp x = \sum_{i=0}^\infty e_ix^{q^i} $. It follows from (\[expfuneq\]) and from $e_0=1$ that for all $ i $ we have $|e_i| = |t|^{-iq^i}$. From this one deduces that for all $i>0$ and all $x$ with $|x|<|t|^{q/(q-1)}$ the inequality $|e_ix^{q^i}| < |x|$ holds. Hence $|\exp x -x|=|\sum_{i>0} e_ix^{q^i} | < |x| $, as claimed.
\[corexpinv\] For all $m\leq 1$ the exponential map restricts to a $k$-linear isomorphism $t^m\OXz \to t^m \OXz$.
We denote its inverse by $\log$.
Near $\infty$
-------------
Let $z\in X\backslash Y$. Consider the $A$-linear map $$\lambda\colon t\OXz\otimes A \to K_z\colon f\otimes a \mapsto a\log f.$$
\[propLie\] The sequence of $A$-modules $$\label{extLie}
0 \longrightarrow
\OXz\otimes A
\overset{\partial}{\longrightarrow}
t\OXz\otimes A
\overset{\lambda}{\longrightarrow}
K_z \longrightarrow
0$$ is exact.
Denote by $\mu$ the multiplication map $$\mu\colon t\OXz\otimes A \to K_z\colon f\otimes a \mapsto af.$$ Using the identity (\[expfuneq\]) one verifies that the diagram $$\begin{CD}
0 @>>> \OXz\otimes A
@>{1\otimes t-t\otimes 1}>>
t\OXz\otimes A
@>{\mu}>> K_z @>>> 0 \\
@. @VV{\exp\otimes\id}V @VV{\exp\otimes\id}V @VV{\id}V @. \\
0 @>>> \OXz\otimes A
@>{\partial}>>
t\OXz\otimes A
@>{\lambda}>> K_z @>>> 0
\end{CD}$$ commutes. The vertical arrows are isomorphisms by Corollary \[corexpinv\] and since the top sequence is exact the same holds for the bottom sequence.
Conclusion
----------
It is now a purely formal matter to deduce Theorem \[thm1\] from Propositions \[propaway\] and \[propLie\]:
Clearly the map $$\O_X\otimes A \overset{\partial}{\longrightarrow}
\O_X(\infty) \otimes A$$ is injective. We need to construct an isomorphism $\operatorname{coker}\partial {\overset{{}_\sim}{\rightarrow}}C$.
For every open $U\subset X$ and every integer $m$ we have an exact sequence $$0 \longrightarrow
\O_X(m\infty)(U) \longrightarrow
\O_X(U\cap Y) \times t^m \prod_{z} \OXz \overset{\delta}{\longrightarrow}
\prod_z K_z$$ where the products range over $z\in U\backslash Y$ and where $\delta(f,g):=f-g$. If moreover $U$ is affine then $\delta$ is surjective and we obtain a short exact sequence which we denote by $E(m)$.
Now, for an affine $U$, consider the map of exact sequences $$\partial\colon E(0)\otimes A \to E(1) \otimes A.$$ It is injective in all three positions. Using (\[away\]) and (\[extLie\]) one sees that the quotient is isomorphic with a short exact sequence $$0 \longrightarrow
(\operatorname{coker}\partial)(U) \longrightarrow
C_0(U\cap Y) \times \prod_z K_z \longrightarrow
\prod_z C_0(K_z) \longrightarrow
0,$$ the last map being $(c,f)\mapsto c-\exp f $. This provides an isomorphism $(\operatorname{coker}\partial)(U) {\overset{{}_\sim}{\rightarrow}}C(U)$ for every affine open $ U\subset X$, and clearly these glue to an isomorphism of sheaves. This proves Theorem \[thm1\].
Proof of Theorem \[thm2\] {#sec2}
=========================
Let $ S $ be a $ k $-scheme. For any $ \O_{S_A} $-module $ \F $ we denote by $\tau$ the canonical isomorphism of $ S_A $-sheaves $$\tau\colon \F \longrightarrow \tau_A^\ast \F$$ which is $ A $-linear but generally *not* $ \O_{S_A} $-linear. If $$\M = \left[
\M \overset{\sigma}{\longrightarrow} \M' \overset{j}{\longleftarrow} \tau^\ast \M
\right]$$ is a shtuka on $ S $ then we denote by $ \M^\bullet $ the complex of $ S_A $-sheaves $$\M \overset{\partial}{\longrightarrow} \M'$$ in degrees $0$ and $1$, with $\partial=\sigma-j\circ\tau$.
The following Proposition can be found implicitly in [@Lafforgue09].
\[proplaff\] For all $ i $ and all $\M$ there are natural isomorphisms $$\Ext^i( \1, \M ) = \HH^i( S_A, \M^\bullet ),$$ functorial in $ \M $ and in $ S $.
Before giving a proof, we first deduce Theorem \[thm2\] from this proposition.
Applying the Proposition to $S=X$ and $\M=\C$ we find $$\Ext^i( \1, \C ) = \HH^i( X_A,
\O_X\boxtimes A \overset{\partial}{\longrightarrow} \O_X(\infty)\boxtimes A ).$$ The latter is isomorphic with $$\HH^i( X, \O_X\otimes A \overset{\partial}{\longrightarrow} \O_X(\infty)\otimes A )$$ which by Theorem \[thm1\] is isomorphic with $\H^{i-1}( X, C ) $.
We will first establish a canonical isomorphism for $ i=0 $, and then conclude the general case by a purely formal argument.
A homomorphism $\1 \to \M $ of shtukas on $ S $ is a commutative diagram $$\begin{CD}
\O_{S_A} @>1>> \O_{S_A} @<1<< \tau^\ast \O_{S_A} \\
@VVfV @VV{f'}V @VV{\tau^\ast f}V \\
\M @>\sigma>> \M' @<j<< \tau^\ast \M
\end{CD}$$ Clearly the homomorphism is uniquely determined by $f \in \Gamma( S_A, \M ) $, and an $ f \in \Gamma( S_A, \M ) $ extends to a homomorphism of shtukas if and only if $ \partial f = 0 $. So we obtain an exact sequence $$0 \longrightarrow \Hom( \1, \M )
\longrightarrow \Gamma( S_A, \M )
\overset{\partial}{\longrightarrow} \Gamma( S_A, \M' )$$ and hence an isomorphism $$\Hom( \1, \M ) = \HH^0( S_A, \M^\bullet ).$$
Now any shtuka $$\I = \left[
\I \overset{\sigma}{\longrightarrow} \I' \overset{j}{\longleftarrow} \tau^\ast \I
\right]$$ with $ \I $ and $ \I' $ injective $\O_{S_A}$-modules is an injective object in the category of shtukas on $ S $. So we can find an injective resolution $ \I^\bullet $ of the shtuka $ \M $ such that the resulting double complex $ \I^{\bullet\bullet} $ is an injective resolution of the complex $ \M^\bullet $. We obtain a canonical isomorphism $$\Ext^i( \1, \M ) = \HH^i( S_A, \M^\bullet )$$ for all $ i $.
|
Hintergrund {#hintergrund .unnumbered}
===========
Die Internationale Raumstation {#die-internationale-raumstation .unnumbered}
------------------------------
Seit 1998 wird die Internationale Raumstation (ISS, Abb. \[f:iss\]) aufgebaut [@loff_dec._2013] und mittels einzelner Module (Abb. \[f:modules\]) ständig erweitert [@zak_after_2017]. Seit 2000 befinden sich in ununterbrochener Reihenfolge Besatzungen an Bord der ISS [@dlr_20_2018]. Ihr Betrieb ist bis mindestens 2024 vorgesehen, wahrscheinlich aber bis 2028 bzw. 2030 möglich [@ulmer_nasa_2015; @sputnik_iss_2016; @foust_house_2018]. Die gesamte Struktur hat eine Masse von . Sie ist lang, breit [@garcia_international_2016] und hoch [@esa_iss:_2014]. Auf einer Bahnhöhe von etwa benötigt die ISS für eine Erdumrundung ungefähr 92 Minuten [@howell_soyuz_2018].
Die Internationale Raumstation (Abb. \[f:iss\]) ist ein internationales Projekt mit derzeit 15 beteiligten Nationen [@garcia_20_2018; @esa_international_2013]. Sie dient als wissenschaftliches Forschungslabor für Fragestellungen, deren experimentelle Untersuchung durch den Einfluss der Gravitation auf der Erde erschwert wird. Man nutzt dafür die an Bord der ISS herrschende Mikrogravitation [@schuttler_mikrogravitationsexperimente_2006], einen Zustand von annähernder Schwerelosigkeit. Neben der Materialforschung und biologischen Studien spielt auch die Medizin eine wichtige Rolle. Der Einfluss der Mikrogravitation auf die Besatzung führt zu Symptomen, die Krankheitsbildern auf der Erde ähneln. Daher hofft man, Erkenntnisse in der kontrollierten Umgebung der Raumstation zu erlangen, die auch bei der Erforschung der Krankheiten helfen und Therapien den Weg ebnen [@buhrke_was_2018]. Ein weiterer Grund besteht darin, langfristige Missionen innerhalb des Sonnensystems vorzubereiten [@ganse_weltraummedizin:_2018].
Manchmal sind Außenbordeinsätze (EVA, Extra-vehicular Activity) erforderlich, um Reparaturen vorzunehmen oder neue Ausrüstung zu montieren. Wie sich die Astronauten darauf vorbereiten, wird im folgenden Kapitel beschrieben.
Astronautentraining {#astronautentraining .unnumbered}
-------------------
Arbeiten außerhalb der schützenden Hülle der ISS sind immer wieder notwendig. Insbesondere der Aufbau der Raumstation war durch zahlreiche EVA gekennzeichnet. Aber auch heute müssen immer wieder Reparaturen durchgeführt oder neue Ausrüstung montiert werden. Für die Ausstiege werden Raumanzüge benötigt, die der Besatzung einen ausreichenden Schutz gegen das Vakuum und die Strahlung im Weltall bieten. Handgriffe, Bewegungen und das zielgerichtete Ansteuern sind mit dieser Hülle schwierig und müssen erlernt werden.
Um die Arbeiten in den Raumanzügen routiniert und sicher ausführen zu können, werden die Aktivitäten auf der Erde in Wasserbecken trainiert [@esa_spacewalk_2013; @gast_glimpse_2009]. Indem man die Astronauten mit ihren Raumanzügen im Wasser zum Schweben bringt, erzeugt man einen Zustand der dem der Schwerelosigkeit auf der ISS recht nahe kommt. Allerdings erzeugt das Wasser einen Widerstand, den man im All nicht spürt. Zudem sind die Astronauten in den Anzügen während des Trainings unter Wasser nicht schwerelos.
Der Raumanzug (EMU, Extra-vehicular Moblity Unit) wiegt inklusive Astronaut ca. [@tate_how_2013; @thomas_u._2011]. Um dem Anzug die nötige Balance aus Auftrieb und Gewicht zu verleihen, wird er mit Schwimmkörpern und Gewichten versehen. Zudem wird der Luftdruck innerhalb der EMU während des Trainings stets auf () über dem Umgebungsdruck gehalten [@hutchinson_swimming_2013]. Dadurch bläht sich der Anzug leicht auf, was ebenfalls zu einem Auftrieb führt. Gleichzeitig wird darauf geachtet, dass während des Trainings Auftrieb und Gewicht so ausbalanciert werden, dass der gesamte Körper in einem neutralen Gleichgewicht ist. Ansonsten könnte es sein, dass sich der Astronaut entweder ständig aufrichtet oder kopfüber im Wasser schwimmt.
Für die europäischen ESA-Astronauten stehen dafür die Neutral Buoyancy Facility am Europäischen Astronautenzentrum (EAC) in Köln sowie das Neutral Buoyancy Lab der NASA an der Sonny Carter Training Facility nahe des Johnson Space Center in Houston, Texas zur Verfügung.
Mit den Planungen für eine Rückkehr zum Mond wird man diese Trainingsmethode auch zur Simulation der verminderten Schwerkraft auf dem Mond nutzen. Studien wie Moondive [@esa_mondspaziergange_2018] ermitteln bereits jetzt, welche Anforderungen für die spezielle Umgebung des Monds erfüllt sein müssen und ob die bereits vorhandenen Trainingseinrichtungen entsprechend angepasst werden können. Nachfolgend wird erläutert, welche physikalischen Kräfte für den Auftrieb und das Schweben unter Wasser verantwortlich sind.
Der hydrostatische Druck {#der-hydrostatische-druck .unnumbered}
------------------------
Um zu verstehen, wodurch der Auftrieb in einem Medium wie dem Wasser erzeugt wird, benötigen wir den Begriff des hydrostatischen Drucks. Dabei handelt es um den Druck, der in einer bestimmten Tiefe des entsprechenden Mediums durch die Gewichtskraft der darüber liegenden Säule erzeugt wird. Zur Verdeutlichung soll Abb. \[f:druck\] betrachtet werden.
Dargestellt ist ein Becherglas mit Wasser. In einer Tiefe $h$ befinde sich eine gedachte Fläche $A$, die zusammen mit $h$ das Volumen $V$ einer Wassersäule (gestrichelt) bildet. Dieses Wasser mit der Dichte $\varrho_w$ übt auf $A$ die Gewichtskraft $F_g$ aus. Die Masse des Wassersäule beträgt $m_w = \varrho_w \cdot V$. Daraus folgt:
$$\begin{aligned}
F_g &= m_w\cdot g \notag \\[5pt]
&= \varrho_w \cdot V \cdot g \\[5pt]
&= \varrho_w \cdot h \cdot A \cdot g \notag \\[5pt]
\Leftrightarrow p &= \frac{F_g}{A} = \varrho_w \cdot h \cdot g\end{aligned}$$
Somit steigt der Druck $p$ proportional mit der Wassertiefe $h$. Tatsächlich ergibt sich der Gesamtdruck aus dem Druck der Wassersäule und dem Luftdruck der darüber liegenden Atmosphäre, bezeichnet mit $p_0$. Somit kann der hydrostatische Druck insgesamt formuliert werden als:
$$p = \varrho \cdot g \cdot h + p_0
\label{e:druck}$$
Der Auftrieb {#der-auftrieb .unnumbered}
------------
Ein Objekt mit der Masse $m$ wird im Schwerefeld der Erde mit dem Ortsfaktor $g$ zum Erdmittelpunkt beschleunigt und erfährt eine Gewichtskraft $F_g = m\cdot g$. Taucht man jedoch dieses Objekt in ein Gefäß mit Wasser, wird die Wirkung von $F_g$ um die Auftriebskraft $F_a$ reduziert. Abbildung \[f:auftrieb\] verdeutlicht die Situation. Sie zeigt einen Körper, der ein Volumen $V$ und eine Grundfläche $A$ aufweist. Er habe die Masse $m$. Die Oberkante der Körpers hat die Tiefe $h_0$, die Unterkante die Tiefe $h_1$. Somit besitzt das Objekt selbst die Höhe $\Delta h = h_1 - h_0$ mit $V = A\cdot \Delta h$. Nun lässt sich der hydrostatische Druck $p$ bei $h_0$ und $h_1$ ermitteln. [ $$\begin{aligned}
p(h_0) &= \varrho_w \cdot g \cdot h_0 + p_0\notag \\[5pt]
p(h_1) &= \varrho_w \cdot g \cdot h_1 + p_0\notag\end{aligned}$$ ]{}
Da $h_1 > h_0$, folgt $p(h_1) > p(h_0)$. Damit ist die auf $A$ wirkende Kraft $F = p\cdot A$ bei $h_1$ ebenfalls größer als bei $h_0$. Somit überwiegt die auf die Unterseite des Körpers wirkende Kraft derjenigen auf seiner Oberseite. Die daraus resultierende Kraftdifferenz wirkt daher nach oben, also entgegen der Richtung der Gravitation. Dies ist die Auftriebskraft $F_a$.
$$\begin{aligned}
F_a &= \Delta p \cdot A = \left(p(h_1)-p(h_0)\right) \cdot A \notag \\[5pt]
&= \left(\varrho_w \cdot g \cdot h_1 + p_0 - \left(\varrho_w \cdot g \cdot h_0 + p_0\right)\right) \cdot A \notag \\[5pt]
&= \varrho_w \cdot g \cdot \left(h_1 - h_0\right) \cdot A \notag \\[5pt]
&= \varrho_w \cdot g \cdot \Delta h \cdot A \notag \\[5pt]
&= \varrho_w \cdot g \cdot V \\[5pt]
&= m_w \cdot g\end{aligned}$$
Daraus folgt das **Archimedische Prinzip**, das besagt:
> “Die Auftriebskraft entspricht der Gewichtskraft des Mediums, das von dem eingetauchten Volumen verdrängt wird.”
Daraus lassen sich drei Fälle ableiten.
- $F_g > F_a$: Der Körper sinkt.
- $F_g < F_a$: Der Körper steigt.
- $F_g = F_a$: Der Körper schwebt im Wasser bzw. schwimmt an der Oberfläche.
Die letztere Situation des Kräftegleichgewichts macht man sich im Astronautentraining zunutze. Bei diesem Gleichgewicht schwebt der Astronaut im Wasserbecken. Doch wie führt man dieses Kräftegleichgewicht herbei? Welche Größe ist hierbei maßgeblich? Eine einfache Rechnung ergibt die Antwort.
$$\begin{aligned}
F_g &= F_a \label{e:Fgg} \\[5pt]
\Leftrightarrow m\cdot g &= m_w \cdot g \notag \\[5pt]
\Leftrightarrow \varrho_k \cdot V \cdot g &= \varrho_w \cdot V \cdot g \notag \\[5pt]
\Leftrightarrow \varrho_k &= \varrho_w\end{aligned}$$
Der Körper schwebt im Wasser, wenn seine Dichte $\varrho_k$ der des Wassers entspricht. Sinkt er, ist seine Dichte zu hoch. Um die Dichte zu verringern, muss man bei gleichbleibender Masse das Volumen erhöhen. Das geschieht im Fall der Astronauten im Trainingsbecken durch Luftpolster. Sie erhöhen das Volumen stärker als im Vergleich die Masse, die sie mit sich tragen. In der Bilanz wird also die mittlere Dichte des Astronauten verringert. Das führt zu einem verstärkten Auftrieb. Nun kann man so lange Luftpolster anbringen, bis im Mittel die Dichte von Wasser erreicht ist. Der Astronaut schwebt.
Aktivität: {#aktivität .unnumbered}
===========
Vorbereitung für Lehrpersonen {#vorbereitung-für-lehrpersonen .unnumbered}
-----------------------------
Beschaffen Sie die auf den Deckblatt angegebenen Materialien in einer Stückzahl, die für Ihre Lerngruppe angemessen ist. Beachten Sie, dass Sie für die Durchführung des Experiments Wasser benötigen.
Bedenken Sie, dass die Experimentierzutaten lediglich als Beispiel anzusehen sind, die der Autor getestet hat. Allerdings können beispielsweise auch kleinere Styroporkugeln verwendet werden, die einen entsprechend geringeren Auftrieb erzeugen. Größere Kugeln sind nicht zu empfehlen, da hierfür deutlich schwerere Gewichte und somit auch ggf. größere Wasserbehälter benötigt werden. Es bietet sich an, vorab das Experiment selbst zu testen, um den Umfang der erforderlichen Gewichte zu ermitteln.
Das Experiment darf nur unter Aufsicht einer Lehrkraft durchgeführt werden. Der Autor und die Projektpartner dieses Materials übernehmen keine Haftung für etwaig auftretende Verletzungen.
Die in der Einführung angegebenen Videos (s. u.) benötigen einen Internetzugang. Falls Sie in der Lehreinrichtung nicht über die notwendige Ausstattung verfügen, weisen Sie die Schülerinnen und Schüler vorab an, sich diese Videos als Vorbereitung zuhause anzusehen. Ermutigen Sie sie, sich davon ausgehend weiter über das Thema zu informieren. Eine gute Seite zur Eigenrecherche ist:
DLR\_next – Raumfahrt\
<https://www.dlr.de/next/desktopdefault.aspx/tabid-6100/>
Thematische Einführung (Vorschlag) {#thematische-einführung-vorschlag .unnumbered}
----------------------------------
Fragen Sie die Schülerinnen und Schüler, ob sie wissen, was ein Weltraumspaziergang ist. Zeigen Sie zur Erläuterung das folgende Video.
DLR\_next – Mit Alex ins All: Der Spacewalk (Dauer: 5:04 min)\
<https://youtu.be/vZUAFdJ3C-4>
Die im Video erwähnten Liveübertragungen der NASA kann man über die folgende Webadresse sehen:\
<https://www.nasa.gov/nasalive>
Bei weiterem Interesse können sich die Schülerinnen und Schüler die folgenden beiden Videos ansehen. Da sie in englischer Sprache produziert sind, ermöglichen Sie zudem eine Verknüpfung mit dem Fach Englisch und schulen das Verständnis gesprochener Sprache.
Alexander Gerst training in Houston (Dauer: 3:26 min, englisch)\
<https://youtu.be/f4cHgAIK79c>
How Astronauts Train Underwater at NASA’s Neutral Buoyancy Lab (Dauer: 7:06 min, englisch)\
<https://youtu.be/BRPb0J8lZcY>
Fragen Sie, wovon es abhängt ob ein Objekt im Wasser schwimmt oder untergeht. Wahrscheinlich wird hier der Begriff Gewicht genannt. Als Gegenbeispiel können Sie den Vergleich eines Steins mit einem riesigen Containerschiff aufstellen. Damit ist das Gewicht bzw. die Masse offenbar nicht die bestimmende Größe. Offenbar ist das Verhältnis von Masse und Volumen maßgeblich, oder in anderen Worten: die Dichte.
Schließlich schauen sich die Schülerinnen und Schüler zur Einführung oder zur Auffrischung des Prinzips des Auftriebs die folgenden Videos an.
Archimedes von Syrakus - Gold oder nicht Gold? (Dauer: 3:24 min)\
<https://youtu.be/e2hcOCSd-xw>
Archimedisches Prinzip – Der Auftrieb (Dauer: 4:45 min)\
<https://youtu.be/ZUWrj-nqkE4>
Auftrieb (Dauer: 1:03 min)\
<https://youtu.be/FPTGbb9p0po>
Fragen Sie zum Schluss, worauf man achten muss, damit die Astronauten während des Trainings im Wasserbecken schweben. Eine korrekte Antwort sollte beinhalten, dass das sich Gewicht und Auftrieb möglichst genau ausgleichen müssen. Ein Objekt schwebt dann im Wasser, wenn es im Mittel dieselbe Dichte wie Wasser hat.
Experiment {#experiment .unnumbered}
----------
[*Hinweis! Der Inhalt des entsprechenden Kapitels im Arbeitsmaterial für die Schülerinnen und Schüler ist gegenüber dieser Version reduziert. Einzelne Zwischenschritte und Lösungen sind aus didaktischen Gründen dort nicht enthalten.*]{}
Das Experiment stellt im Modell nach, was es bedeutet, ein Objekt (z. B. eine Astronautin oder einen Astronauten) im Wasser zum Schweben zu bringen. Eine Styroporkugel dient hierbei als Objekt mit geringer Dichte, das auf der Wasseroberfläche schwimmt. Geeignete Gewichte müssen so lange an einen daran befestigten Haken angehängt werden, bis die Kugel im Wasser schwebt. Dabei wird man feststellen, dass es sehr schwierig ist, Auftrieb und Gewicht exakt zum Ausgleich zu bringen. In dem dargestellten Beispiel werden Schrauben, Muttern und Büroklammern als Gewichte benutzt.
Zu Beginn schwimmt die Kugel auf dem Wasser.
### Erläutere, wie man die Masse abschätzen kann, die benötigt wird, damit die Styroporkugel im Wasser schwebt. {#erläutere-wie-man-die-masse-abschätzen-kann-die-benötigt-wird-damit-die-styroporkugel-im-wasser-schwebt. .unnumbered}
Das Archimedische Prinzip besagt, dass die Kraft des Auftriebs der Gewichtskraft des Wassers entspricht, das durch das Volumen des eingetauchten Körpers verdrängt wird. Daher kann man als erste Abschätzung das Gewicht bzw. die Masse des Wassers innerhalb eines Volumens berechnen, das dem Volumen der Styroporkugel entspricht.
Größe Formelzeichen Zahlenwert und Einheit
----------------------------- ----------------------- ------------------------
Erdbeschleunigung $g$
Atmosphärischer Normaldruck $p_0$ $=$
Dichte von Wasser $\varrho_w$
Dichte von Gold $\varrho_\mathsf{Au}$
Dichte von Silber $\varrho_\mathsf{Ag}$
: Wichtige physikalische Größen und ihre Werte.[]{data-label="t:groessen"}
In diesem Beispiel hat die Kugel einen Radius von . Damit hat sie ein Volumen von oder . Mit der Dichte des Wasser aus Tab. \[t:groessen\] folgt eine Masse von . Zu Beginn sollte die Kugel also so beschwert werden, dass eine Gesamtmasse von etwa diesem Wert erreicht wird (Abb. \[f:exp\_step2\]).
Dieses Gewicht wird nun mit Büroklammern am Haken der Kugel angebracht und im Wasserbecken eingetaucht.
### Beschreibe und erkläre deine Beobachtung. Erkläre, warum die Kugel nicht völlig eingetaucht im Wasser schwimmt. {#beschreibe-und-erkläre-deine-beobachtung.-erkläre-warum-die-kugel-nicht-völlig-eingetaucht-im-wasser-schwimmt. .unnumbered}
Die Kugel schwimmt teilweise eingetaucht immer noch an der Oberfläche (Abb. \[f:exp\_step3\]). Die benötigte Masse wurde offenbar unterschätzt. Der Grund ist, dass das Volumen der Gewichte vernachlässigt wurde. Auch sie erfahren einen Auftrieb.
### Beschwere die Kugel, bis sie im Wasser schwebt. {#beschwere-die-kugel-bis-sie-im-wasser-schwebt. .unnumbered}
Nun beschwert man die Kugel so lange – z. B. mit Büroklammern – bis sich der Schwebezustand einstellt (Abb. \[f:exp\_step4\]).
Dabei wird man feststellen, dass der Schwebezustand einem feinen Gleichgewicht zwischen Auftrieb und Gewicht entspricht. Unter Umständen ist die Gewichtseinheit pro Büroklammer zu grob, so dass die Kugel entweder langsam steigt oder sinkt.
### Begründe, warum es so schwierig ist, eine exakte Balance aus Gewichtskraft und Auftrieb einzustellen. {#begründe-warum-es-so-schwierig-ist-eine-exakte-balance-aus-gewichtskraft-und-auftrieb-einzustellen. .unnumbered}
Jedes Ungleichgewicht führt sofort zu einer resultierenden Kraft in die eine oder andere Richtung. Da in diesem Experiment die kleinste Gewichtseinheit eine Büroklammer ist, können kleinere Gewichte nur dadurch erzielt werden, wenn man Stücke von einer Büroklammer entfernt. Hierzu wird die Zange benutzt.
### Erkläre den Zusammenhang besteht zwischen dem Volumen der Styroporkugel (plus Gewichte) und dem benötigten Gewicht. {#erkläre-den-zusammenhang-besteht-zwischen-dem-volumen-der-styroporkugel-plus-gewichte-und-dem-benötigten-gewicht. .unnumbered}
Der Zusammenhang zwischen dem Volumen der Kugel bzw. den Gewichten und dem benötigten Gewicht besteht in dem Archimedischen Prinzip. Der Auftrieb ist eine Kraft, die dem Gewicht des Wassers entspricht, das von dem eingetauchten Volumen verdrängt wird. Wenn man also weiß, wie groß das Volumen ist, entspricht die Auftriebskraft der Gewichtskraft dieses Wasservolumens. Daher benötigt man exakt so viel Gewicht zum Beschweren, wie das verdrängte Wasser wiegt.
### Der Auftrieb ist in allen Medien vorhanden, so auch in der Luft. Schätze die Stärke des Auftriebs in der Luft im Vergleich zum Wasser ab. Nenne die maßgeblichen Größen. {#der-auftrieb-ist-in-allen-medien-vorhanden-so-auch-in-der-luft.-schätze-die-stärke-des-auftriebs-in-der-luft-im-vergleich-zum-wasser-ab.-nenne-die-maßgeblichen-größen. .unnumbered}
Das Verhältnis des Auftriebs in Luft und Wasser entspricht dem Dichteverhältnis dieser Medien. Mit der Dichte für Luft von $\varrho_L = \unit[1,2]{kg/m^3}$ findet man:
$$\frac{\varrho_w}{\varrho_L} = 831$$
### Optionale Zusatzaufgabe: Ermittle das während des Experiments verdrängte Volumen. {#optionale-zusatzaufgabe-ermittle-das-während-des-experiments-verdrängte-volumen. .unnumbered}
Auch hier lässt sich das Archimedische Prinzip anwenden. Aus der Gesamtmasse folgt über die Dichte des Wassers das verdrängte Volumen. Laut Waage entspricht das Gesamtgewicht einer Masse von (Abb. \[f:exp\_step5\]). Daraus folgt ein Volumen von .
Aufgaben {#aufgaben .unnumbered}
--------
Die nachfolgenden Aufgaben sollen dazu beitragen, die Inhalte durch Rechnungen weiter zu vertiefen.
### 1. Wasserdruck (rechnerisch) {#wasserdruck-rechnerisch .unnumbered}
Der Wasserdruck nimmt proportional zur Tiefe zu. Gemäß Gl. \[e:druck\] lautet der Druck unter einer Wassersäule:
$$p = \varrho_w \cdot g \cdot h + p_0 \nonumber$$
Berechne die Wassertiefe, bei der ein Druck erreicht wird, der zusätzlich zur Atmosphäre über dem Wasser ebenfalls dem Normaldruck von $p-p_0 = p_0 = 101325\,\mathsf{Pa}$ entspricht. Die relevanten Zahlenwerte findest du in Tab. \[t:groessen\].
Beachte, dass gilt:
$$\unit[1]{Pa} = \unit[1]{\frac{N}{m^2}} = \unit[1]{\frac{\frac{kg\cdot m}{s^2}}{m^2}} = \unit[1]{\frac{kg}{m\cdot s^2}}$$
### 2. Wasserdruck (grafisch) {#wasserdruck-grafisch .unnumbered}
Die Gleichung aus Aufgabe 1 entspricht einer mathematischen Funktion der Form:
$$f(x) = a \cdot x + b$$
Es handelt sich dabei um eine Geradengleichung. Ordne die Größen denen der Gleichung aus Aufgabe 1 zu. Achte dabei darauf, bei welchen Größen es sich um Variablen handelt und welche davon konstant sind. Bestimme insbesondere die Steigung der Geraden $a$.
Konstruiere ein Koordinatensystem, das auf der senkrechten Achse den Druck angibt, während die horizontale Achse die variable Größe darstellt. Bestimme den Achsenabschnitt und wähle mit der Maßgabe, dass der Gesamtdruck das Doppelte von $p_0$ annehmen soll, die Skala der senkrechten Achse. Zeichne die Gerade. Ermittle den Wert auf der horizontalen Achse, für den $p = 2\cdot p_0$ beträgt.
### 3. Wasserverdrängung {#wasserverdrängung .unnumbered}
Berechne das Volumen eines Raumanzugs, der mit Astronaut eine Gesamtmasse von hat, wenn er im Wasser schwebt. Benutze das Gleichgewicht der Kräfte der Gravitation und des Auftriebs.
### 4. Das Archimedische Prinzip {#das-archimedische-prinzip .unnumbered}
Das Archimedische Prinzip geht auf eine überlieferte Geschichte vom antiken griechischen Gelehrten Archimedes von Syrakus zurück. Demnach soll König Hieron II von Syrakus ihn beauftragt haben, einen möglichen Betrug zu untersuchen. Hieron hatte einem Goldschmied eine genau abgemessene Masse Gold für die Anfertigung einer Krone übergeben. Er vermutete aber, dass bei der Herstellung ein Teil des Golds durch Silber ersetzt wurde, wobei die Gesamtmasse aber erhalten blieb. Archimedes sollte nun herausfinden, ob der Verdacht gerechtfertigt war, und zwar ohne die Krone zu zerstören.
Der römische Architekt Vitruv, der später über diesen Fall berichtete, gab an, dass Archimedes dem Betrug dadurch auf die Schliche kam, weil eine Legierung von Gold und Silber eine geringere Dichte aufweist und deswegen durch sein höheres Volumen pro Masse auch mehr Wasser verdrängt. Als Archimedes dies durch Eintauchen in Wasser überprüfte, soll im Vergleich zu einem Stück reinem Gold bei der Krone das Wasser übergelaufen sein.
Ist diese Geschichte glaubhaft? Berechne hierzu, wie groß der Unterschied zwischen den Wasserständen für zwei Metallstücke mit unterschiedlichem Goldanteil ist. Nimm hierzu zwei Stücke zu je an. Eines soll zu 100% aus Gold bestehen; das andere soll eine der Masse nach 30%ige Beimischung von Silber haben. Nimm zudem einen Behälter mit Wasser an, dessen kreisrunde Grundfläche einen Radius von aufweist. Bestimme den Wasserstand,
1. wenn der Behälter nur mit Wasser gefüllt ist,
2. wenn das Goldstück vollständig eingetaucht ist,
3. wenn das Stück aus der Gold-Silber-Legierung vollständig eingetaucht ist.
Ermittle damit, ob der Unterschied der Wasserstände präzise genug messbar ist, um einen Unterschied zwischen den Metallstücken nachzuweisen.
Alternativ wird angenommen, dass Archimedes stattdessen das verminderte Gewicht gemessen hat, das der Auftrieb im Wasser erzeugt. Berechne dazu die Auftriebskräfte für die beiden Metallstücke und deren Differenz, wenn sie in Wasser eingetaucht sind. Bestimme die Masse, die dieser Kraftdifferenz entspricht.
Lösungen {#lösungen .unnumbered}
--------
### 1. Wasserdruck (rechnerisch) {#wasserdruck-rechnerisch-1 .unnumbered}
$$h = \frac{p -p_0}{\varrho_w \cdot g} = \frac{101325\,\mathsf{Pa}}{997\,\mathsf{kg/m}^3 \cdot 9,81\,\mathsf{m/s}^2} = 10,4\,\mathsf{m} \nonumber$$
### 2. Wasserduck (grafisch) {#wasserduck-grafisch .unnumbered}
Die Geradengleichung lautet:
$$p(h) = \left(\varrho_w \cdot g\right) \cdot h + p_0$$
Steigung: $\varrho_w \cdot g = \unit[997]{\frac{kg}{m^3} \cdot \unit[9,81]{\frac{m}{s^2}} = \unit[9780,57]{\frac{kg}{m^2\cdot s^2}}}$
Achsenabschnitt: $p_0 = \unit[101325]{Pa} = \unit[101325]{\frac{kg}{m\cdot s^2}}$
Für $p(h) = 2\cdot p_0$ muss gelten:
$$\unit[9780,57]{\frac{kg}{m^2\cdot s^2}} \cdot h = p_0 = \unit[101325]{\frac{kg}{m\cdot s^2}}$$
\[f:graph\]
### 3. Wasserverdrängung {#wasserverdrängung-1 .unnumbered}
Wenn der Astronaut mit Raumanzug im Wasser schwebt, stehen Schwerkraft und Auftriebskraft im Gleichgewicht. Gemäß Gl. \[e:Fgg\] gilt:
$$\begin{aligned}
F_g &= F_a \notag \\[5pt]
\Leftrightarrow m\cdot g &= m_w \cdot g \notag \\[5pt]
\Leftrightarrow m &= \varrho_w \cdot V \notag \\[5pt]
\Leftrightarrow V &= \frac{m}{\varrho_w} \notag \\[5pt]
&= \frac{200\,\mathsf{kg}}{997\,\mathsf{kg/m}^3} = 0,2\,\mathsf{m}^3 \notag\end{aligned}$$
### 4. Das Archimedische Prinzip {#das-archimedische-prinzip-1 .unnumbered}
Aus den Werten in Tab. \[t:groessen\] kann man die Dichte der Legierung berechnen, die einen Silberanteil von 30% hat. $$\varrho_{70/30} = 0,7\cdot\varrho_\mathsf{Au} + 0,3\cdot\varrho_\mathsf{Ag} = 1672\,\mathsf{kg/m}^3$$
Metallstücke aus Gold, Silber und der Legierung von je besitzen die folgenden Volumina.
$$\begin{aligned}
V_\mathsf{Au} &= \frac{m}{\varrho_\mathsf{Au}} = \frac{1000\,\mathsf{kg}}{1939\,\mathsf{kg/m}^3} = 0,5157\,\mathsf{m}^3 \notag \\[5pt]
V_\mathsf{Ag} &= \frac{m}{\varrho_\mathsf{Ag}} = \frac{1000\,\mathsf{kg}}{1049\,\mathsf{kg/m}^3} = 0,9533\,\mathsf{m}^3 \notag \\[5pt]
V_{70/30} &= \frac{m}{\varrho_{70/30}} = \frac{1000\,\mathsf{kg}}{1672\,\mathsf{kg/m}^3} = 0,5981\,\mathsf{m}^3 \notag\end{aligned}$$
Im relativen Vergleich $\frac{V_{70/30}}{V_\mathsf{Au}} = 1,16$ scheint der Unterschied sehr wohl messbar zu sein. Für ein Wasservolumen von in einem Becher mit dem Radius erhält man folgende Wasserstände.
$$h=\frac{V}{A}=\frac{V}{\pi\cdot r^2}$$
1. $V_w = 25000\,\mathsf{cm}^3 \Rightarrow h = 12,73\,\mathsf{cm}$
2. $V_w+V_\mathsf{Au} = 25051,57\,\mathsf{cm}^3 \Rightarrow h = 12,76\,\mathsf{cm}$
3. $V_w+V_{70/30} = 25059,81\,\mathsf{cm}^3 \Rightarrow h = 12,76\,\mathsf{cm}$
Der Wasserstand verändert sich kaum. Daher ist es unwahrscheinlich, dass Archimedes den Betrug auf diese Weise aufgedeckt hat.
Die Kraft des Auftriebs ist definiert als:
$$F_a = \varrho_w\cdot V \cdot g = \varrho_w\cdot \frac{m}{\varrho_\mathsf{Metall}} \cdot g = \frac{\varrho_w}{\varrho_\mathsf{Metall}} \cdot m \cdot g$$
Somit erhält man für das Verhältnis der Auftriebskräfte:
$$\frac{F_{a,70/30}}{F_{a,\mathsf{Au}}} = \frac{V_{70/30}}{V_\mathsf{Au}} = 1,16$$
Die Auftriebskräfte für Gold und der Legierung lauten:
$$\begin{aligned}
F_{a,70/30} &= \frac{997\,\mathsf{kg/m}^3}{1672\,\mathsf{kg/m}^3} \cdot 1\,\mathsf{kg} \cdot 9,81\,\mathsf{m/s}^2 = 5,850\,\mathsf{N}\\[5pt]
F_{a,\mathsf{Au}} &= \frac{997\,\mathsf{kg/m}^3}{1939\,\mathsf{kg/m}^3} \cdot 1\,\mathsf{kg} \cdot 9,81\,\mathsf{m/s}^2 = 5,044\,\mathsf{N}\end{aligned}$$
Die Differenz beträgt $F_{a,70/30} - F_{a,\mathsf{Au}} = 0,806\,\mathsf{N}$. Das entspricht einer Masse von . Dies war auch zu Zeiten von Archimedes gut messbar.
Danksagung {#danksagung .unnumbered}
==========
Der Autor bedankt sich bei den Lehrern Matthias Penselin, Florian Seitz und Martin Wetz für ihre wertvollen Hinweise, Kommentare und Änderungsvorschläge, die in die Erstellung dieses Materials eingeflossen sind. Weiterer Dank gilt Herrn Dr. Volker Kratzenberg-Annies für seine gewissenhafte Durchsicht.
Diese Unterrichtsmaterialien sind im Rahmen des Projekts [*Raum für Bildung*]{} am Haus der Astronomie in Heidelberg entstanden. Experimente dürfen nur unter Aufsicht einer Lehrkraft durchgeführt werden. Der Autor und die Projektpartner übernehmen keine Haftung für etwaig auftretende Verletzungen. Weitere Materialien des Projekts finden Sie unter:
<http://www.haus-der-astronomie.de/raum-fuer-bildung> und <http://www.dlr.de/next>
Das Projekt findet in Kooperation mit dem Deutschen Zentrum für Luft- und Raumfahrt statt und wird von der Joachim Herz Stiftung gefördert.
{height="1.5cm"} {height="1.5cm"} {height="1.5cm"}
|
---
abstract: |
The classical Yao principle states that the complexity $R_\epsilon(f)$ of an optimal *randomized* algorithm for a function $f$ with success probability $1-\epsilon$ equals the complexity $\max_\mu D_\epsilon^\mu(f)$ of an optimal *deterministic* algorithm for $f$ that is correct on a fraction $1-\epsilon$ of the inputs, weighed according to the hardest distribution $\mu$ over the inputs. In this paper we investigate to what extent such a principle holds for quantum algorithms. We propose two natural candidate quantum Yao principles, a “weak” and a “strong” one. For both principles, we prove that the quantum bounded-error complexity is a lower bound on the quantum analogues of $\max_\mu D_\epsilon^\mu(f)$. We then prove that equality cannot be obtained for the “strong” version, by exhibiting an exponential gap. On the other hand, as a positive result we prove that the “weak” version holds up to a constant factor for the query complexity of all symmetric Boolean functions.\
[**Keywords:**]{} Quantum computing, computational complexity.
author:
- Mart de Graaf
- Ronald de Wolf
title: 'On Quantum Versions of the Yao Principle[^1]'
---
Introduction
============
Motivation
----------
In classical computing, the *Yao principle* [@yao:unified] gives an equivalence between two kinds of randomness in algorithms: randomness inside the algorithm itself, and randomness on the inputs. Let us fix some model of computation for computing a Boolean function $f$, like query complexity, communication complexity, etc. Let $R_\epsilon(f)$ be the minimal complexity among all *randomized* algorithms that compute $f(x)$ with success probability at least $1-\epsilon$, for all inputs $x$. Let $D_\epsilon^\mu(f)$ be the minimal complexity among all *deterministic* algorithms that compute $f$ correctly on a fraction of at least $1-\epsilon$ of all inputs, weighed according to a distribution $\mu$ on the inputs. The Yao principle now states that these complexities are equal if we look at the “hardest” input distribution $\mu$: $$R_\epsilon(f) = \max_\mu D_\epsilon^\mu(f).$$ This is a special case of Von Neumann’s minimax theorem in game theory [@neumann47theory; @owen:gametheory].
Since its introduction, the Yao principle has been an extremely useful tool in computational complexity analysis. In particular, it allows us to derive lower bounds on randomized algorithms from lower bounds on deterministic algorithms: choose some “hard” input distribution $\mu$, prove a lower bound on deterministic algorithms that compute $f$ correctly for “most” inputs, weighted according to $\mu$, and then use $R_\epsilon(f)\geq D_\epsilon^\mu(f)$ to get a lower bound on $R_\epsilon(f)$. This method is used very often, because it is usually much easier to analyze deterministic algorithms than to analyze randomized ones.
In recent years *quantum computation* received a lot of attention. Here quantum mechanical principles are employed to realize more efficient computation than is possible with a classical computer. Famous examples are Shor’s polynomial-time factoring algorithm [@shor:factoring] and Grover’s search algorithm [@grover:search]. However, the field is still young and open questions are abundant. In particular, there has been a search for good techniques to provide lower bounds on quantum algorithms. Most of these lower bounds are in the *query model*, where the complexity of an algorithm is measured by the number of queries it needs in order to compute some function (we will provide formal definitions of this and other concepts in the next section). Two general methods in this direction are the *polynomial method* introduced by Beals, Buhrman, Cleve, Mosca, and de Wolf [@bbcmw:polynomials] and the method of *quantum adversaries* of Ambainis [@ambainis:lowerbounds]. In this paper we investigate the possibility of a third method, a *quantum Yao principle*. It is our hope that such a principle will prove itself useful as a link between techniques for lower bounds on exact and bounded-error quantum algorithms.
The first difficulty one runs into when investigating a quantum version of the Yao principle, is the question what the proper quantum counterparts of $R_\epsilon(f)$ and $D_\epsilon^\mu(f)$ are. Let us fix the error probability at $\epsilon=\frac{1}{3}$ here (any other value in $(0,\frac{1}{2})$ would do as well). The quantum analogue of $R_{1/3}(f)$ is straightforward: let $Q_2(f)$ denote the minimal complexity among all *quantum* algorithms that compute $f(x)$ with probability at least $\frac{2}{3}$, for all inputs $x$. However, the inherently “random” nature of quantum algorithms prohibits a straightforward definition of “deterministic” quantum algorithms in analogy of deterministic classical algorithms. We therefore propose two different definitions, a weak and a strong one. In the following, let $f:D\rightarrow\01$ be some function that we want to compute, with $D\subseteq\01^N$. If $D={\{ 0,1 \}}^N$ then $f$ is a *total* function, otherwise $f$ is a *promise* function. Let $A$ be a quantum algorithm, $P_A(x)$ the acceptance probability of $A$ on input $x$ (the probability of outputting 1 on input $x$, and $\mu:D\rightarrow[0,1]$ a probability distribution over the inputs.
$A$ is *weakly $\frac{2}{3}$-exact* for $f$ with respect to $\mu$ iff $\mu({\{ x \mid P_A(x)=f(x) \}}) \geq \frac{2}{3}$.
$A$ is *strongly $\frac{2}{3}$-exact* for $f$ with respect to $\mu$ iff $A$ is weakly $\frac{2}{3}$-exact for $f$ with respect to $\mu$ and $P_A(x) \in {\{ 0,1 \}}$ for all inputs $x \in {\{ 0,1 \}}^N$.
Informally, in the second definition we require the algorithm to output the same output on the same input, even on inputs $x\in D$ where the algorithm fails and even on $x\in{\{ 0,1 \}}^N\backslash D$ (similar to a classical deterministic algorithm). In the first definition, we only require this “input-determines-output” behavior to occur for a $\mu$-fraction of at least $\frac{2}{3}$ of the inputs where the algorithm gives the correct output $f(x)$. Note that a strongly $\frac{2}{3}$-exact algorithm for $f$ with respect to $\mu$ actually computes some total function $g:{\{ 0,1 \}}^N \rightarrow\01$ with success probability 1, namely the function $g(x)=P_A(x)$. This $g$ will agree with $f$ on at least $\frac{2}{3}$ of the inputs.
These two definitions lead to a weak and a strong quantum counterpart to the classical distributional complexity $D^\mu_{1/3}(f)$: let $Q_{WE}^\mu(f)$ and $Q_{SE}^\mu(f)$ denote the minimal complexity among all weakly and strongly $\frac{2}{3}$-exact algorithms for $f$ with respect to $\mu$, respectively. We can now state two potential quantum versions of the Yao principle:
- Strong quantum Yao principle: $\displaystyle Q_2(f) \stackrel{?}{=} \max_\mu Q_{SE}^\mu(f)$
- Weak quantum Yao principle: $\displaystyle Q_2(f) \stackrel{?}{=} \max_\mu Q_{WE}^\mu(f)$
In this paper we investigate to what extent these two quantum Yao principles hold.
Results
-------
Our results are threefold. Firstly, we prove that both of these principles hold in the ‘$\leq$’-direction, for all $f$:
- $\displaystyle Q_2(f) \leq \max_\mu Q_{SE}^\mu(f)$
- $\displaystyle Q_2(f) \leq \max_\mu Q_{WE}^\mu(f)$
Clearly, the second inequality implies the first, since $Q_{WE}^\mu(f)\leq Q_{SE}^\mu(f)$ for all $f$ and $\mu$. The proof is similar to the classical game-theoretic proof, with a bit more technical complication. We emphasize that this result is perfectly general, and applies to all computational models to which the classical Yao principle applies.
In order to investigate to what extent the ‘$\geq$’-directions of these two quantum Yao principles hold, we instantiate our complexity measures to the query complexity setting. Our second result is an exponential gap between $Q_2(f)$ and $Q_{SE}^\mu(f)$ for the query complexity of Simon’s problem [@simon:power]:
- There exist $f$ and $\mu$ such that $Q_2(f)$ is exponentially smaller than $Q_{SE}^\mu(f)$.
This shows that the strong quantum Yao principle is false. Thirdly, we prove that the weak quantum Yao principle holds up to a constant factor for the query complexity of all *symmetric* functions:
- $\displaystyle Q_2(f)=\Theta\left(\max_\mu Q_{WE}^\mu(f)\right)$ for all symmetric $f$
For this result we first construct a quantum algorithm that can determine the $N$-bit input $x$ *with certainty* in $O(\sqrt{kN})$ queries if $k$ is a known upper bound on the Hamming weight of $x$. We then use that algorithm to construct, for every symmetric function $f$ and distribution $\mu$, a quantum algorithm that computes $f(x)$ with certainty for “most” inputs $x$. In addition to this result for symmetric functions, we also show that for a particular *monotone* non-symmetric function $f$, the $\max_\mu Q_{WE}^\mu(f)$ complexity lies in between the best known bounds for $Q_2(f)$.
Preliminaries
=============
In this section we formalize the notion of query complexity, define several complexity measures, state Von Neumann’s minimax theorem and derive the classical Yao principle from it.
Query Complexity
----------------
We assume familiarity with classical computation theory and briefly sketch the basics of quantum computation; an extensive introduction may be found in the book by Nielsen and Chuang . Quantum algorithms operate on *qubits* as opposed to bits in classical computers. The state of an $m$-qubit quantum system can be written as $${\vert \phi \rangle} = \sum_{i \in {\{ 0,1 \}}^m} \alpha_i {\vert i \rangle},$$ where ${\vert i \rangle}$ denotes the basis state $i$, which is a classical $m$-bit string. The $\alpha_i$’s are complex numbers known as the *amplitudes* of the basis states ${\vert i \rangle}$ and we require $\sum_{i\in {\{ 0,1 \}}^m} |\alpha_i|^2 = 1$. Mathematically, the state of a system is thus described by a $2^m$-dimensional complex unit vector. If we measure the value of ${\vert \phi \rangle}$, then we will see the basis state ${\vert i \rangle}$ with probability $|\alpha_i|^2$, after which the system collapses to ${\vert i \rangle}$. Operations which are not measurements on a system of qubits correspond to *unitary transformations* on the vector of amplitudes.
In the *query model* of computation, the goal is to compute some function $f:D \to {\{ 0,1 \}}$ on an input $x\in D\subseteq{\{ 0,1 \}}^N$, using as few accesses (“queries”) to the $N$ input bits as possible. In quantum algorithms, it is by now standard to formalize a query as an application of a unitary transformation $O$ that acts as follows: $$O{\vert i,b,z \rangle} = {\vert i,b \oplus x_i,z \rangle}.$$ Here $i \in {\{ 1,\ldots,N \}}$, $b\in{\{ 0,1 \}}$, $\oplus$ denotes the exclusive-or function, and $z$ denotes the workspace of the algorithm, which is not affected by $O$. A $T$-query quantum algorithm $A$ then has the form $$A=U_TOU_{T-1}O\cdots U_1OU_0,$$ with each $U_i$ a fixed unitary transformation independent of the input $x$. $A$ is assumed to start in the all-zero state ${\vert 0\ldots0 \rangle}$, and its output (0 or 1) is obtained by measuring the rightmost bit of its final state $A{\vert 0\ldots0 \rangle}$. The *acceptance probability* $P_A(x)$ of a quantum algorithm $A$ is defined as the probability of getting output $1$ on input $x$. Its *success probability* $S_A(x)$ is the probability of getting the correct output $f(x)$ on input $x$.
A quantum algorithm $A$ computes a function $f:D\rightarrow{\{ 0,1 \}}$ *exactly* if $S_A(x)=1$ for all inputs $x\in D$. Algorithm $A$ computes $f$ with *bounded-error* if $S_A(x) \geq \frac{2}{3}$ for all $x\in D$. We use $Q_E(f)$ and $Q_2(f)$ to denote the minimal number of queries required by exact and bounded-error quantum algorithms for $f$, respectively. These complexities are the quantum versions of the classical deterministic and bounded-error decision tree complexities $D(f)$ and $R_2(f)$, respectively. For completeness, we repeat our two alternative quantum versions of the classical distributional complexity $D^\mu(f)$ from the introduction. Let $\mu$ be a probability distribution on the set of all possible inputs. An algorithm $A$ is *weakly $\frac{2}{3}$-exact for $f$ with respect to $\mu$* if $\mu({\{ x \mid P_A(x) = f(x) \}}) \geq \frac{2}{3}$, and $A$ is *strongly $\frac{2}{3}$-exact for $f$ with respect to $\mu$* if $A$ is weakly $\frac{2}{3}$-exact for $f$ with respect to $\mu$ and $P_A(x) \in {\{ 0,1 \}}$ for all $x\in{\{ 0,1 \}}^N$. By $Q_{SE}^\mu(f)$ and $Q_{WE}^\mu(f)$ we denote the minimal number of queries needed by strongly and weakly $\frac{2}{3}$-exact quantum algorithms for $f$ with respect to $\mu$, respectively. Note that $Q_{WE}^\mu(f)\leq Q_{SE}^\mu(f)$ for all $f$ and $\mu$, hence in particular $\max_\mu Q_{WE}^\mu(f)\leq\max_\mu Q_{SE}^\mu(f)$.
One of the first quantum algorithms operating in the query model is Grover’s search algorithm [@grover:search; @bhmt:countingj]. If $t=|x|>0$ then the algorithm uses $\frac{\pi}{4}\sqrt{N/t}$ queries and with high probability outputs an $i$ such that $x_i=1$. Here we use $|x|$ to denote the Hamming weight (number of 1’s) in $x$, and $x_i$ to denote the $i$th bit of $x$. If $|x|=0$ then the algorithm outputs ‘no solutions’. Brassard, Høyer, Mosca, and Tapp [@bhmt:countingj] give an exact version of Grover’s algorithm that can accomplish the same task with probability 1 if $t$ (the number of 1’s in the input) is known.
For $\emph{total}$ functions $f:{\{ 0,1 \}}^N \to {\{ 0,1 \}}$, Beals, Buhrman, Cleve, Mosca, and de Wolf [@bbcmw:polynomials] proved that classical deterministic query complexity $D(f)$ is polynomially related to the exact and bounded-error quantum complexities: $D(f) = O(Q_E(f)^4)$ and $D(f) = O(Q_2(f)^6)$.
A function $f:{\{ 0,1 \}}^N \to {\{ 0,1 \}}$ is *symmetric* if its value $f(x)$ depends only on $|x|$. For such $f$, define $f_k = f(x)$ where $|x|=k$. In [@bbcmw:polynomials] it is proven that $Q_2(f) = \Theta(\sqrt{N(N-\Gamma(f))})$, where $\Gamma(f) =
\min {\{ |2k-N-1| \mid f_k \ne f_{k+1} \mathrm{~and~} 0 \leq k \leq N-1 \}}$. Informally, the quantity $\Gamma(f)$ measures the length of the interval around Hamming weight $\frac{N}{2}$ where $f$ is constant. A symmetric function $f$ is a *threshold* function if there is a $0 < t \leq N$, such that $f(x)=1$ iff $|x| \geq t$. Note that for $t\leq N/2$ we have $Q_2(f)=\Theta(\sqrt{tN})$ as a direct consequence of the bound for symmetric functions. A function $f:\01^n \to \01$ is *monotone* if $(\forall i\ x_i \leq y_i) \Rightarrow f(x) \leq f(y)$.
The Classical Yao Principle
---------------------------
Consider the following game-theoretic setting: player 1 has a choice between some $m$ “pure” strategies and player 2 has a choice between $n$ “pure” strategies. If player 1 plays $i$ and player 2 plays $j$, then player 1 receives “payoff” $P_{ij}$. Player 1 wants to maximize the payoff, player 2 wants to minimize. Viewing $P$ as an $m\times n$ matrix, and using $e_i$ and $e_j$ to denote the appropriate unit column vectors with a 1 in place $i$, respectively $j$, the payoff corresponds to the matrix product $e_i^TPe_j$. However, the players may also use “mixed” strategies (probability distributions over “pure” strategies) to further their goals. Mixed strategies of players 1 and 2 correspond to $m$- and $n$-dimensional column vectors $\rho$ and $\mu$, respectively, of non-negative reals that sum to 1. Now the *expected* payoff is $\rho^TP\mu$. Note that if player 1 can choose his strategy $\rho$ knowing player 2’s strategy $\mu$, then he would choose $\rho$ to maximize the payoff $\rho^TP\mu$; in this situation player 2 would do best to choose $\mu$ to minimize $\max_{\rho}\rho^TP\mu$, giving expected payoff $\min_\mu\max_\rho \rho^TP\mu$. Conversely, if player 2 could choose his strategy knowing player 1’s strategy, then the expected payoff would be $\max_\rho\min_\mu\rho^TP\mu$. Von Neumann’s famous minimax theorem [@neumann47theory; @owen:gametheory] tells us that these two quantities are in fact equal: $$\min_\mu \max_\rho \rho^TP\mu = \max_\rho \min_\mu \rho^TP\mu.$$ It is not hard to see that without loss of generality the “inner” choices can be assumed to be pure strategies, so as an easy consequence we also have $$\min_\mu \max_i e_i^TP\mu = \max_\rho \min_j \rho^TPe_j.$$ Yao [@yao:unified] was the first to interpret this result in computational terms. We will sketch the computational interpretation below. Fix some classical model of computation for which the set of deterministic algorithms of complexity $\leq c$ is finite, for every $c$. Examples of such models are query complexity, communication complexity, etc. Player 1 chooses an algorithm to compute $f:D\rightarrow{\{ 0,1 \}}$ and player 2 chooses an input $x$ that is hard for player 1. The pure strategies for player 1 are all *deterministic* classical algorithms of complexity $\leq c$ and hence his mixed strategies are all *randomized* classical algorithms of complexity $\leq c$. The pure strategies for player 2 are the inputs in $D$ and his mixed strategies are all probability distributions $\mu$ over $D$. We define the payoff matrix such that $P_{ix}=1$ if algorithm $i$ computes $f$ correctly on input $x$, and $P_{ix}=0$ otherwise. In this setting, the minimax theorem states $$\min_\mu \max_i e_i^TP\mu = \max_\rho \min_x \rho^TPe_x.$$ Let us interpret both sides of this equation. On the left, the quantity $e_i^TP\mu$ is the fraction of inputs on which deterministic algorithm $i$ is correct, weighed according to $\mu$, and $\max_i e_i^TP\mu$ denotes this fraction for the optimal deterministic algorithm of complexity $\leq c$. Thus the left-hand-side of the equation gives this optimal correct fraction for the hardest distribution $\mu$ achievable by deterministic complexity-$c$ algorithms. On the other hand, $\rho^TPe_x$ is the success probability on input $x$ achieved by the randomized algorithm given by probability distribution $\rho$ over deterministic algorithms, and $\min_x \rho^TPe_x$ is its success probability on the hardest input. Thus the right-hand-side gives the highest worst-case success probability achievable by randomized complexity-$c$ algorithms. Since these two quantities are equal for all $c$, we obtain the classical Yao principle: $$R_\epsilon(f) = \max_\mu D_\epsilon^\mu(f).$$
Proof of One Half of the Quantum Yao Principle
==============================================
As a first result we prove that $Q_2(f) \leq \max_{\mu}Q_{WE}^{\mu}(f)$. The proof is similar to the derivation of the classical Yao principle above, but the details are a bit more messy.
\[thm:Q2lowerbound\] For all $f:D \to {\{ 0,1 \}}$, with $D$ finite, $\displaystyle Q_2(f) \leq \max_{\mu}Q_{WE}^{\mu}(f)$.
Consider the (infinite) set of all quantum algorithms of complexity $\leq \max_{\mu}Q_{WE}^\mu(f)$. Let $i$ be any algorithm from this set, and $x \in D$ an input. Consider the quantity $\lfloor S_i(x) \rfloor$, which is 1 if algorithm $i$ computes $f(x)$ with success probability 1, and which is 0 otherwise. Call algorithms $i$ and $j$ *similar* if $\lfloor S_i(x) \rfloor=\lfloor S_j(x) \rfloor$ for all $x\in D$. In this way, similarity is an equivalence relation on the set of all quantum algorithms of complexity $\leq \max_{\mu}Q_{WE}^\mu(f)$. Note that this relation has at most $2^{|D|}$ equivalence classes. From each equivalence class, we choose as a representative an algorithm from that class with the least complexity.
Now consider the game in which player 1 wants to compute $f$, and as pure strategies he has available the (finite) set of representatives of the equivalence classes. Player 2 is an adversary that tries to make life as hard as possible for player 1 by choosing hard inputs $x\in D$ to $f$. Let $S$ be the matrix of success probabilities ($S_{ix}=S_i(x)$). Define the payoff matrix as $P_{ix} = \lfloor S_{ix} \rfloor$. Now consider the quantity $\max_i e_i^TP\mu$. This represents the $\mu$-fraction of inputs on which the best weakly $\frac{2}{3}$-exact quantum algorithm for $f$ with respect to that $\mu$ is correct. By construction, this quantity is at least $\frac{2}{3}$ for all $\mu$. Using the minimax theorem, we now obtain: $$\frac{2}{3} \leq \min_\mu \max_i e_i^T P\mu =
\max_\rho \min_x \rho^T P e_x \leq \max_\rho \min_x \rho^T S e_x.$$ Here the last term can be interpreted as the success probability of a quantum algorithm formed by a probability distribution $\rho$ over the set of representatives of the equivalence classes. By the above inequality, this algorithm has success probability $\geq\frac{2}{3}$ for all inputs $x\in D$. Since it is a probability distribution over algorithms of complexity $\leq \max_{\mu}Q_{WE}^\mu(f)$, its complexity is at most $\max_{\mu}Q_{WE}^\mu(f)$. Hence
\[cor:Q2lowerbound\] For all $f:D \to {\{ 0,1 \}}$, with $D$ finite, $\displaystyle Q_2(f) \leq \max_{\mu}Q_{SE}^{\mu}(f)$.
Note that although we restrict our attention to the query model of computation, the proofs of Theorem \[thm:Q2lowerbound\] and Corollary \[cor:Q2lowerbound\] also work for the other models of complexity where the classical Yao principle applies.
A Counterexample for the Strong Quantum Yao Principle
=====================================================
In this section we prove that the strong quantum Yao principle does not hold. There exists a problem $f$ such that for a suitable distribution $\mu$, $Q_2(f)$ is exponentially smaller than $Q_{SE}^\mu(f)$. This exponential gap follows from a known result about the classical and quantum complexity of *Simon’s problem* [@simon:power], and the fact that classical deterministic and quantum exact complexity are polynomially related for total problems [@bbcmw:polynomials Theorem 5.4].
\[thm:exponentialGap\] There exist a problem $f$ and a distribution $\mu$ such that $Q_2(f)
=O(n^2)$ and $Q_{SE}^{\mu}(f)=\Omega(2^{\frac{n}{8}})$.
Consider Simon’s problem: given a function $\phi:\{0,1\}^n \to \{0,1\}^n$ with the promise that there is an $s \in \{0,1\}^n$ such that $\phi(a)=\phi(b)$ iff $a \oplus b =s$, decide whether $s=0$ or not. This function $\phi$ is given as an input $x$ of $N=n2^n$ bits, using $n$ 1-bit entries for each function value $\phi(\cdot)$. The input bits can be queried in the usual way. Using Simon’s bounded-error quantum algorithm, this problem can be solved in $O(n^2)$ queries, and hence $Q_2(Simon) =
O(n^2)$. Now define a distribution $\mu$ which uniformly places half the total weight on inputs with $s=0$ and half the total weight on inputs with $s \ne 0$: $$\mu(x) = \left\{ \begin{array}{ll}
\frac{1}{2(2^n)!} & \textrm{if $s = 0$} \\
\frac{1}{2(2^n-1){2^n \choose 2^{n-1}}(2^{n-1})!} &
\textrm{if $s \ne 0$}\\
0 & \textrm{else.}
\end{array} \right.$$ Simon proved that under this distribution, any classical algorithm that is correct on a fraction $\geq \frac{2}{3}$ requires $\Omega(\sqrt{2^n})$ queries. Now take any strongly $\frac{2}{3}$-exact quantum algorithm $A$ that solves this problem and makes $T$ queries, then $A$ computes some *total* function $g$. Since $D(g) = O(Q_E(g)^4)$, this implies that there exists a deterministic classical algorithm that computes $g$ using $O(T^4)$ queries. But this classical algorithm is then exact on a $\mu$-fraction $\frac{2}{3}$ of all Simon inputs. Simon’s lower bound on classical algorithms now implies that $O(T^4)=\Omega(\sqrt{2^n})$, and hence $Q_{SE}^\mu(Simon) = \Omega(2^{\frac{n}{8}})$.
A Positive Result for the Weak Quantum Yao Principle
====================================================
In this section we show that the weak quantum Yao principle holds for all symmetric functions. This section is divided into three subsections, in the first we prove the result for threshold functions, in the second subsection, we extend it to symmetric functions. In the third subsection we investigate the weak quantum Yao principle for the uniform 2-level AND-OR tree, which is monotone and non-symmetric.
Equality up to a Constant Factor for Threshold Functions
--------------------------------------------------------
For every distribution $\mu$, we will exhibit a weakly $\frac{2}{3}$-exact quantum algorithm that computes threshold function $f$ with threshold $t$ in time $O(\sqrt{tN})$. This, together with Theorem \[thm:Q2lowerbound\] and the (known) fact that $Q_2(f)=\Theta(\sqrt{tN})$ for threshold functions $f$ [@bbcmw:polynomials], gives the desired result.
Note that given a threshold function $f:{\{ 0,1 \}}^N \to {\{ 0,1 \}}$ with threshold $t$, in order to be sure that $f(x)=1$, one will have to find at least $t$ 1’s in the input. The crucial idea behind our algorithm is that if the number of 1’s in the input is large enough, then for each distribution $\mu$ over the inputs, we can pick a substantially smaller part of the input such that there are between $t$ and $100t$ 1’s in this subpart for a large $\mu$-fraction of the inputs. This idea is formally stated in the following technical lemma.[^2]
\[lem:probability\] Let $t$ be a threshold, $\mu$ a probability distribution over the $x \in {\{ 0,1 \}}^N$, and $i$ an integer such that $10 \leq i \leq \log N - \log t - 1$. Denote the event $t2^i \leq |x| \leq t2^{i+1}$ by $I$, and let $x \wedge y$ denote the bitwise AND of $x$ and $y$. There is a $y \in \{0,1\}^N$ with $|y|=\min {\{ \frac{10N}{2^i},N \}}$, such that $\mathrm{Pr}_{\mu}[t \leq |x \wedge y| \leq 100t \mid I] > 0.7$.
Fix an $x \in \{0,1\}^N$ with $t2^i \leq |x| \leq t2^{i+1}$ and assume that $\frac{10N}{2^i} \leq N$, for otherwise the lemma trivially holds. We claim that if we pick a $y \in \{0,1\}^N$ with $|y|=\frac{10N}{2^i}$ uniformly at random, then $\mathrm{Pr}[t \leq |x \wedge y| \leq 100t \mid I] > 0.7$. To prove this claim, note that $$\mathrm{Pr}[|x \wedge y| = k \mid I] =
\frac{{|x| \choose k}{N-|x| \choose |y| - k}}
{{N \choose |y|}}.$$ This means that $|x\wedge y|$ is hypergeometrically distributed, with expected value $E(|x\wedge y|) = \frac{|x||y|}{N}$. Note that in this case $10t \leq E(|x\wedge y|) \leq 20t$. By Markov’s inequality, it then follows directly that $\mathrm{Pr}[|x \wedge y| > 100t \mid I] \leq 0.2$.
We can approximate the above distribution with a binomial distribution since the number of draws is small compared to the size of the sample space, see e.g. [@nicholson:normal], and we shall henceforth treat $|x \wedge y|$ as if it were binomially distributed, with success probability $\theta = \frac{|x|}{N}$ and number of draws $n=|y|$. To bound $\mathrm{Pr}[|x \wedge y| < t \mid I]$, we use the Chernoff bound as explained in [@motwani:ra pp.67-73]: $$\mathrm{Pr}[|x \wedge y| < (1-\delta)E(|x\wedge y|) \ \mid \ I] <
e^{\frac{-\delta^2E(|x\wedge y|)}{2}}.$$ Choosing $\delta = \frac{9}{10}$, we obtain $\mathrm{Pr}[
|x \wedge y| < t \mid I] < e^{-\frac{810t}{200}} < 0.1$. Combining the previous two inequalities, it then follows that $\mathrm{Pr}[t \leq |x \wedge y| \leq 100t \mid I] > 0.7$. This proves the above claim.
Now imagine a matrix whose rows are indexed by the $x$ satisfying $t2^i \leq |x| \leq t2^{i+1}$ and whose columns are indexed by the $M = {N \choose |y|}$ different $y$ of weight $|y| = \frac{10N}{2^i}$. We give the $(x,y)$ entry of this matrix value $\mu(x|I)$ if $t \leq |x\wedge y| \leq 100t$ and value 0 otherwise. By the above claim, each $x$ row will contain at least 70% non zero entries, so the sum of the entries of each $x$ row is at least $0.7M\mu(x|I)$. Hence, the sum of all entries in the matrix is equal to $\sum_x 0.7M\mu(x|I)=0.7M$. But then there must be a column with $\mu$-weight at least 0.7. The $y$ corresponding to this column is the $y$ we are looking for in this lemma.
We will use the fact stated in the previous lemma to successively search for $t$ 1’s in exponentially smaller parts of the inputs, assuming the presence of increasingly more 1’s in the original input. The following lemma states that this searching can be done efficiently:
\[lem:count\] There exists a quantum algorithm that can find all the 1’s in an input $x$ of size $N$ with probability 1, using at most $\frac{\pi}{2}\sqrt{kN}$ queries, if $k$ is a known upper bound on the number of 1’s in $x$.
Consider Algorithm \[alg:ronald\]. It is easily proven that this algorithm indeed finds all 1’s, as follows. Assume an upper bound $k \geq |x|$ on the number of 1’s in $x$. If the exact version of Grover’s algorithm finds an index of a 1 bit, then we set this index to 0 in the search space. Because $k$ is an upper bound on the number of 1’s in $x$, we can lower $k$ each time we find a 1, without $k$ ever becoming less than the actual number of 1’s in $x$. If it does not find a 1, then we know that our upper bound was too high and again we can safely lower it by 1. Using these facts, it is easily proven by induction on $k$ that the algorithm indeed works as claimed.
Apply Grover’s exact search algorithm assuming there are $i$ solutions. mark its index as a zero in the search space output the positions of all solutions found
The number of queries made by this algorithm is at most:
$$\sum_{i=1}^k \frac{\pi}{4} \sqrt{\frac{N}{i}}
\leq \frac{\pi}{4}\sqrt{N} \int_{0}^{k} \frac{\mathrm{d}i}{\sqrt{i}}
= \frac{\pi}{2}\sqrt{kN}.$$
We are now ready to prove an upper bound on $Q_{WE}^\mu(f)$ for threshold functions.
\[lem:threshold\] For threshold function $f$ with threshold $t$, and for every distribution $\mu$, we have $Q_{WE}^{\mu}(f) = O(\sqrt{tN})$.
Fix a distribution $\mu$. Invoking Lemmas \[lem:probability\] and \[lem:count\], our algorithm is as follows. First we count the number of 1’s in the input using Algorithm \[alg:ronald\], assuming an upper bound of $2^{10}t$ 1’s. If after that we haven’t found at least $t$ 1’s yet, then we successively assume that there are between $t2^i$ and $t2^{i+1}$ 1’s in the input, with $i$ going up from 10 to $\log N - \log t - 1$. For each of these assumptions, we search a smaller part of the input. If we have reached the $i$ for which $t2^i \leq |x| \leq t2^{i+1}$, then Lemma \[lem:probability\] guarantees us that for a large $\mu$-fraction of the inputs we can find a small subpart containing between $t$ and $100t$ 1’s. We then count the number of 1’s in this subpart using Algorithm \[alg:ronald\]. Algorithm \[alg:thresh\] is the actual algorithm we will use.
Count the number of 1’s in the input using Algorithm \[alg:ronald\], assuming an upper bound of $2^{10}t$ 1’s output 1 Let $y^{(i)} \in \{0,1\}^N$ be a string of weight $\min {\{ N,\frac{10N}{2^i} \}}$ satisfying Lemma \[lem:probability\] Using Algorithm \[alg:ronald\], count the number of solutions in the subpart of the input induced by $y^{(i)}$, assuming an upper bound of $100t$ 1’s. output 1 output 0
This algorithm will be correct on all inputs $x$ with $|x| < t$ and will produce a correct answer on at least a $\mu$-fraction 0.7 of all inputs $x$ with $|x| \geq t$ as guaranteed by Lemma \[lem:probability\]. Hence it will produce a correct answer on a $\mu$-fraction of at least: $$\mu(\{x \mid |x| < t\}) + 0.7 (1 - \mu(\{x \mid |x| < t \}) \geq 0.7.$$ Furthermore, its query complexity is equal to: $$O(\sqrt{tN})+\sum_{i=10}^{\log N - \log t -1} O\left(\sqrt{\frac{tN}{2^i}}\right)
= O(\sqrt{tN}),$$ where the first term corresponds to the cost of searching the entire space once with a small upper bound, and the summation corresponds to searching consecutively smaller subparts $y^{(i)}$.
Recall that for threshold functions $f:{\{ 0,1 \}}^N \to {\{ 0,1 \}}$ with threshold $t$, $Q_2(f)=\Theta(\sqrt{tN})$. By Theorem \[thm:Q2lowerbound\] it then follows that $\max_\mu Q_{WE}^\mu(f) = \Omega(\sqrt{tN})$. In combination with Lemma \[lem:threshold\], this yields:
\[lem:eqthreshold\] For all threshold functions $f:\{0,1\}^N \to \{0,1\}$ with threshold $t$, $$Q_2(f) = \Theta\left(\max_\mu Q_{WE}^\mu(f)\right) = \Theta\left(\sqrt{tN}\right).$$
Equality up to a Constant Factor for Symmetric Functions.
---------------------------------------------------------
With the result about threshold functions in mind, we can easily prove that the quantum Yao principle holds for all symmetric functions as well.
\[thm:symmetric\] For all symmetric functions $f:\{0,1\}^N \to \{0,1\}$ $$Q_2(f) = \Theta\left(\max_\mu Q_{WE}^\mu(f)\right) = \Theta\left(\sqrt{N(N-\Gamma(f))}\right).$$
From [@bbcmw:polynomials] we know that $Q_2(f)=\Theta(\sqrt{N(N-\Gamma(f))})$. Also, Theorem \[thm:Q2lowerbound\] tells us that $Q_2(f) \leq \max_\mu Q_{WE}^\mu(f)$. It remains to show that for every distribution $\mu$, $Q_{WE}^\mu(f) = O(\sqrt{N(N-\Gamma(f))})$.
Fix a probability distribution $\mu$ over the set of all inputs. Note that $\Gamma(f)$ measures the length of the interval around Hamming weight $\frac{N}{2}$ where $f$ is constant, so in order to compute $f(x)$, it suffices to know $|x|$ exactly if $|x| \in
[0,\frac{N-\Gamma(f)}{2}) = I_1$ or $|x| \in (\frac{N+\Gamma(f)-2}{2},N]
= I_3$, or to know that $|x| \in [\frac{N-\Gamma(f)}{2},
\frac{N+\Gamma(f)-2}{2}] = I_2$.
We can use the threshold algorithm of the previous section to determine whether $x\in I_1$ (with $\mu$-error probability reduced to $1/6$). We can use another threshold algorithm to determine whether $x\in I_3$ (with the role of 0’s and 1’s reversed, and also with error $\leq 1/6$). Both threshold algorithms take $O(\sqrt{N(N-\Gamma(f))})$ queries. Now for at least $2/3$ of the inputs $x$, weighed according to $\mu$, *both* of these threshold algorithms will give the correct answer. For all such $x$ we can determine $f(x)$ with certainty: if we know $|x| \in I_2$ then we are done, because $f$ is constant in this interval. If $|x|\in I_1$ or $|x|\in I_3$ then we use Algorithm \[alg:ronald\] to count $|x|$, using $O(\sqrt{N(N-\Gamma(f))})$ queries. Thus we have a weakly $\frac{2}{3}$-exact quantum algorithm for $f$ with respect to $\mu$, using $O(\sqrt{N(N-\Gamma(f))})$ queries in total.
A Result for the AND-OR Tree {#ssecandor}
----------------------------
Above we proved that the weak quantum Yao principle holds (up to a constant factor) for all *symmetric* functions. A similar result might be provable for all *monotone* functions. Recall that a Boolean function $f$ is monotone if the function value cannot change from 1 to 0 if we change some input bits from 0 to 1. In this section we prove a preliminary result in this direction, namely that the known upper and lower bounds on the $Q_2(f)$-complexity of the 2-level *AND-OR tree* carry over to weakly $\frac{2}{3}$-exact quantum algorithms. This monotone but non-symmetric function is the AND of $\sqrt{N}$ independent ORs of $\sqrt{N}$ variables each. In the sequel, we use AO to denote this $N$-bit AND-OR tree.
No tight characterization of $Q_2(AO)$ is known, but Buhrman, Cleve, and Widgerson [@BuhrmanCleveWigderson98] proved $Q_2(AO)=O(\sqrt{N}\log N)$ via a recursive application of Grover’s algorithm. Using a result about efficient error-reduction in quantum search from [@bcwz:qerror], this upper bound can be improved to $Q_2(AO)=O(\sqrt{N\log N})$. This nearly matches Ambainis’ lower bound of $\Omega(\sqrt{N})$ [@ambainis:lowerbounds]. Note that Ambainis’ bound together with our Theorem \[thm:Q2lowerbound\] immediately gives the lower bound $\max_\mu Q_{WE}^\mu(AO)= \Omega(\sqrt{N})$. Below we show that also the best known *upper* bound carries over to weakly $\frac{2}{3}$-exact algorithms: $Q_{WE}^\mu(AO)= O(\sqrt{N\log N})$ for all $\mu$.
To prove this result, we first show that we can efficiently reduce the error in weakly $\frac{2}{3}$-exact quantum search algorithms, in analogy with [@bcwz:qerror]. For every $\mu$ and $\epsilon$, we will construct a quantum search algorithm that uses $O\left(\sqrt{N\log(1/\epsilon)}\right)$ queries and solves the search problem *with certainty* for $1-\epsilon$ of all inputs, weighed by $\mu$. We first need the following lemma, which states that if an input contains many 1’s, then we can deterministically reduce its size to a smaller search space which will probably still contain at least one 1.
\[lem:reduce\] For all probability distributions $\mu$ on $\01^N$ and integers $c$, there exists a $y\in\01^N$ with $|y|=\min\{\frac{cN}{t},N\}$, such that $\mathrm{Pr}_\mu[|x\land y|\geq 1\mid |x|>t]\geq 1- e^{-c}$.
If $\frac{cN}{t}\geq N$ then obviously the lemma holds (pick $y=1^N$), so assume $\frac{cN}{t}<N$. Fix an $x\in\01^N$ with $|x|>t$. If we pick a $y\in\01^N$ with $|y|=\frac{cN}{t}$ uniformly at random, then $$\begin{aligned}
\mathrm{Pr}[|x \land y|=0\mid |x|>t] & = & \frac{{N - |y| \choose |x|}}{{N \choose |x|}}
= \frac{(N-|x|)\cdot (N-|x|-1)\cdots (N-|x|-|y|+1)}{N(N-1)\cdots(N-|y|+1)} \\
& \leq & \left( 1 - \frac{|x|}{N}\right)^{|y|} \leq e^{-|x|\cdot |y|/N}
\leq e^{-c}.\end{aligned}$$ Hence $\mathrm{Pr}[|x \land y|\geq 1\mid |x|>t] \geq 1 - e^{-c}$. By exactly the same averaging argument as in the proof of Lemma \[lem:probability\], we can show that for every distribution $\mu$, there exists a $y$ such that $\mathrm{Pr}_{\mu}[|x \land y|\geq 1\mid |x|>t]
\geq 1 - e^{-c}$.
With Lemma \[lem:reduce\] at our disposal, we can now prove that we can “cheaply” reduce the error of weakly $\frac{2}{3}$-exact quantum search algorithms to small $\epsilon$.
\[lem:weaksearch\] For every $\epsilon>0$ and every probability distribution $\mu$ over $\01^N$, there exists a weakly $(1-\epsilon)$-exact quantum search algorithm with respect to $\mu$ that uses $O\left(\sqrt{N\log(1/\epsilon)}\right)$ queries.
Fix an error bound $\epsilon$ and distribution $\mu$. Our $(1-\epsilon)$-exact search algorithm is inspired by [@bcwz:qerror]. Let $t_0=\log(1/\epsilon)$ (assume for simplicity that this is an integer). First we run the exact version of Grover’s algorithm on the input $x$ assuming that $|x|=1$, then we run it again assuming that $|x|=2$, and so on until $|x|=t_0$. This takes $$\sum_{i=1}^{t_0}\frac{\pi}{4}\sqrt{\frac{N}{i}}=
O(\sqrt{Nt_0})=O\left(\sqrt{N\log(1/\epsilon)}\right)$$ queries, and finds a 1 with certainty whenever $1\leq |x|\leq t_0$.
It remains to find a 1 for “most” of the inputs $x$ that have $|x|>t_0$. Let $\mu_1$ be the probability distribution $\mu$ restricted to the $x$ with $|x|>t$. By Lemma \[lem:reduce\], we know there exists a $y\in\01^N$ with $|y|=O(N/t_0)$ such that $\mathrm{Pr}_{\mu_1}[|x\land y|\geq 1]=\mathrm{Pr}_\mu[|x\land y|\geq 1\mid |x|>t_0]
\geq\frac{5}{6}$. Now we use a $\frac{2}{3}$-exact quantum search algorithm with respect to $\mu_1$ to search the subpart of $x$ indicated by $y$. This subpart has size $O(N/t_0)$, and Lemma \[lem:threshold\] guarantees us that there is $\frac{2}{3}$-exact algorithm with $O(\sqrt{N/t_0})$ queries. Thus we find a 1 with certainty for a $\mu_1$-fraction (and hence also $\mu$-fraction) of at least $\frac{5}{6}-\frac{1}{3}=\frac{1}{2}$ of the inputs with $|x|>t_0$. Now we repeat this idea to “catch” $\frac{1}{2}$ of the remaining inputs. Let $\mu_2$ be $\mu_1$ restricted to the inputs with $|x|>t_0$ where the previous algorithm did not find a 1 with certainty. Using another $\frac{2}{3}$-exact algorithm (this time with respect to $\mu_2$) for another $y$, we can catch $\frac{1}{2}$ of the remaining inputs. We repeat this $t_0$ times and eventually catch $$1-\left(\frac{1}{2}\right)^{t_0}= 1-\epsilon$$ of the inputs (weighed according to $\mu$) in this way. If we still have not found a 1 after all this, we stop and output ‘no solutions’, which ensures that our algorithm is always correct on the all-0 input. Note that the second part of the algorithm uses $$t_0\cdot O(\sqrt{N/t_0})=O\left(\sqrt{N\log(1/\epsilon)}\right)$$ queries, so our overall query complexity is $O\left(\sqrt{N\log(1/\epsilon)}\right)$, as promised.
Using Lemma \[lem:weaksearch\] we now show that the best known upper bound for $Q_2(AO)$ also holds for weakly $\frac{2}{3}$-exact quantum algorithms.
For every distribution $\mu$ on $\{0,1\}^N$ we have $Q_{WE}^\mu(AO)=O(\sqrt{N\log N})$.
Fix some distribution $\mu$. We will sketch a $\frac{2}{3}$-exact quantum algorithm for AO with respect to $\mu$, along the lines of the recursive-Grover of [@BuhrmanCleveWigderson98]. For each of the $1\leq i\leq\sqrt{N}$ OR functions at the “bottom” of the tree, let $\mu_i:\{0,1\}^{\sqrt{N}}\rightarrow[0,1]$ be the distribution over its $\sqrt{N}$ input bits induced by $\mu$, i.e., $\mu_i(y)$ is the sum of $\mu(x)$ over all $x\in\{0,1\}^N$ where the $i$th block of $\sqrt{N}$ variables takes value $y$. Let $A_i$ be a weakly $\left(1-\frac{1}{6\sqrt{N}}\right)$-exact quantum algorithm with respect to $\mu_i$ for the $i$th OR. By Lemma \[lem:weaksearch\], each $A_i$ takes $O\left(\sqrt{\sqrt{N}\log N}\right)$ queries. Note that now for $\frac{5}{6}$ of the inputs, weighed according to $\mu$, *all* $A_i$ deliver the correct answer with certainty. By standard techniques (copying the answer and reversing the computation afterwards ) we can “clean up” these computations, setting the workspace back to the initial state and just retaining the answer bit.
We now want to run a $\frac{5}{6}$-exact quantum algorithm for AND on top of these $\sqrt{N}$ subtrees to compute the AND-OR tree. Let $\mu':\{0,1\}^{\sqrt{N}}\rightarrow[0,1]$ be the induced input distribution for the top-AND, i.e., $\mu'(y)$ is the sum of $\mu(x)$ over all $x\in\{0,1\}^N$ where the $i$th OR takes the value $y_i$, for all $1\leq i\leq \sqrt{N}$. Let $A$ be a weakly $\frac{5}{6}$-exact quantum algorithm for the $\sqrt{N}$-variable AND with respect to $\mu'$. By Lemma \[lem:threshold\] such an algorithm makes $O(N^{1/4})$ queries. If we replace, in $A$, a query to the $i$th bit by a call to $A_i$, then we obtain an $O(\sqrt{N\log N})$-query algorithm that is correct with certainty on a $\mu$-fraction at least $\frac{5}{6}-\frac{1}{6}=\frac{2}{3}$ of all inputs.
Summary and Open Problems
=========================
In this paper we investigated to what extent quantum versions of the classical Yao principle hold. We formulated a strong and a weak version of the quantum Yao principle, showed that both hold in one direction, falsified the other direction for the strong version, and proved the weak version for the query complexity of all symmetric functions.
The main question left open by this research is the general validity of the weak quantum Yao principle. On the one hand, we may be able to find a counterexample to the weak principle as well, perhaps based on the query complexity of the *order-finding problem*. Shor showed that the order-finding problem can be solved by a bounded-error quantum algorithm using $O(\log N)$ queries [@shor:factoring]. Using Cleve’s $\Omega(N^{1/3}/\log N)$ lower bound on classical algorithms for order-finding [@cleve:orderfinding], we can exhibit a $\mu$ such that any strongly $\frac{2}{3}$-exact quantum algorithm for $f$ with respect to $\mu$ requires $N^{\Omega(1)}$ queries (in the same way as Theorem \[thm:Q2lowerbound\]). This gives another counterexample to the strong quantum Yao principle. The same problem may even provide a counterexample to the *weak* quantum Yao principle, as it seems hard to construct even weakly $\frac{2}{3}$-exact quantum algorithms for this problem.
On the other hand, we may try to extend the class of functions for which we know the weak quantum Yao principle *does* hold. A good starting point here might be the class of all *monotone* functions. We discussed one such function, the 2-level AND-OR tree, in Section \[ssecandor\]. Unfortunately, at the time of writing no general characterization of the $Q_2(f)$-complexity of all monotone functions is known, in contrast to the case of symmetric functions. Also, in this direction it might be a fruitful idea to further explore the rapidly growing field of *quantum game theory* (see for example [@meyer:games]) and the possible connections between that area and our work.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Harry Buhrman for initiating this research, for coming up with the counterexample of Theorem \[thm:exponentialGap\], and for useful comments on a preliminary version of this paper. We thank him and Peter Høyer for their contributions to an initial proof of the weak quantum Yao principle for the OR function, which forms the basis for the current proof of Theorem \[thm:symmetric\]. We also thank Leen Torenvliet for useful discussions.
[10]{}
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[^1]: CWI, INS4, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands. Email: $\mathtt{\{mgdgraaf,rdewolf\}@cwi.nl}$. Partially supported by the EU fifth framework project QAIP, IST–1999–11234. Mart de Graaf is also supported by grant 612.055.001 from the Netherlands Organization for Scientific Research (NWO). Ronald de Wolf is also supported by NWO TALENT grant S 62-565.
[^2]: We need the condition $i \geq 10$ in this lemma in order to be able to approximate the hypergeometric distribution by a binomial distribution with sufficient accuracy.
|
---
abstract: 'We consider a class of Mean Field Games in which the agents may interact through the statistical distribution of their states and controls. It is supposed that the Hamiltonian behaves like a power of its arguments as they tend to infinity, with an exponent larger than one. A monotonicity assumption is also made. Existence and uniqueness are proved using a priori estimates which stem from the monotonicity assumptions and Leray-Schauder theorem. Applications of the results are given.'
author:
- 'Z. Kobeissi[^1]'
bibliography:
- 'MFGbiblio.bib'
title: Mean Field Games with monotonous interactions through the law of states and controls of the agents
---
Introduction {#sec:MonoIntro}
============
The theory of Mean Field Games (MFG for short) aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. It has been introduced in the independent works of J.M. Lasry and P.L. Lions [@MR2269875; @MR2271747; @MR2295621], and of M.Y. Huang, P.E. Caines and R.Malham[é]{} [@MR2352434; @MR2346927]. The agents are supposed to be rational (given a cost to be minimized, they always choose the optimal strategies), and indistinguishable. Furthermore, the agents interact via some empirical averages of quantities which depend on the state variable.
The most common Mean Field Game systems, in which the agents may interact only through their states can often be summarized by a system of two coupled partial differential equations which is named the MFG system. On the one hand, the optimal value of a generic agent at some time $t$ and state $x$ is denoted by $u(t, x)$ and is defined as the lowest cost that a representative agent can achieve from time $t$ to $T$ if it is at state $x$ at time $t$. The value function satisfies a Hamilton-Jacobi-Bellman equation posed backward in time with a terminal condition involving a terminal cost. On the other hand, there is a Fokker-Planck-Kolmogorov equation describing the evolution of the statistical distribution m of the state variable; this equation is a forward in time parabolic equation, and the initial distribution at time $t = 0$ is given. Here we take a finite horizon time $T>0$, and we only consider second-order nondegenerate MFG systems. In this case, the MFG system is often written as: $$\label{eq:tradMFG}
{\left\{}
\newcommand{\rc}{\right\}}\begin{aligned}
&-{{\partial_{t}}}u(t,x)
- \nu\Delta u(t,x)
+ H(t,x, \nabla_xu(t,x))
= f(t,x,m(t))
&\text{ in }
(0,T)\times{ \mathbb{R}}^d,\\
& {{\partial_{t}}}m(t,x)
- \nu\Delta m(t,x)
-\operatorname{div}(H_p(t,x,\nabla_xu(t,x))m)
=0
&\text{ in }
(0,T)\times{ \mathbb{R}}^d,\\
&u(T,x)
=g(x,m(T))
&\text{ in }
{ \mathbb{R}}^d,\\
&m(0,x)
=m_0(x)
&\text{ in }
{ \mathbb{R}}^d.
\end{aligned}
\right.$$ We refer the reader to [@MR3967062] for some theoretical results on the convergence of the $N$-agent Nash equilibrium to the solutions of the MFG system. For a thorough study of the well-posedness of the MFG system, see the videos of P.L. Lions’ lecture at the Coll[è]{}ge de France, and the lecture notes [@Cardaliaguet_notes_on_MFG].
In this paper we are considering a class of Mean Field Games in which agents may interact through their states and controls. To underline this, we choose to use the terminology [*Mean Field Games of Controls (MFGCs)*]{}; this terminology was introduced in [@MR3805247].
Since the agents are assumed to be indistinguishable, a representative agent may be described by her state, which is a random process with value in ${ \mathbb{R}}^d$ denoted by $(X_t)_{t\in[0,T]}$ and satisfying the following stochastic differential equation, $$\label{eq:SDEb}
dX_t
=
b{\left(}t,X_t,\aa_t,\mu_{\aa}(t){\right)}dt
+\sqrt{2\nu}dW_t,$$ where $X_0$ is a random process whose law is denoted by $m_0$, ${\left(}W_t{\right)}_{t\in[0,T]}$ is a Brownian motion on ${ \mathbb{R}}^d$ independefn with $X_0$, and $\aa_t$ is the control chosen by the agent at time $t$. The diffusion coefficient $\nu$ is assumed to be uncontrolled, constant and positive. The drift $b$ naturally depends on the control, and may also depend on the time, the state, and on the mean field interactions of all agents through $\mu_{\aa}$ the joint distribution of states and controls. At the equilibrium $\mu_{\aa}$ should be the law of the state and the control of the representative agent, i.e. $\mu_{\aa}(t)={{\mathcal L}}{\left(}X_t,\aa_t{\right)}$, for $t\in[0,T]$. The aim of an agent is to minimize the functional given by, $${{\mathbb E}}{\left[}\int_0^T L{\left(}t,X_t,\aa_t,\mu_{\aa}(t){\right)}+f{\left(}t, X_t,m(t){\right)}dt
+g{\left(}X_T,m(T){\right)}{\right]},$$ where $m(t)$ is the distribution of agents at time $t$, which should satisfy $m(t)={{\mathcal L}}{\left(}X_t{\right)}$ at the equilibrium. The coupling function $f$ and the terminal cost $g$ depend on $m$ in a nonlocal manner. From $L$ the Lagrangian and $b$ the drift, we define $H$ the Hamiltonian by, $$\label{eq:defHMono}
H{\left(}t,x,p,\mu_{\aa}{\right)}=
\sup_{\aa\in{ \mathbb{R}}^d}
-p\cdot b{\left(}t,x,\aa,\mu_{\aa}{\right)}-L{\left(}t,x,\aa,\mu_{\aa}{\right)},$$ for $(t,x)\in[0,T]\times{ \mathbb{R}}^d$, $p\in{ \mathbb{R}}^d$ and $\mu_{\aa}\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$, where ${{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ is the set of probability measures on ${ \mathbb{R}}^d\times{ \mathbb{R}}^d$. Under some assumptions on $b$ and $L$ that will be introduced later, there exists a unique $\aa$ which achieves the supremum in the latter equality and it also satisfies, $$b{\left(}t,x,\aa,\mu_{\aa}{\right)}=
-H_p{\left(}t,x,p,\mu_{\aa}{\right)}.$$ In an attempt to keep this paper easy to read, we introduce $\mu_b$ as the joint law of states and drifts defined by $$\label{eq:mubmuaa}
\mu_b(t)
=
\Bigl[
{\left(}x,\aa{\right)}\mapsto
{\left(}x,
b{\left(}t,x,\aa,\mu_{\aa}(t){\right)}{\right)}\Bigr]{\#}\mu_{\aa}(t).$$ We believe that the fixed point relation satisfied by $\mu_{\aa}$ at equilibrium is more clear if we distinguish $\mu_{b}$ from $\mu_{\aa}$. We assume that $b$ is invertible with respect to $\aa$ in such a way that its inverse map can be expressed in term of $\mu_b$ instead of $\mu_{\aa}$, see Assumption \[hypo:binvert\] below. This allows us to define $\aa^*:[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}\to{ \mathbb{R}}^d$ such that $${{\widetilde b}}=
b{\left(}t,x,\aa^*{\left(}t,x,{{\widetilde b}},\mu_b{\right)},\mu_{\aa}{\right)}$$ for any $(t,x)\in[0,T]\times{ \mathbb{R}}^d$, ${{\widetilde b}}\in{ \mathbb{R}}^d$ and $\mu_{\aa},\mu_b\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ satisfying . Conversely, for any $\aa\in{ \mathbb{R}}^d$ we have $$\aa
=
\aa^*{\left(}t,x,b{\left(}t,x,\aa,\mu_{\aa}{\right)},\mu_{b}{\right)},$$ since $b{\left(}t,x,\cdot,\mu_{\aa}{\right)}$ is injective. This implies that the equality can be inverted to express $\mu_{\aa}$ in term of $\mu_b$ and we obtain below. Within this framework, the usual MFG system is replaced by the following Mean Field Game of Controls (MFGC for short) system,
\[eq:MFGCb\]
[align]{} \[eq:HJBb\] &-[[\_[t]{}]{}]{}u(t,x) - u(t,x) + H[(]{}t,x, \_xu(t,x),\_[å]{}(t)[)]{} = f(t,x,m(t)) & (0,T)[ ]{}\^d,\
\[eq:FPKb\] & [[\_[t]{}]{}]{}m(t,x) - m(t,x) -[(]{}H\_p[(]{}t,x,\_xu(t,x),\_[å]{}(t)[)]{}m[)]{} = 0 & (0,T)[ ]{}\^d,\
\[eq:muaa\] &\_[å]{}(t) = \_[b]{}(t) & \[0,T\],\
\[eq:mub\] &\_b(t) = ( I\_d, -H\_p[(]{}t,,\_xu(t,),\_[å]{}(t)[)]{} )[\#]{}m(t) & \[0,T\],\
\[eq:CFub\] &u(T,x) =g(x,m(T)) & [ ]{}\^d,\
\[eq:CImb\] &m(0,x) =m\_0(x) & [ ]{}\^d.
The structural assumption under which we prove existence and uniqueness of the solution to is that $L$ satisfies the following inequality, $$\int_{{ \mathbb{R}}^d\times{ \mathbb{R}}^d}
{\left(}L{\left(}t, x,\a,\mu^1 {\right)}-L{\left(}t, x,\a,\mu^2{\right)}{\right)}d{\left(}\mu^1-\mu^2 {\right)}(x,\aa)
\geq
0.$$ for any $t\in[0,T]$, $\mu^1, \mu^2 \in
{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times { \mathbb{R}}^d {\right)}$. This is the Lasry-Lions monotonicity assumption extended to MFGC that will be referred to as \[hypo:LMono\]. This assumption is particularly adapted to applications in economics or finance.
This work follows naturally the analysis in [@2019arXiv190411292K] in which a MFGC system in the $d$-dimensional torus and with $b=\aa$ is considered. In [@2019arXiv190411292K], the monotonicity assumption is replaced by another structural assumption, namely that the optimal control $-H_p$ is a contraction with respect to the second marginal of $\mu$ (when the other arguments and the first marginal are fixed) and that it is bounded by a quantity that depends linearly on the second marginal of $\mu$ with a coefficient smaller than $1$.
Related works {#related-works .unnumbered}
-------------
Monotonicity assumptions for MFGC like \[hypo:LMono\] have already been discussed in [@MR3805247; @MR3752669; @MR3112690]. In [@MR3112690], the authors proved uniqueness of the solution to with $b=\aa$ and $\nu=0$ when it exists. In [@MR3752669] Section $4.6$, existence and uniqueness are proved in the quadratic case with a uniformly convex Lagrangian and under an additional linear growth assumption on $H_x$. In [@MR3805247], the existence of weak solutions to a MFGC system with a possibly degenerate diffusion operator is proved assuming that the inequalities satisfied by $H$, its derivatives or the optimal control (here defined in \[hypo:binvert\] as $\aa^*$), are uniform with respect to the joint law of states and controls $\mu_{\aa}$.
A particular application of MFGC satisfying \[hypo:LMono\], namely the Bertrand and Cournot competition for exhaustible ressource described in paragraph \[subsec:Monoexhau\], has been broadly investigated in the literature. Let us mention a non exhaustive list of such works: [@2019arXiv190205461F; @MR3359708; @MR3755719; @MR2762362; @MR4064472]. Its mean field version has been introduced in [@MR2762362], and obtained from the $N$-agent game in [@MR3359708] in the case of a linear supply-demand function. A generalization to the multi-dimensional case is discussed in [@2019arXiv190205461F], and an extension to negatively correlated ressources is addressed in [@2019arXiv190411292K]. A class of MFGC in which the Lagrangian depends separately on $\aa$ and $\mu$, has been investigated in [@MR3325272] and [@MR3752669]. In this case, \[hypo:LMono\] is naturally satisfied since the left-hand side of the inequality is identically equal to $0$. An existence result is proved in [@MR3325272] under the additional assumption that the set of admissible controls is compact. The existence of solution is also proved in [@MR3752669] Theorem $4.65$ when the dependency of $L$ upon $\mu$ is uniformly bounded with respect to $\mu$.
The non-monotone case has been studied in [@MR3160525; @2019arXiv190411292K]. In [@MR3160525], an existence result is proved in the stationnary setting and under the assumption that the dependence of $H$ on $\mu$ is small. In [@2019arXiv190411292K], the existence of solutions to the MFGC system in the $d$-dimensional torus and with $b=\aa$ is discussed under similar growth assumptions as here. By and large, existence of solutions to a MFGC system posed on the $d$-dimensional torus and with $b=\aa$ was proved in [@2019arXiv190411292K] in any of the following cases:
- short time horizon,
- small enough parameters,
- weak dependency of $H$ upon $\mu$,
- weak dependency of $H_x$ upon $\mu$,
and uniqueness is proved only for a short time horizon. Indeed without a monotonicity assumption, it is unlikely that uniqueness holds in general, numerical examples of non-uniqueness of solutions to discrete approximations of with $b=\aa$ and in a bounded domain are showed in [@achdou2020mean].
Organization of the paper {#organization-of-the-paper .unnumbered}
-------------------------
In Section \[sec:MonoAssumptions\], the notations and the assumptions are described, the case when the control is equal to the drift is discussed. The main results of the paper, namely the existence and uniqueness of solution to , are stated in paragraph \[subsec:mainMono\]. We give some insights on our strategy for proving the main results in paragraph \[subsec:outlineMono\]. Two applications of the MFGC system are presented in Section \[sec:appli\]. Section \[sec:MonoFPV\] is devoted to solving the fixed point relation in the joint law of state and control in the particular case when the drift is equal to the control. Section \[sec:aprioriMono\] consists in giving a priori estimates for a MFGC system posed on the $d$-dimensional torus. In Section \[sec:MainMono\], we prove existence and uniqueness of the solution to and of an intermediate MFGC system.
Assumptions {#sec:MonoAssumptions}
===========
Notations {#subsec:notaMono}
---------
The spaces of probability measures are equipped with the weak\* topology. We denote by ${{\mathcal P}}_2{\left(}{ \mathbb{R}}^d{\right)}$ the subset of ${{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)}$ of probability measures with finite second moments, and ${{\mathcal P}}_{\infty}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ the subset of measures $\mu$ in ${{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ with a second marginal compactly supported. For $\mu\in{{\mathcal P}}_{\infty}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ and ${{\widetilde q}}\in[1,\infty)$, we define the quantities $\LL_{{{\widetilde q}}}(\mu)$ and $\LL_{\infty}(\mu)$ by, $$\label{eq:defL}
\begin{aligned}
\LL_{{{\widetilde q}}}(\mu)
&=
{\left(}\int_{{ \mathbb{R}}^d\times{ \mathbb{R}}^d}
{\left|}\aa{\right|}^{{{\widetilde q}}}d\mu{\left(}x,\alpha{\right)}{\right)}^{\frac1{{{\widetilde q}}}},
\\
\LL_{\infty}(\mu)
&=
\sup{\left\{}
\newcommand{\rc}{\right\}}{\left|}\aa{\right|},
(x,\aa)\in\operatorname{\mathop{\rm supp}}\mu\rc.
\end{aligned}$$ For $R>0$, we denote by ${{\mathcal P}}_{\infty,R}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ the subset of measures $\mu$ in ${{\mathcal P}}_{\infty}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ such that $\LL_{\infty}{\left(}\mu{\right)}\leq R$. The probability measures $\mu_{\aa}$ and $\mu_b$ involved in , have a particular form, since they are the images of a measure $m$ on ${ \mathbb{R}}^d$ by ${\left(}I_d,\aa{\right)}$ and ${\left(}I_d,b{\right)}$ respectively, where $\aa$ and $b$ are bounded measurable functions from ${ \mathbb{R}}^d$ to ${ \mathbb{R}}^d$; in particular they are supported on the graph of $\aa$ and $b$ respectively. For $m\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)}$, we call ${{\mathcal P}}_m{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ the set of such measures. For $\mu\in{{\mathcal P}}_m{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$, we set $\aa^{\mu}$ to be the unique element of $L^{\infty}{\left(}m{\right)}$ such that $\mu={\left(}I_d,\aa^{\mu}{\right)}\#m$. Here, $\LL_{{{\widetilde q}}}(\mu)$ and $\LL_{\infty}(\mu)$ defined in are given by $$\label{eq:defLbis}
\begin{aligned}
\LL_{q'}(\mu)
&=
{{\left\| \aa^{\mu} \right\|_{L^{q'}(m)}^{}}},
\\
\LL_{\infty}(\mu)
&=
{{\left\| \aa^{\mu} \right\|_{L^{\infty}(m)}^{}}}.
\end{aligned}$$ Let $C^{0}{\left(}[0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}^n{\right)}$ be the set of bounded continuous functions from $[0,T]\times{ \mathbb{R}}^d$ to ${ \mathbb{R}}^n$, for $n$ a positive integer. We define $C^{0,1}{\left(}[0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}{\right)}$ as the set of the functions $v\in C^0{\left(}[0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}{\right)}$ differentiable at any point with respect to the state variable, and whose its gradient $\nabla_xv$ is in $C^{0}{\left(}[0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}^d{\right)}$ the set of continuous functions from $[0,T]\times{ \mathbb{R}}^d$ to ${ \mathbb{R}}^d$. We shall have the use of the parabolic spaces of Hölder continuous functions $C^{\frac{\bb}2,\bb}([0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}^n)$ defined for any $\bb\in(0,1)$ and $n\geq 1$ by, $$C^{\frac{\bb}2,\bb}{\left(}[0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}^n{\right)}= {\left\{}
\newcommand{\rc}{\right\}}\begin{aligned}
v& \in C^0([0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}^n),
\exists C>0 \text{ s.t. }
\forall (t_1,x_1),(t_2,x_2)\in [0,T]\times{ \mathbb{R}}^d, \\
&|v(t_1,x_1)-v(t_2,x_2)|\leq
C{\left(}|x_1-x_2|^2 +|t_1-t_2|{\right)}^{\frac{\bb}2}
\end{aligned}
\rc.$$ This is a Banach space equipped with the norm, $${{\left\| v \right\|_{C^{\frac{\bb}2,\bb}}^{}}}
=
{{\left\| v \right\|_{\infty}^{}}}
+\sup_{(t_1,x_1)\neq(t_2,x_2)}
\frac{|v(t_1,x_1)-v(t_2,x_2)|}
{{\left(}|x_1-x_2|^2 +|t_1-t_2|{\right)}^{\frac{\bb}2}}.$$ Then we introduce the Banach space $C^{\frac{1+\bb}2,1+\bb}([0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}})$ for $\bb\in(0,1)$ as the set of the functions $v\in C^{0,1}([0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}})$ such that $\nabla_xv\in C^{\frac{\bb}2,\bb}{\left(}[0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}^n{\right)}$ and which admits a finite norm defined by, $${{\left\| v \right\|_{C^{\frac{1+\bb}2,1+\bb}}}}
=
{{\left\| v \right\|_{\infty}^{}}}
+{{\left\| \nabla_xv \right\|_{C^{\frac{\bb}2,\bb}}^{}}}
+ \sup_{(t_1,x)\neq (t_2,x)\in [0,T]\times{ \mathbb{R}}^d}
\frac{|v(t_1,x)-v(t_2,x)|}{|t_1-t_2|^{\frac{1+\bb}2}}.$$
When the drift $b$ is equal to the control $\aa$, can be simplified in the following system,
\[eq:MFGCrd\]
[align]{} \[eq:HJBrd\] &-[[\_[t]{}]{}]{}u(t,x) - u(t,x) + H[(]{}t,x, \_xu(t,x),(t)[)]{} = f(t,x,m(t)) & (0,T)[ ]{}\^d,\
\[eq:FPKrd\] & [[\_[t]{}]{}]{}m(t,x) - m(t,x) -[(]{}H\_p[(]{}t,x,\_xu(t,x),(t)[)]{}m[)]{} = 0 & (0,T)[ ]{}\^d,\
\[eq:murd\] &(t) = ( I\_d, -H\_p[(]{}t,,\_xu(t,),(t)[)]{} )[\#]{}m(t) & \[0,T\],\
\[eq:CFurd\] &u(T,x) =g(x,m(T)) & [ ]{}\^d,\
\[eq:CImrd\] &m(0,x) =m\_0(x) & [ ]{}\^d.
Here, making a distinction between $\mu_{\aa}$ and $\mu_b$ is pointless since they coincide. Therefore, we simply use the notation $\mu$. For the system , the Hamiltonian is defined as the Legendre transform of $L$, $$\label{eq:defHrd}
H{\left(}t,x,p,\mu{\right)}=
\sup_{\aa\in{ \mathbb{R}}^d}
-p\cdot\aa-L{\left(}t,x,\aa,\mu{\right)}.$$
\[def:sol\] We say that ${\left(}u,m,\mu_{\aa},\mu_b{\right)}$ is a solution to if
- $u\in C^{0,1}{\left(}[0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}{\right)}$ is a solution to the heat equation in the sense of distributions with a right-hand side equal to $(t,x)\mapsto f(t,x,m(t))- H{\left(}t,x,\nabla_xu,\mu(t){\right)}$, and satisfies the terminal condition ,
- $m\in C^0{\left(}[0,T];{{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)}{\right)}$ is a solution to in the sense of distributions, and satisfies the initial condition ,
- $\mu_{\aa}(t),\mu_b(t)\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ satisfy and for any $t\in[0,T]$.
We say that ${\left(}u,m,\mu{\right)}$ is a solution to if $u$ and $m$ respectively satisfy the first two points of the latter definition with instead of , and if $\mu(t)\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ satisfies for any $t\in[0,T]$.
Hypothesis {#subsec:hypoMono}
----------
The monotonicity assumption made in this paper concerns the Lagrangian. For this reason and the fact that sometimes it may be hard to obtain an explicit form of the Hamiltonian (like in the example of paragraph \[subsec:abncrowd\] below), all the assumptions will be formulated in term of the Lagrangian and never in term of the Hamiltonian. In particular, working with the Lagrangian gives more flexibility in the arguments of the proofs. The constants entering the assumptions are $C_0$ a positive constant, $q\in(1,\infty)$ an exponent, $q'=\frac{q}{q-1}$ its conjugate exponent, and $\bb_0\in(0,1)$ a Hölder exponent.
1. \[hypo:Lreg\] $L:[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}\to{ \mathbb{R}}$ is differentiable with respect to ${\left(}x,\aa{\right)}$; $L$ and its derivatives are continuous on $[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times{{\mathcal P}}_{\infty,R}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ for any $R>0$; we recall that ${{\mathcal P}}_{\infty,R}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ is endowed with the weak\* topology on measures; we use the notation $L_x$, $L_{\aa}$ and $L_{(x,\aa)}$ for respectively the first-order derivatives of $L$ with respect to $x$, $\aa$ and $(x,\aa)$ .
2. \[hypo:Lbconv\] The maximum in is achieved at a unique $\aa\in{ \mathbb{R}}^d$.
3. \[hypo:LMono\] $L$ satisfies the following monotonicity condition, $$\int_{{ \mathbb{R}}^d\times{ \mathbb{R}}^d}
{\left(}L{\left(}t, x,\a,\mu^1 {\right)}-L{\left(}t, x,\a,\mu^2{\right)}{\right)}d{\left(}\mu^1-\mu^2 {\right)}(x,\aa)
\geq
0.$$ for any $t\in[0,T]$, $\mu^1, \mu^2 \in
{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times { \mathbb{R}}^d {\right)}$.
4. \[hypo:Lcoer\] $L(t,x,\aa,\mu)
\geq
C_0^{-1}|\aa|^{q'}
-C_0{\left(}1+\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)}$, where $\LL_{q'}$ is defined in ,
5. \[hypo:Lbound\] ${\left|}L(t,x,\aa,\mu){\right|}\leq
C_0{\left(}1+|\aa|^{q'}
+\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)}$, and ${\left|}L_x(t,x,\aa,\mu){\right|}\leq
C_0{\left(}1+|\aa|^{q'}
+\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)}$,
6. \[hypo:Monogregx\] $\int_{{ \mathbb{R}}^d}|x|^2dm^0(x)\leq C_0$, ${{\left\| m^0 \right\|_{C^{\bb_0}}^{}}}\leq C_0$, ${{\left\| f(t,\cdot,m) \right\|_{C^{1}}^{}}}\leq C_0$, ${{\left\| g(\cdot,m) \right\|_{C^{2+\bb_0}}^{}}}\leq C_0$, for any $t\in[0,T]$ and $m\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)}$.
Assumption \[hypo:LMono\] can be interpreted as a natural extension of the Lasry-Lions monotonicity condition to MFGC. Roughly speaking, the Lasry-Lions monotonicity condition in the case of MFG without interaction through controls, translates the fact that the agents have aversion for crowed regions of the state space. In the case of MFGC, the monotonicity condition implies that the agents favor moving in a direction opposite to the mainstream. Such an assumption is adapted to models of agents trading goods or financial assets. Indeed in most of the models coming from economics or finance, a buyer may prefer to buy when no one else is buying, and conversely a seller may prefer to sell when no one else is selling.
Assumptions \[hypo:Lcoer\] and \[hypo:Lbound\] imply that at least asymptotically when $\aa$ tends to infinity, $L$ behaves like a power of $\aa$ of exponent $q'$.
Under the monotonicity assumption \[hypo:LMono\], uniqueness is in general easier to obtain than existence. For uniqueness, we assume that $f$ and $g$ are also monotonous, this is the purpose of the following assumption.
1. \[hypo:gMono\] For $m^1,m^2\in {{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)}$, and $t\in[0,T]$, assume that, $$\begin{aligned}
\int_{{ \mathbb{R}}^d}
{\left(}f{\left(}t,x,m^1 {\right)}-f{\left(}t,x,m^2{\right)}{\right)}d{\left(}m^1-m^2 {\right)}(x)
\geq
0,
\\
\int_{{ \mathbb{R}}^d}
{\left(}g{\left(}x,m^1 {\right)}-g{\left(}x,m^2{\right)}{\right)}d{\left(}m^1-m^2 {\right)}(x)
\geq
0.
\end{aligned}$$
In fact, assuming that $f$ satisfies the inequality in \[hypo:gMono\], implies that we can take $f=0$ up to replacing $L$ by $L+f$ and $H$ by $H-f$. However, since \[hypo:gMono\] is not assumed for proving the existence of solutions, we have chosen to write this assumption explicitly, and keeping $f\neq0$ is not pointless.
Let us now make assumptions on the drift function $b$, which concern the system ,
1. \[hypo:binvert\] There exists a function $\aa^*:[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}\rightarrow{ \mathbb{R}}^d$ such that for any ${\left(}t,x,{{\widetilde b}},\mu_{\aa}{\right)}\in[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$, $${{\widetilde b}}=
b{\left(}t,x,\aa^*{\left(}t,x,{{\widetilde b}},\mu_b{\right)},\mu_{\aa}{\right)},$$ where $\mu_b$ is defined by $\mu_b
=
\Bigl[
{\left(}x,\aa{\right)}\mapsto
{\left(}x,
b{\left(}t,x,\aa,\mu_{\aa}{\right)}{\right)}\Bigr]{\#}\mu_{\aa}$. Moreover $\aa^*$ is differentiable with respect to $x$ and ${{\widetilde b}}$; $\aa^*$ and its derivatives are continuous on $[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times{{\mathcal P}}_{\infty,R}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ for any $R>0$.
2. \[hypo:bbound\] $b$ and $\aa^*$ satisfy $$\begin{aligned}
&{\left|}b{\left(}t,x,\aa,\mu_\aa{\right)}{\right|}\leq
C_0{\left(}1+ |\aa|^{q_0}
+\LL_{q'}{\left(}\mu_{\aa}{\right)}^{q_0}{\right)},
\\
&{\left|}\aa^*{\left(}t,x,{{\widetilde b}},\mu_{b}{\right)}{\right|}^{q_0}
+{\left|}\aa^*_x{\left(}t,x,{{\widetilde b}},\mu_{b}{\right)}{\right|}^{q_0}
\leq
C_0{\left(}1+ |{{\widetilde b}}|
+\mathbf{1}_{q_0\neq 0}\LL_{\frac{q'}{q_0}}{\left(}\mu_{b}{\right)}{\right)},
\end{aligned}$$ for some exponent $q_0$ such that $0\leq q_0\leq q'$.
Roughly speaking \[hypo:binvert\] means that $b$ is invertible with respect to $\aa$ in such a way that its inverse map can be expressed in term of $\mu_b$ instead of $\mu_{\aa}$. Conversely, if \[hypo:binvert\] holds, then for any ${\left(}t,x,\aa,\mu_{b}{\right)}\in[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$, $$\aa
=
\aa^*{\left(}t,x,b{\left(}t,x,\aa,\mu_{\aa}{\right)},\mu_{b}{\right)}.$$ where $\mu_{\aa}$ is defined by $\mu_{\aa}
=
\Bigl[
{\left(}x,{{\widetilde b}}{\right)}\mapsto
{\left(}x,
\aa^*{\left(}t,x,{{\widetilde b}},\mu_{b}{\right)}{\right)}\Bigr]{\#}\mu_{b}$. The inequalities in \[hypo:bbound\] means that $|b|$ behaves asymptotically like a power of $\aa$ with exponent $q_0$, when $|\aa|$ is large.
To our knowledge, such a general assumption on the class of drift functions has not been made in the MFGC literature.
Main results {#subsec:mainMono}
------------
The two main results in this work are Theorems \[thm:UniMonob\] and \[thm:bpower\] below, which respectively state the existence and uniqueness of the solution to .
\[thm:UniMonob\] Under assumptions \[hypo:Lreg\]-\[hypo:LMono\], \[hypo:gMono\] and \[hypo:binvert\] there is at most one solution to .
Uniqueness results for MFGC systems with a monotonicity assumption have been proved in [@MR3112690] and [@MR3752669]. In [@MR3112690], uniqueness is proved when the diffusion coefficient is equal to $0$ and the drift is equal to the control, i.e. $\nu=0$ and $b=\aa$. In [@MR3752669] Section $4.6$, the authors stated uniqueness in the quadratic case. Theorem \[thm:UniMonob\] is new in the sense that it yields uniqueness for a large new class of Lagrangians and drift functions. Indeed, beside the monotonicity assumption \[hypo:LMono\] and \[hypo:gMono\], we only assume that $L$ satisfies \[hypo:Lreg\] and \[hypo:Lbconv\], and that the drift $b$ is invertible in the sense of \[hypo:binvert\].
\[thm:bpower\] Under assumptions \[hypo:Lreg\]-\[hypo:Monogregx\] and \[hypo:binvert\]-\[hypo:bbound\], there exists a solution to .
The existence of solutions of the MFGC system is in general much more demanding than for MFG systems without interactions through the controls. Under monotonicity assumptions similar to \[hypo:LMono\], existence has been proved in [@MR3752669] Section $4.6$, for quadratic and uniform convex Lagrangians with a growth condition on the derivatives of the Hamiltonian. In [@MR3805247], the existence of weak solutions of the monotonous MFGC system is discussed with a possibly degenerate diffusion operator, under assumptions which are uniform with respect to the joint law of states and controls.
Here, we prove existence of solutions of the monotonous MFGC system for a large class of Lagrangians and the drifts. Namely, we assume that the Lagrangians and drifts behave asymptotically like a power of $\aa$; we allow them to have a growth in the law of the controls of at most the same order as the order of dependency upon $\aa$. Before starting the discussion on existence of solutions to the MFGC systems and , we introduce a new MFGC system set in the torus, so that the solutions should have more compactness properties. We define ${{\mathbb{T}}}^d_a={ \mathbb{R}}^d/{\left(}a{ \mathbb{Z}}^d{\right)}$ the $d$-dimensional torus of radius $a>0$. Namely, we consider:
\[eq:MFGCttda\]
[align]{} \[eq:HJBttda\] &-[[\_[t]{}]{}]{}u(t,x) - u(t,x) + H[(]{}t,x, \_xu(t,x),(t)[)]{} = f(t,x,m(t)) & (0,T)\^d\_a,\
\[eq:FPKttda\] & [[\_[t]{}]{}]{}m(t,x) - m(t,x) -[(]{}H\_p[(]{}t,x,\_xu(t,x),(t)[)]{}m[)]{} = 0 & (0,T)\^d\_a,\
\[eq:muttda\] &(t) = ( I\_d, - H\_p[(]{}t,,\_xu(t,),(t)[)]{} )[\#]{}m(t) & \[0,T\],\
\[eq:CFuttda\] &u(T,x) = g(x,m(T)) & \^d\_a,\
\[eq:CImttda\] &m(0,x) = m\_0(x) & \^d\_a.
All the assumptions in paragraph \[subsec:hypoMono\] are stated in ${ \mathbb{R}}^d$. When considering that $L:[0,T]\times{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d\times{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}\to{ \mathbb{R}}$ (like in ) satisfies one of those assumptions, we shall simply replace ${ \mathbb{R}}^d$ by ${{\mathbb{T}}}^d_a$ as the state set in the chosen assumption. The fixed point satisfied by the joint law of states and controls, namely -, or , may be a difficult issue for MFGC systems. Here, using mainly the monotonicity assumption \[hypo:LMono\] and the compactness of the state space of , we prove in Section \[sec:MonoFPV\] the following lemma which states well-posedness for the fixed point , and ensures continuity with respect to time.
\[lem:FPmucont\] Assume \[hypo:Lreg\]-\[hypo:Lbound\]. Let $p\in C^0{\left(}[0,T]\times{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$ and $m\in C^0{\left(}[0,T];{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a{\right)}{\right)}$ be such that $t\mapsto p(t,\cdot)$ is continuous with respect to the topology of the local uniform convergence and $m(t)$ admits a finite second moment uniformly bounded with respect to $t\in[0,T]$. For any $t\in[0,T]$, there exists a unique $\mu(t)\in{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$ such that $$\mu(t)
=
{\left(}I_d,
-H_p{\left(}t,\cdot,p(t,\cdot),\mu(t){\right)}{\right)}\# m(t).$$ Moreover, the map $t\mapsto \mu(t)$ is continuous where ${{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$ is equipped with the weak\* topology.
The next step in our strategy for proving existence is to look for a priori estimates for the solutions of the MFGC systems and obtain compactness results to use a fixed point theorem. In section \[sec:aprioriMono\], we prove the a priori estimates stated in the following lemma for solutions to .
\[lem:Monoapriorith1\] Assume \[hypo:Lreg\]-\[hypo:Monogregx\]. If $(u,m,\mu)$ is a solution to , then ${{\left\| u \right\|_{\infty}^{}}}$, ${{\left\| \nabla_xu \right\|_{\infty}^{}}}$ and $\displaystyle{\sup_{t\in[0,T]}} \LL_{\infty}{\left(}\mu(t){\right)}$ are uniformly bounded by a constant independent of $a$.
Let us mention that the a priori estimates of Lemma \[lem:Monoapriorith1\] rely on the monotonicity assumption on $L$ and a Bernstein method introduced in [@2019arXiv190411292K]. To our knowledge, these are the first results in the literature of MFGC which use the monotonicity assumption for getting a priori estimates. They are the key ingredients of the proof of the existence of solutions to in the following theorem, proved in paragraph \[subsec:Exittda\].
\[thm:ExiMonottda\] Under assumptions \[hypo:Lreg\]-\[hypo:Monogregx\], there exists a solution to system .
Therefore, for any $a>0$ we can construct a solution to which satisfies uniform estimates with respect to $a$. This allows us to construct a compact sequence of approximating solutions to . Passing to the limit for a subsequence allows us to generalize the conclusion of Theorem \[thm:ExiMonottda\] to system . This leads to the following theorem proved in paragraph \[subsec:rd\].
\[thm:Exird\] Under assumptions \[hypo:Lreg\]-\[hypo:Monogregx\], there exists a solution to .
Uniqueness relies on the monotonicity assumptions \[hypo:LMono\] and \[hypo:gMono\], the following theorem is proved in paragraph \[subsec:UniMono\].
\[thm:UniMono\] Under assumptions \[hypo:Lreg\]-\[hypo:LMono\] and \[hypo:gMono\], there is at most one solution to or .
The idea to pass from to , is to change the optimization problem in $\aa$ into a new optimization problem expressed in term of $b$. In paragraph \[subsec:Exib\], we prove the equivalence between the solutions of these two optimization problems. A first existence results for is stated in Corollary \[thm:ExiMonob\] which uses this equivalence. Theorem \[thm:bpower\] is a consequence of Corollary \[thm:ExiMonob\] with more tractable assumptions. Let us mention that for proving Theorem \[thm:bpower\], the structure of the Lagrangian should be invariant when passing from one optimization problem to the other. In particular, one may figure out that the assumptions on the Lagrangian behaving asymptotically like a power of $\aa$ are preserved under our assumptions on the drift function $b$.
Finally, Theorem \[thm:UniMonob\] is a consequence of Theorem \[thm:UniMono\] and the above-mentionned equivalence between the two optimization problems.
1. If the Lagrangian admits the following form, $$L{\left(}t,x,\aa,\mu{\right)}=
L^0{\left(}t,x,\aa{\right)}+L^1{\left(}t,\mu{\right)},$$ we say that the Lagrangian is separated. Then \[hypo:LMono\] is automatically satisfied since the left-hand side of the inequality is identically equal to $0$. In this case, the assumptions on $L$ are satisfied if $L^0$ behaves asymptotically like a power of $\aa$ of exponenent $q'$, and $L^1$ at most involves $\Lambda_{q_0}(\mu)^{q'}$.
Here, we do not provide an explicit application in which the Lagrangian is separated, however this is a general hypothesis in the MFGC literature. Therefore, our framework in the present paper can be seen as an extension of the case when $L$ is separated.
2. All our assumptions are uniform with respect to the state variable $x$. In particular, we restrain from considering more general functions $f$ and $g$ since this topic has been investigated in the literature devoted to MFG systems without interaction through controls; we believe that the same tools can be applied to the present case, and that our results may be extended so.
3. We did not address the case without diffusion, i.e. $\nu=0$. However, all the a priori estimates of Sections \[sec:MonoFPV\] and \[sec:aprioriMono\] are uniform with respect to $\nu$. Here, the diffusion is used to easily obtain compactness results which are central for proving our existence results since the proofs rely on a fixed point theorem and approximating sequences of solutions. Using weaker topological spaces and tools from the literature devoted to weak solutions of systems of MFGs without interaction through controls, we believe that we can extend our results to weak solutions to MFGC systems without diffusion or with possibly degenerate diffusion operators. We plan to address this question in forthcoming works.
General outline {#subsec:outlineMono .unnumbered}
---------------
The present work aims at proving Theorems \[thm:UniMonob\] and \[thm:bpower\]. We list below the main steps of our analysis to make it easier for the reader to understand the structure of the proofs.
1. \[step:FPmu\] We solve the fixed point in $\mu$, which proves Lemma \[lem:FPmucont\], in three steps:
1. \[step:FPmuapriori\] in Lemma \[lem:apriorimu\] we state a priori estimates for a solution of ;
2. \[step:FPmugen\] using the Leray-Schauder fixed point theorm (Theorem \[thm:Leray\]), we solve the fixed point at any time $t\in[0,T]$, in Lemma \[lem:FPmuLS\];
3. \[step:FPmucont\] we prove that the fixed point $\mu(t)$ defined at any $t\in[0,T]$ in step \[step:FPmugen\], is continuous with respect to time (Lemma \[lem:contmu\]); this implies lemma \[lem:FPmucont\].
2. \[step:ttda\] We prove the existence of a solution to , stated in Theorem \[thm:ExiMonottda\], in two steps:
1. \[step:ttdaapriori\] we obtain a priori estimates for solutions to (Lemmas \[lem:Monoapriorith1\] and \[lem:Monoapriori\]);
2. \[step:ttdaexi\] in paragraph \[subsec:Exittda\], we use Leray-Schauder fixed point theorem (Theorem \[thm:Leray\]) and the estimates of step \[step:ttdaapriori\] to conclude.
3. \[step:MFGCrd\] We prove existence and uniqueness of the solution to (Theorems \[thm:Exird\] and \[thm:UniMono\]):
1. \[step:MFGCrdexi\] the proof of Theorem \[thm:Exird\] is given in paragraph \[subsec:rd\];
2. \[step:MFGCrduni\] the proof of Theorem \[thm:UniMono\] is given in paragraph \[subsec:UniMono\];
4. \[step:MFGCb\] The proof of existence and uniqueness of the solution to (Theorems \[thm:UniMonob\] and \[thm:bpower\]) is given in paragraph \[subsec:Exib\].
Contribution {#contribution .unnumbered}
------------
An important novelty in the present work comes from the assumptions we are considering. On the one hand, we consider a general class of monotonous Lagrangians which behave asymptotically like a power of $\aa$ with any exponent in $(1,\infty)$ (while most of the results in the literature only address the quadratic case with uniformly convex Lagrangian); they may depend on moments of $\mu_{\aa}$ at most of the same order as the above-mentioned exponent of $L$ in $\aa$; we do not require them to depend separately on ${\left(}x,\aa{\right)}$ and $\mu_{\aa}$. On the other hand, the drift functions are also general since we allow them to behave like power functions and to be not separated too. See the assumptions in paragraph \[subsec:hypoMono\] for more details.
Moreover, most contributions focus on MFG systems stated on ${{\mathbb{T}}}^d$ for simplicity. Here, we introduce a method to extend an existence result for a MFGC system stated on the torus to its counterpart on the whole Euclidean space. In particular, this method holds for MFG system without interaction through controls and the proof becomes easier. See paragraph \[subsec:rd\]. We also introduce a method to extend the well-posedness of MFGC (or MFG) systems to general drift functions, see paragraph \[subsec:Exib\]. We would like to insist on the fact that our techniques are designed in order to preserve the structure of the Lagrangian when passing from one setting to another. Here, namely it preserves the monotonicity assumption \[hypo:LMono\]. Furthermore, these methods apply to the conclusions of [@2019arXiv190411292K] and consequently generalize them.
Properties of the Lagrangian and the Hamiltonian in and
--------------------------------------------------------
Here, we write the results and the proofs for the Lagrangian and Hamiltonian involved in system . However, none of the arguments below is specific to the domain ${ \mathbb{R}}^d$, therefore the conclusions hold and the proofs can be repeated for the Lagrangian and Hamiltonian involved in .
We start by proving that under the assumptions of paragraph \[subsec:hypoMono\], when $b=\aa$, $L$ is strictly convex.
\[lem:Lstrconv\] If $L$ is coercive and differentiable with respect to $\aa$, and $b=\aa$, assuming that $L$ is strictly convex is equivalent to \[hypo:Lbconv\].
If $L$ is stricly convex and coercive, it is straightforward to check \[hypo:Lbconv\].
Conversely, we take ${\left(}t,x,\mu{\right)}\in[0,T]\times{ \mathbb{R}}^d\times{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$. We set $\ell(\aa)=L{\left(}t,x,\aa,\mu{\right)}$. It is sufficient to prove that $\ell$ is strictly convex.
*First step: proving that $\ell$ is convex.*
We define $\ell^{**}$ as the biconjugate of $\ell$, $\ell^{**}$ is in particular the Legendre transform of $H{\left(}t,x,\cdot,\mu{\right)}$. The map $\ell^{**}$ is convex and continuous since $\ell$ is coercive, and it satisfies $\ell^{**}\leq \ell$. In what follows, we will prove that $\ell^{**}=\ell$.
We assume by contradiction that $\ell^{**}\neq \ell$: there exists $\aa^0\in{ \mathbb{R}}^d$ such that $\ell^{**}{\left(}\aa^0{\right)}<\ell{\left(}\aa^0{\right)}$. We recall that $\ell$ and $\ell^{**}$ admit the same convex envelope, therefore by Carathéorthéodory’s theorem, there exists ${\left(}\aa^i{\right)}_{1\leq i\leq d+1}\in{\left(}{ \mathbb{R}}^d{\right)}^{d+1}$ and ${\left(}\ll^i{\right)}_{1\leq i\leq d+1}\in{\left(}{ \mathbb{R}}_+{\right)}^{d+1}$ such that $$\aa^0
=
\sum_{i=1}^{d+1}\ll^i\aa^i,
\;\;
\ell^{**}{\left(}\aa^0{\right)}=
\sum_{i=1}^{d+1}\ll^i\ell{\left(}\aa^i{\right)},
\;\;\text{ and }\;\;
\sum_{i=1}^{d+1}\ll^i
=
1.$$ Using the inequality $\ell^{**}\leq \ell$, we obtain that $$\ell^{**}(\aa)
=
\sum_{i=1}^{d+1}\ll^i\ell{\left(}\aa^i{\right)}\geq
\sum_{i=1}^{d+1}\ll^i\ell^{**}{\left(}\aa^i{\right)}.$$ This inequality is in fact an equality since $\ell^{**}$ is convex, which implies that $\ell^{**}{\left(}\aa^i{\right)}=\ell{\left(}\aa^i{\right)}$ for any $i\in{\left\{}
\newcommand{\rc}{\right\}}1,2,\dots,d+1\rc$. Take $p\in\partial \ell^{**}{\left(}\aa^0{\right)}$, where $\partial \ell^{**}{\left(}\aa^0{\right)}$ is the subdifferential of $\ell^{**}$ at $\aa^0$. For $i\in{\left\{}
\newcommand{\rc}{\right\}}1,\dots,d+1\rc$, this implies $$\ell{\left(}\aa^i{\right)}=
\ell^{**}{\left(}\aa^i{\right)}\geq
\ell^{**}{\left(}\aa^0{\right)}+p\cdot{\left(}\aa^i-\aa^0{\right)}.$$ Multiplying the latter inequality by $\ll^i$ and taking the sum over $i$, yield that it is in fact an equality. Then, it is straightforward to check that $p\in\partial \ell^{**}{\left(}\aa^i{\right)}$ for any $i$; this implies that $p=\nabla_{\aa}\ell{\left(}\aa^i{\right)}$, since $\ell^{**}{\left(}\aa^i{\right)}=\ell{\left(}\aa^i{\right)}$ and $\ell$ is differentiable with respect to $\aa$. The maximum in the definition of $H(t,x,-p,\mu)$ is achieved at any $\aa^i$, this is a contracdition with \[hypo:Lbconv\]. Therefore $\ell=\ell^{**}$ and $\ell$ is convex.
*Second step: $\ell$ is striclty convex.*
By definition of the subdifferential of a convex function, $\aa\in{ \mathbb{R}}^d$ achieves the maximum in the definition of $H{\left(}t,x,-\nabla_{\aa}\ell{\left(}\aa{\right)},\mu{\right)}$. Using the fact that this maximum is unique by \[hypo:Lbconv\], we obtain the strict convexity of $\ell$, and the one of $L$ with respect to $\aa$.
In paragraph \[subsec:hypoMono\], we assume that $L$ behaves at infinity as a power of $\aa$ of exponent $q'$. Because of the conjugacy relation between $L$ and $H$, it implies that $H$ behaves at infinity like a power of $p$ of exponent $q$.
\[lem:Hbound\] Under assumptions \[hypo:Lreg\], \[hypo:Lbconv\], \[hypo:Lcoer\] and \[hypo:Lbound\], the map $H$, defined in \[eq:defHrd\], is differentiable with respect to $x$ and $p$, $H$ and its derivatives are continuous on $[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times{{\mathcal P}}_{\infty,R}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ for any $R>0$. Moreover there exists ${{\widetilde C}}_0>0$ a constant which only depends on $C_0$ and $q$ such that $$\begin{aligned}
\label{eq:Hpbound}
{\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}&\leq
{{\widetilde C}}_0{\left(}1+|p|^{q-1}
+\LL_{q'}{\left(}\mu{\right)}{\right)},
\\
\label{eq:Hbound}
{\left|}H{\left(}t,x,p,\mu{\right)}{\right|}&\leq
{{\widetilde C}}_0{\left(}1+|p|^{q}
+\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)},
\\
\label{eq:Hcoer}
p\cdot H_p{\left(}t,x,p,\mu{\right)}-H{\left(}t,x,p,\mu{\right)}&\geq
{{\widetilde C}}_0^{-1}|p|^q
-{{\widetilde C}}_0{\left(}1
+\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)},
\\
\label{eq:Hxbound}
{\left|}H_x{\left(}t,x,p,\mu{\right)}{\right|}&\leq
{{\widetilde C}}_0{\left(}1+|p|^{q}
+\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)},
\end{aligned}$$ for any $(t,x)\in[0,T]\times{ \mathbb{R}}^d$, $p\in{ \mathbb{R}}^d$ and $\mu\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$.
Up to replacing $C_0$ with $\max(C_0,{{\widetilde C}}_0)$, we can assume that the inequalities in Lemma \[lem:Hbound\] are satisfied with $C_0$ instead of ${{\widetilde C}}_0$.
Let us notice that it is possible and not more difficult to extend the results stated in Lemma \[lem:Hbound\] to the Hamiltonian used in and defined in , however we will not have any use of such results in the present paper.
*First step: differentiability of $H$ in $p$, and continuity of $H$ and $H_p$.*
For $(t,x,\mu)\in[0,T]\times{ \mathbb{R}}^d\times{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$, the map $\aa\mapsto L{\left(}t,x,\aa,\mu{\right)}$ is stricly convex by Lemma \[lem:Lstrconv\] and coercive by \[hypo:Lcoer\]; Theorem $26.6$ in [@MR0274683] implies that $H$ is differentiable with respect to $p$, the map $\aa\mapsto -L_{\aa}{\left(}t,x,\aa,\mu{\right)}$ is invertible; its iverse map is $p\mapsto -H_{p}{\left(}t,x,p,\mu{\right)}$ by [@MR0274683] Theorem $26.5$. Theorem $26.6$ in [@MR0274683] also implies that the maximum in \[eq:defHrd\] is achieved by a unique control given by $-H_p{\left(}t,x,p,\mu{\right)}$. In the next step, we prove \[eq:Hpbound\] which implies that $H_p$ maps the bounded subsets of $[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times{{\mathcal P}}_{\infty,R}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ for $R>0$ into relatively compact subspaces of ${ \mathbb{R}}^d$; we recall that $L_{\aa}$ is continuous on $[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times{{\mathcal P}}_{\infty,R}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$; therefore $H_p$ is likewise continuous on the same space. We recall that $H$ satisfies $$H(t,x,p,\mu)
=
p\cdot H_p{\left(}t,x,p,\mu{\right)}-L{\left(}t,x, -H_p{\left(}t,x,p,\mu{\right)},\mu{\right)},$$ therefore $H$ is also continuous on the same spaces.
*Second step: proving the first three inequalities of the Lemma.* Using the growth assumptions on $L$, we first prove . On the one hand we have that $$H{\left(}t,x,p,\mu{\right)}\geq
-L{\left(}t,x,0,\mu{\right)}\geq
-C_0{\left(}1+\LL_{q'}(\mu)^{q'}{\right)},$$ by \[hypo:Lcoer\] and the condition of optimality in . On the other hand, \[hypo:Lbound\], the fact that $-H_p{\left(}t,x,p,\mu{\right)}$ satisfies the optimality condition in , and the Young inequality $y\cdot z\leq
\frac{|y|^{q}}{q}
+\frac{|z|^{q'}}{q'}$ for $y,z\in{ \mathbb{R}}^d$, yield that, $$\begin{aligned}
H{\left(}t,x,p,\mu{\right)}&=
p\cdot H_p{\left(}t,x,p,\mu{\right)}-L{\left(}t,x,-H_p{\left(}t,x,p,\mu{\right)},\mu{\right)}\\
&\leq
\frac1{q'C_0}{\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}^{q'}
+\frac{C_0^{\frac{q}{q'}}}{q}{\left|}p{\right|}^{q}
-C_0^{-1}{\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}^{q'}
+C_0{\left(}1+\LL_{q'}(\mu)^{q'}{\right)}\\
&\leq
-\frac1{qC_0}{\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}^{q'}
+\frac{C_0^{\frac{q}{q'}}}{q}{\left|}p{\right|}^{q}
+C_0{\left(}1+\LL_{q'}(\mu)^{q'}{\right)}.
\end{aligned}$$ Therefore, using the latter two chains of inequalities, and the fact that $\frac{q}{q'}=q-1$, we obtain that, $$\label{eq:Hpq'bound}
\frac1{qC_0}{\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}^{q'}
\leq
\frac{C_0^{q-1}}{q}{\left|}p{\right|}^{q}
+2C_0{\left(}1+\LL_{q'}(\mu)^{q'}{\right)}.$$ This and the inequality $|y+z|^{\frac1{q'}}
\leq
|y|^{\frac1{q'}}
+|z|^{\frac1{q'}}$ for $y,z\in{ \mathbb{R}}$, imply that $${\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}\leq
C_0^{q-1}{\left|}p{\right|}^{q-1}
+{\left(}2qC_0^2{\right)}^{\frac1{q'}}
{\left(}1+\LL_{q'}(\mu){\right)}.$$ From \[hypo:Lbound\] and , we obtain that, $$\begin{aligned}
{\left|}H{\left(}t,x,p,\mu{\right)}{\right|}&=
{\left|}p\cdot H_p{\left(}t,x,p,\mu{\right)}-L{\left(}t,x,-H_p{\left(}t,x,p,\mu{\right)},\mu{\right)}{\right|}\\
&\leq
\frac{|p|^q}q
+\frac{{\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}^{q'}}{q'}
+C_0{\left(}1+
{\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}^{q'}
+\LL_{q'}(\mu)^{q'}{\right)}\\
&\leq
{\left(}\frac1q+\frac{C_0^q}{q'}{\right)}|p|^q
+C_0{\left(}1+2qC_0{\left(}\frac1{q'}+C_0{\right)}{\right)}{\left(}1+\LL_{q'}(\mu)^{q'}{\right)}.
\end{aligned}$$ We still have to prove . Let $\ee$ be a positive constant depending only on $C_0$ such that $\ee-C_0\ee^{q'}\geq\frac{\ee}2$, by the optimality condition in used with $\aa=-\ee|p|^{q-2}p$, we have, $$\begin{aligned}
H{\left(}t,x,p,\mu{\right)}&\geq
\ee |p|^q
-L{\left(}t,x, -\ee|p|^{q-2}p,\mu{\right)}\\
&\geq
\ee |p|^q
-C_0{\left(}1+\ee^{q'}|p|^{(q-1)q'}+\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)}\\
&\geq
\frac{\ee}2 |p|^q
-C_0{\left(}1+\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)}.
\end{aligned}$$ Then from \[hypo:Lbound\], $$\begin{aligned}
H{\left(}t,x,p,\mu{\right)}&=
p\cdot H_p{\left(}t,x,p,\mu{\right)}-L{\left(}t,x,-H_p{\left(}t,x,p,\mu{\right)},\mu{\right)}\\
&\leq
\frac{\ee}4 |p|^q
+\frac{{\left(}4q\ee^{-1}{\right)}^{\frac{q'}q}}{q'}
{\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}^{q'}
+C_0{\left(}1+
{\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}^{q'}
+\LL_{q'}(\mu)^{q'}{\right)}.
\end{aligned}$$ Combining the latter two chains of inequalities, there exists $C$ a positive constant depending only on $C_0$ such that $${\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}^{q'}
\geq
C^{-1}|p|^q
-C{\left(}1+ \LL_{q'}{\left(}\mu{\right)}^{q'}{\right)}.$$ This and \[hypo:Lcoer\] yield that $$\begin{aligned}
p\cdot H_p{\left(}t,x,p,\mu{\right)}-H{\left(}t,x,p,\mu{\right)}&=
L{\left(}t,x,-H_p{\left(}t,x,p,\mu{\right)},\mu{\right)}\\
&\geq
C_0^{-1}
{\left|}H_p{\left(}t,x,p,\mu{\right)}{\right|}^{q'}
-C_0{\left(}1+ \LL_{q'}{\left(}\mu{\right)}^{q'}{\right)}\\
&\geq
{\left(}C_0C{\right)}^{-1}|p|^q
-{\left(}C_0+C_0^{-1}C{\right)}{\left(}1+\LL_{q'}(\mu)^{q'}{\right)}.
\end{aligned}$$
*Third step: the smothness properties and the last inequality.*
From , $-H_p(t,x,p,\mu)$ is locally uniformly bounded, therefore we can reduce the set of admissible controls $\aa$ in from ${ \mathbb{R}}^d$ to a compact subset of ${ \mathbb{R}}^d$. Within this framework, the envelop theorem states that $H$ is differentiable in $x$ and its derivatives are defined by, $$H_x{\left(}t,x,p,\mu{\right)}=
-L_x{\left(}t,x,-H_p{\left(}t,x,p,\mu{\right)},\mu{\right)}.$$ The continuity property of $H_x$ relies on the ones of $L_x$ and $H_p$. Moreover, from \[hypo:Lbound\] and , we obtain $${\left|}H_x{\left(}t,x,p,\mu{\right)}{\right|}\leq
C_0^{q+1}|p|^{q}
+C_0{\left(}1+2qC_0^2{\right)}{\left(}1+\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)}.$$ This concludes the proof.
Applications {#sec:appli}
============
Exhaustible ressource model {#subsec:Monoexhau}
---------------------------
This model is often referred to as Bertrand and Cournot competition model for exhaustible ressources, introduced in the independent works of Cournot [@cournot1838recherches] and Bertrand [@Bertrand]; its mean field game version in dimension one was introduced in [@MR2762362] and numerically analyzed in [@MR3359708]; for theoretical results see [@2019arXiv190205461F; @MR3755719; @MR4064472; @MR3888969]. We consider a continuum of producers selling exhaustible ressources. The production of a representative agent at time $t\in[0,T]$ is denoted by $q_t\geq 0$; the agents differ in their production capacity $X_t\in{ \mathbb{R}}$ (the state variable), that satifies, $$dX_t
=
-q_tdt
+\sqrt{2\nu} dW_t,$$ where $\nu>0$ and $W$ is a $d$-dimentional Brownian motion. Each producer is selling a different ressource and has her own consumers. However, the ressources are substitutable and any consumer may change her mind and buy from a competitor depending on the degree of competition in the game (which stands for $\ee$ in the linear demand case below for instance). Therefore the selling price per unit of ressource that a producer can make when she sales $q$ units of ressource, depends naturally on $q$ and on the quantity produced by the other agents. The price satisfies a supply-demand relationship, and is given by $P{\left(}q,{{\overline q}}{\right)}$, where ${{\overline q}}$ is the accumulated demand which depends on the overall distribution of productions of the agents. A producer tries to maximize her profit, or equivalently to minimize the following quantity, $${{\mathbb E}}{\left[}\int_0^T -P(q_t,{{\overline q}}_t)\cdot q_tdt
+g{\left(}X_T{\right)}{\right]},$$ where $g$ is a terminal cost which often penalizes the producers who have non-zero production capacity at the end of the game. In the Cournot competition, see [@cournot1838recherches], the producers are controling their production $q$. Like in the formulation of the MFG arising in such a problem in [@MR3359708], here we consider the Bertrand’s formulation [@Bertrand], where an agent directly controls her selling price $\aa=P(q,{{\overline q}})$. Then after inverting the latter equality, the production can be viewed as a function of the price and the mean field. Mathematically this corresponds to writing $q=Q{\left(}\aa,{{\overline \aa}}{\right)}$.
In [@MR3359708], the authors considered a linear demand system depending on ${{\overline q}}_{\text{lin}}=\int_{{ \mathbb{R}}}q(x)dm(x)$, and a price satisfying $\aa=P_{\text{lin}}(q,{{\overline q}}_{\text{lin}})=1-q-\ee {{\overline q}}_{\text{lin}}$. In this case, $L^{\text{lin}}$ the running cost, and $H_{\text{lin}}$ its Legendre transform are defined by $$\begin{aligned}
L^{\text{lin}}{\left(}\aa,\mu{\right)}&=
\aa^2
+\frac{\ee}{1+\ee}\aa{{\overline \aa}}-\frac1{1+\ee}\aa,
\\
H^{\text{lin}}{\left(}p,\mu{\right)}&=
\frac14{\left(}p+\frac{\ee}{1+\ee}{{\overline \aa}}-\frac1{1+\ee}{\right)}^2,\end{aligned}$$ where $\aa,p\in{ \mathbb{R}}$, $\mu\in{{\mathcal P}}{\left(}{ \mathbb{R}}\times{ \mathbb{R}}{\right)}$ and ${{\overline \aa}}$ is defined by ${{\overline \aa}}=\int_{{ \mathbb{R}}\times{ \mathbb{R}}}{{\widetilde \aa}}d\mu(y,{{\widetilde \aa}})$. Therefore the system of MFGC has the following form, $$\label{eq:exhaulin}
{\left\{}
\newcommand{\rc}{\right\}}\begin{aligned}
&-{{\partial_{t}}}u
-\nu\Delta u
+
\frac14{\left(}\nabla_xu+\frac{\ee}{1+\ee}{{\overline \aa}}-\frac1{1+\ee}{\right)}^2
=
0,
\\
&{{\partial_{t}}}m
-\nu\Delta m
-\operatorname{div}{\left(}\frac12{\left(}\nabla_xu+\frac{\ee}{1+\ee}{{\overline \aa}}-\frac1{1+\ee}{\right)}m{\right)}=
0,
\\
&{{\overline \aa}}(t)
=
-\int_{{ \mathbb{R}}^d}
\frac12{\left(}\nabla_xu+\frac{\ee}{1+\ee}{{\overline \aa}}(t)-\frac1{1+\ee}{\right)}dm(t,x),
\\
&u(T,x)
=
g(x),
\\
&m(0,x)
=
m_0(x),
\end{aligned}
\right.$$ for $(t,x)\in[0,T]\times{ \mathbb{R}}$. Roughly speaking, $\ee=0$ corresponds to a monopolist who does not suffer from competition, and she plays as if she was alone in the game. Conversely, $\ee=\infty$ stands for all the producers selling the same ressource and the consumers not having any a priori preference.
Let us consider the following generalization of the latter system to the $d$-dimensional case with a more general Hamiltonian and interaction through controls, $$\label{eq:exhau}
{\left\{}
\newcommand{\rc}{\right\}}\begin{aligned}
&-{{\partial_{t}}}u
-\nu\Delta u
+
H{\left(}t,x,\nabla_xu+\vp(x)^TP(t){\right)}=
f(t,x,m(t)),
\\
&{{\partial_{t}}}m
-\nu\Delta m
-\operatorname{div}{\left(}H_p{\left(}t, x,\nabla_xu+\vp(x)^TP(t){\right)}m{\right)}=
0,
\\
&P(t)
=
\Psi{\left(}t,
-\int_{{ \mathbb{R}}^d}\vp(x)
H_p{\left(}t,x,\nabla_xu+\vp(x)^TP(t){\right)}dm(t,x){\right)},
\\
&u(T,x)
=
g(x,m(T)),
\\
&m(0,x)
=
m_0(x),
\end{aligned}
\right.$$ where $\vp:{ \mathbb{R}}^d\mapsto{ \mathbb{R}}^{d\times d}$ and $\Psi:{ \mathbb{R}}^d\mapsto{ \mathbb{R}}^{d\times d}$ are given functions. The counterpart of the latter system posed on ${{\mathbb{T}}}^d$ has been introduced in [@2019arXiv190205461F]. Theorem \[thm:UniMonob\] and \[thm:bpower\] provide the existence and the uniqueness respectively of the solution of this MFGC system.
\[prop:uniexhau\] Assume \[hypo:Lreg\], \[hypo:Lbconv\], \[hypo:gMono\]. If the function $\Psi$ is continuous, $\Psi(t,\cdot)$ is monotone, locally Lipschitz continuous, and admits at most a power-like growth of exponent $q'-1$ with a coefficient uniform in $t\in[0,T]$, there exists at most one solution to .
\[prop:exiexhau\] Assume \[hypo:Lreg\], \[hypo:Lbconv\], \[hypo:Lcoer\]-\[hypo:Monogregx\], and that $\Psi$ satisfies the same assumptions as in Proposition \[prop:uniexhau\]. There exists a solution to .
Take the drift function as $b=\aa$. We define the Lagrangian $\ell$ by $$\ell{\left(}t,x,\aa,\mu{\right)}=
L{\left(}t,x,\aa{\right)}+\vp(x)\aa\cdot P(t,\mu)
+f(t,x,m),$$ where $L$ is the Legendre transform of the map $H$ in , and $P(t,\mu)$ is defined by $P(t,\mu)=\Psi{\left(}t,
\int_{{ \mathbb{R}}^d\times{ \mathbb{R}}^d}\vp(x)\aa d\mu(x,\aa){\right)}$, for $(t,x,\aa,m,\mu)\in
[0,T]\times{ \mathbb{R}}^d\times{ \mathbb{R}}^d\times
{{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)}\times
{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ such that $m$ is the first marginal of $\mu$. We take $h$ as the Legendre transform of $\ell$ with respect to $\aa$.
If $\Psi$ satisfies the assumptions in \[prop:uniexhau\], any of the assumptions \[hypo:Lreg\], \[hypo:Lbconv\], \[hypo:Lcoer\], or \[hypo:Lbound\] is preserved by replacing $L$ by $\ell$. Moreover, a straightforward calculation yields that $$\begin{gathered}
\int_{{ \mathbb{R}}^d\times{ \mathbb{R}}^d}
{\left(}\ell{\left(}t, x,\a,\mu^1 {\right)}-\ell{\left(}t, x,\a,\mu^2{\right)}{\right)}d{\left(}\mu^1-\mu^2 {\right)}(x,\aa)
\\
=
{\left(}P{\left(}t,\mu^1{\right)}-P{\left(}t,\mu^2{\right)}{\right)}\cdot
\int_{{ \mathbb{R}}^d\times{ \mathbb{R}}^d}\vp(x)\aa d{\left(}\mu^1-\mu^2{\right)}(x,\aa),
\end{gathered}$$ for $t\in[0,T]$ and $\mu^1,\mu^2\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$. This and the monotonicity of $\Psi$ implies that $\ell$ satisfies \[hypo:LMono\]. Therefore, Propositions \[prop:uniexhau\] and \[prop:exiexhau\] are direct consequences of Theorems \[thm:UniMonob\] and \[thm:bpower\] respectively.
In [@2019arXiv190205461F], similar existence and uniqueness results for the counterpart of posed on ${{\mathbb{T}}}^d$ are given in the quadratic setting, with a uniformly convex Lagrangian and $\Psi$ being the gradient of a convex map. Here, we generalize their results to a wider class of Lagrangians and functions $\Psi$.
For an extension of this model to the case when $\Psi$ is non-monotone, see [@2019arXiv190411292K].
A model of crowd motion {#subsec:abncrowd}
-----------------------
This model of crowd motion has been introduced in [@2019arXiv190411292K] in the non-monotone setting. It has been numerically studied in [@achdou2020mean] in the quadratic non-monotone case. For $\mu\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ we define $V(\mu)$ the average drift by, $$V(\mu)
=
\frac1{Z(\mu)}
\int_{{ \mathbb{R}}^d\times{ \mathbb{R}}^d}
\aa k(x) d\mu(x,\aa),$$ where $Z(\mu)$ is a normalization constant defined by $Z(\mu)
=
{\left(}\int_{{ \mathbb{R}}^d\times{ \mathbb{R}}^d}
k(x)^{q_1} d\mu(x,\aa){\right)}^{\frac1{q_1}}$, for some constant $q_1\in[q,\infty]$ where $q$ is defined below. To be consistent with the notations used in [@2019arXiv190411292K], $k:{ \mathbb{R}}^d\rightarrow{ \mathbb{R}}_+$ is a non-negative kernel. By convention, if $Z{\left(}\mu{\right)}=0$, we take $V{\left(}\mu{\right)}=0$. The state of a representative agent is given by her position $X_t\in{ \mathbb{R}}^d$ which she controls through her velocity $\aa$ via the following stochastic differential equation, $$dX_t
=
\aa_t dt
+\sqrt{2\nu}dW_t.$$ Her objective is to minimize the cost functional given by, $${{\mathbb E}}{\left[}\int_0^T
\frac{\th}{2}{\left|}\aa_t+\ll V{\left(}\mu(t){\right)}{\right|}^{2}
+\frac{1-\th}{a'}{\left|}\aa_t{\right|}^{a'}
+f(t,X_t,m(t))dt
+g(X_T,m(T)){\right]},$$ where $\ll\geq0$ and $0\leq\th\leq1$ are two constants standing for the intensity of the preference of an individual to have an opposite control as the stream one, and $a'> 1$ is an exponent. In this model we define the Lagrangian $L$ by, $$L{\left(}x,\aa,\mu{\right)}=
\frac{\th}{2}{\left|}\aa+\ll V(\mu){\right|}^{2}
+\frac{1-\th}{a'}{\left|}\aa{\right|}^{a'},$$ and the Hamiltonian $H$ as its Legendre transform. The map $H$ does not admit an explicit form for every choice of the parameters $a'$. We take $q'=\max{\left(}2,a'{\right)}$, and $q=\frac{q'}{q'-1}$ its conjugate exponent.
Here, since the control is equal to the drift, the MFGC system is of the form of . Therefore, the following proposition is a consequence of Theorems \[thm:UniMonob\] and \[thm:bpower\].
Under assumption \[hypo:Monogregx\], there exists a solution to the above MFGC system of crowd motion.
Under assumption \[hypo:gMono\], this solution is unique.
The proof is straightforward, it consists in checking that $L$ satisfies \[hypo:Lreg\]-\[hypo:Lbound\].
For existence results of the MFGC system of this model with $\ll<0$, see [@2019arXiv190411292K].
The fixed point and the proof of Lemma \[lem:FPmucont\] {#sec:MonoFPV}
=======================================================
This section is devoted to step \[step:FPmu\]. In paragraph \[subsec:FPmuMono\], we state a priori estimates on a fixed point of (Lemma \[lem:apriorimu\]); then we we use these estimates and Leray-Schauder fixed point theorem (Theorem \[thm:Leray\]) and obtain the existence of a fixed point at any time $t\in[0,T]$ (Lemma \[lem:FPmuLS\]). We address the continuity with respect to time of the fixed point, i.e. step \[step:FPmucont\], in Lemma \[lem:contmu\].
In this section and the next one, we work on ${{\mathbb{T}}}^d_a={ \mathbb{R}}^d/{\left(}a{ \mathbb{Z}}^d{\right)}$ ${{\mathbb{T}}}^d_a$ the $d$-dimensional torus of radius $a>0$. Here we take $L:[0,T]\times{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d\times{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}\to { \mathbb{R}}$. All the assumptions in paragraph \[subsec:hypoMono\] are stated in ${ \mathbb{R}}^d$, but, when considering that $L$ satisfies one of those assumptions, we shall simply replace ${ \mathbb{R}}^d$ by ${{\mathbb{T}}}^d_a$ as the state set in the chosen assumption (note that we keep ${ \mathbb{R}}^d$ as the set of admissible controls). The initial distribution $m_0$ is now in ${{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a{\right)}$. The Hamiltonian $H$ is still defined as the Legendre transform of $L$, i.e. it satisfies .
Leray-Schauder Theorem for solving the fixed point in $\mu$ {#subsec:FPmuMono}
-----------------------------------------------------------
We start by stating a priori estimates for solutions of the fixed point in $\mu$ , involving $\LL_{q'}{\left(}\mu{\right)}$ and $\LL_{\infty}{\left(}\mu{\right)}$ defined in .
\[lem:apriorimu\] Assume that $L$ satisfies \[hypo:Lreg\]-\[hypo:Lbound\] For any $t\in[0,T]$, $m\in {{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a{\right)}$ and $p\in C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$, if there exists $\mu\in{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$ such that $$\label{eq:MonoFPmu}
\mu
=
{\left(}I_d,-H_p{\left(}t, \cdot,p(\cdot),\mu{\right)}{\right)}\# m,$$ then it satisfies $$\begin{aligned}
\label{eq:aprioriLq'}
\LL_{q'}{\left(}\mu{\right)}^{q'}
&\leq
4C_0^2
+\frac{{\left(}q'{\right)}^{q-1}{\left(}2C_0{\right)}^q}{q}
{{\left\| p \right\|_{L^q(m)}^{q}}},
\\
\label{eq:aprioriLinf}
\LL_{\infty}{\left(}\mu{\right)}&\leq
C_0{\left(}1+{{\left\| p \right\|_{\infty}^{}}}+\LL_{q'}{\left(}\mu{\right)}{\right)}.
\end{aligned}$$
We use \[hypo:LMono\] with $m\otimes\delta_0$ and $\mu$ satisfying , $$\label{eq:LMono_apriori}
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
{\left(}L{\left(}t,x,\aa, \mu {\right)}-L{\left(}t,x,\aa, m\otimes\delta_0 {\right)}{\right)}d\mu(x,\aa)
+
\int_{{{\mathbb{T}}}^d_a}
{\left(}L{\left(}t,x,0, m\otimes\delta_0 {\right)}-L{\left(}t,x,0, \mu{\right)}{\right)}dm(x)
\geq
0.$$ From \[hypo:Lbound\], we obtain $$\label{eq:Lbounded}
\int_{{{\mathbb{T}}}^d_a}
L{\left(}t,x,0, m\otimes\delta_0 {\right)}dm(x)
\leq
C_0.$$ The latter two inequalities, \[hypo:Lcoer\] and the convexity of $L$ (stated in Lemma \[lem:Lstrconv\]) yield $$\begin{aligned}
C_0^{-1}
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
{\left|}\aa{\right|}^{q'}d\mu(x,\aa)
-C_0
&\leq
C_0+
\int_{{{\mathbb{T}}}^d_a}
{\left(}L{\left(}t,x,\aa^{\mu}(x), \mu {\right)}-L{\left(}t,x,0, \mu {\right)}{\right)}dm(x)
\\
&\leq
C_0+
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
\aa\cdot L_{\aa}{\left(}t,x,\aa, \mu {\right)}d\mu(x,\aa),
\end{aligned}$$ where $\aa^{\mu}$ is defined in paragraph \[subsec:notaMono\]. We recall that $p(x)=-L_{\aa}{\left(}t,x,\aa^{\mu}(x),\mu{\right)}$. Using the inequality $yz\leq
\frac{y^{q'}}{c^{q'}q'}
+\frac{c^qz^{q}}{q}$ which holds for any $y,z,c>0$, we obtain $$\frac1{C_0}\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
{\left|}\aa{\right|}^{q'}d\mu(x,\aa)
\leq
2C_0
+\frac{{\left(}2C_0q'{\right)}^{\frac{q}{q'}}}q
\int_{{{\mathbb{T}}}^d_a}{\left|}p(x){\right|}^q dm(x)
+\frac1{2C_0}
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
{\left|}\aa{\right|}^{q'}d\mu(x,\aa).$$ This and $\frac{q}{q'}+1=q$ imply . This and \[eq:Hpbound\] implies \[eq:aprioriLinf\], we recall that we assume $C_0={{\widetilde C}}_0$.
Here, we shall use Leray-Schauder fixed point theorem as stated in [@MR1814364] Theorem $11.6$.
\[thm:Leray\] Let ${{\cal B}}$ be a Banach space and let $\Psi$ be a compact mapping from $[0,1]\times{{\cal B}}$ into ${{\cal B}}$ such that $\Psi(0,x)=0$ for all $x\in{{\cal B}}$. Suppose that there exists a constant $C$ such that $${{\left\| x \right\|_{{{\cal B}}}^{}}}\leq C,$$ for all ${\left(}\th,x{\right)}\in[0,1]\times{{\cal B}}$ satisfying $x=\Psi(\th,x)$. Then the mapping $\Psi(1,\cdot)$ of ${{\cal B}}$ into itself has a fixed point.
From Lemma \[lem:apriorimu\] and Theorem \[thm:Leray\], we obtain the following existence result for a fixed point .
\[lem:FPmuLS\] Assume \[hypo:Lreg\]-\[hypo:Lbound\]. For $t\in[0,T]$, $m\in{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a{\right)}$ and $p\in C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$, there exists a unique $\mu\in{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$ such that $\mu
=
{\left(}I_d,
-H_p{\left(}t,\cdot,p(\cdot),\mu{\right)}{\right)}\# m$. Moreover, $\mu$ satisfies the inequality stated in Lemma \[lem:apriorimu\].
In the following proof, we will take advantage of the flexibily offered when making all assumptions on the Lagrangian, instead of the Hamiltonian. We will introduce a sequence of new Lagrangians. The associated Hamiltonians may not admit explicit form; therefore it would be difficult to check assumptions on them. Here on the one hand, checking the assumptions on the new Lagrangians is straightforward. On the other hand, we obtain the same conclusions on the new Hamiltonian as stated in Lemma \[lem:Hbound\].
Take $(t,{{\overline p}},m)$ satisfying the same assumptions as $(t,p,m)$ in Lemma \[lem:FPmuLS\]. In order to use the Leray-Schauder fixed point theorem later, we introduce the following family of Lagrangians indexed by $\ll\in[0,1]$, $$\label{eq:Lpo}
L^{{{\overline p}},\ll}{\left(}x,\aa,\mu{\right)}=
\ll
L{\left(}t,x,\aa,\mu{\right)}+(1-\ll)
{\left(}\frac{{\left|}\aa{\right|}^{q'}}{q'}
-\aa\cdot {{\overline p}}(x){\right)},$$ for ${\left(}x,\aa,\mu{\right)}\in{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d\times{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$. We denote by $H^{{{\overline p}},\ll}$ the Legendre transform of $L^{{{\overline p}},\ll}$. For $\ll=0$ it satisfies $$\label{eq:Hpo0}
H^{{{\overline p}},0}{\left(}x,p,\mu{\right)}=
\frac1q{\left|}p-{{\overline p}}(x){\right|}^q.$$ From Young inequality, we obtain that $$\begin{aligned}
{\left|}\aa\cdot{{\overline p}}(x){\right|}\leq
\frac{{\left|}\aa{\right|}^{q'}}{2q'}
+
\frac{2^{q-1}}{q}{{\left\| p \right\|_{\infty}^{}}}.
\end{aligned}$$ Therefore, up to changing $C_0$ into $\max{\left(}\frac1{2q'},\frac{2^{q-1}}{q}{{\left\| p \right\|_{\infty}^{}}},C_0{\right)}$, we may assume that $L^{{{\overline p}},\ll}$ satisfies \[hypo:Lreg\]-\[hypo:Lbound\], with the same constant $C_0$ for any $\ll\in[0,1]$. The map ${\left(}\ll,x,p,\mu{\right)}\mapsto -H^{{{\overline p}},\ll}_p{\left(}x,p,\mu{\right)}$ is continuous on $[0,1]\times{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d\times{{\mathcal P}}_{\infty,R}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$, for any $R>0$, by the same arguments as in the proof of Lemma \[lem:Hbound\].
For $\aa\in C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$, we set $\mu={\left(}I_d,\aa{\right)}\#m\in{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$ and ${{\overline \aa}}(x)=-H_p^{{{\overline p}},\ll}{\left(}x,{{\overline p}}(x),\mu{\right)}$, for $x\in{{\mathbb{T}}}^d_a$. We define the map $\Psi$, from $[0,1]\times C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$ to $C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$, by $\Psi{\left(}\ll,\aa{\right)}={{\overline \aa}}$. If $\aa$ is a fixed point of $\Psi(1,\cdot)$, then $\mu$ defined as above satisfies the fixed point in Lemma \[lem:FPmuLS\]. Conversely, if $\mu$ satisfies the fixed point in Lemma \[lem:FPmuLS\], then $\aa^{\mu}$ (defined in paragraph \[subsec:notaMono\]) is a fixed point of $\Psi(1,\cdot)$.
The map $\Psi$ is continuous by the continuity of ${\left(}\ll,x,p,\mu{\right)}\mapsto -H^{{{\overline p}},\ll}_p{\left(}x,p,\mu{\right)}$. For $R>0$, the set $A_R$, defined by $A_R=[0,1]\times{{\mathbb{T}}}^d_a\times B_{{ \mathbb{R}}^d}{\left(}0,R{\right)}\times{{\mathcal P}}_{\infty,R}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$, is compact. By Heine theorem, the map ${\left(}\ll,x,p,\mu{\right)}\mapsto -H^{{{\overline p}},\ll}_p{\left(}x,p,\mu{\right)}$ is uniformly continuous on $A_R$. Here, note that we use the fact that ${{\mathcal P}}_{\infty,R}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$ is a metric space since the weak\* topology coincides with the topology induced by the $1$-Wassertein distance on ${{\mathcal P}}_{\infty,R}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$. Heine theorem also implies that ${{\overline p}}$ is uniformly continuous. Therefore, $\Psi$ is a compact mapping from $[0,1]\times C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$ to $C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$, i.e. it maps bounded subsets of $[0,1]\times C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$ into relatively compact subsets of $C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$: this comes from the latter observation and Arzelà-Ascoli theorem.
Take $\aa$ a fixed point of $\Psi(\ll,\cdot)$, for $\ll\in[0,1]$, Lemma \[lem:apriorimu\] implies that ${{\left\| \aa \right\|_{\infty}^{}}}$ is bounded by a constant $C$ which does not depend on $\ll$.
Moreover, it is straightforward to check that $\Psi(0,\cdot)=0$. Leray-Schauder Theorem \[thm:Leray\] implies that there exists a fixed point of the map $\aa\mapsto\Psi{\left(}1,\aa{\right)}$, which concludes the existence part of the proof.
The proof of uniqueness relies on \[hypo:LMono\] and the strict convexity of $L$, see [@MR3805247] Lemma $5.2$ for the detailed proof.
The continuity of the fixed point in time {#subsec:FPmugen}
-----------------------------------------
The fixed point result stated in Lemma \[lem:FPmuLS\] yields the existence of a map $(t,p,m)\mapsto \mu$. The continuity of this map is addressed in the following lemma:
\[lem:contmu\] Assume \[hypo:Lreg\]-\[hypo:Lbound\]. Let ${\left(}t^n,m^n,p^n{\right)}_{n\in{ \mathbb{N}}}$ be a sequence in $[0,T]\times
{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a{\right)}\times
C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$. Assume that
- $t^n\to_{n\to\infty}t\in[0,T]$,
- ${\left(}p^n{\right)}_{n\in{ \mathbb{N}}}$ is uniformly convergent to $p\in C^0{\left(}{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$,
- ${\left(}m^n{\right)}_{n\in{ \mathbb{N}}}$ tends to $m$ in the weak\* topology.
We define $\mu^n$ and $\mu$ as the unique solutions of the fixed point relation of Lemma \[lem:FPmuLS\] respectively associated to ${\left(}t^n,m^n,p^n{\right)}$ and ${\left(}t,m,p{\right)}$, for $n\in{ \mathbb{N}}$. Then the sequence ${\left(}\mu^n{\right)}_{n\in{ \mathbb{N}}}$ tends to $\mu$ in ${{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$ equipped with the weak\* topology.
The sequence ${\left(}p^{n}{\right)}_{n\in{ \mathbb{N}}}$ it is uniformly bounded in the norm ${{\left\| \cdot \right\|_{\infty}^{}}}$. Therefore ${\left(}\mu^n{\right)}_{n\in{ \mathbb{N}}}$ is uniformly compactly supported by Lemma \[lem:apriorimu\]. Thus we can extract a subsequence ${\left(}\mu^{\vp(n)}{\right)}_{n\in{ \mathbb{N}}}$ convergent to some limit ${{\widetilde \mu}}\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ in the weak\* toplogy on measures.
We recall that $\aa^{\mu}$ is defined in paragraph \[subsec:notaMono\]. Here, since $\mu^{\vp(n)}$ and $\mu$ are fixed points like in Lemma \[lem:FPmuLS\], they satisfy: $$\begin{aligned}
\aa^{\mu^{\vp(n)}}(x)
&=
-H_p{\left(}t^{\vp(n)},x,p^{\vp(n)}(x),\mu^{\vp(n)}{\right)},
\\
\aa^{\mu}(x)
&=
-H_p{\left(}t,x,p(x),\mu{\right)},
\end{aligned}$$ for $x\in{{\mathbb{T}}}^d_a$ and $n\in{ \mathbb{N}}$. From the continuity of $H_p$ stated in Lemma \[lem:Hbound\], ${\left(}\aa^{\vp(n)}{\right)}_{n\in{ \mathbb{N}}}$ tends uniformly to the function ${{\widetilde \aa}}:x\mapsto-H_p{\left(}t,x,p,{{\widetilde \mu}}{\right)}$. Then ${\left(}{\left(}I_d,\aa^{\vp(n)}{\right)}\#m^n{\right)}_{n\in{ \mathbb{N}}}$ tends to ${\left(}I_d,{{\widetilde \aa}}{\right)}\#m$ in the weak\* topology. Hence ${{\widetilde \mu}}$ satisfies the same fixed point relation as $\mu$; by uniqueness we deduce that ${{\widetilde \mu}}=\mu$. This implies that all the convergent subsequences of ${\left(}\mu^{n}{\right)}_{n\in{ \mathbb{N}}}$ have the same limit $\mu$, thus the whole sequence converges to $\mu$.
Lemma \[lem:FPmuLS\] states that for all time the fixed point has a unique solution. Then Lemma \[lem:contmu\] yields the continuity of the map defined by the fixed point under suitable assumptions. Therefore, the conclusion of step \[step:FPmucont\] and the Lemma \[lem:FPmucont\] are straightforward consequences of these two lemmas.
All the conclusions of this section hold when we relax Assumption \[hypo:LMono\], assuming that the inequality holds only when $\mu^1$ and $\mu^2$ have the same first marginal. Some applications of MFGC do not satisfy \[hypo:LMono\], but satisfy the above-mentioned relaxed monotonicity assumption. This is the case of the MFG version of the Almgren and Chriss’ model for price impact and high-frenquency trading, discussed in [@MR3805247; @MR3752669; @MR3325272; @2019arXiv190411292K].
However, the a priori estimates in the next section do not hold under this relaxed monotonicity assumption. We refer to [@2019arXiv190411292K] for estimates which do not rely on \[hypo:LMono\] (Assumptions [**FP1**]{} and [**FP2**]{} in [@2019arXiv190411292K] are unnecessary if $L$ satisfies the relaxed monotonicity assumption).
A priori estimates for the solutions to {#sec:aprioriMono}
========================================
In order to use the Leray-Schauder fixed point theorem later, we introduce the following family of Lagrangians indexed by $\th\in(0,1]$, $$\label{eq:Lth}
L^{\th}{\left(}t,x,\aa,\mu{\right)}=
\th
L{\left(}t,x,\th^{-1}\aa,\Th(\mu){\right)},$$ where the map $\Th:{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}\rightarrow{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$ is defined by $\Th(\mu)={\left(}I_d\otimes\th^{-1}I_d{\right)}\#\mu$. Then the Hamiltonian defined as the Legendre transform of $L^{\th}$ is given by $$\label{eq:Hth}
H^{\th}{\left(}t,x,p,\mu{\right)}=
\th
H{\left(}t,x,p,\Th(\mu){\right)}.$$ The definition of the Hamiltonian can naturally be extended to $\th=0$ by $H^0=0$, the associated Lagrangian is $L^0=0$ if $\aa=0$ and $L^0=\infty$ otherwise. We introduce the following system of MFGC,
\[eq:MFGCth\]
[align]{} \[eq:HJBth\] &-[[\_[t]{}]{}]{}u(t,x) - u(t,x) + H\^[(]{}t,x, \_xu(t,x),(t)[)]{} = f(t,x,m(t)) & (0,T)\^d\_a,\
\[eq:FPKth\] & [[\_[t]{}]{}]{}m(t,x) - m(t,x) -[(]{}H\^\_p[(]{}t,x,\_xu(t,x),(t)[)]{}m[)]{} = 0 & (0,T)\^d\_a,\
\[eq:muth\] &(t) = ( I\_d, - H\^\_p[(]{}t,,\_xu(t,),(t)[)]{} )[\#]{}m(t) & \[0,T\],\
\[eq:CFuth\] &u(T,x) = g(x,m(T)) & \^d\_a,\
\[eq:CImth\] &m(0,x) = m\_0(x) & \^d\_a.
When $\th=1$, the latter system coincides with . When $\th=0$, consists in a situation in which the state of a representative agent satisfies a non-controlled stochastic differential equation. Alternatively it can be interpreted as a game in which the agents pay an infinite price as soon as they try to use a control different than $0$. In particular the case $\th=0$ is specific and easier than the case when $\th>0$. Therefore, in the rest of this section, we only consider $\th\in(0,1]$.
Let us mention that assumptions \[hypo:Lreg\]-\[hypo:LMono\] are preserved when replacing $L$ and $H$ by $L^{\th}$ and $H^{\th}$ respectively. Moreover the inequalities from \[hypo:Lcoer\], \[hypo:Lbound\], become respectively $$\begin{aligned}
\label{eq:Lthcoer}
L^{\th}(t,x,\aa,\mu)
&\geq
C_0^{-1}\th^{1-q'}|\aa|^{q'}
-C_0\th
-C_0\th^{1-q'}\LL_{q'}{\left(}\mu{\right)}^{q'},
\\
\label{eq:Lthbound}
{\left|}L^{\th}(t,x,\aa,\mu){\right|}&\leq
C_0\th
+C_0\th^{1-q'}{\left(}|\aa|^{q'}
+\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)},\end{aligned}$$ since $\LL_{q'}{\left(}\Th(\mu){\right)}=\th^{-1}\LL_{q'}{\left(}\mu{\right)}$. Furthermore, the conclusions of Lemma \[lem:Hbound\] hold and the inequalities become respectively $$\begin{aligned}
\label{eq:Hthpbound}
{\left|}H^{\th}_p{\left(}t,x,p,\mu{\right)}{\right|}&\leq
C_0\th{\left(}1+|p|^{q-1}{\right)}+C_0\LL_{q'}{\left(}\mu{\right)},
\\
\label{eq:Hthbound}
{\left|}H^{\th}{\left(}t,x,p,\mu{\right)}{\right|}&\leq
C_0\th{\left(}1+|p|^{q}{\right)}+C_0\th^{1-q'}\LL_{q'}{\left(}\mu{\right)}^{q'},
\\
\label{eq:Hthcoer}
p\cdot H^{\th}_p{\left(}t,x,p,\mu{\right)}-H^{\th}{\left(}t,x,p,\mu{\right)}&\geq
C_0^{-1}\th|p|^q
-C_0\th
-C_0\th^{1-q'}\LL_{q'}{\left(}\mu{\right)}^{q'},
\\
\label{eq:Hthxbound}
{\left|}H^{\th}_x{\left(}t,x,p,\mu{\right)}{\right|}&\leq
C_0\th{\left(}1+{\left|}p{\right|}^{q}{\right)}+C_0\th^{1-q'}\LL_{q'}{\left(}\mu{\right)}^{q'}.\end{aligned}$$ We recall that without loss of generality, we assumed ${{\widetilde C}}_0=C_0$ where ${{\widetilde C}}_0$ is defined in Lemma \[lem:Hbound\].
Instead of proving Lemma \[lem:Monoapriorith1\] and step \[step:ttdaapriori\], we address the more general following lemma which provides a priori estimates not only for solutions to but also for solutions to . This will help to use the Leray-Schauder theorem in the next section.
\[lem:Monoapriori\] Under assumptions \[hypo:Lreg\]-\[hypo:Monogregx\], there exists a positive constant $C$ which only depends on the constants in the assumptions and not on $a$ or $\th$, such that the solution to satisfies: ${{\left\| u \right\|_{\infty}^{}}}
\leq
C\th$, ${{\left\| \nabla_xu \right\|_{\infty}^{}}}
\leq
C\th^{\frac12}$ and $\displaystyle{\sup_{t\in[0,T]}} \LL_{\infty}{\left(}\mu(t){\right)}\leq
C\th$.
*First step: controlling $\int_0^T\LL_{q'}{\left(}\mu(t){\right)}^{q'} dt$*
Let us take $(X,\aa)$ defined by $${\left\{}
\newcommand{\rc}{\right\}}\begin{aligned}
\aa_t
&=
\aa^{\mu(t)}(t,X_t)
=
-H_p^{\th}{\left(}t, X_t,\nabla_xu(t,X_t),\mu(t){\right)},
\\
dX_t
&=
\aa_tdt
+\sqrt{2\nu}dB_t,
\\
X_0
&=
\xi\sim m_0,
\end{aligned}
\right.$$ where ${\left(}B_t{\right)}_{t\in[0,T]}$ is a Brownian motion independent of $\xi$.
The function $u$ is the value function of an optimization problem, i.e. the lowest cost that a representative agent can achieve from time $t$ to $T$ if $X_t=x$, when the probability measures $m$ and $\mu$ are fixed, i.e. $$\label{eq:HJBopt}
\aa_{|s\in[t,T]}
=
\operatorname{\mathop{\rm argmin}}_{\aa'}
{{\mathbb E}}{\left[}\int_t^T
L^{\th}{\left(}s, X^{\aa'}_s,\a'_s, \mu(s) {\right)}+\th f{\left(}s,X^{\aa'}_s,m(s){\right)}ds
+\th g{\left(}X^{\aa'}_T,m(T){\right)}{\right]},$$ where for a control $\aa'$, we define $${\left\{}
\newcommand{\rc}{\right\}}\begin{aligned}
dX^{\aa'}_t
&=
\aa'_tdt
+\sqrt{2\nu}dB'_t, \\
X^{\aa'}_0
&=
\xi'\sim m_0,
\end{aligned}
\right.$$ and ${\left(}B'_t{\right)}_{t\in[0,T]}$ is a Brownian motion independent of $\xi'$. Let us recall that for any $t\in[0,T]$, $m(t)$ is the law of $X_t$, and $\mu(t)$ is the law of ${\left(}X_t,\aa_t{\right)}$. We introduce ${{\widetilde X}}$ the stochastic process defined by $${\left\{}
\newcommand{\rc}{\right\}}\begin{aligned}
d{{\widetilde X}}_t
&=
\sqrt{2\nu}dB_t, \\
{{\widetilde X}}_0
&=
\xi\sim m_0.
\end{aligned}
\right.$$ We set ${{\widetilde m}}(t)={{\mathcal L}}({{\widetilde X}}_t)$ and ${{\widetilde \mu}}(t)={{\mathcal L}}({{\widetilde X}}_t)\otimes\dd_0$ for $t\in[0,T]$. For the strategy consisting in taking $\alpha'=0$, yields the inequality: $$\begin{gathered}
\int_0^T
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
L^{\th}{\left(}t,x,\aa, \mu(t) {\right)}d\mu(t,x,\aa) dt
+\int_0^T\int_{{{\mathbb{T}}}^d_a}\th f{\left(}t,x,m(t){\right)}dm(t,x)dt
+\int_{{{\mathbb{T}}}^d_a}\th g{\left(}x,m(T){\right)}dm(T,x)
\\
\leq
\int_0^T
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
L^{\th}{\left(}t,x,\aa, \mu(t) {\right)}d{{\widetilde \mu}}(t,x,\aa)dt
+\int_0^T\int_{{{\mathbb{T}}}^d_a}\th f{\left(}t,x,m(t){\right)}d{{\widetilde m}}(t,x)dt
+\int_{{{\mathbb{T}}}^d_a}\th g{\left(}x,m(T){\right)}d{{\widetilde m}}(T,x).
\end{gathered}$$ This and \[hypo:Monogregx\] imply that, $$\label{eq:apriori_mono1}
\int_0^T
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
L^{\th}{\left(}t,x,\aa, \mu(t){\right)}d\mu(t,x,\aa) dt
\leq
\int_0^T
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
L^{\th}{\left(}t,x,\aa, \mu(t){\right)}d{{\widetilde \mu}}(t,x,\aa)dt
+2C_0\th{\left(}1+T{\right)}.$$ Assumption \[hypo:LMono\] with $(\mu(t),{{\widetilde \mu}}(t))$ yields $$\begin{gathered}
\label{eq:apriori_mono2}
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
L^{\th}{\left(}t,x,\aa, \mu(t) {\right)}d{{\widetilde \mu}}(t,x,\aa)+
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
L^{\th}{\left(}t,x,\aa, {{\widetilde \mu}}(t) {\right)}d\mu(t,x,\aa) \\
\leq
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
L^{\th}{\left(}t,x,\aa, \mu(t) {\right)}d\mu(t,x,\aa)+
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
L^{\th}{\left(}t,x,\aa, {{\widetilde \mu}}(t){\right)}d{{\widetilde \mu}}(t,x,\aa).
\end{gathered}$$ Moreover, from \[hypo:Lbound\] we obtain that, $$\label{eq:apriori_mono3}
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
L^{\th}{\left(}t,x,\a, {{\widetilde \mu}}(t) {\right)}{{\widetilde \mu}}(t,d(x,\a))
=
\int_{{{\mathbb{T}}}^d_a}
\th L{\left(}t,x,0, {{\widetilde \mu}}(t) {\right)}{{\widetilde m}}(t,dx)
\leq
C_0\th.$$ Therefore from , , and , we obtain, $$\begin{aligned}
\int_0^T
\int_{{{\mathbb{T}}}^d_a}
{\left(}C_0^{-1}
\th^{1-q'}{\left|}\aa{\right|}^{q'}
-C_0\th{\right)}d\mu(t,x,\aa)dt
&\leq
\int_0^T
\int_{{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d}
L^{\th}{\left(}t,x,\aa, {{\widetilde \mu}}(t) {\right)}d\mu(t,x,\aa)dt
\\
&\leq
C_0\th(2+3T).
\end{aligned}$$ This implies $$\label{eq:apriori_intmu}
\int_0^T
\LL_{q'}{\left(}\mu(t){\right)}^{q'} dt
\leq
2C_0^{2}
\th^{q'}
(1+2T).$$
*Second step: the uniform estimate on ${{\left\| u \right\|_{\infty}^{}}}$*
Let us rewrite in the following way, $$-{{\partial_{t}}}u
- \nu\Delta u
+ {\left[}\int_0^1 H^{\th}_p(t,x,s\nabla_x u,\mu(t))ds{\right]}\cdot \nabla_x u
=
H^{\th}(t,x, 0,\mu(t))
+\th f(t,x,m(t)),$$ for $(t,x)\in(0,T)\times{{\mathbb{T}}}^d_a$. The maximum principle for second-order parabolic equation, \[hypo:Monogregx\], and yield that $${{\left\| u \right\|_{\infty}^{}}}
\leq
C_0\th(1+2T)
+C_0\th^{1-q'}\int_0^T
\LL_{q'}{\left(}\mu(t){\right)}^{q'} dt,$$ which implies that $u$ is uniformly bounded using the conclusion of the previous step.
*Third step: the uniform estimate on ${{\left\| \nabla_xu \right\|_{\infty}^{}}}$.*
The proof of this step relies on the same Bernstein-like method introduced in [@2019arXiv190411292K] Lemma $6.5$. We refer to the proof of the latter results for more details in the derivation of the equations below.
Let us introduce $\rho\in C^{\infty}{\left(}{\left[}-\frac{a}2,\frac{a}2{\right)}^d {\right)}$ a nonnegative mollifier such that $\rho(x)=0$ if $|x|\geq \frac{a}4$ and $\int_{{\left[}-\frac{a}2,\frac{a}2{\right)}^d}\rho(x)dx=1$. For any $0<\dd<1$ and $t\in [0,T]$, we introduce $\rho^{\dd}=\dd^{-d}\rho{\left(}\frac{\cdot}{\dd}{\right)}$ and $u^{\dd}(t)=\rho^{\dd}\star u(t)$ with $\star$ being the convolution operator with respect to the state variable.
Possibly after modifying the constant $C$ appearing in the first step, we can assume that ${{\left\| u \right\|_{\infty}^{}}}+{\left(}1+C_0{\right)}\th^{1-q'}\int_{0}^T\LL_{q'}{\left(}\mu(s){\right)}^{q'}ds\leq C$ using the first two steps in such a way that $C$ depends only on the constants in the assumptions, and not on $\th$. Then we introduce $\vp:[-C,C]\rightarrow{ \mathbb{R}}_+^*$ and $w^{\dd}$ defined by $$\label{eq:defphiw}
\begin{aligned}
&\vp(v)
=
\exp{\left(}\exp{\left(}- v{\right)}{\right)},
\\
&w^{\dd}(t,x)
=
\vp{\left(}u^{\dd}(T-t,x)
+{\left(}1+C_0{\right)}\th^{1-q'}\int_{T-t}^T\LL_{q'}{\left(}\mu(s){\right)}^{q'}ds{\right)}{\left|}\nabla_x u^{\dd}{\right|}^2(T-t,x),
\end{aligned}$$ for $(t,x)\in[0,T]\times{{\mathbb{T}}}^d_a$, $v\in B_{{ \mathbb{R}}^d}{\left(}0, C{\right)}$. In particular $\vp'<0$, and $\vp$, $1/\vp$, $-\vp'$ and $-1/\vp'$ are uniformly bounded. We refer to the proof of Lemma $6.5$ in [@2019arXiv190411292K] for the derivation of the following partial differential equation satisfied by $w^{\dd}$, $$\begin{gathered}
\label{eq:PDEw}
{{\partial_{t}}}w^{\dd} -\nu\Delta w^{\dd}
+ \nabla_xw^{\dd}\cdot H^{\th}_p{\left(}x,\nabla_x u^{\dd}, \mu{\right)}-2\nu\frac{\vp'}{\vp} \nabla_xw^{\dd}\cdot \nabla_xu^{\dd}
+2\nu\vp{\left|}D^2_{x,x}u^{\dd}{\right|}^2
\\
=
\frac{\vp'}{\vp}w^{\dd}{\left[}\nabla_xu^{\dd}\cdot H^{\th}_p{\left(}x,\nabla_xu^{\dd},\mu{\right)}-H^{\th}{\left(}x,\nabla_xu^{\dd},\mu{\right)}+{\left(}1+C_0{\right)}\th^{1-q'} \LL_{q'}{\left(}\mu{\right)}^{q'}{\right]}\\
-\nu\frac{\vp''\vp-2{\left(}\vp'{\right)}^2}{\vp^3}{\left(}w^{\dd}{\right)}^2
-2\vp\nabla_xu^{\dd}\cdot H^{\th}_x{\left(}x,\nabla_xu^{\dd},\mu{\right)}+2\th\vp\nabla_xu^{\dd}\cdot f_x^{\dd}{\left(}x,m{\right)}+R^{\dd}(t,x)
\end{gathered}$$ in which $H^{\th}$, $f$, $f^{\dd}$, $u$, $u^{\dd}$ and $\mu$ are taken at time $T-t$ and $w^{\dd}$ at time $t$, and where $f^{\dd}$ and $R^{\dd}$, are defined by, $$\begin{aligned}
f^{\dd}(x,m)
=&
\rho^{\dd}\star
{\left(}f(\cdot, m){\right)}(x),
\\
R^{\dd}(t,x)
=&
-\vp'{\left|}\nabla_xu^{\dd}{\right|}^2
{\left[}\rho^{\dd}\star{\left(}H^{\th}{\left(}\cdot , \nabla_x u, \mu{\right)}{\right)}(x)
-H^{\th}{\left(}x,\nabla_xu^{\dd} ,\mu{\right)}{\right]}\\
&-2\vp\nabla_xu^{\dd}\cdot{\left[}{\left(}\rho^{\dd}\star
H^{\th}_x{\left(}\cdot, \nabla_xu(\cdot),\mu{\right)}{\right)}(x)
-H^{\th}_x{\left(}x,\nabla_xu^{\dd},\mu{\right)}{\right]},
\\
&+2\vp\nabla_xu^{\dd}\cdot{\left[}D_{x,x}^2u^{\dd}H^{\th}_p{\left(}x,\nabla_x u^{\dd},\mu{\right)}-\rho^{\dd}\star {\left(}D^2_{x,x}uH^{\th}_p{\left(}\cdot,\nabla_x u,\mu{\right)}{\right)}{\right]}.
\end{aligned}$$ From , we obtain that $$\nabla_xu^{\dd}\cdot H^{\th}_p{\left(}x,\nabla_xu^{\dd},\mu{\right)}-H^{\th}{\left(}x,\nabla_xu^{\dd},\mu{\right)}+{\left(}1+C_0{\right)}\th^{1-q'} \LL_{q'}{\left(}\mu{\right)}^{q'}
\geq
C_0^{-1}\th{\left|}\nabla_xu^{\dd}{\right|}^q
+\th^{1-q'}\LL_{q'}{\left(}\mu{\right)}^{q'}
-C_0\th.$$ Therefore, using \[hypo:Monogregx\], , , the facts that $\vp'<0$, that $\vp''\vp-2{\left(}\vp'{\right)}^2\geq 0$, that $\vp$, $\vp^{-1}$, $\vp'$, ${\left(}\vp'{\right)}^{-1}$ are bounded, and the latter inequality, we get $$\begin{gathered}
\label{eq:PDIw}
{{\partial_{t}}}w^{\dd} -\nu\Delta w^{\dd}
+ \nabla_xw^{\dd}\cdot H^{\th}_p{\left(}x,\nabla_x u^{\dd}, \mu{\right)}-2\nu\frac{\vp'}{\vp} \nabla_xw^{\dd}\cdot \nabla_xu^{\dd}
\\
\leq
-C^{-1}{\left(}\th{\left(}w^{\dd}{\right)}^{\frac{q}2}
+\th^{1-q'}\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)}w^{\dd}
\\
+C{\left(}w^{\dd}{\right)}^{\frac12}
{\left[}\th+\th{\left(}w^{\dd}{\right)}^{\frac12}
+\th{\left(}w^{\dd}{\right)}^{\frac{q}2}
+\th^{1-q'}\LL_{q'}{\left(}\mu{\right)}^{q'}{\right]}+{{\left\| R^{\dd} \right\|_{\infty}^{}}},
\end{gathered}$$ up to updating $C$. We notice that the terms with the highest exponents in $w^{\dd}$ and $\LL_{q'}{\left(}\mu{\right)}^{q'}$ in the right-hand side of the latter inequality is non-positive. Let us use Young inequalities and obtain $$\begin{aligned}
&{\left(}w^{\dd}{\right)}^{\frac12}
\LL_{q'}{\left(}\mu{\right)}^{q'}
\leq
\ee w^{\dd}
\LL_{q'}{\left(}\mu{\right)}^{q'}
+
\frac1{4\ee}
\LL_{q'}{\left(}\mu{\right)}^{q'},
\\
&{\left(}w^{\dd}{\right)}^{{{\widetilde q}}}
\leq
\ee{\left(}w^{\dd}{\right)}^{1+\frac{q}2}
+\frac{q+2-2{{\widetilde q}}}{q+2}
{\left(}\frac{\ee (q+2)}{2{{\widetilde q}}}{\right)}^{-\frac2{q+2-2{{\widetilde q}}}},
\end{aligned}$$ for any ${{\widetilde q}}<1+\frac{q}2$ and $\ee>0$. Using systematically these two inequalities in and taking $\ee$ small enough we finally obtain, $${{\partial_{t}}}w^{\dd} -\nu\Delta w^{\dd}
+ \nabla_xw^{\dd}\cdot H^{\th}_p{\left(}x,\nabla_x u^{\dd}, \mu{\right)}-2\nu\frac{\vp'}{\vp} \nabla_xw^{\dd}\cdot \nabla_xu^{\dd}
\\
\leq
C_{\ee}{\left(}\th+
\th^{1-q'}\LL_{q'}{\left(}\mu{\right)}^{q'}{\right)}+{{\left\| R^{\dd} \right\|_{\infty}^{}}},$$ where $C_{\ee}$ is a constant which depends on $\ee$ and the constants in the assumtions. From \[hypo:Monogregx\], the initial condition of $w^{\dd}$ is bounded. Therefore the maximum principle for second-order parabolic equations implies that $$\label{eq:ineqwdd}
{{\left\| w^{\dd} \right\|_{\infty}^{}}}
\leq
C_{\ee}{\left(}\th+ \th T+
\th^{1-q'}\int_0^T\LL_{q'}{\left(}\mu(t){\right)}^{q'}dt{\right)}+T{{\left\| R^{\dd} \right\|_{\infty}^{}}}.$$ Let us point out that $\nabla_xu$ is the solution of the following backward $d$-dimensional parabolic equation, $$-{{\partial_{t}}}\nabla_xu
-\nu\Delta\nabla_xu
+D^2_{x,x}u H_p{\left(}x,\nabla_xu,\mu{\right)}=
\nabla_xf(x,m)
-H_x{\left(}x,\nabla_xu,\mu{\right)},$$ which has bounded coefficients and right-hand side, and a terminal condition in $C^{1+\bb_0}{\left(}{{\mathbb{T}}}^d_a{\right)}$. Theorem $6.48$ in [@MR1465184] states that $\nabla_xu$ and $D^2_{x,x}u$ are continuous. This and the continuity of $H^{\th}$ and $H^{\th}_x$ stated in Lemma \[lem:Hbound\] imply that $R^{\dd}$ is uniformly convergent to $0$ when $\dd$ tends to $0$. We conclude this step of the proof by passing to the limit in as $\dd$ tends to $0$, using the estimate on $\int_0^T\LL_{q'}{\left(}\mu(t){\right)}^{q'}dt$ computed in the first step. We obtain that $\nabla_xu$ is uniformly bounded by a constant which depends on the constants in the assumptions, and depends linearly on $\th^{\frac12}$.
*Fourth step: obtaining uniform estimates on $\LL_{q'}{\left(}\mu{\right)}$ and $\LL_{\infty}{\left(}\mu{\right)}$.*
Repeating the calculation in the proof of Lemma \[lem:apriorimu\] with $L$ satisfying and , we obtain: $$\label{eq:apriorimuLq'}
\LL_{q'}{\left(}\mu(t){\right)}^{q'}
\leq
4C_0^2\th^{q'}
+\frac{{\left(}q'{\right)}^{q-1}{\left(}2C_0{\right)}^q}{q}
\th^{q'}{{\left\| \nabla_xu(t) \right\|_{L^q(m(t))}^{q}}}.$$ This and the third step of this proof yield that $\sup_{t\in[0,T]}\LL_{q'}{\left(}\mu(t){\right)}\leq C\th$ for some $C$ depending only on the constants of the assumptions. We conclude that $\sup_{t\in[0,T]}\LL_{\infty}{\left(}\mu(t){\right)}$ satisfies a similar inequality using .
Existence and Uniqueness Results {#sec:MainMono}
================================
Paragraph \[subsec:Exittda\] is devoted to proving the existence of solutions to , which is step \[step:ttdaexi\]. In paragraph \[subsec:rd\], we propose a method to extend the existence result to system which is stated on ${ \mathbb{R}}^d$; this concludes step \[step:MFGCrdexi\]. This method relies on compactness results using the uniform estimates of $\nabla_xu$ that we obtained in Lemma \[lem:Monoapriori\]. In paragraph \[subsec:UniMono\], we prove step \[step:MFGCrduni\], namely the uniqueness of the solution to and . Then the main results of the paper and step \[step:MFGCb\] are addressed in paragraph \[subsec:Exib\]. We introduce a one-to-one correspondance between solutions to and , which allows us to obtain directly the existence and the uniqueness of the solution to from the ones to .
Proof of Theorem \[thm:ExiMonottda\]: existence of solutions to {#subsec:Exittda}
----------------------------------------------------------------
We will use the a priori estimates stated in Section \[sec:aprioriMono\] and the latter fixed point theorem, in order to achieve step \[step:ttdaexi\] and prove the existence of solutions to .
We would like to use the Leray-Schauder theorem \[thm:Leray\] on a map which takes a flow of measures ${\left(}{{\widetilde m}}_t{\right)}_{t\in[0,T]}\in{\left(}{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a{\right)}{\right)}^{[0,T]}$ as an argument. However, ${{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a{\right)}$ is not a Banach space. A way to go through this difficulty is to compose the latter map with a continuous map from a convenient Banach space to the set of such flows of measures. Here, we consider the map introduced in [@2019arXiv190205461F], namely $\rho:C^0{\left(}[0,T]\times{{\mathbb{T}}}^d_a;{ \mathbb{R}}{\right)}\to C^0{\left(}[0,T]\times{{\mathbb{T}}}^d_a;{ \mathbb{R}}{\right)}$ defined by $$\rho({{\widetilde m}})(t,x)
=
\frac{{{\widetilde m}}_+(t,x)- a^{-d}\int{{\widetilde m}}_+(t,y)dy}{\max{\left(}1,\int {{\widetilde m}}_+{\left(}t,y{\right)}dy{\right)}}
+a^{-d},
$$ where ${{\widetilde m}}_+(t,x)=\max{\left(}0,{{\widetilde m}}(t,x){\right)}$. We will also have the use of ${{\widetilde m}}^0$ defined as the unique weak solution of $$\label{eq:defmt0}
{{\partial_{t}}}{{\widetilde m}}^0
-\nu\Delta {{\widetilde m}}^0
=
0
\text{ on }
(0,T)\times{{\mathbb{T}}}^d_a,
\quad
\text{ and }
{{\widetilde m}}^0(0,\cdot)=m^0.$$ We are now ready to construct the map $\Psi$ on which we will use the Leray-Schauder theorem \[thm:Leray\]. Take $\th\in[0,1]$, $u\in C^{0,1}{\left(}[0,T]\times{{\mathbb{T}}}^d_a;{ \mathbb{R}}{\right)}$ and ${{\widetilde m}}\in C^{0}{\left(}[0,T]\times{{\mathbb{T}}}^d_a;{ \mathbb{R}}{\right)}$. We define $m=\rho{\left(}{{\widetilde m}}+{{\widetilde m}}^0{\right)}$ and ${\left(}\mu,\aa{\right)}\in C^0{\left(}[0,T];{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}{\right)}\times C^0{\left(}[0,T]\times{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$ by, $$\begin{aligned}
\aa(t,x)
&=
-H^{\th}_p{\left(}t,x,\nabla_xu(t,x),\mu(t){\right)}\\
\mu(t)
&=
{\left(}I_d,\aa(t,\cdot){\right)}\# m(t).
\end{aligned}$$ This definition comes from the conclusions of Lemma \[lem:FPmucont\] when $\th>0$. For $\th=0$, it simply consists in taking $\aa=0$ and $\mu(t)=m(t)\otimes\dd_0$. Here we can repeat the calculation and obtain inequality . This and implies that ${{\left\| \aa \right\|_{\infty}^{}}}$ is bounded by $C\th$ for some constant $C>0$ which depends on ${{\left\| \nabla_xu \right\|_{\infty}^{}}}$ and is independent of $\th$ and $a$.
Then we define ${{\overline m}}$ the solution in the sense of distributions of $${{\partial_{t}}}{{\overline m}}-\nu\Delta {{\overline m}}+\operatorname{div}{\left(}\aa{{\overline m}}{\right)}=
0,$$ supplemented with the initial condition ${{\overline m}}(0,\cdot)=m^0$, with $m^0$ being $\bb_0$-Hölder continuous. Theorem $2.1$ section $V.2$ in [@MR0241822] states that ${{\overline m}}$ is uniformly bounded by a constant which depends on ${{\left\| m_0 \right\|_{\infty}^{}}}$ and ${{\left\| \aa \right\|_{\infty}^{}}}$. Theorem $6.29$ in [@MR1465184] yields that $m\in C^{\frac{\bb}2,\bb}{\left(}[0,T]\times{{\mathbb{T}}}^d_a{\right)}$ for $\bb\in(0,\bb_0)$, and that its associated norm can be estimated from above by a constant which depends on ${{\left\| \nabla_xu \right\|_{\infty}^{}}}$, $\bb$, $a$ and the constants in the assumptions. The same arguments applied to ${{\widetilde m}}^0$ defined in imply that ${{\widetilde m}}^0$ is in $C^{\frac{\bb}2,\bb}{\left(}[0,T]\times{{\mathbb{T}}}^d_a{\right)}$ and its associated norm is bounded.
Then we take ${{\overline{\mu}}}(t)={\left(}I_d,\aa(t,\cdot){\right)}\#{{\overline m}}(t)$ for any $t\in[0,T]$, and ${{\overline u}}\in C^{0,1}{\left(}[0,T]\times{{\mathbb{T}}}^d_a;{ \mathbb{R}}{\right)}$ the unique solution in the sense of distributions of the following heat equation with bounded right-hand side, $$-{{\partial_{t}}}{{\overline u}}-\nu\Delta {{\overline u}}=
-H^{\th}{\left(}t,x,\nabla_xu,{{\overline{\mu}}}(t){\right)}+\th f(x,{{\overline m}}(t)),$$ supplemented with the terminal condition ${{\overline u}}(T,\cdot)=\th g{\left(}\cdot,{{\overline m}}(T){\right)}$ which is in $C^{1+\bb_0}{\left(}{{\mathbb{T}}}^d_a{\right)}$. Classical results (see for example Theorem $6.48$ in [@MR1465184]) state that $u$ is in $C^{\frac12+\frac{\bb}2,1+\bb}$ and its associated norm is bounded by a constant which depends on ${{\left\| \nabla_xu \right\|_{\infty}^{}}}$, $\bb$, $a$ and the constants in the assumptions.
We can now construct the map $\Psi:{\left(}\th, u,{{\widetilde m}}{\right)}\mapsto{\left(}{{\overline u}},{{\overline m}}-{{\widetilde m}}^0{\right)}$, from $C^{0,1}{\left(}[0,T]\times{{\mathbb{T}}}^d_a;{ \mathbb{R}}{\right)}\times C^{0}{\left(}[0,T]\times{{\mathbb{T}}}^d_a;{ \mathbb{R}}^d{\right)}$ into itself. This map is continuous and compact, it satisfies $\Psi{\left(}0,u,{{\widetilde m}}{\right)}=0$ for any ${\left(}u,{{\widetilde m}}{\right)}$. In particular, the fact that ${{\left\| \aa \right\|_{\infty}^{}}}\leq C\th$ in the previous paragraph, implies that ${{\overline m}}$ tends to ${{\widetilde m}}^0$ and ${{\overline u}}$ tends to $0$ as $\th$ tends to $0$. This gives the continuity of $\Psi$ at $\th=0$. Moreover the fixed points of $\Psi(\th)$ are exactly the solutions to , which are uniformly bounded by Lemma . Therefore, by the Leray-Schauder fixed point theorem \[thm:Leray\], there exists a solution to .
Proof of Theorem \[thm:Exird\]: passing from the torus to ${ \mathbb{R}}^d$ {#subsec:rd}
---------------------------------------------------------------------------
The purpose of this paragraph is to extend the existence result to the system and achieve step \[step:MFGCrdexi\].
*First step: constructing a sequence of approximate solutions.*
For $a>0$ we define ${{\widetilde m}}^{0,a}=\pi^a\#m^0$, where $\pi^a:{ \mathbb{R}}^d\to{{\mathbb{T}}}_a^d$ is the quotient map. Let $\chi^a:{{\mathbb{T}}}^1_a\to{ \mathbb{R}}$ be the canonical injection from the one-dimensional torus of radius $a$ to ${ \mathbb{R}}$, which image is ${\left[}-\frac{a}2,\frac{a}2{\right)}$. Take ${{\widetilde{\psi}}}\in C^{2}{\left(}{ \mathbb{R}};{ \mathbb{R}}{\right)}$ periodic with a period equal to $1$ and such that, $$\label{eq:defpsi}
\begin{aligned}
{{\widetilde{\psi}}}(x)
&=
x,
\quad
\text{ if }
|x|\leq\frac{1}4,
\\
{\left|}{{\widetilde{\psi}}}(x){\right|}&\leq
{\left|}x{\right|},
\quad
\text{ for any }
x\in{\left[}-\frac12,\frac12{\right]},
\end{aligned}$$ We define $\psi^a:{{\mathbb{T}}}^d_a\to{ \mathbb{R}}^d$ by $\psi^a(x)_i
=a{{\widetilde{\psi}}}{\left(}a^{-1}\chi^a(x_i){\right)}$ for $i=1,\dots,d$, this is a $C^2$ function. Since ${{\widetilde{\psi}}}{\left(}\frac{\cdot}a{\right)}$ has a period of $a$, the function $\psi^a\circ\pi^a:{ \mathbb{R}}^d\to{ \mathbb{R}}^d$ satisfies $$\label{eq:psipi}
\psi^a\circ\pi^a(x)_i=a{{\widetilde{\psi}}}{\left(}\frac{x_i}a{\right)},$$ for $i=1,\dots,d$ and $x\in{ \mathbb{R}}^d$, and is a $C^2$ function.
We are ready to construct periodic approximations of $L$, $f$ and $g$ defined by, $$\begin{aligned}
L^a{\left(}t,x,\aa,\mu{\right)}&=
L{\left(}t,\psi^a(x),\aa,{\left(}\psi^a\otimes I_d{\right)}\# \mu{\right)},
\\
f^a{\left(}t,x,m{\right)}&=
f{\left(}t,\psi^a(x),\psi^a\# m{\right)},
\\
g^a{\left(}x,m{\right)}&=
g{\left(}\psi^a(x),\psi^a\# m{\right)},\end{aligned}$$ for $(t,x)\in[0,T]\times{{\mathbb{T}}}^d_a$, $\aa\in{ \mathbb{R}}^d$, $\mu\in{{\mathcal P}}{\left(}{{\mathbb{T}}}^d_a\times{ \mathbb{R}}^d{\right)}$. Let $H^a$ be the periodic Hamiltonian associated with $L^a$ by the Legendre transform: $$H^a{\left(}t,x,p,\mu{\right)}=
H{\left(}t,\psi^a(x),p,{\left(}\psi^a\otimes I_d{\right)}\# \mu{\right)}.$$ Let us point out that the fact that $L$, $H$, $f$ and $g$ satisfy \[hypo:Lreg\]-\[hypo:Monogregx\], implies that $L^{a}$, $H^a$, $f^a$ and $g^a$ satisfy these assumptions too with $C_0{{\left\| {{\widetilde{\psi}}}' \right\|_{\infty}^{}}}$ instead of $C_0$.
So we can define ${\left(}{{\widetilde u}}^a,{{\widetilde m}}^a,{{\widetilde \mu}}^a{\right)}$ a solution to with $H^a$, $f^a$, $g^a$ and ${{\widetilde m}}^{0,a}$ instead of $H$, $f$, $g$ and $m^0$. We define $u^a\in C^0{\left(}[0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}{\right)}$, $m^a\in C^0{\left(}[0,T];{{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)}{\right)}$ and $\mu^a\in C^0{\left(}[0,T];{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}{\right)}$ respectively by $$\begin{aligned}
u^a(t,x)
=
{{\widetilde u}}^a{\left(}t,\pi^a(x){\right)},
\quad
m^a(t)
=
\psi^a\#{{\widetilde m}}^a(t),
\text{ and }
\quad
\mu^a(t)
=
{\left(}\psi^a\otimes I_d{\right)}\#{{\widetilde \mu}}^a(t),\end{aligned}$$ for $(t,x)\in[0,T]\times{ \mathbb{R}}^d$.
*Second step: Proving that $m^a$ is compact.*
We are going to use the Arzelà-Ascoli Theorem on $C^0{\left(}[0,T];{\left(}{{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)},W_1{\right)}{\right)}$ (${{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)}$ is endowed with the $1$-Wassertein distance). First we prove that for any $t\in[0,T]$, the sequence ${\left(}m^a(t){\right)}_{a>1}$ is compact with the $1$-Wassertein distance, by proving that $\int_{{ \mathbb{R}}^d}|x|^2dm^{a}(t,x)$ is uniformly bounded in $a$. At time $t=0$, we have $$\int_{{ \mathbb{R}}^d}{\left|}x{\right|}^2 dm^a(0,x)
=
\int_{{{\mathbb{T}}}^d_a}{\left|}\psi^a(x){\right|}^2 d{{\widetilde m}}^{a,0}(x)
=
\int_{{ \mathbb{R}}^d}{\left|}\psi^a\circ\pi^a(x){\right|}^2 dm^0(x)
\leq
\int_{{ \mathbb{R}}^d}{\left|}x{\right|}^2 dm^0(x)
\leq
C_0,$$ using , and \[hypo:Monogregx\]. Let us differentiate $\int_{{ \mathbb{R}}^d}|x|^2dm^{a}(t,x)$ with respect to time, perform some integrations by part and obtain that $$\begin{aligned}
\frac{d}{dt}
\int_{{ \mathbb{R}}^d}|x|^2dm^{a}(t,x)
&=
\frac{d}{dt}
\int_{{{\mathbb{T}}}^d_a}{\left|}\psi^a(x){\right|}^2d{{\widetilde m}}^{a}(t,x)
\\
&=
\int_{{{\mathbb{T}}}^d_a}
{\left|}\psi^a(x){\right|}^2
{\left(}\nu\Delta{{\widetilde m}}^{a}(t,x)
-\operatorname{div}{\left(}\aa^{{{\widetilde \mu}}^a(t)}(x){{\widetilde m}}^a(t,x){\right)}{\right)}dx
\\
&\!\begin{multlined}[t][10.5cm]
=
2\int_{{{\mathbb{T}}}^d_a}
\sum_{i=1}^d
{\left[}\nu{{\widetilde{\psi}}}''{\left(}\frac{\chi^a(x^i)}a{\right)}{{\widetilde{\psi}}}{\left(}\frac{\chi^a(x^i)}a{\right)}+\nu{{\widetilde{\psi}}}'{\left(}\frac{\chi^a(x^i)}a{\right)}^2
\right.
\\
\left.
+\psi^a{\left(}x{\right)}{{\widetilde{\psi}}}'{\left(}\frac{\chi^a(x^i)}a{\right)}\aa^{{{\widetilde \mu}}^a(t),i}(x)
{\right]}d{{\widetilde m}}^a(t,x)
\end{multlined}
\\
&\leq
2\nu d{{\left\| {{\widetilde{\psi}}}'' \right\|_{\infty}^{}}}{{\left\| {{\widetilde{\psi}}}\right\|_{\infty}^{}}}
+2\nu d{{\left\| {{\widetilde{\psi}}}' \right\|_{\infty}^{2}}}
+{{\left\| \psi' \right\|_{\infty}^{2}}}{{\left\| \aa^{{{\widetilde \mu}}^a(t)} \right\|_{\infty}^{2}}}
+\int_{{{\mathbb{T}}}^d_a}{\left|}\psi^a(x){\right|}^2d{{\widetilde m}}^{a}(t,x)
\\
&\leq
2\nu d{{\left\| {{\widetilde{\psi}}}'' \right\|_{\infty}^{}}}{{\left\| {{\widetilde{\psi}}}\right\|_{\infty}^{}}}
+2\nu d{{\left\| {{\widetilde{\psi}}}' \right\|_{\infty}^{2}}}
+{{\left\| {{\widetilde{\psi}}}' \right\|_{\infty}^{2}}}{{\left\| \aa^{{{\widetilde \mu}}^a(t)} \right\|_{\infty}^{2}}}
+\int_{{ \mathbb{R}}^d}{\left|}x{\right|}^2dm^{a}(t,x).\end{aligned}$$ We recall that $(t,x)\mapsto\aa^{{{\widetilde \mu}}^a(t)}(x)$ is uniformly bounded with respect to $t$ and $a$ by Lemma \[lem:Monoapriori\]. Therefore, the latter two inequalities and a comparison principle for ordinary differential equation imply that $\int_{{ \mathbb{R}}^d}|x|^2dm^{a}(t,x)$ is uniformly bounded with respect to $a$ and $t$.
We define ${{\overline X}}^a$ a random process on ${ \mathbb{R}}^d$ by $$d{{\overline X}}^a_t
=
\aa^{\mu^a(t)}{\left(}\pi^a{\left(}{{\overline X}}^a_t{\right)}{\right)}dt
+\sqrt{2\nu}dB_t,
\;
\text{ and }
{{\mathcal L}}{\left(}{{\overline X}}^a_0{\right)}=
m^0,$$ where $B$ is a Brownian motion on ${ \mathbb{R}}^d$ independent of ${{\overline X}}^a_0$. For $t,s\in[0,T]$, we have that, $$\begin{aligned}
{{\mathbb E}}{\left[}{\left|}{{\overline X}}^a_t-{{\overline X}}^a_s{\right|}{\right]}&\leq
{{\mathbb E}}{\left[}{\left|}{{\overline X}}^a_t-{{\overline X}}^a_s{\right|}^2{\right]}^{\frac12}
\\
&\leq
{{\mathbb E}}{\left[}{\left|}\int_s^t\sqrt{2\nu} dW_r{\right|}^2{\right]}^{\frac12}
+{{\mathbb E}}{\left[}{\left|}\int_s^t\aa^{\mu^a(r)}dr{\right|}^2{\right]}^{\frac12}
\\
&\leq
\sqrt{2\nu d}|t-s|^{\frac12}
+|t-s|\sup_{r\in[0,T]}{{\left\| \aa^{\mu^a(r)} \right\|_{\infty}^{}}}.\end{aligned}$$ We define ${{\widetilde X}}^a_t=\pi^a{\left(}{{\overline X}}^a_t{\right)}\in{{\mathbb{T}}}^d_a$ and $X^a_t=\psi^a{\left(}{{\widetilde X}}^a_t{\right)}\in{ \mathbb{R}}^d$, for $t\in[0,T]$. One may check that the law of ${{\widetilde X}}^a_t$ satisfies the same Fokker-Planck equation in the sense of distributions as ${{\widetilde m}}^a(t)$ by testing it with $C^{\infty}{\left(}(0,T)\times{{\mathbb{T}}}^d{\right)}$ test functions. Therefore, the law of ${{\widetilde X}}^a_t$ is ${{\widetilde m}}^a(t)$ and the law of $X^a_t$ is $m^a(t)$. By definition of the $1$-Wassertein distance, we obtain $$\begin{aligned}
W_1{\left(}m^a(t),m^a(s){\right)}&\leq
{{\mathbb E}}{\left[}{\left|}X^a_t-X^a_s{\right|}{\right]}\\
&\leq
{{\mathbb E}}{\left[}{\left|}\psi^a\circ\pi^a{\left(}{{\overline X}}^a_t{\right)}-\psi^a\circ\pi^a{\left(}{{\overline X}}^a_s{\right)}{\right|}{\right]}\\
&\leq
{{\left\| {{\widetilde{\psi}}}' \right\|_{\infty}^{}}}
{{\mathbb E}}{\left[}{\left|}{{\overline X}}^a_t-{{\overline X}}^a_s{\right|}{\right]}\\
&\leq
{{\left\| {{\widetilde{\psi}}}' \right\|_{\infty}^{}}}
{\left(}\sqrt{2\nu d}|t-s|^{\frac12}
+|t-s|\sup_{r\in[0,T]}{{\left\| \aa^{\mu^a(r)} \right\|_{\infty}^{}}}{\right)},\end{aligned}$$ where we used and the mean value theorem to pass from the second to the third line in the latter chain of inequalities. Therefore by the Arzelà-Ascoli theorem, ${\left(}m^{a}{\right)}_{a>0}$ is relatively compact in $C^0{\left(}[0,T];{\left(}{{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)},W_1{\right)}{\right)}$.
*Third Step: passing to the limit for a subsequence.*
We recall that ${{\widetilde u}}^a$ and $\nabla_x{{\widetilde u}}^a$ are uniformly bounded with respect to $a$, so are $u^a$ and $\nabla_xu^a$. Moreover $u^a$ satisfies the following PDE, $$-{{\partial_{t}}}u^a
-\nu\Delta u^a
+H{\left(}t,\psi^a\circ\pi^a(x),\nabla_xu(t,x),\mu^a(t){\right)}=
f{\left(}t,\psi^a\circ\pi^a(x),m^a(t){\right)},$$ for $(t,x)\in (0,T)\times B_{{ \mathbb{R}}^d}{\left(}0,a{\right)}$, we recall that $\psi^a\circ\pi^a(x)=x$ if $|x|\leq\frac{a}4$. For $a_0>0$, we choose $a$ such that $a>4{\left(}a_0+1{\right)}$, this implies that $u^a$ satisfies a backward heat equation on $B_{{ \mathbb{R}}^d}{\left(}0,a_0+1{\right)}$ with a bounded right-hand side, a bounded terminal condition, and bounded boundary conditions. Classical results on the heat equation (see for example Theorem $6.48$ in [@MR1465184]) state that $u^a$ is in $C^{\frac12+\frac{\bb}2,1+\bb}{\left(}[0,T]\times B_{{ \mathbb{R}}^d}{\left(}0,a_0{\right)};{ \mathbb{R}}{\right)}$ and that its associated norm is bounded by a constant which depends on the constants in the assumptions and $a_0$, but not on $a$. Therefore ${\left(}u^a_{|B_{{ \mathbb{R}}^d}{\left(}0,a_0{\right)}}{\right)}_{a>1}$ is a compact sequence in $C^{0,1}{\left(}[0,T]\times B_{{ \mathbb{R}}^d}{\left(}0,a_0{\right)};{ \mathbb{R}}{\right)}$ for any $a_0>0$. Then by a diagonal extraction method, there exists $a_n$ an increasing sequence tending to $+\infty$ in ${ \mathbb{R}}_+$ such that $$\begin{aligned}
m^{a_n}
&\to
m
\quad
&\text{ in } C^0{\left(}[0,T],{\left(}{{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)},W_1{\right)}{\right)},
\\
u^{a_n}
&\to
u
\quad
&\text{ locally in }
C^{0,1},\end{aligned}$$ for some $(u,m)\in
C^{0,1}{\left(}[0,T]\times{ \mathbb{R}}^d;{ \mathbb{R}}{\right)}\times
C^0{\left(}[0,T];{\left(}{{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)},W_1{\right)}{\right)}$. Let us prove that for $t\in[0,T]$, $\mu^{a_n}(t)$ converges to a fixed point of when $n$ tends to infinity; indeed we notice that $$\begin{aligned}
\mu^{a_n}(t)
&=
{\left(}\psi^{a_n}\otimes I_d{\right)}\#{{\widetilde \mu}}^{a_n}(t)
\\
&=
{\left(}\psi^{a_n}\otimes I_d{\right)}\#
{\left[}{\left(}I_d,
-H^{a_n}_p{\left(}t,\cdot,\nabla_x{{\widetilde u}}^{a_n}{\left(}t,\pi^{a_n}\circ\psi^{a_n}(\cdot){\right)},{{\widetilde \mu}}^{a_n}(t){\right)}{\right)}\#{{\widetilde m}}^{a_n}{\right]}\\
&=
{\left(}\psi^{a_n},
-H_p{\left(}t,\psi^{a_n}(\cdot),\nabla_xu^{a_n}{\left(}t,\psi^{a_n}(\cdot){\right)},\mu^{a_n}(t){\right)}{\right)}\#{{\widetilde m}}^{a_n}
\\
&=
{\left(}I_d,
-H_p{\left(}t,\cdot,\nabla_xu^{a_n}{\left(}t,\cdot{\right)},\mu^{a_n}(t){\right)}{\right)}\#m^{a_n}.\end{aligned}$$ In particular, $\aa^{{{\widetilde \mu}}^{a_n}(t)}=\aa^{\mu^{a_n}(t)}\circ\psi^{a_n}$ so ${{\left\| \aa^{\mu^{a_n}(t)} \right\|_{L^{\infty}{\left(}m{\right)}}^{}}}$ is not larger than ${{\left\| \aa^{{{\widetilde \mu}}^{a_n}(t)} \right\|_{L^{\infty}{\left(}{{\widetilde m}}{\right)}}^{}}}$ since the support of $m^{a_n}$ is contained in the image of the support of ${{\widetilde m}}^{a_n}$ by $\psi^{a_n}$. We proved in the previous step that ${\left(}m^a(t){\right)}_{a\geq 1}$ is compact in ${\left(}{{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)},W_1{\right)}$, and so is ${\left(}\mu^{a_n}(t){\right)}_{n\geq1}$ in ${\left(}{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)},W_1{\right)}$, since they are the pushforward measures of ${\left(}m^{a_n}(t){\right)}_{n\geq1}$ by ${\left(}I_d,\aa^{\mu^{a_n}(t)}{\right)}$. Let $\mu(t)\in{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)}$ be the limit of a convergent subsequence of ${\left(}\mu^{a_n}(t){\right)}_{n\geq1}$. Passing to the limit in the weak\* topology in the latter chain of equalities implies that $$\mu(t)=
{\left(}I_d,
-H_p{\left(}t,\cdot,\nabla_xu{\left(}t,\cdot{\right)},\mu(t){\right)}{\right)}\#m(t).$$ Moreover, the uniqueness of the fixed point \[eq:murd\] holds here, see [@MR3805247] Lemma $5.2$ for the proof. We obtained that there exists a unique fixed point satisfying \[eq:murd\], and that it is the limit of any convergent subsequence of ${\left(}\mu^{a_n}(t){\right)}$. This implies that the whole sequence ${\left(}\mu^{a_n}(t){\right)}_{n\geq1}$ tends to $\mu(t)$ in ${\left(}{{\mathcal P}}{\left(}{ \mathbb{R}}^d\times{ \mathbb{R}}^d{\right)},W_1{\right)}$.
Let us point out that $m^{a_n}$ satisfies $${{\partial_{t}}}m^{a_n}
-\nu\Delta m^{a_n}
-\operatorname{div}{\left(}H_p{\left(}t,x,\nabla_xu^{a_n},\mu^{a_n}{\right)}m^{a_n}{\right)}=
0$$ in the sense of distributions on $(0,T)\times B{\left(}0,\frac{a_n}4{\right)}$, by the definitions of $\psi^a$ and ${{\widetilde{\psi}}}$. Furthermore, at time $t=0$ we know that $m^{a_n}(0)={\left(}\psi^{a_n}\circ\pi^{a_n}{\right)}\#m^0$. We recall that $\psi^{a_n}\circ\pi^{a_n}(x)=x$ for $x\in B_{{ \mathbb{R}}^d}{\left(}0,\frac{a_n}4{\right)}$. This implies that $m^{a_n}(0)$ tends to $m^0$ in the weak\* topology of ${{\mathcal P}}{\left(}{ \mathbb{R}}^d{\right)}$. Finally we obtain that $(u,m,\mu)$ is a solution to , by passing to the limit as $n$ tends to infinity in the equations satisfied by ${\left(}u^{a_n},m^{a_n},\mu^{a_n}{\right)}$.
In the above proof, we obtain that there exists a unique fixed point satisfying \[eq:murd\]. We have thereby extended the conclusions of Lemma \[lem:FPmuLS\] to system \[eq:MFGCrd\]. Similarly, one may extend the conclusions of Lemma \[lem:FPmucont\] to system .
Proof of Theorem \[thm:UniMono\]: uniqueness of the solutions to and {#subsec:UniMono}
---------------------------------------------------------------------
Step \[step:MFGCrduni\], namely the uniqueness of the solution to , is obtained from the monotonicity assumptions \[hypo:LMono\] and \[hypo:gMono\], and the same arguments as in the case of MFG without interaction through controls.
Here, we write the proof for the system . However, none of the arguments below is specific to the domain ${ \mathbb{R}}^d$, therefore this proof can be repeated for .
We suppose that $(u^1,m^1,\mu^1)$ and $(u^2,m^2,\mu^2)$ are two solutions to . Now standard arguments (see [@MR2295621]) lead to $$\begin{gathered}
\label{eq:MonoUni}
0
=
\int_0^T\int_{{ \mathbb{R}}^d}
{\left[}\nabla_x(u^1-u^2)
\cdot H_p{\left(}t, x,\nabla_xu^1,\mu^1{\right)}- H{\left(}t,x,\nabla_xu^1,\mu^1{\right)}+H{\left(}t,x,\nabla_xu^2,\mu^2{\right)}{\right]}dm^1(t,x)
\\
+ \int_0^T\int_{{ \mathbb{R}}^d}
{\left[}\nabla_x(u^2-u^1)
\cdot H_p{\left(}t,x,\nabla_xu^2,\mu^2{\right)}- H{\left(}t,x,\nabla_xu^2,\mu^2{\right)}+H{\left(}t,x,\nabla_xu^1,\mu^1{\right)}{\right]}dm^2(t,x)
\\
+ \int_0^T\int_{{ \mathbb{R}}^d}
{\left(}f(t,x,m^1(t))-f(t,x,m^2(t)){\right)}d(m^1(t,x)-m^2(t,x))dt
\\
+ \int_{{ \mathbb{R}}^d}
{\left(}g(x,m^1(T))-g(x,m^2(T)){\right)}d(m^1(T,x)-m^2(T,x)).
\end{gathered}$$ Recall that $$\label{eq:conjugacy}
\begin{aligned}
&L{\left(}t,x,\aa^{\mu^i},\mu^i{\right)}=
\nabla_xu^i
\cdot H_p{\left(}t,x,\nabla_xu^i,\mu^i{\right)}- H{\left(}t,x,\nabla_xu^i,\mu^i{\right)},
\\
&\nabla_xu^i
=
-L_{\aa}{\left(}t,x,\aa^{\mu^i},\mu^i{\right)},
\end{aligned}$$ because $L$ is the Legendre tranform of $H$. From \[hypo:gMono\], and , we obtain that, $$\begin{gathered}
\label{eq:UniMono_aux2}
0
\geq
\int_0^T\int_{{ \mathbb{R}}^d}
{\left[}L{\left(}t,x,\aa^{\mu^1},\mu^1{\right)}-L{\left(}t,x,\aa^{\mu^2},\mu^2{\right)}-{\left(}\aa^{\mu^1}-\aa^{\mu^2}{\right)}\cdot L_{\aa}{\left(}t,x,\aa^{\mu^2},\mu^2{\right)}{\right]}dm^1(t,x)dt \\
+\int_0^T\int_{{ \mathbb{R}}^d}
{\left[}L{\left(}t,x,\aa^{\mu^2},\mu^2{\right)}-L{\left(}t,x,\aa^{\mu^1},\mu^1{\right)}-{\left(}\aa^{\mu^2}-\aa^{\mu^1}{\right)}\cdot L_{\aa}{\left(}t,x,\aa^{\mu^1},\mu^1{\right)}{\right]}dm^2(t,x)dt
\end{gathered}$$ The function $L$ is strictly convex in $\aa$ by Lemma \[lem:Lstrconv\], which implies that, $$\label{eq:Lconvex}
\begin{aligned}
L{\left(}t,x,\aa^{\mu^1},\mu^2{\right)}-L{\left(}t,x,\aa^{\mu^2},\mu^2{\right)}-{\left(}\aa^{\mu^1}-\aa^{\mu^2}{\right)}\cdot L_{\aa}{\left(}t,x,\aa^{\mu^2},\mu^2{\right)}\geq
0,
\\
L{\left(}t,x,\aa^{\mu^2},\mu^1{\right)}-L{\left(}t,x,\aa^{\mu^1},\mu^1{\right)}-{\left(}\aa^{\mu^2}-\aa^{\mu^1}{\right)}\cdot L_{\aa}{\left(}t,x,\aa^{\mu^1},\mu^1{\right)}\geq
0,
\end{aligned}$$ and turn to identities if and only if $\aa^{\mu^1}=\aa^{\mu^2}$. The latter inequalities and yield $$\begin{aligned}
0
&\geq
\int_0^T\int_{{ \mathbb{R}}^d}
{\left[}L{\left(}t,x,\aa^{\mu^1},\mu^1{\right)}-L{\left(}t,x,\aa^{\mu^1},\mu^2{\right)}{\right]}dm^1dt
+\int_0^T\int_{{ \mathbb{R}}^d}
{\left[}L{\left(}t,x,\aa^{\mu^2},\mu^2{\right)}-L{\left(}t,x,\aa^{\mu^2},\mu^1{\right)}{\right]}dm^2dt \\
&=
\int_0^T\int_{{ \mathbb{R}}^d\times{ \mathbb{R}}^d}
{\left[}L{\left(}t,x,\aa,\mu^1{\right)}-L{\left(}t,x,\aa,\mu^2{\right)}{\right]}d{\left(}\mu^1-\mu^2{\right)}(t,x,\aa)dt.
\end{aligned}$$ Assumption \[hypo:LMono\] turns the latter inequality into an equality. This, the case of equality in and the continuity of $\aa^{\mu^1}$ and $\aa^{\mu^2}$ yield that $\aa^{\mu^1}=\aa^{\mu^2}$. This implies that $m^1=m^2$ by the uniqueness of the solution to , . Therefore, we obtain $\mu^1=\mu^2$, and then $u^1=u^2$ by the uniqueness of the solution to ,\[eq:CFurd\].
Theorems \[thm:UniMonob\] and \[thm:bpower\]: existence and uniqueness of the solution to {#subsec:Exib}
------------------------------------------------------------------------------------------
So far, no distinction has been made between $\mu_b$ and $\mu_{\aa}$, because they coincide for and . Now they may differ since the drift function and the control may be different. In this case $\mu_b$ defined by $\mu_b(t)
=
\Bigl[
{\left(}x,\aa{\right)}\mapsto
{\left(}x,
b{\left(}t,x,\aa,\mu_{\aa}(t){\right)}{\right)}\Bigr]{\#}\mu_{\aa}(t)$ is naturally the joint law of the states and the drifts. The idea here to pass from to , is to assume that $b$ is invertible with respect to $\aa$, which changes the optimization problem in $\aa$ into a new optimization problem expressed in term of $b$. This consists in changing the Lagrangian from $L{\left(}t,x,\aa,\mu_{\aa}{\right)}$ into $$L^b{\left(}t,x,b,\mu_{b}{\right)}=
L{\left(}t,x,\aa^*{\left(}t,x,b,\mu_b{\right)},
\Bigl[
{\left(}x,{{\widetilde b}}{\right)}\mapsto
{\left(}x,
\aa^*{\left(}t,x,{{\widetilde b}},\mu_{b}{\right)}{\right)}\Bigr]{\#}\mu_{b}{\right)}.$$ The Hamiltonian $H^b$ defined as the Legendre transform of $L^b$ is given by $$\label{eq:defHb}
H^b{\left(}t,x,p,\mu_{b}{\right)}=
H{\left(}t,x,p,
\Bigl[
{\left(}x,{{\widetilde b}}{\right)}\mapsto
{\left(}x,
\aa^*{\left(}t,x,{{\widetilde b}},\mu_{b}{\right)}{\right)}\Bigr]{\#}\mu_{b}{\right)}.$$ Conversely, we can obtain $L$ and $H$ from $L^b$ and $H^b$ with the following relations, $$\begin{aligned}
L{\left(}t,x,\aa,\mu_{\aa}{\right)}&=
L^b{\left(}t,x,b{\left(}t,x,\aa,\mu_{\aa}{\right)},
\Bigl[
{\left(}x,\aa{\right)}\mapsto
{\left(}x,
b{\left(}t,x,\aa,\mu_{\aa}{\right)}{\right)}\Bigr]{\#}\mu_{\aa}{\right)},
\\
H{\left(}t,x,p,\mu_{\aa}{\right)}&=
H^b{\left(}t,x,p,
\Bigl[
{\left(}x,\aa{\right)}\mapsto
{\left(}x,
b{\left(}t,x,\aa,\mu_{\aa}{\right)}{\right)}\Bigr]{\#}\mu_{\aa}{\right)}.\end{aligned}$$ Now we can state the following lemma which allows us to pass from to , or vice versa.
\[lem:equisol\] Under assumption \[hypo:binvert\], ${\left(}u,m,\mu_{\aa},\mu_{b}{\right)}$ is a solution to if and only if ${\left(}u,m,\mu_{b}{\right)}$ is a solution to with $H^b$ instead of $H$.
The proof is straightforward and only consists in checking on the one hand, that and are respectively equivalent to and with $H^b$ instead of $H$; on the other hand, that and are equivalent to with $H^b$, where we take $\mu_b=\mu$ and $\mu_{\aa}$ defined by .
The following existence theorem is a direct consequence of Lemma \[lem:equisol\], and Theorem \[thm:Exird\].
\[thm:ExiMonob\] If $L^{b}$ satisfies \[hypo:Lreg\]-\[hypo:Monogregx\], and $b$ satisfies \[hypo:binvert\], there exists a solution to .
Theorem \[thm:bpower\], i.e. the existence part of step \[step:MFGCb\], is a consequence of the latter existence result in which the assumptions on $L^b$ are stated on $L$ instead, which makes them more tractable. However, we have to make the additional \[hypo:bbound\].
If $L$ and $b$ satisfy the assumptions of Theorem \[thm:bpower\], it is straightforward to check that $L^b$ satisfies \[hypo:Lreg\]-\[hypo:Lbound\]. Therefore, Theorem \[thm:bpower\] is a consequence of Corollary \[thm:ExiMonob\]. Finally, Theorem \[thm:UniMonob\] and the uniqueness part of step \[step:MFGCb\] are direct consequences of Theorem \[thm:UniMono\] and Lemma \[lem:equisol\].
I wish to express my gratitude to Y. Achdou and P. Cardaliaguet for technical advices, insightful comments and corrections. This research was partially supported by the ANR (Agence Nationale de la Recherche) through MFG project ANR-16-CE40-0015-01.
[^1]: Laboratoire Jacques-Louis Lions, Univ. Paris Diderot, Sorbonne Paris Cité, UMR 7598, UPMC, CNRS, 75205, Paris, France. zkobeissi@math.univ-paris-diderot.fr
|
---
abstract: |
In this paper we introduce a new notion of convergence of sparse graphs which we call Large Deviations or LD-convergence and which is based on the theory of large deviations. The notion is introduced by ”decorating” the nodes of the graph with random uniform i.i.d. weights and constructing random measures on $[0,1]$ and $[0,1]^2$ based on the decoration of nodes and edges. A graph sequence is defined to be converging if the corresponding sequence of random measures satisfies the Large Deviations Principle with respect to the topology of weak convergence on bounded measures on $[0,1]^d, d=1,2$. We then establish that LD-convergence implies several previous notions of convergence, namely so-called right-convergence, left-convergence, and partition-convergence. The corresponding large deviation rate function can be interpreted as the limit object of the sparse graph sequence. In particular, we can express the limiting free energies in terms of this limit object.
Finally, we establish several previously unknown relationships between the formerly defined notions of convergence. In particular, we show that partition-convergence does not imply left or right-convergence, and that right-convergence does not imply partition-convergence.
author:
- '[Christian Borgs]{}[^1]'
- '[Jennifer Chayes]{}[^2]'
- '[David Gamarnik]{}[^3]'
bibliography:
- 'LD-bibliography.bib'
date: 'February 10, 2013'
title: 'Convergent sequences of sparse graphs: A large deviations approach'
---
Introduction
============
The theory of convergent graph sequences for dense graphs is by now a well-developed subject, with many [*a priori*]{} distinct notions of convergence proved to be equivalent. For sparse graph sequences (which in this paper we take to be sequences with bounded maximal degree), much less is known. While several of the notions defined in the context of dense graphs were generalized to the sparse setting, and some other interesting notions were introduced in the literature, the equivalence between the different notions was either unknown or known not to hold. In some cases, it was not even known whether one is stronger than the other or not. In this paper, we introduce a new natural notion of convergence for sparse graphs, which we call Large Deviations Convergence, and relate it to other notions of convergence. We hope that this new notion will also ultimately allow us to relate many of the previous notions to each other.
Let ${\mathbb{G}}_n$ be a sequence graphs such that the number of vertices in ${\mathbb{G}}_n$ goes to infinity. The question we address here is the notion of convergence and limit of such a sequence.
For dense graphs, i.e., graphs for which the average degree grows like the number of vertices, this question is by now well understood [@BorgsChayesCountingGraphHom],[@BorgsChayesEtAlGraphLimitsI],[@BorgsChayesEtAlGraphLimitsII]. These works introduced and showed the equivalence of various notions of convergence: [*left-convergence*]{}, defined in terms of homomorphisms from a small graph ${\mathbb{F}}$ into ${\mathbb{G}}_n$; [*right-convergence*]{}, defined in terms of homomorphisms from ${\mathbb{G}}_n$ into a weighted graph $\mathbb H$ with strictly positive edge weights; convergence in terms of the so-called [*cut metric*]{}; and several other notions, including convergence of the set of [*quotients*]{} of the graphs ${\mathbb{G}}_n$, a notion which will play a role in this paper as well. The existence and properties of the limit object were established in [@LovaszSzegedy], and its uniqueness was proved in [@borgs2010moments]. Some lovely follow up work on the dense case gave alternative proofs in terms of exchangeable random variables [@diaconis2007graph], and provided applications using large deviations [@ChatterjeeVaradhan].
For sparse graph sequences much less is known. In [@BorgsChayesKahnLovasz], *left-convergence* was defined by requiring that the limit $$\label{L-convergent}
t({\mathbb{F}})=\lim_{n\to \infty}\frac 1{|V({\mathbb{G}}_n)|} \hom({\mathbb{F}},{\mathbb{G}}_n)$$ exists for every connected, simple graph ${\mathbb{F}}$, where $\hom({\mathbb{F}},{\mathbb{G}})$ is used to denote the number of homomorphisms from ${\mathbb{F}}$ into ${\mathbb{G}}$. Here and in the rest of this paper $V({\mathbb{G}})$ denotes the vertex set of ${\mathbb{G}}$. Also $E({\mathbb{G}})$ denotes the edge set of ${\mathbb{G}}$ and for $u,v\in V({\mathbb{G}})$ we write $u\sim v$ if $(u,v)\in E({\mathbb{G}})$. It is easy to see that for sequences of graphs with bounded degrees, this notion is equivalent to an earlier notion of convergence, the notion of *local convergence* introduced by Benjamini and Schramm [@benjamini_schramm].
To define right-convergence, one again considers homomorphisms, but now from ${\mathbb{G}}_n$ into a small “target” weighted graph ${\mathbb{H}}$. Each homomorphism is equipped with the weight induced by ${\mathbb{H}}$. Since the total weight of all homomorphisms denoted by $\hom({\mathbb{G}}_n,{\mathbb{H}})$ is at most exponential in $|V({\mathbb{G}}_n)|$, it is natural to consider the sequence $$\begin{aligned}
\label{eq:rightlimit}
f({\mathbb{G}}_n,{\mathbb{H}})=-\frac 1{|V({\mathbb{G}}_n)|}\log \hom({\mathbb{G}}_n,{\mathbb{H}}),\end{aligned}$$ which in the terminology of statistical physics is the sequence of free energies. In [@BorgsChayesKahnLovasz] a sequence ${\mathbb{G}}_n$ was defined to be right-convergent with respect to a given graph ${\mathbb{H}}$ if the free energies $f({\mathbb{G}}_n,{\mathbb{H}})$ converge as $n\to\infty$. It was shown that a sequence ${\mathbb{G}}_n$ that is right-convergent on all simple graphs ${\mathbb{H}}$ is also left-convergent. But the converse was shown not to be true. The counterexample is simple: Let $C_n$ be the cycle on $n$ nodes, and let ${\mathbb{H}}$ be a bipartite graph. Then $\hom(C_n,{\mathbb{H}})=0$ for odd $n$, while $\hom(C_n,{\mathbb{H}})\geq 1$ for even $n$.
This example, however, relies crucially on the fact that ${\mathbb{H}}$ encodes so-called *hard-core* constraints, meaning that some of the maps $\phi:V({\mathbb{G}}_n)\to V({\mathbb{H}})$ do not contribute to $\hom({\mathbb{G}}_n,{\mathbb{H}})$. As we discuss in Section 4, such hard-core constraints are not very natural in many respects, one of them being that the existence or value of the limit of the sequence can change if we remove a sub-linear number of edges from ${\mathbb{G}}_n$ (in the case of $C_n$, just removing a single edge makes the sequence convergent, as can be seen by a simple sub-additivity argument). Hard-core constraints are also not very natural from a point of view of applications in physics, where hard-core constraints usually represent an idealized limit like the zero-temperature limit, which is a mathematical idealization of something that in reality can never be realized.
We therefore propose to *define* right-convergence by restricting ourselves to so-called soft-core graphs, i.e., graphs ${\mathbb{H}}$ with strictly positive edge weights. In other words, we [define]{} ${\mathbb{G}}_n$ to be *right-convergent* iff the free energy defined in has a limit for every soft-core graph ${\mathbb{H}}$. This raises the question of how to relate the existence of a limit for models with hard-core constraints to those on soft-core graphs, a question we will address in Section \[subsection:right-convergence\] below, but we do not make this part of the definition of right-convergence. Given our relaxed definition of the right-convergence, the question then remains whether left-convergence possibly implies right-convergence. The answer turns out to be still negative, and involves an example already considered in [@BorgsChayesKahnLovasz], see Section \[sec:left-not-right\] below.
Next let us turn to the notion of partition-convergence of sparse graphs, a notion which was introduced by Bollobas and Riordan [@BollobasRiordanMetrics], motivated by a similar notion for dense graphs from [@BorgsChayesEtAlGraphLimitsI], [@BorgsChayesEtAlGraphLimitsII], where it was called convergence of quotients. We start by introducing quotients for sparse graphs. Fix a graph ${\mathbb{G}}$ and let $\sigma=(V_1,\dots,V_k)$ be a partition of $V({\mathbb{G}})$ into disjoint subsets (where some of the $V_i$’s may be empty). We think of this partition as a (non-proper) coloring $\sigma:V({\mathbb{G}})\rightarrow [k]$ with $V_i=\sigma^{-1}(i), 1\le
i\le k$. The partition $\sigma$ then induces a weighted graph ${\mathbb{F}}={\mathbb{G}}/\sigma$ on $[k]=\{1,\dots,k\}$ (called a $k$-quotient) by setting $$\label{k-quotient}
x_i(\sigma)=\frac {|V_i|} {|V|}
\qquad\text{and}\qquad
X_{ij}(\sigma)=\frac {1} {|V|} \left|\{u\in V_i, v\in V_j\colon u\sim v\}\right|.$$ Here $x(\sigma)=(x_i(\sigma),~1\le i\le k)$ and $X(\sigma)=(X_{i,j}(\sigma),~1\le i,j\le k)$ are the set of node and edge weights of ${\mathbb{F}}$, respectively. Then for a given $k$, the set of all possible $k$-quotients $(x(\sigma),X(\sigma))$ is a discrete subset ${\mathcal
S}_k({\mathbb{G}})\subset{\mathbb{R}}_+^{(k+1)k}$. Henceforth ${\mathbb{R}}({\mathbb{R}}_+)$ denotes the set of all (non-negative) reals, and ${\mathbb{Z}}({\mathbb{Z}}_+)$ denotes the set of all (non-negative) integers. Motivated by the corresponding notion from dense graphs, one might want to study whether the sequence of $k$-quotients ${\mathcal S}_k({\mathbb{G}}_n)$ converges to a limiting set $\mathcal
S_k\subset{\mathbb{R}}_+^{(k+1)k}$ with respect to some appropriate metric on the subsets of ${\mathbb{R}}_+^{(k+1)k}$. This leads to the notion of partition-convergence, a notion introduced in [@BollobasRiordanMetrics]. Since it is not immediate whether this notion of convergence implies, for example, left-convergence (in fact, one of the results in this paper is that it does not), Bollobas and Riordan [@BollobasRiordanMetrics] also introduced a stronger notion of colored-neighborhood-convergence. This notion was further studied by Hatami, Lov[á]{}sz and Szegedi [@HatamiLovaszSzegedy]. An implication of the results of these and earlier papers (details below) is that colored-neighborhood-convergence is strictly stronger than the notion of both partition- and left-convergence, but it is not known whether it implies right-convergence.
We now discuss the main contribution of this paper: the definition of convergent sparse graph sequences using the formalism of large deviations theory. We define a graph sequence to be Large Deviations (LD)-convergent if for every $k$, the weighted factor graphs ${\mathbb{F}}=G/\sigma$ defined above viewed as a vector $(x_i({\mathbb{F}}), 1\le i\le k)$ and matrix $(X_{i,j}({\mathbb{F}}), 1\le i,j\le k)$, satisfy the large deviation principle in ${\mathbb{R}}^k$ and ${\mathbb{R}}^{k\times k}$, when the $k$-partition $\sigma$ is chosen uniformly at random.[^4] Intuitively, the large deviations rate associated with a given graph ${\mathbb{F}}$ provides the limiting exponent for the number colorings $\sigma$ such that the corresponding factor graphs ${\mathbb{G}}/\sigma$ is approximately ${\mathbb{F}}$. It turns out that the large deviations rates provide enough information to ”read off” the limiting partition sets $\mathcal{S}_k$. Those are obtained as partitions with finite large deviations rate. Similarly, one can ”read off” the limits of free energies (\[eq:rightlimit\]). As a consequence, the LD-convergence implies both the partition and right-convergence.
The deficiency of the definition above is that it requires that the large deviations principle holds for a infinite collection of probability measures associated with the choice of $k$. It turns out that there is a more elegant unifying way to introduce LD-convergence by defining just one large deviations rate, but on the space of random measures rather than the space of random weighted graphs ${\mathbb{G}}_n/\sigma$. This is done as follows. Given a sequence of sparse graphs ${\mathbb{G}}_n$, suppose the nodes of each graph are equipped with values $\left(\sigma(u), u\in V({\mathbb{G}})\right)$ chosen independently uniformly at random from $[0,1]$. We may think of these values as real valued ”colors”. From these values we construct a one-dimensional measure $\rho_n=\sum_{u\in V({\mathbb{G}})}
|V({\mathbb{G}}_n)|^{-1}\delta(\sigma(u))$ on $[0,1]$ and a two-dimensional measure $\mu_n=\sum_{u,v\colon u\sim v}
|V({\mathbb{G}}_n)|^{-1}\delta(\sigma(u),\sigma(v))$ on $[0,1]^2$. Here $\delta(x)$ is a measure with unit mass at $x$ and zero elsewhere. As such we obtain a sequence of measures $(\rho_n,\mu_n)$. We define the graph sequence to be LD-convergent if this sequence of random measures satisfy the Large Deviations Principle on the space of measures on $[0,1]^d, d=1,2$ equipped with the weak convergence topology. One of our first result is establishing the equivalence of two definitions of the LD-convergence. To distinguish the two, the former mode of convergence is referred to as $k$-LD-convergence, and the latter simply as LD-convergence.
Our most important result concerning the new definition of graph convergence is that LD-convergence implies right (and therefore left) as well as partition-convergence. We conjecture that it also implies colored-neighborhood-convergence but we do not have a proof at this time. Finally, we conjecture that LD-convergence holds for a sequence of random $D$-regular graphs with high probability (w.h.p.), but at the present stage we are very far from proving this conjecture and we discuss important implications to the theory of spin glasses should this convergence be established.
Let us finally return to a question already alluded to when we defined right-convergence. From several points of view, it seems natural to consider two sequence ${\mathbb{G}}_n$ and $\tilde {\mathbb{G}}_n$ on the same vertex set $V_n=V({\mathbb{G}}_n)=V(\tilde{\mathbb{G}}_n)$ to be equivalent if the edge sets $E({\mathbb{G}}_n)$ and $E(\tilde{\mathbb{G}}_n)$ differ on a set of $o(|V_n|)$ edges. One natural condition one might require of all definitions of convergence is the condition that convergence of ${\mathbb{G}}_n$ implies that of $\tilde{\mathbb{G}}_n$ and vice versa, whenever ${\mathbb{G}}_n$ and $\tilde{\mathbb{G}}_n$ are equivalent. We show that this holds for all notions of convergence considered in the paper, namely left-convergence, our modified version of right-convergence, partition-convergence, colored-neighborhood-convergence, and LD-convergence. We also show that for all right-convergent sequences ${\mathbb{G}}_n$, one can find an equivalent sequence $\tilde{\mathbb{G}}_n$ such that on $\tilde{\mathbb{G}}_n$, the limiting free energy $f({\mathbb{H}})=\lim_{n\to\infty} f(\tilde{\mathbb{G}}_n,{\mathbb{H}})$ exists even when ${\mathbb{H}}$ contains hard-core constraints, see Theorem \[theorem:RightConvergenceEquivalence\] below. Our alternative definition of right-convergence can therefore be reexpressed by the condition that there exists a sequence $\tilde{\mathbb{G}}_n$ equivalent to ${\mathbb{G}}_n$ such that the free energies $f(\tilde{\mathbb{G}}_n,{\mathbb{H}})$ converge for all weighted graphs ${\mathbb{H}}$.
The organization of this paper is as follows. In the next section we introduce notations and definitions and provide the necessary background on the large deviations theory. In Section \[section:LDconvergence\] we introduce LD-convergence, establish some basic properties and consider two examples of LD-convergent graph sequences. Other notions of convergence are discussed in Section \[section:OtherConvergence\]. In the same section we prove Theorem \[theorem:RightConvergenceEquivalence\] relating free energy limits with respect to hard-core and soft models, discussed above. The relationship between different notions of convergence is considered in Section \[section:ConvergenceRelations\]. The main result of this section is Theorem \[theorem:ConvergenceRelations\], which establishes relationships between different notions of convergence. Figures \[figure:ConvergenceRelations\] and \[figure:ConvergenceRelationsColorNeighb\] are provided to illustrate the relations between the notions of convergence, both those known earlier and those established in this paper. In the last Section \[sec:discussion\] we compare the expressions for the limiting free energy to those in the dense case (pointing out that the large-deviations rate function plays a role quite similar to the role the limiting graphon played in the dense case) and discuss some open questions.
Notations and definitions {#sec:notation}
=========================
For the convenience of the reader, we repeat some of the notations already used in the introduction. We consider in this paper finite simple undirected graphs ${\mathbb{G}}$ with node and edge sets denoted by $V({\mathbb{G}})$ and $E({\mathbb{G}})$ respectively. We write $u\sim v$ for nodes $u$ and $v$ when $(u,v)\in E({\mathbb{G}})$. For every node $u\in
V({\mathbb{G}})$, $\mathcal{N}(u)$ denotes the set of neighbors of $u$, namely all $v\in V({\mathbb{G}})$ such that $u\sim v$. Sometimes we will write $\mathcal{N}_{{\mathbb{G}}}(u)$ in order to emphasize the underlying graph. The number $\Delta_{\mathbb{G}}(u)$ of vertices in $\mathcal{N}_{{\mathbb{G}}}(u)$ is called the degree of $u$ in ${\mathbb{G}}$. We use $\Delta_{\mathbb{G}}$ to denote the maximum degree $\max_u
\Delta_G(u)$.
A path of length $r$ is a sequence of nodes $u_{0},\ldots,u_r$ such that $u_i\sim u_{i+1}$ for all $i={0},1,2,\ldots,r-1$. The distance between nodes $u$ and $v$ is the length of the shortest path $u_0,\ldots,u_r$ with $u_0=u$ and $u_r=v$. The distance is assumed to be infinite if $u$ and $v$ belong to different connected components of ${\mathbb{G}}$. Given $u$ and $r$, let $B(u,r)=B_{{\mathbb{G}}}(u,r)$ be the subgraph of ${\mathbb{G}}$ induced by nodes with distance at most $r$ from $u$. Given a set of nodes $A\subset V$ and $r\ge 1$, $B(A,r)$ is the graph induced by the set of nodes $v$ with distance at most $r$ from some node $u\in
A$.
Given two graphs ${\mathbb{G}},{\mathbb{H}}$ a map $\sigma:V({\mathbb{G}})\rightarrow
V({\mathbb{H}})$ is a graph homomorphism if $(u,v)\in E({\mathbb{G}})$ implies $(\sigma(u),\sigma(v))\in E({\mathbb{H}})$ for every $u,v\in V({\mathbb{G}})$. Namely, $\sigma$ maps edges into edges. $\sigma:{\mathbb{G}}_1\rightarrow
{\mathbb{G}}_2$ is called a graph isomorphism if it is a bijection and $(u,v)\in E({\mathbb{G}}_1)$ if and only if $(\sigma (u), \sigma(v))\in
E({\mathbb{G}}_2)$, i.e., $\sigma$ is an isomorphism which maps edges into edges, and non-edges into non-edges. Two graphs are called isomorphic if there is a graph isomorphism mapping them into each other. Two rooted graphs $({\mathbb{G}}_1,x)$ and $({\mathbb{G}}_2,y)$ are called isomorphic if there is an isomorphism $\sigma:{\mathbb{G}}_1\rightarrow {\mathbb{G}}_2$ such that $\sigma(x)=y$.
The edit distance between two graphs ${\mathbb{G}}$ and $\tilde{\mathbb{G}}$ with the same number of vertices is defined as $$\delta_{\text{edit}}({\mathbb{G}},\tilde{\mathbb{G}})=
{\min}_{{\mathbb{G}}'}\Bigl(|E({\mathbb{G}}')\setminus E(\tilde{\mathbb{G}})| + |E(\tilde{\mathbb{G}})\setminus E({\mathbb{G}}')|\Bigr)$$ where the [minimum]{} goes over all graph ${\mathbb{G}}'$ isomorphic to ${\mathbb{G}}$.
Given two graphs ${\mathbb{G}},{\mathbb{H}}$, we use ${\ensuremath{\operatorname{Hom}}\xspace}({\mathbb{G}},{\mathbb{H}})$ to denote the set of all homomorphisms from ${\mathbb{G}}$ to ${\mathbb{H}}$, and $\hom({\mathbb{G}},{\mathbb{H}})=|{\ensuremath{\operatorname{Hom}}\xspace}({\mathbb{G}},{\mathbb{H}})|$ to denote the number of homomorphisms from ${\mathbb{G}}$ to ${\mathbb{H}}$. If ${\mathbb{H}}$ is weighted, with node weights given by a vector $\left(\alpha_i, i\in
V({\mathbb{H}})\right)$ and edge weights given by a symmetric matrix $\left(A_{ij}, (i,j)\in E({\mathbb{H}})\right)$ (which we denote by $\alpha=\alpha({\mathbb{H}})$ and $A=A({\mathbb{H}})$ respectively), then $\hom({\mathbb{G}},{\mathbb{H}})$ is defined by $$\begin{aligned}
\label{eq:WeightedHom}
\hom({\mathbb{G}},{\mathbb{H}})=\sum_{\sigma:V({\mathbb{G}})\rightarrow V({\mathbb{H}})}\prod_{u\in V({\mathbb{G}})}\alpha_{\sigma(u)}\prod_{(u,v)\in E({\mathbb{G}})}A_{\sigma(u),\sigma(v)}.\end{aligned}$$ Clearly, if $\alpha$ is a vector of ones and $A$ is a matrix consisting of zeros and ones with zeros on the diagonal, then the definition reduces to the unweighted case where $A$ denotes the adjacency matrix of a simple undirected graph ${\mathbb{H}}$.
A graph sequence ${\mathbb{G}}_n=(V({\mathbb{G}}_n),E({\mathbb{G}}_n)), n\ge 1$ is defined to be sparse if $\Delta_{{\mathbb{G}}_n}\le D$ for some finite $D$ for all $n$. When dealing with graph sequences we write $V_n$ and $E_n$ instead of $V({\mathbb{G}}_n)$ and $E({\mathbb{G}}_n)$, respectively. Two graph sequences ${\mathbb{G}}_n$ and $\tilde{\mathbb{G}}_n$ are defined to be equivalent, in which case we write ${\mathbb{G}}_n\sim \tilde{\mathbb{G}}_n$, if $\tilde{\mathbb{G}}_n$ has the same number of nodes as ${\mathbb{G}}_n$ and $\delta_{\text{edit}}({\mathbb{G}}_n,\tilde{\mathbb{G}}_n)=o(|V({\mathbb{G}}_n)|)$ as $n\rightarrow\infty$. Observe that $\sim$ defines an equivalency relationship on the set of sequences of graphs.
Next we review some concepts from large deviations theory. Given a metric space $S$ equipped with metric $d$, a sequence of probability measures ${\mathbb{P}}_n$ on Borel sets of $S$ is said to satisfy the Large Deviations Principle (LDP) with rate $\theta_n>0, \lim_n\theta_n=\infty$, if there exists a lower semi-continuous function $I:S\rightarrow {\mathbb{R}}_+\cup \{\infty\}$ such that for every (Borel) set $A\subset S$ $$\begin{aligned}
\label{eq:LDprinciple}
-\inf_{x\in A^o}I(x)\le \liminf_n {\log{\mathbb{P}}_{n}(A^o)\over \theta_n}\le \limsup_n {\log {\mathbb{P}}_{n}(\bar A)\over \theta_n} \le -\inf_{x\in \bar A}I(x),\end{aligned}$$ where $A^o$ and $\bar A$ denote the interior and the closure of the set $A$, respectively. In this case, we say that $({\mathbb{P}}_n)$ obeys the LDP with rate function $I$. Typically $\theta_n=n$ is used in most cases, but in our case we will be considering the normalization $\theta_n=|V({\mathbb{G}}_n)|$ corresponding to some sequence of graphs ${\mathbb{G}}_n$, and it is convenient not to assume that $n$ is the number of nodes in ${\mathbb{G}}_n$.
The definition above immediately implies that that the rate function $I$ is uniquely recovered as $$\begin{aligned}
\label{eq:Identity1}
I(x)=-\lim_{\epsilon\rightarrow 0}\liminf_n {\log{\mathbb{P}}_{n}(B(x,\epsilon))\over \theta_n}=
-\lim_{\epsilon\rightarrow 0}\limsup_n {\log{\mathbb{P}}_{n}(B(x,\epsilon))\over \theta_n},\end{aligned}$$ for every $x\in S$, where $B(x,\epsilon)=\{y\in S:d(x,y)\le
\epsilon\}$. Indeed, from lower semi-continuity we have $$\begin{aligned}
\label{eq:Ikepsilon}
I(x)=\lim_{\epsilon\rightarrow 0}\inf_{y\in B^o(x,\epsilon)}I(y)=\lim_{\epsilon\rightarrow 0}\inf_{y\in \bar B(x,\epsilon)}I(y),\end{aligned}$$ from which the claim follows. In fact the existence and the equality of double limits in (\[eq:Identity1\]) is also sufficient for the LDP to hold when $S$ is compact. We provide a proof here for completeness.
\[prop:LDReverse\] Given a sequence of probability measures ${\mathbb{P}}_n$ on a compact metric space $S$, and given a sequence $\theta_n>0,
\lim_n\theta_n=\infty$, suppose the limits in the second and third term in (\[eq:Identity1\]) exist and are equal for every $x$. Let $I(x)$ be defined as the negative of these terms for every $x\in S$. Then the LDP holds with rate function $I$ and normalization $\theta_n$.
We first establish that $I$ is lower semi-continuous. Fix $x,x_m\in S$ such that $x_m\rightarrow x$. For every $\epsilon$, find $m_0$ such that $d(x,x_m)\le \epsilon/2$ for $m\ge m_0$. We have for all $m\ge m_0$, $$\begin{aligned}
-I(x_m)\le \limsup_n{1\over \theta_n}\log{\mathbb{P}}_{n}\left(B(x_m,\epsilon/2)\right)\le \limsup_n{1\over \theta_n}\log{\mathbb{P}}_{n}\left(B(x,\epsilon)\right),\end{aligned}$$ implying $$\begin{aligned}
\limsup_m (-I(x_m))\le \lim_{\epsilon\rightarrow 0}\limsup_n{1\over \theta_n}\log{\mathbb{P}}_{n}\left(B(x,\epsilon)\right)=-I(x).\end{aligned}$$ Therefore $I$ is lower semi-continuous.
It remains to verify (\[eq:LDprinciple\]). Fix an arbitrary closed (and therefore compact) $A\subset S$, and let $\delta>0$. Then $$\begin{aligned}
\limsup_n {1\over \theta_n}\log{\mathbb{P}}_{n}(A)\le
{1\over \theta_n}\log{\mathbb{P}}_{n}(A) +\delta\end{aligned}$$ for an infinite number of values for $n$. Let $N_0$ be the set of $n$ for which this holds. Further, given $m\in {\mathbb{Z}}_+$, we can find $x_1,\ldots,x_{N(m)} \in A$ such that $A\subset \cup_{i\le
N(m)}B(x_i,1/m)$. Let $i(m,n)\le N(m)$ be the index corresponding to a largest value of ${\mathbb{P}}_{n}(B(x_i,1/m))$. There exists an $i(m)$ such that $i(m,n)=i(m)$ for an infinite number of values $n\in N_0$. Denote the set of $n$ for which this holds by $N_m$, and assume without loss of generality that $\theta_n^{-1}\log N(m)\leq
\delta$ for all $n\in N_m$. For such $n$, we then have $$\begin{aligned}
{\mathbb{P}}_{n}(A)\le \sum_{i\le N(m)}{\mathbb{P}}_{n}(B(x_i,1/m))\le N(m){\mathbb{P}}_{n}(B(x_{i(m)},1/m))\end{aligned}$$ and hence $$\begin{aligned}
\limsup_n {1\over \theta_n}\log{\mathbb{P}}_{n}(A)\le 2\delta+
{1\over \theta_n}\log{\mathbb{P}}_{n}(B(x_{i(m)},1/m)).\end{aligned}$$
Find any limit point $x\in A$ of $x_{i(m)}$ as $m\rightarrow\infty$. Fix $\epsilon>0$. Find $m_1$ such that $B(x_{i(m_1)},1/m_1)\subset B(x,\epsilon)$, and hence $$\begin{aligned}
\limsup_n {1\over \theta_n}\log{\mathbb{P}}_{n}(A)\le 2\delta+
{1\over \theta_n}\log{\mathbb{P}}_{n}(B(x,\epsilon))\end{aligned}$$ whenever $n\in N_{m_1}$. Since $N_{m_1}$ contains infinitely many $n\in {\mathbb{Z}}_+$, we conclude that $$\begin{aligned}
\limsup_n {1\over \theta_n}\log{\mathbb{P}}_{n}(A)\le 2\delta+
\limsup_n {1\over \theta_n}\log{\mathbb{P}}_{n}(B(x,\epsilon)).\end{aligned}$$ Since this holds for every $\delta$ and $\epsilon$, we may apply (\[eq:Identity1\]) to obtain $$\begin{aligned}
\limsup_n {1\over \theta_n}\log{\mathbb{P}}_{n}(A)\le -I(x)\le -\inf_{y\in A}I(y),\end{aligned}$$ thus verifying the upper bound in (\[eq:LDprinciple\]).
For the lower bound, suppose $A\subset S$ is open. Fix $\epsilon>0$ and find $x\in A$ such that $I(x)\le \inf_{y\in
A}I(y)+\epsilon$. Find $\delta$ such that $B(x,\delta)\subset
A$. We have $$\begin{aligned}
\liminf_n {1\over \theta_n}\log{\mathbb{P}}_{n}(A)&\ge \liminf_n {1\over \theta_n}\log{\mathbb{P}}_{n}\left(B(x,\delta)\right)\\
&\ge -I(x)\\
&\ge -\inf_{y\in A}I(y)-\epsilon.\end{aligned}$$ Since $\epsilon$ is arbitrary, this proves the lower bound.
We finish this section with some basic notions of weak convergence of measures. Let ${{\bf \mathcal{M}}}^d$ denote the space of Borel measures on $[0,1]^d$ bounded by some constant $D>0$. The space ${{\bf \mathcal{M}}}^d$ is equipped with the topology of weak convergence which can be realized using the so-called Prokhorov metric denoted by $d$ defined as follows (see [@Billingsley] for details). For any two measures $\mu,\nu\in {{\bf \mathcal{M}}}^d$, their distance $d(\mu,\nu)$ is the smallest $\tau$ such that for every measurable $A\subset [0,1]^d$ we have $\mu(A)\le
\nu(A^\tau)+\tau$ and $\nu(A)\le \mu(A^\tau)+\tau$. Here $A^\tau$ is the set of points with distance at most $\tau$ from $A$ in the $\|\cdot\|_\infty$ norm on $[0,1]^d$ (although the choice of norm on $[0,1]^d$ is not relevant). It is known that that the metric space ${{\bf \mathcal{M}}}^d$ is compact with respect to the topology of weak convergence. We will focus in this paper on the product ${{\bf \mathcal{M}}}^1\times{{\bf \mathcal{M}}}^2$ which we equip with the following metric (which we also denote by $d$ with some abuse of notation): for every two pairs of measures $(\rho_j,\mu_j)\in{{\bf \mathcal{M}}}^1\times{{\bf \mathcal{M}}}^2,~j=1,2$, the distance is $d\left((\rho_1,\mu_1),(\rho_2,\mu_2)\right)=\max\left(d(\rho_1,\rho_2),d(\mu_1,\mu_2)\right)$, where $d$ is the metric on ${{\bf \mathcal{M}}}^1$ and ${{\bf \mathcal{M}}}^2$.
LD-convergence of sparse graphs {#section:LDconvergence}
===============================
Definition and basic properties {#subsection:DefProperties}
-------------------------------
Given a graph ${\mathbb{G}}=(V,E)$ and any mapping $\sigma:V\rightarrow
[0,1]$, construct $(\rho(\sigma),\mu(\sigma))\in {{\bf \mathcal{M}}}^1\times
{{\bf \mathcal{M}}}^2$ as follows. For every $u\in V$ we put mass $1/|V|$ on $\sigma(u)\in [0,1]$. This defines $|V|$ points on $[0,1]$ with the total mass $1$. The resulting measure is denoted by $\rho(\sigma)\in{{\bf \mathcal{M}}}^1$. Similarly, we put mass $1/|V|$ on $(\sigma(u),\sigma(v))\in [0,1]^2$ and $(\sigma(v),\sigma(u))\in [0,1]^2$ for every edge $(u,v)\in E$. The resulting measure is denoted by $\mu(\sigma)\in {{\bf \mathcal{M}}}^2$. Formally $$\label{rho-sigma}
\begin{aligned}
\rho(\sigma)&=\sum_{u\in V}|V|^{-1}\delta_{\sigma(u)}, \\
\mu(\sigma)&=\sum_{(u,v)\in E}|V|^{-1}\delta_{\sigma(u),\sigma(v)},
\end{aligned}$$ where the sum goes over oriented edges, i.e., ordered pairs $(u,v)$ such that $u$ is connected to $v$ by an edge in $E$, and $\delta_x$ denotes a unit mass measure on $x\in{\mathbb{R}}^d$. For example, consider a graph with $4$ nodes $1,2,3,4$ and edges $(1,2),(1,3),(2,3),(3,4)$. Suppose a realization of $\sigma$ is $\sigma(1)=.4,\sigma(2)=.8,\sigma(3)=.15,\sigma(4)=.5$, as shown on Figure \[figure:graph\]. The corresponding $\rho(\sigma)$ and $\mu(\sigma)$ are illustrated on Figure \[figure:rhomu\].
-48pt
Now given a sparse graph sequence ${\mathbb{G}}_n=(V_n,E_n)$ with a uniform degree bound $\max_n \Delta_{{\mathbb{G}}_n}\le D$, let $\sigma_n:V_n\rightarrow [0,1]$ be chosen independently uniformly at random with respect to the nodes $u\in V_{n}$. Then we obtain a random element $(\rho(\sigma_n),\mu(\sigma_n))\in {{\bf \mathcal{M}}}^1\times {{\bf \mathcal{M}}}^2$. We will use notations $\rho_n$ and $\mu_n$ instead of $\rho(\sigma_n)$ and $\mu(\sigma_n)$. Observe that $\rho_n([0,1])=1$ and $0\le
\mu_n([0,1]^2)\le D$. In particular, $\mu_n([0,1]^2)=0$ if and only if ${\mathbb{G}}_n$ is an empty graph. We now introduce the notion of LD-convergence. In this definition and the Definition \[definition:GraphConvergenceLDk\] below we assume $\theta_n=|V_n|$.
\[definition:GraphConvergenceLDGeneral\] A graph sequence ${\mathbb{G}}_n$ is defined to be LD-convergent if the sequence $(\rho(\sigma_n),\mu(\sigma_n))$ satisfies LDP in the metric space ${{\bf \mathcal{M}}}^1\times {{\bf \mathcal{M}}}^2$.
Given a sparse graph sequence ${\mathbb{G}}_n$ with degree bounded by $D$ and a positive integer $k$, let $\sigma_n:V_n\rightarrow [k]$ be chosen uniformly at random. Consider the corresponding random weighted $k$-quotients ${\mathbb{F}}_n={\mathbb{G}}_n/\sigma_n$ defined via (\[k-quotient\]) with random node weights $x(\sigma_n)=(x_i(\sigma_n), 1\le i\le k)$ and random edge weights $X(\sigma_n)=(X_{ij}(\sigma_n), 1\le i,j\le k)$. Note that $\sum_i x_i(\sigma_n)=1$ and $\sum_{i,j}
x_{ij}(\sigma_n)\le D$. Assuming without loss of generality that $D\ge 1$, it it will be convenient to view $(x(\sigma_n),X(\sigma_n))$ as a $(k+1)\times k$ matrix with elements bounded by $D$, namely an element in ${[0,D]^{(k+1)\times k}}$. We consider ${[0,D]^{(k+1)\times k}}$ equipped with the ${\mathbb{L}}_\infty$ norm. That is the distance between $(x,X)$ and $(y,Y)$ is the maximum of $\max_{1\le
i\le k}|x_i-y_i|$ and $\max_{1\le i,j\le k}|X_{i,j}-Y_{i,j}|$.
\[definition:GraphConvergenceLDk\] A graph sequence ${\mathbb{G}}_n$ is defined to be $k$-LD-convergent if for every $k$ the sequence $(x(\sigma_n),X(\sigma_n))$ satisfies the LDP in ${[0,D]^{(k+1)\times k}}$ with some rate function $I_k:{[0,D]^{(k+1)\times k}}\rightarrow {\mathbb{R}}_+\cup \{\infty\}$.
The intuition behind this definition is as follows. Given $(x,X)$ suppose there exists a $k$-coloring of a graph $G_n$ such that the corresponding $k$-quotient weighted graph is ”approximately” equal to $(x,X)$ to within some additive error $\epsilon$. Then in fact there are exponentially in $|V_n|$ many $k$-colorings of ${\mathbb{G}}_n$ which achieve ”nearly” the same quotient graph up to say an $2\epsilon$ additive error (by arbitrarily recoloring $\epsilon |V_n|/D$ many nodes). In this case $k$-LD-convergence of the graph sequence ${\mathbb{G}}_n$ means that the exponent of the number of colorings achieving the quotient $(x,X)$ is well defined and is given by $\log
k-I(x,X)$. In other words, the total number of $k$-coloring achieving the quotient $(x,X)$ is approximately $\exp\left(|V_n|(\log k-I(x,X))\right)$. In case $(x,X)$ is not ”nearly” achievable by any coloring, we have $I(x,X)=\infty$, meaning that there are simply no coloring achieving the target quotient $(x,X)$.
While Definition \[definition:GraphConvergenceLDk\] might be more intuitive than Definition \[definition:GraphConvergenceLDGeneral\], the latter is more powerful in the sense that it does not require a ”for all $k$” condition, namely it is introduced through having the LDP on *one* as opposed to infinite (for every $k$) sequence of probability spaces. It is also more powerful, in the sense that the measure $(\rho_n,\sigma_n)$ completely defines the underlying graph, whereas this is not the case for the $k$-quotients. Since the space ${[0,D]^{(k+1)\times k}}$ being a subset of a Euclidian space is a much ”smaller” than the space of measures ${{\bf \mathcal{M}}}^d$, it is not too surprising that the LD-convergence implies the $k$-LD-convergence. But it is perhaps surprising that the converse is true.
\[theorem:LDconvergenceEquivalence\] A graph sequence ${\mathbb{G}}_n$ is LD-convergent if and only if it is $k$-LD-convergent for every $k$.
\[cor:Ik-from-I\] Given the rate function $I$ for LD-convergence, the rate function $I_k:{[0,D]^{(k+1)\times k}}\to {\mathbb{R}}_+\cup \{\infty\}$ for $k$-LD-convergence is easy to calculate, and will be given by $$I_k(x,X)=\inf_{(\rho,\mu)} I(\mu,\rho)$$ where the $\inf$ goes over all $(\rho,\mu)\in {{\bf \mathcal{M}}}^1\times{{\bf \mathcal{M}}}^2$ such that $$\label{phi-k-def}
\begin{aligned}
x_{i+1}&=\rho\left(\Delta_{i,k}\right), \quad& 0\le i\le k-1;\\
X_{i+1,j+1}&=\mu\left(\Delta_{i,k}\times \Delta_{j,k}\right), \quad& 0\le i,j\le k-1,
\end{aligned}$$ with $\Delta_{i,k}$ denoting the open interval $\Delta_{i,k}=\left({i\over k},{(i+1)\over k}\right)$.
We will also give an explicit construction for the rate function $I$ from the rate functions $I_k$, see Corollary \[cor:Ikhat\] below.
Before proving the theorem, we reformulate the notion of $k$-LD-convergence in terms of random measures with piecewise constant densities. We need some notation. Given a positive integer $k$, let $O_k^d\subset [0,1]^d$ be the open set of all points such that no coordinate is of the form $0,1/k,2/k,\ldots,1$. Let ${{\bf \mathcal{M}}}_k^d\subset {{\bf \mathcal{M}}}^d$ be the set of measures supported on $O_k^d$, and let ${{\bf \mathcal{N}}}^d_k\subset {{\bf \mathcal{M}}}^d_k$ be the set of measures in ${{\bf \mathcal{M}}}^d_k$ which have constant density on rectangles $$\begin{aligned}
\Delta_{i_1,k}\times \cdots \Delta_{i_d,k}, \qquad 0\le i_1,\ldots, i_d\le k-1.\end{aligned}$$ We define a projection operator $T_k:{{\bf \mathcal{M}}}^1\times{{\bf \mathcal{M}}}^2\to
{{\bf \mathcal{N}}}_k^1\times{{\bf \mathcal{N}}}_k^2$ by mapping a pair of measures $(\rho,\mu)\in {{\bf \mathcal{M}}}^1\times{{\bf \mathcal{M}}}^2$ into $(\rho_k,\mu_k)=T_k(\rho,\mu)$ defined as the pair of measures in ${{\bf \mathcal{N}}}_k^1\times{{\bf \mathcal{N}}}_k^2$ for which $$\label{rhok-muk}
\begin{aligned}
\rho_k(\Delta_{i,k})&=\rho(\Delta_{i,k})\quad\text{for}\quad 0\leq i\leq k-1
\\
\mu_k\left(\Delta_{i,k}\times \Delta_{j,k}\right)&=\mu\left(\Delta_{i,k}\times \Delta_{j,k}\right)
\quad\text{for}\quad 0\leq i,j,\leq k-1.
\end{aligned}$$ Given a real valued coloring $\sigma:V_n\to [0,1]$ let $\sigma_k=t_k(\sigma)$ be the coloring $\sigma_k:V_n\to [k]$ defined by $\sigma_k(u)=\lceil k\sigma(k)\rceil$ if $\sigma(u)>0$ and $\sigma_k(u)=1$ if $\sigma(u)=0$. Note that $t_k(\sigma_n)$ is an i.i.d. uniformly random coloring with colors in $[k]$ if $\sigma_n$ is chosen to be an i.i.d.uniformly random coloring with colors in $[0,1]$.
\[lemma:GraphConvergenceLDk\] Let ${\mathbb{G}}_n=(V_n,E_n)$, and let $\sigma_n:V_n\to [0,1]$ be an i.i.d. uniformly random coloring of $V_n$. Then ${\mathbb{G}}_n$ is $k$-LD-convergent if an only if $T_k(\rho(\sigma_n),\mu(\sigma_n))$ obeys a LDP on ${{\bf \mathcal{N}}}^1_k\times {{\bf \mathcal{N}}}^2_k$.
Let $\hat\sigma_n:V_n\to [k]$ be an i.i.d uniformly random coloring of $V_n$ with colors in $[k]$, and define $F_k:{[0,D]^{(k+1)\times k}}\rightarrow {{\bf \mathcal{N}}}^1_k\times {{\bf \mathcal{N}}}^2_k$ as follows: given $(x,X)\in {[0,D]^{(k+1)\times k}}$ with $x=(x_i, 1\le i\le k)$ and $X=(X_{ij}, 1\le
i,j\le k)$, let $F_k((x,X)):=(\rho,\mu)\in {{\bf \mathcal{N}}}_k^1\times{{\bf \mathcal{N}}}_k^2$ be the piece-wise constant measures with densities $kx_{i}$ on $\Delta_{i-1,k}$ and $k^2X_{i,j}$ on $\Delta_{i-1,k}\times
\Delta_{j-1,k}$, respectively. Then $F_k$ is an invertible continues function with continuous inverse. Thus, applying the Contraction Principle, $(\rho_k(\hat\sigma_n),\mu_k(\hat\sigma_n))=F_k((x(\hat\sigma_n),X(\hat\sigma_n)))$ satisfies the LDP on ${{\bf \mathcal{N}}}^1_k\times {{\bf \mathcal{N}}}^2_k$ if and only if $(x(\hat\sigma_n),X(\hat\sigma_n))$ satisfies the LDP on ${[0,D]^{(k+1)\times k}}$.
To conclude the proof, we only need to show that $T_k(\rho(\sigma_n),\mu(\sigma_n))$ has the same distribution as $(\rho_k(\hat\sigma_n),\mu_k(\hat\sigma_n))$. But this follows immediately from the facts that $t_k(\sigma_n)$ has the same distribution as $\hat\sigma_n$ and that $T_k(\rho(\sigma_n),\mu(\sigma_n))$ depends only on the total mass in the intervals $\Delta_{k,i}$, so that $T_k(\rho(\sigma_n),\mu(\sigma_n))=(\rho_k(t_k(\sigma_n)),\mu_k(t_k(\sigma_n)))$.
### Proof of Theorem \[theorem:LDconvergenceEquivalence\], Part 1: LD-convergence implies $k$-LD-convergence
In view of the last lemma, it seems quite intuitive that LD-convergence should imply $k$-LD-convergence, given that the set of measures ${{\bf \mathcal{N}}}^1_k\times {{\bf \mathcal{N}}}^2_k$ is much smaller than the set of measures ${{\bf \mathcal{M}}}^1\times {{\bf \mathcal{M}}}^2$. More formally, one might hope to define a continuous map from ${{\bf \mathcal{M}}}^1\times {{\bf \mathcal{M}}}^2$ to ${{\bf \mathcal{N}}}^1_k\times {{\bf \mathcal{N}}}^2_k$ (or to ${[0,D]^{(k+1)\times k}}$) and use the Contraction Principle to prove the desired $k$-LD-convergence.
It turns out that we can’t quite do that due to point masses on points of the form $i/k, 1\le i\le k$. To address this issue, we will first prove that we can restrict ourselves to ${{\bf \mathcal{M}}}_k^1\times{{\bf \mathcal{M}}}_k^2$, and then define a suitable continuous function from ${{\bf \mathcal{M}}}_k^1\times{{\bf \mathcal{M}}}_k^2$ to ${[0,D]^{(k+1)\times k}}$ to apply the Contraction Principle. We need the following lemma.
\[lemma:PositiveMassMeasure\] Suppose ${\mathbb{G}}_n$ is LD-convergent with rate function $I$. Suppose $\rho$ is such that $\rho(\{x\})>0$ for some $x\in
[0,1]$. Then for every $\mu\in{{\bf \mathcal{M}}}^2, I(\rho,\mu)=\infty$. Similarly, suppose $\mu$ is such that $\mu(\{x\}\times
[0,1])>0$ or $\mu([0,1]\times \{x\})>0$ for some $x\in [0,1]$. Then for every $\rho\in{{\bf \mathcal{M}}}^1, I(\rho,\mu)=\infty$.
Suppose $\rho$ is such that $\rho(\{x\})=\alpha>0$ for some $x\in [0,1]$. Fix $\delta<\alpha/2$. Suppose $\rho'$ is such that $d(\rho,\rho')\le \delta$. Then, by definition of the Prokhorov metric, $$\begin{aligned}
\alpha=\rho(\{x\})\le \rho'(x-\delta,x+\delta)+\delta,\end{aligned}$$ implying $\rho'(x-\delta,x+\delta)\ge \alpha/2$. Thus the event $(\rho(\sigma_n),\mu(\sigma_n))\in B((\rho,\mu),\delta)$ implies the event $\rho(\sigma_n)(x-\delta,x+\delta)\ge
\alpha/2$. We now estimate the probability of this event. For this event to occur we need to have at least $\alpha/2$ fraction of values $\sigma_n(u), u\in V_n$ to fall into the interval $(x-\delta,x+\delta)$. This occurs with probability at most $$\begin{aligned}
\sum_{i\ge (\alpha/2)|V_n|} {|V_n| \choose i}(2\delta)^{i}\le 2^{|V_n|}(2\delta)^{(\alpha/2)|V_n|}.\end{aligned}$$ Therefore $$\begin{aligned}
\limsup_n |V_n|^{-1}{\mathbb{P}}\left(\rho(\sigma_n)(x-\delta,x+\delta)\ge \alpha/2\right)\le \log 2+(\alpha/2)\log(2\delta),\end{aligned}$$ implying $$\begin{aligned}
\lim_{\delta\rightarrow 0}\limsup_n |V_n|^{-1}{\mathbb{P}}\left((\rho(\sigma_n),\mu(\sigma_n))\in B((\rho,\mu),\delta) \right)=-\infty,\end{aligned}$$ and thus $I(\rho,\mu)=\infty$ utilizing property (\[eq:Identity1\]). The proof for the case $\mu(\{x\}\times
[0,1])>0$ or $\mu([0,1]\times \{x\})>0$ is similar.
We now establish that LD-convergence implies $k$-LD-convergence. To this end, we first observe that for an i.i.d. uniformly random coloring $\sigma_n:V_n\to [0,1]$, we have that $\rho(\sigma_n)\in {{\bf \mathcal{M}}}^1_k$ and $\mu(\sigma_n)\in{{\bf \mathcal{M}}}^2_k$ almost surely (as the probability of hitting one of the points $i/k$ is zero).
Next we claim that since $(\rho(\sigma_n),\mu(\sigma_n))$ satisfies the LDP in ${{\bf \mathcal{M}}}^1\times {{\bf \mathcal{M}}}^2$, it also does so in ${{\bf \mathcal{M}}}_k^1\times {{\bf \mathcal{M}}}_k^2$ with the same rate function. For this we will show that (\[eq:LDprinciple\]) holds when closures and interiors are taken with respect to the metric space ${{\bf \mathcal{M}}}_k^1\times {{\bf \mathcal{M}}}_k^2$ as opposed to ${{\bf \mathcal{M}}}^1\times {{\bf \mathcal{M}}}^2$. Indeed, fix any set $A\subset{{\bf \mathcal{M}}}_k^1\times {{\bf \mathcal{M}}}_k^2$. Its closure in ${{\bf \mathcal{M}}}_k^1\times {{\bf \mathcal{M}}}_k^2$ is a subset of its closure in ${{\bf \mathcal{M}}}^1\times
{{\bf \mathcal{M}}}^2$ and the set-theoretic difference between the two consists of measures $(\rho,\mu)$ such that $\rho$ assigns a positive mass to some point $i/k$, or $\mu$ assigns a positive mass to some segment $\{i/k\}\times [0,1]$ or segment $[0,1]\times
\{j/k\}$. By Lemma \[lemma:PositiveMassMeasure\] the large deviations rate $I$ at these points $(\rho,\mu)$ is infinite, so that the value $\inf I(\rho,\mu)$ over the closure of $A$ in ${{\bf \mathcal{M}}}_k^1\times {{\bf \mathcal{M}}}_k^2$ and ${{\bf \mathcal{M}}}^1\times {{\bf \mathcal{M}}}^2$ is the same. Similarly, the interior of every set $A\subset {{\bf \mathcal{M}}}_k^1\times
{{\bf \mathcal{M}}}_k^2$ is a superset of its interior in ${{\bf \mathcal{M}}}^1\times {{\bf \mathcal{M}}}^2$, and again the set theoretic difference between the two interiors consists of points with infinite rate $I$. This proves the claim.
Consider $\phi_k:{{\bf \mathcal{M}}}_k^1\times {{\bf \mathcal{M}}}_k^2\rightarrow {[0,D]^{(k+1)\times k}}$, where $\phi_k$ maps every pair of measures $\rho,\mu$ into total measure assigned to interval $\Delta_{i,k}$ by $\rho$ and assigned to rectangle $\Delta_{i,k}\times \Delta_{j,k}$ by $\mu$, for $i,j=0,1,\ldots,k-1$. Namely $\phi_k(\rho,\mu)=(x,X)$ where $(x,X)$ is given by .
We claim that $\phi_k$ is continuous with respect to the respective metrics. Indeed, suppose $(\rho_n,\mu_n)\rightarrow (\rho,\mu)$ in ${{\bf \mathcal{M}}}_k^1\times {{\bf \mathcal{M}}}_k^2$. Since ${{\bf \mathcal{M}}}^d$ is equipped with the weak topology, then for every closed (open) set $A\subset
[0,1]$, we have $\limsup_n\rho_n(A)\le \rho(A)
~~(\liminf_n\rho_n(A)\ge \rho(A))$. The same applies to $\mu_n,\mu$. Setting $A$ to be $[i/k,(i+1)/k]$ first and then $(i/k,(i+1)/k)$, and using the fact that $\rho_n$ and $\rho$ are supported on $O_k^1$, namely $\rho_n(i/k)=\rho(i/k)=0$ for all $i$, we obtain $\rho_n(i/k,(i+1)/k)\rightarrow
\rho(i/k,(i+1)/k)$. A similar argument applies to $\mu_n,\mu$. This proves the claim.
Since $\phi_k$ is continuous, by the Contraction Principle [@demzei98], the image of $(\rho(\sigma_n),\mu(\sigma_n))$ under $\phi_k$ satisfies the LDP as well. But this means precisely that we have $k$-LD-convergence with rate function $I_k$ as defined in Remark \[cor:Ik-from-I\] (strictly speaking, the Contraction Principle gives a rate function where the $\inf$ is taken over all $(\rho,\mu)\in {{\bf \mathcal{M}}}_k^1\times {{\bf \mathcal{M}}}_k^2$, but by Lemma \[lemma:PositiveMassMeasure\] we can extend the $\inf$ to that larger set ${{\bf \mathcal{M}}}^1\times {{\bf \mathcal{M}}}^2$ without changing the value of $I_k$).
### Proof of Theorem \[theorem:LDconvergenceEquivalence\], Part 2: $k$-LD-convergence implies LD-convergence
In order to prove that $k$-LD-convergence for all $k$ implies LD-convergence, we would like to use the rate functions $I_k:{{\bf \mathcal{N}}}_k^1\times{{\bf \mathcal{N}}}_k^2\to {\mathbb{R}}_+\cup\{\infty\}$ associated with the LDP for $T_k(\rho(\sigma_n),\mu(\sigma_n))$ from Lemma \[lemma:GraphConvergenceLDk\] to construct a suitable rate function $I:{{\bf \mathcal{M}}}^1\times{{\bf \mathcal{M}}}^2\to {\mathbb{R}}_+\cup\{\infty\}$ for the random variables $(\rho(\sigma_n),\mu(\sigma_n))$. The existence of such a rate function will follow from a suitable monotonicity property, see Lemma \[lemma:Ikkhat\] and Corollary \[cor:Ikhat\] below.
Before stating Lemma \[lemma:Ikkhat\], we state and prove a more technical lemma which we will use at several places in this subsection. For two pairs of measures $(\rho,\mu)),(\rho',\mu')\in{{\bf \mathcal{N}}}_k^1\times {{\bf \mathcal{N}}}_k^2$ we define $$d_{var}((\rho,\mu)),(\rho',\mu'))=
\max
\Bigl\{
\max_{A\subset [0,1]}\left|\rho(A)-\rho'(A)\right|,
\max_{B\subset [0,1]^2}
\left|\mu(B)-\mu'(B)\right|
\Bigr\}.$$
\[lem:Tk-1/k-bound\]
1\) If $(\rho,\mu)\in {{\bf \mathcal{M}}}^1_k\times {{\bf \mathcal{M}}}^2_k$, then $$\label{tk-1/k-bound}
d((\rho,\mu),T_k(\rho,\mu))\leq \frac 1k.$$
2\) If $(\rho,\mu),(\rho',\mu')\in{{\bf \mathcal{N}}}_k^1\times {{\bf \mathcal{N}}}_k^2$, then $$\label{dvar-dProc}
d((\rho,\mu),(\rho',\mu'))
\leq
d_{var}((\rho,\mu)),(\rho',\mu'))\leq (4kD+1) d((\rho,\mu),(\rho',\mu')).$$
3\) If $(\rho,\mu),(\rho',\mu')\in{{\bf \mathcal{N}}}_{2k}^1\times {{\bf \mathcal{N}}}_{2k}^2$, then $$\label{dvar-Tk}
d_{var}(T_k(\rho,\mu)),T_k(\rho',\mu'))
\leq
d_{var}((\rho,\mu)),(\rho',\mu')).$$
1\) Set $(\rho_k,\mu_k)=T_k(\rho,\mu)$, and let $A\subset
[0,1]$. Define $A_k$ to be the set that is obtained by replacing every non-empty set $A\cap \Delta_{i,k}$ with the entire interval $\Delta_{i,k}$. Then $A\subset A_k\subset
A^{1\over k}$. We have $\rho(A\cap \Delta_{i,k})\le
\rho(\Delta_{i,k})=\rho_k(\Delta_{i,k})$ by the definition of $\rho_k$. Let $\alpha=\{i: A\cap \Delta_{i,k}\ne\emptyset\}$. Then $$\rho(A)=\sum_{i\in \alpha}\rho(A\cap \Delta_{i,k})
\le\sum_{i\in \alpha}\rho_k(\Delta_{i,k})
=\rho_k(A_k)
\le\rho_k(A^{1\over k}).$$ Conversely, $$\rho_k(A)\le \rho_k(A_k)=\rho(A_k)\le \rho(A^{1\over k}).$$ A similar argument is used for $\mu$.
2\) The lower bound follows immediately from the definitions of $d$ and $d_{var}$. To prove the upper bound, we first show that $$\max_{A\subset [0,1]}\left|\rho(A)-\rho'(A)\right|\leq (2kD+1)d(\rho,\rho').$$ To this end, we note the maximum is obtained when $A$ is of the form $A=\bigcup_{i\in \alpha}\Delta_{i,k}$ for some $\alpha\subset \{0,\dots,k-1\}$. On the other hand, for sets $A$ of this form, we have that $\rho(A^\epsilon)\leq\rho(A)+2k\epsilon\rho(A^c)\leq
\rho(A)+2k\epsilon D$, a bound which follows from the fact that $A^c=[0,1]\setminus A$ is a union of intervals of length at least $1/k$. Choosing $\epsilon=d(\rho,\rho')$ we then have $\rho(A')\leq \rho(A^\epsilon)+\epsilon\leq \rho(A)
+(2kD+1)\epsilon$, which implies $\rho'(A)-\rho(A)\leq
(2kD+1)\epsilon=(2kD+1)d(\rho,\rho')$. Exchanging the roles of $\rho$ and $\rho'$ gives $\rho(A)-\rho'(A)\leq
(2kD+1)d(\rho,\rho')$, which completes the claim. In a similar way, one proves that $$\max_{B\subset [0,1]^2}\left|\mu(B)-\mu'(B)\right|\leq (4kD+1)d(\mu,\mu').$$ The proof uses that for sets $B$ which are unions of sets of the form $\Delta_{i,k}\times\Delta_{j,k}$ we have $\mu(B^{\epsilon})\leq\mu(B)+4k\epsilon D$.
3\) Let $(\rho_k,\mu_k)=T_k(\rho,\mu)$ and $(\rho_k^\prime,\mu_k^\prime)=T_k(\rho',\mu')$. We need to show that $d_{var}(\rho_k,\rho_k^\prime)\leq
d_{var}(\rho,\rho^\prime)$ and $d_{var}(\mu_k,\mu_k^\prime)\leq
d_{var}(\mu,\mu^\prime)$. The first bound follows from the definition of $T_k$ and the fact that the maximum in the definition $ d_{var}(\rho_k,\rho_k^\prime)= \max_{A\subset
[0,1]}|\rho_k(A)-\rho_k^\prime(A)| $ is obtained when $A$ is of the form $A=\bigcup_{i\in \alpha}\Delta_{i,k}$ for some $\alpha\subset\{0,\dots,k-1\}$. Indeed, for such an $A$, one easily shows that $|\rho_k(A)-\rho_k^\prime(A)|=|\rho(A)-\rho^\prime(A)|$, which in turn implies that $d_{var}(\rho_k,\rho_k^\prime)\leq
d_{var}(\rho,\rho^\prime)$. The proof of the bound $d_{var}(\mu_k,\mu_k^\prime)\leq d_{var}(\mu,\mu^\prime)$ is similar.
\[lemma:Ikkhat\] Suppose ${\mathbb{G}}_n$ is $k$-LD-convergent, and let $I_k:{{\bf \mathcal{N}}}_k^1\times{{\bf \mathcal{N}}}_k^2\to {\mathbb{R}}_+\cup\{\infty\}$ be the rate function associated with the LDP for $T_k(\rho(\sigma_n),\mu(\sigma_n))$. Then $I_{k}(T_k(\rho,\mu))\le I_{2k}(\rho,\mu)$ for every $(\rho,\mu)\in{{\bf \mathcal{N}}}^1_{2k}\times {{\bf \mathcal{N}}}^2_{2k}$.
Set $(\rho_k,\mu_k)=T_k(\rho,\mu)$, fix $\epsilon>0$, and set $\epsilon'=(4kD+1)\epsilon$. By the bounds and , we have that $$\label{in-B-eps}
(\rho_k^\prime,\mu_k^\prime)=T_{k}(\rho',\mu')\in B(T_k(\rho,\mu),\epsilon')$$ whenever $(\rho',\mu')\in B((\rho,\mu),\epsilon)$. On the other hand, $T_k(\rho(\sigma_n),\mu(\sigma_n))=
T_k(T_{2k}(\rho(\sigma_n),\mu(\sigma_n)))$. By , this implies that $$\begin{aligned}
{\mathbb{P}}\Big( T_k(\rho(\sigma_n),\mu(\sigma_n))\in B\Bigl(T_k(\rho,\mu),(4kD+1)\epsilon\Bigr)\Big)
\geq
{\mathbb{P}}\Big( T_{2k}(\rho(\sigma_n),\mu(\sigma_n))\in B\Bigl((\rho,\mu),\epsilon\Bigr)\Big).\end{aligned}$$ Applying we obtain the claim of the lemma.
\[cor:Ikhat\] Suppose ${\mathbb{G}}_n$ is $k$-LD-convergent for all $k$, let $I_k:{{\bf \mathcal{N}}}_k^1\times{{\bf \mathcal{N}}}_k^2\to {\mathbb{R}}_+\cup\{\infty\}$ be the rate function associated with the LDP for $T_k(\rho(\sigma_n),\mu(\sigma_n))$, let $(\rho,\mu)\in
{{\bf \mathcal{M}}}^1\times{{\bf \mathcal{M}}}^2$, and let $B_k$ be the ball $B((\rho,\mu),2k^{-1})\cap {{\bf \mathcal{N}}}^1_{k}\times{{\bf \mathcal{N}}}^2_{k}$. Then the limit $$I(\rho,\mu)=\lim_{\ell\to\infty}\inf_{(\rho',\mu')\in B_{2^\ell}}I_{2^\ell}(\rho',\mu')$$ exists.
Let $k\in{\mathbb{Z}}_+$. By Lemma \[lem:Tk-1/k-bound\], we have that $T_k(\rho',\mu')\in B_k$ whenever $(\rho',\mu')\in B_{2k}$. Combined with Lemma \[lemma:Ikkhat\], this proves that $$\inf_{(\rho',\mu')\in B_{2k}}I_{2k}(\rho',\mu')\geq
\inf_{(\rho',\mu')\in B_{2k}}I_{k}(T_k(\rho',\mu'))
\geq
\inf_{(\rho'',\mu'')\in B_{k}}I_{k}(T_k(\rho'',\mu'')).$$ The claim now follows by monotonicity.
We are now ready to prove that $k$-LD-convergence for all $k$ implies LD-convergence. Let $\sigma_n:V_n\to [0,1]$ be an i.i.d. uniformly random coloring of $V_n$. Recalling Lemma \[lemma:GraphConvergenceLDk\], we will assume that $T_k(\rho(\sigma_n),\mu(\sigma_n))$ obeys a LDP with some rate function $I_k$. We will want to prove that $(\rho(\sigma_n),\mu(\sigma_n))$ obeys a LDP with the rate function $I$ defined in Corollary \[cor:Ikhat\].
Fix $(\rho,\mu)\in {{\bf \mathcal{M}}}^1\times {{\bf \mathcal{M}}}^2$. By the Lemma \[lem:Tk-1/k-bound\], the fact that $(\rho(\sigma_n),\mu(\sigma_n))\in{{\bf \mathcal{M}}}_{k}^1\times {{\bf \mathcal{M}}}^2_{k}$ almost surely, and the triangle inequality, we have that $$\begin{aligned}
\liminf_n |V_n|^{-1}&\log{\mathbb{P}}\left((\rho(\sigma_n),\mu(\sigma_n))\in B((\rho,\mu),3k^{-1}\right)\\
&\ge \liminf_n |V_n|^{-1}\log{\mathbb{P}}\left(T_{k}(\rho(\sigma_n),\mu(\sigma_n))\in B((\rho,\mu),2k^{-1})\right)\\
&\ge-\inf I_{k}(\rho',\mu'),\end{aligned}$$ where the infimum is taken over $(\rho',\mu')\in
B((\rho,\mu),2k^{-1})\cap {{\bf \mathcal{N}}}^1_{k}\times {{\bf \mathcal{N}}}^2_{k}$. Similarly, $$\begin{aligned}
\limsup_n |V_n|^{-1}&\log{\mathbb{P}}\left((\rho(\sigma_n),\mu(\sigma_n))\in B((\rho,\mu),k^{-1})\right)\\
&\le \limsup_n |V_n|^{-1}\log{\mathbb{P}}\left(T_{k}(\rho(\sigma_n),\mu(\sigma_n))\in B((\rho,\mu),2k^{-1})\right)\\
&\leq-\inf I_k(\rho',\mu'),\end{aligned}$$ where the infimum is again taken over $(\rho',\mu')\in
B((\rho,\mu),2k^{-1})\cap {{\bf \mathcal{N}}}^1_{k}\times {{\bf \mathcal{N}}}^2_{k}$.
To complete the proof we apply Proposition \[prop:LDReverse\] in conjunction with Corollary \[cor:Ikhat\], yielding that $(\rho(\sigma_n),\mu(\sigma_n))$ obeys a LDP with rate function $I$.
Basic properties and examples
-----------------------------
We begin by showing that the definition of LD-convergence is robust with respect to the equivalency relationship $\sim$ on graphs.
\[theorem:LDrobust\] If ${\mathbb{G}}_n$ is LD-convergent and $\tilde{\mathbb{G}}_n\sim{\mathbb{G}}_n$, then $\tilde{\mathbb{G}}_n$ is also LD-convergent.
We apply Theorem \[theorem:LDconvergenceEquivalence\] and show that if ${\mathbb{G}}_n$ is $k$-LD-convergent then $\tilde{\mathbb{G}}_n$ is also $k$-LD-convergent. Fix any $k$, any $(x,X)\in{[0,D]^{(k+1)\times k}}$ and $\epsilon>0$. We denote by $\tilde{\mathbb{P}}$ the probability measure associated with $( \tilde x(\sigma_n), \tilde X(\sigma_n))= \tilde{\mathbb{G}}_n/\sigma_n$ when $\sigma_n:V_n\rightarrow [k]$ is a random uniformly chosen map on $V_n$. The definition of $\sim$ implies that for all sufficiently large $n$ $$\begin{aligned}
\tilde{\mathbb{P}}\Big((\tilde x(\sigma_n),\tilde X(\sigma_n))\in B\left((x,X),\epsilon/2\right)\Big)\le
{\mathbb{P}}\Big((x(\sigma_n),X(\sigma_n))\in B\left((x,X),\epsilon\right)\Big),\end{aligned}$$ implying $$\begin{aligned}
\lim_{\epsilon\rightarrow 0}\limsup_n |V_n|^{-1}&\log\tilde{\mathbb{P}}\Big((\tilde x(\sigma_n),\tilde X(\sigma_n))\in B\left((x,X),\epsilon/2\right)\Big) \\
&\le \lim_{\epsilon\rightarrow 0}\limsup_n |V_n|^{-1}\log{\mathbb{P}}\Big((x(\sigma_n),X(\sigma_n))\in B\left((x,X),\epsilon\right)\Big).\end{aligned}$$ Similarly, we establish that $$\begin{aligned}
\lim_{\epsilon\rightarrow 0}\liminf_n |V_n|^{-1}&\log\tilde{\mathbb{P}}\Big((\tilde x(\sigma_n),\tilde X(\sigma_n))\in B\left((x,X),\epsilon\right)\Big) \\
&\ge \lim_{\epsilon\rightarrow 0}\liminf_n |V_n|^{-1}\log{\mathbb{P}}\Big((x(\sigma_n),X(\sigma_n))\in B\left((x,X),\epsilon/2\right)\Big).\end{aligned}$$ Since ${\mathbb{G}}_n$ is $k$-LD-convergent, then from (\[eq:Identity1\]) we have $$\begin{aligned}
\lim_{\epsilon\rightarrow 0}\limsup_n |V_n|^{-1}\log\tilde{\mathbb{P}}(\cdot)=\lim_{\epsilon\rightarrow 0}\liminf_n |V_n|^{-1}\log\tilde{\mathbb{P}}(\cdot).\end{aligned}$$ Thus the same identity applies to $\tilde{\mathbb{P}}$. Applying Proposition \[prop:LDReverse\] we conclude that $\tilde{\mathbb{P}}$ is LD-convergent.
Let us give some examples of LD-convergent graph sequences. As our first example, consider a sequence of graphs which consists of a disjoint union of copies of a fixed graph.
\[theorem:UnionConverging\] Let ${\mathbb{G}}_0$ be a fixed graph and let ${\mathbb{G}}_n$ be a disjoint union of $n$ copies of ${\mathbb{G}}_0$. Then ${\mathbb{G}}_n$ is LD-convergent.
We use Theorem \[theorem:LDconvergenceEquivalence\] and prove that ${\mathbb{G}}_n$ is $k$-LD-convergent for every $k$. Fix $k$, let $\Sigma_k$ be the space of all possible $k$-colorings of ${\mathbb{G}}_0$, and let $M=k^{|V({\mathbb{G}}_0)|}$ denote the size of $\Sigma_k$. Every $k$-coloring $\sigma_n:V_n\rightarrow [k]$ of the nodes of ${\mathbb{G}}_n$ can then be encoded as $\sigma_n=(\sigma_i^n)_{1\le i\le n}$, where $\sigma_i^n\in \Sigma_k$ is a $k$-coloring of the $i$-th copy of ${\mathbb{G}}_0$ in ${\mathbb{G}}_n$. Consider the $M$-dimensional simplex $S_M$, i.e., the set of vectors $(z_1,\ldots,z_{M})$ such that $\sum_{i\le M}z_i=1,
z_i\ge 0$. For every $m=1,\ldots,M$ and $\sigma_n=(\sigma_i^n)_{1\le i\le n}$, let $z_m(\sigma_n)$ be the number of $\sigma_i^n$-s which are equal to the $m$-element of $\Sigma_k$ divided by $n$, and let $z(\sigma_n)=(z_m(\sigma_n))_{ 1\le m\le M}\in
S_{M}$. By Sanoff’s Theorem, the sequence $z(\sigma_n),
n\ge 1$ satisfies the LDP with respect to the metric space $S_{M}$. Now consider a natural mapping from $S_{M}$ into ${[0,D]^{(k+1)\times k}}$, where each $z=(z_m)_{1\leq m\le M}$ is mapped into $(x,X)\in {[0,D]^{(k+1)\times k}}$ as follows. For each element of $\Sigma_k$ encoded by some $m\le M$ and each $i\le k$, let $x(i,m)$ be the number of nodes of ${\mathbb{G}}_0$ colored $i$ according to $m$, divided by $|V({\mathbb{G}}_0)|$. In particular $\sum_ix(i,m)=1$. Similarly, let $X(i,j,m)$ be $2$ times the number of edges in ${\mathbb{G}}_0$ with end node colors $i$ and $j$ according to $m$, divided by $|V({\mathbb{G}}_0)|$. In particular $\sum_{i,j}X(i,j,m)=2|E({\mathbb{G}}_0)|/|V({\mathbb{G}}_0)|$. Setting $x_i=\sum_m x(i,m)z_m, X_{i,j}=\sum_m X(i,j,m)z_m$, this defines the mapping $S_{M}$ into ${[0,D]^{(k+1)\times k}}$. This mapping is continuous. Observe that the composition $\sigma_n\rightarrow
z(\sigma_n)\rightarrow (x,X)$ is precisely the construction of the factor graph ${\mathbb{F}}={\mathbb{G}}_n/\sigma_n$ via (\[k-quotient\]). Applying the Contraction Principle, we obtain LD-convergence of the sequence ${\mathbb{G}}_n$.
Perhaps a more interesting example is the case of subgraphs of the $d$-dimensional lattice ${\mathbb{Z}}^d$, more precisely the case of graph sequences $L_{d,n}$ with vertex sets $V_{d,n}=\{-n,-n+1,\dots, n\}^d$. It is known that the free energy of statistical mechanics systems on these graphs has a limit [@SimonLatticeGases], [@GeorgyGibbsMeasure]. In our terminology this means that the sequence $(L_{d,n})$ is right-convergent (see Section \[section:OtherConvergence\]). Here we show that this sequence is also LD-convergent.
\[theorem:Lattice\] Let $d\geq 1$, let $V_{d,n}=\{-n,-n+1,\dots, n\}^d$, let $E_{n,d}$ be the set of pairs $\{x,y\}\subset V_{d,n}$ of $\ell_1$ distance $1$, and let $L_{d,n}=(V_{d,n},E_{d,n})$. Then the sequence $(L_{d,n})$ is LD-convergent.
We again apply Theorem \[theorem:LDconvergenceEquivalence\] and establish $k$-LD-convergence for every $k$. Fix $(x,X)\in
{[0,D]^{(k+1)\times k}}$ and let $$\begin{aligned}
\bar I_k(x)=-\lim_{\delta\rightarrow 0}\limsup_n {\log {\mathbb{P}}\Bigl((x(\sigma_n),X(\sigma_n))\in B((x,X),\delta)\Bigr)\over |V_{d,n}|},\end{aligned}$$ where $\sigma_n:V_{d,n}\rightarrow [k]$ is chosen uniformly at random. The limit $\lim_{\delta\rightarrow 0}$ exists by monotonicity. Fix an arbitrary $\delta>0$. Then again by monotonicity $$\begin{aligned}
\limsup_n {\log {\mathbb{P}}\Bigl((x(\sigma_n),X(\sigma_n))\in B((x,X),\delta/2)\Bigr)\over |V_{d,n}|}\ge -\bar I_k(x,X),\end{aligned}$$ and we can find $n_0$ so that $$\begin{aligned}
\label{eq:barIkdelta}
{\log {\mathbb{P}}\Big((x(\sigma_{n_0}),X(\sigma_{n_0}))\in B((x,X),\delta/2)\Bigr)\over |V_{d,n_0}|}\ge -\bar I_k(x,X)-\delta.\end{aligned}$$ Consider arbitrary $n\ge n_0$, and set $q=\lfloor\frac{2n+1}{2n_0+1}\rfloor$, $M=q^d$ and $m=(2n+1)^d-M(2n_0+1)^d$. The set of vertices $V_{d,n}$ can then be written as a disjoint union of $M$ copies of $V_{d,n_0}$ and $m$ sets consisting of one node each. Let ${\mathbb{H}}_1,\ldots,{\mathbb{H}}_M$ be the induced subgraphs on the copies of $V_{d,n_0}$, and let $\tilde L_{d,n}=(\tilde V_{d,n},\tilde
E_{d,n})$ be the union of ${\mathbb{H}}_1,\ldots,{\mathbb{H}}_M$. Then $$|\tilde V_{d,n}|=|V_{d,n}|-m =|V_{d,n}|(1-O(n_0/n))$$ and $$|\tilde E_{d,n}|=
|E_{d,n}|-O(Mn_0^{d-1})- O(m)
=|E_{d,n}|-O(n^d n_0^{-1})-O(n^{d-1}n_0).$$
A uniformly random coloring $\sigma_n:V_{d,n}\rightarrow [k]$ then induces a uniformly random coloring $\tilde
\sigma_n:\tilde V_{d,n}\to [k]$. Due to the above bounds, the corresponding quotients $L_{d,n}/\sigma_n=(x(\sigma_n),X(\sigma_n))$ and $\tilde
L_{d,n}/\tilde \sigma_n=(\tilde x(\tilde\sigma_n),\tilde
X(\tilde\sigma_n))$ are close to each other. More precisely, $$\begin{aligned}
d\Bigl((x(\sigma_n),X(\sigma_n))\,,\,(\tilde x(\tilde \sigma_n),\tilde X(\tilde\sigma_n))\Bigr)= O(n_0/n)+O(1/n_0).\end{aligned}$$ Then we can find $n_0$ large enough so that (\[eq:barIkdelta\]) still holds, such that for all large enough $n$, $$\begin{aligned}
d\Bigl((x(\sigma_n),X(\sigma_n))\,,\,(\tilde x(\tilde \sigma_n),\tilde X(\tilde\sigma_n))\Bigr)\le \delta/2.\end{aligned}$$ Then $$\begin{aligned}
{\mathbb{P}}\left((x(\sigma_n),X(\sigma_n)\in B((x,X),\delta))\right)
\ge {\mathbb{P}}\left((\tilde x(\tilde \sigma_n),\tilde X(\tilde \sigma_n))\in B((x,X),\delta/2)\right).\end{aligned}$$ On the other hand, since $\tilde X_{i,j}(\tilde\sigma_n)$ are counted over a disjoint union of graphs identical to $L_{d,n_0}$, then if $\tilde\sigma_n$ is such that within each graph ${\mathbb{H}}_r$ we have $d((x,X), (\tilde x(\tilde
\sigma_n),\tilde X(\tilde\sigma_n)))\le \delta/2$, then the same applies to the overall graph $\tilde L_{d,n}$. Namely, $$\begin{aligned}
{\mathbb{P}}\left((\tilde x(\tilde\sigma_n),\tilde X(\tilde\sigma_n))\in B(x,\delta/2)\right)
\ge
\Big({\mathbb{P}}\left((x(\sigma_{n_0}),X(\sigma_{n_0}))\in B(x,X,\delta/2)\right)\Big)^{ M},\end{aligned}$$ where the second expectation is with respect to uniformly random coloring of $L_{d,n_0}$. Since $M\le
|V_{d,n}|/|V_{d,n_0}|$, we obtain $$\begin{aligned}
{1\over |V_{d,n}|}\log {\mathbb{P}}\Bigl(x({\sigma_n}), X(\sigma_n))\in B(x,\delta)\Bigr)
&\ge {1\over |V_{d,n_0}|}\log{\mathbb{P}}\Bigl((x(\sigma_{n_0}),X(\sigma_{n_0}))\in B(x,X,\delta/2)\Bigr)\\
&\ge -\bar I_k(x,X)-\delta,\end{aligned}$$ where the second inequality follows from (\[eq:barIkdelta\]). Since this holds for all large enough $n$, then from the bound above we obtain $$\begin{aligned}
\lim_{\delta\rightarrow 0}\liminf_n{1\over |V_{d,n}|}\log {\mathbb{P}}\Bigl((x(\sigma_n),X(\sigma_n))\in B((x,X),\delta)\Bigr)\ge -\bar I_k(x,X).\end{aligned}$$ We conclude that the relation (\[eq:Identity1\]) holds.
Other notions of convergence and comparison with LD-convergence {#section:OtherConvergence}
===============================================================
In this section we consider other types of convergence discussed in the earlier literature and compare them with LD-convergence. Later we will show that every other mode of convergence, except for colored-neighborhood-convergence, is implied by LD-convergence. We begin with the notion of left-convergence.
Left-convergence
----------------
We start with the definition of left-convergence as introduced in [@BorgsChayesKahnLovasz]:
\[defi:leftGraph-converging\] A sequence of graphs ${\mathbb{G}}_n$ with uniformly bounded degrees is left-convergent if the limit $$\begin{aligned}
\label{eq:Left-limit}
\lim_{n\rightarrow\infty}|V_n|^{-1}\hom({\mathbb{H}},{\mathbb{G}}_n)\end{aligned}$$ exists for every connected, finite graph ${\mathbb{H}}$.
As already noted in [@BorgsChayesKahnLovasz], this notion is equivalent to the notion of Benjamini-Schramm convergence of the sequence ${\mathbb{G}}_n=(V_n,E_n)$. To define the latter, consider a fixed positive integer $r$, and a rooted graph ${\mathbb{H}}$ of diameter at most $r$. Define $p_n({\mathbb{H}},r)$ to be the probability that the $r$-ball around a vertex $U$ chosen uniformly at random from $V_n$ is isomorphic to ${\mathbb{H}}$. The sequence is called Benjamini-Schramm convergent if the limit $p({\mathbb{H}},r)=\lim_n p_n({\mathbb{H}},r)$ exists for all $r$ and all routed graphs ${\mathbb{H}}$ of diameter $r$ or less.
Recall our definition of relation $\sim$ between graph sequences. It is immediate that if ${\mathbb{G}}_n$ is left-convergent and $\tilde{\mathbb{G}}_n\sim{\mathbb{G}}_n$ then $\tilde{\mathbb{G}}_n$ is also left-convergent with the same set of limits $p({\mathbb{H}},r)$. Thus left-convergence can be defined on the equivalency classes of graph sequences.
Right-convergence {#subsection:right-convergence}
-----------------
For sequences of bounded degree graphs, the notion of right-convergence for a fixed target graph ${\mathbb{H}}$ was defined in [@BorgsChayesKahnLovasz]. In this paper, we define right-convergence without referring to a particular target graph, and require instead the existence of limits for all soft-core graphs ${\mathbb{H}}$. Recall that for a weighted graph ${\mathbb{H}}$ with vertex and edge weights given by the vector $\alpha=(\alpha_1({\mathbb{H}}),\dots,\alpha_k({\mathbb{H}}))$ and the matrix $A=(A_{ij}({\mathbb{H}}))_{1\leq i,j\leq k}$, respectively, the homomorphism number $\hom({\mathbb{G}},{\mathbb{H}})$ is given by (\[eq:WeightedHom\]). Without loss of generality we assume $\alpha_i({\mathbb{H}})>0$ for all $i$. We say that ${\mathbb{H}}$ is soft-core if also $A_{ij}({\mathbb{H}})>0$ for all $i,j$.
To motivate our definition of right-convergence, let us note that for a soft-core graph ${\mathbb{H}}$ on $k$ nodes, $$\alpha_{\min}^{(D+1)|V_n|}
\leq
\alpha_{\min}^{|V_n|+|E_n|}
\leq
k^{-|V_n|}\hom({\mathbb{G}}_n,{\mathbb{H}})
\leq
\alpha_{\max}^{|V_n|+|E_n|}
\leq
\alpha_{\max}^{(D+1)|V_n|}$$ where $$\label{eq:alphamax}
\alpha_{\max}=\max\left(1,\max\alpha_i,\max A_{ij}\right)
\qquad\text{and}\qquad
\alpha_{\min}=\min\left(1,\min\alpha_i,\min A_{ij}\right).$$ Thus $\log \hom({\mathbb{G}}_n,{\mathbb{H}})$ grows linearly with $|V_n|$. We will define a sequence to be right-convergent if the coefficient of proportionality converges for all soft-core graphs.
\[defi:right-graphs converging\] A graph sequence ${\mathbb{G}}_n$ is defined to be right-convergent if for every soft-core graph ${\mathbb{H}}$, the limit $$\begin{aligned}
\label{eq:LimitPartitionFunctin}
-f({\mathbb{H}})\triangleq \lim_{n\rightarrow\infty}{\log \hom({\mathbb{G}}_n,{\mathbb{H}})\over |V_n|},\end{aligned}$$ exists.
In the language of statistical physics, the quantity $\hom({\mathbb{G}},{\mathbb{H}})$ is usually called the partition function of the soft-core model with interaction ${\mathbb{H}}$ on ${\mathbb{G}}$, and the quantity $f({\mathbb{H}})$ is called its free energy. Sometimes we will write $f_{({\mathbb{G}}_n)}({\mathbb{H}})$ instead of $f({\mathbb{H}})$ to emphasize the dependence on the underlying graph sequence $G_n$.
Observe that our definition is robust with respect to the equivalency notion $\sim$ on graph sequences. Namely if $\tilde{\mathbb{G}}_n\sim{\mathbb{G}}_n$, then the limits (\[eq:LimitPartitionFunctin\]) are the same. Indeed, deleting or adding one edge in ${\mathbb{G}}_n$ increases the partition function by a factor at most $\max A_{ij}$ and decreases it by a factor at most $\min A_{ij}$. Thus adding/deleting $o(|V_n|)$ edges changes the partition function of a soft-core model by a factor of $\exp(o(|V_n|)$.
By contrast, the partition function of a hard-core model can be quite sensitive to adding or deleting edges, as we already demonstrated in the introduction, where we discussed the case of cycles. The same observation can be extended to the case when ${\mathbb{G}}_n$ is a $d$-dimensional cylinder or torus, again parity of $n$ determining the two-colorability property, see [@BorgsChayesKahnLovasz] for details. An even more trivial example is the following: let ${\mathbb{G}}_n$ be a graph on $n$ isolated node (no edges) when $n$ is even, and $n-2$ isolated nodes plus one edge when $n$ is odd. Then there exists $1$-coloring of ${\mathbb{G}}_n$ if and only if $n$ is even, again leading to the conclusion that ${\mathbb{G}}_n$ is not converging on all simple graphs ${\mathbb{H}}$.
We adapted Definition \[defi:right-graphs converging\] to avoid these pathological examples. Nevertheless, we still would like to be able to define the limits (\[eq:LimitPartitionFunctin\]) for hard-core graphs ${\mathbb{H}}$, and do it in a way which is insensitive to changing from ${\mathbb{G}}_n$ to some equivalent sequence $\tilde{\mathbb{G}}_n\sim{\mathbb{G}}_n$. There are two ways to achieve this which are equivalent, as we establish below. Given $\lambda>0$ and a (not necessarily soft-core graph) ${\mathbb{H}}$ with edge weights given by a matrix $A=(A_{ij}({\mathbb{H}}))$, let $A_\lambda$ be the matrix with positive entries $$(A_\lambda)_{ij}=\max\{\lambda,A_{ij}({\mathbb{H}})\}$$ and let ${\mathbb{H}}_\lambda$ be the corresponding soft-core graph.
\[theorem:RightConvergenceEquivalence\] Given a right-convergent sequence ${\mathbb{G}}_n$ and a weighted graph ${\mathbb{H}}$, the following limit exists $$\begin{aligned}
\label{eq:hepsilon}
f_{({\mathbb{G}}_n)}({\mathbb{H}})\triangleq \lim_{\lambda\rightarrow 0} f_{({\mathbb{G}}_n)}({\mathbb{H}}_\lambda).\end{aligned}$$ Furthermore, there exists a graph sequence $\tilde{\mathbb{G}}_n\sim
{\mathbb{G}}_n$ such that for every graph ${\mathbb{H}}$ $$\begin{aligned}
\label{eq:HardSoftEqual}
-f_{({\mathbb{G}}_n)}({\mathbb{H}})=\lim_n{\log\hom(\tilde{\mathbb{G}}_n,{\mathbb{H}})\over |V_n|}\ge
\sup_{\hat{\mathbb{G}}_n\sim{\mathbb{G}}_n}\limsup_n{\log\hom(\hat{\mathbb{G}}_n,{\mathbb{H}})\over |V_n|},\end{aligned}$$ where the supremum is over all graphs sequences $\hat{\mathbb{G}}_n$ which are equivalent to ${\mathbb{G}}_n$. In particular, the maximizing graph sequence $\tilde{\mathbb{G}}_n$ exists and can be chosen in such a way that the $\limsup$ is actually a limit.
The intuition of the definition above is that we define $f_{({\mathbb{G}}_n)}({\mathbb{H}})$ by slightly softening the “non-edge” requirement of zero elements of the weight matrix $A$. This turns out to be equivalent to the possibility of removing $o(|V_n|)$ edges in the underlying graph in trying to achieve the largest possible limit within the equivalency classes of graph sequences. One can further give a statistical physics interpretation of this definition as defining the free energy at zero temperature as a limit of free energies at positive temperatures.
The existence of the limit (\[eq:hepsilon\]) follows immediately by monotonicity: note that $\hom({\mathbb{G}},{\mathbb{H}})$ is monotonically non-decreasing in every element of the weight matrix $A$.
The proof of the second part is more involved, even though the intuition behind it is again simple: First, one easily shows that monotonicity in $\lambda$ and the fact that the limit is not changed when we change $o(|V_n|)$ edges implies that for every $\tilde{\mathbb{G}}_n\sim{\mathbb{G}}_n$ $$\begin{aligned}
\label{eq:limsuptildeGn}
\limsup_n{\log \hom(\tilde{\mathbb{G}}_n,{\mathbb{H}})\over |V_n|}\le -f_{({\mathbb{G}}_n)}({\mathbb{H}}).\end{aligned}$$ To prove that $f_{({\mathbb{G}}_n)}({\mathbb{H}})$ is achieved asymptotically by some $\tilde{\mathbb{G}}_n\sim {\mathbb{G}}_n$ we may assume without loss of generality that $f_{({\mathbb{G}}_n)}({\mathbb{H}})<\infty$, since otherwise the identity holds trivially. Consider a “typical configuration” $\sigma:V_n\rightarrow V({\mathbb{H}}_\lambda)$ contributing to $\hom({\mathbb{G}}_n,{\mathbb{H}}_\lambda)$, and let $E_0(\sigma)$ be the set of edges $\{u,v\}\in E_n$ such that $A_{\sigma(u),\sigma(v)}({\mathbb{H}})=0$, $$\label{E_0-def}
E_{0}(\sigma)=\{(u,v)\in E_n: A_{\sigma(u),\sigma(v)}({\mathbb{H}})=0\}.$$ For small $\lambda$, we expect the size of this set to grow slowly with $|V_n|$, since otherwise $\hom({\mathbb{G}}_n,{\mathbb{H}}_\lambda)$ would become too small to be consistent with $\lim_{\lambda\to
0}f_{({\mathbb{G}}_n)}({\mathbb{H}}_\lambda)<\infty$. We might therefore hope that we can find a subset $E_{0,n}\subset E_n$ such that (i) $|E_{0,n}|=o(|V_n|)$ and (ii) removing the set of edges $E_{0,n}$ from $E_n$ leads to a graph sequence $\tilde G_n$ such that $|V_n|^{-1}\log \hom(\tilde{\mathbb{G}}_n,{\mathbb{H}})$ is close to $-f_{({\mathbb{G}}_n)}({\mathbb{H}})$.
It will require a little bit of work to make this rather vague argument precise. Before doing so, let us give the proof of , which is much simpler: Indeed, by monotonicity, for for every $\lambda>0$ we have $$\begin{aligned}
\limsup_n{\log \hom(\tilde{\mathbb{G}}_n,{\mathbb{H}})\over |V_n|}&\le \limsup_n{\hom(\tilde{\mathbb{G}}_n,{\mathbb{H}}_\lambda)\over |V_n|} \\
&=\limsup_n{\log \hom({\mathbb{G}}_n,{\mathbb{H}}_\lambda)\over |V_n|}\\
&=\lim_n{\log \hom({\mathbb{G}}_n,{\mathbb{H}}_\lambda)\over |V_n|} \\
&=-f_{({\mathbb{G}}_n)}({\mathbb{H}}_\lambda),\end{aligned}$$ where the first equality follows since ${\mathbb{H}}_\lambda$ is a soft-core graph. Passing to the limit $\lambda\rightarrow
0$, we obtain , which in turn implies $$\begin{aligned}
\sup_{\tilde{\mathbb{G}}_n\sim{\mathbb{G}}_n}\limsup_n{\log \hom(\tilde{\mathbb{G}}_n,{\mathbb{H}})\over |V_n|}\le -f_{({\mathbb{G}}_n)}({\mathbb{H}}).\end{aligned}$$
We now show that $f_{({\mathbb{G}}_n)}({\mathbb{H}})$ is achieved asymptotically by some $\tilde{\mathbb{G}}_n\sim {\mathbb{G}}_n$. Our main technical result leading to the claim is as follows.
\[lemma:Gnhatepsilon\] For every $\epsilon\in(0,1)$, every graph ${\mathbb{H}}$, and for all large enough $n$, there exists a graph $\tilde {\mathbb{G}}_n$ which is obtained from ${\mathbb{G}}_n$ by deleting at most $\epsilon|V_n|$ edges such that $$\begin{aligned}
\label{eq:Hgplusepsilon}
\hom(\tilde {\mathbb{G}}_n,{\mathbb{H}})&\ge
\exp\left(-(f_{({\mathbb{G}}_n)}({\mathbb{H}})+\epsilon)|V_n|\right).\end{aligned}$$
We first show that this lemma implies the required claim. It relies on a standard diagonalization argument. Note that it suffices to establish the result for graphs ${\mathbb{H}}$ with rational (possibly zero) weights. Let ${\mathbb{H}}_1,{\mathbb{H}}_2,\ldots$ be an arbitrary enumeration of graphs with rational weights. For every $m=1,2,\ldots~$, we find large enough $n(m)$ such that for $n\ge n(m)$, the claim in lemma holds for $\epsilon=1/m^2$ and for all ${\mathbb{H}}={\mathbb{H}}_1,\ldots,{\mathbb{H}}_m$. Namely, for every $n\ge n(m)$, graphs $\tilde{\mathbb{G}}_{n,1},\ldots,\tilde{\mathbb{G}}_{n,m}$ can be obtained from the graph ${\mathbb{G}}_n$ each be deleting at most $n/m^2$ of edges in ${\mathbb{G}}_n$ such that (\[eq:Hgplusepsilon\]) holds for $\epsilon=1/m^2$. Without loss of generality we may assume that $n(m)$ is strictly increasing in $m$. Let $\tilde E_{n,i}$ be the set of edges deleted from ${\mathbb{G}}_n$ to obtain ${\mathbb{G}}_{n,i}$ for $1\le i\le m$ and $\tilde{\mathbb{G}}_n^m$ be the graph obtained from ${\mathbb{G}}_n$ by deleting the edges in $\cup_{1\le i\le
m}E_{n,i}$. The number of deleted edges is at most $m(n/m^2)=n/m$. By monotonicity, (\[eq:Hgplusepsilon\]) implies that $$\begin{aligned}
\hom(\tilde {\mathbb{G}}_n^m,{\mathbb{H}}_i)&\ge
\exp\left(-(f_{({\mathbb{G}}_n)}({\mathbb{H}}_i)+1/m^2)|V_n|\right), \qquad i=1,2,\ldots,m.\end{aligned}$$ We now construct the graph sequence $\tilde G_n$. For all $n<n(1)$ we simply set $\tilde G_n={\mathbb{G}}_n$. For all other $n$, we find a unique $m_n$ such that $n(m_n)\le n<n(m_n+1)$ and set $\tilde{\mathbb{G}}_n=\tilde{\mathbb{G}}_n^{m_n}$. Then $\tilde{\mathbb{G}}_n\sim{\mathbb{G}}_n$ since the number of deleted edges is at most $|V_n|/m_n=o(|V_n|)$ as $n\rightarrow\infty$. For this sequence we have $$\begin{aligned}
|V_n|^{-1}\log\hom(\tilde {\mathbb{G}}_n,{\mathbb{H}}_i)&\ge
-f_{({\mathbb{G}}_n)}({\mathbb{H}}_i)-1/m_n^2, \qquad i=1,2,\ldots,m_n.\end{aligned}$$ Fixing $i$ and taking $\liminf_n$ of each side we obtain for each $i$ $$\begin{aligned}
\liminf_n|V_n|^{-1}\log\hom(\tilde {\mathbb{G}}_n,{\mathbb{H}}_i)&\ge
-f_{({\mathbb{G}}_n)}({\mathbb{H}}_i).\end{aligned}$$ Combining with (\[eq:limsuptildeGn\]) we conclude that for each $i$ $$\begin{aligned}
\lim_n|V_n|^{-1}\log\hom(\tilde {\mathbb{G}}_n,{\mathbb{H}}_i)&=
-f_{({\mathbb{G}}_n)}({\mathbb{H}}_i).\end{aligned}$$ This completes the proof of the theorem.
For the ease of exposition we write $f({\mathbb{H}})$ for $f_{({\mathbb{G}}_n)}({\mathbb{H}})$. If $f({\mathbb{H}})=\infty$, there is nothing to prove, so we assume $f({\mathbb{H}})<\infty$.
Let $$\begin{aligned}
\alpha_\lambda(\sigma)\triangleq \prod_{u\in V_n}\alpha_{\sigma(u)}\prod_{(u,v)\in E_n}\max(A_{\sigma(u),\sigma(v)},\lambda)\end{aligned}$$ and let $\Sigma(E)$ be the set of $\sigma:V_n\to[k]$ such that $E_0(\sigma)=E$. Then $$\begin{aligned}
\hom({\mathbb{G}}_n,{\mathbb{H}}_\lambda)=
\sum_{E\subset E_n}
\sum_{\sigma\in\Sigma(E)}\alpha_\lambda(\sigma).\end{aligned}$$ We will prove that for $\lambda$ sufficiently small and $n$ sufficiently large, we can find a set $E_{0}\subset E_n$ such that $|E_0|\le \epsilon |V_n|$ and $$\sum_{\sigma\in\Sigma(E_{0})}\alpha_\lambda(\sigma)\ge
\exp\left(-(f({\mathbb{H}})+\epsilon)|V_n|\right).
\label{eq:E0}$$ Note that this bound immediately implies the claim of the lemma. Indeed, let $\tilde{\mathbb{G}}_n$ be the graph obtained from ${\mathbb{G}}_n$ by deleting the edges in $E_{0}$ and assume with out loss of generality that $\lambda$ is smaller than the smallest non-zero entry of $A$. Then every $\sigma\in \Sigma(E_{0})$ satisfies $\alpha_\lambda(\sigma)=\lambda^{|E_0|}\tilde\alpha(\sigma)$, where $\tilde\alpha$ is the weight with respect to the graph $\tilde {\mathbb{G}}_n$, vector $\alpha$ and the matrix $A$. This implies $$\begin{aligned}
\hom(\tilde {\mathbb{G}}_n,{\mathbb{H}})&\ge \sum_{\sigma\in\Sigma(E_0)}\tilde\alpha(\sigma)
=\lambda^{-|E_0|}\sum_{\sigma\in\Sigma(E_0)}\alpha_\lambda(\sigma) \\
&\ge \sum_{\sigma\in\Sigma(E_0)}\alpha_\lambda(\sigma)
\geq \exp\left(-(f({\mathbb{H}})+\epsilon)|V_n|\right),\end{aligned}$$ as required. The proof of the lemma is therefore reduced to the proof of .
Given $0<\epsilon<1$ we chose $0<\hat\lambda<\epsilon$ such that $$\begin{aligned}
\label{eq:deltahat}
3\hat\lambda+\hat\lambda\log D+\hat\lambda\log(1/\hat\lambda)^{-1}<\epsilon,\end{aligned}$$ and $0<\lambda<\hat\lambda$ such that $$\begin{aligned}
\label{eq:deltaepsilon}
{f({\mathbb{H}})+2+(D+1)\log\alpha_{\max}+\log k\over \log(\lambda^{-1})}<\hat\lambda.\end{aligned}$$ Here we note that $-f({\mathbb{H}})\le \log k+(D+1)\log\alpha_{\max}$, and thus the numerator in the left-hand side is positive. Let $n_0=n_0(\lambda)$ be large enough so that $$\log
(D|V_n|) \le \lambda|V_n|
\quad\text{and}\quad
\hom({\mathbb{G}}_n,{\mathbb{H}}_\lambda)\ge \exp\left((-f({\mathbb{H}}_\lambda)-\lambda)|V_n|\right)$$ for all $n\geq n_0(\lambda)$. Note that by monotonicity in $\lambda$, the second bound implies that $$\begin{aligned}
\hom({\mathbb{G}}_n,{\mathbb{H}}_\lambda)&\ge \exp\left((-f({\mathbb{H}})-\lambda)|V_n|\right). \label{eq:n0}\end{aligned}$$
Fix an arbitrary such $n\geq n_0$, and let $\Sigma(r)$ be the set of mappings $\sigma:V_n\rightarrow
V({\mathbb{H}})$ such that precisely $r$ edges of $V_n$ are mapped into non-edges of ${\mathbb{H}}$, $$\Sigma(r)=\{\sigma:V_n\rightarrow
V({\mathbb{H}})\colon
\left| E_0(\sigma)\right|=r\}.$$ For all $\lambda$ which are smaller than the smallest positive element of $A$ we have $$\begin{aligned}
\hom({\mathbb{G}}_n,{\mathbb{H}}_\lambda)=\sum_{0\le r\le |E_n|} \sum_{\sigma\in \Sigma(r)}\alpha_\lambda(\sigma).\end{aligned}$$ Let $r_0$ be such that $$\begin{aligned}
\sum_{\sigma\in \Sigma(r_0)}\alpha_\lambda(\sigma)&\ge |E_n|^{-1}\hom({\mathbb{G}}_n,{\mathbb{H}}_\lambda).
\label{eq:r0}\end{aligned}$$ We claim that $ r_0\le \hat\lambda |V_n| $. Indeed, since for every $\sigma\in\Sigma(r_0)$ we have $\alpha_\lambda(\sigma)\le
{ \lambda^{r_0}}\alpha_{\max}^{|V_n|+|E_n|} \le {
\lambda^{r_0}}\alpha_{\max}^{(D+1)|V_n|}$, it follows from (\[eq:r0\]) that $$\begin{aligned}
{k}^{|V_n|}\lambda^{r_0}\alpha_{\max}^{(D+1)|V_n|}
\ge |E_n|^{-1}\hom({\mathbb{G}}_n,{\mathbb{H}}_\lambda)
\ge \exp\left((-f({\mathbb{H}})-2\lambda)|V_n|\right),\end{aligned}$$ where the second inequality follows from (\[eq:n0\]) and the fact that for $n\geq n_0$, we have $|E_n|\leq D|V_n|\leq
e^{\lambda |V_n|}$. Rearranging, we get that $$\begin{aligned}
r_0\le (\log(1/\lambda))^{-1}\Big(|V_n|\big(f({\mathbb{H}})+{2}\lambda+{ (D+1)}\log\alpha_{\max}+\log { k}\big)\Big).\end{aligned}$$ Applying (\[eq:deltaepsilon\]), this in turn implies that $$\begin{aligned}
r_0&\le (\log\lambda^{-1})^{-1}|V_n|\left(f({\mathbb{H}})+2\lambda+(D+1)\log\alpha_{\max}+\log k\right)\notag \\
&\leq (\log\lambda^{-1})^{-1}|V_n|\left(f({\mathbb{H}})+2+(D+1)\log\alpha_{\max}+\log k\right)\notag\\
&\leq \hat\lambda |V_n|\label{eq:bound2 r_0},\end{aligned}$$ as claimed. Next we observe that $$\begin{aligned}
\sum_{\sigma\in \Sigma(r_0)}\alpha_\lambda(\sigma)=\sum_{E\subset E_n:\atop |E|=r_0}
\sum_{\sigma\in\Sigma(E)}\alpha_\lambda(\sigma).\end{aligned}$$ Since the number of subsets of the edges of ${\mathbb{G}}_n$ with cardinality $r_0$ is ${|E_n| \choose r_0}$, we can find a subset $E_0\subset E_n, |E_0|=r_0$ such that $$\begin{aligned}
\sum_{\sigma\in\Sigma(E_0)}\alpha_\lambda(\sigma)&\ge {|E_n| \choose r_0}^{-1}\sum_{\sigma\in \Sigma(r_0)}\alpha_\lambda(\sigma) \notag
\\
&\ge
{|E_n| \choose r_0}^{-1}\exp\left((-f({\mathbb{H}})-2\lambda)|V_n|\right)\label{E0-bd}\end{aligned}$$ where the second inequality follows from (\[eq:r0\]). Next, we note that $ {N\choose k} \leq \Bigl(\frac{Ne}k\Bigr)^k, $ a fact which can easily be proved by induction on $k$. Combining this bound with the fact that $|E_n|\leq D|V_n|$ and the bounds and , we get $$\begin{aligned}
{|E_n| \choose r_0}\le {D|V_n| \choose r_0}\le {D|V_n| \choose {\hat\lambda} |V_n|}
\le \left(\frac {eD}{\hat\lambda}\right)^{\hat\lambda |V_n|}
= e^{((1+\log D)\hat\lambda +\hat\lambda\log(\hat\lambda^{-1}))|V_n|}
\leq e^{(\epsilon-2\hat\lambda)|V_n|}.\end{aligned}$$ Combined with this implies the bound .
Partition-convergence and colored-neighborhood-convergence {#subsection:partitionConvergence}
----------------------------------------------------------
The notions of partition-convergence and colored-neighborhood-convergence were introduced in Bollobas and Riordan [@BollobasRiordanMetrics]. Recall the notion of a graph quotient ${\mathbb{F}}={\mathbb{G}}/\sigma$ defined for every graph ${\mathbb{G}}$ and mapping $\sigma:V({\mathbb{G}})\rightarrow [m]$. Then ${\mathbb{F}}$ is a weighted graph on $m$ nodes with node and edge weights $(x(\sigma),X(\sigma))\in {[0,D]^{(m+1)\times m}}$ defined by (\[k-quotient\]). When ${\mathbb{G}}$ is the $n$-th element of a graph sequence ${\mathbb{G}}_n$, we will use notations $x_n(\sigma)$ and $X_n(\sigma)$. Let $\Sigma_n(m)$ be the set of all pairs $(x_n(\sigma),X_n(\sigma))$, when we vary over all maps $\sigma$. As such $\Sigma_n(m)\in\mathcal{P}\left({[0,D]^{(m+1)\times m}}\right)$, where $\mathcal{P}(A)$ is the set of closed subsets of $A$. For every set $A\subset{[0,D]^{(m+1)\times m}}$, let $A^\epsilon$ be the set of points in ${[0,D]^{(m+1)\times m}}$ with distance at most $\epsilon$ to $A$ with respect to the ${\mathbb{L}}_\infty$ norm on ${[0,D]^{(m+1)\times m}}$ (the actual choice of metric is not relevant). Define a metric $\rho_m$ on closed sets in $\mathcal{P}\left({[0,D]^{(m+1)\times m}}\right)$ via $$\begin{aligned}
\label{eq:DugunjiMetric}
\rho(A,B)=\inf_{\epsilon>0}\{B\subset A^\epsilon,A\subset B^\epsilon\}.\end{aligned}$$
A graph sequence ${\mathbb{G}}_n$ is partition-convergent if for every $m$, the sequence $\Sigma_n(m), n\ge 1$ is a Cauchy sequence in the metric space $\mathcal{P}({[0,D]^{(m+1)\times m}})$.
It is known that $\mathcal{P}({[0,D]^{(m+1)\times m}})$ is a compact (thus closed) metric space, which implies that $\Sigma_n(m)$ has a limit $\Sigma(m)\in \mathcal{P}({[0,D]^{(m+1)\times m}})$. Loosely speaking $\Sigma(m)$ describes the limiting set of graph quotients achievable through the coloring of nodes by $m$ different colors. Thus, if for example some pair $(x,X)$ belong to $\Sigma(m)$, this means that there is a sequence of partitions of ${\mathbb{G}}_n$ into $m$ colors such that the normalized number of nodes colored $i$ converges to $x_i$ for each $i$, and the normalized number of edges between colors $i$ and $j$ converges to $X_{i,j}$ for each $i,j$.
The definition above raises the question whether partition-convergence implies convergence of neighborhoods, namely left-convergence. We will show in the next section that this is not the case: partition-convergence does not imply left-convergence (whereas, as we mentioned above, right-convergence does imply the left-convergence). In order to address this issue Bollobas and Riordan extended their definition of partition-convergence [@BollobasRiordanMetrics] to a richer notion called *colored-neighborhood-convergence*. This notion was further studied by Hatami, Lov[á]{}sz and Szegedi [@HatamiLovaszSzegedy] under the name *local-global* convergence, to stress the fact that this notion captures both global and local properties of the sequence ${\mathbb{G}}_n$. We prefer the original name, given that both right and LD-convergence capture local and global properties as well.
To define colored-neighborhood-convergence, we fix positive integers $m$ and $r$. Let $\mathcal{H}_{m,r}\subset\mathcal{H}$ denote the finite set of all rooted colored graphs with radius at most $r$ and degree at most $\Delta$, colored using colors $1,\ldots,m$. For every graph ${\mathbb{G}}$ with degree $\le\Delta$ and every node $u\in V({\mathbb{G}})$, every coloring $\sigma:V({\mathbb{G}})\rightarrow [m]$ produces an element ${\mathbb{H}}\in
\mathcal{H}_{m,r}$, which is the colored $r$-neighborhood $B_{\mathbb{G}}(u,r)$ of $u$ in ${\mathbb{G}}$. For each ${\mathbb{H}}\in
\mathcal{H}_{m,r}$, let $N({\mathbb{G}},m,r,\sigma,{\mathbb{H}})$ be the number of nodes $u$ such that there exists a color matching isomorphism from $B_{\mathbb{G}}(u,r)$ to ${\mathbb{H}}$. Namely, it is the number of times that ${\mathbb{H}}$ appears as a colored neighborhood $B_{\mathbb{G}}(u,r)$ when we vary $u$. Let $\pi({\mathbb{G}},m,r,\sigma)$ be the vector $\left(|V({\mathbb{G}})|^{-1}N({\mathbb{G}},m,r,\sigma,{\mathbb{H}}), {\mathbb{H}}\in
\mathcal{H}_{m,r}\right)$. Namely, it is the vector of frequencies of observing graphs ${\mathbb{H}}$, as neighborhoods $B_{\mathbb{G}}(u,r)$ when we vary $u$. Then for every $m$, we obtain a finite and therefore closed set $\pi({\mathbb{G}},m,r)\subset
[0,1]^{|\mathcal{H}_{m,r}|}$ when we vary over all $m$-colorings $\sigma$ in vectors $\pi({\mathbb{G}},m,r,\sigma)$. As such, $\pi({\mathbb{G}},m,r)\in\mathcal{P}([0,1]^{|\mathcal{H}_{m,r}|})$. We consider $\mathcal{P}([0,1]^{|\mathcal{H}_{m,r}|})$ as a metric space with metric given by (\[eq:DugunjiMetric\]).
\[Defi:Local-GlobalConvergence\] A graph sequence ${\mathbb{G}}_n$ is defined to be colored-neighborhood-convergent if the sequence $\pi({\mathbb{G}}_n,m,r)$ is convergent in $\mathcal{P}([0,1]^{|\mathcal{H}_{m,r}|})$ for every $m,r$.
It is not too hard to see that colored-neighborhood-convergence implies partition-convergence and left-convergence. As we show in the next section, the converse does not hold for either partition- or left-convergence.
Relationship between different notions of convergence {#section:ConvergenceRelations}
=====================================================
In this section we explore the relations between different notions of convergence. As we will see, most of the definitions are non-equivalent and many of them are not comparable in strength. Our findings are summarized in the following theorem.
\[theorem:ConvergenceRelations\] The notions of left, right, partition, colored-neighborhood and LD-convergence satisfy the relations described on Figures \[figure:ConvergenceRelations\] and \[figure:ConvergenceRelationsColorNeighb\], where the solid arrow between $A$ and $B$ means type $A$ convergence implies type $B$ convergence, and the striped arrow between $A$ and $B$ means type $A$ convergence does not imply type $B$ convergence.
We have deliberately described the relations in two figures, since at this stage we do not know how LD and colored-neighborhood-convergence are related to each other. As a step towards this it would be interesting to see whether colored-neighborhood-convergence implies right-convergence. We will discuss this question along with some other open questions in Section \[sec:discussion\].
Note that it was already proved in [@BorgsChayesKahnLovasz] that right-convergence implies left-convergence, and that it is obvious that colored-neighborhood-convergence implies partition and left-convergence. To prove the Theorem, it will therefore be enough to show that LD-convergence implies both right and partition-convergence, that left-convergence does not imply right-convergence, that partition-convergence does not imply left-convergence, and that right-convergence does not imply partition-convergence (all other arrows are implied). We will start with the negative implications, which are all proved by giving counter-examples.
Left-convergence does not imply right-convergence {#sec:left-not-right}
-------------------------------------------------
As discussed earlier left-convergence does not imply right-convergence on all hard-core graphs ${\mathbb{H}}$. But our modified definition of right-convergence involving only soft-core graphs raises the question whether with our new definition, left-convergence does imply right-convergence. It turns out that this is not the case since right-convergence implies convergence of certain global properties like the convergence of the maximal cut in a graph, a property which is not captured by the notion of left-convergence.
Indeed, let ${\mathbb{G}}_n$ be right-convergent, let $\text{MaxCut}({\mathbb{G}}_n)$ denotes the size of the maximal cut in ${\mathbb{G}}_n$, and let ${\mathbb{H}}$ be a graph consisting of a single edge with weights $\alpha=(1,1)$ and $A_{1,2}=A_{2,1}=e^\beta,
A_{1,1}=A_{2,2}=1$. Then $$e^{\beta\text{MaxCut}({\mathbb{G}}_n)}
\leq \hom({\mathbb{G}}_n,{\mathbb{H}})
\leq 2^ne^{\beta\text{MaxCut}({\mathbb{G}}_n)}.$$ Sending $\beta\to\infty$, we see that right-convergence implies convergence of $\frac 1{|V({\mathbb{G}}_n)|}\text{MaxCut}({\mathbb{G}}_n)$.
As an immediate consequence, we obtain that left-convergence does not implies right-convergence.
\[Lem:left-not-right\] Let $\Delta$ be a positive integer, let ${\mathbb{G}}(n,\Delta)$ denote a random $\Delta$-regular graph on $n$ nodes, and let $B_{n,\Delta}$ denote a random $\Delta$-regular bi-partite graph on $n$ nodes. Let ${\mathbb{G}}_n$ be equal to ${\mathbb{G}}(n,\Delta)$ when $n$ is odd and equal to $B_{n,\Delta}$ when $n$ is even. Then ${\mathbb{G}}_n$ is left-convergent for all $\Delta$, but for $\Delta$ sufficiently large, it is not right-convergent.
${\mathbb{G}}_n$ is left-convergent since for every $r$, with high probability $B(U,r)$ is a $\Delta$-Kelly (regular) tree truncated at depth $r$, where $U$ is a random uniformly chosen node in ${\mathbb{G}}_n$. We now show that $G_n$ is not right-convergent for large $\Delta$. Indeed, $\text{MaxCut}({\mathbb{G}}_n)=\text{MaxCut}(B_{n,\Delta})=\frac 12\Delta
n $ for even $n$, while $\text{MaxCut}({\mathbb{G}}_n)=\text{MaxCut}({\mathbb{G}}(n,\Delta))=\frac 14\Delta
n +O(\sqrt\Delta n)$ for odd $n$ [@BCP]. This implies that for large $\Delta$, the sequence ${\mathbb{G}}_n$ is not right-convergent.
Right-convergence does not imply partition-convergence
------------------------------------------------------
In order to construct our counter example, we need the following lemma. We recall that the edge expansion constant of a sequence of graphs ${\mathbb{G}}_n$ is the largest constant $\gamma$ such that the number of edges between $W$ and $V({\mathbb{G}}_n)\setminus
W$ is larger than $\gamma |W|$ whenever $|W|\leq |V({\mathbb{G}}_n)|/2$.
Let $\Delta\geq 3$. Then there exists a $\gamma>0$ and a right-convergent sequence of expanders ${\mathbb{G}}_n$ with edge expansion constant $\gamma$ and maximal degree bounded by $\Delta$.
Consider an arbitrary sequence ${\mathbb{G}}_n=(V_n,E_n)$ of expanders with edge expansion $\gamma>0$ and maximal degree at most $\Delta$, e.g., a sequence or 3-regular random graphs. Construct a right-convergent subsequence of this sequence by the following, standard diagonalization argument: enumerate the countable collection of weighted graphs with rational positive weights: ${\mathbb{H}}_1,{\mathbb{H}}_2,\ldots~$. For each $m=1,2,\ldots,$ construct a nested subsequences $n^{(m)}\subset n^{(m-1)}$, such that ${\mathbb{G}}_n$ is right-convergent with respect ${\mathbb{H}}_1,\ldots,{\mathbb{H}}_m$ along the subsequence $n^{(m)}$, and then take the diagonal subsequence $n^{(m)}_m$ - the $m$-element of the $m$-th sequence. The constructed sequence of graphs is right-convergent with respect to every weighted graph ${\mathbb{H}}$ with rational coefficients. A straightforward argument shows that then the sequence is right-convergent with respect to all graphs with positive real weights, namely it is right-convergent.
Given a right-convergent sequence of expanders ${\mathbb{G}}_n$ with bounded maximal degree and edge expansion $\gamma>0$, let $\tilde {\mathbb{G}}_n={\mathbb{G}}_m$ if $n=2m$ is even, and let $\tilde {\mathbb{G}}_n$ be a disjoint union of two copies of ${\mathbb{G}}_m$ if $n=2m+1$ is odd. Then $(\tilde G_n)$ is right-convergent but not partition-convergent.
Let $G_n=(V_n,E_n)$. For every weighted graph ${\mathbb{H}}$, $$\begin{aligned}
|\tilde V_{2n+1}|^{-1}\log Z(\tilde {\mathbb{G}}_{2n+1},{\mathbb{H}})&=(2|V_n|)^{-1}(2\log Z({\mathbb{G}}_n,{\mathbb{H}}))\\
&=|V_n|^{-1}\log Z({\mathbb{G}}_n,{\mathbb{H}})\\
&=|\tilde V_{2n}|^{-1}\log Z(\tilde {\mathbb{G}}_{2n},{\mathbb{H}}).\end{aligned}$$ Since ${\mathbb{G}}_n$ is right-convergent, $\tilde{\mathbb{G}}_n$ is right-convergent as well.
On the other hand, we claim that $\tilde{\mathbb{G}}_n$ is not partition-convergent, since every cut of $\tilde {\mathbb{G}}_n={\mathbb{G}}_m$ into two equal parts has size at least $\gamma |V_n|/2$ if $n$ is even, while $\tilde{\mathbb{G}}_n$ has a cut into equal parts with zero edges when $n$ is odd. Indeed, denoting the node set of $\tilde{\mathbb{G}}_n$ by $\tilde
V_n$, consider the set $\Sigma_n(2)$ of pairs $(x_n(\sigma),X_n(\sigma))$ of vectors $x_n(\sigma)=(x^n_1(\sigma),x^n_2(\sigma))$ and matrices $X_n(\sigma,2)=(X^n_{i,j}(\sigma), 1\le i,j\le 2)$ when we vary over $\sigma:\tilde V_n\rightarrow [2]$. Consider the projection of $\Sigma_n(2)$ onto $[0,D]^3$ corresponding to $(x^n_1(\sigma),x^n_2(\sigma),X^n_{1,2}(\sigma))$. Consider the point $a=(1/2,1/2,0)\in [0,D]^3$ Observe that for each $n$, there exists a $\sigma$ such that $(x^{2n+1}_1(\sigma),x^{2n+1}_2(\sigma),X^{2n+1}_{1,2}(\sigma))=a$, since we can split the graph $\tilde{\mathbb{G}}_{2n+1}$ into two parts with the same number of nodes in each part and no edges in between the parts. On the other hand for each graph $\tilde{\mathbb{G}}_{2n}$ and each $\sigma$, the distance between $(x^{2n}_1(\sigma),x^{2n}_2(\sigma),X^{2n}_{1,2}(\sigma))$ and $a$ is bounded from below my $\min\{1/4,\gamma/4\}$ (indeed, if $|x^{2n}_1(\sigma)-1/2|<
1/4$, then by the expansion property of $G_n$, $X^{2n}_{1,2}(\sigma) > \gamma/4$). This shows that the sets $\{(x^n_1(\sigma),x^n_2(\sigma),X^n_{1,2}(\sigma))\}$ obtained by varying $\sigma$ are not converging as $n\rightarrow\infty$ and thus the sequence $\tilde{\mathbb{G}}_n$ is not partition-convergent.
Partition-convergence does not imply left-convergence {#subsection:PartitionDoesNotImplyLeft}
-----------------------------------------------------
We now exhibit an example of a graph sequence which is partition-convergent but not left-convergent.
Let ${\mathbb{G}}_n$ be a disjoint union of $n$ $4$-cycles when $n$ is even and a disjoint union of $n$ $6$-cycles when $n$ is odd. Then ${\mathbb{G}}_n$ is partition-convergent and not left-convergent.
${\mathbb{G}}_n$ is clearly not left-convergent. Let us show that it is partition-convergent. For this purpose set $D=2$ and consider arbitrary $k$. Let $\Sigma(k)$ be the set of all $(x,X)\in {[0,D]^{(k+1)\times k}}$ such that $\sum_{1\le i\le k} x_{i}=1$, $X_{i,j}=X_{j,i}$ for all $i,j$, and $$\begin{aligned}
\label{eq:xijsum}
\sum_{j=1}^kX_{i,j}
=2x_{i}
\quad\text{for all }i.\end{aligned}$$ We will prove that $\Sigma_n(k)\rightarrow \Sigma(k)$ which shows that the sequence is partition-convergent with $\Sigma(k)$ being the limit.
To this end, we first show that $\Sigma_n(k)\subset \Sigma(k)$ for all $n$. Indeed, let $\sigma:V_n\to [k]$. By the definition , we have that $$\sum_{j=1}^k X_{i,j}(\sigma)=
\frac 1{|V_n|}\sum_{u\in V_i}\sum_{v\in V_n:\atop (u,v)\in E_n} 1
= \frac 1{|V_n|}\sum_{u\in V_i} 2 =2x_i(\sigma),$$ where we used that the degree of all vertices in ${\mathbb{G}}_n$ is two. This proves that holds for all $(x,X)\in \Sigma_n(k)$.
We now claim that for every element $(x,X)\in \Sigma(k)$ and every $n$ there exists $\sigma_n:V_n\rightarrow [k]$ such that the ${\mathbb{L}}_\infty$ distance between $(x,X)$ and $(x(\sigma_n),X(\sigma_n))$ is at most $k/n$. This is clearly enough to complete the proof, since it implies that for every $\epsilon>0$ and $n$ large enough, we have $\Sigma(k)\subset (\Sigma_n(k))^\epsilon$.
To prove the claim, we first consider even $n$ so that ${\mathbb{G}}_n$ is a disjoint union of $n$ $4$-cycles. To this end, let $n_{i,j}$, $i,j\in [k]$, be such that $$\Bigl|n_{i,j}-\frac n2 X_{i,j}\Bigr|\leq 1
\quad\text{for all }i,j\in [k]$$ and $$\sum_{i,j\in [k]}n_{i,j}
=\sum_{i,j\in [k]}\frac n2 X_{i,j}=n.$$ We consider the following coloring $\sigma_n$ of ${\mathbb{G}}_n$: For each $i\ne j$ we take $n_{i,j}$ cycles and color their $4$ vertices alternating between color $i$ and $j$ as we go around these cycles, while for $i=j$, we take $n_{i,i}$ cycles and color them with only one color, $i=j$. For this coloring, we have $$X_{i,j}(\sigma)=X_{j,i}(\sigma)=
4\frac{n_{i,j}+n_{j,i}}{4n},$$ so by our choice of $n_{i,j}$ we have that $$\Bigl|
X_{i,j}-X_{i,j}(\sigma)
\bigr|
\leq\frac 2n.$$ Since $\Sigma_n(k)\subset \Sigma(k)$, we have $$|x_i-x_i(\sigma)|= \frac12\left|\sum_{j=1}^k\Bigl(X_{i,j}-X_{i,j}(\sigma)\Bigr)\right|
\leq \frac kn,$$ completing the proof of the claim when $n$ is even. A similar result is established for odd $n$, when ${\mathbb{G}}_n$ is a disjoint union of $n$ $6$-cycles.
LD-convergence implies right-convergence {#section:LD-implies-Right}
----------------------------------------
Let the sequence ${\mathbb{G}}_n$ be LD-convergent. Applying Theorem \[theorem:LDconvergenceEquivalence\] it is also $k$-LD-convergent with rate functions $I_k, k\ge 1$. Consider an arbitrary $k$-node weighted graph ${\mathbb{H}}$ with strictly positive node weights $\alpha=(\alpha_i, 1\le i\le k)$ and strictly positive edge weights $A=(A_{i,j}, 1\le i,j\le k)$. We define an “energy functional” $\mathcal E_{{\mathbb{H}}}$ on ${[0,D]^{(k+1)\times k}}$ by $$\label{E-def}
\mathcal E_{{\mathbb{H}}}(x,X)=
-\sum_{1\le i\le k}x_{i}\log \alpha_{i}
-\frac 12\sum_{1\le i,j\le k}X_{i,j}\log A_{i,j}.$$ We also introduce an “entropy functional” $\mathcal S_k$ on ${[0,D]^{(k+1)\times k}}$ by setting $$\label{S-def}
\mathcal S_k(x,X)=\log k - I_k(x,X).$$
\[thm:LD-implies-Right\] Suppose the sequence of graphs ${\mathbb{G}}_n$ is LD-convergent. Then it is also right-convergent and $$\begin{aligned}
\label{F-limit-sparse}
-\lim_{n\to\infty}{1\over |V_n|}\log Z({\mathbb{G}}_n,{\mathbb{H}})
=\inf_{(x,X)\in {[0,D]^{(k+1)\times k}}}\Bigl(\mathcal E_{{\mathbb{H}}}(x,X)-\mathcal S_k(x,X)\Bigr).\end{aligned}$$
By the lower semi-continuity of $I_k$, the infimum over $(x,X)\in{[0,D]^{(k+1)\times k}}$ is attained for some $(x^*,X^*)\in{[0,D]^{(k+1)\times k}}$. The right hand side can then be rewritten as $\mathcal
E_{{\mathbb{H}}}(x^*,X^*)-\mathcal S_k(x^*,X^*)$. The theorem thus states that the limiting free energy exists, and is given as the energy minus entropy, an identity which is usually assumed axiomatically in traditional thermodynamics. Here it is a consequence of LD-convergence.
By Remark \[cor:Ik-from-I\], we can express the limiting free energy directly in terms of the rate function $I$. Defining $\hat{\mathcal E}_{{\mathbb{H}}}(\rho,\mu)=\mathcal E_{{\mathbb{H}}}(x(\rho),X(\mu))$ where $x(\rho)$ and $X(\mu)$ are given by , this gives $$-\lim_{n\to\infty}{1\over |V_n|}\log Z({\mathbb{G}}_n,{\mathbb{H}})
=\inf_{(\rho,\mu)\in {{\bf \mathcal{M}}}^1\times{{\bf \mathcal{M}}}^2}\Bigl(\hat{\mathcal E}_{{\mathbb{H}}}(\rho,\mu)+I(\rho,\mu)-\log k\Bigr).$$
Fix $\delta>0$. Recalling the definition , let $\Sigma(x,X)$ be the set of colorings $\sigma:V_n\rightarrow [k]$ such that $$\begin{aligned}
x_{i}-\delta \le
x_i(\sigma)
\le x_{i}+\delta.\end{aligned}$$ for every $i=1,\ldots,k$ and $$\begin{aligned}
X_{i,j}-\delta \le
X_{i,j}(\sigma)
\le X_{i,j}+\delta.\end{aligned}$$ for every $i,j=1,2,\ldots,k$. Note that we have the identity $$\begin{aligned}
|\Sigma(x,X)|=k^{|V_n|}{\mathbb{P}}_{k,n}\Bigl(B((x,X),\delta)\Bigr),\end{aligned}$$ where we recall that $B((x,X),\delta)$ denotes the closed $\delta$-ball around $(x,X)$ with respect to the ${\mathbb{L}}_\infty$ norm, see Subsection \[subsection:DefProperties\]. Note also that $$\Bigl| \mathcal E_{{\mathbb{H}}}(x(\sigma),X(\sigma))-\mathcal E_{{\mathbb{H}}}(x,X)\Bigr|\leq K\delta,\qquad K=\Bigl(k+\frac{k^2}2\Bigr)\max\{\log\alpha_i,\log A_{i,j}\},$$ whenever $\sigma\in\Sigma(x,X)$.
Define $\Gamma_\delta$ to be the set of pairs $(x,X)\in{[0,D]^{(k+1)\times k}}$ such that every $x_i$ and every $X_{i,j}$ belongs to the set $\lbrace
0,\delta,2\delta,\ldots,\lceil D/\delta\rceil \delta\rbrace$. Since $X_{i,j}(\sigma)\leq D$ for all $\sigma:V_n\to
[k]$, every $\sigma:V_n\to [k]$ lies in $\cup_{(x,X)\in\Gamma_\delta}\Sigma(x,X)$. As a consequence, $$\begin{aligned}
Z({\mathbb{G}}_n,{\mathbb{H}})
&\le
\sum_{(x,X)\in\Gamma_\delta}\sum_{\sigma\in\Sigma(x,X)}
\prod_{u\in V_n}\alpha_{\sigma(u)}
\prod_{(u,v)\in E_n}A_{\sigma(u),\sigma(v)}\\
&=
\sum_{(x,X)\in\Gamma_\delta}\sum_{\sigma\in\Sigma(x,X)}
e^{-\mathcal E_{{\mathbb{H}}}(x(\sigma),\,X(\sigma))|V_n|}
\\
&\le
\sum_{(x,X)\in\Gamma_\delta}k^{|V_n|}{\mathbb{P}}_{k,n}\left(B((x,X),\delta)\right)
e^{-\mathcal E_{{\mathbb{H}}}(x,\,X)|V_n|+K\delta|V_n|}
\\
&\le
|\Gamma_\delta|\max_{(x,X)\in\Gamma_\delta}
\left(k^{|V_n|}{\mathbb{P}}_{k,n}\Bigl(B((x,X),\delta)\Bigr)
e^{-\mathcal E_{{\mathbb{H}}}(x,\,X)|V_n|+K\delta|V_n|}\right).\end{aligned}$$ We obtain $$\begin{aligned}
{1\over |V_n|}\log Z({\mathbb{G}}_n,{\mathbb{H}})
&\le
{1\over |V_n|}\log|\Gamma_\delta|+\log k
\\
&+
\max_{(x,X)\in\Gamma_\delta}
\left({1\over |V_n|}\log {\mathbb{P}}_{k,n}\Bigl(B((x,X),\delta)\Bigr)
-\mathcal E_{{\mathbb{H}}}(x,\,X)+K\delta\right),\end{aligned}$$ and hence $$\begin{aligned}
\limsup_n&{1\over |V_n|}\log Z({\mathbb{G}}_n,{\mathbb{H}})-\log k
\\
&\le
\limsup_n\max_{(x,X)\in\Gamma_\delta}
\left({1\over |V_n|}\log {\mathbb{P}}_{k,n}\Bigl(B((x,X),\delta)\Bigr)
-\mathcal E_{{\mathbb{H}}}(x,\,X)\right)+K\delta
\\
&=
\max_{(x,X)\in\Gamma_\delta}
\left(\limsup_n{1\over |V_n|}\log {\mathbb{P}}_{k,n}\Bigl(B((x,X),\delta)\Bigr)
-\mathcal E_{{\mathbb{H}}}(x,\,X)\right)+K\delta
\\
&\le
\max_{(x,X)\in\Gamma_\delta}
\left(-\inf_{(y,Y)\in B((x,X),\delta)}I_k(y,Y)
+\mathcal E_{{\mathbb{H}}}(x,\,X)\right)+K\delta
\\
&\le \max_{(x,X)\in\Gamma_\delta}
\left(-\inf_{(y,Y)\in B((x,X),\delta)}
\Bigl(I_k(y,Y)+\mathcal E_{{\mathbb{H}}}(y,\,Y)\Bigr)\right)+2K\delta
\\
&\le \max_{(x,X)\in\Gamma_\delta}
\left(-\inf_{(y,Y)\in {[0,D]^{(k+1)\times k}}}\Bigl(I_k(y,Y)+\mathcal E_{{\mathbb{H}}}(y,\,Y)\Bigr)\right)+2K\delta
\\
&=
-\inf_{(y,Y)\in {[0,D]^{(k+1)\times k}}}
\Bigl(I_k(y,Y)+\mathcal E_{{\mathbb{H}}}(y,\,Y)\Bigr)+2K\delta.\end{aligned}$$ Since this inequality holds for arbitrary $\delta$, we obtain $$\begin{aligned}
\limsup_n{1\over |V_n|}\log Z({\mathbb{G}}_n,{\mathbb{H}})\leq -\inf_{(x,X)\in {[0,D]^{(k+1)\times k}}}\Bigl(\mathcal E_{{\mathbb{H}}}(x,X)-\mathcal S_k(x,X)\Bigr).\end{aligned}$$ We now establish the matching lower bound. Fix $\delta>0$ and let $(x^*,X^*)$ be the minimizer of $\mathcal
E_{{\mathbb{H}}}-\mathcal S_k$. We have $$\begin{aligned}
Z({\mathbb{G}}_n,{\mathbb{H}})&\ge\sum_{\sigma\in\Sigma(x^*,X^*)}
e^{-\mathcal E_{{\mathbb{H}}}(x(\sigma),\,X(\sigma))|V_n|}
\\
&\ge k^{|V_n|}{\mathbb{P}}_{k,n}\Bigl(B((x^*,X^*),\delta)\Bigr)e^{-\mathcal E_{{\mathbb{H}}}(x^*,\,X^*)|V_n|-K\delta|V_n|},\end{aligned}$$ and thus $$\begin{aligned}
\liminf_n{1\over |V_n|}
&\log Z({\mathbb{G}}_n,{\mathbb{H}})-\log k\geq
\\
&\ge
\liminf_n{1\over |V_n|}\log {\mathbb{P}}_{k,n}\Bigl(B((x^*,X^*),\delta)\Bigr)
-\mathcal E_{{\mathbb{H}}}(x^*,\,X^*)-K\delta
\\
&\ge
\liminf_n{1\over |V_n|}\log {\mathbb{P}}_{k,n}\Bigl(B^o((x^*,X^*),\delta)\Bigr)
-\mathcal E_{{\mathbb{H}}}(x^*,\,X^*)-K\delta
\\
&\ge
-\inf_{(y,Y)\in B^o((x^*,X^*),\delta)}I_k((y,Y))
-\mathcal E_{{\mathbb{H}}}(x^*,\,X^*)-K\delta
\\
&\ge
-I_k(x^*,X^*)
-\mathcal E_{{\mathbb{H}}}(x^*,\,X^*)-K\delta.\end{aligned}$$ Letting $\delta\rightarrow 0$ we obtain the result.
LD-convergence implies partition-convergence
--------------------------------------------
We prove this result by identifying the limiting sets $\Sigma(k), k\ge 1$. In order to do this, we recall that by Theorem \[theorem:LDconvergenceEquivalence\] every LD-convergent sequence is k-LD-convergent for all $k$. We denote the corresponding functions by $I_k$.
\[theorem:PartitionIsLDInfty\] Let ${\mathbb{G}}_n$ be a sequence of graphs which is LD-convergent. Then ${\mathbb{G}}_n$ is partition-convergent. Moreover, for each $k$, the sets $\Sigma_n(k)$ converge to $$\begin{aligned}
\label{eq:SigmakInfinite}
\Sigma(k)=\{(x,X)\in{[0,D]^{(k+1)\times k}}: I_k(x,X)<\infty\}.\end{aligned}$$
The intuition behind this result is as follows. $\Sigma(k)$ is the set of ”approximately achievable” partitions $(x,X)$. If some partition is approximately realizable as factor graph ${\mathbb{G}}_n/\sigma$ for some $\sigma:V_n\rightarrow [k]$, then there are exponentially many such realizations, since changing a small linear in $|V_n|$ number of assignments of $\sigma$ does not change the factor graph significantly. Therefore, a uniform random partition $\sigma$ has at least an inverse exponential probability of creating a factor graph ${\mathbb{G}}_n/\sigma$ near $(x,X)$ and, as a result, the LD rate associated with $(x,X)$ is positive.
On the other hand if $(x,X)$ is $\epsilon$-away from any factor graph ${\mathbb{G}}_n/\sigma$, then the likelihood that a uniform random partition $\sigma$ results in a factor graph close to $(x,X)$ is zero, implying that the LD rate associated with $(x,X)$ is infinite. We now formalize this intuition.
We define $\Sigma(k)$ as in (\[eq:SigmakInfinite\]) and prove that it is the limit of $\Sigma_n(k)$ as $n\rightarrow\infty$ for each $k$. We assume the contrary. This means that either there exist $k$ and $\epsilon_0>0$ such that for infinitely many $n$, $\Sigma_n(k)$ is not a subset of $(\Sigma(k))^{\epsilon_0}$, or there exist $k$ and $\epsilon_0>0$ such that for infinitely many $n$, $\Sigma(k)$ is not a subset of $(\Sigma_n(k))^{\epsilon_0}$.
We begin with the first assumption. Then, for infinitely many $n$ we can find $(x_n,X_n)\in\Sigma_n(k)$ such that $(x_n,X_n)$ has distance at least $\epsilon_0$ from $\Sigma(k)$. Denote this subsequence by $(x_{n_r},X_{n_r})$ and find any limit point $(\bar x,\bar X)$ of this sequence. For notational simplicity we assume that in fact $(x_{n_r},X_{n_r})$ converges to $(\bar x,\bar X)$ as $r\rightarrow\infty$ (as opposed to taking a further subsequence of $n_r$). Then the distance between $(\bar x,\bar X)$ and $\Sigma(k)$ is at least $\epsilon_0$ as well. We now show that in fact $I_k(\bar x,\bar
X)<\infty$, implying that in fact $(\bar x,\bar X)$ belongs to $\Sigma(k)$, which is a contradiction.
To prove the claim we fix $\delta>0$. We have that ${\mathbb{L}}_\infty$ distance between $(x_{n_r},X_{n_r})$ and $(\bar
x,\bar X)$ is at most $\delta$ for all large enough $r$. Since $(x_{n_r},X_{n_r})\in\Sigma_{n_r}(k)$, then for all $r$ there exists some $\hat\sigma_{n_r}:V_{n_r}\rightarrow [k]$ such that $(x_{n_r},X_{n_r})$ arises from the partition $\hat\sigma_{n_r}$. Namely, $x_{n_r}=x({\mathbb{G}}_{n_r}/\hat\sigma_{n_r})$ and $X_{n_r}=X({\mathbb{G}}_{n_r}/\hat\sigma_{n_r})$. Since a randomly uniformly chosen map $\sigma_{n_r}:V_{n_r}\rightarrow [k]$ coincides with $\hat\sigma_{n_r}$ with probability exactly $k^{-n_r}$, we conclude that $$\begin{aligned}
{\mathbb{P}}\left((x(\sigma_{n_r}),X(\sigma_{n_r}))\in B((\bar x,\bar X),\delta)\right)\ge k^{-n_r},\end{aligned}$$ for all sufficiently large $r$. This implies $$\begin{aligned}
\limsup_n \frac 1n&\log{\mathbb{P}}\left((x(\sigma_{n}),X(\sigma_{n}))\in B((\bar x,\bar X),\delta)\right)\\
&\ge\limsup_r \frac 1{n_r}\log{\mathbb{P}}\left((x(\sigma_{n_r}),X(\sigma_{n_r}))\in B((\bar x,\bar X),\delta)\right)\\
&\ge -\log k.\end{aligned}$$ Applying property (\[eq:Identity1\]) we conclude that $I_k(\bar x,\bar X)\le \log k<\infty$, and the claim is established.
Now assume that $k$ and $\epsilon_0>0$ are such that for infinitely many $n$, $\Sigma(k)$ is not a subset of $(\Sigma_n(k))^{\epsilon_0}$. Thus there is a subsequence of points $(x_{n_r},X_{n_r})\in \Sigma(k)$ which are at least $\epsilon_0$ away from $\Sigma_{n_r}(k)$ for each $r$. Let $(\bar x,\bar X)\in {[0,D]^{(k+1)\times k}}$ be any limit point of $(x_{n_r},X_{n_r}), r\ge 1$. By lower semi-continuity we have that the set $\Sigma(k)$ is closed, and thus compact. Therefore, we also have $(\bar x,\bar X)\in\Sigma(k)$. Again, without loss of generality, we assume that in fact $(x_{n_r},X_{n_r})$ converges to $(\bar x,\bar X)$. Fix any $\delta<\epsilon_0/2$. We have $(x_{n_r},X_{n_r})\in B((\bar
x,\bar X),\delta)$ for all sufficiently large $r$. Then for all sufficiently large $r$ we have that the distance from $(\bar
x,\bar X)$ to $\Sigma_{n_r}(k)$ is strictly larger than $\delta$ (since otherwise the distance from $(x_{n_r},X_{n_r})$ to this set is less than $\epsilon_0$). This means that for all sufficiently large $r$, $$\begin{aligned}
{\mathbb{P}}\left((x(\sigma_{n_r}),X(\sigma_{n_r}))\in B((\bar x,\bar X),\delta)\right)=0,\end{aligned}$$ implying $$\begin{aligned}
\liminf_n \frac 1n&\log{\mathbb{P}}\left((x(\sigma_{n}),X(\sigma_{n}))\in B((\bar x,\bar X),\delta)\right)\\
&\le\liminf_r \frac 1{n_r}\log{\mathbb{P}}\left((x(\sigma_{n_r}),X(\sigma_{n_r}))\in B((\bar x,\bar X),\delta)\right)\\
&=-\infty.\end{aligned}$$ Applying property (\[eq:Identity1\]) we conclude that $I_k(\bar x,\bar X)=\infty$, contradicting the fact that $(\bar
x,\bar X)\in \Sigma(k)$. This concludes the proof.
Discussion {#sec:discussion}
==========
Right-convergence for dense vs. sparse graphs
---------------------------------------------
It is instructive to compare our results from Section \[section:LD-implies-Right\], in particular the expression for the limiting free energy from Theorem \[thm:LD-implies-Right\], to the corresponding results from the theory of convergent sequences for dense graphs.
For sequences ${\mathbb{G}}_n$ of dense graphs, i.e., graphs with average degree proportional to the number of vertices, the weighted homomorphism numbers $\hom({\mathbb{G}}_n,{\mathbb{H}})$ typically either grow or decay exponentially in $|V({\mathbb{G}}_n)|^2$, implying that the limit of an expression of the form would be either $\infty$ or $-\infty$. To avoid this problem, we adopt the usual definition from statistical physics, where the free energy is defined by $$\begin{aligned}
\label{F-def-dense}
f({\mathbb{G}}_n,{\mathbb{H}})=-\frac 1{|V({\mathbb{G}}_n)|}\log Z({\mathbb{G}}_n,{\mathbb{H}}),\end{aligned}$$ where $Z({\mathbb{G}}_n,{\mathbb{H}})$ is the [*partition function*]{}[^5] $$Z({\mathbb{G}}_n,{\mathbb{H}})=\sum_{\sigma:V({\mathbb{G}}_n)\rightarrow V({\mathbb{H}})}\prod_{u\in V({\mathbb{G}}_n)}\alpha_{\sigma(u)}\prod_{(u,v)\in E({\mathbb{G}}_n)}\left(A_{\sigma(u),\sigma(v)}\right)^{1/|V({\mathbb{G}}_n)|},$$ with $\alpha$ and $A$ denoting the vector and matrix of the node and edges weights of ${\mathbb{H}}$, respectively.
It was shown in [@BorgsChayesEtAlGraphLimitsII] that for a left-convergent sequence of dense graphs, the limit of exists, and can be expressed in terms of the limiting graphon. We refer the reader to the literature [@LovaszSzegedy] for the definition of the limiting graphon; for us, it is only important that it is a measurable, symmetric function $W:[0,1]^2\to [0,1]$ which can be thought of as the limit of a step function representing the adjacency matrices of ${\mathbb{G}}_n$ (see Corollary 3.9 of [@BorgsChayesEtAlGraphLimitsI] for a precise version of this statement).
To express the limit of the free energies in terms of $W$, we need some notation. Assume $|V({\mathbb{H}})|=k$, and let $\Omega_k$ be the set of measurable functions $\rho=(\rho_1,\dots,\rho_k):[0,1]\to [0,1]^k$ such that $\rho_i$ is non-negative for all $i\in [k]$ and $\sum_i\rho_i(x)=1$ for all $x\in [0,1]$. Setting $
x_i(\rho)=\int_0^1 \rho_i(x)dx
$ and $
X_{i,j}(\rho)=\int_0^1\int_0^1 \rho_i(x)\rho_j(y)W(x,y)dxdy,
$ we define the energy and entropy of a “configuration” $\rho\in \Omega_k$ as $$\mathcal E_{{\mathbb{H}}}(\rho) = -\sum_{1\leq i\leq k} x_i(\rho)\log\alpha_i
-\frac 12\sum_{1\le i,j\le k}X_{i,j}(\rho)
\log A_{i,j}$$ and $$\mathcal S_k(\rho)=-\int_0^1 dx\sum_{1\leq i\leq k}\rho_i(x)\log\rho_i(x),$$ respectively. Then $$\lim_{n\to\infty}f({\mathbb{G}}_n,{\mathbb{H}})
=\inf_{\rho\in\Omega_k}\Bigl(\mathcal E_{{\mathbb{H}}}(\rho) - \mathcal S_k(\rho)\Bigr).$$
Note the striking similarity to the expressions , and . There is, however, one important difference: in the dense case, the limit object $W$ describes an effective edge-density and enters into the definition of the energy, while in the sparse case, the limit object is a large-deviation rate and enters into the entropy. By contrast, the entropy in the dense case is trivial, and does not depend on the sequence ${\mathbb{G}}_n$, similar to the energy in the sparse case, which does not depend on ${\mathbb{G}}_n$.
As we will see in the next subsection, this points to the fact that right-convergence is a much richer concept for sparse graphs than for dense graphs. Indeed, for many interesting sequences of dense graphs, one can explicitly construct the limiting graphon $W$, reducing free energy calculations to simple variational problems. By contrast, we do not even know how to calculate the rate functions for the simplest random sparse graph, the Erdös-Renyi random graph. We discuss this, and how it is related to spin glasses, in the next section.
Conjectures and open questions
------------------------------
While Theorem \[theorem:ConvergenceRelations\] establishes a complete picture regarding the relationships between LD, left, right and partition-convergence, several questions remain open regarding colored-neighborhood-convergence. In particular, not only do we not know whether LD implies colored-neighborhood-convergence or vice verse, but we not even know whether colored-neighborhood-convergence implies right-convergence.
Since colored-neighborhood-convergence does not appear to take into account the counting measures involved in the definition of right-convergence, we conjecture that in fact colored-neighborhood-convergence *does not* imply right-convergence. Should this be the case it would also imply that colored-neighborhood-convergence does not imply LD-convergence, via Theorem \[theorem:ConvergenceRelations\]. At the same time we conjecture that the LD-convergence implies the colored-neighborhood-convergence. It is an interesting open problem to resolve these conjectures.
Perhaps the most important outstanding conjecture regarding the theory of convergent sparse graph sequences concerns convergence of random sparse graphs. Consider two natural and much studied random graphs sequences, the sequence of Erdös-Renyi random graphs ${\mathbb{G}}(n,c/n)$, and the sequence of random regular graphs ${\mathbb{G}}(n,\Delta)$. As usual, ${\mathbb{G}}(n,c/n)$ is the random subgraph of the complete graph on $n$ nodes where each edge is kept independently with probability $c/n$, where $c$ is an $n$-independent constant, and ${\mathbb{G}}(n,\Delta)$ is a $\Delta$-regular graph on $n$ nodes, chosen uniformly at random from the family of all $n$-node $\Delta$-regular graphs. We ask whether these graph sequences are convergent with respect to any of the definitions of graph convergence discussed above[^6].
It is well known and easy to show that these graphs sequences are left-convergent, since the $r$-depth neighborhoods of a random uniformly chosen point can be described in the limit as $n\rightarrow\infty$ as the first $r$ generations of an appropriate branching process. The validity for other notions of convergence, however, presents a serious challenge as here we touch on the rich subject of mathematical aspects of the spin glass theory. The reason is that right-convergence of sparse graph sequences is nothing but convergence of normalized partition functions of the associated Gibbs measures. Such convergence means the existence of the associated thermodynamic limit and this existence question is one of the outstanding open areas in the theory of spin glasses.
Substantial progress has been achieved recently due to the powerful interpolation method introduced by Guerra and Toninelli [@GuerraTon] in the context of the Sherrington-Kirkpatrick model, and further by Franz and Leone [@FranzLeone], Panchenko and Talagrand [@PanchenkoTalagrand], Bayati, Gamarnik and Tetali [@BayatiGamarnikTetali], Abbe and Montanari [@AbbeMontanari] and Gamarnik [@GamarnikRightConvergence]. In particular, a broad class of weighted graphs ${\mathbb{H}}$ is identified, with respect to which the random graph sequence ${\mathbb{G}}(n,c/n)$ is right-convergent w.h.p. (in some appropriate sense). Some extensions of these results are also known for random regular graph sequences ${\mathbb{G}}(n,\Delta)$. A large general class of such weighted graphs ${\mathbb{H}}$ is described in [@GamarnikRightConvergence] in terms of suitable positive semi-definiteness properties of the graph ${\mathbb{H}}$. Another class of graphs ${\mathbb{H}}$ with respect to which convergence is known and in fact the limit can be computed is when ${\mathbb{H}}$ corresponds to the ferromagnetic Ising model and some extensions [@DemboMontanariIsing],[@DemboMontanariSun]. We note that for the latter models, it suffices that the graph sequence is left-convergent to some locally tree-like graphs satisfying some mild additional properties.
Still, the question whether the two most natural random graph sequences ${\mathbb{G}}(n,c/n)$ and ${\mathbb{G}}(n,\Delta)$ are right-convergent (with respect to all ${\mathbb{H}}$) is still open. The same applies to all other notions of convergence discussed in this paper (except for left-convergence). We conjecture that the answer is yes for all of them. Thanks to Theorem \[theorem:ConvergenceRelations\] this would follow from the following conjecture.
The random graph sequences ${\mathbb{G}}(n,c/n)$ and ${\mathbb{G}}(n,\Delta)$ are almost surely LD-convergent.
The quantifier ”almost sure” needs some elaboration. Here we consider the product $\prod_n \mathcal{G}_n$, where $\mathcal{G}_n$ is the set of all simple graphs on $n$ nodes. We equip this product space with the natural product probability measure, arising from having each $\mathcal{G}_n$ induced with random graph ${\mathbb{G}}(n,c/n)$ measure. Our conjecture states that, almost surely with respect to this product probability space $\mathcal{G}$, the graph sequence realization is LD-convergent.
Acknowledgements {#acknowledgements .unnumbered}
================
We gratefully acknowledge fruitful discussions with Laci Lov[á]{}sz and wish to thank him for providing the key ideas for the counterexample in Subsection \[subsection:PartitionDoesNotImplyLeft\]. The third author also wishes to thank Microsoft Research New England for providing exciting and hospitable environment during his visit in 2012.
[^1]: Microsoft Research New England; e-mail: [borgs@microsoft.com]{}.
[^2]: Microsoft Research New England; e-mail: [jchayes@microsoft.com]{}.
[^3]: MIT; e-mail: [gamarnik@mit.edu]{}.Research supported by the NSF grants CMMI-1031332.
[^4]: Some necessary background on Large Deviations Principle and weak convergence of measures will be provided in Section \[section:LDconvergence\].
[^5]: Note that the only difference with respect to is the exponent $1/|V({\mathbb{G}}_n)|$, which ensures that $\log Z({\mathbb{G}}_n,{\mathbb{H}})$ is of order $O(|V_n|)$.
[^6]: In contrast to ${\mathbb{G}}(n,\Delta)$, the graphs ${\mathbb{G}}(n,c/n)$ do not have uniformly bounded degrees, which means that most of the theorems proved in this paper do not hold. Nevertheless, we can still pose the question of convergence of these graphs with respect to various notions.
|
---
abstract: 'In this paper a model incorporating chiral symmetry breaking and dynamical soft-wall AdS/QCD is established. The AdS/QCD background is introduced dynamically as suggested by Wayne de Paula etc al and chiral symmetry breaking is discussed by using a bulk scalar field including a cubic term. The mass spectrum of scalar, vector and axial vector mesons are obtained and a comparison with experimental data is presented.'
author:
- 'Qi Wang$^{1}$[^1], An Min Wang$^{2}$[^2]\'
title: 'Chiral Symmetry Breaking in the Dynamical Soft-Wall Model'
---
Introduction {#secIntro}
============
Quantum Chromodynamics (QCD) is an non-abelian gauge theory of the strong interactions. It has a coupling constant that highly depends on the energy scale, and the perturbation theory doesn’t work at low energies. The Anti-de Sitter/conformal field theory (AdS/CFT) correspondence [@Maldacena:1997re] provides a novel way to relate QCD with a 5-D gravitational theory and calculate many observables more efficiently. This leads to two types of AdS/QCD models: the top-down models that are constructed by branes in string theory and the bottom-up models by the phenomenological aspects. In the top-down model, mesons are identified as open strings with both ends on flavor branes [@Karch:2002sh]. The most successful top-down model in reproducing various facts of low energy QCD dynamics is D4-D8 system of Sakai and Sugimoto [@Sakai:2004cn] [@Sakai:2005yt].
The bottom-up approach assumes that QCD has suitable 5D dual and construct the dual model from a small number of operators that are influential. The first bottom-up model is the so-called hard-wall model [@EKSS] [@DP]; it simply place an infrared cutoff on the fifth dimension to break the chiral symmetry. But in hard-wall model the spectrum of mass square $m_n^2$ grow as $n^2$, which is contradict with the experimental data [@Shifman-Regge]. The analysis of experimental data indicates a Regge behavior of the highly resonance $m_n^2\sim n$. To compensate this difference between QCD and AdS/QCD, soft-wall model was proposed [@Karch:2006pv]. In this model the spacetime smoothly cap off instead of the hard-wall infrared cutoff by introducing a background dilaton field $\Phi$: $$S_{5}=-\int d^{5}x
\sqrt{-g}\,e^{-\Phi(z)}\mathscr{L},$$where the background field is parametrized to reproduce Regge-like mass spectrum. In spite of success in reproducing Regge-like mass spectrum, chiral symmetry breaking in this model is not QCD-like. Some further researches try to incorporate chiral symmetry breaking and confinement in this model by using high order terms in the potential for scalar field [@Gherghetta:2009ac] [@zhangpeng:2010prd].
But the dilaton field and scalar field in the soft-wall model are imposed by hand to and cannot be derived from any equation of motion. Some works on the dilaton-gravity coupled model show that a linear confining background is possible as a solution of the dilation-gravity coupled equations and Regge trajectories of meson spectrum is obtained [@dePaulaPRD09] [@dynamicalmetric].
In this paper, we try to incorporate soft-wall AdS/QCD and dynamical dilaton-gravity model. We introduce the meson sector action with a cubic term of the bulk field in the bulk scalar potential under the dynamical metric background. We derive a nonlinear differential equation related the vacuum expectation value(VEV) of bulk field, wrap factor and dilaton field. An analysis of the asymptotic behavior of the VEV is presented. We obtain that the VEV is a constant in the IR limit that suggests chiral symmetry restoration exists. The coupling constant can be determined by other parameters in this model.
This paper proceeds as follow: A brief review of dynamical gravity-dilaton background is presented in Sec. \[secMetric\]. In Sec. \[model\] we introduce the bulk field and meson sector action under dynamical metric background and obtain the differential equation of the bulk field VEV. An analysis of asymptotic behavior is presented and a parametrized solution is given. Using the parametrized solution we calculate the scalar, vector and axial vector mass spectrums in Sec. \[massspectrum\]. A conclusion is given in Sec. \[conclusion\].
Background Equations {#secMetric}
====================
We start from the Einstein-Hilbert action of five-dimensional gravity coupled to a dilaton $\Phi $ as proposed by de Paula, Frederico, Forkel and Beyer [@dePaulaPRD09]:
$$S=\frac{1}{2\kappa ^{2}}\int d^{5}x\sqrt{-g}\left( -%
\emph{R}+\frac{1}{2}g^{MN}\partial _{M}\Phi \partial _{N}\Phi
-V(\Phi )\right) \label{actiongd}$$
where $\kappa $ is the five-dimensional Newton constant and $V$ is a still general potential for the scalar field. Then we restrict the metric to the form $$g_{MN}=e^{-2A(z)}\eta_{MN} \label{metric}$$ where $\eta_{MN}$ is the Minkowski metric. We write the warp factor as $$A(z)=\ln z+C\left( z\right) \label{wrapfactor}$$
Variation of the action (\[actiongd\]) leads to the Einstein-dilaton equations for the background fields $A$ and $\Phi$: $$\begin{aligned}
6A^{\prime }{}^{2}-\frac{1}{2}\Phi ^{\prime 2}+e^{-2A}V(\Phi ) &=&0
\label{einsteinzz} \\
3A^{\prime \prime }-3A^{\prime }{}^{2}-\frac{1}{2}\Phi ^{\prime
2}-e^{-2A}V(\Phi ) &=&0 \label{einstein00} \\
\Phi ^{\prime \prime }-3A^{\prime }\Phi ^{\prime
}-e^{-2A}\frac{dV}{d\Phi } &=&0 \label{dilatonequation}\end{aligned}$$ Here the prime denotes the derivative with respect to $z$.
By adding the two Einstein equation ones can get the dilaton field directly: $$\Phi ^{\prime }=\sqrt{3}\sqrt{A^{\prime
}{}^{2}+A^{\prime \prime }}~ \label{constrain}$$
and by substituting (\[constrain\]) in (\[einsteinzz\]) or (\[einstein00\]): $$V(\Phi \left( z\right) )=\frac{3e^{2A\left( z\right) }}{2}\left[
A^{\prime \prime }\left( z\right) -3A^{\prime }{}^{2}\left( z\right)
\right] \label{vz}$$
If $A(z)=\log(z)+z^\lambda$ is set for simplicity, the asymptotic behavior of dilation field $\Phi$ in UV and IR limits can be obtained
$$\Phi(z)=\left\{
\begin{array}{ll}
\underrightarrow{z\rightarrow0} & 2\sqrt{3(1+\lambda^{-1})}z^{\frac{\lambda}{2}}\\
\underrightarrow{z\rightarrow\infty} & \sqrt{3}z^\lambda
\end{array}
\right.$$
The Model {#model}
=========
In this section, we will establish the soft-wall AdS/QCD model first introduced in [@Karch:2006pv] under the dynamical background mentioned in Sec. \[secMetric\]. The background geometry is chosen to be 5D AdS space with the metric $$ds^{2}=g_{MN}dx^{M}dx^{N}=e^{-2A}\left(
\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}\right)~ \label{background}$$ where $A$ is the warp factor, and $\eta_{\mu\nu}$ is Minkowski metric. Unlike [@Karch:2006pv] we introduce a background dilaton field $\Phi$ that is coupled with $A$ and $\Phi$ can be determined in terms of $A$ as shown in Sec. \[secMetric\].
In order to describe chiral symmetry breaking, a bifundamental scalar field $X$ is introduced and we add an cubic term potential instead of the quartic term as shown in [@Gherghetta:2009ac] $$S_{5}=-\int d^{5}x \sqrt{-g}\,e^{-\Phi(z)}{\rm
Tr}[|DX|^{2}+m_{X}^{2} |X|^{2} \nonumber$$ $$-\lambda|X|^{3}+\frac{1}{4g_{5}^{2}}(F_{L}^{2}+F_{R}^{2})]
\label{mesonaction}$$ where $m_{X}=-3$, $\lambda$ is a constant and $g_5^2=12\pi^2/N_c=4\pi^2$. The fields $F_{L,R}$ are defined by $$F_{L,R}^{MN} = \partial^{M} A_{L,R}^{N} - \partial^{N} A_{L,R}^{M} -
i [A_{L,R}^{M},A_{L,R}^{N}], \nonumber$$ here $A_{L,R}^{MN} = A_{L,R}^{MN} t^{a}$, $Tr[t^{a} t^{b}] =
\frac{1}{2} \delta^{ab}$, and the covariant derivative is $D^M
X=\partial^M X-i A_L^MX+iXA_R^M$.
Bulk scalar VEV solution and dilaton field
------------------------------------------
The scalar field $X$ is assumed to have a z-dependent VEV $$<X>=\frac{v(z)}{2}\left(\begin{array}{cc}1 & 0\\
0 & 1
\end{array}\right) \label{xvev}$$
We can obtain a nonlinear equation related $A$, $\Phi$ and $v(z)$ from the action (\[mesonaction\]) $$\partial_z( e^{-3A} e^{-\Phi} \partial_z v(z))+ e^{-5A} e^{-\Phi} (3 v(z)+\frac{3}{4}\lambda v^2(z))=0 \label{VEVequation}$$
If $A$ and $\Phi$ are given, this equation can be simplified to a second order differential equation of $v(z)$ $$v''(z)-(\Phi'+3A')v'(z)+e^{-2A}(3 v(z)+\frac{3}{4}\lambda v^2(z))=0
\label{VEVequation1}$$ where $ \Phi ^{\prime }=\sqrt{3}\sqrt{A^{\prime }{}^{2}+A^{\prime
\prime }}~ \label{constrain} $.
Firstly, we consider the limit $z\rightarrow \infty$. If wrap factor $C(z)$ has a asymptotic behavior of $z^2$ as $z\rightarrow \infty$, we can determine that $v(z)\rightarrow$ constant. This means that chiral symmetry restoration in the mass spectrum. But there are many controversies on whether such a restoration really exists [@Cohen:2005am] [@Wagenbrunn:2006cs].
Secondly, we analyze the asymptotic behavior of $v(z)$ as $z\rightarrow 0$. As shown in [@Witten:1998qj] [@Klebanov:1999tb], the VEV as $z\rightarrow 0$ is required to be $$v(z)=m_q\zeta z+\frac{\sigma}{\zeta}z^3$$ where $m_q$ is the quark mass, $\sigma$ is the chiral condensate and $\zeta=\frac{\sqrt{3}}{2\pi}$.
We assume that $C(z)$ takes the asymptotic form as $z\rightarrow 0$ $$C(z)=\alpha z^2+\beta z^3$$ Then we can get the asymptotic form of $\Phi'$ in the IR limit. Considering the coefficients of $\frac{1}{z}$ and constant terms at the left side of Equation (\[VEVequation1\]) should be eliminated as $z\rightarrow 0$, we can conclude that the cubic term of bulk field is necessary and a relation between $\alpha$, $\beta$ and $\lambda$ can be obtained $$\lambda m_q\zeta=4\sqrt{2\alpha}. \label{lambdarelation}$$
A parametrized solution and parameter setting
---------------------------------------------
The equation (\[VEVequation1\]) is a nonlinear differential equation and it is difficult to solve the VEV $v(z)$, so we choose to select a parametrized form of $v(z)$ that satisfies the asymptotic constraint instead of solving the equation directly.
We set the wrap factor as suggested in [@dePaulaPRD09] $$A(z)=\log z+\frac{(\xi\Lambda_{QCD} z)^2}{1+\exp(1-\xi\Lambda_{QCD}
z)} \label{wrapfactor}$$ where $\xi$ is a scale factor and $\Lambda_{QCD}=0.3\textrm{GeV}$ is the QCD scale.
Also we assume the VEV $v(z)$ asymptotically behaves as discussed in previous section $$v(z)=\left\{
\begin{array}{ll}
\underrightarrow{z\rightarrow\infty} & \gamma\\
\underrightarrow{z\rightarrow0} & m_q\zeta z+\frac{\sigma}{\zeta}z^3
\end{array}
\right. \label{VEVasymptotic}$$
A simple parametrized form for $v(z)$ that satisfies asymptotic constraint (\[VEVasymptotic\]) can be chosen as $$v(z)=\frac{z(A+Bz^2)}{\sqrt{1+(Cz)^6}}$$ The relations between $m_q$, $\sigma$, $\gamma$ and $A$, $B$, $C$ are: $$A=m_q\zeta, B=\frac{\sigma}{\zeta}, \frac{B}{C^3}=\gamma$$
Using the data of meson mass data, pion mass and pion decay constant we can get a good fitting of $A$, $B$, $C$, $\lambda$ as follows:
$$A=1.63\textrm{MeV}, B=(157\textrm{MeV})^3, C=10,
\lambda=\frac{4\sqrt{2}\xi \Lambda_{QCD}}{m_q\zeta\sqrt{1+e}}$$
In next section we will use this parametrized solution to calculate scalar, vector and axial-vector meson mass spectra and compare them with experimental data.
Meson Mass Spectrum {#massspectrum}
===================
Starting from the action (\[mesonaction\]), we can derive the Schr$\ddot{o}$dinger-like equations that describe the scalar, vector and axial vector mesons: $$-\partial_z^2S_n(z)+(\frac{1}{4}\omega'^2-\frac{1}{2}\omega''-e^{-2A}(3+\frac{3}{2}\lambda
v(z)))S_n(z)=m^2_{S_n}S_n(z) \nonumber$$
$$-\partial_z^2V_n(z)+(\frac{1}{4}\omega'^2-\frac{1}{2}\omega'')V_n(z)=m^2_{V_n}V_n
\nonumber$$
$$-\partial_z^2A_n(z)+(\frac{1}{4}\omega'^2-\frac{1}{2}\omega''+g_5^2e^{-2A}v^2(z))A_n(z)=m^2_{A_n}A_n(z)
\label{mesonquation}$$
where the prime denotes the derivative with respect to $z$ and $\omega=\Phi(z)+3A(z)$ for scalar mesons, $\omega=\Phi(z)+A(z)$ for vector and axial vector mesons.
The meson mass spectrum is obtained through calculating the eigenvalues of these equations. We will calculate the eigenvalues of these equations numerically using the parametrized solution introduced in Sec. \[model\].
Since the potential is complicated, we must solve the eigenvalue equation numerically. As suggest in [@dePaulaPRD09], we set $\xi=0.58$ for scalars and the numerical results are shown in Table I.
n $f_0 (Exp)$ $f_0 (Model)$
--- -- ------------- ---------------
0 550 897
1 980 1135
2 1350 1348
3 1505 1540
4 1724 1717
5 1992 1881
6 2103 2034
7 2134 2178
: Scalar mesons spectra in MeV.
For vectors, the value of $\xi$ is chosen to be 0.88 and the numerical results are listed in Table II.
n $\rho (Exp)$ $\rho (Model)$
--- -- -------------- ----------------
0 775.5 988.9
1 1465 1348
2 1720 1650
3 1909 1913
4 2149 2147
5 2265 2359
: Vector mesons spectra in MeV.
Sine $\Delta m^2=m^2_{A^2_n}-m^2_{V^2_n}=g^2_5e^{-2A}v^2(z)$ tends to zero as $z\rightarrow 0$, it means the mass of vector and axial-vector mesons are equal at high values.
Conclusion
==========
In this paper we incorporated chiral symmetry breaking into the soft-wall AdS/QCD model with a dynamical background. Then a discussion about the asymptotic behavior of the VEV of the bulk field was presented and a chiral symmetry restoration was suggested in the IR limit. We also introduced a parametrized solution of the VEV of the bulk field and fitted the parameters by using the meson mass spectra and pion mass and its decay constant. The numerical solution of meson mass spectra and comparison with experimental data was also considered. The agreement between the theoretical calculation and experimental data is good.
There are some issues worthy of further consideration. The differential equation of the VEV $v(z)$ needs further study, numerically and analytically, to obtain a more precise form of $v(z)$. The mass spectra of nucleons and some more complicated properties of mesons and nucleons in this model will be our following work.
Acknowledgement
===============
This work has been supported by the National Natural Science Foundation of China under Grant No. 10975125.
[11]{}
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W. de Paula, T. Frederico, H. Forkel and M. Beyer, Phys. Rev. D **79** (2009) 075019; PoS LC2008 (2008) 046.
B. Batell, T. Gherghetta, Phys. Rev. D **78**(2008)026002 \[arXiv:0807.1054 \[hep-ph\]\]
A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D [**74**]{}, 015005 (2006).
T. Gherghetta, J. I. Kapusta and T. M. Kelley, Phys. Rev. D [**79**]{} (2009) 076003 \[arXiv:0902.1998 \[hep-ph\]\]. Peng Zhang, Phys. Rev. D [**82**]{} (2010) 094013 \[arXiv:1007.2163 \[hep-ph\]\].
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[^1]: Email:anomalyfree@gmail.com
[^2]: Email:anmwang@ustc.edu.cn
|
---
abstract: 'Results of observations of the Galactic bulge X-ray source GX 9+9 by the All-Sky Monitor (ASM) and Proportional Counter Array (PCA) onboard the *Rossi X-ray Timing Explorer* are presented. The ASM results show that the 4.19 hour X-ray periodicity first reported by Hertz and Wood in 1987 was weak or not detected for most of the mission prior to late 2004, but then became strong and remained strong for approximately 2 years after which it weakened considerably. When the modulation at the 4.19 hour period is strong, it appears in folded light curves as an intensity dip over $\lesssim30$% of a cycle and is distinctly nonsinusoidal. A number of PCA observations of GX 9+9 were performed before the appearance of strong modulation; two were performed in 2006 during the epoch of strong modulation. Data obtained from the earlier PCA observations yield at best limited evidence of the presence of phase-dependent intensity changes, while the data from the later observations confirm the presence of flux minima with depths and phases compatible with those apparent in folded ASM light curves. Light curves from a *Chandra* observation of GX 9+9 performed in the year 2000 prior to the start of strong modulation show the possible presence of shallow dips at the predicted times. Optical observations performed in 2006 while the X-ray modulation was strong do not show an increase in the degree of modulation at the 4.19 hour period. Implications of the changes in modulation strength in X-rays and other observational results are considered.'
author:
- 'Robert J. Harris, Alan M. Levine, Martin Durant, Albert K. H. Kong, Phil Charles, and Tariq Shahbaz'
title: 'GX 9+9: Variability of the X-ray Orbital Modulation'
---
Introduction
============
The bright Galactic X-ray source GX 9+9 has the characteristics of a low-mass X-ray binary (LMXB) system and particularly of the atoll-type LMXB subclass that is defined by the behavior of the source X-ray energy and power density spectra [@mayer70; @schulz89; @hasvdk89; @2006csxs]. A periodicity of $4.19
\pm 0.02$ hours in its X-ray flux was discovered using [*HEAO*]{} A-1 data by @hw88. The periodic component was weak; its strength, in terms of the amplitude of the best-fit sinusoid, was approximately $4\%$ of the average flux. @hw88 interpreted the modulation as indicative of the orbital period of the binary system. Under the assumption that the system consists of a 1.4 M$_\sun$ neutron star and a Roche-lobe-filling lower-main-sequence star, they inferred that the latter is an early M dwarf with mass $M \approx$ 0.2-0.45 M$_\sun$ and radius $R\approx$ 0.3-0.6 R$_\sun$. They used these values to put an upper bound on the orbital inclination of $i
\lesssim 63 ^\circ$ but, as noted by @s90, apparently they calculated $R_2/a$ incorrectly; the correct bound on the inclination is $i \lesssim 77 ^\circ$.
@s90 found variations of the brightness of the optical counterpart of GX 9+9 at essentially the same period, viz. $4.198 \pm
0.009$ hours. The modulation amplitude was about 0.19 mag peak to peak in the B band in 1987 and appeared to be somewhat larger in 1988. In the latter observations, the amplitude was roughly the same in each of the $B, V,$ and $R$ bands.
@kong06 carried out nearly simultaneous observations of GX 9+9 in 1999 in the optical at the Radcliffe Telescope of the South African Astronomical Observatory and in X-rays with the Proportional Counter Array (PCA) instrument on the [*Rossi X-ray Timing Explorer*]{} ([*RXTE*]{}). They found that modulation at the presumed orbital period was clearly present in the optical, but no modulation at that period was apparent in either the 2-3.5 keV or the 9.7-16 keV photon energy bands. @kong06 also reviewed archival data from a 14.4 hour observation of GX 9+9 made with *EXOSAT* in 1983, a 6.6 hour observation made with *ASCA* in 1994, and a 6.4 hour observation made with *BeppoSAX* in 2000, but did not find significant evidence of the 4.19 hour periodicity.
@corn_etal07 obtained phase-resolved spectra of the optical counterpart in 2004 May and found emission lines which comprise components with differing radial velocity variations that must originate in different places in the binary system, one of which is likely to be the X-ray-illuminated face of the secondary. The 4.19 hour period is clearly evident in the radial velocity variations. Thus, these observations strongly confirm that this is the orbital period. The phenomenon was most clear in the He II $\lambda$4686 line but was also evident in the Bowen blend and around H$\beta$. Following early reports of the detection of the 4.19-h period in the [*RXTE*]{} All-Sky Monitor (ASM) data and variation of the strength of the modulation [@lev06; @levcor06], @corn_etal07 used the ASM data to estimate the period and time of X-ray minimum. They were then able to relate the time of X-ray minimum, the phases of the radial velocity variations, and the variation of the brightness of the optical counterpart in the blue continuum to each other. They also derived model-dependent lower and upper limits on the binary mass ratio and on the orbital velocity of the secondary.
Herein we present the evidence that the X-ray photometric signature of the (presumed) binary orbit of GX 9+9 has undergone dramatic changes over the $\sim$12 years that the source has been monitored with the ASM. In addition to describing the results of our analysis of ASM data, we present results of analysis of a number of PCA observations, of an observation of GX 9+9 performed in 2000 with the [*Chandra X-ray Observatory*]{}, and of optical observations performed in 2006. In Section 2 we describe the instrumentation, observations, and the results from the observations. In Section 3, we summarize our results, suggest for the first time in the literature, to our knowledge, that the observed modulation is closely related to the dipping seen in other LMXBs, and discuss in general the possible implications of this investigation.
Instrumentation, Observations, and Results
==========================================
ASM Observations and Results
----------------------------
The ASM consists of three Scanning Shadow Cameras (SSCs) mounted on a rotating Drive Assembly [@asm96]. Approximately 50,000 measurements of the intensity of GX 9+9, each from a 90-s exposure with a single SSC, were obtained from the beginning of the [*RXTE*]{} mission in early 1996 through 2007. A single exposure yields intensity estimates in each of three spectral bands which nominally correspond to photon energy ranges of 1.5-3, 3-5, and 5-12 keV with sensitivity of a few SSC counts s$^{-1}$ (the Crab Nebula produces intensities of 27, 23, and 25 SSC counts s$^{-1}$ in the 3 bands, respectively). For the analyses presented herein, we use observation times that have been adjusted so as to represent times at the barycenter of the Solar System.
{height="6.0in"}
The properties of the three SSCs have evolved over the $\sim 12$ years of operation of the ASM. SSC 1 (of SSCs 1 - 3) has a slow gas leak that has resulted in the photon energy to pulse height conversion gain increasing by about 10% yr$^{-1}$ or about a factor of three over 12 years. A number of the proportional counter cells in SSCs 2 and 3 have become permanently inoperational at various times since the beginning of the mission because of catastrophic high voltage breakdown events that removed the thin carbon layer on the carbon-coated quartz fiber anodes of those cells. In all three detectors the properties of the carbon coatings of the anodes have gradually changed. These effects are, to first order, removed in the course of the standard analysis procedure of the raw ASM data by the ASM team that produces the light curve files. The procedure is adjusted so that the observed intensities of the Crab Nebula in the four energy bands are more or less stable and close to the intensities seen in SSC 1 in March 1996. The results in the 5-12 keV energy band from SSC 1 at late times are corrected by the largest factors since most events due to photons in this energy range in this SSC are not usable because they are saturated in the amplifiers. Thus, in preparing light curves from GX9+9 for this paper, we compared the results from SSC 1 with those from SSCs 2 and 3. We found that the SSC 1 light curve deviated negligibly from that produced from SSCs 2 and 3 before MJD 54200 and after that time gradually increased up to $\sim 1$ SSC ct s$^{-1}$ above the intensities derived from SSCs 2 and 3 by MJD 54600.
The ASM light curve of GX 9+9 is shown in Figure \[fig:lc\]. It was made using data from all three SSCs for times prior to MJD 54200 and from only SSCs 2 and 3 for times after MJD 54200. It shows a relatively strong source that varies on time scales of years. The intensity changes include a distinct sinusoidal-like component superimposed on a slowly increasing baseline. No spectral changes are apparent in the ASM light curves beyond small differences that are likely to be of instrumental origin.
To obtain a quantitative description of the long-period sinusoidal-like component, we fit the light curve shown in Figure \[fig:lc\] with the simple function $$F_x = a + b(t-50100) + c \sin(2\pi t/P_{\rm long}) + d \cos(2\pi t/P_{\rm long})$$ where $F_x$ is the model 1.5-12 keV X-ray intensity, $t$ is the time as a Modified Julian Date, $P_{\rm long}$ is the to-be-determined period of the sinusoid, and $a, b, c$ and $d$ are also parameters to be determined. The best fit curve, with $P_{\rm long} = 1473$ days, is shown superimposed on the ASM light curve in Figure \[fig:lc\]. The fit is not good, i.e., the value of the reduced $\chi^2$ statistic is much greater than one. Given this fit quality and the small number of cycles of the sinusoid, we cannot confidently conclude that the long-term light curve contains a periodic component that is coherent over more than $\sim$3 cycles. Nonetheless, the form of the light curve suggests that some type of quasiperiodic long-term variability may be present.
In the early stages of a search for periodicities in ASM data using advanced analysis techniques, @shiv05 applied the first of two primary strategies of improving the sensitivity of the search to high frequency variations, and detected the 4.19 hour periodicity in the ASM light curve of GX 9+9. The first strategy involves the use of appropriate weights such as the reciprocals of the variances in Fourier and other types of analysis since the individual ASM measurements have a wide range of associated uncertainties. The second strategy stems from the fact that the observations of the source are obtained with a low duty cycle, i.e., the window function is sparse (and complex). The properties of the window function, in combination with the presence of slow variations of the source intensity, act to hinder the detection of variations on short time scales. The window function power density spectrum (PDS) has substantial power at high frequencies, e.g., 1 cycle d$^{-1}$ and 1 cycle per spacecraft orbit ($\sim95$ minute period). Since the data may be regarded as the product of a (hypothetical) continuous set of source intensity measurements with the window function, a Fourier transform of the data is equivalent to the convolution of a transform of a continuous set of intensity measurements with the window function transform. The high frequency structure in the window function transform acts to spread power at low frequencies in the source intensity to high frequencies in the calculated transform (or, equivalently, the PDS). This effectively raises the noise level at high frequencies.
In our analysis, the sensitivity to high frequency variations is enhanced by subtracting a smoothed version of the light curve from the unsmoothed light curve. To perform the smoothing, we do not simply convolve a box function with the binned light curve data since that would not yield any improvement in the noise level at high frequencies. Rather, we ignore bins which do not contain any actual measurements and we use weights based on estimates of the uncertainties in the individual measurements to compute the smoothed light curve. The “box” used in the smoothing had a length of 0.9 day, so the smoothed light curve displayed only that variability with Fourier components at frequencies below $\sim 1$ day$^{-1}$. The smoothed light curve was subtracted from the unsmoothed light curve, and the difference light curve was Fourier transformed. The results are illustrated in Figure \[fig:pds\]. The center frequency and width (half-width at half-maximum) of the peak correspond to a period of $4.19344 \pm 0.00007$ hours ($0.1747267 \pm 0.0000029$ days).
![Power density spectrum (PDS) of a 1.5-12 keV ASM light curve of GX 9+9 that had been modified to remove variability on time scales longer than $\sim1$ day (see text). The original FFT was oversampled and had approximately 2 million frequencies from 0 cycles day$^{-1}$ to the Nyquist frequency of 144 cycles day$^{-1}$. (upper panel) A rebinned PDS in which the number of frequency bins was reduced by a factor of 300 by using the maximum power in each contiguous set of 300 frequency bins in the original PDS as the power of the corresponding bin in the rebinned PDS. (lower panel) A portion of the original PDS. In both panels, the power is normalized relative to the PDS-wide average. \[fig:pds\]](f2.eps){height="3.4in"}
To investigate the time variability of the orbital modulation, we folded the barycenter-corrected light curves for each of 30 equal time intervals at the 4.19344 hour period; see Figure \[fig:lcpcs\]. The figure shows that the modulation of GX 9+9 was weak during the first 5 years of the mission. It then seems to have appeared at a somewhat detectable level for a brief time in late 2001 and/or early 2002 (panel 15). In late 2004 or early 2005 the modulation grew stronger, and it stayed strong until late in 2006 when it began to decrease in strength. In the final two panels the modulation is not present at a significant level. We note that the interval of relatively strong modulation, i.e., approximately from MJD 53300 to MJD 54100, corresponds to a time interval in which the overall intensity is below the levels given by an interpolation from earlier to later times (see Fig. \[fig:lc\]).
{height="6.0in"}
[lll]{} 1 & 50135.0 & 1996 Feb 22\
2 & 50279.4 & 1996 Jul 15\
3 & 50423.8 & 1996 Dec 6\
4 & 50568.2 & 1997 Apr 30\
5 & 50712.6 & 1997 Sep 21\
6 & 50857.0 & 1998 Feb 13\
7 & 51001.4 & 1998 Jul 7\
8 & 51145.8 & 1998 Nov 28\
9 & 51290.2 & 1999 Apr 22\
10 & 51434.6 & 1999 Sep 13\
11 & 51579.0 & 2000 Feb 5\
12 & 51723.4 & 2000 Jun 28\
13 & 51867.8 & 2000 Nov 19\
14 & 52012.2 & 2001 Apr 13\
15 & 52156.6 & 2001 Sep 4\
16 & 52301.0 & 2002 Jan 27\
17 & 52445.4 & 2002 Jun 20\
18 & 52589.8 & 2002 Nov 11\
19 & 52734.2 & 2003 Apr 5\
20 & 52878.6 & 2003 Aug 27\
21 & 53023.0 & 2004 Jan 19\
22 & 53167.4 & 2004 Jun 11\
23 & 53311.8 & 2004 Nov 2\
24 & 53456.2 & 2005 Mar 27\
25 & 53600.6 & 2005 Aug 18\
26 & 53745.0 & 2006 Jan 10\
27 & 53889.4 & 2006 Jun 3\
28 & 54033.8 & 2006 Oct 25\
29 & 54178.2 & 2007 Mar 19\
30 & 54322.6 & 2007 Aug 10\
31 & 54467.0 & 2008 Jan 2
In Figure \[fig:latefold\] we show folded light curves for the 1.5-3, 3-5, and 5-12 keV photon-energy bands as well as for the overall 1.5-12 keV band for the time interval of MJD 53300 through MJD 54075 when the modulation was strongest. The modulation appears to be largely energy-independent and, in particular, the spectrum of GX 9+9 does not significantly change during the dips. This is confirmed by the hardness ratios computed from these folded light curves by taking bin-wise ratios as shown in the right-hand side of Fig. \[fig:latefold\].
We have used the folded light curves to determine the epoch of X-ray minimum, i.e., the time corresponding to the centroid of the dip-like feature. The best linear fit to the times of the minima is given by: $$T_{\rm min} = 53382.959 \pm 0.003 + (0.1747267 \pm 0.0000029)N$$ where the times are Modified Julian Dates in the UTC time system and apply at the barycenter of the Solar System. One should note that, per Fig. \[fig:latefold\], dip activity may occur within $\pm0.15$ cycles ($= \pm0.63$ hours) of the times of minimum. It must be pointed out that this ephemeris is not consistent with that of @corn_etal07; their time of minimum occurs at $0.24 \pm 0.04$ cycles relative to a time of minimum given by this equation.
PCA Observations and Results
----------------------------
The PCA consists of 5 mechanically-collimated large area Proportional Counter Units (PCUs) and is primarily sensitive to 2-60 keV photons [@pca06 and references therein]. A total of 61 observations of GX 9+9 were carried out with the PCA from 1996 through 2004. Only 29 of these were over one hour in duration; 17 were over 4 hours; the longest was $\sim 15$ hours. Twelve of the 29 observations were performed in the time period 1996-1999; 17 were performed between 2002 and 2004. In addition, per our request a pointed observation of GX 9+9 was carried out on June 20, 2006. One further observation of the source was carried out on September 1, 2006. GX 9+9 typically produced 500 to 700 cts s$^{-1}$ PCU$^{-1}$ in the 2-12 keV energy band in these observations.
For our periodicity search and X-ray “color” analyses using PCA data we have used data accumulated in Standard Mode 2 which provides counts in each of 128 pulse height channels for each PCU for 16 s time intervals. Data accumulated in various event and single-bit modes that provided a time resolution of 500 $\mu$s or better were used for our fast-timing analyses. Background has been neglected for most of the timing analyses. However, in the analyses involving X-ray colors, the data have been corrected for both the background as well as gain drifts over the course of the mission. The times of the data have been corrected to the Solar System barycenter for all of the analyses presented herein.
In response to the discovery of the strengthening of the orbital modulation in the ASM data [@lev06], we searched the pre-2006 archival PCA data for evidence of the 4.19 h periodicity. In our search we used only the 29 observations that were longer than 1 hour. The total exposure obtained during these observations was $\sim460$ ks. Since the ASM data show no significant modulation during the times of the observations, we expected to find no evidence of modulation in the PCA data.
We folded the barycenter-corrected data from the 29 observations at each of a grid of closely spaced periods centered on 4.19344 hrs. We did this for the energy bands 2-5, 5-12, and 12-30 keV. For each folded light curve we computed a $\chi^2$ statistic based on the hypothesis of no variation as a function of phase and plotted the resulting values against folding period. No significant peaks were found. We also found no significant changes in the color of the source that correlated with the phase of the folds.
This search for periodic modulation was done against a background of variability which tends to be stronger at higher than at lower photon energies. On timescales of 256 sec, the RMS variability seen in the PCA light curves, expressed as a fraction of the mean count rate, is $3.1 \%$ in the 2-6 keV band, $6.4 \%$ in the 6-10 keV band, and $9.6
\%$ in the 10-16 keV band. On timescales of 24 min, the RMS is $3.3
\%$ in the 2-6 keV band, $5.1 \% $ in the 6-10 keV band, and $6.9 \%$ in the 10-16 keV band. These results agree roughly with the ASM light curves, which yield a $\sim 4 \%$ RMS intrinsic variability in each of the A, B, and C bands. This should be compared to the original variability estimate of @hw88 who calculated a $4.6 \%$ intrinsic source variability in addition to the $3.8 \%$ sinusoidal modulation at the orbital period (4.19 hrs) in the 0.5-20 keV [*HEAO*]{} A-1 lightcurve.
While the folding analysis did not yield any evidence for persistent modulation that was stable in phase, upon examination of the individual pre-2006 PCA observations, dip-like events were evident on rare occasions; they are illustrated in Figures \[fig:pcamay02\] and \[fig:pca3other\]. In the light curve from the observation on 2002 May 1 (Fig. \[fig:pcamay02\]), the most well-defined dip-like event is seen near 12.5 hours. A weaker dip-like event is seen near 8.3 hours; it is not much more prominent than some other dip-like events in this observation. In the light curve from the observation on 2002 June 6 (top panel of Fig. \[fig:pca3other\]), narrow dips are evident at about 0.6 and 5.3 hours. The times, orbital phases, and depths of these dips are given in Table \[tbl-2\]. Prominent dips are not seen in the light curves from 2002 June 11, merely 5 days later, nor are they evident in the light curves from 2004 April 29. The dips seen on 2002 May 1 and on 2002 June 6 are shorter than the dips seen in the folded ASM light curves. The time of one of these dips is not centered on a time given by eq. (2), but the phases of all four may be consistent with the extent in phase of the dips seen in the folded ASM light curves. We did not find any other particularly noticeable dips in the light curves from any of the other pre-2006 PCA observations that were one or more hours in duration.
{height="3.8in"}
![PCA count rates in the 2-6 keV band during 3 observations of GX 9+9. (top) The observation beginning at 9:06:20 on 2002 June 6 (MJD 52431; observation ID 70022-02-05-00). (middle) That beginning at 10:58:36 on 2002 June 11 (MJD 52436; ID 70022-02-06-00. (bottom) That beginning at 17:03:08 on 2004 April 29 (MJD 53124; ID 70022-02-09-00). The horizontal bars indicate the times of X-ray minimum according to eq. (2). \[fig:pca3other\]](f6.eps){height="3.5in"}
Per our request to the [*RXTE*]{} Mission Scientist for new observations, GX 9+9 was observed for 10 h with the PCA (and HEXTE) on 2006 June 20 (MJD 53906). The results are shown in three different energy bands in Figure \[fig:pcajune06\]. According to eq. (2); we find that the times of minimum flux should occur at $6.37 \pm 0.22$ hours and $10.56 \pm 0.22$ hours (TT) on 2006 June 20. Indeed, prominent dips in the intensity can be seen close to these predicted times in the two lower energy bands and at the earlier of the two times in the 10-18 keV band. Estimates of the times, orbital phases, and depths of these two dips are given in Table \[tbl-2\] wherein the depths are seen to be more or less independent of energy. It is not clear whether the second of these two dips is less prominent in the 10-18 keV band (Fig. \[fig:pcajune06\]) because of generally stronger variability at higher than at lower energies or, alternatively, that it actually is less deep and the uncertainty on the depth given in Table \[tbl-2\] is underestimated.
Another pointed observation was done on 2006 September 1 (MJD 53979). Like that on 2006 June 20, this one also took place during an interval (Fig. \[fig:lcpcs\]) in which the source exhibited strong modulation. With respect to the time reference used for the right half of Fig. \[fig:pcajune06\], eq. (2) predicts times of minimum flux at $11.42 \pm 0.25$ and $15.61 \pm 0.25$ hours. In this observation, dips are evident near 11.7 and 15.9 hours. They are most apparent in the 2-6 keV band. The first dip is hardly evident in the 6-10 and 10-18 keV bands, perhaps because of a flare in which the spectral hardness undergoes a large increase. The dip near 15.9 hours is quite evident in both higher energy bands; times, phases, and fractional depths are given in Table \[tbl-2\]. The depths of this dip are, like those seen on 2006 June 20, also more or less independent of energy. We have not determined the depths or phase of the dip near 11.7 hours because of its proximity to the flare.
The two dips seen in the 2006 June observation and the second dip seen in the 2006 September observation are comparable in terms of fractional depth with those seen in the folded ASM light curves (see interval 27 in Fig. \[fig:lcpcs\]), and are, as noted immediately above, energy independent to a good approximation. They are narrower ($\sim 0.05$ orbital cycles) than the dips in the folded ASM light curves. The phases of the three dips fall within the extent in phase of the dips in the folded ASM light curves.
[ccccccc]{} 5 & 70022-02-01-01 & 2-6 & $8.363 \pm 0.02$ & 52395.9195 & $-0.048 \pm 0.095$ & $0.031 \pm 0.004$\
5 & 70022-02-01-01 & 2-6 & $12.548 \pm 0.01$ & 52396.0939 & $-0.050 \pm 0.095$ & $0.041 \pm 0.002$\
6 & 70022-02-05-00 & 2-6 & $0.575 \pm 0.01$ & 52431.4034 & $0.034 \pm 0.092$ & $0.026 \pm 0.006$\
6 & 70022-02-05-00 & 2-6 & $5.328 \pm 0.01$ & 52431.6014 & $0.168 \pm 0.092$ & $0.067 \pm 0.008$\
7(left) & 92415-01-01-00 & 2-6 & $6.201 \pm 0.01$ & 53906.2584 & $-0.041 \pm 0.053$ & $0.137 \pm 0.003$\
7(left) & 92415-01-01-00 & 6-10 & $6.212 \pm 0.01$ & 53906.2588 & $-0.038 \pm 0.053$ & $0.172 \pm 0.004$\
7(left) & 92415-01-01-00 & 10-18 & $6.204 \pm 0.02$ & 53906.2585 & $-0.040 \pm 0.053$ & $0.171 \pm 0.007$\
7(left) & 92415-01-01-00 & 2-6 & $10.461 \pm 0.02$ & 53906.4359 & $-0.025 \pm 0.053$ & $0.102 \pm 0.003$\
7(left) & 92415-01-01-00 & 6-10 & $10.461 \pm 0.01$ & 53906.4359 & $-0.025 \pm 0.053$ & $0.114 \pm 0.003$\
7(left) & 92415-01-01-00 & 10-18 & $10.443 \pm 0.05$ & 53906.4351 & $-0.029 \pm 0.054$ & $0.095 \pm 0.004$\
7(right) & 92415-01-02-00 & 2-6 & $15.937 \pm 0.01$ & 53979.6640 & $0.076 \pm 0.059$ & $0.098 \pm 0.007$\
7(right) & 92415-01-02-00 & 6-10 & $15.939 \pm 0.01$ & 53979.6641 & $0.077 \pm 0.059$ & $0.117 \pm 0.009$\
7(right) & 92415-01-02-00 & 10-18 & $15.940 \pm 0.01$ & 53979.6642 & $0.077 \pm 0.059$ & $0.108 \pm 0.013$
After searching the PCA observations for evidence of persistent modulation, we attempted to determine whether or not the characteristics of the source’s high frequency PDS had changed since the increase of modulation. We also attempted to see if any change in the X-ray color related phenomenology had occurred in tandem with the modulation strength increase. We report on both the fast-timing and color analyses below.
Fast-Timing Analysis
--------------------
Quasi-periodic oscillations (QPOs) are often seen in the X-ray intensities of neutron-star LMXBs (see the review by @2006csxs and references therein). Theoretical considerations suggest that these oscillations are generated in the accretion disks, although there is no consensus on the details of the physical mechanisms causing the oscillations. GX 9+9 has never been reported to exhibit such QPOs [see, e.g., @wij98]. In this respect it is like the other bright atoll-type GX sources [@2006csxs]. Because of the discovery of the modulation-strength increase in the source, we reviewed all available PCA observations, but did not find statistically significant QPOs. In particular, the source did not exhibit statistically significant QPOs in the frequency range $\sim 0.01 - 1000$ Hz during either the June 2006 or September 2006 observations.
Color Analysis
--------------
GX 9+9, like the other atoll-type GX sources, generally occupies the upper and lower banana regions in a soft color - hard color diagram [@schulz89; @wij98; @2006csxs]. We wondered whether the presence of strong modulation would have any impact on the details of the color-color diagram. We used data only from PCU 3 (of PCUs 1-5) for this analysis. The data were corrected for background and gain changes, and then used to make color-color diagrams for both the June 2006 and September 2006 observations. We then compared these with a color-color diagram made from all data from previous PCA observations of GX 9+9. We found no significant differences among the diagrams.
Chandra Observations
--------------------
The *Chandra X-ray Observatory* ([*CXO*]{}) comprises a high angular resolution X-ray telescope and two focal plane cameras along with a pair of transmission diffraction gratings each of which may be placed in the X-ray beam just behind the mirror assembly for spectral observations. The ACIS-S focal plane array, which was used for the observation described below, is composed of 4 front-illuminated (FI) and 2 back-illuminated (BI) CCDs configured in a 6 by 1 array. It provides a field of approximately 8 arcminutes by 48 arcminutes. The on-axis effective area for either the FI or BI CCDs is 340 cm$^2$ at 1 keV, and the spectral resolution is $E/dE =$ 20-50 (1-6 keV) for the FI CCDs and $E/dE =$ 9-35 (1-6 keV) for the BI CCDs. See @chand02 for further information.
![Intensity as a function of time of GX 9+9 as shown by counts in 200-s time bins obtained from the *Chandra* ACIS-HETG observation. Time is referenced to the beginning of the observation at 2000 August 22 05:20:21 (MJD 51778.2225). The horizontal bars indicate the times of X-ray minimum according to eq. (2). \[fig:chandra\]](f8.eps){height="3.3in"}
A 20 ks long observation of GX 9+9 was carried out with the Observatory on 2000 August 22 beginning at 05:20:21 UTC (MJD 51778.2225); the ACIS-S instrument was used as the focal plane camera, and the High-Energy Transmission Grating (HETG) was in the X-ray beam. The zero-order image of GX 9+9 was placed on the back-illuminated CCD S3. The count rate of good events in the dispersed spectra, i.e., not including the counts in the zero-order image, is shown in Figure \[fig:chandra\]. Only small changes in the rate are evident. Among these changes are two shallow minima with depths of approximately 4% of the mean count rate that are approximately 15 ks apart. These times are consistent with the times of X-ray minimum according to eq. (2).
Optical Observations
--------------------
We observed the field of GX 9+9 in June 2006 in white light, i.e., with no filter, with the CCD imaging camera on the 1.9m Radcliffe Telescope of the South African Astronomical Observatory (SAAO). In order to achieve rapid read-out, we only used a 127$\times$96pixel data area, giving a read-out cycle of 6s. The observing run was seven nights, of which two were clouded out and two partially affected by poor weather. Seeing was variable, but under 1 for substantial periods of the three good nights.
These observations were performed not long after the PCA observations of GX 9+9 on 2006 June 20 while the orbital modulation in X-rays was strong (see Figs. \[fig:lcpcs\] and \[fig:pcajune06\]).
For each image, we subtracted a bias frame and divided by a flat field image prepared for each night. Since GX 9+9 is located between a bright star and two close fainter stars, aperture photometry does not yield reliable results in the presence of seeing variability. We therefore used the DAOphot-II software package [@stet87; @stet90] to fit a point spread function (PSF) to each stellar image; the image of the brightest star served as the PSF and photometric reference. Another star of similar distance from the reference star and similar brightness to GX 9+9 is used as a test of the photometric systematics. It shows a typical uncertainty of 0.01mag (1$\sigma$) during stable conditions. The photometric results were further selected by rejecting those values which deviated by more than 0.05 mag from a 10 m local average and were then averaged in 1 m time bins. The resulting light curves from the three nights with the best conditions are shown in Figure \[fig:optlcs\].
The light curves are plotted as a function of X-ray phase in Fig. \[fig:optfold\]. A marked decline in the intensity of the counterpart is apparent late in the observations particularly on the nights of June 23-24 and 24-25. Thus the tails of the light curves from these nights (shown in red and blue in Fig. \[fig:optfold\]) fall significantly below the other measurements at similar phases. This could easily be an atmospheric effect since the counterpart of GX 9+9 has a relatively blue spectrum compared to most stars deep in the Galactic plane and, thus, may have more blue or UV flux than the comparison star. Furthermore, the field was on the meridian at $\sim0.075$ days before $0^h$ UTC during these observations, and was lowest in the sky near the end of the observations, especially on the nights of June 23-24 and 24-25 (cf. Fig. \[fig:optlcs\]). This also suggests that the use of the comparison star to normalize the flux may not have removed other slow variations in the intensity of the counterpart of GX 9+9. If we discount the tails of the red and blue curves where they fall below the other photometric results, we find that the variability has a strong periodic component. The light curves roughly suggest that optical minimum comes near X-ray phase 0.2, but the non-periodic aspects of the light curves do not allow a firm conclusion to be drawn.
From a Lomb-Scargle periodogram of the data, we find the source varies roughly sinusoidally with a period 4.17$\pm$0.11h, with shorter-term variability superposed. The amplitude of the 4.19 h modulation is roughly $\pm15$%. This is close to or slightly smaller than that seen in previous observations.
Discussion
==========
The long-term ASM light curve of GX 9+9 shows that its intensity has changed slowly over the 12 years that it has been observed. The form of the variations suggest the presence of a periodicity with a period of 1400 to 1600 days, but the light curve is not well-fit by a simple linear function plus a sinusoid. Perhaps the variability is quasiperiodic in nature. Possible causes of the changes include 1) variation in the accretion rate due to the donor star being a long-period variable star that is undergoing small changes in size, 2) the presence of a third star in the system in a $\sim 16$-day orbit around the center of mass of the other two stars which produces the variation by dynamical effects on the close pair [e.g., @maz79; @ford2000; @zdz07], and 3) the presence of a tilted, precessing accretion disk (although the long period would be hard to understand if this is the correct explanation). In any case, it would certainly be of interest to extend the X-ray light curve by monitoring the intensity for many additional years. It would also be of interest to determine whether similar variation is also manifest in the optical band.
A detailed analysis of the ASM data has revealed significant changes in the amplitude of the orbital (4.19 h) modulation in the X-ray intensity. When strong, the modulation is characterized by energy-independent reductions in intensity that are limited in phase; the light curves are definitely nonsinusoidal in form.
We used the ASM data to determine an ephemeris for the times of X-ray minimum. Our epoch differs from that of @corn_etal07; the epoch in the latter report occurs at a phase of $\sim 0.24$ cycles according to eq. (2). In response to our query, we were informed (R. Cornelisse, private communication) that the epoch of the time of minimum that is explicitly stated in §3.1 of @corn_etal07 is indeed incorrect but the X-ray phases used for the analyses and figures in that paper actually had been calculated using the correct epoch.
While the modulation is strong (MJD 53300-54100), the intensity of GX 9+9 is reduced in comparison to that based on an interpolation from the 2003-2004 time interval to the 2007 time frame. However, the degree of modulation during two similar reduced-intensity time intervals (MJD 50300-51100 and MJD 51700-52500) was generally low (with the exception of a small part of the latter interval when it was of moderate strength as seen in interval 15 in Fig. \[fig:lcpcs\]).
PCA data obtained during the two-year interval when the X-ray modulation was strong show evidence of dip-like intensity reductions at orbital phases more or less consistent with the ASM folded light curves. The energy independence of the folded ASM light curves (Fig. \[fig:latefold\]) is confirmed by the results in Table \[tbl-2\] which show that the depths of the dip-like events seen in the PCA data are roughly independent of photon energy. The intensity of GX 9+9 is generally less variable at low X-ray energies, e.g., below 6 keV, than at higher energies. This random variability tends to mask the dip-like reductions at the higher energies. Thus, the reductions are more evident at low X-ray energies. PCA and [*Chandra*]{} data obtained prior to this two-year interval show that if dip-like events were present, they tended to be rather shallow. The dip-like intensity reductions, regardless if they occurred during the two year time interval when the modulation was strong on average or at other times, are relatively limited in phase; they generally lasted less than 0.1 orbital periods. On the whole, the PCA and [*Chandra*]{} data suggest that the form of the light curves seen in the ASM data is the result of the superposition of many dips with preferred but varying phases, depths, and widths.
The amplitude of the 4.19 h modulation in the 2006 optical observations is comparable to that seen in 1999 by @kong06, to that seen in 2004 by @corn_etal07, and also to that seen in 1987 and 1988 by @s90 even though the X-ray modulation strength was much greater around the time of the 2006 observations than at the time of the 1999 or 2004 observations.
The light curves from the 2006 optical observations, which covered just over three orbital cycles, indicate that the time of minimum optical flux is likely around 0.2 cycles after the time of minimum X-ray flux given by eq. (2). However, this conclusion about the optical-to-x-ray phase relationship is not secure because of deviations in the light curves from one cycle to another. The light curves obtained by @kong06 from just over four orbital cycles in 1999 present a similar picture; phase zero in their Figures 5 and 6 corresponds (by coincidence) to phase $0.01 \pm 0.20$ according to our X-ray ephemeris. Note that all four nightly folded light curves in their Fig. 6 have minima near phase 0.2. In contrast, @corn_etal07 obtained light curves over three orbital cycles in 2004 and concluded that the time of minimum optical flux (in the continuum) corresponded closely to the time of minimum X-ray flux. @corn_etal07 did not show the individual light curves nor do we know the precise time they used as the time of minimum X-ray flux. It is possible that the X-ray and optical light curves of GX 9+9 do not maintain a strict phase relationship. Extensive further observations will be needed to resolve this question.
The X-ray intensity reductions in GX 9+9 are grossly similar to the dips in 10 or more LMXBs such as 4U1915-05, X1254-690, EXO 0748-676, X1755-33, or X1746-370 that appear at certain (limited) orbital phases as absorption-like events. In such dipping sources the dips do not always occur at precisely the same orbital phases, often exhibit complicated structure, and often are stronger at low energies in a manner that suggests photoelectric absorption. The dips occur with a wide range of depths and can have durations up to as much as 25% or 30% of an orbital period. At one extreme, the source 4U1915-05 has exhibited dips that were essentially 100% deep, showed evidence of photoelectric absorption, and were, in some cases, temporally complex [@walt82; @whtswk82; @chdot97]. At the other extreme, the dips seen in X1755-33 were not much more than 30% deep, had essentially no energy dependence, and, at least in some cases, were relatively simple in their degree of structure [@whpar84; @ch1993]. It is common for the characteristics of the dips seen in a given source to vary; see, e.g., @bcs04 for a discussion of changes in the dipping amplitude for X1746-370, @parm86 and @homan03 for reports on dipping activity in EXO 0748-676, and @sw99 for a report on the entire cessation of dipping for a time in X1254-690.
No evidence of enhanced absorption at low energy is apparent in the folded ASM light curves that show the GX 9+9 dips. This is confirmed by a few dips seen in PCA light curves which also show that the intensity reductions do not have the degree of structure seen in individual orbital cycles in many dippers. However, the characteristics of dips seen in various sources are rather diverse, and the diplike events in GX 9+9 are reminiscent of the dips that were evident in the [*EXOSAT*]{} observations of X1755-33 [@whpar84; @ch1993].
The dipping phenomenon has been explained at a general level as due to the occultation of the X-ray source by localized regions of enhanced density relatively far from the orbital plane around the place where the stream of gas from the donor impacts the accretion disk [@lubow76; @whtswk82; @walt82]. According to this general idea, dips will only be evident in those LMXBs whose orbital planes are inclined to the line of sight in a relatively narrow range, namely those not at such high inclinations that our line of sight to the neutron star always intercepts the high density parts of the disk nor those at such low inclinations that the line of sight is never intercepted by the thick structures in the gas stream/disk collision region. Since the present results on GX 9+9 are similar to those previously seen in X1755-33, they do not suggest extensions of the conventional picture, but they do indicate that GX 9+9 is viewed at relatively high inclination like the other dippers. The energy independence of the dips seen in GX9+9 may then be the product of different degrees of energy-dependent absorption of multiple emission components such as discussed by @ch1993 in the context of X1755-33 rather than the product of high ionization or elemental abundance anomalies [@whpar84]. However, one should note that there is not at this time a full consensus on the interpretations of the observations of dips especially on the locations and sizes of the emission components and the degree of ionization of the absorbers [e.g., @chbal04; @chetal05; @diaz06 and references therein].
We thank an anonymous referee for helpful comments. ATS and MD acknowledge support from the Spanish Ministry of Science and Technology under the grant AYA2004–02646 and AYA2007–66887.
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---
abstract: |
Given a semiring with a preorder subject to certain conditions, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is a compact Hausdorff space together with a map from the semiring to the ring of continuous functions, which contains all information required to asymptotically compare large powers of the elements.
Compactness of the asymptotic spectrum is closely tied with a boundedness condition assumed in Strassen’s work. In this paper we present a generalization that relaxes this condition while still allowing asymptotic comparison via continuous functions on a locally compact Hausdorff space.
author:
- Péter Vrana
bibliography:
- 'refs.bib'
title: 'A generalization of Strassen’s spectral theorem'
---
Introduction
============
Motivated by the study of asymptotic tensor rank and more generally relative bilinear complexity, in [@strassen1988asymptotic] Strassen developed the theory of asymptotic spectra of preordered semirings (see also [@zuiddam2018algebraic Chapter 2] for a recent exposition). The spectrum of a (commutative, unital) semiring $S$ with respect to a preorder ${\preccurlyeq}$ is the set $\Delta(S,{\preccurlyeq})$ of ${\preccurlyeq}$-monotone homomorphisms $S\to\mathbb{R}_{\ge 0}$, i.e. maps $f$ satisfying $f(x+y)=f(x)+f(y)$, $f(xy)=f(x)f(y)$, $f(1)=1$ and $x{\preccurlyeq}y\implies f(x)\le f(y)$. Strassen introduces the asymptotic preorder ${\precsim}$ as $x{\succsim}y$ iff there is a sublinear nonnegative integer sequence $(k_n)_{n\in\mathbb{N}}$ such that for all $n$ the inequality $2^{k_n}x^n{\succcurlyeq}y^n$ holds (see Section \[sec:asymptoticpreorder\] for precise definitions). Every element $f\in\Delta(S,{\preccurlyeq})$ satisfies $x{\succsim}y\implies f(x)\ge f(y)$. The key insight is that under an additional boundedness assumption the converse also holds in the sense that $\left(\forall f\in\Delta(S,{\preccurlyeq}):f(x)\ge f(y)\right)\implies x{\succsim}y$ (see the precise statement below).
In the main application in [@strassen1988asymptotic] the role of $S$ is played by the set of (equivalence classes of) tensors over a fixed field and of fixed order but arbitrary finite dimension. The operations are the direct sum and the tensor (Kronecker) product of tensors and the preorder is given by tensor restriction (alternatively: degeneration). Asymptotic tensor rank can be characterized in terms of the resulting asymptotic restriction preorder, and as a consequence also in terms of the asymptotic spectrum.
More recently, the theory of asymptotic spectra has been applied to a range of other problems as well. In [@zuiddam2019asymptotic] Zuiddam introduced the asymptotic spectrum of graphs and found a dual characterization of the Shannon capacity. Continuing this line of research, Li and Zuiddam [@li2018quantum] studied the quantum Shannon capacity and entanglement-assisted quantum capacity of graphs as well as the entanglement-assisted quantum capacity of noncommutative graphs via the asymptotic spectra of suitable semirings of graphs and noncommutative graphs. In the context of quantum information theory, tensors in Hilbert spaces represent entangled pure states, and the relevant preorder is given by local operations and classical communication. This again gives rise to a preordered semiring [@jensen2019asymptotic] which refines the tensor restriction preorder and which also fits into Strassen’s framework. In this application, the asymptotic spectrum provides a characterization of converse error exponents for asymptotic entantanglement transformations.
At the same time, it became clear that the boundedness condition in Strassen’s theorem is too restrictive for some purposes. To motivate the need for a relaxation of this condition, note first that the concept of asymptotic preorder does not make use of the additive structure and in the applications one is mainly interested in the ordered commutative monoid $S\setminus\{0\}$ with multiplication and the preorder. Such objects have been studied in [@fritz2017resource] as a mathematical model for resource theories. In this context, asymptotic properties such as the asymptotic preorder above and more generally, rate formulas are of central interest. [@fritz2017resource Theorem 8.24.] implies that if an ordered commutative monoid has an element $u$ such that for every $x$ there is $k$ such that $u^k{\succcurlyeq}x$ and $u^kx{\succcurlyeq}1$, then $u^{o(n)}x^n{\succcurlyeq}y^n$ for all large $n$ iff for all monotone homomorphisms $f$ into the semigroup $\mathbb{R}_{\ge0}$ (with multiplication and the usual order) the inequality $f(x)\ge f(y)$ holds. At this point the relation to Strassen’s result should be clear: if the monoid in question is the multiplicative monoid of a preordered semiring and the above condition is satisfied with $u=2$, then one can restrict to monotones that preserve both operations. One may wonder if $u=2$ is necessary for this stronger conclusion to hold, but simple examples show that without this assumption the set of monotone semiring homomorphisms can be too small to characterize the asymptotic preorder. In this spirit, Fritz proved a generalization of Strassen’s theorem [@fritz2018generalization], listing conditions that are equivalent to $\forall f\in\Delta(S,{\preccurlyeq}):f(x)\ge f(y)$ and generalize the asymptotic preorder in different ways, emphasizing also the similarity to Positivstellensätze.
Our main result is a sufficient condition under which the spectrum does characterize the asymptotic preorder as above and which generalizes Strassen’s condition. Before stating our main theorem, let us recall Strassen’s result in a form that eases comparison. We say that an element $u\in S$ is power universal [@fritz2018generalization Definition 2.7.] if for every $x\in S\setminus\{0\}$ there is a $k\in\mathbb{N}$ such that $u^kx{\succcurlyeq}1$ and $u^k{\succcurlyeq}x$. If $S$ contains a power universal element than $S$ is said to be of polynomial growth. The asymptotic preorder is defined as $x{\succsim}y$ iff there is a sublinear nonnegative integer sequence $(k_n)_{n\in\mathbb{N}}$ such that for all $n$ the inequality $u^{k_n}x^n{\succcurlyeq}y^n$ holds. Note that the asymptotic preorder does not depend on the choice of the power universal element (see Lemma \[lem:changepoweruniversal\] below). With these definitions Strassen’s theorem on asymptotic spectra can be stated as follows.
\[thm:Strassen\] Let $(S,{\preccurlyeq})$ be a preordered semiring such that the canonical map $\mathbb{N}\hookrightarrow S$ is an order embedding, and suppose that $u=2$ is power universal. Then for every $x,y\in S$ we have $$x{\succsim}y\iff\forall f\in\Delta(S,{\preccurlyeq}):f(x)\ge f(y).$$
$\Delta(S,{\preccurlyeq})$ is a nonempty compact Hausdorff space.
Our main result the following (for $s\in S$, $\operatorname{ev}_s:\Delta(S,{\preccurlyeq})$ denotes the evaluation map $f\mapsto f(s)$).
\[thm:main\] Let $(S,{\preccurlyeq})$ be a preordered semiring of polynomial growth such that the canonical map $\mathbb{N}\hookrightarrow S$ is an order embedding, and let $M\subseteq S$ and $S_0$ the subsemiring generated by $M$. Suppose that
1. \[it:invertibleuptobounded\] for all $s\in S\setminus\{0\}$ there exist $m\in M$ and $n\in\mathbb{N}$ such that $1{\preccurlyeq}nms$ and $ms{\preccurlyeq}n$
2. \[it:boundedev\] for all $m\in M$ such that $\operatorname{ev}_m:\Delta(S_0,{\preccurlyeq})\to\mathbb{R}_{\ge 0}$ is bounded there is an $n\in\mathbb{N}$ such that $m{\preccurlyeq}n$.
Then for every $x,y\in S$ we have $$x{\succsim}y\iff\forall f\in\Delta(S,{\preccurlyeq}):f(x)\ge f(y).$$
$\Delta(S,{\preccurlyeq})$ is a nonempty locally compact Hausdorff space and if $u$ is power universal then $\operatorname{ev}_u:\Delta(S,{\preccurlyeq})\to\mathbb{R}_{\ge 0}$ is proper.
This result is a generalization of Theorem \[thm:Strassen\]: if $u=2$ is power universal then one can choose $M=\{1\}$ for which both conditions are easily verified, whereas the topological part follows from the fact that $\operatorname{ev}_u$ is a constant map in this case.
It may happen that \[it:invertibleuptobounded\] is not satisfied by any subset $M$ of $S$ but can be satisfied after localizing at a suitable multiplicative set $T$. We will see that localization does not affect the asymptotic preorder, which leads to the following corollary, a somewhat more flexible version of our main result:
\[cor:main\] Let $(S,{\preccurlyeq})$ be a preordered semiring of polynomial growth such that $\mathbb{N}\hookrightarrow S$ is an order embedding. Let $M\subseteq S$ and $T\subseteq S\setminus\{0\}$ be a multiplicative set containing $1$, and let $S_0$ be the subsemiring generated by $M\cup T$. Suppose that
1. \[it:fracinvertibleuptobounded\] for all $s\in S\setminus\{0\}$ there exist $m\in M$, $t_1,t_2\in T$ and $n\in\mathbb{N}$ such that $t_2{\preccurlyeq}n mt_1s$ and $mt_1s{\preccurlyeq}n t_2$
2. \[it:fracboundedev\] for all $m\in M$ and $t_1,t_2\in T$ such that $\operatorname{ev}_m\frac{\operatorname{ev}_{t_1}}{\operatorname{ev}_{t_2}}:\Delta(S_0,{\preccurlyeq})\to\mathbb{R}_{\ge 0}$ is bounded there is an $n\in\mathbb{N}$ such that $mt_1{\preccurlyeq}n t_2$. Then for every $x,y\in S$ we have $$\label{eq:maincor}
x{\succsim}y\iff\forall f\in\Delta(S,{\preccurlyeq}):f(x)\ge f(y).$$
In particular, choosing $M=\{1\}$ and $T=S\setminus\{0\}$ (effectively repacing $S$ with its semifield of fractions) guarantees \[it:fracinvertibleuptobounded\], but then verifying \[it:fracboundedev\] (which is also a necessary condition in this case) may require a fairly detailed knowledge of $\Delta(S,{\preccurlyeq})$. In practice it is desirable to choose $M$ and $T$ in such a way that $S_0$ is as simple as possible. Note however that having a complete classification of the elements of $\Delta(S_0,{\preccurlyeq})$ is not a prerequisite for verifying \[it:fracboundedev\].
Under the conditions of Theorem \[thm:Strassen\], the asymptotic spectrum has a certain uniqueness (or minimality) property [@strassen1988asymptotic Corollary 2.7.]. To state the version that is true in the more general setting of Theorem \[thm:main\], note first that the map $s\mapsto\operatorname{ev}_s$ is a semiring homomorphism into $C(\Delta(S,{\preccurlyeq}))$, the space of continuous functions on $\Delta(S,{\preccurlyeq})$, monotone with respect to the pointwise partial order. Given a preordered semiring $(S,{\preccurlyeq})$ of polynomial growth, let us call a pair $(X,\Phi)$ an abstract asymptotic spectrum for $(S,{\preccurlyeq})$ if $X$ is a locally compact topological space, $\Phi:S\to C(X)$ a semiring homomorphism such that $\Phi(S)$ separates the points of $X$, the image of every power universal element is a proper map, and $\forall x,y\in S:x{\succsim}y\iff\Phi(x)\ge\Phi(y)$.
\[prop:uniqueness\] Let $(X,\Phi)$ be an abstract asymptotic spectrum for $(S,{\preccurlyeq})$. Then there is a unique homeomorphism $h:X\to\Delta(S,{\preccurlyeq})$ such that $\forall s\in S:\Phi(s)=\operatorname{ev}_s\circ h$.
The paper is structured as follows. Section \[sec:asymptoticpreorder\] studies basic properties of preordered semirings, including the asymptotic preorder and its relation to localization. In Section \[sec:spectrum\] we define the spectrum as a topological space and study the continuous maps between spectra induced by monotone homomorphisms between preordered semirings. Section \[sec:mainproof\] contains the proof of our main result, Theorem \[thm:main\], and of Proposition \[prop:uniqueness\].
Asymptotic preorder {#sec:asymptoticpreorder}
===================
By a semiring we mean a set equipped with binary operations $+$ and $\cdot$ that are commutative and associative and have neutral elements $0$ and $1$ such that $0\cdot a=a$ and $a(b+c)=ab+ac$ for any elements $a,b,c$. A semiring homomorphism $\varphi:S_1\to S_2$ is a map satisfying $\varphi(0)=0$, $\varphi(1)=1$, $\varphi(a+b)=\varphi(a)+\varphi(b)$ and $\varphi(ab)=\varphi(a)\varphi(b)$ for $a,b\in S_1$.
A preordered semiring is a pair $(S,{\preccurlyeq})$ where $S$ is a semiring, ${\preccurlyeq}$ is a transitive and reflexive relation on $S$ such that $0{\preccurlyeq}1$ and when $a,b,c\in S$ satisfy $a{\preccurlyeq}b$ then $a+c{\preccurlyeq}b+c$ and $ac{\preccurlyeq}bc$.
Let $(S_1,{\preccurlyeq}_1)$ and $(S_2,{\preccurlyeq}_2)$ be preordered semirings. A semiring homomorphism $\varphi:S_1\to S_2$ is monotone if $a,b\in S_1$, $a{\preccurlyeq}_1 b$ implies $\varphi(a){\preccurlyeq}_2\varphi(b)$.
Let $(S,{\preccurlyeq})$ be a preordered semiring. An element $u\in S$ is power universal if $u{\succcurlyeq}1$ and for every $x\in S$ there is a $k\in\mathbb{N}$ such that $x{\preccurlyeq}u^k$ and $u^kx{\succcurlyeq}1$.
If such an element exists then $S$ is said to be of polynomial growth.
It is clear that any element larger than a power universal one is also power universal. More generally, if $u'{\succcurlyeq}1$ and $u{\preccurlyeq}(u')^k$ for some $k\in\mathbb{N}$ then $u'$ is also power universal.
With the help of a power universal element we can define a generalization of the asymptotic preorder [@strassen1988asymptotic (2.7)] as follows.
\[def:asymptoticge\] Let $(S,{\preccurlyeq})$ be a preordered semiring and $u\in S$ a power universal element. The asymptotic preorder ${\precsim}_u$ is defined as $x{\succsim}_u y$ iff there is a sequence $(k_n)_{n\in\mathbb{N}}$ such that $$\lim_{n\to\infty}\frac{k_n}{n}=0$$ and $$\label{eq:asymptoticgedef}
\forall n\in\mathbb{N}:u^{k_n}x^n{\succcurlyeq}y^n.$$
Since $u{\succcurlyeq}1$, we may assume whenever convenient that $(k_n)_{n\in\mathbb{N}}$ is nondecreasing. By multiplying the inequalities we may replace $k_n$ with a subadditive sequence, or even require the inequality in only for infinitely many $n$.
The asymptotic preorder is defined in terms of a power universal element. However, changing to a different power universal element does not affect the resulting preorder, as the following lemma shows.
\[lem:changepoweruniversal\] If $u_1$ and $u_2$ are power universal elements of $S$ then ${\precsim}_{u_1}={\precsim}_{u_2}$.
By symmetry it is enough to prove ${\precsim}_{u_1}\supseteq{\precsim}_{u_2}$. Let $k\in\mathbb{N}$ such that $u_2{\preccurlyeq}u_1^k$. Suppose that $x{\succsim}_{u_2}y$. This means there is a sublinear sequence $(k_n)_{n\in\mathbb{N}}$ of natural numbers such that holds with $u_2$. Then we have $u_1^{kk_n}x^n{\succcurlyeq}u_2^{k_n}x^n{\succcurlyeq}y^n$ for all $n$ and $kk_n/n\to 0$, therefore $x{\succsim}_{u_1}y$.
By Lemma \[lem:changepoweruniversal\] the asymptotic preorder is determined by the preordered semiring of polynomial growth even without specifying a power universal element. For this reason we will drop the subscript from the notation and write ${\precsim}$ for the asymptotic preorder.
The following pair of lemmas show basic properties of the asymptotic preorder. Some of these are analogous to those in [@zuiddam2018algebraic Lemma 2.3., Lemma 2.4.], with nearly identical proofs.
\[lem:asymptoticproperties\] Let $(S,{\preccurlyeq})$ be a preordered semiring and $u$ a power universal element. Then
(i) \[it:asymptoticlarger\] ${\preccurlyeq}\subseteq{\precsim}$.
(ii) \[it:asymptoticpreorderedsemiring\] $(S,{\precsim})$ is a preordered semiring.
(iii) \[it:asymptoticpolygrowth\] $u$ is power universal with respect to the asymptotic preorder.
(iv) \[it:asymptoticcancellation\] Suppose that $x,y\in S$ and there is an $s\in S\setminus\{0\}$ such that $sx{\succcurlyeq}sy$. Then $x{\succsim}y$.
(v) \[it:asymptoticsmallfactors\] Let $x,y,s,t\in S\setminus\{0\}$ and suppose that for all $n\in\mathbb{N}$ the inequality $sx^n{\succcurlyeq}ty^n$ holds. Then $x{\succsim}y$.
\[it:asymptoticlarger\]: If $x{\succcurlyeq}y$ then is satisfied with $k_n=0$. \[it:asymptoticpreorderedsemiring\]: Reflexivity and $0{\precsim}1$ follows from \[it:asymptoticlarger\]. To prove transitivity let $x{\succsim}y$ and $y{\succsim}z$. Choose sublinear sequences $(k_n)_{n\in\mathbb{N}}$ and $(l_n)_{n\in\mathbb{N}}$ such that $u^{k_n}x^n{\succcurlyeq}y^n$ and $u^{l_n}y^n{\succcurlyeq}z^n$ for all $n$. Then $n\mapsto k_n+l_n$ is also sublinear and $u^{k_n+l_n}x^n{\succcurlyeq}u^{l_n}y^n{\succcurlyeq}z^n$, therefore $x{\succsim}z$. We prove compatibility with the operations. Let $x{\succsim}y$ and $z\in S$, and choose $(k_n)_{n\in\mathbb{N}}$ sublinear and nondecreasing such that $u^{k_n}x^n{\succcurlyeq}y^n$ for all $n$. Then also $u^{k_n}(xz)^n{\succcurlyeq}(yz)^n$ for all $n$, therefore $xz{\succsim}yz$. For the sum we use $$\begin{split}
u^{k_n}(x+z)^n
& = u^{k_n}\sum_{m=0}^n\binom{n}{m}x^mz^{n-m} \\
& {\succcurlyeq}\sum_{m=0}^n\binom{n}{m}u^{k_m}x^mz^{n-m} \\
& {\succcurlyeq}\sum_{m=0}^n\binom{n}{m}y^mz^{n-m} \\
& = (y+z)^n,
\end{split}$$ therefore $x+z{\succsim}y+z$. \[it:asymptoticpolygrowth\] is an immediate consequence of \[it:asymptoticlarger\]. \[it:asymptoticcancellation\]: Let $k\in\mathbb{N}$ be such that $u^ks{\succcurlyeq}1$ and $u^k{\succcurlyeq}s$. By induction, $sx^n{\succcurlyeq}sx^{n-1}y{\succcurlyeq}\ldots{\succcurlyeq}sy^n$, and therefore $u^kx^n{\succcurlyeq}y^n$ for all $n$. This implies $x{\succsim}y$. \[it:asymptoticsmallfactors\]: Let $k,l\in\mathbb{N}$ be such that $u^kt{\succcurlyeq}1$ and $u^l{\succcurlyeq}s$. Then $u^{k+l}x^n{\succcurlyeq}u^ksx^n{\succcurlyeq}u^kty^n{\succcurlyeq}y^n$ for all $n$, therefore is satisfied with $k_n=k+l$.
By \[it:asymptoticpreorderedsemiring\] and \[it:asymptoticpolygrowth\] of Lemma \[lem:asymptoticproperties\] we can iterate the construction and form the asymptotic preorder ${\precapprox}$ that compares large powers via ${\precsim}$. However, the following lemma tells us that this does not give anything new.
\[lem:aasymptotic\] ${\precapprox}={\precsim}$.
We have ${\precsim}\subseteq{\precapprox}$ by \[it:asymptoticlarger\] of Lemma \[lem:asymptoticproperties\]. Let $x{\succapprox}y$ and choose $(k_n)_{n\in\mathbb{N}}$ sublinear such that $u^{k_n}x^n{\succsim}y^n$ for all $n$. Let $(l_{n,m})_{n,m\in\mathbb{N}}$ be such that for all $n$ $\lim_{m\to\infty}l_{n,m}/m=0$ and for all $n,m$ $$u^{l_{n,m}+mk_n}x^{nm}=u^{l_{n,m}}(u^{k_n}x^n)^m{\succcurlyeq}(y^n)^m=y^{nm}$$ holds. Choose a sequence $n\mapsto m_n$ such that $l_{n,m_n}\le m_n$ (e.g. let $m_n$ be the first index such that $l_{n,m_n}\le m_n$). Then $$\lim_{n\to\infty}\frac{l_{n,m_n}+m_nk_n}{nm_n}\le\lim_{n\to\infty}\frac{1}{n}+\frac{k_n}{n}=0,$$ therefore $x{\succsim}y$.
Our next goal is to understand how the asymptotic preorder interacts with localization. Let $T\subseteq S\setminus\{0\}$ be a multiplicative subset (i.e. $t_1,t_2\in T\implies t_1t_2\in T$) containing $1$. The localization of $S$ at $T$, denoted $T^{-1}S$ is the semiring $S\times T$ modulo the equivalence relation $(s_1,t_1)\sim(s_2,t_2)$ iff there exists $r\in T$ such that $rs_1t_2=rs_2t_1$, equipped with the operations $(s_1,t_1)+(s_2,t_2)=(s_1t_2+s_2t_1,t_1t_2)$, $(s_1,t_1)\cdot(s_2,t_2)=(s_1s_2,t_1t_2)$. We denote the equivalence class of $(s,t)$ by $\frac{s}{t}$. There is a canonical homomorphism $S\to T^{-1}S$ that sends $s$ to $\frac{s}{1}$. The additive and multiplicative neutral elements are $\frac{0}{1}$ and $\frac{1}{1}$.
\[lem:localizationpreorderwelldefined\] Let $s_1,s_2,s_1',s_2'\in S$ and $t_1,t_2,t_1',t_2'\in T$ such that $\frac{s_1}{t_1}=\frac{s_1'}{t_1'}$ and $\frac{s_2}{t_2}=\frac{s_2'}{t_2'}$. Then $\exists r\in T:rs_1t_2{\succcurlyeq}rs_2t_1$ iff $\exists r'\in T:r's_1't_2'{\succcurlyeq}r's_2't_1'$.
The roles of the primed and unprimed elements are symmetric, therefore it is enough to show that $rs_1t_2{\succcurlyeq}rs_2t_1$ with $r\in T$ implies $\exists r'\in T:r's_1't_2'{\succcurlyeq}r's_2't_1'$. Let $q_1,q_2\in T$ such that $q_1s_1t_1'=q_1s_1't_1$ and $q_2s_2t_2'=q_2s_2't_2$. Let $r'=(q_1q_2s_1t_2r)$. Then $$\begin{split}
r's_1't_2'
& = q_1q_2s_1t_2rs_1't_2' \\
& = q_1q_2s_1't_2'(rs_1t_2) \\
& {\succcurlyeq}q_1q_2s_1't_2'(rs_2t_1) \\
& = (q_1t_1s_1')(q_2s_2t_2')r \\
& = (q_1t_1's_1)(q_2s_2't_2)r \\
& = (q_1q_2s_1t_2)rs_2't_1' \\
& = r's_2't_1'.
\end{split}$$
According to Lemma \[lem:localizationpreorderwelldefined\] we can define a relation ${\preccurlyeq}$ on $T^{-1}S$ as $$\label{eq:localizationpreorderdef}
\frac{s_1}{t_1}{\succcurlyeq}\frac{s_2}{t_2}\iff\exists r\in T:rs_1t_2{\succcurlyeq}rs_2t_1.$$
(i) \[it:localizationpreorderedsemiring\] $(T^{-1}S,{\preccurlyeq})$ is a preordered semiring.
(ii) \[it:localizationmapmonotone\] The canonical homomorphism $s\mapsto\frac{s}{1}$ is monotone.
(iii) \[it:localizationpoweruniversal\] If $u\in S$ is power universal then $\frac{u}{1}$ is power universal in $T^{-1}S$.
\[it:localizationpreorderedsemiring\]: For $\frac{s}{t}\in T^{-1}S$ we have $st{\succcurlyeq}st$, therefore $\frac{s}{t}{\succcurlyeq}\frac{s}{t}$, i.e. ${\preccurlyeq}$ is reflexive. Suppose that $\frac{s_1}{t_1}{\succcurlyeq}\frac{s_2}{t_2}$ and $\frac{s_2}{t_2}{\succcurlyeq}\frac{s_3}{t_3}$. This means $rs_1t_2{\succcurlyeq}rs_2t_1$ and $qs_2t_3{\succcurlyeq}qs_3t_2$ for some $r,q\in T$. Therefore $$(qrt_2)s_1t_3=q(rs_1t_2)t_3 {\succcurlyeq}q(rs_2t_1)t_3=rt_1(qs_2t_3)= {\succcurlyeq}qrs_3t_1t_2=(qrt_2)s_3t_1,$$ which implies $\frac{s_1}{t_1}{\succcurlyeq}\frac{s_3}{t_3}$, i.e. ${\preccurlyeq}$ is transtive. We have $\frac{0}{1}{\preccurlyeq}\frac{1}{1}$ since $0{\preccurlyeq}1$. We prove compatibility with the operations. Suppose that $\frac{s_1}{t_1}{\succcurlyeq}\frac{s_2}{t_2}$ and let $\frac{s'}{t'}\in T^{-1}S$. This means $rs_1t_2{\succcurlyeq}rs_2t_1$ for some $r\in T$. Then $r(s_1t'+s't_1)t_2{\succcurlyeq}r(s_2t'+s't_2)t_1$, therefore $$\frac{s_1}{t_1}+\frac{s'}{t'}=\frac{s_1t'+s't_1}{t_1t'}{\succcurlyeq}\frac{s_2t'+s't_2}{t_2t'}=\frac{s_2}{t_2}+\frac{s'}{t'},$$ and $rs_1t_2s't'{\succcurlyeq}rs_2t_1s't'$, therefore $$\frac{s_1}{t_1}\frac{s'}{t'}=\frac{s_1s'}{t_1t'}{\succcurlyeq}\frac{s_2s'}{t_2t'}=\frac{s_2}{t_2}\frac{s'}{t'}.$$ \[it:localizationmapmonotone\]: If $x,y\in S$ and $x{\succcurlyeq}y$ then $\frac{x}{1}{\succcurlyeq}\frac{y}{1}$ since is satisfied for $r=1$. \[it:localizationpoweruniversal\]: Let $u\in S$ be power universal, $x=\frac{s}{t}\in T^{-1}S\setminus\{\frac{0}{1}\}$. Choose $k_s,k_t$ such that the inequalities $s{\preccurlyeq}u^{k_s}$, $u^{k_s}s{\succcurlyeq}1$, $t{\preccurlyeq}u^{k_t}$, $u^{k_t}t{\succcurlyeq}1$ hold. Then $s{\preccurlyeq}u^{k_s}{\preccurlyeq}u^{k_s+k_t}t$, therefore $$\frac{s}{t}{\preccurlyeq}\left(\frac{u}{1}\right)^{k_s+k_t},$$ and $t{\preccurlyeq}u^{k_t}{\preccurlyeq}u^{k_s+k_t}s$, therefore $$\frac{1}{1}{\preccurlyeq}\left(\frac{u}{1}\right)^{k_s+k_t}\frac{s}{t}.$$
It should be noted that even though we use the same symbol for the preorder on $S$ and that induced on $T^{-1}S$, it is in general not true that $x{\succcurlyeq}y$ iff $\frac{x}{1}{\succcurlyeq}\frac{y}{1}$. When $S$ is of polynomial growth, we denote by ${\succsim}$ the asymptotic preorder on $T^{-1}S$. Note that this could mean two different relations, depending on whether we form the asymptotic preorder on the localization or consider the preorder induced on $T^{-1}S$ by the asymptotic preorder on $S$. However, these two relations turn out to be the same. If $rs_1t_2{\succsim}rs_2t_1$ then $r^nu^{o(n)}s_1^nt_2^n{\succcurlyeq}r^ns_2^nt_1^n$ for all $n$, i.e. $$\label{eq:localizationasymptoticpreorder}
\left(\frac{u}{1}\right)^{o(n)}\left(\frac{s_1}{t_1}\right)^{n}{\succcurlyeq}\left(\frac{s_2}{t_2}\right)^{n}$$ Conversely, if holds, then there is a sequence $(r_n)_{n\in\mathbb{N}}$ in $T$ such that $r_nu^{o(n)}(s_1t_2)^n{\succcurlyeq}r_n(s_2t_1)^n$, which implies $s_1t_2{\succsim}s_2t_3$ by \[it:asymptoticcancellation\] of Lemma \[lem:asymptoticproperties\] and Lemma \[lem:aasymptotic\]. A similar argument in the following lemma shows that the asymptotic preorder is essentially the same on $S$ and on its image in $T^{-1}S$.
\[lem:localizationasymptotic\] For $x,y\in S$ we have $x{\succsim}y$ iff $\frac{x}{1}{\succsim}\frac{y}{1}$.
Suppose that $x{\succsim}y$, i.e. $u^{k_n}x^n{\succcurlyeq}y^n$ for some sublinear sequence $k_n$ and all $n$. This implies $\left(\frac{u}{1}\right)^{k_n}\left(\frac{x}{1}\right)^{n}{\succcurlyeq}\left(\frac{y}{1}\right)^{n}$, therefore $\frac{x}{1}{\succsim}\frac{y}{1}$.
Suppose that $\frac{x}{1}{\succsim}\frac{y}{1}$. This means there is a sublinear sequence $k_n$ and elements $r_n\in S\setminus\{0\}$ such that $$r_nu^{k_n}x^n{\succcurlyeq}r_ny^n$$ for all $n\in\mathbb{N}$. By \[it:asymptoticcancellation\] of Lemma \[lem:asymptoticproperties\] and Lemma \[lem:aasymptotic\] this implies $x{\succsim}y$.
According to Lemma \[lem:localizationasymptotic\], we may safely identify $x$ with $\frac{x}{1}$ for the purposes of studying the asymptotic preorder, even though the canonical homomorphism $S\to T^{-1}S$ is in general not injective.
We conclude this section with a construction that enlarges a preorder by forcing an ordering on certain pairs of elements.
\[def:Rpreorderdef\] Let $(S,{\preccurlyeq})$ be a preordered semiring and $R\subseteq S\times S$ a relation. We define the relation ${\preccurlyeq}_R$ on $S$ as $a{\preccurlyeq}_R b$ iff there is an $n\in\mathbb{N}$ and $s_1,\ldots,s_n,x_1,\ldots,x_n,y_1,\ldots,y_n\in S$ such that $$\forall i\in[n]:(x_i,y_i)\in R$$ and $$\label{eq:Rpreorderdef}
a+s_1y_1+\cdots+s_ny_n{\preccurlyeq}b+s_1x_1+\cdots+s_nx_n.$$
\[lem:Rpreorderproperties\]
(i) \[it:Rpreorderlarger\] ${\preccurlyeq}\subseteq{\preccurlyeq}_R$
(ii) \[it:RpreordercontainsR\] $R\subseteq{\preccurlyeq}_R$
(iii) \[it:Rpreordersemiring\] $(S,{\preccurlyeq}_R)$ is a preordered semiring.
(iv) \[it:Rpreorderunion\] If $R_1,R_2\subseteq S\times S$ then ${\preccurlyeq}_{R_1\cup R_2}=({\preccurlyeq}_{R_1})_{R_2}$.
(v) \[it:Rpreorderfinite\] $\displaystyle{\preccurlyeq}_R=\bigcup_{\substack{R'\subseteq R \\ |R'|<\infty}}{\preccurlyeq}_{R'}$.
\[it:Rpreorderlarger\]: If $a{\preccurlyeq}b$ then we may choose $n=0$ in the definition, therefore $a{\preccurlyeq}_R b$. \[it:RpreordercontainsR\]: If $(x,y)\in R$ then we can choose $n=1$, $s_1=1$, $x_1=x$, $y_1=y$ so that becomes $x+y{\preccurlyeq}y+x$, therefore $x{\preccurlyeq}_R y$. \[it:Rpreordersemiring\]: By \[it:Rpreorderlarger\] the relation ${\preccurlyeq}_R$ is reflexive. We prove transitivity. Let $a{\preccurlyeq}_R b$ and $b{\preccurlyeq}_R c$. Then there are $n,m\in\mathbb{N}$ and elements $x_1,\ldots,x_n,y_1,\ldots,y_n,z_1,\ldots,z_m,w_1,\ldots,w_m\in S_0$ and $s_1,\ldots,s_n,t_1,\ldots,t_m\in S$ such that $(x_i,y_i)\in R$ and $(z_i,w_i)\in R$ for all $i$ in $[n]$, respectively $[m]$, and $$\begin{aligned}
a+s_1y_1+\cdots+s_ny_n & {\preccurlyeq}b+s_1x_1+\cdots+s_nx_n \\
b+t_1w_1+\cdots+t_nw_m & {\preccurlyeq}c+t_1z_1+\cdots+t_nz_m.\end{aligned}$$ After adding $t_1w_1+\cdots+t_nw_m$ to both sides of the first inequality and $s_1x_1+\cdots+s_nx_n$ to both sides of the second one we can chain the inequalities and conclude $a{\preccurlyeq}_R c$.
The compatibility with addition and multiplication can be seen directly by adding $c$ to (respectively multiplying by $c$) both sides of . \[it:Rpreorderunion\]: Expanding the definition, we see that $a({\preccurlyeq}_{R_1})_{R_2}b$ means that there are natural numbers $n,n'$ and elements $s_1,\ldots,s_n,x_1,\ldots,x_n,y_1,\ldots,y_n S$ and $s'_1,\ldots,s'_{n'},x'_1,\ldots,x'_{n'},y'_1,\ldots,y'_{n'}\in S$ such that for all $i$ $(x_i,y_i)\in R_1$ and $(x'_i,y'_i)\in R_2$ and $$(a+s_1y_1+\cdots+s_ny_n)+s'_1y'_1+\cdots+s'_{n'}y'_{n'}{\preccurlyeq}(b+s_1x_1+\cdots+s_nx_n)+s'_1x'_1+\cdots+s'_{n'}x'_{n'},$$ which is clearly equivalent to $a{\preccurlyeq}_{R_1\cup R_2}$. \[it:Rpreorderfinite\] follows from the fact that involves only finitely many pairs $(x_i,y_i)\in R$.
Spectrum {#sec:spectrum}
========
In this section we define and study the spectrum of a preordered semiring, i.e. the set of monotone semiring homomorphisms with the topology generated by the evaluation maps. We begin with properties related to compactness, and show that semirings of polynomial growth have locally compact spectra. Then we show that neither replacing the preorder with its asymptotic preorder nor localization affects the spectrum. Finally we prove that the inclusion of a certain type of subsemiring induces a surjective map on the spectra.
Let $(S,{\preccurlyeq})$ be a preordered semiring. The spectrum $\Delta(S,{\preccurlyeq})$ is the set of monotone semiring homomorphisms $S\to\mathbb{R}_{\ge 0}$ equipped with the initial topology with respect to the family of evaluation maps $\operatorname{ev}_s:\Delta(S,{\preccurlyeq})\to\mathbb{R}_{\ge 0}$ $(s\in S)$. For a subset $X\subseteq S$ we define $\operatorname{ev}_X:\Delta(S,{\preccurlyeq})\to\mathbb{R}_{\ge 0}^X$ as $f\mapsto(f(x))_{x\in X}$. Elements of the spectrum will be referred to as spectral points.
Let $(S_1,{\preccurlyeq}_1)$ and $(S_2,{\preccurlyeq}_2)$ be preordered semirings and let $\varphi:S_1\to S_2$ be a monotone homomorphism. We define the map $\Delta(\varphi):\Delta(S_2,{\preccurlyeq}_2)\to\Delta(S_1,{\preccurlyeq}_1)$ as $f\mapsto f\circ\varphi$.
\[lem:spectrumproperties\] Let $(S,{\preccurlyeq})$ be a preordered semiring.
(i) \[it:closedembedding\] $\operatorname{ev}_S$ is a closed embedding. In particular, $\Delta(S,{\preccurlyeq})$ is Tychonoff.
(ii) \[it:locallycompact\] Suppose that $S$ is generated by a single element $u$. Then $\operatorname{ev}_u$ is proper and $\Delta(S,{\preccurlyeq})$ is locally compact.
Let $(S_1,{\preccurlyeq}_1)$ and $(S_2,{\preccurlyeq}_2)$ be preordered semirings and $\varphi:S_1\to S_2$ a monotone semiring homomorphism.
1. \[it:continuous\] $\Delta(\varphi)$ is continuous.
2. \[it:proper\] If for every $x_2\in S_2$ there is an $x_1\in S_1$ such that $x_2{\preccurlyeq}_2\varphi(x_1)$ then $\Delta(\varphi)$ is proper.
\[it:closedembedding\]: The evaluation map $\operatorname{ev}_S$ clearly separates the points of $\Delta(S,{\preccurlyeq})$, thus gives an embedding into the Tychonoff space $\mathbb{R}_{\ge 0}^S$. The property of being a monotone semiring homomorphism is preserved by pointwise limits, therefore the embedding is closed. \[it:locallycompact\]: $S$ consists of polynomials in $u$ with coefficients from $\mathbb{N}$ and $\operatorname{ev}_{p(u)}=p(\operatorname{ev}_u)$. This means that $\operatorname{ev}_u$ already separates the points and its continuity is equivalent to the continuity of all evaluation maps. Therefore $\operatorname{ev}_u:\Delta(S,{\preccurlyeq})\to\mathbb{R}_{\ge 0}$ is a closed embedding. This implies that $\operatorname{ev}_u$ is proper and $\Delta(S,{\preccurlyeq})$ is locally compact. \[it:continuous\]: ${\left\{\operatorname{ev}_s^{-1}(U)\middle|s\in S_1,U\subseteq\mathbb{R}_{\ge0}\text{ open}\right\}}$ is a subbasis for the topology of $\Delta(S_1,{\preccurlyeq}_1)$. The preimage of such a set under $\Delta(\varphi)$ is $\operatorname{ev}_{\varphi(s)}^{-1}(U)$, which is open. \[it:proper\]: Let $C_1\subseteq\Delta(S_1,{\preccurlyeq}_1)$ be compact and let $C_2=\Delta(\varphi)^{-1}(C_1)\subseteq\Delta(S_2,{\preccurlyeq}_2)$. $C_1$ is a compact subset of a Hausdorff space, therefore closed, which implies that $C_2$ is also closed. For all $x_2\in S_2$ let $$B_{x_2}=\inf_{\substack{x_1\in S_1 \\ x_2{\preccurlyeq}_2\varphi(x_1)}}\sup_{f\in C_1}f(x_1).$$ The infimum is by assumption over a nonempty set and the supremum is of a continuous function ($\operatorname{ev}_{x_1}$) over a compact set, therefore $B_{x_2}$ is finite. If $f\in C_2$ then $f\circ\varphi\in C_1$ so $f(x_2)\in[0,B_{x_2}]$. Thus $\operatorname{ev}_{S_2}$ embeds $C_2$ as a closed subset of the compact space $\prod_{x_2\in S_2}[0,B_{x_2}]$.
\[prop:polygrowthlocallycompact\] Let $(S,{\preccurlyeq})$ be a preordered semiring of polynomial growth and suppose that $u\in S$ is power universal. Then $\Delta(S,{\preccurlyeq})$ is locally compact and $\operatorname{ev}_u:\Delta(S,{\preccurlyeq})\to\mathbb{R}_{\ge0}$ is proper.
Let $S_0$ be the subsemiring generated by $u$ and $i:S_0\to S$ the inclusion. By \[it:locallycompact\] and \[it:proper\] of Lemma \[lem:spectrumproperties\], $\Delta(i):\Delta(S,{\preccurlyeq})\to\Delta(S_0,{\preccurlyeq})$ is a proper map into a locally compact space $\Delta(S_0,{\preccurlyeq})$, thus $\Delta(S,{\preccurlyeq})$ is also locally compact.
$\operatorname{ev}^{S}_u=\operatorname{ev}^{S_0}_u\circ\Delta(i)$ (note that $u$ may be regarded as an element of both semirings and the domain of the evaluation map is the spectrum of the semiring indicated in the superscript) is a composition of proper maps, therefore also proper.
\[lem:spectrumofasymptoticpreorder\] Let $(S,{\preccurlyeq})$ be a preordered semiring of polynomial growth. Let $j:(S,{\preccurlyeq})\to(S,{\precsim})$ be the monotone homomorphism with underlying homomorphism the identity. Then $\Delta(j):\Delta(S,{\precsim})\to\Delta(S,{\preccurlyeq})$ is a homeomorphism.
$\Delta(j)$ is injective since for $f\in\Delta(S,{\precsim})$ we have $\Delta(j)(f)(x)=f(j(x))=f(x)$ for all $x\in S$. To see that $\Delta(j)$ is surjective we show that every $f\in\Delta(S,{\preccurlyeq})$ is also monotone under ${\precsim}$. Suppose that $x{\succsim}y$. Let $(k_n)_{n\in\mathbb{N}}$ be as in Definition \[def:asymptoticge\], i.e. $k_n/n\to 0$ and for all $n\in\mathbb{N}$ $u^{k_n}x^n\ge y^n$. Then for every $f\in\Delta(S,{\preccurlyeq})$ we have $$f(u)^{k_n}f(x)^n\ge f(y)^n.$$ After taking roots and letting $n$ go to infinity we get $f(x)\ge f(y)$.
Thus $\Delta(j)$ is a bijection and can be used to identify the two spectra as sets. Under this identification the evaluation maps are the same in both cases, and so are the topologies they generate. Therefore $\Delta(j)$ is a homeomorphism.
\[lem:spectrumlocalization\] Let $(S,{\preccurlyeq})$ be a preordered semiring of polynomial growth and $T\subseteq S\setminus\{0\}$ a multiplicative set containing $1$. Let $j:S\mapsto T^{-1}S$ be the canonical map $x\mapsto\frac{x}{1}$. Then $\Delta(j):\Delta(T^{-1}S,{\preccurlyeq})\to\Delta(S,{\preccurlyeq})$ is a homeomorphism.
We prove that $\Delta(j)$ is injective. Let $\tilde{f}_1,\tilde{f}_2\in\Delta(T^{-1}S,{\preccurlyeq})$ be different. Then there are $s\in S$ and $t\in T$ such that $\tilde{f}_1(\frac{s}{t})\neq \tilde{f}_2(\frac{s}{t})$. Since $\frac{1}{1}=\frac{t}{1}\frac{1}{t}$ we have $\tilde{f}_1\left(\frac{t}{1}\right)\neq 0$ and $\tilde{f}_2\left(\frac{t}{1}\right)\neq 0$. If $\tilde{f}_1\left(\frac{t}{1}\right)\neq\tilde{f}_2\left(\frac{t}{1}\right)$ then $\Delta(j)(\tilde{f}_1)$ and $\Delta(j)(\tilde{f}_2)$ are different at $t$. Otherwise we have $$\tilde{f}_1\left(\frac{s}{1}\right)=\tilde{f}_1\left(\frac{s}{t}\right)\tilde{f}_1\left(\frac{t}{1}\right)\neq\tilde{f}_2\left(\frac{s}{t}\right)\tilde{f}_2\left(\frac{t}{1}\right)=\tilde{f}_2\left(\frac{s}{1}\right),$$ and therefore $\Delta(j)(\tilde{f}_1)$ and $\Delta(j)(\tilde{f}_2)$ are different at $s$.
We prove that $\Delta(j)$ is surjective. Let $f\in\Delta(S,{\preccurlyeq})$. Let $u\in S$ be power universal. For $s\in S\setminus\{0\}$ there is a $k\in\mathbb{N}$ such that $1{\preccurlyeq}u^ks$. Applying $f$ to both sides we get $1\le f(u)^kf(s)$, which implies $f(s)>0$. If $\frac{s_1}{t_1}=\frac{s_2}{t_2}$ then by definition $rs_1t_2=rs_2t_1$ for some $r\in T$, therefore $f(r)f(s_1)f(t_2)=f(r)f(s_2)f(t_1)$, i.e. $f(s_1)f(t_2)=f(s_2)f(t_1)$. This means that the equality $$\tilde{f}\left(\frac{s}{t}\right)=\frac{f(s)}{f(t)}$$ gives a well-defined function on $T^{-1}S$. We claim that $\tilde{f}\in\Delta(T^{-1}S,{\preccurlyeq})$ and $\Delta(j)(\tilde{f})=f$. For $\frac{s_1}{t_1},\frac{s_2}{t_2}\in T^{-1}S$ we have $$\begin{split}
\tilde{f}\left(\frac{s_1}{t_1}+\frac{s_2}{t_2}\right)
& = \tilde{f}\left(\frac{s_1t_2+s_2t_1}{t_1t_2}\right) \\
& = \frac{f(s_1t_2+s_2t_1)}{f(t_1t_2)}=\frac{f(s_1)}{f(t_1)}+\frac{f(s_2)}{f(t_2)}=\tilde{f}\left(\frac{s_1}{t_1}\right)+\tilde{f}\left(\frac{s_1}{t_1}\right)
\end{split}$$ and $$\tilde{f}\left(\frac{s_1}{t_1}\cdot\frac{s_2}{t_2}\right)=\tilde{f}\left(\frac{s_1s_2}{t_1t_2}\right)=\frac{f(s_1s_2)}{f(t_1t_2)}=\frac{f(s_1)}{f(t_1)}\frac{f(s_2)}{f(t_2)}=\tilde{f}\left(\frac{s_1}{t_1}\right)\tilde{f}\left(\frac{s_1}{t_1}\right).$$ Let $\frac{s_1}{t_1}{\succcurlyeq}\frac{s_2}{t_2}$. This means there exists $r\in T$ such that $rs_1t_2{\succcurlyeq}rs_2t_1$. Since $f$ is monotone and $f(r)>0$, this implies $$\tilde{f}\left(\frac{s_1}{t_1}\right)=\frac{f(s_1)}{f(t_1)}\ge\frac{f(s_2)}{f(t_2)}=\tilde{f}\left(\frac{s_2}{t_2}\right).$$ If $x\in S$ then clearly Clearly $\tilde{f}(j(x))=\tilde{f}(\frac{x}{1})=f(x)$, thus $\tilde{f}$ extends $f$. In particular, $\tilde{f}(\frac{0}{1})=0$ and $\tilde{f}(\frac{1}{1})=1$.
Thus we may identify the two spectra via $\Delta(j)$ as sets. Since for $t\in T$ the map $\operatorname{ev}_t$ vanishes nowhere, the topology generated by the maps $(\operatorname{ev}_s)_{s\in S}$ and $(\frac{\operatorname{ev}_s}{\operatorname{ev}_t})_{s\in S,t\in T}$ is the same, therefore $\Delta(j)$ is a homeomorphism.
Our goal in the following is to relate the spectral points of a semiring to those of a subsemiring. We will make use of a relaxed preorder whose restriction to the subsemiring is total.
Let $(S,{\preccurlyeq})$ be a preordered semiring, $S_0\le S$ a subsemiring and $f\in\Delta(S_0,{\preccurlyeq})$. We define the relation ${\preccurlyeq}_f:={\preccurlyeq}_{R_f}$ (see Definition \[def:Rpreorderdef\]) where $R_f$ is the relation $$R_f={\left\{(x,y)\in S_0\times S_0\middle|f(x)\le f(y)\right\}}\subseteq S\times S.$$
Since ${\preccurlyeq}\subseteq{\preccurlyeq}_f$ by Lemma \[lem:Rpreorderproperties\], we may identify $\Delta(S,{\preccurlyeq}_f)$ with a subset of $\Delta(S,{\preccurlyeq})$.
\[lem:inclusionpreimage\] Let $(S,{\preccurlyeq})$ be a preordered semiring, $S_0\le S$ a subsemiring and $f\in\Delta(S_0,{\preccurlyeq})$. Let $i:S_0\hookrightarrow S$ be the inclusion. Then $\Delta(S,{\preccurlyeq}_f)=\Delta(i)^{-1}(f)$.
Let $\tilde{f}\in\Delta(S,{\preccurlyeq}_f)$ and $x\in S_0$. For all $n\in\mathbb{N}$ we have $\lfloor f(nx)\rfloor\le f(nx)\le\lceil f(nx)\rceil$, therefore by \[it:RpreordercontainsR\] of Lemma \[lem:Rpreorderproperties\] also $$\lfloor f(nx)\rfloor{\preccurlyeq}_f nx{\preccurlyeq}_f\lceil f(nx)\rceil.$$ Apply $\tilde{f}$ to both sides, divide by $n$ and let $n\to\infty$ to get $$f(x)=\lim_{n\to\infty}\frac{\lfloor f(nx)\rfloor}{n}\le\tilde{f}(x)\le\lim_{n\to\infty}\frac{\lceil f(nx)\rceil}{n}=f(x).$$ Therefore $\tilde{f}$ agrees with $f$ on $S_0$, which means $\tilde{f}\in\Delta(i)^{-1}(f)$.
Let $\tilde{f}\in\Delta(i)^{-1}(f)$. Then $\tilde{f}$ is a semiring homomorphism and we need to show that it is monotone with respect to ${\preccurlyeq}_f$. Let $a{\preccurlyeq}_f b$. This means there are $x_1,\ldots,x_n,y_1,\ldots,y_n\in S_0$ and $s_1,\ldots,s_n\in S$ for some $n\in\mathbb{N}$ such that $\forall j\in[n]:f(x_j)\le f(y_j)$ and $$a+s_1y_1+\cdots+s_ny_n{\preccurlyeq}b+s_1x_1+\cdots+s_nx_n.$$ Apply $\tilde{f}$ to both sides and rearrange as $$\tilde{f}(s_1)\left(\tilde{f}(y_1)-\tilde{f}(x_1)\right)+\cdots+\tilde{f}(s_n)\left(\tilde{f}(y_n)-\tilde{f}(x_n)\right)\le \tilde{f}(b)-\tilde{f}(a).$$ Since $$\tilde{f}(y_j)-\tilde{f}(x_j)=f(y_j)-f(x_j)\ge 0$$ for all $j$, this implies $\tilde{f}(b)\ge\tilde{f}(a)$, i.e. $\tilde{f}$ is monotone with respect to ${\preccurlyeq}_f$.
\[lem:Rpreorderspectrum\] Let $(S,{\preccurlyeq})$ be a preordered semiring and $S_0\le S$ a subsemiring such that $\forall s\in S\setminus\{0\}\exists r,q\in S_0$ such that $1{\preccurlyeq}rs{\preccurlyeq}q$. Let $R\subseteq S_0\times S_0\subseteq S\times S$ be an arbitrary relation. Then $$\Delta(S_0,\left.{\preccurlyeq}_R\right|_{S_0})={\left\{f\in\Delta(S_0,\left.{\preccurlyeq}\right|_{S_0})\middle|\forall(x,y)\in R:f(x)\le f(y)\right\}}.$$
We emphasize that $\left.{\preccurlyeq}_R\right|_{S_0}$ is in general not the same preorder as $(\left.{\preccurlyeq}\right|_{S_0})_R$, since the latter would only allow elements $s_1,\ldots,s_n\in S_0$ in Definition \[def:Rpreorderdef\].
First note that the condition $\forall s\in S\setminus\{0\}\exists r,q\in S_0$ such that $1{\preccurlyeq}rs{\preccurlyeq}q$ implies that every $f\in\Delta(S_0,\left.{\preccurlyeq}\right|_{S_0})$ and $s\in S_0\setminus\{0\}$ satisfies $f(s)\neq 0$. To see this choose $r\in S_0$ such that $1{\preccurlyeq}rs$ and apply $f$ to both sides.
We prove $\Delta(S_0,\left.{\preccurlyeq}_R\right|_{S_0})\subseteq{\left\{f\in\Delta(S_0,\left.{\preccurlyeq}\right|_{S_0})\middle|\forall(x,y)\in R:f(x)\le f(y)\right\}}$. The preorders satisfy ${\preccurlyeq}\subseteq{\preccurlyeq}_R$, therefore $\Delta(S_0,\left.{\preccurlyeq}_R\right|_{S_0})\subseteq\Delta(S_0,\left.{\preccurlyeq}\right|_{S_0})$. Let $f\in\Delta(S_0,\left.{\preccurlyeq}_R\right|_{S_0})$ and $(x,y)\in R$. \[it:RpreordercontainsR\] of Lemma \[lem:Rpreorderproperties\] implies that $x{\preccurlyeq}_R y$, therefore $f(x)\le f(y)$.
We prove $\Delta(S_0,\left.{\preccurlyeq}_R\right|_{S_0})\supseteq{\left\{f\in\Delta(S_0,\left.{\preccurlyeq}\right|_{S_0})\middle|\forall(x,y)\in R:f(x)\le f(y)\right\}}$. By \[it:Rpreorderfinite\] of Lemma \[lem:Rpreorderproperties\] it is enough to prove the statement for $|R|<\infty$. We prove by induction on $|R|$. If $R=\emptyset$ then ${\preccurlyeq}_R={\preccurlyeq}$ and there is nothing to prove. Otherwise choose $(x,y)\in R$ and let $R_1=R\setminus\{(x,y)\}$ and $R_2=\{(x,y)\}$. Suppose that $$\begin{split}
f
& \in{\left\{f'\in\Delta(S_0,\left.{\preccurlyeq}\right|_{S_0})\middle|\forall(x',y')\in R:f'(x')\le f'(y')\right\}} \\
& ={\left\{f'\in\Delta(S_0,\left.{\preccurlyeq}_{R_1}\right|_{S_0})\middle|f'(x)\le f'(y)\right\}}
\end{split}$$ Let $a,b\in S_0$ such that $a{\preccurlyeq}_Rb$. This means $$\label{eq:basecase}
a+sy{\preccurlyeq}_{R_1}b+sx$$ for some $s\in S$ (\[it:Rpreorderunion\] of Lemma \[lem:Rpreorderproperties\]). If $sx=sy=0$ then apply $f$ to both sides to get $f(a)\le f(b)$. Otherwise $s\neq 0$ and at least one of $x,y$ is nonzero. We prove $$\label{eq:toprove}
a\left(\sum_{m=0}^n x^my^{n-m}\right)+sy^{n+1}{\preccurlyeq}_{R_1} b\left(\sum_{m=0}^n x^my^{n-m}\right)+sx^{n+1}$$ by induction on $n$. The $n=0$ base case is . Suppose holds with $n-1$ instead of $n$. Then $$\begin{split}
a\left(\sum_{m=0}^n x^my^{n-m}\right)+sy^{n+1}
& = ax^n+y\left[a\left(\sum_{m=0}^{n-1} x^my^{n-1-m}\right)+sy^{n}\right] \\
& {\preccurlyeq}_{R_1} ax^n+y\left[b\left(\sum_{m=0}^{n-1} x^my^{n-1-m}\right)+sx^{n}\right] \\
& = x^n(a+sy)+b\left(\sum_{m=0}^{n-1} x^my^{n-m}\right) \\
& {\preccurlyeq}_{R_1} x^n(b+sx)+b\left(\sum_{m=0}^{n-1} x^my^{n-m}\right) \\
& = b\left(\sum_{m=0}^n x^my^{n-m}\right)+sx^{n+1},
\end{split}$$ where the first inequality uses the induction hypothesis and the second one uses .
Let $r,q\in S_0$ such that $1{\preccurlyeq}rs{\preccurlyeq}q$. Then $$ra\left(\sum_{m=0}^n x^my^{n-m}\right)+y^{n+1}{\preccurlyeq}rb\left(\sum_{m=0}^n x^my^{n-m}\right)+qx^{n+1}.$$ Apply $f$ to both sides and rearrange to get $$f(r)\left(\sum_{m=0}^n f(x)^m f(y)^{n-m}\right)\left(f(a)-f(b)\right)\le f(q)f(x)^{n+1}-f(y)^{n+1}$$
Divide by the coefficient of $f(a)-f(b)$ (nonzero since $x$ or $y$ is nonzero and $r\neq 0$). Then we use $f(x)\le f(y)$ and $f(q)\ge 1$ to get $$\begin{split}
f(a)-f(b)
& \le \frac{f(q)f(x)^{n+1}-f(y)^{n+1}}{f(r)\left(\sum_{m=0}^n f(x)^m f(y)^{n-m}\right)} \\
& \le \frac{(f(q)-1)f(x)^{n+1}}{f(r)\left(\sum_{m=0}^n f(x)^m f(y)^{n-m}\right)} \\
& \le \frac{(f(q)-1)f(x)^{n+1}}{f(r)(n+1)f(x)^n} \\
& = \frac{(f(q)-1)f(x)}{f(r)}\frac{1}{n+1}.
\end{split}$$ This inequality is true for every $n\in\mathbb{N}$, therefore $f(a)\le f(b)$.
\[prop:surjective\] Let $(S,{\preccurlyeq})$ be a preordered semiring of polynomial growth and $S_0\le S$ a subsemiring satisfying $\forall s\in S\setminus\{0\}\exists r,q\in S_0$ such that $1{\preccurlyeq}rs{\preccurlyeq}q$. Let $i:S_0\hookrightarrow S$ be the inclusion. Then $\Delta(i)$ is surjective.
We can assume that the inclusion $\mathbb{N}\hookrightarrow S$ is an order embedding (otherwise both spectra are empty).
By Lemma \[lem:inclusionpreimage\] it is enough to show that for every $f\in\Delta(S_0,{\preccurlyeq})$ the set $\Delta(S,{\preccurlyeq}_f)$ is nonempty. Let $u\in S$ be power universal. By the assumption on $S_0$ there is an $u'\in S_0$ such that $u'{\succcurlyeq}u$, and any such $u'$ is also power universal in $S$. By \[it:RpreordercontainsR\] of Lemma \[lem:Rpreorderproperties\], $$u'{\preccurlyeq}_f 2^{\lceil\log_2 f(u')\rceil},$$ therefore $2$ is also power universal for $(S,{\preccurlyeq}_f)$.
Lemma \[lem:Rpreorderspectrum\] implies that $\Delta(S_0,\left.{\preccurlyeq}_f\right|_{S_0})=\{f\}$. In particular, for $n,m\in\mathbb{N}$ we have $n{\preccurlyeq}_f m$ iff $n\le m$. Therefore we can apply Theorem \[thm:Strassen\] to the preordered semiring $(S,{\preccurlyeq}_f)$ and conclude $\Delta(S,{\preccurlyeq}_f)\neq\emptyset$.
Proof of main result and uniqueness {#sec:mainproof}
===================================
Now we have all the technical tools to prove Theorem \[thm:main\]. In the setting of that theorem, we introduce the following notations: $$\begin{aligned}
S_+ & = {\left\{s\in S\middle|\exists n\in\mathbb{N}:ns{\succcurlyeq}1\right\}}\cup\{0\} \\
S_- & = {\left\{s\in S\middle|\exists n\in\mathbb{N}:n{\succcurlyeq}s\right\}} \\
S_b & = S_+\cap S_-.\end{aligned}$$ We will see that all three are subsemirings, and the definition of $S_b$ ensures that it satisfies the conditions of Theorem \[thm:Strassen\]. Given a pair of elements in $S$, we can multiply both with the same element of $M$ to get elements of $S_-$, at least one of them in $S_b$. Here Strassen’s theorem ensures that the collection of monotone homomorphisms characterize the asymptotic preorder. An essential part of the proof is to show that most spectral points of $S_b$ admit an extension to $S$. Informally, an extension exists unless it would evaluate to $\infty$ on the power universal element. More precisely, by assumption \[it:invertibleuptobounded\] we can form an “inverse up to bounded elements” $\bar{u}$ of $u$, and $f\in\Delta(S_b,{\preccurlyeq})$ extends to $S$ iff $f(1+\bar{u})>1$.
\[lem:plusminusproperties\]
(i) \[it:plusminussubsemiring\] $S_+$, $S_-$ and $S_b$ are subsemirings of $S$.
(ii) \[it:boundedtominusextension\] Let $i:S_b\to S_-$ denote the inclusion. Then $\Delta(i):\Delta(S_-,{\preccurlyeq})\to\Delta(S_b,{\preccurlyeq})$ is a homeomorphism.
\[it:plusminussubsemiring\]: $0$ and $1$ are clearly contained both in $S_+$ and $S_-$. Let $s_1,s_2\in S_+$. This means that there are $n_1,n_2\in\mathbb{N}$ such that $n_1s_1{\succcurlyeq}1$ and $n_2s_2{\succcurlyeq}1$. Then $n_1(s_1+s_2){\succcurlyeq}n_1s_1{\succcurlyeq}1$ and $n_1n_2(s_1s_2){\succcurlyeq}1\cdot 1=1$, therefore $s_1+s_2\in S_+$ and $s_1s_2\in S_+$. Let $s_1,s_2\in S_-$. This means that there are $n_1,n_2\in\mathbb{N}$ such that $n_1{\succcurlyeq}s_1$ and $n_2{\succcurlyeq}s_2$. Then $n_1+n_2{\succcurlyeq}s_1+s_2$ and $n_1n_1{\succcurlyeq}s_1s_2$, therefore $s_1+s_2\in S_-$ and $s_1s_2\in S_-$. $S_b=S_+\cap S_-$ is the intersection of subsemirings, therefore it is also a subsemiring.
\[it:boundedtominusextension\]: We prove that $\Delta(i)$ is injective. Let $\tilde{f}\in\Delta(S_-,{\preccurlyeq})$. If $x\in S_-$ then $1+x\in S_b$ and therefore $\tilde{f}(x)=\tilde{f}(1+x)-1=\Delta(i)(\tilde{f})(1+x)-1$, so $\tilde{f}$ can be reconstructed from its restriction $\Delta(i)(\tilde{f})$.
We prove that $\Delta(i)$ is surjective. Let $f\in\Delta(S_b,{\preccurlyeq})$ and let $\tilde{f}:S_-\to\mathbb{R}$ be defined as $\tilde{f}(x)=f(1+x)-1$. Since $1{\preccurlyeq}1+x$, we have $\tilde{f}(x)\ge f(1)-1=0$. We show that $\tilde{f}$ is a monotone semiring homomorphism and $\Delta(i)(\tilde{f})=f$. Clearly $\tilde{f}(0)=f(1+0)-1=1-1=1$ and $\tilde{f}(1)=f(1+1)-1=2-1=1$.
We prove additivity. $$\begin{split}
\tilde{f}(x+y)
& = f(1+x+y)-1 \\
& = f(1+x+1+y)-1-1 \\
& = f(1+x)-1+f(1+y)-1 \\
& = \tilde{f}(x)+\tilde{f}(y)
\end{split}$$
We prove multiplicativity. $$\begin{split}
\tilde{f}(xy)
& = f(1+xy)-1 \\
& = f(1+x+1+y+1+xy)-f(1+x)-f(1+y)-1 \\
& = f(1+x+y+xy)-f(1+x)-f(1+y)+1 \\
& = f((1+x)(1+y))-f(1+x)-f(1+y)+1 \\
& = (f(1+x)-1)(f(1+y)-1) \\
& = \tilde{f}(x)\tilde{f}(y)
\end{split}$$
We prove that $\tilde{f}$ is monotone. Let $x,y\in S_-$ and suppose that $x{\preccurlyeq}y$. Then $1+x{\preccurlyeq}1+y$, therefore $$\tilde{f}(x)=f(1+x)\le f(1+y)=\tilde{f}(y).$$
If $x\in S_b$ then $\tilde{f}(x)=f(1+x)-1=f(1)+f(x)-1=f(x)$, so $\Delta(i)(\tilde{f})=f$.
Finally, from the equality $\tilde{f}(x)=\tilde{f}(1+x)-1$ we see that pointwise convergence in $\Delta(S_-,{\preccurlyeq})$ is equivalent to pointwise convergence of the restrictions to $S_b$.
\[lem:extensionfromminus\] Let $u$ be power universal in $S$ and suppose that there is a $\bar{u}\in S\setminus\{0\}$ such that $u\bar{u}\in S_b$. Then $\bar{u}\in S_-$ and for any $f\in\Delta(S_-,{\preccurlyeq})$ the following are equivalent:
(i) \[it:extends\] $f$ has an extension $\tilde{f}:S\to\mathbb{R}_{\ge 0}$ such that $\tilde{f}\in\Delta(S,{\preccurlyeq})$
(ii) \[it:onminusnonzero\] $\forall x\in S_-\setminus\{0\}:f(x)\neq 0$.
(iii) \[it:ubarnonzero\] $f(\bar{u})\neq 0$
When an extension exists, it is unique.
Since $1{\preccurlyeq}u$ and $u\bar{u}\in S_b$, there is an $n\in\mathbb{N}$ such that $\bar{u}{\preccurlyeq}u\bar{u}{\preccurlyeq}n$, therefore $\bar{u}\in S_-$.
\[it:extends\]$\implies$\[it:onminusnonzero\]: Let $x\in S_-\setminus\{0\}$ and choose $k\in\mathbb{N}$ such that $1{\preccurlyeq}u^kx$. Then $1\le\tilde{f}(u^kx)=\tilde{f}(u)^kf(x)$, therefore $f(x)\neq 0$.
\[it:onminusnonzero\]$\implies$\[it:ubarnonzero\]: We have seen that $\bar{u}\in S_-$. It is also nonzero since there is an $n$ such that $1{\preccurlyeq}nu\bar{u}$.
\[it:ubarnonzero\]$\implies$\[it:extends\]: Let $x\in S$. There is a $k\in\mathbb{N}$ such that $x{\preccurlyeq}u^k$, and therefore $\bar{u}^k x{\preccurlyeq}(u\bar{u})^k$, which implies $\bar{u}^k x\in S_-$. From this we see that if an extension $\tilde{f}$ exists, it must satisfy $\tilde{f}(x)=f(\bar{u})^{-k}f(\bar{u}^k x)$, which proves uniqueness. We prove that this expression is well defined. If $\bar{u}^{k_1} x\in S_-$ and $\bar{u}^{k_2} x\in S_-$ with $k_1<k_2$ then $$\begin{split}
f(\bar{u})^{-k_2}f(\bar{u}^{k_2} x)
& = f(\bar{u})^{-k_2}f(\bar{u}^{k_2-k_1}\bar{u}^{k_1} x) \\
& = f(\bar{u})^{-k_2}f(\bar{u}^{k_2-k_1})f(\bar{u}^{k_1} x)=f(\bar{u})^{-k_1}f(\bar{u}^{k_1} x).
\end{split}$$ $\tilde{f}$ extends $f$ since for $x\in S_-$ one can take $k=0$ above.
We show that $\tilde{f}$ is monotone. Let $x_1,x_2\in S$, $x_1{\preccurlyeq}x_2$. Choose $k\in\mathbb{N}$ such that $\bar{u}^kx_1\in S_-$ and $\bar{u}^kx_2\in S_-$. Then $\bar{u}^kx_1{\preccurlyeq}\bar{u}^kx_2$, therefore $$\tilde{f}(x_1)=f(\bar{u})^{-k}f(\bar{u}^k x_1)\le f(\bar{u})^{-k}f(\bar{u}^k x_2)=\tilde{f}(x_2).$$
We show that $\tilde{f}$ is multiplicative. Let $x_1,x_2\in S$ and choose $k_1,k_2\in\mathbb{N}$ such that $\bar{u}^{k_1}x_1\in S_-$ and $\bar{u}^{k_2}x_2\in S_-$. Then $\bar{u}^{k_1+k_2}x_1x_2\in S_-$ and $$\begin{split}
\tilde{f}(x_1x_2)
& = f(\bar{u})^{-{k_1+k_2}}f(\bar{u}^{k_1+k_2}x_1x_2) \\
& = f(\bar{u})^{-k_1}f(\bar{u}^{k_1} x)f(\bar{u})^{-k_2}f(\bar{u}^{k_2} x)=\tilde{f}(x_1)\tilde{f}(x_2).
\end{split}$$
We show that $\tilde{f}$ is additive. Let $x_1,x_2\in S$ and choose $k_1,k_2\in\mathbb{N}$ such that $\bar{u}^{k_1}x_1\in S_-$ and $\bar{u}^{k_2}x_2\in S_-$. Then $\bar{u}^{k_1+k_2}(x_1+x_2)\in S_-$ and $$\begin{split}
\tilde{f}(x_1+x_2)
& =f(\bar{u})^{-{k_1+k_2}}f(\bar{u}^{k_1+k_2}(x_1+x_2)) \\
& =f(\bar{u})^{-{k_1+k_2}}\left(f(\bar{u}^{k_1+k_2}x_1)+f(\bar{u}^{k_1+k_2}x_2)\right)=\tilde{f}(x_1)+\tilde{f}(x_2).
\end{split}$$
According to Lemma \[lem:plusminusproperties\] and Lemma \[lem:extensionfromminus\] we may make the identification $\Delta(S,{\preccurlyeq})\subseteq\Delta(S_-,{\preccurlyeq})=\Delta(S_b,{\preccurlyeq})$.
The implication $x{\succsim}y\implies\forall f\in\Delta(S,{\preccurlyeq}):f(x)\ge f(y)$ follows from Lemma \[lem:spectrumofasymptoticpreorder\].
For the other direction, suppose that $x,y\in S$ satisfy $\forall f\in\Delta(S,{\preccurlyeq}):f(x)\ge f(y)$. Let $m_1\in M$ be such that $m_1y\in S_b\setminus\{0\}$ and let $m_2\in M$ be such that $m_2m_1x\in S_b\setminus\{0\}$. Then for all $f\in\Delta(S,{\preccurlyeq})$ we have $$f(m_2)=\frac{f(m_2m_1x)}{f(m_1)f(x)}\le\frac{f(m_2m_1x)}{f(m_1)f(y)}=\frac{f(m_2m_1x)}{f(m_1y)}.$$ The numerator of the right hand side is bounded from above and the denominator is bounded away from $0$, therefore $\operatorname{ev}_{m_2}$ is bounded on $\Delta(S,{\preccurlyeq})$. By Proposition \[prop:surjective\] it is also bounded on $\Delta(S_0,{\preccurlyeq})$, and thus by assumption \[it:boundedev\] we have $m_2\in S_-$.
Let $t=m_1m_2$ so that $tx\in S_b$ and $ty\in S_-$ (see \[it:plusminussubsemiring\] of Lemma \[lem:plusminusproperties\]). Let $u$ be power universal and $\bar{u}\in M$ such that $u\bar{u}\in S_b\setminus\{0\}$ (possible by \[it:invertibleuptobounded\]). Let $k\in\mathbb{N}$ such that $k{\succcurlyeq}\bar{u}$ (Lemma \[lem:extensionfromminus\]) and let $$\delta=\min_{f\in\Delta(S_b,{\preccurlyeq})}f(tx).$$ For every $f\in\Delta(S,{\preccurlyeq})$ we have $$\begin{split}
f((k+1)\lceil\delta^{-n}\rceil(tx)^n)
& \ge f(\bar{u})\lceil\delta^{-n}\rceil f(tx)^n+1 \\
& \ge f(\bar{u})\lceil\delta^{-n}\rceil f(ty)^n+1 \\
& = f(\bar{u}\lceil\delta^{-n}\rceil(ty)^n+1),
\end{split}$$ whereas for every $f\in\Delta(S_b,{\preccurlyeq})\setminus\Delta(S,{\preccurlyeq})$ we have $f(\bar{u})=0$ and therefore $$\begin{split}
f((k+1)\lceil\delta^{-n}\rceil(tx)^n)\ge 1
& = f(\bar{u})\lceil\delta^{-n}\rceil f(ty)^n+1 \\
& = f(\bar{u}\lceil\delta^{-n}\rceil(ty)^n+1).
\end{split}$$ We apply Theorem \[thm:Strassen\] to $S_b$ and infer $$(k+1)\lceil\delta^{-n}\rceil(tx)^n{\succsim}\bar{u}\lceil\delta^{-n}\rceil(ty)^n+1{\succcurlyeq}\bar{u}\lceil\delta^{-n}\rceil(ty)^n,$$ The factors $\lceil\delta^{-n}\rceil$ can be cancelled by \[it:asymptoticcancellation\] of Lemma \[lem:asymptoticproperties\] and Lemma \[lem:aasymptotic\]. Thus we have $(k+1)(tx)^n{\succsim}\bar{u}(ty)^n$ for all $n$, which implies $tx{\succsim}ty$ by \[it:asymptoticsmallfactors\] of Lemma \[lem:asymptoticproperties\]. Finally, we cancel the factors $t$ using \[it:asymptoticcancellation\] of Lemma \[lem:asymptoticproperties\] once more and conclude $x{\succsim}y$.
$\Delta(S,{\preccurlyeq})$ is locally compact and $\operatorname{ev}_u$ is proper on $\Delta(S,{\preccurlyeq})$ by Proposition \[prop:polygrowthlocallycompact\].
Consider the localization $T^{-1}S$ with its asymptotic preorder (see Lemma \[lem:localizationasymptotic\]). Choose $M'={\left\{\frac{mt_1}{t_2}\middle|m\in M,t_1,t_2\in T\right\}}$. Then the semiring generated by $M'$ is ${\left\{\frac{s}{t}\middle|s\in S_0,t\in T\right\}}=T^{-1}S_0$. We use Theorem \[thm:main\] with the subset $M'\subseteq T^{-1}S$.
If $\frac{s}{t}\in T^{-1}S\setminus\{0\}$ (with $s\in S\setminus\{0\}$ and $t\in T$) then let $m\in M$, $t_1,t_2\in T$ and $n\in\mathbb{N}$ such that $t_2{\preccurlyeq}n mt_1s$ and $mt_1s{\preccurlyeq}n t_2$ as in the condition \[it:fracinvertibleuptobounded\]. Then $1{\preccurlyeq}n\frac{mt_1t}{t_2}\frac{s}{t}$ and $\frac{mt_1t}{t_2}\frac{s}{t}{\preccurlyeq}n$, therefore \[it:invertibleuptobounded\] is satisfied by $M'$.
Let $\frac{mt_1}{t_2}\in M'$ (with $m\in M$ and $t_1,t_2\in T$) such that $\operatorname{ev}_{\frac{mt_1}{t_2}}$ is bounded on $\Delta(T^{-1}S_0,{\precsim})$. Then $\operatorname{ev}_m\frac{\operatorname{ev}_{t_1}}{\operatorname{ev}_{t_2}}$ is bounded on $\Delta(S_0,{\preccurlyeq})$ (see Lemma \[lem:spectrumofasymptoticpreorder\] and Lemma \[lem:spectrumlocalization\]), so by \[it:fracboundedev\] we have $mt_1{\preccurlyeq}n t_2$ for some $n\in\mathbb{N}$. This implies $\frac{mt_1}{t_2}{\precsim}n$, therefore \[it:boundedev\] is satisfied.
Using Lemma \[lem:localizationasymptotic\] and $\Delta(S,{\preccurlyeq})=\Delta(S,{\precsim})=\Delta(T^{-1}S,{\precsim})$ (again by Lemma \[lem:spectrumofasymptoticpreorder\] and Lemma \[lem:spectrumlocalization\]), for $x,y\in S$ we conclude $$x{\succsim}y\iff\frac{x}{1}{\succsim}\frac{y}{1}\iff\forall f\in\Delta(S,{\preccurlyeq}):f(x)\ge f(y).$$
We prove existence. Define the map $h(x):S\to\mathbb{R}_{\ge 0}$ by $h(x)(s)=\Phi(s)(x)$. Then $h(x)\in\Delta(S,{\precsim})=\Delta(S,{\preccurlyeq})$, $\Phi(s)=\operatorname{ev}_s\circ h$ and the map $h$ is injective (since $\Phi(S)$ separates points) and continuous (because $\forall s\in S:\Phi(s)$ is continuous). Let $u$ be power universal and consider the set $$A={\left\{a\frac{\operatorname{ev}_s}{\operatorname{ev}_u^{k+1}}-b\frac{\operatorname{ev}_t}{\operatorname{ev}_u^{k+1}}\middle|a,b\in\mathbb{R}_{\ge 0},k\in\mathbb{N},\exists n\in\mathbb{N}\exists s,t\in S,s{\preccurlyeq}nu^k,t{\preccurlyeq}nu^k\right\}}$$ Then $A\subseteq C_0(\Delta(S,{\precsim}))$ is a subalgebra that separates points (if $f_1(u)\neq f_2(u)$ then $\frac{1}{\operatorname{ev}_u}\in A$ separates them, otherwise if $f_1(s)\neq f_2(s)$ then $\frac{\operatorname{ev}_s}{\operatorname{ev}_u^{k+1}}\in A$ for some $k$ does) and vanishes nowhere (e.g. $\frac{1}{\operatorname{ev}_u}\in A$ is nowhere zero), so by the Stone–Weierstrass theorem for locally compact spaces [@deitmar2009principles Theorem A.10.2] it is dense in $C_0(\Delta(S,{\precsim}))$.
Suppose that $h$ is not surjective and let $f_0\in\Delta(S,{\preccurlyeq})\setminus h(X)$. Let $0<\epsilon<f_0(u)$. By continuity, $(\operatorname{ev}_u)^{-1}((f_0(u)-\epsilon,f_0(u)+\epsilon))$ is an open set containing $f_0$. Since $\Phi(u)$ is proper, the subset $h(\Phi(u)^{-1}([f_0(u)-\epsilon,f_0(u)+\epsilon]))\subseteq\Delta(S,{\preccurlyeq})$ is compact in a Hausdorff space and therefore closed, and does not contain $f_0$. The set $$U=(\operatorname{ev}_u)^{-1}((f_0(u)-\epsilon,f_0(u)+\epsilon))\setminus h(\Phi(u)^{-1}([f_0(u)-\epsilon,f_0(u)+\epsilon])).$$ is open, disjoint from $h(X)$, and contains $f_0$. By Urysohn’s lemma for locally compact Hausdorff spaces [@deitmar2009principles Lemma A.8.1], there is a function $g\in C_0(\Delta(S,{\preccurlyeq}))$ that is $1$ at $f_0$ and $0$ outside $U$, i.e. vanishes on $h(X)$.
By the density of $A$, there are exist $N\in\mathbb{N}$, $s,t\in S$, $k\in\mathbb{N}$ such that $s{\preccurlyeq}nu^k,t{\preccurlyeq}nu^k$ for some $n\in\mathbb{N}$ and $$\begin{aligned}
\frac{1}{2N}\left(\frac{f(s)}{f(u)^{k+1}}-\frac{f(t)}{f(u)^{k+1}}\right)
& < \frac{1}{4}\qquad\text{for all $f\in h(X)$} \\
\frac{1}{2N}\left(\frac{f_0(s)}{f_0(u)^{k+1}}-\frac{f_0(t)}{f_0(u)^{k+1}}\right)
& > \frac{3}{4}.\end{aligned}$$ so $$\begin{aligned}
f(s)-f(t+Nu^{k+1})
& <-\frac{N}{2}f(u)^{k+1}\qquad\text{for all $f\in h(X)$} \\
f_0(s)-f_0(t+Nu^{k+1})
& >\frac{N}{2}f_0(u)^{k+1}\end{aligned}$$ This means that $\Phi(t+Nu^{k+1})\ge\Phi(s)$, so by assumption $t+Nu^{k+1}{\succsim}s$. On the other hand, $f_0(t+Nu^{k+1})<f_0(s)$, a contradiction.
We prove uniqueness. Let $h_1,h_2:X\to\Delta(S,{\preccurlyeq})$ be two homeomorphisms satisfying $\forall s\in S:\Phi(s)=\operatorname{ev}_s\circ h_1=\operatorname{ev}_s\circ h_2$. This means that for every $x\in X$ and $s\in S$ the equality $h_1(x)(s)=h_2(x)(s)$ holds. Since $S$ separates the points of $\Delta(S,{\preccurlyeq})$, this implies $h_1(x)=h_2(x)$ for all $x\in X$, i.e. $h_1=h_2$.
Acknowledgement {#acknowledgement .unnumbered}
===============
I thank Tobias Fritz for providing useful feedback. This work was supported by the ÚNKP-19-4 New National Excellence Program of the Ministry for Innovation and Technology and the Bolyai János Research Fellowship of the Hungarian Academy of Sciences. I acknowledge support from the Hungarian National Research, Development and Innovation Office (NKFIH) within the Quantum Technology National Excellence Program (Project Nr. 2017-1.2.1-NKP-2017-00001) and via the research grants K124152, KH129601.
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---
abstract: 'Learning value functions off-policy is at the core of modern Reinforcement Learning (RL). Traditional off-policy actor-critic algorithms, however, only approximate the true policy gradient, since the gradient $\nabla_{\theta} Q^{\pi_{\theta}}(s,a)$ of the action-value function with respect to the policy parameters is often ignored. We introduce a class of value functions called Parameter-based Value Functions (PVFs) whose inputs include the policy parameters. PVFs can evaluate the performance of any policy given a state, a state-action pair, or a distribution over the RL agent’s initial states. We show how PVFs yield exact policy gradient theorems. We derive off-policy actor-critic algorithms based on PVFs trained using Monte Carlo or Temporal Difference methods. Preliminary experimental results indicate that PVFs can effectively evaluate deterministic linear and nonlinear policies, outperforming state-of-the-art algorithms in the continuous control environment Swimmer-v3. Finally, we show how recurrent neural networks can be trained through PVFs to solve supervised and RL problems involving partial observability and long time lags between relevant events. This provides an alternative to backpropagation through time.'
author:
- |
Francesco Faccio & Jürgen Schmidhuber\
The Swiss AI Lab, IDSIA\
USI & SUPSI, Manno-Lugano\
Switzerland\
`francesco@idsia.ch juergen@idsia.ch`
bibliography:
- 'biblio.bib'
title: '**Parameter-based Value Functions**'
---
Introduction {#sec:intro}
============
Value functions are central to Reinforcement Learning (RL). For a given policy, they estimate the value of being in a specific state (or of choosing a particular action in a given state). Many RL breakthroughs were achieved through improved estimates of such values, which can be used to find optimal policies [@tesauro1995temporal; @mnih2015human]. Value functions are usually learned by Monte Carlo (MC) [@MET49] or Temporal Difference (TD) [@sutton1988learning] methods. However, learning value functions of arbitrary policies without observing their behavior in the environment is not trivial. Such off-policy learning requires to correct the mismatch between the distribution of updates of the policy used to collect the data and the one we want to learn. Importance sampling [@hesterberg1988advances; @mcbook] techniques are often used to produce unbiased estimates. They typically suffer from large variance [@cortes2010learning; @metelli2018policy]. Here we address the problem of generalization across many value functions in the off-policy setting. We introduce a class of [**“parameter-based value functions”**]{} (PVFs) defined for any policy. PVFs are value functions whose inputs include the policy parameters.\
The parameter-based state-value function (PSVF), $V(s, \theta) = {\operatorname*{\mathbb{E}}}[R_t|s_t=s, \theta]$, is defined as the expected return for being in state $s$ and following a policy parameterized by $\theta$. Similarly, the parameter-based action-value function (PAVF) $Q(s,a, \theta) = {\operatorname*{\mathbb{E}}}[R_t|s_t=s, a_t=a, \theta]$ is defined as the expected return for being in state $s$, choosing action $a$, and following a policy parameterized by $\theta$. Unlike standard value functions $V^{\pi_{\theta}}(s)$ and $Q^{\pi_{\theta}}(s,a)$, PVFs are defined for any policy.\
The PSVF $V(s, \theta)$ can be learned using Gradient Monte Carlo (MC) [@MET49] or Temporal Difference (TD) [@sutton1988learning] methods. In standard RL methods [@Sutton:2018:RLI:3312046] a model of the environment is required to obtain a policy from the state-value function. In our setting, a policy improvement step can be obtained model-free by finding the policy maximizing the value function over a set of states. However, computing $\operatorname*{arg\,max}_{\theta \in \Theta}\operatorname*{\mathbb{E}}_s [V(s, \theta)]$ over a continuous set $\Theta$ is problematic. Actor-critic methods [@sutton1984temporal; @kondaactorcritic; @Peters:2008:NA:1352927.1352986; @Bhatnagar:2007:INA:2981562.2981576] can avoid this computation by incrementally updating the policy following the gradient of the value function. We conducted experiments in the MuJoCo environment Swimmer-v3 using linear and nonlinear deterministic policies. When V is learned using MC or TD updates, our method can outperform other well-known continuous control algorithms [@lillicrap2015continuous; @softactorcritic; @td3] and closely related methods [@harb2020policy].\
When applied to POMDPs or supervised learning tasks involving long time lags between relevant events, PVFs can help to avoid the vanishing gradient problem [@Hochreiter:91]. In supervised learning tasks, the PVF $V(\theta)$ can be trained to map the weights of an RNN to its loss. The policy improvement step then consists in finding $\theta^* = \operatorname*{arg\,min}_{\theta}V(\theta)$. Preliminary results for a simple copy task show that it is possible to train an LSTM [@Hochreiter:1997:LSM:1246443.1246450] using only the directions of improvement suggested by V.\
The PAVF $Q(s, a, \theta)$ leads to a novel stochastic and deterministic policy gradient theorem. Previous off-policy policy gradients [@Degris2012; @Silver2014] are only approximations of the true gradient, since they do not estimate or compute the gradient of the action-value function with respect to the policy parameters $\nabla_{\theta} Q^{\pi_{\theta}}(s,a)$. With PAVFs, we can directly compute this contribution to the gradient. This yields an exact policy gradient theorem. Based on these results, we develop two actor-critic methods, which are strongly related to Off-PAC [@Degris2012] and DPG [@Silver2014; @schmidhuber1990making] when using stochastic and deterministic policies respectively.\
The rest of this paper is structured as follows: Section \[sec:mdp\] introduces the standard MDP setting; Section \[sec:pbvf\] formally presents PVFs; Sections \[sec:v\] and \[sec:q\] derive algorithms for $V(s,\theta)$ and $Q(s,a,\theta)$; Section \[sec:pomdp\] discusses algorithms for POMDPs and supervised learning; Section \[sec:exp\] describes preliminary experiments; Sections \[sec:related\] and \[sec:todo\] discuss related and future work; Appendix \[sec:impl\] presents implementation details.
Background {#sec:mdp}
==========
We consider a Markov Decision Process (MDP) [@puterman2014markov] $\mathcal{M}=(\mathcal{S},\mathcal{A},P,R,\gamma,\mu_0)$ where at each step an agent observes a state $s \in \mathcal{S}$, chooses action $a \in \mathcal{A}$, transitions into state $s'$ with probability $P(s' | s, a)$ and receives a reward $R(s,a)$. The agent starts from an initial state, chosen with probability $\mu_0(s)$. It is represented by a parametrized stochastic policy $\pi_{\theta}: \mathcal{S} \rightarrow \Delta(\mathcal{A})$, which provides the probability of performing action $a$ in state $s$. A policy is deterministic if for each state $s$ there exists an action $a$ such that $\pi_{\theta}(a|s)=1$. The return $R_t$ is defined as the cumulative discounted reward from time step t: $R_t = \sum_{k=0}^{T-t-1}\gamma^k R(s_{t+k+1}, a_{t+k+1})$, where T denotes the time horizon and $\gamma$ a real-valued discount factor. The performance of the agent is measured by the cumulative discounted expected reward (expected return), defined as $J(\pi_{\theta})={\operatorname*{\mathbb{E}}}_{\pi_{\theta}}[R_0].$\
Given a policy $\pi_{\theta}$, the state-value function $V^{\pi_{\theta}}(s) = {\operatorname*{\mathbb{E}}}_{\pi_{\theta}}[R_t|s_t=s]$ is defined as the expected return for being in a state $s$ and following policy $\pi_{\theta}$. The value function can also be defined in a recursive way, using Bellman’s Expectation Equation: $$V^{\pi_{\theta}}(s) = \int_{\mathcal{A}} \pi_{\theta}(a|s) \left(R(s,a) + \gamma \int_{\mathcal{S}} P(s'|s,a)V^{\pi_{\theta}}(s') {\,\mathrm{d}}s' \right) {\,\mathrm{d}}a.$$ We define by $d^{\pi_{\theta}}(s')$ the discounted state distribution $d^{\pi_{\theta}}(s') = \int_{\mathcal{S}}\sum_{t=1}^{\infty} \gamma^{t-1} \mu_0(s) P(s \rightarrow s', t, \pi) {\,\mathrm{d}}s $, where $P(s \rightarrow s', t, \pi_{\theta})$ is the probability of transitioning to $s'$ after t time steps, starting from s and following policy $\pi$. By integrating over the state space $\mathcal{S}$, we can express the maximization of the expected cumulative reward in terms of the state-value function: $$J(\pi_{\theta})=\int_{\mathcal{S}} d^{\pi_{\theta}}(s) V^{\pi_{\theta}}(s) {\,\mathrm{d}}s.$$ In off-policy policy optimization, we seek to find the parameters of the policy maximizing the performance index $ J(\pi_{\theta})$ using data collected from a different behavioral policy $\pi_b$. Formally, we want to find: $$\max_{\theta} J(\pi_{\theta})=\max_{\theta} \int_{\mathcal{S}} d^{\pi_b}(s) V^{\pi_{\theta}}(s) {\,\mathrm{d}}s,$$ where the value function is averaged over the distribution of the states under the behavioral policy.\
The action-value function $Q^{\pi_{\theta}}(s,a)$, which is defined as the expected return for performing action $a$ in state $s$, and following the policy $\pi_{\theta}$, is: $$Q^{\pi_{\theta}}(s,a) = {\operatorname*{\mathbb{E}}}_{\pi_{\theta}}[R_t|s_t=s,a_t=a],$$ which can also be expressed recursively, using Bellman’s Expectation Equation: $$Q^{\pi_{\theta}}(s,a) = R(s,a) + \gamma \int_{\mathcal{S}} P(s'|s,a) V^{\pi_{\theta}}(s') {\,\mathrm{d}}s'.$$ The action-value function and state-value function are related by: $$V^{\pi_{\theta}}(s) = \int_{\mathcal{A}} \pi_{\theta}(a|s) Q^{\pi_{\theta}}(s,a) {\,\mathrm{d}}a.$$
Parameter-based Value Functions (PVFs) {#sec:pbvf}
======================================
We augment the state and action-value functions, allowing them to receive as an input also the weights of a parametric policy. The parameter-based state-value function (PSVF) $V(s, \theta) = {\operatorname*{\mathbb{E}}}[R_t|s_t=s, \theta]$ is defined as the expected return for being in state $s$ and following policy parameterized by $\theta$. Similarly, the parameter-based action-value function (PAVF) $Q(s,a, \theta) = {\operatorname*{\mathbb{E}}}[R_t|s_t=s, a_t=a, \theta]$ is defined as the expected return for being in state s, taking action $a$ and following policy parameterized by $\theta$. Bellman equations hold for PVFs: $$Q(s,a,\theta) = R(s,a) + \gamma \int_{\mathcal{S}} P(s'|s,a) V(s', \theta) {\,\mathrm{d}}s'.$$
$$V(s, \theta) = \int_{\mathcal{A}} \pi_{\theta}(a|s) Q(s,a,\theta) {\,\mathrm{d}}a.$$
In the off-policy setting, the objective to be maximized becomes: $$J(\pi_{\theta^{*}})=\max_{\theta}\int_{\mathcal{S}} d^{\pi_b}(s) V(s, \theta) {\,\mathrm{d}}s = \max_{\theta}\int_{\mathcal{S}} \int_{\mathcal{A}} d^{\pi_b}(s) \pi_{\theta}(a|s) Q(s,a,\theta) {\,\mathrm{d}}a {\,\mathrm{d}}s.$$ Sections \[sec:v\] and \[sec:q\] will take the gradient of the performance $J$ with respect of the policy parameters $\theta$ to obtain novel policy gradient theorems. We will call $V(s,\theta)$ the state-value function (instead of PSVF) and $Q(s,a,\theta)$ the action-value function (instead of PAVF) when there is no danger of confusion.
The state-value function $V(s, \theta)$ {#sec:v}
=======================================
In this section we present two main methods for training an agent using the value function $V(s,\theta)$: Gradient Monte Carlo and Temporal Difference learning. Given the performance index $J$ and taking the gradient with respect to $\theta$, we obtain: $$\nabla_{\theta}J(\pi_{\theta})=\int_{\mathcal{S}} d^{\pi_b}(s) \nabla_{\theta} V(s, \theta) {\,\mathrm{d}}s = {\operatorname*{\mathbb{E}}}_{s \sim d^{\pi_b}(s)}[\nabla_{\theta} V(s, \theta)].$$ Standard RL methods require a model of the environment to obtain a policy from the state-value function. In our setting, we can directly optimize the policy following the gradient of the performance of the PSVF, obtaining a policy improvement step in a model-free way.\
The performance index $J$ can also be defined only with respect to the distribution of the initial state of the agent: $$J_{\mu_0}(\pi_{\theta})=\int_{\mathcal{S}} \mu_0(s) V^{\pi}(s, \theta) {\,\mathrm{d}}s.$$ This notation allows us to omit the dependence on the state in the PVF and learn a function $V(\theta)$ which is an approximation of $J(\pi_{\theta})$: $$V(\theta) := {\operatorname*{\mathbb{E}}}_{s \sim \mu_0(s)}[V(s, \theta)] = \int_{\mathcal{S}} \mu_0(s) V(s, \theta) {\,\mathrm{d}}s = J_{\mu_0}(\pi_{\theta}),$$ where the gradient becomes: $$\nabla_{\theta}J_{\mu_0}(\pi_{\theta})=\int_{\mathcal{S}} \mu_0(s)\nabla_{\theta} V(s, \theta) {\,\mathrm{d}}s = {\operatorname*{\mathbb{E}}}_{s \sim \mu_0(s)}[\nabla_{\theta} V(s, \theta)] = \nabla_{\theta}V(\theta).$$ Since s and $\theta$ are continuous, we need to use a function approximator ${V}_{\textbf{w}}(s, \theta) \approx V(s, \theta)$ and ${V}_{\textbf{w}}(\theta) \approx V(\theta)$. In both cases, the policy improvement step can be very expensive, due to the computation of the $\operatorname*{arg\,max}$ over a continuous space $\Theta$. Actor-critic methods can be derived to solve this optimization problem, where the critic $V(s,\theta)$ or $V(\theta)$ can be learned using TD or MC methods respectively, while the actor is updated following the gradient with respect to the critic.
Actor-critic with Monte Carlo prediction for $V(\theta)$
--------------------------------------------------------
In Algorithm \[alg:GMCV\], the critic $V_{\textbf{w}}(\theta)$ is learned via Gradient Monte Carlo to estimate the value of any policy $\theta$. The actor is then updated following the direction of improvement suggested by the critic. Although the algorithm can be used with stochastic policies, removing the stochasticity of the action-selection process can facilitate learning the value function. Therefore, we use deterministic policies. The algorithm makes use of a replay buffer:
**Input**: Differentiable critic $V_{\textbf{w}}: \Theta \rightarrow \mathcal{R}$ with parameters $\textbf{w}$; deterministic actor $\pi_{\theta}$ with parameters $\theta$; empty replay buffer $D$\
**Output** : Learned $V_{\textbf{w}} \approx V(\theta) \forall \theta$, learned $\pi_{\theta} \approx \pi_{\theta^*}$
Initialize critic and actor weights $\textbf{w}, \theta$ : Generate an episode $s_{0}, a_{0}, r_{1}, s_{1}, a_{1}, r_{2}, \dots, s_{T-1}, a_{T-1}, r_{T}$ with policy $\pi_{\theta}$ Compute return $r = \sum_{k=1}^T r_k$ Store $(\theta, r)$ in the replay buffer $D$ : Sample a batch $B = \{(r, \theta)\}$ from $D$ Update critic by stochastic gradient descent: $\nabla_{\textbf{w}} \operatorname*{\mathbb{E}}_{(r, \theta) \in B} [r - V_{\textbf{w}}(\theta)]^2$ : Update actor by gradient ascent: $\nabla_{\theta} V_{\textbf{w}}(\theta)$
Actor-critic with TD prediction for $V(s, \theta)$
--------------------------------------------------
Learning the value function using Monte Carlo approaches can be difficult due to the high variance of the estimate. Furthermore, episode-based algorithms like Algorithm \[alg:GMCV\] are unable to credit good actions in bad episodes. Gradient methods based on Temporal Difference updates provide a biased estimate of $V(s,\theta)$ with much lower variance and can credit actions at each time step. Algorithm \[alg:GTDVS\] uses the actor-critic architecture, where the critic is learned via Temporal Difference:
**Input**: Differentiable critic $V_{\textbf{w}}: \mathcal{S} \times \Theta \rightarrow \mathcal{R}$ with parameters $\textbf{w}$; deterministic actor $\pi_{\theta}$ with parameters $\theta$; empty replay buffer $D$\
**Output** : Learned $V_{\textbf{w}} \approx V(s, \theta)$, learned $\pi_{\theta} \approx \pi_{\theta^*}$
Initialize critic and actor weights $\textbf{w}, \theta$ : Observe state s, take action $a=\pi_{\theta}(s)$, observe reward $r$ and next state $s'$ Store $(s, \theta, r, s')$ in the replay buffer $D$ : : Sample a batch $B = \{(s, \theta, r, s')\}$ from $D$ Update critic by stochastic gradient descent: $\nabla_{\textbf{w}} \frac{1}{|B|} \operatorname*{\mathbb{E}}_{(s, \theta, r, s') \in B} [V_{\textbf{w}}(s, \theta) - (r + \gamma V_{\textbf{w}}(s', \theta))]^2$ Update actor by gradient ascent: $\nabla_{\theta} \frac{1}{|B|} \operatorname*{\mathbb{E}}_{s \in B}[ V_{\textbf{w}}(s, \theta)]$
The action-value function $Q(s, a, \theta)$ {#sec:q}
===========================================
The introduction of the PAVF $Q(s, a, \theta)$ allows us to derive new policy gradients methods [@Sutton1999] when using a stochastic or deterministic policy.
Stochastic policy gradients
---------------------------
We want to use data collected from some stochastic behavioral policy $\pi_{b}$ in order to learn the action-value of a target policy $\pi_{\theta}$. Traditional off-policy actor-critic algorithms only approximate the true policy gradient, since they do not estimate the gradient of the action-value function with respect to the policy parameters $\nabla_{\theta} Q^{\pi_{\theta}}(s,a)$ [@Degris2012; @Silver2014]. With PVFs, we can directly compute this contribution to the gradient. This yields an exact policy gradient theorem:
[thr]{}[Off-Policy Policy Gradient Theorem]{} For any Markov Decision Process, the following holds: $$\nabla_{\theta} J(\pi_{\theta}) = {\operatorname*{\mathbb{E}}}_{s \sim d^{\pi_{b}}(s), a \sim \pi_{b}(.|s)}\left[\frac{\pi_{\theta}(a|s)}{\pi_{b}(a|s)} \left(Q(s,a,\theta) \nabla_{\theta} \log\pi_{\theta}(a|s) + \nabla_{\theta}Q(s,a,\theta)\right)\right]$$
$$\begin{aligned}
\nabla_{\theta}J(\pi_{\theta}) & = \nabla_{\theta} \int_{\mathcal{S}} d^{\pi_{b}}(s) V(s,\theta) {\,\mathrm{d}}s \\
& = \nabla_{\theta} \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) Q(s,a,\theta) {\,\mathrm{d}}a {\,\mathrm{d}}s \\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} [Q(s,a,\theta) \nabla_{\theta} \pi_{\theta}(a|s) + \pi_{\theta}(a|s) \nabla_{\theta}Q(s,a,\theta)] {\,\mathrm{d}}a {\,\mathrm{d}}s\\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \frac{\pi_{b}(a|s)}{\pi_{b}(a|s)}\pi_{\theta}(a|s) [Q(s,a,\theta) \nabla_{\theta} \log\pi_{\theta}(a|s) + \nabla_{\theta}Q(s,a,\theta)] {\,\mathrm{d}}a {\,\mathrm{d}}s\\
& = {\operatorname*{\mathbb{E}}}_{s \sim d^{\pi_{b}}(s), a \sim \pi_{b}(.|s)}\left[\frac{\pi_{\theta}(a|s)}{\pi_b(a|s)} \left(Q(s,a,\theta) \nabla_{\theta} \log\pi_{\theta}(a|s) + \nabla_{\theta}Q(s,a,\theta)\right)\right]
\end{aligned}$$
Let us compare this to the standard off-policy policy gradient [@Degris2012]: $$\nabla_{\theta}J(\pi_{\theta}) \approx {\operatorname*{\mathbb{E}}}_{s \sim d^{\pi_{b}}(s), a \sim \pi_{b}(.|s)}\left[ \frac{\pi_{\theta}(a|s)}{\pi_{b}(a|s)} \left(Q^{\pi_{\theta}}(s,a) \nabla_{\theta} \log\pi_{\theta}(a|s)\right)\right],$$ We notice that our expectation contains the derivative of the value function with respect to the policy. Algorithm \[alg:SACTDQ\] uses an actor-critic architecture and can be seen as an extension of Off-PAC [@Degris2012] to PVFs:
**Input**: Differentiable critic $Q_{\textbf{w}}: \mathcal{S} \times \mathcal{A} \times \Theta \rightarrow \mathcal{R}$ with parameters $\textbf{w}$; stochastic differentiable actor $\pi_{\theta}$ with parameters $\theta$; empty replay buffer $D$\
**Output** : Learned $Q_{\textbf{w}} \approx Q(s, a, \theta)$, learned $\pi_{\theta} \approx \pi_{\theta^*}$
Initialize critic and actor weights $\textbf{w}, \theta$ : Observe state s, take action $a=\pi_{\theta}(s)$, observe reward $r$ and next state $s'$ Store $(s, a, \theta, r, s')$ in the replay buffer $D$ : : Sample a batch $B = \{(s, a, \Tilde{\theta}, r, s')\}$ from $D$ Update critic by stochastic gradient descent: $\nabla_{\textbf{w}} \frac{1}{|B|} \operatorname*{\mathbb{E}}_{(s, a, \Tilde{\theta}, r, s') \in B} [Q_{\textbf{w}}(s, a, \Tilde{\theta}) - (r + \gamma Q_{\textbf{w}}(s',a' \sim \pi_{\Tilde{\theta}}(s'),\Tilde{\theta}))]^2$ Update actor by stochastic gradient ascent: $ \frac{1}{|B|} \operatorname*{\mathbb{E}}_{(s,a, \Tilde{\theta}) \in B} \left[ \frac{\pi_{\theta}(a|s)}{\pi_{\Tilde{\theta}}(a|s)} \left(Q(s,a,\theta) \nabla_{\theta} \log\pi_{\theta}(a|s) + \nabla_{\theta}Q(s,a,\theta)\right)\right]$
Deterministic policy gradients
------------------------------
Estimating $V(s,\theta)$ is in general a difficult problem due to the stochasticity of the policy. Deterministic policies of the form $\pi: \mathcal{S} \rightarrow \mathcal{A}$ can help improving the efficiency in learning value functions, since the expectation over the action space is no longer required. Using PVFs, we can write the performance of a policy $\pi_{\theta}$ as:
$$J(\pi_{\theta})=\int_{\mathcal{S}} d^{\pi_{b}}(s) V(s, \theta) {\,\mathrm{d}}s = \int_{\mathcal{S}} d^{\pi_{b}}(s) Q(s,\pi_{\theta}(s),\theta) {\,\mathrm{d}}s$$
Taking the gradient with respect to $\theta$ we obtain a deterministic policy gradient theorem:
[thr]{}[Deterministic Off-Policy Policy Gradient Theorem]{} Under standard regularity assumptions [@Silver2014], for any Markov Decision Process, the following holds: $$\nabla_{\theta} J(\pi_{\theta}) = {\operatorname*{\mathbb{E}}}_{s \sim d^{\pi_{b}}(s)} \left[ \nabla_{a} Q(s,a,\theta)|_{a=\pi_{\theta}(s)} \nabla_{\theta} \pi_{\theta}(s) + \nabla_{\theta} Q(s,a,\theta)|_{a=\pi_{\theta}(s)} \right]$$
$$\begin{aligned}
\nabla_{\theta}J(\pi_{\theta}) & = \int_{\mathcal{S}} d^{\pi_{b}}(s) \nabla_{\theta} Q(s,\pi_{\theta}(s),\theta) {\,\mathrm{d}}s \\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) \left[ \nabla_{a} Q(s,a,\theta)|_{a=\pi_{\theta}(s)} \nabla_{\theta} \pi_{\theta}(s) + \nabla_{\theta} Q(s,a,\theta)|_{a=\pi_{\theta}(s)} \right] {\,\mathrm{d}}s \\
& = {\operatorname*{\mathbb{E}}}_{s \sim d^{\pi_{b}}(s)} \left[ \nabla_{a} Q(s,a,\theta)|_{a=\pi_{\theta}(s)} \nabla_{\theta} \pi_{\theta}(s) + \nabla_{\theta} Q(s,a,\theta)|_{a=\pi_{\theta}(s)} \right]
\end{aligned}$$
Algorithm \[alg:DACTDQ\] uses an actor-critic architecture and and can be seen as an extension of DPG [@Silver2014] to PVFs:
**Input**: Differentiable critic $Q_{\textbf{w}}: \mathcal{S} \times \mathcal{A} \times \Theta \rightarrow \mathcal{R}$ with parameters $\textbf{w}$; differentiable deterministic actor $\pi_{\theta}$ with parameters $\theta$; empty replay buffer $D$\
**Output** : Learned $Q_{\textbf{w}} \approx Q(s, a, \theta)$, learned $\pi_{\theta} \approx \pi_{\theta^*}$
Initialize critic and actor weights $\textbf{w}, \theta$ : Observe state s, take action $a=\pi_{\theta}(s)$, observe reward $r$ and next state $s'$ Store $(s, a, \theta, r, s')$ in the replay buffer $D$ : : Sample a batch $B = \{(s, a, \theta, r, s')\}$ from $D$ Update critic by stochastic gradient descent: $\nabla_{\textbf{w}} \frac{1}{|B|} \operatorname*{\mathbb{E}}_{(s, a, \theta, r, s') \in B} [Q_{\textbf{w}}(s, a, \theta) - (r + \gamma Q_{\textbf{w}}(s',\pi_{\theta}(s'),\theta))]^2$ Update actor by stochastic gradient ascent: $ \frac{1}{|B|} \operatorname*{\mathbb{E}}_{s \in B}[ \nabla_{\theta} \pi_{\theta}(s) \nabla_a Q_{\textbf{w}}(s,a,\theta)|_{a=\pi_{\theta}(s)} + \nabla_{\theta} Q_{\textbf{w}}(s, a, \theta)|_{a=\pi_{\theta}(s)}]$
POMDPs {#sec:pomdp}
======
In Partially Observable MDPs (POMDPs), the learning agent can only observe a part of the environmental state. To deal with such non-Markovian situations, many agents learn a mapping from the history of past observations to a probability distribution over actions. Recurrent Neural Networks (RNNs) such as the Long Short-Term Memory (LSTM) [@Hochreiter:1997:LSM:1246443.1246450] can be used in a model-free setting to represent a value function predicting the cumulative expected reward for receiving an observation and taking a certain action [@Bakker:03robot]. An LSTM can also directly represent the policy of the agent [@recurrentpg]. The traditional way of training RNNs by backpropagation through time (BPTT) [@werbos1990backpropagation] or similar methods [@Werbos:88gasmarket; @WilliamsZipser:92; @RobinsonFallside:87tr] is based on the runtime view of what the RNN’s weight matrix (considered as a program [@schmidhuber1990making]) does while it operates on a sequence of inputs. In case of long time lags between relevant events, this leads to well-known issues (e.g., vanishing gradients [@Hochreiter:91]) at the heart of all of deep learning [@888]. PVFs avoid BPTT and allow to view the program as a static object and edit it directly, changing some code here and there, like a programmer would, based on earlier acquired knowledge about typical effects of code.\
The algorithms obtained in the previous section for learning $V(\theta)$ can be adapted to this context to solve supervised learning or RL problems with partial observability. In this setting, $V(\theta)$ is trained to predict the loss of the parameters $\theta$ of an RNN. The RNN is then trained to minimize the loss predicted by $V(\theta)$. The iterative process is reported in Algorithm \[alg:POMDPGMCV\]:
**Input**: Set of training sequences $\{S_t\}_{t=1}^T$; differentiable critic $V_{\textbf{w}}: \Theta \rightarrow \mathcal{R}$ with parameters $\textbf{w}$; RNN $f_{\theta}(s, h)$ with parameters $\theta$, input $s$ and hidden state $h$; empty replay buffer $D$\
**Output** : Learned $V_{\textbf{w}} \approx V(\theta) \forall \theta$, learned $f_{\theta}(s, h)$
Initialize critic and actor weights $\textbf{w}, \theta$ : Given a batch of sequences $\{S_t\}_{t=1}^K$, run the forward pass of the RNN $f_{\theta}$ Compute the average loss $l$ over the batch Store $(\theta, l)$ in the replay buffer $D$ : Sample a batch $B = \{(l, \theta)\}$ from $D$ Update critic by stochastic gradient descent: $\nabla_{\textbf{w}} \operatorname*{\mathbb{E}}_{(l, \theta) \in B} [l - V_{\textbf{w}}(\theta)]^2$ Update RNN by gradient descent: $\nabla_{\theta} V_{\textbf{w}}(\theta)$
A similar algorithm can be derived for learning $V(s,\theta)$. In this case there is no need to wait until the end of the sequence to update the RNN and the value function, but V needs to be recurrent to be trained by Temporal Difference.
Experiments {#sec:exp}
===========
Here we describe preliminary results for Algorithms \[alg:GMCV\] and \[alg:GTDVS\] in the continuous control environment Swimmer-v3, using deterministic linear and nonlinear policies. We also test Algorithm \[alg:POMDPGMCV\] on a simple copy task.\
Applying Algorithms \[alg:GMCV\], \[alg:GTDVS\] and \[alg:POMDPGMCV\] directly can be problematic due to the lack of exploration. In our experiments, we observed that the algorithms are likely to converge prematurely to local optima, which are hard to escape from without some sort of exploration strategy. To encourage exploration, we add noise to the vector of policy parameters before interacting with the environment, using $\pi_{\Tilde{\theta}}$ with $\Tilde{\theta} = \theta + \epsilon, \epsilon \sim \mathcal{N}(0, \sigma I)$ instead of $\pi_{\theta}$ and then storing $\Tilde{\theta}$ in the replay buffer. We leave the implementation details to Appendix \[sec:impl\].\
Learning with $V(\theta)$ and $V(s, \theta)$
--------------------------------------------
In Figure \[fig:sub1\] we plot the average performance of Algorithms \[alg:GMCV\] and \[alg:GTDVS\] when training deterministic linear and nonlinear policies. Surprisingly, our policies are able to exploit the information contained in the PVF and PSVF, outperforming by sample efficiency and final performance many well-known continuous control algorithms [@lillicrap2015continuous; @softactorcritic; @td3] and closely related methods [@harb2020policy]. The results show that the algorithm is able to achieve a final average return of 354 when linear policies are used. We observe that nonlinear policies need more samples to reach optimality. Therefore they exhibit lower performance, achieving a final average return of 338 and 297 when using PVFs and PSVFs respectively. The baseline results for algorithms DDPG [@lillicrap2015continuous], SAC [@softactorcritic], TD3 [@td3] which use policies based on deep neural networks are taken from <https://spinningup.openai.com/en/latest/spinningup/bench.html> (PyTorch version).
![Learning curves of Algorithms \[alg:GMCV\] and \[alg:GTDVS\] (PVF and PSVF trained with policies represented by linear and nonlinear function approximators), representing the average return of the policy $\pi_{\theta}$ as a function of the number of episodes in the environment used for learning (average across 10 runs, 95% c.i.). For the baselines, the final return achieved in 3000 episodes is reported, averaged across 10 runs. For PEN [@harb2020policy], the final return after 1M policy evaluations is reported, averaged across 5 runs.[]{data-label="fig:sub1"}](final_plot.pdf){width="0.4\paperwidth"}
Training policies from scratch
------------------------------
In order to test if the value function is generalizing across the policy space, we perform the following experiment using PVFs and linear policies: every 100 episodes, we stop training and randomly initialize 5 policies. Then, without interacting with the environment, we train these policies offline, in a zero-shot manner, following only the direction of improvement suggested by $\nabla_{\theta} V_{\textbf{w}}(\theta)$, whose weights $\textbf{w}$ remain frozen. In Figure \[fig:sub2\], we compare policies learned from scratch to the policy $\pi_{\theta}$ learned with Algorithm \[alg:GMCV\] and the perturbed policy used to collect data from the environment $\pi_{\Tilde{\theta}}$. We observe that the policies learned from scratch obtain a return comparable to the highest return seen during the training of V. We note also that, during training, an improvement can occur in two ways: the perturbed policy randomly achieves a better return, which is then stored in the replay buffer and used to learn $V_{\textbf{w}}(\theta)$ and consequently the policy $\pi_{\theta}$; the policy $\pi_{\theta}$ achieves a return better than any return seen during training, only by following the direction of the gradient of $V_{\textbf{w}}(\theta)$. This suggests that the value function is pointing in some direction of improvement, generalizing to unseen values in the parameter space. When the policy learned during the iterative procedure reaches an optimal value, we note that policies learned from scratch are also able to reach optimal performance. They tend to converge to policies that are different from any other policy seen during training (here two policies $\pi_{\theta_i}$ and $\pi_{\theta_j}$ are different if $||\theta_i - \theta_j||_2$ is relatively large).
![Policies learned from scratch during training. The plot in the center represents the return of the agent learning while interacting with the environment using Algorithm \[alg:GMCV\]. In particular, we compare the noisy policy $\pi_{\Tilde{\theta}}$ used for exploration to the policy $\pi_{\theta}$ learned through the critic. The learning curves in the small plots represent the return obtained by policies trained from scratch following the fixed critic $V_{\textbf{w}}(\theta)$.[]{data-label="fig:sub2"}](scratch_run.pdf){width="0.8\paperwidth"}
POMDPs {#pomdps}
------
We apply Algorithm \[alg:POMDPGMCV\] to a simple copy task where it is important to remember events that happened a long time ago. We compare its performance to the one of a standard LSTM trained by BPTT. The training test consists of sequences of variable length. The first element for each sequence is a real number $x \in [-0.5, 0.5]$, while all the other elements of the sequences are zeros. The goal is to remember the first number of the sequence and to provide it as an output after the whole sequence is processed, possibly after a long time. In our experiment we use sequences of lengths between 1 and 10.\
Consider Figure \[fig:sub5\]. Although Algorithm \[alg:POMDPGMCV\] is able to learn to make good predictions, it is still outperformed by LSTM trained by BPTT. The final L1 norm of the difference between the prediction of the LSTM and the target is 0.0187 for our method and 0.002 for the standard LSTM. We hypothesize that the benefits of our approach can be observed only when the time lag is substantially larger, while for short sequences it is better to have access to the true gradient with respect to the loss. However, here our focus is on showing that this algorithm can work at all as an alternative to BPTT (although it does not achieve state-of-the-art performance in this illustrative example). Future experiments will analyze our approach in the presence of very long time lags.
![Learning curves representing the average loss of the LSTM $f_{\theta}$ in the test set as a function of the number of iterations used for learning (average across 10 runs, 95% c.i.). The loss reported is the L1 norm of the difference between the LSTM’s prediction and the target. The performance of the LSTM trained by Algorithm \[alg:POMDPGMCV\] is plotted in blue, the one of LSTM trained by BPTT in green.[]{data-label="fig:sub5"}](loss_copy_sl=10.pdf){width="0.35\paperwidth"}
Related work {#sec:related}
============
Policy Evaluation Networks [@harb2020policy]
--------------------------------------------
Recently we learned about Policy Evaluation Networks (PENs) [@harb2020policy] which are closely related to our work and share the same motivation. PENs focus on the simplest PVF $V(\theta)$. Like in some of our experiments, the authors show how following the direction of improvement suggested by $V(\theta)$ leads to policies that outperform any policy used to train V. They also suggest to explore in future work a more complex setting where a PSVF $V(s, \theta)$ is learned using an actor-critic architecture.\
Our work directly introduces the PSVF $V(s,\theta)$ and PAVF $Q(s,a,\theta)$, with applications to both MDPs and POMDPs as well as to standard supervised learning tasks where long time lags are involved. We view the learning of PVF $V(\theta)$ [@harb2020policy] as a particular case where $V(s,\theta)$ is averaged over the initial state distribution and trained via Monte Carlo instead of Temporal Difference. We also derive novel policy gradient theorems for PAVFs when stochastic or deterministic policies are used.\
There are many differences between our approach to learning $V(\theta)$ and theirs [@harb2020policy]. For example, we do not use a fingerprint mechanism [@harb2020policy] for embedding the weights of complex policies. Instead, we simply parse all the policy weights as inputs to the value function, even in the nonlinear case. However, fingerprinting [@harb2020policy] may be important for representing nonlinear policies without losing information about their structure and for saving memory required to store the weights.\
Our training procedures differ as well. Harb et al. [@harb2020policy] first collect a batch of randomly initialized policies and perform rollouts to collect reward from the environment. The PVF $V(\theta)$ is then trained using the data collected. Once V is trained, many gradient ascent steps through V yield new, unseen, randomly initialized policies in a zero-shot manner, exhibiting improved performance. In our setting, however, we use an actor-critic architecture, incrementally and iteratively training the critic and the actor simultaneously.\
Other differences concern the optimization problem: we do not predict a bucket index for discretized reward, but perform a regression task. Therefore our loss is simply the mean squared error between the prediction of $V(\theta)$ and the reward obtained by $\pi_{\theta}$, while their loss [@harb2020policy] is the KL divergence between the predicted and target distributions. Both approaches optimize the undiscounted objective when learning $V(\theta)$.\
Harb et al. [@harb2020policy] train their value function on Swimmer-v3, evaluating 2000 deterministic policies on 500 episodes each (1 million policy evaluations). They achieve a final expected return of $\approx180$ on new policies trained from scratch through V and a maximum observed return of 250. On the other hand, after only 3000 policy evaluations, our actor-critic architecture obtains expected returns of 354 and 338, using linear and nonlinear policies respectively. Although Harb et al. [@harb2020policy] use Swimmer-v3 “to scale up their experiments”, our results suggest that Swimmer-v3 does not conclusively demonstrate possible benefits of their policy embedding.\
Let us emphasize though that these results are not directly comparable: our method learns incrementally and iteratively rather than in a zero-shot way.
Other related work
------------------
There are two main classes of similar algorithms performing a search in policy parameter space. Evolutionary algorithms [@salimans2017evolution; @wierstra2014natural; @mania2018simple] iteratively estimate a fitness function evaluating the performance of a population of policies and then perform gradient ascent in parameter space, often estimating the gradient using finite difference approximation. By replacing the performance of a population through a likelihood estimation, evolutionary algorithms become a form of Parameter Exploring Policy Gradients [@sehnke_parameterexploring_2010; @sehnkepgpecontrol]. Our methods are similar to evolution since our value function can be seen as a fitness. Unlike evolution, however, our approach allows for obtaining directly the gradient of the fitness. It is more suitable for reusing past data. While direct $V(\theta)$ optimization is strongly related to evolution, our more informed algorithms optimize $V(s,\theta)$. That is, ours both perform a search in policy parameter space AND train the value function and the policy online, without having to wait for the ends of trials or episodes.\
The second related class of methods involves surrogate functions [@box1951; @Booker1998; @NIPS1995_1124; @Jones2001]. They often use local optimizers for generalizing across fitness functions. In particular, Bayesian Optimization [@snoek2012practical; @snoekscalable] uses a surrogate function to evaluate the performance of a model over a set of hyperparameters and follows the uncertainty on the surrogate to query the new data to sample. Unlike Bayesian Optimization, we do not build a probabilistic model and we use the gradient of the value function instead of a sample from the posterior to decide which policy parameters to use next in the policy improvement step.\
The vanishing gradient problem in POMDPS was first discovered and analyzed in 1991 [@Hochreiter:91]. The LSTM [@Hochreiter:1997:LSM:1246443.1246450] architecture was introduced to alleviate this problem, but still may suffer from long time lags between relevant events. Several ways of avoiding BPTT have been proposed. One is known as Reservoir Computing through Echo State networks [@jaegerecho] or Liquid State Machines [@maassliquid]. Here the weights of an RNN are randomly initialized and only a linear output layer is trained. More recently, a related class of learning algorithms based on training the hidden units through evolution (EVOLINO) [@Schmidhuber:07nc; @Schmidhuber:05ijcai; @Wierstra:05geccoevolino] was able to solve previously unsolvable tasks.\
In 1990, adaptive critics trained by TD were used to predict the gradients of an RNN from its activations, avoiding BPTT and producing an algorithm local in time and space [@schmidhuber1990networks]. This idea was later used to update the weights of a neural network asynchronously [@jaderberg2017decoupled]. In our work, the critic is predicting errors instead of gradients and BPTT is avoided by viewing the program (the parameters) of an RNN as a static object and editing it directly. The static program of a NN can also be generated by another NN [@schmidhuber1992learning], or modified directly using a self-referential mechanism, where the NN is able to inspect its own policy-changing algorithm [@schmidhuber1993self].\
The possibility of augmenting the value functions with auxiliary parameters was already considered in work on General Value Functions [@Sutton:2011:HSR:2031678.2031726], where the return is defined with respect to an arbitrary reward function. The main idea is to use a collection of value functions, one for each agent, representing chunks of knowledge about the environment. Universal Value Function Approximators [@Schaul:2015:UVF:3045118.3045258] used the same approach to learn a single value function representing the value, given possible goals of an agent–compare early work (1990) on goal-conditional RL [@SchmidhuberHuber:91]. However, General and Universal Value Functions have not been applied to learn a single value function for every possible policy.\
Recent work [@unterthiner2020predicting] shows how to map the weights of a trained Convolutional Neural Network (CNN) to its accuracy. Experiments show how these predictions allow for ranking the performance of neural networks on new unseen tasks. These maps are either learned by taking the flattened weights as input or using simple statistics. Despite the promising prediction capabilities of their method, they do not use these maps to guide the training process of CNNs.\
Gradient Temporal Difference [@gtd; @fastgd; @convergentnonlinear; @offpolicycontrol; @maei2011gradient] and Emphatic Temporal Difference methods [@sutton2016emphatic] were developed to address convergence under on-policy and off-policy [@Precup2001OffpolicyTD] learning with function approximation. The first attempt to obtain a stable off-policy actor-critic algorithm [@kondaactorcritic; @Bhatnagar:2007:INA:2981562.2981576; @Peters:2008:NA:1352927.1352986] under linear function approximation was called Off-PAC [@Degris2012], where the critic is updated using GTD($\lambda$) algorithm [@maei2011gradient] to estimate the state-value function. This algorithm has promising theoretical guarantees when the learned policy is tabular. In the general case, however, the actor does not follow the true gradient descent direction. Only recently it was possible to prove convergence of the off-policy actor-critic architecture under linear function approximation [@maei2018convergent]. A paper on Deterministic Policy Gradients [@Silver2014] extended the Off-PAC policy gradient theorem [@Degris2012] to deterministic policies. This was coupled with a deep neural network to solve continuous control tasks through Deep Deterministic Policy Gradients [@lillicrap2015continuous]. The idea of using actor-critic methods with deterministic policies was first proposed in 1990 [@schmidhuber1990making]. Here the critic was trained by TD to predict possibly multi-dimensional rewards, although no policy gradient theorem was formally derived.\
All our algorithms are based on the off-policy actor-critic architecture. The two algorithms based on $Q(s,a,\theta)$ can be viewed as analogous to Off-PAC and DPG where the critic is defined for all policies and the actor is updated following the true gradient with respect to the critic.\
Limitations and future work {#sec:todo}
===========================
Algorithms based on $V(\theta)$ and $V(s,\theta)$:
--------------------------------------------------
One limitation of our work is the way we parse the parameters of the policy as an input to the value function. Embeddings similar to those used in Policy Evaluation Networks [@harb2020policy] may be useful not only for saving memory and computational time, but also for preventing the loss of useful information when deep neural networks are used as policies. We intend to experimentally evaluate the benefits of such embeddings in our actor-critic architecture.\
Another interesting direction of research concerns the convergence analysis of our off-policy actor-critic when $V_w(\theta)$ or $V_w(s,\theta)$ are assumed to be linear.
Algorithms based on $Q(s,a,\theta)$
-----------------------------------
So far we have not yet experimentally analyzed Algorithms \[alg:SACTDQ\] and \[alg:DACTDQ\]. We plan to compare them to Off-PAC and DPG, to assess potential benefits of following exact policy gradients. In Appendix \[sec:comp\] we derive a class of functions for Q compatible with the policy parameterization. This can be used to derive theoretical guarantees for these algorithms under certain assumptions.
Algorithms for POMDPs
---------------------
In POMDPs, policy embeddings might greatly improve learning and dealing with the non-linearity of RNNs. We plan to experimentally analyze the potential benefits and drawbacks of avoiding BPTT.
### Acknowledgments {#acknowledgments .unnumbered}
We thank Paulo Rauber, Imanol Schlag, Louis Kirsch, Róbert Csordás, Aleksandar Stanić, Anand Gopalakrishnan and Julius Kunze for their feedback. This work was supported by the ERC Advanced Grant (no: 742870). We also thank NVIDIA Corporation for donating a DGX-1 as part of the Pioneers of AI Research Award and to IBM for donating a Minsky machine.
Implementation details {#sec:impl}
======================
Here we present implementation details of Algorithms \[alg:GMCV\], \[alg:GTDVS\] and \[alg:POMDPGMCV\].
Algorithm \[alg:GMCV\]
----------------------
The policy is represented by a Feedforward Neural Network (FNN) consisting of one hidden layer and 32 neurons with tanh activation. Linear policies are represented by a linear deterministic function from states to actions. The value function is represented by an FNN consisting of two hidden layers and 512 neurons with ReLu activation [@von1973self] and Layer Normalization [@ba2016layer]. The batch size is 16. Data are sampled uniformly from the replay buffer. The value function and policy are updated offline through 10 gradient steps after every episode. We use the Adam optimizer [@kingma2014adam] for both the policy and the value function with a fixed learning rate $1e-3$. The noise we add to the policy is sampled from a standard gaussian distribution. The policy and the value function are initialized using Xavier uniform initialization [@glorot2010understanding].\
Algorithm \[alg:GTDVS\]
-----------------------
The policy is represented by an FNN consisting of one hidden layer and 32 neurons with tanh activation. Linear policies are represented by a linear deterministic function from states to actions. The value function is represented by a FNN consisting of two hidden layer and 512 neurons with ReLu activation [@von1973self] and Layer Normalization [@ba2016layer]. The batch size is 128. Data are sampled uniformly from the replay buffer, which stores the last 100000 transitions. The value function is updated offline through 500 gradient steps after every 1000 interactions with the environment. Every 25 gradient steps for the value function, the policy is updated through a single gradient step. We use the Adam optimizer [@kingma2014adam] for both the policy and the value function with a fixed learning rate 1e-3. The noise we add to the policy is sampled from a standard gaussian distribution. The discount factor $\gamma$ is set to 0.9999. The policy and the value function are initialized using Xavier uniform initialization [@glorot2010understanding].\
Algorithm \[alg:POMDPGMCV\]
---------------------------
The policy is represented by an LSTM with a single neuron. The value function is represented by an FNN consisting of 4 hidden layers and 128 neurons with ReLu activations [@von1973self] and Layer Normalization [@ba2016layer]. The batch size is 64. Data are sampled choosing half of the batch size uniformly and half from the most recently collected pairs $(\theta, l)$. At each iteration, 100 sequences with different values and lengths are generated. Then a forward pass with LSTM is performed. The average loss is computed and stored in the replay buffer. The value function and the LSTM are updated through one gradient step after every iteration. We use the Adam optimizer [@kingma2014adam] with fixed learning rate of 3e-3 for the policy and 1e-3 for the value function. Every time we process a batch of sequences, we perturb with probability 0.5 the parameters of the LSTM, choosing $\Tilde{\theta} = \theta + \epsilon, \epsilon \sim \mathcal{N}(0, 0.1I)$. The test set consists of 200 sequences independently generated in line with the distribution of the training set. The LSTM trained with BPTT has the same architecture and batch size as the one trained with the parameter-based value function. All the neural nets are initialized using Xavier uniform initialization [@glorot2010understanding].
Compatible function approximation {#sec:comp}
=================================
If we approximate the true $Q(s,a, \theta)$ with a function $Q_{w}(s,a,\theta)$, we need to ensure that the approximator will follow the gradient of the true $Q(s,a,\theta)$. Here we derive a class of functions compatible with this objective.
Stochastic case
---------------
[thr]{}[CFA Stochastic]{} Let $\pi_{\theta}(a|s)$ be stochastic. A function approximator $Q_{w}(s,a,\theta)$ is compatible with the policy parametrization $\pi_{\theta}(a|s)$, i.e. $$\begin{aligned}
\label{grad}
\nabla_{\theta}J(\pi_{\theta}) & = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) [Q(s,a,\theta) \nabla_{\theta} \log\pi_{\theta}(a|s) + \nabla_{\theta}Q(s,a,\theta)] {\,\mathrm{d}}a {\,\mathrm{d}}s \\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) [Q_{w}(s,a,\theta) \nabla_{\theta} \log\pi_{\theta}(a|s) + \nabla_{\theta}Q_{w}(s,a,\theta)] {\,\mathrm{d}}a {\,\mathrm{d}}s
\end{aligned}$$\
if:
1. $$\label{eq1}
Q_{w^*}(s,a,\theta) = \nabla_{\theta} \log \pi_{\theta}(a|s) w_{1}^* + \theta w_{2}^*$$
where $w^* = [w_1^*; w_2^*]$.
2. $w_1^*$ is minimising: $$\label{eq2}
w_1^* = \operatorname*{arg\,min}_{w_1}MSE(\theta, w_1, w_2^*) =\int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s)(Q(s,a,\theta) - Q_{w}(s,a,\theta))^2 {\,\mathrm{d}}a {\,\mathrm{d}}s$$
3. $w_2^*$ is minimising:
$$\label{eq3}
w_2^* = \operatorname*{arg\,min}_{w_2}MSE(\theta, w_1^*, w_2) = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s)(\nabla_{\theta}Q(s,a,\theta) - \nabla_{\theta}Q_{w}(s,a,\theta))^2 {\,\mathrm{d}}a {\,\mathrm{d}}s$$
When the process described by equations \[eq2\] and \[eq3\] converges to a local optimum, we have:
$$\begin{aligned}
0 & = \nabla_{w_1} \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) (Q(s,a,\theta) - Q_{w}(s,a,\theta))^2 {\,\mathrm{d}}a {\,\mathrm{d}}s \\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) (Q(s,a,\theta) - Q_{w}(s,a,\theta)) \nabla_{w_1} Q_{w}(s,a,\theta) {\,\mathrm{d}}a {\,\mathrm{d}}s \\
\label{eqq}
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) (Q(s,a,\theta) - Q_{w}(s,a,\theta)) \nabla_{\theta} \log \pi_{\theta}(a|s) {\,\mathrm{d}}a {\,\mathrm{d}}s
\end{aligned}$$
\
and
$$\begin{aligned}
0 & = \nabla_{w_2} \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) (\nabla_{\theta}Q(s,a,\theta) - \nabla_{\theta} Q_{w}(s,a,\theta))^2 {\,\mathrm{d}}a {\,\mathrm{d}}s \\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) (\nabla_{\theta} Q(s,a,\theta) - \nabla_{\theta} Q_{w}(s,a,\theta)) \nabla_{w_2} \nabla_{\theta} Q_{w}(s,a,\theta) {\,\mathrm{d}}a {\,\mathrm{d}}s \\
\label{ere}
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) (\nabla_{\theta} Q(s,a,\theta) - \nabla_{\theta} Q_{w}(s,a,\theta)) 1 {\,\mathrm{d}}a {\,\mathrm{d}}s
\end{aligned}$$
\
Subtracting \[eqq\] and \[ere\] from \[grad\]:
$$\begin{aligned}
\label{ewwq2}
\nabla_{\theta}J(\pi_{\theta}) & = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) [Q(s,a,\theta) \nabla_{\theta} \log\pi_{\theta}(a|s) + \nabla_{\theta}Q(s,a,\theta)] {\,\mathrm{d}}a {\,\mathrm{d}}s \\
& \;\;\;\; - \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) (Q(s,a,\theta) - Q_{w}(s,a,\theta)) \nabla_{\theta} \log \pi_{\theta}(a|s) {\,\mathrm{d}}a {\,\mathrm{d}}s \\
& \;\;\;\; - \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) (\nabla_{\theta} Q(s,a,\theta) - \nabla_{\theta} Q_{w}(s,a,\theta)) {\,\mathrm{d}}a {\,\mathrm{d}}s \\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) \int_{\mathcal{A}} \pi_{\theta}(a|s) [Q_{w}(s,a,\theta) \nabla_{\theta} \log\pi_{\theta}(a|s) + \nabla_{\theta}Q_{w}(s,a,\theta)] {\,\mathrm{d}}a {\,\mathrm{d}}s
\end{aligned}$$
\
Deterministic case
------------------
[thr]{}[CFA - Deterministic]{} Let $\pi_{\theta}(a|s)$ be deterministic. A function approximator $Q_{w}(s,a,\theta)$ is compatible with the policy parametrization $\pi_{\theta}(a|s)$, i.e. $$\begin{aligned}
\label{grad2}
\nabla_{\theta}J(\pi_{\theta}) & = \int_{\mathcal{S}} d^{\pi_{b}}(s) [\nabla_a Q(s,a,\theta) \nabla_{\theta} \pi_{\theta}(a|s) + \nabla_{\theta}Q(s,a,\theta)] {\,\mathrm{d}}s \\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) [\nabla_a Q_{w}(s,a,\theta) \nabla_{\theta} \pi_{\theta}(a|s) + \nabla_{\theta}Q_{w}(s,a,\theta)] {\,\mathrm{d}}s
\end{aligned}$$\
if:
1. $$Q_{w^*}(s,a,\theta) = \nabla_{\theta} \pi_{\theta}(a|s) a w_{1}^* + \theta w_{2}^*$$
where $w^* = [w_1^*; w_2^*]$.
2. $w_1^*$ is minimising: $$\label{eq22}
w_1^* = \operatorname*{arg\,min}_{w_1}MSE(\theta, w_1, w_2^*) =\int_{\mathcal{S}} d^{\pi_{b}}(s) (\nabla_a Q(s,a,\theta) - \nabla_a Q_{w}(s,a,\theta))^2 {\,\mathrm{d}}s$$
3. $w_2^*$ is minimising: $$\label{eq33}
w_2^* = \operatorname*{arg\,min}_{w_2}MSE(\theta, w_1^*, w_2) = \int_{\mathcal{S}} d^{\pi_{b}}(s) (\nabla_{\theta}Q(s,a,\theta) - \nabla_{\theta}Q_{w}(s,a,\theta))^2 {\,\mathrm{d}}s$$
When the process described by equations \[eq22\] and \[eq33\] converges to a local optimum, we get:
$$\begin{aligned}
0 & = \nabla_{w_1} \int_{\mathcal{S}} d^{\pi_{b}}(s) (\nabla_a Q(s,a,\theta) - \nabla_a Q_{w}(s,a,\theta))^2 {\,\mathrm{d}}s \\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) (\nabla_a Q(s,a,\theta) - \nabla_a Q_{w}(s,a,\theta)) \nabla_{w_1} \nabla_a Q_{w}(s,a,\theta) {\,\mathrm{d}}s \\
\label{eqqqq}
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) (\nabla_a Q(s,a,\theta) - \nabla_a Q_{w}(s,a,\theta)) \nabla_{\theta} \pi_{\theta}(a|s) {\,\mathrm{d}}s
\end{aligned}$$
\
and
$$\begin{aligned}
0 & = \nabla_{w_2} \int_{\mathcal{S}} d^{\pi_{b}}(s) (\nabla_{\theta}Q(s,a,\theta) - \nabla_{\theta} Q_{w}(s,a,\theta))^2 {\,\mathrm{d}}s \\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) (\nabla_{\theta} Q(s,a,\theta) - \nabla_{\theta} Q_{w}(s,a,\theta)) \nabla_{w_2} \nabla_{\theta} Q_{w}(s,a,\theta) {\,\mathrm{d}}a {\,\mathrm{d}}s \\
\label{ereere}
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) (\nabla_{\theta} Q(s,a,\theta) - \nabla_{\theta} Q_{w}(s,a,\theta)) 1 {\,\mathrm{d}}s
\end{aligned}$$
\
Subtracting \[eqqqq\] and \[ereere\] from \[grad2\]:
$$\begin{aligned}
\label{ewwq2}
\nabla_{\theta}J(\pi_{\theta}) & = \int_{\mathcal{S}} d^{\pi_{b}}(s) [\nabla_a Q(s,a,\theta) \nabla_{\theta} \pi_{\theta}(a|s) + \nabla_{\theta}Q(s,a,\theta)] {\,\mathrm{d}}s \\
& \;\;\;\; - \int_{\mathcal{S}} d^{\pi_{b}}(s) (\nabla_a Q(s,a,\theta) - \nabla_a Q_{w}(s,a,\theta)) \nabla_{\theta} \pi_{\theta}(a|s) {\,\mathrm{d}}s \\
& \;\;\;\; - \int_{\mathcal{S}} d^{\pi_{b}}(s) (\nabla_{\theta} Q(s,a,\theta) - \nabla_{\theta} Q_{w}(s,a,\theta)) {\,\mathrm{d}}s \\
& = \int_{\mathcal{S}} d^{\pi_{b}}(s) [\nabla_a Q_{w}(s,a,\theta) \nabla_{\theta} \pi_{\theta}(a|s) + \nabla_{\theta}Q_{w}(s,a,\theta)] {\,\mathrm{d}}s
\end{aligned}$$
\
|
---
abstract: 'We consider chiral liquids, that is liquids consisting of massless fermions and right-left asymmetric. In such media, one expects existence of electromagnetic current flowing along an external magnetic field, associated with the chiral anomaly. The current is predicted to be dissipation-free. We consider dynamics of chiral liquids, concentrating on the issues of possible instabilities and infrared sensitivity. Instabilities arise, generally speaking, already in the limit of vanishing electromagnetic constant, $\alpha_{el}\to~0$. In particular, liquids with non-vanishing chiral chemical potential might decay into right-left asymmetric states containing vortices.'
author:
- 'A. Avdoshkin'
- 'V.P. Kirilin'
- 'A.V. Sadofyev'
- 'V.I. Zakharov'
title: On consistency of hydrodynamic approximation for chiral media
---
Introduction
============
Interest in theory of chiral liquids was originally boosted by the discovery of the quark-gluon plasma (since the light quarks are nearly massless). In theoretical studies, one mostly concentrates, however, on a generic plasma of massless fermions which interact in a chiral-invariant way and possess U(1) charges. A remarkable feature of the chiral materials is that the chiral anomaly, which is a loop, or quantum effect, is predicted to have macroscopic consequences and effectively modifies the Maxwell equations. In particular, in the equilibrium there is an electric current $j_{\mu}^{el}$ proportional to external magnetic field [@vilenkin; @chianov; @kharzeev]: $$\label{cme}
j_{\mu}^{el}~=~\sigma_M B_{\mu}~~.$$ Here $B_{\mu}$ is the magnetic field in the rest frame of the element of the liquid, or $B_{\mu}\equiv~(1/2)\epsilon_{\mu\nu\alpha\beta}u^{\nu}F^{\alpha\beta}$, where $u^{\mu}$ is the 4-velocity of an element of the liquid.
In most recent times, the interest in theory of chiral liquids was triggered by the paper in Ref. [@surowka] where it was demonstrated that the value of $\sigma_M$ is uniquely fixed in the hydrodynamic approximation. In particular, for a single (massless) fermion of charge $e$: $$\sigma_M~=~\frac{e^2\mu_5}{2\pi^2}~~,$$ where $\mu_5$ is the chiral chemical potential, $\mu_5~\equiv~\mu_L-\mu_R$, so that $\mu_5\neq 0$ implies that the medium is not invariant under parity transformation.
Using the hydrodynamic approximation and equations of motion one can demonstrate that the magnetic conductivity is protected against corrections [@surowka]. This non-renormalization of $\sigma_M$ goes back to the Adler-Bardeen theorem. Moreover, one can argue that the current (\[cme\]) is dissipation free [@yee]. Indeed, both the r.h.s. and l.h.s. of (\[cme\]) are odd under time reversal. This is a strong indication that the dynamics behind (\[cme\]) is Hamiltonian and there is no dissipation [@yee]. Analogy to the superconducting current in the London limit, $\vec{j}^{el}~=~m_{\gamma}^2\vec{A}$ where $m_{\gamma}$ is the photon mass and $\vec{A}$ is the vector potential, supports [@zakharov2] the conclusion on the dissipation-free nature of the current (\[cme\]).
Further studies of dynamics of chiral liquids seem desirable. It is worth emphasizing that some chiral effects survive in the limit of the electromagnetic coupling tending to zero, $\alpha_{el}\to~0$. In particular there is so called chiral vortical effect [@surowka] which is the flow of axial current along the liquid’s vorticity, $
j_{\alpha}^5~\sim~\epsilon_{\alpha\beta\gamma\delta}
u^{\beta}\partial^{\gamma}u^{\delta}$, where $u_{\alpha}$ is the 4-velocity of an element of the liquid.
In view of the indications that chiral liquids possess such unusual properties. In this note we will consider dynamics of the chiral liquids, concentrating mostly on possible instabilities and infrared sensitivity. In particular, we argue that the instabilities arise already in the limit $\alpha_{el}\to 0$. Namely, chiral liquids with non-vanishing chemical potential might decay into a right-left asymmetric state containing vortices. The basic element of our analysis is consideration of consequences from conservation of the axial current in the hydrodynamic approximation.
Turn first to the definition of the axial charge on the fundamental, field-theoretic level: $$\label{operator}
Q^A~=~Q^A_{naive} + \frac{e^2}{4\pi^2}\mathcal{H}, ~~~\frac{d}{dt}Q^A~=~0~,$$ where $Q^A_{naive}$ is the axial charge which is conserved according to the classical equations of motion (without account of the anomaly) and $\mathcal{H}$ is the so called magnetic helicity: $$\label{helicity}
\mathcal{H}=\int \vec{A}\cdot \vec{B} d^3x~~,$$ where $\vec{B}$ is the magnetic field and $\vec{A}$ is the corresponding vector potential.
In the approximation of external fields consideration of (\[operator\]) would not bring any new insight compared to the approach of Ref. [@surowka]. However Eq. (\[operator\]) becomes more informative in case of dynamical electromagnetic fields. The point is that, according to intuition based on thermodynamics, all degrees of freedom contributing to the axial charge (\[operator\]) are to be manifested in the equilibrium. This implies that if one starts, for example, with a state $Q^A_{naive}~\neq~0,~
\mathcal{H}~=~0,$ then this state is in fact unstable and a non-vanishing $\mathcal{H}$ would emerge spontaneously. These expectations were verified in [@redlich; @shaposhnikov; @akamatsu; @khaidukov] where the corresponding negative mode was identified explicitly.
Turn now to the hydrodynamic setup, as it is introduced in Ref. [@surowka]. Here, one assumes that there exists a liquid whose constituents interact in a chiral invariant way. The liquid is described by the standard hydrodynamic (relativistic) equations. To probe properties of the liquid one introduces coupling with external electromagnetic fields. Mostly, the electromagnetic field is considered to be not dynamic. Which means, in particular, that one neglects electromagnetic interactions between the constituents as well as excitation of electromagnetic waves in the medium. Within this framework one can send the the electromagnetic coupling $\alpha_{el}$ to zero, $\alpha_{el}\to~0$ without affecting the properties of the medium. The electromagnetic coupling survives only as a coefficient in front of, say, the chiral magnetic current. We will work within this framework when electromagnetic interaction between constituents is absent or overshadowed by another, stronger interaction. Note that this framework differs from, say, magnetohydrodynamics. In the latter case the electromagnetic field is dynamic and this is crucial to establish electromagnetic instabilities [@redlich; @akamatsu; @shaposhnikov; @khaidukov]. We have mentioned these results to introduce the issue of chiral-liquid instabilities. Our central point, on the other hand, is that instabilities of chiral liquids exist also in case of non-dynamic electromagnetic field, or even in case of neutral constituents.
In the hydrodynamic approximation, the axial current corresponding to $Q^A_{naive}$ takes the form $j_{\mu}^A~=~n^Au_{\mu}$ where $n^A\equiv n^L-n^R$ and $n^{L,R}$ are the densities of the left- and right-handed fermionic constituents, respectively. What is much less trivial, is that because of the anomaly the axial current contains further terms. Indeed, already the analysis of Ref. [@surowka] reveals existence of the chiral vortical effect, with axial current being contributed by helical motion of the liquid (see also below).
In more detail, the axial charge in hydrodynamics can be represented as: $$\label{extended}
Q^A_{hydro}~=~Q^A_{naive}~+~Q^A_{mh}~+~Q^A_{mfh}~+~Q^A_{fh}~,$$ where indices $``mh'' , ``fh''$ and $``mfh''$ stand for “magnetic helicity” , “fluid helicity” and mixed “magnetic-fluid helicity”, respectively. Note that we are using here the standard terminology of magneto-hydrodynamics, see, e.g., [@bekenstein; @moffatt], where the fluid and magnetic helicities were considered phenomenologically, without reference to chiral liquids. In particular, $Q^A_{mh}$ stands for $(e^2/2\pi^2)\mathcal{H}$. The fluid helicity, $Q^A_{fh}$ is defined as the charge associated with the current $j^{\alpha}_{fh}$: $$\label{zero}
Q^A_{fh}~=~\frac{1}{4\pi^2}\int d^3 xj^0_{fh}~~,$$ while the current $j^{\alpha}_{fh}$ is given by: $$\label{fluidcurrent}
j^{\alpha}_{fh}~=~2\mu^2 \omega^{\alpha}~~,$$ where $$\label{omega}
\omega^{\alpha}~\equiv~\frac{1}{2}\epsilon^{\alpha\beta\sigma\rho}u_{\beta}\partial_{\sigma}u_{\rho}.$$ An explicit expression for the mixed helicity is given later. Actually, the algorithm of construction of various pieces in (\[extended\]) is readily identified by using analogy with the pure electromagnetic case, as we explain in a moment.
Eq. (\[extended\]) can be substantiated in a number of ways. One possibility is to look into explicit expressions for vector and axial currents obtained via the procedure introduced first in [@surowka]. We will come to this point later, see discussion of Eq. (\[close\]) below. Now, we will follow [@shevchenko] to argue that in the hydrodynamic setup one should substitute the standard electromagnetic potential $eA_{\mu}$ in the expression for the chiral anomaly by the following combination: $$\label{shevchenko}
eA_{\mu}~\to~eA_{\mu}~+~\mu u_{\mu} ~,$$ where $\mu$ is the chemical potential associated with the charge, source of the potential $A_{\mu}$.
The most straightforward way to justify (\[shevchenko\]) is to observe that chemical potential thermodynamically is introduced through an extension of the original Hamiltonian $\hat{H}$: $$\label{hydro}
\hat{H}~\to~\hat{H}~-~\mu\hat{Q}~-~\mu_5\hat{Q}^A~.$$ As far as the chemical potentials $\mu,\mu_5$ are considered to be small, the corresponding change in the Lagrangian $\delta L$ is given by $$\label{deltaL}
\delta L=-\delta H~=~\mu Q +\mu_5Q^A=~\mu\bar{\psi}\gamma_0\psi+
\mu_5\bar{\psi}\gamma_0\gamma_5\psi .$$ The next step is to generalize (\[deltaL\]) to the case of hydrodynamics. The generalization assumes rewriting (\[deltaL\]) in an explicitly Lorentz-covariant way: $$\delta L~=~\mu u^{\alpha}\bar{\psi}\gamma_{\alpha}\psi+
\mu_5u^{\alpha}\bar{\psi}\gamma_{\alpha}\gamma_5\psi~ .$$ In case of $\mu_5=0$, the substitution (\[shevchenko\]) becomes then obvious.
As a result of substitution (\[shevchenko\]) the definition of the conserved axial charge (\[operator\]) is generalized in hydrodynamics to the expression (\[extended\]), and we will consider implications of this extension [^1].
Consider a non-vanishing chiral chemical potential, $\mu_5\neq 0$. Then one expects that in the equilibrium all the degrees of freedom with a non-vanishing axial charge are excited and all the helicities entering (\[extended\]) are non-vanishing. This implies, in turn, that if one starts with the state where the whole axial charge is attributed to a single term in the r.h.s. of (\[extended\]), say, to the charge of elementary constituents, $$Q^A_{naive}~\neq~0,~~Q^A_{mh}~=~Q^A_{fh}~=~Q^A_{mfh}~=0~,$$ then this state is unstable with respect to generation of all types of helicities. The instability with respect to generation of magnetic fields with non-vanishing helicity (\[helicity\]) was considered in detail and in various applications [@redlich; @shaposhnikov; @akamatsu; @khaidukov]. Eq. (\[extended\]) implies that in fact one can expect that in hydrodynamics there are more general instabilities which would result in generation of all possible helicities: $$\label{equal}
Q^A_{naive}~\sim~Q^A_{mh}~\sim~Q^A_{mfh}~\sim~Q^A_{fh}~.$$ In other words, all types of helical motions are excited in chiral plasma on macroscopic scales.
It is amusing that the possibility of the helicity conservation in the ordinary hydrodynamics (without any reference to the chiral liquids) has been studied in great detail, for review see, e.g., [@bekenstein; @moffatt]. The generic conclusion is that the fluid helicities are conserved in the limit of vanishing dissipation. The outline of the paper is as follows. In the next section we address the issue of infrared sensitivity of the definition of axial charge in field theory. Next, we substantiate representation of the axial charge in hydrodynamics as a sum of various vorticities. Then, we argue that the magnetic and fluid helicities are conserved in classical limit in case of absence of dissipation. Next, we introduced various types of instabilities of chiral liquids.
In conclusion, we summarize the results obtained.
Evaluation of axial charge
===========================
In this section we outline evaluation of the anomalous piece in the conserved axial charge (\[operator\]). In particular, we emphasize that the calculation is actually valid only in the limit of exact symmetry. In other words, fermion masses are assigned to be exact zero. Considering this limit is common to the recent papers on the anomalous hydrodynamics, see, e.g. , [@jensen]. Only in this limit the effect of the anomaly is reduced to local terms in the effective action. Moreover, there is no explicit time dependence as if we are discussing [*static*]{} processes. A specific feature of such local, or polynomial terms is that the action is gauge invariant while the density of the action is not gauge invariant. The expression (\[helicity\]) for the magnetic helicity provides the best known example of such a term.
If one introduces explicit violation of the chiral symmetry, say, through the masses of the constituents, then the effect of the anomaly does not reduce to local terms in the effective action. Nevertheless, in a certain kinematic limit the matrix element of the axial current becomes again the same polynomial as in (\[operator\]). We emphasize that this kinematic limit actually assumes non-vanishing [*time-dependent*]{} fields. In particular, the expression (\[operator\]) for the matrix element of the axial charge in the limit of electric fields much stronger than the fermionic masses: $$\label{limit}
m_f^2~\ll~E~\ll~H~~,$$ where $E,H$ are electric and magnetic fields, respectively. The constraint (\[limit\]) is mentioned in [@zakharov2]. Here we present a more detailed derivation of (\[limit\]).
Thus, our aim here is to evaluate the matrix element of the axial charge over a photonic state $\langle\gamma|Q^A|\gamma\rangle$, where $$\label{axialcharge}
Q^A~=~\int d^3x j^A_0(\vec{x},t)~=~
\int d^3x \bar{\psi}\gamma_0\gamma_5\psi$$ and $\psi$ is a massless Dirac field of charge $e$. Moreover, consider temperature-zero case and the photons on mass shell. Then, it is well known that the matrix element of the axial current $j_{\mu}^A$ corresponding to the anomalous triangle graph has a pole. In the momentum space, $$\label{current}
\langle \gamma|j_{\mu}^A|\gamma\rangle~=~\frac{e^2}{2\pi^2}\frac{iq_{\mu}}{q^2}
\epsilon_{\rho\sigma\alpha\beta}e^{(1)}_{\rho}k^{(1)}_{\sigma}e^{(2)}_{\alpha}k^{(2)}_{\beta}~~,$$ where $q_{\mu}$ is the 4-momentum brought in by the axial current, $e_{\rho}^{(1)},k^{(1)}_{\sigma}$ and $e^{(2)}_{\alpha}k^{(2)}_{\beta}$ are the polarization vectors and momenta of the photons.
The matrix element (\[current\]) of the axial current is clearly non-local in nature, by virtue of the Lorentz covariance and gauge invariance. Concentrate, however, on the matrix element of the axial charge (\[axialcharge\]). Since the charge is defined as $Q^A=\int d^3x j_0^A(\vec{x},t)$, evaluating the charge implies considering the kinematical limit $$\vec{q}~\equiv~0,~q_{0}~\to~0 ~~.$$ In this limit the matrix element (\[current\]) reduces to a polynomial: $$\langle\gamma|Q^A|\gamma\rangle~=i\frac{e^2}{4\pi^2} \epsilon^{ijk}
e^{(1)}_ie^{(2)}_j (k^{(1)}-k^{(2)})_k~$$ and we come, indeed, to the standard expression for the magnetic helicity (\[operator\]).
Evaluating the charge (\[axialcharge\]) starting from the non-local expression (\[current\]) for the current has advantages, from the theoretical point of view. In particular, we avoid considering contribution of heavy regulator fields, and our derivation of (\[axialcharge\]) is given entirely in terms of physical, or light (massless) degrees of freedom. On the other hand, the now-standard way of evaluating the magnetic conductivity $\sigma_M$ is to relate it to the spatial correlator of two electromagnetic currents (for review see, e.g., [@landsteiner]). In the momentum space: $$\label{correlator}
\sigma_M~=~\lim_{q_0\equiv 0,{q}_k~\to 0}\epsilon^{ijk}\frac{i}{2q_k}
\langle j^{el}_i,~j^{el}_j\rangle$$ Although taking the limit of $q_k\to 0$ implies, at first sight, that the correlator (\[correlator\]) is sensitive to large distances, $r~\sim~1/|\vec{q}|$, in fact, it depends on the correct definition of the correlator at the coinciding points. Therefore, one has to consider carefully the ultraviolet regularization procedure, for details see [@landsteiner].
Necessity of a careful treatment of the [*time-dependent*]{} fields looks counter-intuitive in view of the fact that (\[correlator\]) relates the magnetic conductivity to a pure spatial correlator. It might, therefore, worth reminding the reader that in the original derivation of the axial anomaly in terms of zero modes in magnetic field [@ninomiya] one evaluates actually the work $W$ produced by an external [*electric*]{} field $\vec{E}$: $$\label{work}
W~\equiv ~\vec{E}\cdot\vec{j}_{el}~=~\vec{E}\cdot \vec{B}\frac{e^2}{2\pi^2}\mu_5~~.$$ This work compensates the energy needed for massless pair production. And only after cancelling the electric field from the both sides of (\[work\]) one arrives at the current (\[cme\]) which apparently depends on the magnetic field alone.
If, on the other hand, one introduces finite fermionic masses then there is no pair production for $E\ll m_f^2$ and the role of the time-dependent electromagnetic potentials is made explicit. In particular, taking the limit $q_0~\to~0~ (\vec{q}~\equiv~0)$ now gives $$\langle \gamma|Q^A|\gamma \rangle_{m_f\neq 0}~=~0~~,$$ since there is no singularity at $q^2=0$ in the matrix element corresponding to the triangle graph.
Axial charge in hydrodynamics
=============================
Probably, the most striking novel feature brought in by consideration of hydrodynamics is the emergence of the chiral vortical effect, see, e.g. [@surowka; @review], or the flow of the axial current along the fluid vorticity: $$\label{vortical}
j_{\alpha}^A~= ~ \sigma_{\omega}
\omega_{\alpha}~,$$ with the vortical conductivity $\sigma_{\omega}~\neq ~0$. The substitution (\[shevchenko\]), in case of both $\mu,\mu_5~\ne ~0$, fixes $\sigma_{\omega}$ as: $$\label{notemperature}
\sigma_{\omega}~=\frac{(\mu^2+\mu_5^2)}{2\pi^2}~~.$$ The reservation in using the substitution (\[shevchenko\]) is that it does not capture temperature dependences. Also, in Eq. (\[notemperature\]) we did not account for possible spatial variations of the chemical potentials.
Although the chiral vortical effect is rooted in the anomaly it survives in the limit of vanishing electromagnetic coupling. If we restore the term proportional to $\sqrt{\alpha_{el}}$ then the axial-vector current looks as follows: $$\begin{aligned}
\label{close}
j^A_{\mu}=n^Au_{\mu}+
\left(\frac{\mu^2+\mu_5^2}{2\pi^2}\right)\omega_{\mu}+
\frac{e\cdot \mu}{2\pi^2}B_{\mu}+O(e^2)\label{Acurrent}\end{aligned}$$ where $n^A$ is the density of constituents and $B_{\mu}$ is defined in Eq. (\[cme\]). The first term in the r.h.s. of Eq. (close) corresponds to $Q^A_{naive}$ in Eq. (\[extended\]), the second term corresponds to $Q^A_{fh}$ and the third term in the r.h.s. of this equation corresponds to the so called mixed magneto-fluid helicity, see, e.g., [@bekenstein]. The corresponding axial charge, $Q^A_{mfh}$ entering (\[extended\]) is proportional to the volume integral of the temporal component of $j^{\alpha}_{mfh}$: $$\label{one}
j_{mfh}^{\alpha}~=~
\frac{1}{2} \epsilon^{\alpha\beta\rho\sigma}A_{\beta}\omega_{\rho\sigma}$$ where $\omega_{\alpha\beta} = \partial_{\alpha}(\mu u_{\beta})-\partial_{\beta}(\mu u_{\alpha})$. There is also an alternative form of $j^{\alpha}_{mfh}$ defined as: $$\label{two}
j^{\alpha}_{mfh}~=~\frac{1}{2}
\epsilon^{\alpha\beta\rho\sigma}(\mu u_{\beta})F_{\rho\sigma}~~.$$ The corresponding charge, $Q^A_{mfh}$ is the same in the both cases of (\[one\]) and (\[two\]).
Eq. (\[close\]) is close to the expression found first in the pioneering paper [@surowka], see also [@oz]. And, as is mentioned in the Introduction, the expression for the current obtained in [@surowka] could be considered as a motivation to introduce (\[extended\]). The main difference is that the coefficient in front of the fluid vorticity $\omega_{\mu}$ in Ref. [@surowka] contains also terms of third power in chemical potentials. Moreover, using the hydrodynamic expansion to higher orders in derivatives, in the spirit of the approach of [@surowka] would bring further corrections to the current $j_{\mu}^A$. The possibility of variations in explicit expressions for the current is rooted in freedom of choosing frames, or precise definition of the fluid velocity $u_{\mu}$. Ref. [@surowka] uses the Landau frame introduced, e.g., in the textbook of Landau & Lifshitz, while Eq. (\[close\]) assumes the use of the so called entropy frame, see, in particular, [@nicolis].
A crucial point is that the expressions for $Q^A_{naive}, Q^A_{fh}, Q^A_{mfh}, Q^A_{mh}$ do not receive further contributions in the expansion in hydrodynamic derivatives. The absence of higher order terms in the hydrodynamic expression (\[close\]) is a reflection of the important property of the chiral anomaly on the fundamental level that it is limited to a single term $\mathcal{H}$.
So far we exploited the substitution (\[shevchenko\]) following the argumentation of [@shevchenko] reproduced above. However, this argumentation by itself is of somewhat heuristic nature and it is worth emphasizing that similar conclusions were reached more recently in a systematic way within the geometric approach to hydrodynamics, see, e.g., [@megias; @jensen].
To derive the chiral effects in hydrodynamics within these approaches one considers motion in both electromagnetic and gravitational backgrounds. This seems to be rooted in the very nature of the hydrodynamics which is entirely determined by conservation laws, of energy-momentum tensor and of relevant currents.
Technically, one way to trace this kind of unification of electromagnetic and gravitational interactions is to start with a covariant action in higher dimensions. One can demonstrate then [@megias] that the mixed gauge-gravity anomaly in higher dimensions generates a 4d action which is responsible for the chiral effects.
Moreover the conductivity $\sigma_{\omega}$ gets related [@megias] to the correlator of components of the energy-momentum tensor and electric current: $$\label{sigmaomega}
\sigma_{\omega}~=~\lim_{q_0\equiv 0,q_k\to 0}\frac{i}{q_k}\epsilon^{ijk}\langle j^A_i,T_{0j}\rangle~.$$ The connection of our procedure to that of [@megias] can readily be established. Indeed, modification of the naively conserved axial charge $Q^A_{naive}$ by the anomaly is in one-to-one correspondence with the non-vanishing correlator (\[correlator\]). The hydrodynamic modification (\[hydro\]) of the field theoretic Hamiltonian implies modification of the $T^{0i}$ component of the energy-momentum tensor. Choosing, for simplicity, $\mu_5=0$, $$(\delta T^{0i})_{hydro}~=~\mu J^i~~,$$ where $J^i~(i=1,2,3)$ are the spatial components of the vector current. Therefore, there arises an anomalous piece in the correlator $$\label{relation}
\epsilon_{ijk}\frac{\langle T^{0i},J^{j}\rangle}{q^k}~=~\mu\epsilon_{ijk}\frac{\langle
J^{i},J^{j}\rangle}{q^k}~.$$ Eqs (\[correlator\]),(\[sigmaomega\]), (\[relation\]) imply that conductivities $\sigma_M,\sigma_{\omega}$ are related to each other and, actually, in exactly the same way as prescribed by the substitution (\[shevchenko\]). Following this logic, we derive the $Q^A_{fh}$ contribution to the axial charge in the hydrodynamic approximation (\[extended\]).
Another geometric approach [@jensen] starts with considering a static metric $$ds^2~=~-\exp{ (2\sigma (\vec{x}))}
\big(dt+a_i(\vec{x})dx^i\big)^2+g_{ij}(\vec{x})dx^idx^j$$ There is also electromagnetic background ${A}_{\mu}(\vec{x})$. Then one can demonstrate that the symmetries of the problem imply that the partition function depends in fact on the combinations $\mathcal{A}_0$, $\mathcal{A}_i$, $$\label{KK}
e{A}_0~=~e\mathcal{A}_0~+~\mu~,~~e{A}_i~=~e\mathcal{A}_i~-~A_0a_i$$ which are Kaluza-Klein gauge invariant and are replacing $e{A}_{\mu}$ in the standard field theoretic expressions.
Moreover, it is quite obvious that the procedure we are using has much in common with the approach [@jensen]. Indeed, the field $a_i$ entering (\[KK\]) is also proportional to $u_i$ and we readily come to the same contributions to the hydrodynamic axial charge as derived above. Note, however, that the approach of [@jensen] applies only in equilibrium while the substitution (\[shevchenko\]) works in general case.
This concludes the derivation of the axial charge in the hydrodynamic limit, see Eq. (\[extended\]). It is worth mentioning again that all the conclusions concerning chiral plasmas are subject to the reservation that, from the microscopical point of view, the underlying field theories are assumed to be infrared stable. Note that within the holographic approach it turns possible in some cases to study dynamics of chiral liquids in infrared. There are indications that the physics in the infrared could be richer than is usually assumed. In particular, new scales can be generated, for a recent study and further references see [@marolf].
Classical conservation of magnetic and fluid helicities
=======================================================
As it is mentioned above, possibility of conservation of the magnetic and fluid helicities was intensely discussed in the context of the magnetohydrodynamics. Here we will reproduce the main results in a way close to Ref. [@bekenstein] Let us begin with the conservation of the fluid helicity. The main tool to be used is the relativistic version of the Euler equation: $$\begin{aligned}
(p + \epsilon)a^{\mu} = (-\partial^{\mu}p-u^{\mu}u^{\nu}\partial_{\nu}p) =
-P^{\mu \nu}\partial_{\nu}p.\end{aligned}$$ where $\epsilon$ and $p$ stand for proper energy density and pressure respectively, $u^{\beta}$ is the 4-velocity normalized as $u_{\beta}u^{\beta}~=
~-1$, $a^{\mu} = u^{\nu}\partial_{\nu}u^{\mu}$ is the acceleration and $P^{\mu\nu}=u^{\mu}u^{\nu}+g^{\mu\nu}$ is the projection operator. Here electric field is switched off what will be justified later. Moreover, we will utilize Gibbs-Duhem relation: $$\begin{aligned}
d \epsilon = \rho d \mu + s dT.\end{aligned}$$ where $\mu$ is the chemical potential conjugated to the charge $\rho$, $T$ is the local temperature and $s$ is proper density of entropy.
Now we investigate the behaviour of different parts of the axial current in a chiral-neutral charged fluid. After some algebra, one can demonstrate that the current $j^{\alpha}_{fh}$ associated with the fluid helicity has the following divergence: $$\begin{aligned}
\label{divergence0}
\partial_{\alpha} j^{\alpha}_{fh}
= \frac{2T^2\mu s}{p+\epsilon}\omega^{\alpha} \partial_{\alpha}\left( \frac{\mu}{T} \right).\end{aligned}$$ where $j^{\alpha}_{fh}$ and $\omega^{\alpha}$ are defined in (\[fluidcurrent\]) and (\[omega\]), respectively. Thus, if $\displaystyle \frac{\mu}{T}=
const$ (e.g. $T\to0$) this contribution to the axial charge (\[zero\]) is conserved individually.
Turn now to the mixed magnetic-fluid helicity in the absence of electric field. The divergence of the corresponding current (\[two\]) is given by: $$\label{divergence1}
(j^{\alpha}_{fmh})_{,\alpha}~=~1/4\epsilon^{\alpha\beta\gamma\delta}
\omega_{\alpha\beta}F_{\gamma\delta}~.$$ The next step is to express $F_{\alpha\beta}$ in terms of $B_{\mu}$ when $E_{\nu}=0$ [@yodzis] : $$F_{\alpha\beta}~=~\epsilon_{\alpha\beta\gamma\delta}B^{\gamma}u^{\delta}~.$$ Using this as an input one comes to: $$(j^{\alpha}_{fmh})_{,\alpha}~=~\frac{T^2\mu s}{p+
\epsilon}B^{\alpha} \partial_{\alpha}\left( \frac{\mu}{T} \right)~~,$$ and the current is again conserved if $T\to0$ or $\displaystyle \frac{\mu}{T}= const$.
Finally, consider the magnetic helicity (\[helicity\]). The corresponding 4d current is defined as $$j^{\alpha}_{mh}~=~
\frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}A_{\beta}F_{\gamma\delta}~~.$$ The divergence of this current is proportional to the product of magnetic and electric fields $B_{\mu}$ and $E_{\mu}$, $$\label{divergence}
(j^{\alpha}_{mh})_{,\alpha}~=~-2B^{\mu}E_{\mu}~,$$ where $B_{\mu}$ is defined in (\[cme\]), $E_{\mu}~=~F_{\mu\nu}u^{\nu}$ and one finds that $Q_{mh}=\frac{e^2 }{4\pi^2}\mathcal{H}$ .
Eq. (\[divergence\]) is pure kinematic in nature. The dynamic input that ensures conservation of the current $j^{\alpha}_{mh}$ is that in the case of zero-temperature perfect magneto-hydrodynamics $E^{\mu}$ is to vanish along with the temperature: $$E_{\mu}~\to~0,~~if~~\sigma_E~\to~\infty~.$$ One can also easily evaluate the dissipation rate of the magnetic helicity in the case of zero temperature and finite conductivity $\sigma_E$ in classical hydrodynamics [@moffatt]: $$\frac{d\mathcal{H}}{dt}~=~\frac{-2}{\sigma_E}
\int d^3x~\vec{B}\cdot\mathbf{curl}~\vec{B}~~,$$ for details of the derivation see, e.g., [@bekenstein].
In the case of arbitrary temperature one can show that the divergence of the sum of the three helicity currents is $$\begin{aligned}
2\partial_{\alpha}(\mu B^{\alpha}+\mu^2\omega^{\alpha})+
\frac{1}{4}\epsilon^{\alpha\beta\delta\gamma}F_{\alpha\beta}F_{\gamma\delta}=\nonumber\\
=-(F^{\alpha\beta}+\omega^{\alpha\beta})u_{\beta}(\tilde{F}_{\alpha\gamma}
+\tilde{\omega}_{\alpha\gamma})u^{\gamma}=\\
= -\frac{sT}{(\epsilon +p)}\left(E^{\alpha}-
TP^{\alpha\beta}\partial_{\beta}\left(\frac{\mu}{T}\right)\right)(\tilde{F}_{\alpha\gamma}
+\tilde{\omega}_{\alpha\gamma})u^{\gamma} \nonumber\end{aligned}$$ and the similar result holds for the thermal part of the flow helicity:
$$\begin{aligned}
\partial_{\mu}(T^2 \omega^{\mu}) =\frac{2 T^2\rho}{\epsilon + p}
\omega^{\mu}\left(E_{\mu}-T\partial_{\mu}\left(\frac{\mu}{T} \right)\right).\end{aligned}$$
Thus charge conservation constraint is fulfilled in dissipationless limit when the conductivity related quantity $E^{\alpha}-TP^{\alpha\beta}
\partial_{\beta}\left(\frac{\mu}{T}\right)$ is zero. Which, in turn, corresponds to $\sigma \to \infty $. It is worth emphasizing that the viscosity also causes helicity to dissipate so as it was above for an ideal liquid there is an extra requirement $\eta \to 0$ for the axial current conservation to hold.
Another feature which unifies various types of helicity is that the corresponding charges are related to linkage of magnetic and fluid vortices. In particular, the fluid helicity is a measure of linkage of vortex lines in the liquid [@moffatt3]. The fluid-magnetic helicity measures the linkage number of closed vortex lines and magnetic flux lines. Finally, the magnetic helicity can be interpreted in terms of the fluxes of linked flux tubes.
This relation of various types of helicities to topology is a source of non-renormalization theorems. In particular, the anomalous term in the charge (\[operator\]) in case of magnetostatics can be rewritten (by using (\[deltaL\])) as a 3d topological photon mass, see, in particular, [@khaidukov] and references therein. Furthermore, consider currents of the form $J_i(x)~=~I\int d\tau\delta^3(\vec{x}-\vec{x}(\tau)\dot{x}_i(\tau)$. Then the interaction term of two current loops is given by: $$\begin{aligned}
\label{linking}
V=\frac{2II'}{\sigma_M} \int_{C} \!\!\int_{C'} dx^i dy^j
\epsilon_{ijk} \frac{(x-y)^{k}}{4 \pi |x-y|^3}~,\end{aligned}$$ where $\sigma_M$ is defined in (\[cme\]). The integral in (\[linking\]) is apparently proportional to the Gauss linking number of the two current circuits. Moreover one can demonstrate [@khaidukov] that the interaction term (\[linking\]) is not renormalized to any order in electromagnetic interactions.
To summarize this section, consideration of the chiral anomaly in the hydrodynamic limit led us to include into the definition of the conserved axial charge fluid, magnetic and mixed helicities. All three types of helicities are conserved in the zero-temperature limit of perfect magnetohydrodynamics. It is amusing that the chiral anomaly unifies all types of helicities which were considered separately so far.
It is tempting to reverse the logic and assume that the chiral anomaly in the hydrodynamic approximation mixes up charges, associated with helical motions, which are separately conserved classically. It is known that $Q^A_{naive}$ is indeed conserved classically. As is discussed above, other contributions to the axial charge (\[extended\]) are conserved in the absence of dissipation. Therefore, following this logic one would predict, in particular, that for chiral liquids $$\label{ratio}
\Big(\frac{\eta}{s}\Big)_{classically}~\to~0~$$ Such a solution has an advantage of naturally incorporating dissipation-free chiral magnetic current (\[cme\]). Note that Eq. (\[ratio\]) is in no contradiction with the famous lower bound [@kovtun] on the same ratio $\eta/s$. Indeed, one invokes the quantum-mechanical uncertainty principle to establish existence of a lower bound on $\eta/s$ [@kovtun].
Instabilities of chiral plasma
==============================
As is mentioned in the Introduction, chiral plasma can be unstable if one starts with a state where, say, $Q^A_{naive}\neq 0$ while all other terms in the expression (\[extended\]) are vanishing. This kind of instabilities has been discussed recently [@shaposhnikov; @akamatsu; @khaidukov], see also [@redlich].
We have a few points to add:
- [The state with $Q^A_{naive}\neq 0, Q^A_{fh}=Q^A_{mh}=Q^A_{fmh}=0$ can decay not only into the domains with non-vanishing magnetic field [@shaposhnikov; @akamatsu; @khaidukov] but also into domains with helical motion of the plasma, so that $Q^A_{fh}\neq 0$. ]{}
- [In particular, we expect that not only primordial magnetic field could be produced from an original right-left asymmetric state [@shaposhnikov], but primordial helical motion could be generated on a cosmological scale as well.]{}
- [It is amusing to observe, again, that transitions among certain kinds of helicities have been discussed in the literature, independent of the issue of chiral media. Starting from conservation of the extended axial charge (\[extended\]) allows to introduced all possible instabilities in a systematic way. In particular, in paper [@zeldovich] there is a rather detailed discussion of generation of magnetic field from the initial helical motion. In our language, this is about the instability: $$Q^A_{fh}~\to~Q^A_{mh}$$]{}
- [The novel point brought by the consideration of chiral media above is the transition of $Q_{naive}\neq 0$ to other components of the conserved axial charge (\[extended\]).]{}
The most interesting novel example is the transition of the axial charge of elementary constituents into right- (or left-) handed vortices: $$\label{vortical}
Q_{naive}~\to~Q_{fh} .$$ Superficially, this transition is similar to the decay of the axial charge of the constituents to decay into the magnetic heleicity, $Q_{naive}~\to~Q_{mh}$ discussed above. On the dynamical level, however, there is an important difference. In case of the electromagnetic instability, $Q_{naive}~\to~Q_{mh}$, one can find the unstable mode explicitly, see [@akamatsu] and references therein.
In case of the vortical instability (\[vortical\]) there is no analytic expression for the unstable mode. The reason is that there is no perturbation theory for vortical modes because of the infrared divergences. This is known since long, for a recent exposition see, e.g., [@endlich] and references therein. Roughly speaking, if in field theory one starts with (an infinite number of) oscillators, in case of hydrodynamics one starts with (an infinite number of) free particles. As a result, dynamics is decided by non-linearities and no analytical methods exist. Thus, one would turn to numerical methods. Numerical estimates seem especially crucial since we are considering a rather unusual transformation of motion of elementary constituents into the motion of macroscopic vortices. One could suspect that the decay rate of the axial charge accumulated in the constituents into vortices is very low.
Recently, an important progress was reached in answering this type of question, see [@burch]. Namely, one starts numerical simulations from a medium of sound waves at a certain temperature. In other words, it is only microscopic degrees of freedom that are excited first ordinary . It is found that vortices are emerging as a result of interactions. A crucial point is that one starts with the effective Lagrangian derived from first principles, see, e.g., [@endlich]. Moreover, there is a general observation on equivalence of an irrotational heated system, with no chemical potential, and (correspondingly adjusted) chemical potential at zero temperature [@rattazzi]. Thus, we can say that there is evidence that a system with a non-zero chemical potential is unstable with respect to generation of macroscopic vortices.
This is not yet the instability we predict. But it shares the basic dynamical feature, namely, transformation of a microscopically chaotic motion into a macroscopically organized motion. As a result of the instability seemingly observed in Ref. [@burch], an equal number of left- and right-handed vortices is produced since the total axial charge of the system is zero. What we predict, is that if one starts with a non-zero axial chemical potential, vortices are produced which are preferably left- (or right-, depending on the sign of the axial potential) handed.
Conclusions.
============
We a argued that chiral irrotational liquids are unstable with respect to spontaneous generation of vortices. Analytically, however, it is not possible to estimate the decay rate since it involves transformation of motion of microscopical degrees of freedom into a macroscopic motion of vortices. Very recent lattice simulations [@burch] indicate that if one starts with a state of perfect fluid with sound waves at a certain temperature and no vortices, vortices are generated dynamically. Thus, the transformation of a microscopic chaotic motion into an organized macroscopic motion is not suppressed in this case. Because of the similarity of underlying mechanisms the spontaneous generation of non-vanishing macroscopic fluid helicity from chiral liquids with $\mu_5\neq 0$ would seemingly be not suppressed either.
Acknowledgments
===============
The authors are grateful to G.E. Volovik for useful discussions. The work on this paper has been partly supported by RFBR grant 14-02-01185 and RFBR grant 14-01-31353-mol\_a.
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[^1]: Note, that generally speaking there is also a thermal contribution to $Q_{fh}$ which is not captured by the substitution (\[shevchenko\]). It can be derived by considering field theory in a non-inertial reference frame [@vil]. In this note we omit this contribution for simplicity and in what follows separate it from the “fh” term.
|
---
abstract: 'We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and in the theory of species. We prove that the composition of two cofree coalgebras is again cofree, and we give sufficient conditions that ensure the composition is a one-sided Hopf algebra. We show these conditions are satisfied when one coalgebra is a graded Hopf operad ${\mathcal{D}}$ and the other is a connected graded coalgebra with coalgebra map to ${\mathcal{D}}$. We conclude by computing the primitive elements for compositions of coalgebras built on the vertices of multiplihedra, composihedra, and hypercubes.'
address:
- |
Department of Theoretical and Applied Mathematics\
The University of Akron\
Akron, OH 44325-4002
- |
Department of Mathematics\
Loyola University of Chicago\
Chicago, IL 60660
- |
Department of Mathematics\
Texas A&M University\
College Station, Texas 77843
author:
- Stefan Forcey
- Aaron Lauve
- Frank Sottile
title: Cofree compositions of coalgebras
---
Introduction {#sec: intro .unnumbered}
============
The Malvenuto-Reutenauer Hopf algebra of ordered trees [@MalReu:1995; @AguSot:2005] and the Loday-Ronco Hopf algebra of planar binary trees [@LodRon:1998; @AguSot:2006] are cofree as coalgebras and are connected by cellular maps from the permutahedra to the associahedra. Closely related polytopes include Stasheff’s multiplihedra [@Sta:1970] and the composihedra [@forcey2], and it is natural to study to what extent Hopf structures may be placed on these objects. The map from permutahedra to associahedra factors through the multiplihedra, and in [@FLS:2010] we used this factorization to place Hopf structures on bi-leveled trees, which correspond to vertices of multiplihedra.
The multiplihedra form an operad module over the associahedra, and this leads to the concept of painted trees, which also correspond to the vertices of the multiplihedra. Moreover, expressing the Hopf structures of [@FLS:2010] in terms of painted trees relates these Hopf structures to the operad module structure. Abstracting this structure leads to the general notion of a composition of coalgebras, which is a functorial construction of a graded coalgebra ${\mathcal{D}}\circ{\mathcal{C}}$ from graded coalgebras ${\mathcal{C}}$ and ${\mathcal{D}}$. We define this composition in Section \[sec: cccc\] and show that it preserves cofreeness. In Section \[sec: hopf\], we suppose that ${\mathcal{D}}$ is a Hopf algebra and give sufficient conditions for the compositions of coalgebras ${\mathcal{D}}\circ{\mathcal{C}}$ and ${\mathcal{C}}\circ{\mathcal{D}}$ to be one-sided Hopf algebras. These also guarantee that these compositions are Hopf modules and comodule algebras over ${\mathcal{D}}$.
The definition of the composition of coalgebras is familiar from the theory of operads. In general, a (nonsymmetric) operad is a monoid in the category of graded sets, with product given by composition (also known as the substitution product). In Section \[sec: hopf\] we show that an operad ${\mathcal{D}}$ in the category of connected graded coalgebras is automatically a Hopf algebra. Those familiar with the theory of species will also recognize our construction. The coincidence is explained in [@AguMah:2010 Appendix B]: species and operads are one-and-the-same.
We conclude in Sections \[sec: painted\], \[sec: weighted\], and \[sec: simplices\] with a detailed look at several compositions of coalgebras that enrich the understanding of well-known objects from category theory and algebraic topology. In particular, we prove that the (one sided) Hopf algebra of simplices in [@ForSpr:2010] is cofree as a coalgebra.
Preliminaries {#sec: prelims}
=============
We work over a fixed field ${\mathbb{K}}$ of characteristic zero. For a graded vector space $V = \bigoplus_n V_n$, we write $|v|=n$ and say $v$ has [[**]{}]{} $n$ if $v\in V_n$.
Hopf algebras and cofree coalgebras {#sec: cofree def}
-----------------------------------
A bialgebra $H$ is a unital associative algebra equipped with two algebra maps: a coproduct homomorphism $\Delta\colon H\to H\otimes H$ that is coassociative and a counit homorphism $\varepsilon\colon H\to {\mathbb{K}}$ which plays the role of the identity for $\Delta$. See [@Mont:1993] for more details. A graded bialgebra $H=(\bigoplus_{n\geq0}H_n,\bm\cdot,\Delta,\varepsilon)$ is [[**]{}]{} if $H_0 = {\mathbb{K}}$. In this case, a result of Takeuchi [@Tak:71 Lemma 14] guarantees the existence of an antipode map for $H$, making it a Hopf algebra.
We recall Sweedler’s coproduct notation for later use. A coalgebra ${\mathcal{C}}$ is a vector space ${\mathcal{C}}$ equipped with a coproduct $\Delta$ and counit $\varepsilon$. Given $c\in{\mathcal{C}}$, the coproduct $\Delta(c)$ is written $\sum_{(c)} c'\otimes c''$. Coassociativity means that $$\sum_{(c),(c')} (c')'\otimes (c')'' \otimes c''\ =\
\sum_{(c),(c'')} c'\otimes (c'')' \otimes (c'')''\ =\
\sum_{(c)} c'\otimes c'' \otimes c''' \,,$$ and the counit condition means that $\sum_{(c)} \varepsilon(c')c'' = \sum_{(c)} c'\varepsilon(c'') = c$.
The [[**]{}]{} on a vector space $V$ has underlying vector space $\Cofree(V):= \bigoplus_{n\geq0}V^{\otimes n}$. Its counit is the projection $\varepsilon\colon\Cofree(V)\to{\mathbb{K}}=V^{\otimes 0}$. Its coproduct is the [[**]{}]{}: writing “$\backslash$” for the tensor product in $V^{\otimes n}$, we have $$\Delta(c_1\backslash \dotsb\backslash c_n)\ =\
\sum_{i=0}^{n} (c_1\backslash \dotsb\backslash c_{i})
\otimes (c_{i+1}\backslash \dotsb\backslash c_n) \,.$$ Observe that $V$ is exactly the set of primitive elements of $\Cofree(V)$. A coalgebra ${\mathcal{C}}$ is [[**]{}]{} by a subspace $V\subset{\mathcal{C}}$ if ${\mathcal{C}}\simeq\Cofree(V)$ as coalgebras. Necessarily, $V$ is the space of primitive elements of ${\mathcal{C}}$. Many of the coalgebras and Hopf algebras arising in combinatorics are cofree. We recall a few key examples.
Cofree Hopf algebras on trees {#sec: Hopf examples}
-----------------------------
We describe three cofree Hopf algebras built on rooted planar binary trees: [[**]{}]{} $\s_n$, [[**]{}]{} $\y_n$, and [[**]{}]{} $\c_n$ on $n$ internal nodes. Let $\s_\bb$ denote the union $\bigcup_{n\geq0} \s_n$ and define $\y_\bb$ and $\c_\bb$ similarly.
### Constructions on trees {#sec: trees}
The nodes of a tree $t\in\y_n$ are a poset (with root maximal) whose Hasse diagram is the internal edges of $t$. An [[**]{}]{} is a linear extension of this node poset of $t$ that we indicate by placing a permutation in the gaps between its leaves. Ordered trees are in bijection with the permutations of $n$. The map $\tau \colon \s_n \to \y_n$ forgets the total ordering of the nodes of an ordered tree $w(t)$ and gives the underlying tree $t$. The map $\kappa \colon \y_n \to \c_n$ shifts all nodes of a tree $t$ to the right branch from the root. We let $\s_0=\y_0=\c_0=\includegraphics{figures/0.eps}$. Note that $|\c_n|=1$ for all $n\geq0$.
Figure \[fig: many trees\] gives some examples from $\s_\bb$, $\y_\bb$, and $\c_\bb$ and indicates the natural maps $\tau$ and $\kappa$ between them. See [@FLS:2010] for more details.
$$\psset{unit=10pt}
\begin{pspicture}(8.5,20)
\thicklines
\put(.5,0){\small ordered trees $\s_\bb$}
\put(4,18.1){ \begin{pspicture}(1,2) \put(0,0){\includegraphics{figures/1.d.eps}}
\put(0,1.1){{{\small\hspace{2pt}$1$}}}
\end{pspicture}}
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\end{pspicture}}
\end{pspicture}
\qquad
\begin{pspicture}(3,20)
\thicklines
\put(0,0){\vector(1,0){3.0}}\put(1,0.4){$\tau$}
\put(0,18.6){\Color{1 0 1 .3}{\vector(1,0){3.0}}}
\put(0,14.9){\Color{1 0 1 .3}{\vector(1,0){3.0}}}
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\put(0, 5.2){\Color{1 0 1 .3}{\vector(1,0){3.0}}}
\end{pspicture}
\qquad
\begin{pspicture}(6.5,20)
\thicklines
\put(0,0){\small binary trees $\y_\bb$}
\put(3,18.1){ \includegraphics{figures/1.d.eps}}
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\qquad
\begin{pspicture}(3,20)
\thicklines
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\qquad
\begin{pspicture}(6.5,20)
\thicklines
\put(0,0){\small left combs $\c_\bb$}
\put(2,18.1){\includegraphics{figures/1.d.eps}}
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\put(0.5,3.8){\includegraphics{figures/4321.d.eps}}
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[[**]{}]{} an ordered tree $w$ along the path from a leaf to the root yields an ordered forest (where the nodes in the forest are totally ordered) or a pair of ordered trees, $$\begin{picture}(52,49)
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\ \raisebox{12pt}{\large$\leadsto$}\
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\ \raisebox{12pt}{$\xrightarrow{\ \curlyvee\ }$}\quad
\raisebox{7.5pt}{$\displaystyle
\raisebox{8pt}{$\biggl(\;$}
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,\
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\raisebox{8pt}{$\;\biggr)$}
\quad \raisebox{5.5pt}{\mbox{or}}\quad
\raisebox{8pt}{$\biggl(\;$}
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,\
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\put(0,16){{{\small\hspace{2pt}$2$}}} \put(10,16){{{\small\hspace{2pt}$1$}}}
\end{picture}
\raisebox{8pt}{$\;\biggr)$}.
$}$$ Write $w\psplit(w_0,w_1)$ when the ordered forest $(w_0,w_1)$ (or pair of ordered trees) is obtained by splitting $w$. (Context will determine how to interpret the result.)
We may [[**]{}]{} an ordered forest $\vec w = (w_0,\dotsc,w_n)$ onto an ordered tree $v\in\s_n$, obtaining the tree $\vec w/v$ as follows. First increase each label of $v$ so that its nodes are greater than the nodes of $\vec w$, and then graft tree $w_i$ onto the $i^{\mathrm{th}}$ leaf of $v$. For example, $$\begin{aligned}
\hbox{if } (\vec w, v) &\ =\
\raisebox{0pt}{$\displaystyle
\raisebox{8pt}{$\Biggl( \biggl(\;$}
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, \
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\ ,
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, \
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\ , \
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\raisebox{8pt}{$\;\biggr)$} ,
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\end{picture}
\raisebox{8pt}{$\;\Biggr)$},
$}
\\[2ex]
\raisebox{30pt}{then $\vec w/v$} & \ \ \raisebox{30pt}{$=$} \ \
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\quad\raisebox{30pt}{$=$}\quad
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\raisebox{30pt}{.}\end{aligned}$$
The notions of splitting and grafting make sense for trees in $\y_\bb$ (simply forget the labels on the nodes). They also work for $\c_\bb$, if after grafting a forest of combs onto the leaves of a comb, $\kappa$ is applied to the resulting planar binary tree to get a new comb.
### Three cofree Hopf algebras {#sec: cofree examples}
Let $\ssym := \bigoplus_{n\geq0} \ssym_n$ be the graded vector space whose $n^{\mathrm{th}}$ graded piece has basis $\{F_w \mid w \in \s_n\}$. Define $\ysym$ and $\csym$ similarly. The set maps $\tau$ and $\kappa$ induce vector space maps $\btau$ and $\bkappa$, $\btau(F_w) = F_{\tau(w)}$ and $\bkappa(F_t)=F_{\kappa(t)}$. Fix $\mathfrak X \in \{\s, \y, \c\}$. For $w\in \mathfrak X_\bb$ and $v\in \mathfrak X_n$, set $$\begin{aligned}
F_w \cdot F_v\ &:=\ \sum_{w\psplit(w_0,\dotsc,w_n)} F_{(w_0,\dotsc,w_n)/v} \,,
\intertext{the sum over all ordered forests obtained by splitting $w$ at a multiset of $n$
leaves, and }
\Delta(F_w) \ &:= \ \sum_{w\psplit(w_0,w_1)} F_{w_0} \otimes F_{w_1} \,,
\end{aligned}$$ the sum over pairs of trees obtained by splitting $w$ at one leaf. The counit $\varepsilon$ is projection onto the $0^\mathrm{th}$ graded piece, which is spanned by the unit element $1=F_{\includegraphics{figures/0.eps}}$ for the multiplication.
\[thm: Hopf examples\] For $(\Delta,\cdot,\varepsilon)$ above, $\ssym$ is the Malvenuto–Reutenauer Hopf algebra of permutations, $\ysym$ is the Loday–Ronco Hopf algebra of planar binary trees, and $\csym$ is the divided power Hopf algebra. Moreover, $\ssym\xrightarrow{\,\btau\,}\ysym$ and $\ysym\xrightarrow{\,\bkappa\,}\csym$ are surjective Hopf algebra maps.
The part of the proposition involving $\ssym$ and $\ysym$ is found in [@AguSot:2005; @AguSot:2006]; the part involving $\csym$ is straightforward and we leave it to the reader.
Typically [@Mont:1993 Example 5.6.8], the divided power Hopf algebra is defined to be ${\mathbb{K}}[x] := \mathrm{span}\{x^{(n)} \mid n\geq0\}$, with basis vectors $x^{(n)}$ satisfying $x^{(m)}\cdot x^{(n)} = \binom{m+n}{n} x^{(m+n)}$, $1=x^{(0)}$, $\Delta(x^{(n)}) = \sum_{i+j=n} x^{(i)} \otimes x^{(j)}$, and $\varepsilon(x^{(n)})=0$ for $n>0$. An isomorphism between ${\mathbb{K}}[x]$ and $\csym$ is given by $x^{(n)}\mapsto F_{c_n}$, where $c_n$ is the unique comb in $\c_n$.
The following result is important for what follows.
\[thm: cofree examples\] The Hopf algebras $\ssym$, $\ysym$, and $\csym$ are cofreely cogenerated by their primitive elements.
The result for $\csym$ is easy. Proposition \[thm: cofree examples\] is proven for $\ssym$ and $\ysym$ in [@AguSot:2005] and [@AguSot:2006] by performing a change of basis—from the [[**]{}]{} $F_w$ to the [[**]{}]{} $M_w$—by means of Möbius inversion in a poset structure placed on $\s_\bb$ and $\y_\bb$. We revisit this in Section \[sec: coalgebra psym\].
Cofree Compositions of Coalgebras {#sec: cccc}
=================================
Cofree composition of coalgebras {#sec: cccc main results}
--------------------------------
Let ${\mathcal{C}}$ and ${\mathcal{D}}$ be graded coalgebras. We form a new coalgebra ${\mathcal{E}}={\mathcal{D}}\circ{\mathcal{C}}$ on the vector space $$\begin{gathered}
\label{eq: E_(n)}
{\mathcal{D}}\circ{\mathcal{C}}\ := \ \bigoplus_{n\geq0} {\mathcal{D}}_n \otimes {\mathcal{C}}^{\otimes(n+1)} \,.
\end{gathered}$$ We write ${\mathcal{E}}=\bigoplus_{n\geq0}{\mathcal{E}}_{{(n)}}$, where ${\mathcal{E}}_{{(n)}}= {\mathcal{D}}_n \otimes {\mathcal{C}}^{\otimes(n+1)}$. This gives a coarse coalgebra grading of ${\mathcal{E}}$ by [[**]{}]{}. There is a finer grading of ${\mathcal{E}}$ by [[**]{}]{}, in which a decomposable tensor $c_0\otimes \dotsb\otimes c_n \otimes d$ (with $d\in{\mathcal{D}}_n$) has total degree $|c_0|+\dotsb+|c_n|+|d|$. Write ${\mathcal{E}}_n$ for the linear span of elements of total degree $n$.
\[ex: painted\] This composition is motivated by a grafting construction on trees. Let $d\times (c_0,\dotsc,c_n) \in {\red\y_n} \times \bigl({\blue\y_\bb}^{{n+1}}\bigr)$. Define $\circ$ by attaching the forest $(c_0,\dots,c_n)$ to the leaves of $d$ while remembering $d$, $$\raisebox{0pt}{$\displaystyle
\includegraphics{figures/4213.red.d.eps}
\raisebox{2pt}{\,\ $\times$\, }
\raisebox{8pt}{$\biggl(\;$}
\includegraphics{figures/12.d.eps}
, \
\includegraphics{figures/0.d.eps}
\ ,
\includegraphics{figures/231.d.eps}
, \
\includegraphics{figures/1.d.eps}
\ , \
\includegraphics{figures/0.d.eps}
\raisebox{8pt}{$\;\biggr)$}
$} \raisebox{3pt}{\ \ \ $\stackrel{\textstyle\circ}{\longmapsto}$\ }
\raisebox{-18pt}{\includegraphics{figures/p756...8110.eps}} .$$ This is a new type of tree ([[**]{}]{} in Section \[sec: painted\]). Applying this construction to the indices of basis elements of ${\mathcal{C}}$ and ${\mathcal{D}}$ and extending by multilinearity gives ${\mathcal{D}}\circ {\mathcal{C}}$.
Motivated by this example, we represent an decomposable tensor in ${\mathcal{D}}\circ{\mathcal{C}}$ as $${{d{\,\circ\,}({c_0{\treeseparatorinline}\dotsb{\treeseparatorinline}c_n})}} \qquad\hbox{or}\qquad {\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c_n}}{d}}$$ to compactify notation.
### The coalgebra ${\mathcal{D}}\circ{\mathcal{C}}$
We define the [[**]{}]{} $\Delta$ for ${\mathcal{D}}\circ{\mathcal{C}}$ on indecomposable tensors and extend multilinearly: if $|d|=n$, put $$\label{eq: delta}
\Delta\left({\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c_n}}{d}}\right)\ =\
\sum_{i=0}^n \sum_{\substack{(d) \\ |d'|=i}}
\sum_{(c_i)} {\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c_{i-1}{\treeseparator}c'_i}}{d'}} \otimes {\frac{{c''_i{\treeseparator}c_{i+1}{\treeseparator}\dotsb{\treeseparator}c_n}}{d''}} \,,$$ where the coproducts in ${\mathcal{C}}$ and ${\mathcal{D}}$ are expressed using Sweedler notation.
The [[**]{}]{} $\varepsilon:{\mathcal{D}}\circ{\mathcal{C}}\to {\mathbb{K}}$ is given by $\varepsilon({{d{\,\circ\,}({c_0{\treeseparatorinline}\dotsb{\treeseparatorinline}c_n})}})=
\varepsilon_{\mathcal{D}}(d) \cdot \prod_j \varepsilon_{\mathcal{C}}(c_j) $. Hence, it is zero off of ${\mathcal{D}}_0 \otimes {\mathcal{C}}_0$.
The reader may check that for the painted trees of Example \[ex: painted\], if $c_0,\dotsc,c_n$ and $d$ are elements of the $F$-basis of $\ysym$, then ${{{d{\,\circ\,}({c_0{\treeseparatorinline}\dotsb{\treeseparatorinline}c_n})}}}$ represents a painted tree $t$ and $\Delta ({{{d{\,\circ\,}({c_0{\treeseparatorinline}\dotsb{\treeseparatorinline}c_n})}}})$ is the sum over all splittings $t\psplit(t',t'')$ of $t$ into a pair of painted trees.
$({\mathcal{D}}\circ{\mathcal{C}},\Delta,\varepsilon)$ is a coalgebra. This composition is functorial, if $\varphi\colon{\mathcal{C}}\to{\mathcal{C}}'$ and $\psi\colon{\mathcal{D}}\to{\mathcal{D}}'$ are morphisms of graded coalgebras, then $${\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c_n}}{d}}\ \longmapsto\
{\frac{{\varphi(c_0){\treeseparator}\dotsb{\treeseparator}\varphi(c_n)}}{\psi(d)}}$$ defines a morphism of graded coalgebras $\varphi\circ\psi\colon {\mathcal{D}}\circ{\mathcal{C}}\to {\mathcal{D}}'\circ{\mathcal{C}}'$.
Let $\id$ be the identity map. Fix $e:={{d{\,\circ\,}({c_0{\treeseparatorinline}\dotsb{\treeseparatorinline}c_n})}} \in ({\mathcal{D}}\circ{\mathcal{C}})_{{(n)}}$. From , we have $$\begin{aligned}
(\Delta\otimes \id)\Delta (e) \ =\ &
\sum_{i=0}^n \sum_{j=0}^{i-1} \sum_{\substack{(d),(d') \\ |d'|=i,|(d')'|=j}}
\sum_{(c_i),(c_j)} {\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c'_j}}{(d')'}} \otimes {\frac{{c''_j{\treeseparator}\dotsb{\treeseparator}c'_i}}{(d')''}}
\otimes {\frac{{c''_i{\treeseparator}\dotsb{\treeseparator}c_n}}{d''}} \\
&\mbox{\ \ }\quad + \sum_{i=0}^n \sum_{\substack{(d),(d') \\ |d'|=i,|(d')''|=0}}
\sum_{(c_i),(c'_i)} {\frac{{c_0{\treeseparator}\dotsb{\treeseparator}(c'_i)'}}{(d')'}} \otimes {\frac{{(c'_i)''}}{(d')''}}
\otimes {\frac{{c''_i{\treeseparator}\dotsb{\treeseparator}c_n}}{d''}} \ .
\intertext{Using coassociativity, this becomes}
& \sum_{i=0}^n \sum_{j=0}^{i-1} \sum_{\substack{(d) \\ |d'|=i,|d''|=j}}
\sum_{(c_i),(c_j)} {\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c'_j}}{d'}} \otimes {\frac{{c''_j{\treeseparator}\dotsb{\treeseparator}c'_i}}{d''}}
\otimes {\frac{{c''_i{\treeseparator}\dotsb{\treeseparator}c_n}}{d'''}} \\
&\mbox{\ \ }\quad+ \sum_{i=0}^n \sum_{\substack{(d) \\ |d'|=i,|d''|=0}}
\sum_{(c_i)} {\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c'_i}}{d'}} \otimes {\frac{{c''_i}}{d''}}
\otimes {\frac{{c'''_i{\treeseparator}\dotsb{\treeseparator}c_n}}{d'''}} \ .\end{aligned}$$ Simplification of $(\id\otimes\Delta)\Delta(e)$ reaches the same expression, proving coassociativity.
For the counital condition, we have $$\begin{aligned}
(\varepsilon\otimes \id)\Delta (e) \ =\ &
\sum_{i=0}^n \sum_{\substack{(d) \\ |d'|=i}}
\sum_{(c_i)} \varepsilon\!\left({\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c_{i-1}{\treeseparator}c'_i}}{d'}}\right)
{\frac{{c''_i{\treeseparator}c_{i+1}{\treeseparator}\dotsb{\treeseparator}c_n}}{d''}} \\
\ =\ &
\sum_{\substack{(d) \\ |d'|=0}}
\sum_{(c_0)} \varepsilon\!\left({\frac{{c'_0}}{d'}}\right)
{\frac{{c''_0{\treeseparator}c_{1}{\treeseparator}\dotsb{\treeseparator}c_n}}{d''}} \,, \\
\intertext{since $\varepsilon_{\mathcal{D}}(d')=0$ unless $|d'|=0$. Continuing, this becomes}
& \sum_{\substack{(d) \\ |d'|=0}}
\sum_{(c_0)} {\frac{{\varepsilon(c'_0)c''_0{\treeseparator}c_{1}{\treeseparator}\dotsb{\treeseparator}c_n}}{\varepsilon(d')d''}} \ = \ e\,,\end{aligned}$$ by the counital conditions in ${\mathcal{C}}$ and ${\mathcal{D}}$. The identity $(\id \otimes \varepsilon)\Delta(e) = e$ is similarly verified, proving the counital condition for ${\mathcal{D}}\circ{\mathcal{C}}$. Lastly, the functoriality is clear.
### The cofree coalgebra ${\mathcal{D}}\circ{\mathcal{C}}$
Suppose that ${\mathcal{C}}$ and ${\mathcal{D}}$ are graded, connected, and cofree. Then ${\mathcal{C}}=\Cofree(P_{\mathcal{C}})$, where $P_{\mathcal{C}}\subset{\mathcal{C}}$ is its space of primitive elements. Likewise, ${\mathcal{D}}=\Cofree(P_{\mathcal{D}})$. As in Section \[sec: cofree def\], we use “$\backslash$” for internal tensor products.
\[thm: cc cofree\] If ${\mathcal{C}}$ and ${\mathcal{D}}$ are cofree coalgebras then ${\mathcal{D}}\circ{\mathcal{C}}$ is also a cofree coalgebra. Its space of primitive elements is spanned by indecomposible tensors of the form $$\label{eq: primitives}
{\frac{{1{\treeseparator}c_1{\treeseparator}\dotsb{\treeseparator}c_{n-1}{\treeseparator}1}}{\delta}} \qquad\hbox{ and }\qquad {\frac{{\,\gamma\,}}{1}} \,$$ where $\gamma,c_i\in{\mathcal{C}}$ and $\delta\in{\mathcal{D}}_n$ with $\gamma,\delta$ primitive.
Let ${\mathcal{E}}= {\mathcal{D}}\circ{\mathcal{C}}$ and let $P_{\mathcal{E}}$ denote the vector space spanned by the vectors in . We compare the compositional coproduct $\Delta$ to the deconcatenation coproduct $\Delta_{\Cofree}$ on the space $\Cofree(P_{\mathcal{E}})$. We define a vector space isomorphism $\varphi\colon {\mathcal{E}}\to \Cofree(P_{\mathcal{E}})$ and check that $\Delta_{\Cofree}\,\varphi(e) = (\varphi\otimes\varphi)\Delta(e)$ for all $e\in{\mathcal{E}}$.
Let $e = {{d{\,\circ\,}({c_0{\treeseparatorinline}\dotsb{\treeseparatorinline}c_n})}}$. Define $\varphi$ recursively as follows:
- If $d=1$ and $c_0 = c'_{0}\backslash c''_{0}$, put $\displaystyle \varphi\left({\frac{{c_{0}}}{1}}\right) =
\varphi\left({\frac{{c'_{0}}}{1}}\right) \Big\backslash
\varphi\left({\frac{{c''_{0}}}{1}}\right)$.
- If $|c_0|>0$, put $\displaystyle \varphi(e) = \varphi\left({\frac{{c_0}}{1}}\right) \Big\backslash \varphi\left({\frac{{1{\treeseparator}c_1{\treeseparator}\dotsb{\treeseparator}c_n}}{d}}\right)$.
- If $|c_n|>0$, put $\displaystyle \varphi(e) = \varphi\left({\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c_{n-1}{\treeseparator}1}}{d}}\right)\Big\backslash \varphi\left({\frac{{c_n}}{1}}\right)$.
- If $d=d'\backslash d''$ with $|d'|=i$, then put $\displaystyle \varphi(e) = \varphi\left({\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c_i}}{d'}}\right) \Big\backslash \varphi\left({\frac{{1{\treeseparator}c_{i+1}{\treeseparator}\dotsb{\treeseparator}c_n}}{d''}}\right)$.
We illustrate $\varphi$ with an example from ${\mathcal{E}}_{{(5)}}$: $${\frac{{a'\backslash a'' \treeseparator b \treeseparator c \treeseparator u'\backslash u'' \treeseparator v \treeseparator w}}{d'\backslash d''}}
\ \ \stackrel{\varphi}{\longmapsto} \ \
{\frac{{a'}}{1}} \,\Big\backslash\, {\frac{{a''}}{1}} \,\Big\backslash\,
{\frac{{1 \treeseparator b \treeseparator c \treeseparator 1}}{d'}} \,\Big\backslash\,
{\frac{{u'}}{1}} \,\Big\backslash\, {\frac{{u''}}{1}} \,\Big\backslash\,
{\frac{{1 \treeseparator v \treeseparator 1}}{d''}} \,\Big\backslash\, {\frac{{w}}{1}} \,.$$ Here $|d'|=3$ and all variables belong to $P_{\mathcal{C}}\cup P_{\mathcal{D}}$.
To see that $\varphi$ is a coalgebra map, notice that locations to deconcatenate $\varphi(e)$, $$t_1 \backslash \dotsb \backslash t_N \ \longmapsto \
t_1 \backslash \dotsb \backslash t_i \otimes t_{i_1} \backslash \dotsb \backslash t_N \,,$$ are in bijection with pairs of locations: a place to deconcatenate $d$ and a place to deconcatenate an accompanying $c_i$. These are exactly the choices governing , given that $d$ and each $c_i$ belong to tensor powers of $P_{\mathcal{D}}$ and $P_{\mathcal{C}}$, respectively.
Examples of cofree compositions of coalgebras {#sec: cccc examples}
---------------------------------------------
The graded Hopf algebras of ordered trees $\ssym$, planar trees $\ysym$, and divided powers $\csym$ are all cofree, and so their compositions are cofree. We have the surjective Hopf algebra maps $$\ssym\ \longrightarrow\ \ysym\ \longrightarrow\ \csym$$ giving a commutative diagram of nine cofree coalgebras (Figure \[fig: diamond\]), as the composition $\circ$ is functorial.
(270,212) (100,200)[$\ssym\circ\ssym$]{} (128,194)[(-1,-1)[30]{}]{} (146,194)[(1,-1)[30]{}]{} (50,150)[$\ssym\circ\ysym$]{} (150,150)[$\ysym\circ\ssym$]{} ( 96,144)[(1,-1)[30]{}]{} (178,144)[(-1,-1)[30]{}]{} ( 78,144)[(-1,-1)[30]{}]{} (196,144)[(1,-1)[30]{}]{} (0,100)[$\ssym\circ\csym$]{} (200,100)[$\csym\circ\ssym$]{} (100,100)[$\ysym\circ\ysym$]{} ( 46,94)[(1,-1)[30]{}]{} (128,94)[(-1,-1)[30]{}]{} (146,94)[(1,-1)[30]{}]{} (228,94)[(-1,-1)[30]{}]{} (50,50)[$\ysym\circ\csym$]{} (150,50)[$\csym\circ\ysym$]{} ( 96,44)[(1,-1)[30]{}]{} (178,44)[(-1,-1)[30]{}]{} (100,0)[$\csym\circ\csym$]{}
In Section \[sec: hopf\], we use operads to further analyze eight of these nine (all except $\ssym \circ \ssym$). We show that these eight are one-sided Hopf algebras. The algebra $\psym$ of painted trees appears in the center of this $3\times 3$ grid. We discuss $\psym$ further in Section \[sec: painted\], the algebra $\ysym\circ\csym$ in Section \[sec: weighted\], and the algebra $\csym\circ\csym$ in Section \[sec: simplices\].
Enumeration {#S:enumeration}
-----------
We enumerate the graded dimension of many examples from Section \[sec: cccc examples\]. Set ${\mathcal{E}}:={\mathcal{D}}\circ{\mathcal{C}}$ and let $C_n$ and $E_n$ be the dimensions of ${\mathcal{C}}_n$ and ${\mathcal{E}}_n$, respectively.
When ${\mathcal{D}}_n$ has a basis indexed by combs with $n$ internal nodes we have the recursion $$E_0\ =\ 1\,,\quad\mbox{and for \ $n>0$,}\quad
E_n\ =\ C_n\ +\ \sum_{i=0}^{n-1}C_i E_{n-i-1}\,.$$
The first term in the expression for $E_n$ counts elements in ${\mathcal{E}}_n$ of the form $\includegraphics{figures/0.eps}\circ c$. Removing the root node of $d$ from ${{{d{\,\circ\,}({c_0{\treeseparatorinline}\dotsb{\treeseparatorinline}c_k})}}}$ gives a pair ${{{\includegraphics{figures/0.eps}}{\,\circ\,}(c_0)}}$ and ${{d'{\,\circ\,}({c_1{\treeseparatorinline}\dotsb{\treeseparatorinline}c_k})}}$ with $c_0\in{\mathcal{C}}_i$, whose dimensions are enumerated by the terms $C_iE_{n-i-1}$ of the sum.
For combs over a comb, $E_n=2^n$. For trees over a comb, $E_n$ are the Catalan numbers. For permutations over a comb, we have the recursion $$E_0\ =\ 1\,,\quad\mbox{and for \ $n>0$,}\quad
E_n\ =\ n! + \sum_{i=0}^{n-1}i! E_{n-i-1}\,,$$ which begins $1, 2, 5, 15, 54, 235,\dotsc$, and is sequence A051295 in the On-line Encyclopedia of Integer Sequences (OEIS) [@Slo:oeis]. (This is the invert transform [@BernSloan] of the factorial numbers.)
When ${\mathcal{D}}_n$ has a basis indexed by $\y_n$, then we have the recursion $$E_0\ =\ 1\,,\quad\mbox{and for \ $n>0$,}\quad
E_n\ =\ C_n + \sum_{i=0}^{n-1}E_i E_{n-i-1} \,.$$
Again, the first term in the expression for $E_n$ is the number of basis elements of ${\mathcal{C}}_n,$ since each of these trees is grafted on to the unit element of ${\mathcal{D}}$. The sum accounts for the possible pairs of trees obtained from removing root nodes in ${\mathcal{D}}$. In this case, each subtree from the root is another tree in ${\mathcal{E}}$.
For example, combs over a tree are enumerated by the binary transform of the Catalan numbers [@forcey2]. Trees over a tree are enumerated by the Catalan transform of the Catalan numbers [@forcey1]. Permutations over a tree are enumerated by the recursion $$E_0\ =\ 1\,,\quad\mbox{and for \ $n>0$,}\quad
E_n\ =\ n! + \sum_{i=0}^{n-1}E_i E_{n-i-1}\,,$$ which begins $1, 2, 6, 22, 92, 428,\dotsc$ and is not a recognized sequence in the OEIS [@Slo:oeis].
For ${\mathcal{E}}= \ssym \circ {\mathcal{C}}$, we do not have a recursion, but do have a formula from direct inspection of the possible trees ${{d{\,\circ\,}({c_0{\treeseparatorinline}\dotsb{\treeseparatorinline}c_k})}}$ with $|d|=k$ (since $|\s_k| = k!$) $$E_n\ =\ \sum_{k=0}^{n} \,k! \! \sum_{(\gamma_0,\dotsc,\gamma_k)} C_{\gamma_0} \dotsb C_{\gamma_k}\,,$$ the sum over all weak compositions $\gamma = (\gamma_0,\dotsc,\gamma_k)$ of $n{-}k$ into $k{+}1$ parts ($\gamma_i \geq0$). Since the number of such weak compositions is $\binom{(n-k)+(k+1)-1}{(k+1)-1}=\binom{n}{k}$, when ${\mathcal{C}}=\csym$ so that $C_n=1$, this formula becomes $$E_n\ =\ \sum_{k=0}^n k! \tbinom{n}{k}\ =\
\sum_{k=0}^n n!/k!\,,$$ which is sequence A000522 in the OEIS [@Slo:oeis].
Composition of Coalgebras and Hopf Modules {#sec: hopf}
==========================================
We give conditions ensuring that a composition of coalgebras is a one-sided Hopf algebra, interpret these in the language of operads, and then investigate which compositions of Section \[sec: cccc examples\] are one-sided Hopf algebras.
Module coalgebras {#sec: covering}
-----------------
Let ${\mathcal{D}}$ be a connected graded Hopf algebra with product $m_{{\mathcal{D}}}$, coproduct $\Delta_{{\mathcal{D}}}$, and unit element $1_{{\mathcal{D}}}$.
A map $f: {\mathcal{E}}\to {\mathcal{D}}$ of graded coalgebras is a [[**]{}]{} on ${\mathcal{D}}$ if ${\mathcal{E}}$ is a ${\mathcal{D}}$–module coalgebra, $f$ is a map of ${\mathcal{D}}$-module coalgebras, and ${\mathcal{E}}$ is connected. That is, ${\mathcal{E}}$ is an associative (left or right) ${\mathcal{D}}$-module whose action (denoted $\star$) commutes with the coproducts, so that $\Delta_{{\mathcal{E}}}(e \star d) = \Delta_{{\mathcal{E}}}(e) \star \Delta_{{\mathcal{D}}}(d)$, for $e\in {\mathcal{E}}$ and $d \in {\mathcal{D}}$, *and* the coalgebra map $f$ is also a module map, so that for $e\in{\mathcal{E}}$ and $d\in{\mathcal{D}}$ we have $$(f\otimes f )\,\Delta_{{\mathcal{E}}}(e)\ =\ \Delta_{{\mathcal{D}}}\, f(e)
\qquad\mbox{and}\qquad
f(e\star d) \ = \ m_{{\mathcal{D}}} \,( f(e)\otimes d)\,.$$ We may sometimes use subscripts ($f_l$ or $f_r$) on a connection $f$ to indicate that the action is a left- or right-module action.
\[cover\_implies\_Hopf\] If ${\mathcal{E}}$ is a connection on ${\mathcal{D}}$, then ${\mathcal{E}}$ is also a Hopf module and a comodule algebra over ${\mathcal{D}}$. It is also a one-sided Hopf algebra with one-sided unit $\Blue{1_{{\mathcal{E}}}}:=f^{-1}(1_{{\mathcal{D}}})$ and antipode.
Suppose ${\mathcal{E}}$ is a right ${\mathcal{D}}$-module. Define the product $m_{{\mathcal{E}}}:{\mathcal{E}}\otimes{\mathcal{E}}\to {\mathcal{E}}$ via the ${\mathcal{D}}$-action: $m_{{\mathcal{E}}}:=\star\circ(1\otimes f)$. The one-sided unit is $1_{{\mathcal{E}}}$. Then $\Delta_{\mathcal{E}}$ is an algebra map. Indeed, for $e,e' \in {\mathcal{E}}$, we have $$\Delta_{\mathcal{E}}(e\cdot e') \ =\ \Delta_{\mathcal{E}}(e\star f(e'))
\ =\ \Delta_{\mathcal{E}}e \star \Delta_{\mathcal{D}}f(e')
\ =\ \Delta_{\mathcal{E}}e \star (f\otimes f)(\Delta_{\mathcal{E}}e')
\ =\ \Delta_{\mathcal{E}}e \cdot \Delta_{\mathcal{E}}e'\,.$$ As usual, $\varepsilon_{\mathcal{E}}$ is just projection onto ${\mathcal{E}}_0$. The unit $1_{{\mathcal{E}}}$ is one-sided, since $$e\cdot 1_{{\mathcal{E}}}
\ =\ e\star f(1_{{\mathcal{E}}})
\ =\ e\star f(f^{-1}(1_{{\mathcal{D}}}))
\ =\ e\star 1_{{\mathcal{D}}}
\ =\ e \,,$$ but $1_{\mathcal{E}}\cdot e = 1_{\mathcal{E}}\star f(e)$ is not necessarily equal to $e$. As ${\mathcal{E}}$ is a graded bialgebra, the antipode $S$ may be defined recursively to satisfy $m_{\mathcal{E}}(S\otimes \id)\Delta_{\mathcal{E}}= \varepsilon_{\mathcal{E}}$, see \[thm: painted is hopf\]. (If instead ${\mathcal{E}}$ is a left ${\mathcal{D}}$-module, then it has a left-sided unit and right-sided antipode.)
Define $\rho \colon{\mathcal{E}}\to{\mathcal{E}}\otimes{\mathcal{D}}$ by $\rho := (1\otimes f)\, \Delta_{{\mathcal{E}}}$. This gives a coaction so that ${\mathcal{E}}$ is a Hopf module and a comodule algebra over ${\mathcal{D}}$.
Operads and operad modules {#sec: operad}
--------------------------
Composition of coalgebras is the same product used to define operads internal to a symmetric monoidal category [@AguMah:2010 Appendix B]. A [[**]{}]{} in a category with a product $\bullet$ is an object ${\mathcal{D}}$ with a morphism $\gamma \colon {\mathcal{D}}\bullet{\mathcal{D}}\to {\mathcal{D}}$ that is associative. An [[**]{}]{} is a monoid in the category of graded sets with an analog of the composition product $\circ$ defined in Section \[sec: cccc main results\].
The category of connected graded coalgebras and coalgebra maps is a symmetric monoidal category with the composition $\circ$ of coalgebras. A [[**]{}]{} ${\mathcal{D}}$ is a monoid in this category. That is, ${\mathcal{D}}$ has associative composition maps $\gamma\colon{\mathcal{D}}\circ {\mathcal{D}}\to {\mathcal{D}}$ obeying $$\Delta_{\mathcal{D}}\gamma(a)\ =\
(\gamma\otimes\gamma)\left(\Delta_{{\mathcal{D}}\circ{\mathcal{D}}}(a)\right) \
\hbox{ \ for all \ }a\in {\mathcal{D}}\circ{\mathcal{D}}\,.$$ By Theorem \[thm: inthopf\], ${\mathcal{D}}$ is a Hopf algebra; this explains our nomenclature.
A [[**]{}]{} is an operad module (left or right) over ${\mathcal{D}}$ and a graded coassociative coalgebra whose module action is compatible with its coproduct. Write $\mu_l:{\mathcal{D}}\circ {\mathcal{E}}\to {\mathcal{E}}$ and $\mu_r:{\mathcal{E}}\circ {\mathcal{D}}\to{\mathcal{E}}$ for the left and right actions, which obey, e.g., $$\Delta_{\mathcal{E}}\mu_r(b)\ =\ (\mu_r\otimes\mu_r)\Delta_{{\mathcal{E}}\circ{\mathcal{D}}} b
\qquad \mbox{for all }b\in{\mathcal{E}}\circ{\mathcal{D}}\,.$$
$\ysym$ is an operad in the category of vector spaces. The action of $\gamma$ on ${{F_t{\,\circ\,}({F_{t_0}{\treeseparatorinline}\dotsb{\treeseparatorinline}F_{t_n}})}}$ grafts the indexing trees $t_0,\dotsc,t_n$ onto the tree $t$ and, unlike in Example \[ex: painted\], forgets which nodes of the resulting tree came from $t$. This is associative in the appropriate sense. The same action $\gamma$ makes $\ysym$ an operad in the category of connected graded coalgebras, and thus a graded Hopf operad. Finally, operads are operad modules over themselves, so $\ysym$ is also graded Hopf operad module.
This notion differs from that of Getzler and Jones [@GetJon:1], who defined a Hopf operad ${\mathcal{D}}$ to be an operad where each component ${\mathcal{D}}_n$ is a coalgebra.
\[thm: inthopf\] A graded Hopf operad ${\mathcal{D}}$ is also a Hopf algebra with product $$\begin{gathered}
\label{eq: operad to algebra}
a\cdot b\ :=\ \gamma( b \otimes \Delta^{(n)} a)\end{gathered}$$ where $b\in {\mathcal{D}}_n$ and $\Delta^{(n)}$ is the iterated coproduct from ${\mathcal{D}}$ to ${\mathcal{D}}^{\otimes(n+1)}.$
If we swap the roles of $a$ and $b$ on the right-hand side of , we also obtain a Hopf algera, for $H^{\mathrm{op}}$ is a Hopf algebra whenever $H$ is one. Our choice agrees with the description (Section \[sec: Hopf examples\]) of products in $\ysym$ and $\csym$.
Before we prove Theorem \[thm: inthopf\], we restate an old result in the language of operads.
\[prop:tree and comb\] The well-known Hopf algebra structures of $\ysym$ and $\csym$ are induced by their structure as graded Hopf operads.
The operad structure on $\ysym$ is the operad of planar, rooted, binary trees, where composition $\gamma$ is grafting. The operad structure on $\csym$ is the terminal operad, which has a single element in each component. Representing the single element of degree $n$ as a comb of $n$ leaves, the composition $\gamma$ becomes grafting and combing all branches of the result.
We check that these compositions $\gamma$ are coalgebra maps. For ${\mathcal{D}}=\ysym$, the coproduct $\Delta_{{\mathcal{D}}\circ{\mathcal{D}}}$ in the $F$-basis is the sum over possible splittings of the composite trees. Then splitting an element of $e\in {\mathcal{D}}\circ {\mathcal{D}}$ and grafting both resulting trees (via $\gamma\otimes\gamma$) yields the same result as first grafting ($e\to \gamma(e)$), then splitting the resulting tree. When ${\mathcal{D}}=\csym$, virtually the same analysis holds, with the proviso that graftings are always followed by combing all branches to the right.
Finally, we note that the product in $\ysym$ in terms of the $F$-basis is simply $a\cdot b = \gamma(b\otimes \Delta^{(|b|)}a)$. The same holds for $\csym$, again with the proviso that $\gamma$ is grafting, followed by combing.
We have $\gamma(1\otimes1)=1$ and $\gamma(b\otimes 1^{\otimes|b|+1}) = b$ by construction, since ${\mathcal{D}}$ is connected. Thus $1=1_{\mathcal{D}}$ is the unit in ${\mathcal{D}}$.
The image of $ \id\otimes\Delta^{(n)}$ lies in ${\mathcal{D}}\circ{\mathcal{D}}$. As $\gamma$ is a map of graded coalgebras, $\Delta(a\cdot b) = \Delta a \cdot \Delta b$. Indeed, for $b\in{\mathcal{D}}$ homogeneous, $$\begin{aligned}
\Delta(a\cdot b) \ =\ \Delta (\gamma(b\otimes \Delta^{(|b|)}a))
& =\ (\gamma \otimes \gamma)(\Delta_{{\mathcal{D}}\circ{\mathcal{D}}}(b\otimes \Delta^{(|b|)}a))\\
& =\ (\gamma \otimes \gamma)(( \Delta b\otimes \Delta^{(|b|)}\Delta a) )
\ =\ \Delta a \cdot \Delta b\,.
\end{aligned}$$ Associativity of the product follows, since for $b,c$ homogeneous elements of ${\mathcal{D}}$, we have $$\begin{aligned}
\nonumber a\cdot(b\cdot c)
\ =\ a\cdot\gamma(c\otimes\Delta^{(|c|)} b )\
&=\ \gamma\left(\gamma(c\otimes\Delta^{(|c|)}b )\otimes\Delta^{(|b|+|c|)}a\right)\\
\label{eq: assoc}
&=\ \gamma\left(c\otimes
\gamma^{\otimes(|c|+1)}(\Delta^{(|c|)}b\otimes\Delta^{(|b|+|c|)}a) \right)\\
\nonumber
&=\ \gamma\left(c \otimes(\Delta^{(|c|)}a\cdot\Delta^{(|c|)}b )\right)\\
\label{eq: alg-map}
&=\ \gamma(c\otimes\Delta^{(|c|)}(a\cdot b) )
\ =\ (a\cdot b)\cdot c\,.\end{aligned}$$ Here, is by the associativity of composition $\gamma$ in an operad, where we assume the isomorphism ${\mathcal{D}}\circ ({\mathcal{D}}\circ {\mathcal{D}}) \cong ({\mathcal{D}}\circ {\mathcal{D}}) \circ {\mathcal{D}}$. The step follows as ${\mathcal{D}}$ is a bialgebra ($\Delta^{(n)}$ is an algebra map since $\Delta=\Delta^{(1)}$ is one).
\[lem:opmod\] If ${\mathcal{C}}$ is a graded coalgebra and ${\mathcal{D}}$ is a graded Hopf operad, then ${\mathcal{D}}\circ {\mathcal{C}}$ is a (left) graded Hopf operad module and ${\mathcal{C}}\circ {\mathcal{D}}$ is a (right) graded Hopf operad module.
We grade ${\mathcal{D}}\circ {\mathcal{C}}$ and ${\mathcal{C}}\circ {\mathcal{D}}$ by total degree. An operad module of vector spaces is a sequence of vector spaces acted upon by the operad. The action $\mu_l:{\mathcal{D}}\circ({\mathcal{D}}\circ {\mathcal{C}})\to({\mathcal{D}}\circ {\mathcal{C}})$ is given by $$\mu_l\left(d\otimes {\frac{{c_{0_0}{\treeseparator}\dotsb{\treeseparator}c_{i_0}}}{d_0}}\otimes\dots\otimes
{\frac{{c_{0_n}{\treeseparator}\dotsb{\treeseparator}c_{i_n}}}{d_n}}\right)\
=\ {\frac{{c_{0_0}{\treeseparator}\dotsb{\treeseparator}c_{i_n}}}{\gamma(d\otimes d_0\otimes\dots\otimes d_n)}}\ .$$ Associativity of $\gamma$ implies that this action is associative. The action $\mu_r:({\mathcal{C}}\circ {\mathcal{D}})\circ{\mathcal{D}}\to({\mathcal{C}}\circ {\mathcal{D}})$ is given by $$\begin{gathered}
\qquad
\mu_r\left({\frac{{d_{0}{\treeseparator}\dotsb{\treeseparator}d_{m}}}{c}}\otimes d_{0_0}\otimes\dots \otimes d_{j_m}\right)\\
= \
{\frac{{\gamma( d_0\otimes d_{0_0}\otimes\dots\otimes d_{j_0}){\treeseparator}\dotsb{\treeseparator}\gamma(d_m\otimes d_{0_m}\otimes\dots\otimes d_{j_m})}}{c}}\ .\end{gathered}$$ Associativity of $\gamma$ implies that this action is associative as well. We leave the reader to check that $\Delta\mu_l = (\mu_l\otimes \mu_l)\Delta$ and $\Delta\mu_r = (\mu_r\otimes \mu_r)\Delta$.
\[lem:coal\] A graded Hopf operad module ${\mathcal{E}}$ over a graded Hopf operad ${\mathcal{D}}$ is also a module coalgebra for the Hopf algebra ${\mathcal{D}}$.
Fix $e\in {\mathcal{E}}$ and $d\in{\mathcal{D}}$ to be homogeneous elements. If ${\mathcal{E}}$ is a right operad module over ${\mathcal{D}}$ then define a left action by $d\star e :=\mu_r(e \otimes \Delta^{(|e|)} d)$. If ${\mathcal{E}}$ is a left operad module over ${\mathcal{D}}$ then $e\star d := \mu_l(d \otimes \Delta^{(|d|)} e)$ defines a right action.
Checking that either case defines an associative action and a module coalgebra uses the same reasoning as for the proof of Theorem \[thm: inthopf\], with $\mu$ replacing $\gamma$.
\[thm:suffcover\] Given a coalgebra map $\lambda \colon {\mathcal{C}}\to {\mathcal{D}}$ from a connected graded coalgebra ${\mathcal{C}}$ to a graded Hopf operad ${\mathcal{D}}$, the maps $f_r = \gamma\circ(\id\circ\lambda)\colon {\mathcal{D}}\circ{\mathcal{C}}\to {\mathcal{D}}$ and $f_l = \gamma\circ(\lambda\circ \id)\colon {\mathcal{C}}\circ{\mathcal{D}}\to {\mathcal{D}}$ give connections on ${\mathcal{D}}$.
By Theorem \[thm: inthopf\] and Lemmas \[lem:opmod\] and \[lem:coal\], ${\mathcal{D}}\circ{\mathcal{C}}$ and ${\mathcal{C}}\circ{\mathcal{D}}$ are connected graded module coalgebras over ${\mathcal{D}}$. We need only show that the maps $f_r$ and $f_l$ are coalgebra maps and module maps. In terms of decomposable tensors, the maps take the form, $$\begin{gathered}
f_r\left({\frac{{c_0{\treeseparator}\dotsb{\treeseparator}c_n}}{d}}\right)\ :=\
\gamma\left({\frac{{\lambda(c_0){\treeseparator}\dotsb{\treeseparator}\lambda(c_n)}}{d}}\right)
\quad\hbox{and}\quad
f_l\left({\frac{{d_0{\treeseparator}\dotsb{\treeseparator}d_n}}{c}}\right)\ := \
\gamma\left({\frac{{d_0{\treeseparator}\dotsb{\treeseparator}d_n}}{\lambda(c)}}\right).\end{gathered}$$
These are coalgebra maps since both $\lambda$ and $\gamma$ are coalgebra maps. The associativity of $\gamma$ implies that $f_r$ and $f_l$ are maps of right and left ${\mathcal{D}}$-modules, respectively.
Examples of module coalgebra connections {#sec: Hopf op examples}
----------------------------------------
Eight of the nine compositions of coalgebras from Section \[sec: cccc examples\] are connections on one or both of the factors ${\mathcal{C}}$ and ${\mathcal{D}}$.
\[thm:eightex\] For ${\mathcal{C}}\in\{\ssym,\ysym,\csym\}$, the coalgebra compositions ${\mathcal{C}}\circ\csym$ and $\csym\circ{\mathcal{C}}$ are connections on $\csym$. For ${\mathcal{C}}\in \{\ssym,\ysym, \csym\}$, the coalgebra compositions ${\mathcal{C}}\circ\ysym$ and $\ysym\circ{\mathcal{C}}$ are connections on $\ysym$.
By Theorem \[thm:suffcover\] and Proposition \[prop:tree and comb\], we need only show the existence of coalgebra maps from ${\mathcal{C}}$ to ${\mathcal{D}}$, for ${\mathcal{C}}\in \{\ssym,\ysym, \csym\}$ and ${\mathcal{D}}\in\{\ysym,\csym\}$.
For ${\mathcal{D}}= \csym$, the maps $\bkappa\btau$, $\bkappa$, and $\id$ are all coalgebra maps to $\csym$ (Proposition \[thm: Hopf examples\]). For ${\mathcal{D}}= \ysym$, the maps $\btau$ and $\id$ are coalgebra maps to $\ysym$. Lastly, combs are binary trees, and the induced inclusion map $\csym \hookrightarrow \ysym$ is a coalgebra map.
Note that in particular, $\ysym\circ\csym$ is a connection on both $\csym$ and $\ysym$. This yields two distinct one-sided Hopf algebra structures on $\ysym\circ\csym$. Likewise, $\ysym\circ\ysym$ is a connection on $\ysym$ in two distinct ways (again leading to two distinct one-sided Hopf structures). In the remaining sections, we discuss three of the compositions of Section \[sec: cccc examples\] which have appeared previously.
A Hopf Algebra of Painted Trees {#sec: painted}
===============================
Our motivating example is the self-composition $\psym := \ysym\circ \ysym$. Elements of the fundamental basis of $\psym$ are $F_p = {{d{\,\circ\,}({c_0{\treeseparatorinline}\dotsb{\treeseparatorinline}c_{|d|}})}}$, where $c_0,\dotsc,c_{|d|}$ and $d$ are elements of the fundamental basis of $\ysym$. The indexing trees of $c_1,\dotsc,c_{|d|}$ and $d$ may be combined to form painted trees as in Example \[ex: painted\]. We describe the topological origin of painted trees and their relation to the multiplihedron, and we relate the Hopf structures of $\psym$ to the Hopf structures of $\msym$ developed in [@FLS:2010].
Painted binary trees in topology.
---------------------------------
A [[**]{}]{} is a planar binary tree $t$, together with a (possibly empty) upper order ideal of its node poset. We indicate this ideal by painting on top of a representation of $t$. For clarity, we stop our painting in the middle of edges. Here are a few simple examples, $$\label{Eq:PT}
\raisebox{-15pt}{ \includegraphics{figures/p0.d.eps}\, {,} \qquad
\includegraphics{figures/p21.d.eps}\!\! {,} \qquad
\includegraphics{figures/p12.d.eps} \!\! {,} \qquad
\includegraphics{figures/p213.d.eps} \!\! {,} \qquad
\includegraphics{figures/p2143.d.eps}\!\! {,} \qquad
\includegraphics{figures/p436521.d.eps} \! {.} }$$
An [[**]{}]{} is a topological $H$-space with a weakly associative multiplication of points [@Sta:1963]. (Products are represented by planar binary trees as these distinguish between possible choices of associations.) Maps between $A_{n}$-spaces preserve the multiplicative structure only up to homotopy. Stasheff [@Sta:1963] described these maps combinatorially using cell complexes called multiplihedra, while Boardman and Vogt [@BoaVog:1973], used spaces of painted trees. Both the spaces of trees and the cell complexes are homeomorphic to convex polytope realizations of the multiplihedra as shown in [@forcey1].
If $f\colon ( X,\Blue{{\raisebox{.09ex}{\footnotesize $\bullet$}}}) \to (Y,\Red{\ast})$ is a map of $A_n$-spaces, then the different ways to multiply and map $n$ points of $X$ are represented by painted trees. Unpainted nodes are multiplications in $X$, painted nodes are multiplications in $Y$, and the beginning of the painting indicates that $f$ is applied to a given point in $X$. See Figure \[fig: maps to painted\].
$$f({\blue a})\,{\red\bm\ast}\,\bigl(f({\blue b\,{\raisebox{.09ex}{\footnotesize $\bullet$}}\,c})\,
{\red\bm\ast}\,f({\blue d})\bigr)
\ \longleftrightarrow \
{\raisebox{-.5\height}{\includegraphics{figures/p2413.d.eps}}}$$
Figure \[fig:\_multi\_paint\] shows the three-dimensional multiplihedron with its vertices labeled by painted trees having three internal nodes. This picture of the multiplihedron also shows that the vertices are the elements of a lattice whose Hasse diagram is the one-skeleton of the polytope in the view shown. See [@FLS:2010] for an explicit description of the covering relations in terms of *bi-leveled trees*.
Painted trees as bi-leveled trees
---------------------------------
A [[**]{}]{} is a planar binary tree $t$ together with an order ideal ${{\sf T}}$ of its node poset which contains the leftmost node, but none of its children. We display bi-leveled trees corresponding to the painted trees of , circling the nodes in ${{\sf T}}$. $$\includegraphics{figures/m1.d.eps}\, {,} \qquad
\includegraphics{figures/m21.d.eps}\!\! {,}\qquad
\includegraphics{figures/m12.d.eps}\!\! {,}\qquad
\includegraphics{figures/m213.d.eps}\!\! {,}\qquad
\includegraphics{figures/m2143.d.eps}\!\! {,}\qquad
\includegraphics{figures/m436521.d.eps}\! {.}$$ Bi-leveled trees having $n{+}1$ internal nodes are in bijection with painted trees having $n$ internal nodes, the bijection being given by pruning: Remove the leftmost branch and node from a bi-leveled tree to get a tree whose order ideal is the order ideal of the bi-leveled tree, minus the leftmost node. For an illustration of this and the inverse mapping, see Figure \[fig: painted to bi-leveled\].
$$\includegraphics{figures/p2413.d.eps}
\ \xleftrightarrow{\raisebox{-12pt}{\
\includegraphics{figures/pm2413.d.eps}}\ \ }
\
\includegraphics{figures/m2413.d.eps} $$
Let $\m_n$ be the set of bi-leveled trees with $n$ internal nodes. In [@FLS:2010] we developed several algebraic structures on the graded vector space $\msym$ with basis $F_b$ indexed by bi-leveled trees $b$, graded by the number of internal nodes of $b$. We also placed a $\ysym$–Hopf module structure on $\msym_+$, the positively graded part of $\msym$. We revisit this structure in Section \[sec: add structs\].
The coalgebra of painted trees. {#sec: coalgebra psym}
-------------------------------
Let $\p_n$ be the poset of painted trees on $n$ internal nodes, with partial order inherited from the identification with bi-leveled trees $\m_{n+1}$. We show $\p_3$ in Figure \[fig:\_multi\_paint\].
$$\begin{picture}(260,238)(2,2)
\put(2,5){\includegraphics[height=240pt]{figures/M4.eps}}
\put(119,224){\includegraphics{figures/p4321.s.eps}}
\put(56,204){\includegraphics{figures/p4312.s.eps}}
\put(182,203){\includegraphics{figures/p3421.s.eps}}
\put(19,163){\includegraphics{figures/p4213.s.eps}}
\put(74,163){\includegraphics{figures/p4231.s.eps}}
\put(149,171){\includegraphics{figures/p3412.s.eps}}
\put(225,164){\includegraphics{figures/p2431.s.eps}}
\put( 4,116.5){\includegraphics{figures/p4123.s.eps}}
\put(58,121){\includegraphics{figures/p3214.s.eps}}
\put(105,130.7){\includegraphics{figures/p3241.s.eps}}
\put(160.8,130.5){\includegraphics{figures/p2413.s.eps}}
\put(235,118){\includegraphics{figures/p1432.s.eps}}
\put(21, 70){\includegraphics{figures/p3124.s.eps}}
\put(80.3, 74){\includegraphics{figures/p2143.s.eps}}
\put(121, 84){\includegraphics{figures/p2314.s.eps}}
\put(155.5, 90){\includegraphics{figures/p2341.s.eps}}
\put(228, 72){\includegraphics{figures/p1423.s.eps}}
\put(49, 32){\includegraphics{figures/p2134.s.eps}}
\put(128.5, 38.5){\includegraphics{figures/p1243.s.eps}}
\put(185, 24){\includegraphics{figures/p1324.s.eps}}
\put(121, 0){\includegraphics{figures/p1234.s.eps}}
\end{picture}$$
We refer to [@FLS:2010] for a description of the order on $\m_{n+1}$. (Note that the map from $\p_\bb$ to $\m_\bb$ actually lands in $\m_+$, which consists of the trees in $\m_\bb$ with one or more nodes.)
We reproduce the compositional coproduct defined in Section \[sec: cccc main results\].
\[def: painted is cccc\] Given a painted tree $p$, define the coproduct in the fundamental basis $\bigl\{ F_p \mid p \in \p_\bb \bigr\}$ by $$\begin{gathered}
\Delta(F_p)\ =\ \sum_{p \psplit (p_0,p_1)} F_{p_0} \otimes F_{p_1} ,
\end{gathered}$$ where the painting in $p$ is preserved in the splitting $p\psplit(p_0,p_1)$.
The counit $\varepsilon$ is projection onto $\psym_0$, which is spanned by $F_{\includegraphics{figures/p0.eps}}$.
Theorem \[thm: cc cofree\] describes the primitive elements of $\psym = \ysym \circ \ysym$ in terms of the primitive elements of $\ysym$. We recall the description of primitive elements of $\ysym$ as given in [@AguSot:2006]. The set of trees $\y_n$ forms a poset whose covering relation is obtained by moving a child node of a given node from the left to the right branch above the given node. Thus $$\raisebox{-7pt}{\includegraphics{figures/123-dot.eps}}
{\color{red} \ \longrightarrow\ }
\raisebox{-7pt}{\includegraphics{figures/213-dot.eps}}
{\color{blue} \ \longrightarrow\ }
\raisebox{-7pt}{\includegraphics{figures/312-dot.eps}}
{\color{red} \ \longrightarrow\ }
\raisebox{-7pt}{\includegraphics{figures/321-dot.eps}}$$ is an increasing chain in $\y_3$ (the moving vertices are marked with dots).
Let $\mu$ be the Möbius function of $\y_n$ which is defined by $\mu(t,s)=0$ unless $t\leq s$, $$\mu(t,t)\ =\ 1\,, \qquad\mbox{and}\qquad
\mu(t,r)\ =\ -\sum_{t\leq s< r} \mu(t,s)\,.$$ We define a new basis for $\ysym$ using the Möbius function. For $t\in\y_n$, set $$M_t\ :=\ \sum_{t\leq s} \mu(t,s) F_s\,.$$ Then the coproduct for $\ysym$ with respect to this $M$-basis is still given by splitting of trees, but only at leaves emanating directly from the right branch above the root: $$\Delta( M_{\;\includegraphics{figures/42513.eps}})\ =\
1 \otimes M_{\;\includegraphics{figures/42513.eps}} \ +\
M_{\;\includegraphics{figures/213.eps}} \otimes M_{\;\includegraphics{figures/12.eps}}\ +\
M_{\;\includegraphics{figures/42513.eps}}\otimes 1\,.$$ A tree $t\in\y_n$ is [[**]{}]{} if it has no branching along the right branch above the root node. A consequence of the description of the coproduct in this $M$-basis is Corollary 5.3 of [@AguSot:2006] that the set $\{M_t \mid t \hbox{ is progressive}\}$ is a linear basis for the space of primitive elements in $\ysym$.
Thus according to Theorem \[thm: cc cofree\] the cogenerating primitives in $\psym$ are of two types: $${\frac{{1{\treeseparator}c_1{\treeseparator}\dotsb{\treeseparator}c_{n-1}{\treeseparator}1}}{M_t}} \qquad\hbox{ and }\qquad {\frac{{\,M_t\,}}{1}} \,,$$ where $t$ is a progressive tree.
Here are some examples of the first type, $$\begin{aligned}
M_{\;\includegraphics{figures/p24135.eps}} \ &:= \
{\frac{{1\treeseparator F_{\;\includegraphics{figures/1.eps}}\treeseparator 1
\treeseparator 1}}{M_{\;\includegraphics{figures/213.eps}}}}
\ = \
F_{\;\includegraphics{figures/p24135.eps}}\ - \
F_{\;\includegraphics{figures/p25134.eps}}\ ,\\
M_{\;\includegraphics{figures/p1324.eps}}\ &:= \
{\frac{{1\treeseparator 1\treeseparator 1\treeseparator
1}}{M_{\;\includegraphics{figures/213.eps}}}}
\ = \
F_{\;\includegraphics{figures/p1324.eps}} -
F_{\;\includegraphics{figures/p1423.eps}} \,, \rule{0pt}{20pt}
\\
M_{\;\includegraphics{figures/p2314.eps}}\ &:=\
{\frac{{1\treeseparator F_{\;\includegraphics{figures/1.eps}}\treeseparator 1}}{M_{\;\includegraphics{figures/12.eps}}}}
\ = \
F_{\;\includegraphics{figures/p2314.eps}} -
F_{\;\includegraphics{figures/p2413.eps}}\ , \rule{0pt}{20pt}
\intertext{and one of the second type,}
M_{\;\includegraphics{figures/p4213.eps}}\ &:= \
{\frac{{M_{\;\includegraphics{figures/213.eps}}}}{1}}
\ = \
F_{\;\includegraphics{figures/p4213.eps}} -
F_{\;\includegraphics{figures/p4312.eps}} \,. \rule{0pt}{20pt}
\end{aligned}$$
The primitives can be described in terms of Möbius inversion on certain subintervals of the multiplihedra lattice. For the first type, the subintervals are those with a fixed unpainted forest of the form $(\includegraphics{figures/0.eps}, t, \dotsc, s,\includegraphics{figures/0.eps})$. For the second type, the subinterval consists of those trees whose painted part is trivial, $\includegraphics{figures/p0.eps}$. Each subinterval of the first type is isomorphic to $\y_m$ for some $m\leq n$, and the second subinterval is isomorphic to $\y_{n}$. Figure \[fig:\_multi\_sub\] shows the multiplihedron lattice $\p_3$, with these subintervals highlighted.
$$\begin{picture}(260,238)(2,2)
\put(2,5){\includegraphics[height=240pt]{figures/M4.shade.eps}}
\put(119,224){\includegraphics{figures/p4321.s.eps}}
\put(56,204){\includegraphics{figures/p4312.s.eps}}
\put(182,203){\includegraphics{figures/p3421.s.eps}}
\put(19,163){\includegraphics{figures/p4213.s.eps}}
\put(74,163){\includegraphics{figures/p4231.s.eps}}
\put(151,173){\includegraphics{figures/p3412.s.eps}}
\put(224,164){\includegraphics{figures/p2431.s.eps}}
\put( 4,116.5){\includegraphics{figures/p4123.s.eps}}
\put(58,121){\includegraphics{figures/p3214.s.eps}}
\put(105,130.7){\includegraphics{figures/p3241.s.eps}}
\put(160.8,130.5){\includegraphics{figures/p2413.s.eps}}
\put(235,118){\includegraphics{figures/p1432.s.eps}}
\put(21, 70){\includegraphics{figures/p3124.s.eps}}
\put(80.3, 74){\includegraphics{figures/p2143.s.eps}}
\put(121, 84){\includegraphics{figures/p2314.s.eps}}
\put(159, 88){\includegraphics{figures/p2341.s.eps}}
\put(228, 72){\includegraphics{figures/p1423.s.eps}}
\put(49, 32){\includegraphics{figures/p2134.s.eps}}
\put(128.5, 38.5){\includegraphics{figures/p1243.s.eps}}
\put(185, 24){\includegraphics{figures/p1324.s.eps}}
\put(121, 0){\includegraphics{figures/p1234.s.eps}}
\end{picture}$$
Hopf structures on painted trees. {#sec: add structs}
---------------------------------
As determined in the proof of Theorem \[thm:eightex\], the identity map $\id:\ysym\to\ysym$ yields a connection $f_r \colon \psym \to \ysym$. In particular (Theorem \[cover\_implies\_Hopf\]), $\psym$ is a one-sided Hopf algebra, a $\ysym$–Hopf module, and a $\ysym$–comodule algebra. We discuss these structures, and relate them to structures placed on $\msym$ in [@FLS:2010].
Let $p,q$ be painted trees with $|q|=n$. In terms of the $F$-basis, $f_r$ simply forgets the painting level, e.g., $f_r(F_{\;\includegraphics{figures/p231.eps}}) =F_{\;\includegraphics{figures/21.eps}}$. Thus Theorem \[cover\_implies\_Hopf\] describes the product $F_p\cdot F_q$ in $\psym$ as $$F_p\cdot F_q\ =\ \sum_{p\psplit (p_0,p_1,\dots,p_n)} F_{(p_0,p_1,\dots,p_n)/q^+} \,,$$ where the painting in $p$ is preserved in the splitting $(p_0,p_1,\dots,p_n)$, and $q^+$ signifies that $q$ is painted completely before grafting. Here is an example of the product, $$F_{\;\includegraphics{figures/p3412.eps}}\cdot
F_{\;\includegraphics{figures/p21.eps}}
\ =\
F_{\;\includegraphics{figures/p35412.eps}}\ +\
F_{\;\includegraphics{figures/p34512.eps}}\ +\
F_{\;\includegraphics{figures/p35241.eps}}\ +\
F_{\;\includegraphics{figures/p34125.eps}}\,.$$
The painted tree  with $0$ nodes is only a right unit: for all $q\in\p_\bb$, $$F_{\includegraphics{figures/p0.eps}} \cdot F_q\ =\
F_{q^+} \qquad\hbox{and}\qquad F_q\cdot F_{\includegraphics{figures/p0.eps}}\ =\ F_q \,.$$
Although the antipode is guaranteed to exist, we include a proof for purpose of exposition.
\[thm: painted is hopf\] There are unit and antipode maps $\eta\colon {\mathbb{K}}\to \psym$ and $S \colon \psym \to \psym$ making $\psym$ a one-sided Hopf algebra.
We just observed that $\eta\colon1\mapsto F_{\includegraphics{figures/p0.eps}}$ is a right unit for $\psym$. We verify that a *left antipode* exists. That is, there exists a map $S\colon\psym\to\psym$ such that $S(F_{\includegraphics{figures/p0.eps}})=F_{\includegraphics{figures/p0.eps}}$, and for $p\in\p_+$, we have $$\label{Eq:antipode_def}
\sum_{p \psplit (p_0,p_1)} S(F_{p_0})\cdot F_{p_1} \ =\ 0\,.$$ Since $\psym$ is graded, and $|p|=|p_0|+|p_1|$ whenever $p \psplit (p_0,p_1)$, we may recursively construct $S$ using induction on $|p|$. First, set $S(F_{\includegraphics{figures/p0.eps}})=F_{\includegraphics{figures/p0.eps}}$. Then, given any painted tree $p$, the only term involving $S(F_q)$ in with $|q|=|p|$ is $S(F_p)\cdot F_{\includegraphics{figures/p0.eps}}=S(F_p)$, and so we may solve for $S(F_p)$ to obtain $$S(F_p)\ :=\ -\!\sum_{\substack{p \psplit (p_0,p_1)\\
\rule{0pt}{1.4ex}|p_0|,|p_1|>0}} S(F_{p_0})\cdot F_{p_1} \ -\
S(F_{\includegraphics{figures/p0.eps}})\cdot F_{p}\,,$$ expressing $S(F_p)$ in terms of previously defined values $S(F_q)$.
For example, $$\begin{aligned}
S(F_{\;\includegraphics{figures/p21.eps}})\
&=\ -S(F_{\includegraphics{figures/p0.eps}})\cdot
F_{\;\includegraphics{figures/p21.eps}}\ =\
-F_{\;\includegraphics{figures/p12.eps}}\,,\qquad\mbox{and}\\
S(F_{\;\includegraphics{figures/p213.eps}})\
&=\ -S(F_{\;\includegraphics{figures/p21.eps}})\cdot F_{\;\includegraphics{figures/p12.eps}}
\ -\ S(F_{\;\includegraphics{figures/p0.eps}})\cdot
F_{\;\includegraphics{figures/p213.eps}}\ =\
F_{\;\includegraphics{figures/p12.eps}}\cdot F_{\;\includegraphics{figures/p12.eps}}
\ -\ F_{\;\includegraphics{figures/p123.eps}}\\
&=\ F_{\;\includegraphics{figures/p123.eps}}
\ +\ F_{\;\includegraphics{figures/p132.eps}}
\ -\ F_{\;\includegraphics{figures/p123.eps}}\ =\
F_{\;\includegraphics{figures/p132.eps}} \,.\end{aligned}$$
\[rem: one-sided\] One may be tempted to artificially adjoin a true unit $e$ to $\psym$, but this only pushes the problem to the antipode map: $S(F_{\includegraphics{figures/p0.eps}})$ cannot be defined if $\eta(1) = e$.
The $\ysym$-Hopf module structure on $\psym$ of Theorem \[cover\_implies\_Hopf\] has coaction, $$\rho(F_p)\ =\ \sum_{p \psplit (p_0,p_1)} F_{p_0} \otimes F_{f(p_1)}\,,$$ where the painting in $p$ is preserved in the first half of the splitting $(p_0,p_1)$, and forgotten in the second half.
Under the bijection between $\p_\bb$ and $\m_+$ that grows an extra node as in Figure \[fig: painted to bi-leveled\], the splittings and graftings on $\psym$ map to the restricted splittings $\rsplit$ and graftings defined in [@FLS:2010 Section 4.1]. Moreover, we can split and graft before or after the bijection to achieve the same results. These facts allow the following corollary.
\[thm: msym+ is Hopf\] The $\ysym$ action and coaction defined in [@FLS:2010 Section 4.1] make $\msym_+$ into a Hopf module isomorphic to the Hopf module $\psym$.
The [[**]{}]{} of a Hopf module $\rho\colon{\mathcal{E}}\to{\mathcal{E}}\otimes{\mathcal{D}}$ are elements $e\in{\mathcal{E}}$ such that $\rho(e) = e \otimes 1$. The coinvariants for the action of Corollary \[thm: msym+ is Hopf\] were described explicitly in [@FLS:2010 Corollary 4.5]. In contrast to the discussion in Section \[sec: coalgebra psym\], Möbius inversion in the entire lattice $\m_\bb$ helps to find the coinvariants.
A Hopf Algebra of Weighted Trees {#sec: weighted}
================================
The composition of coalgebras $\ysym\circ\csym$ has fundamental basis indexed by forests of combs attached to binary trees, which we will call weighted trees. By the first statement of Theorem \[thm:eightex\], it has a connection on $\csym$ that gives it the structure of a one-sided Hopf algebra. We examine this Hopf algebra in more detail.
Weighted trees in topology
--------------------------
In a forest of combs attached to a binary tree, the combs may be replaced by corollae or by a positive [[**]{}]{} counting the number of leaves in the comb. These all give [[**]{}]{}. $$\label{Eq:composite_trees}
\raisebox{-17pt}{${\displaystyle
\includegraphics{figures/combTree.eps} \quad\raisebox{17pt}{=}\quad
\includegraphics{figures/corollaTree.eps} \quad\raisebox{17pt}{=}\quad
\begin{picture}(50,37.5)(1,-3)
\put(3,-3){\includegraphics{figures/compTree.eps}}
\put( 0,28){$2$} \put(15,28){$3$}
\put(30,28){$1$} \put(45,28){$2$}
\end{picture}}$}$$ Let $\ck_n$ denote the weighted trees with weights summing to $n{+}1$. These index the vertices of the $n$-dimensional [[**]{}]{}, $\ck(n{+}1)$ [@forcey2]. This sequence of polytopes parameterizes homotopy maps between strictly associative and homotopy associative $H$-spaces. Figure \[F:comp\] gives a picture of the composihedron $\ck_3$.
$$\begin{picture}(210,192)(0,2)
\put( 8, 5){\includegraphics{figures/CK4_skeleton.eps}}
\put( 77, 177){\includegraphics{figures/p4321.s.eps}}
\put(135, 164){\includegraphics{figures/p3421.s.eps}}
\put( 3, 136){\includegraphics{figures/p3214.s.eps}}
\put( 75, 130){\includegraphics{figures/p3241.s.eps}}
\put(117, 133){\includegraphics{figures/p2413.s.eps}}
\put(170, 129){\includegraphics{figures/p2431.s.eps}}
\put( 34, 75){\includegraphics{figures/p2143.s.eps}}
\put( 77, 84){\includegraphics{figures/p2314.s.eps}}
\put(116, 92){\includegraphics{figures/p2341.s.eps}}
\put(183, 88){\includegraphics{figures/p1432.s.eps}}
\put( 3, 47){\includegraphics{figures/p2134.s.eps}}
\put( 83, 37){\includegraphics{figures/p1243.s.eps}}
\put(135, 15){\includegraphics{figures/p1324.s.eps}}
\put(168, 48){\includegraphics{figures/p1423.s.eps}}
\put(75, 0){\includegraphics{figures/p1234.s.eps}}
\end{picture}$$
For small values of $n$, the composihedra $\ck(n)$ also appear as the commuting diagrams in enriched bicategories [@forcey2]. These diagrams appear in the definition of pseudomonoids [@AguMah:2010 Appendix C].
A Hopf algebra of weighted trees {#a-hopf-algebra-of-weighted-trees}
--------------------------------
We describe the key definitions of Section \[sec: cccc main results\] and Section \[sec: hopf\] for $\Blue{\cksym}:=\ysym\circ\csym$. In the fundamental basis $\bigl\{ F_p \mid p \in \ck_\bb \bigr\}$ of $\cksym$, the coproduct is $$\begin{gathered}
\Delta(F_p)\ =\ \sum_{p \psplit (p_0,p_1)} F_{p_0} \otimes F_{p_1}\,,
\end{gathered}$$ where the painting in $p\in\ck_\bb$ is preserved in the splitting $p\psplit(p_0,p_1)$. The counit $\varepsilon$ is projection onto $\cksym_0$, which is spanned by $F_{\includegraphics{figures/p0.eps}}.$ Here is an example in terms of weighted trees,
$$\Delta(F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 2} \put(6,10){\tiny {\hspace{-0.3pt}}1}\put(12,10){\tiny 2} \end{picture}}})\ =\
F_{{\begin{picture}(5.2,11)\put(1.2,0){\includegraphics{figures/0.eps}}
\put(1.2,8.5){\tiny 1}\end{picture}}}\otimes F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 2} \put(6,10){\tiny {\hspace{-0.3pt}}1}\put(12,10){\tiny 2} \end{picture}}} \ +\
F_{{\begin{picture}(5.2,11)\put(1.2,0){\includegraphics{figures/0.eps}}
\put(1.2,8.5){\tiny 2}\end{picture}}}\otimes F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 1} \put(6,10){\tiny 1}\put(12,10){\tiny 2} \end{picture}}} \ +\
F_{{\begin{picture}(11.5,12)(-1,0)\put(1.5,0){\includegraphics{figures/1.eps}}
\put(0,9){\tiny 2} \put(8,9){\tiny 1} \end{picture}}}\otimes F_{{\begin{picture}(11.5,12)(-1,0)\put(1.5,0){\includegraphics{figures/1.eps}}
\put(0,9){\tiny 1} \put(8,9){\tiny 2} \end{picture}}} \ +\
F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 2} \put(6,10){\tiny 1}\put(12,10){\tiny 1} \end{picture}}}\otimes F_{{\begin{picture}(5.2,11)\put(1.2,0){\includegraphics{figures/0.eps}}
\put(1.2,8.5){\tiny 2}\end{picture}}} \ +\
F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 2} \put(6,10){\tiny {\hspace{-0.3pt}}1}\put(12,10){\tiny 2} \end{picture}}}\otimes F_{{\begin{picture}(5.2,11)\put(1.2,0){\includegraphics{figures/0.eps}}
\put(1.2,8.5){\tiny 1}\end{picture}}} \,.$$
The primitive elements of $\cksym=\ysym\circ\csym$ have the form $$F_{\;{\begin{picture}(5.2,11)\put(1.2,0){\includegraphics{figures/0.eps}}
\put(1.2,8.5){\tiny 2}\end{picture}}}\ =\ {\frac{{\,F_{\;\includegraphics{figures/1.eps}}\,}}{1}}
\qquad\mbox{ and }\qquad
{\frac{{1{\treeseparator}c_1{\treeseparator}\dotsb{\treeseparator}c_{n-1}{\treeseparator}1}}{M_t}}\,,$$ where $t$ is a progressive tree with $n$ nodes and $c_1,\dotsc,c_{n-1}$ are any elements of $\csym$. In terms of weighted trees, the indices of the second type are weighted progressive trees with weights of 1 on their leftmost and rightmost leaves.
Let $f_l\colon\cksym \to \csym$ be the connection given by Theorem \[thm:suffcover\] (built from the coalgebra map $\bkappa$). Then Theorem \[cover\_implies\_Hopf\] gives the product $$F_p \cdot F_q\ :=\ f_l(F_p)\star F_q\,,\qquad
\mbox{where }p,q\in\ck_\bb\,.$$ In terms of the $F$-basis, $f_l$ acts on indices, sending a weighted tree $p$ to the unique comb $f_l(p)$ with the same number of nodes as $p$. The action $\star$ in the $F$-basis is given as follows: split $f_l(p)$ in all ways to make a forest of $|q|{+}1$ combs; graft each splitting onto the leaves of the forest of combs in $q$; comb the resulting forest of trees to get a new forest of combs. We illustrate one term in the product. Suppose that $p={\begin{picture}(11.5,12)(-1,0)\put(1.5,0){\includegraphics{figures/1.eps}}
\put(0,9){\tiny 2} \put(8,9){\tiny 1} \end{picture}}=\includegraphics{figures/p213.eps}$ and $q = {\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 1} \put(6,10){\tiny 2}\put(12,10){\tiny 1} \end{picture}}=\includegraphics{figures/p2314.eps}$. Then $f_l(p) =\includegraphics{figures/21.eps}$ and one way to split $f_l(p)$ gives the forest $(\includegraphics{figures/0.eps}\;,\;
\includegraphics{figures/1.eps}\;,\;
\includegraphics{figures/0.eps}\;,\;
\includegraphics{figures/1.eps}\;)$. Grafting this onto $q$ gives , which after combing the forest yields the term $\raisebox{-3pt}{\includegraphics{figures/p2314.graft_comb.eps}}
={\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 1} \put(6,10){\tiny 3}\put(12,10){\tiny 2} \end{picture}}$ in the product $p\cdot q$. Doing this for the other nine splittings of $f_l(p)$ gives $$F_{{\begin{picture}(11.5,12)(-1,0)\put(1.5,0){\includegraphics{figures/1.eps}}
\put(0,9){\tiny 2} \put(8,9){\tiny 1} \end{picture}}}\cdot F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 1} \put(6,10){\tiny 2}\put(12,10){\tiny 1} \end{picture}}} \ =\
F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 3} \put(6,10){\tiny 2}\put(12,10){\tiny 1} \end{picture}}}\ +\ 3 F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 1} \put(6,10){\tiny 4}\put(12,10){\tiny 1} \end{picture}}}\ +\
F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 1} \put(6,10){\tiny 2}\put(12,10){\tiny 3} \end{picture}}}\ +\ 2 F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 2} \put(6,10){\tiny 3}\put(12,10){\tiny 1} \end{picture}}}\ +\
F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 2} \put(6,10){\tiny 2}\put(12,10){\tiny 2} \end{picture}}}\ +\ 2 F_{{\begin{picture}(15.5,13)(-1.2,0)\put(1.5,0){\includegraphics{figures/12.eps}}
\put(0,10){\tiny 1} \put(6,10){\tiny 3}\put(12,10){\tiny 2} \end{picture}}}\,.$$
Composition trees and the Hopf algebra of simplices {#sec: simplices}
===================================================
The simplest composition of Section \[sec: cccc examples\] is $\csym\circ\csym$. As shown in Section \[S:enumeration\], the graded component of total degree $n$ has dimension $2^n$, indexed by trees with $n$ interior nodes obtained by grafting a forest of combs to the leaves of a comb (which is painted). Analogous to , these are weighted combs. As these are in bijection with compositions of $n{+}1$, we refer to them as [[**]{}]{}. $$\includegraphics{figures/cc3214.d.eps}\
\raisebox{10pt}{$\ =\ $}\
\begin{picture}(28,24)
\put(2.5,0){\includegraphics{figures/321.d.eps}}
\put( 0,17.5){\small 3} \put( 8,17.5){\small 2}
\put(15.5,17.5){\small 1} \put(24,17.5){\small 4}
\end{picture}\
\raisebox{10pt}{$\ = \ (3,2,1,4)$\,.}$$
Hopf algebra structures on composition trees
--------------------------------------------
The coproduct may again be described via splitting. Since the composition tree $(1,3)$ has the four splittings $$\label{Eq:comp_split}
\raisebox{-3pt}{\includegraphics{figures/p3421.d.eps}}\
\raisebox{2pt}{$\xrightarrow{\ \curlyvee\ }$}\
\Bigl(\;\raisebox{-3pt}{\includegraphics{figures/p0.d.eps}}\,,
\raisebox{-3pt}{\includegraphics{figures/p3421.d.eps}}\Bigr)
\,,\quad
\Bigl(\raisebox{-3pt}{\includegraphics{figures/p12.d.eps}}\,,
\raisebox{-3pt}{\includegraphics{figures/p321.d.eps}}\;\Bigr)
\,,\quad
\Bigl(\;\raisebox{-3pt}{\includegraphics{figures/p231.d.eps}}\,,
\raisebox{-3pt}{\includegraphics{figures/p21.d.eps}}\Bigr)
\,,\quad
\Bigl(\;\raisebox{-3pt}{\includegraphics{figures/p3421.d.eps}},\;
\raisebox{-3pt}{\includegraphics{figures/p0.d.eps}}\;\Bigr)\,,$$ we have $\Delta(F_{1,3}) = F_{1}\otimes F_{1,3}+F_{1,1}\otimes F_{3}+
F_{1,2}\otimes F_{2}+ F_{1,3}\otimes F_{1}$.
The identity map on $\csym$ gives two connections $\csym\circ\csym \to
\csym$ (using either $f_l$ or $f_r$ from Theorem \[thm:suffcover\]). This gives two new one-sided Hopf algebra structures on compositions.
### Hopf structure induced by $f_l$
Let $p,q$ be composition trees and consider the product $
F_p \cdot F_q\ :=\ f_l(F_p)\star F_q\,.
$ At the level of indices in the $F$-basis, the connection $f_l$ sends the composition tree $p$ to the unique comb $f_l(p)$ with the same number of vertices as $p$. The action $f_l(F_p) \star F_q$ may be described as follows: split the comb $f_l(p)$ into a forest of $|q|{+}1$ combs in all possible ways; graft each splitting onto the leaves of the forest in $q$; comb the resulting forest of trees to get a new forest of combs. For example, $F_{1,3}\cdot F_{1,1}=F_{1,4}+F_{2,3}+F_{3,2}+F_{4,1}$, or alternatively, $$\label{Eq:secondprodCC}
F_{\;\includegraphics{figures/p3421.eps}} \cdot F_{\;\includegraphics{figures/p12.eps}}
\ =\
F_{\;\includegraphics{figures/p45321.eps}}\ +\ F_{\;\includegraphics{figures/p43521.eps}}\ +\
F_{\;\includegraphics{figures/p43251.eps}}\ +\ F_{\;\includegraphics{figures/p43215.eps}}\,.$$ This may be seen by unpainting and grafting the splittings onto the tree . Likewise, $F_{1,3}\cdot F_2=4F_{4}$, for no matter which of the four splittings of $f_l(1,3)$ is chosen, the grafting onto  and subsequent combing will yield the same tree .
### Hopf structure induced by $f_r$
Let $p,q$ be composition trees and consider the product $
F_p \cdot F_q\ :=\ F_p \star f_r(F_q)\,.
$ At the level of indices in the $F$-basis, the connection $f_r$ sends a composition tree $q$ to the unique comb $f_r(q)$ with $|q|$ vertices. The action $F_p \star f_r(F_q)$ may be described as follows: first paint the comb $f_r(q)$; next split the composition tree $p$ into a forest of $|q|{+}1$ composition trees in all possible ways; finally, graft each forest onto the leaves of the painted tree $f_r(q)$ and comb the resulting painted subtree (which comes from the nodes of $q$ and the painted nodes of $p$). For example, $F_{1,3}\cdot F_2=2F_{1,1,3}+F_{1,2,2}+F_{1,3,1}$, or alternatively, $$\label{Eq:prodCC}
F_{\;\includegraphics{figures/p3421.eps}} \cdot F_{\;\includegraphics{figures/p21.eps}}
\ =\
F_{\;\includegraphics{figures/p35421.eps}}\ +\ F_{\;\includegraphics{figures/p35421.eps}}\ +\
F_{\;\includegraphics{figures/p35241.eps}}\ +\ F_{\;\includegraphics{figures/p35214.eps}}\,.$$ This may be seen by grafting the splittings onto the tree $f_r(\includegraphics{figures/p21.eps}) = \includegraphics{figures/1.eps}$.
Composition trees in topology
-----------------------------
A one-sided Hopf algebra $\delsym$ was defined in [@ForSpr:2010 Section 7.3] whose $n$th graded piece had a basis indexed by the faces of the ($n{-}1$)-dimensional simplex. We recount the product and coproduct introduced there. (The notation $\tilde{\delsym}$ was used for this algebra in [@ForSpr:2010] to distinguish it from an algebra based only on the vertices of the simplex.) Faces of the ($n{-}1$)-dimensional simplex correspond to subsets ${{\sf S}}$ of $[n]:=\{1,\dotsc,n\}$, so this is a Hopf algebra whose $n$th graded piece also has dimension $2^n$, with fundamental basis $F^{[n]}_{{{\sf S}}}$.
An ordered decomposition $n=p+q$ gives a splitting of $[n]$ into two pieces $[p]$ and $\iota_p([q]):=\{p{+}1,\dotsc,n\}$. Any subset ${{\sf S}}\subseteq[n]$ gives a pair of subsets ${{\sf S}}'\subseteq[p]$ and ${{\sf S}}''\subseteq[q]$, $${{\sf S}}'\ :=\ {{\sf S}}\cap[p]
\qquad\mbox{and}\qquad
{{\sf S}}''\ :=\ \iota_p^{-1}({{\sf S}}\cap\{p{+}1,\dotsc,n\})\,.$$ Then the coproduct is $$\Delta(F^{[n]}_{{{\sf S}}})\ =\
\sum_{p+q=n} F^{[p]}_{{{\sf S}}'}\otimes F^{[q]}_{{{\sf S}}''}\,.$$ For example, the coproduct on the basis element corresponding to $\{1\}\subseteq[3]$ is $$\Delta\bigl(F^{[3]}_{\{1\}}\bigr) \ = \
F^{\emptyset}_\emptyset \otimes F^{[3]}_{\{1\}} +
F^{[1]}_{\{1\}} \otimes F^{[2]}_{\emptyset} +
F^{[2]}_{\{1\}} \otimes F^{[1]}_{\emptyset} +
F^{[3]}_{\{1\}} \otimes F^{\emptyset}_{\emptyset} \,.$$ This was motivated by constructions based on certain tubings of graphs. In terms of tubings on an edgeless graph with three nodes, the coproduct takes the form $$\label{Eq:DS_coprod}
\Delta\raisebox{-2pt}{\includegraphics{figures/1.3.g.eps}}\ =\
\raisebox{-2pt}{\includegraphics{figures/0.0.g.eps}} \otimes
\raisebox{-2pt}{\includegraphics{figures/1.3.g.eps}} +
\raisebox{-2pt}{\includegraphics{figures/1.1.g.eps}} \otimes
\raisebox{-2pt}{\includegraphics{figures/0.2.g.eps}} +
\raisebox{-2pt}{\includegraphics{figures/1.2.g.eps}} \otimes
\raisebox{-2pt}{\includegraphics{figures/0.1.g.eps}} +
\raisebox{-2pt}{\includegraphics{figures/1.3.g.eps}} \otimes
\raisebox{-2pt}{\includegraphics{figures/0.0.g.eps}}\,.$$ We leave it to the reader to make the identification (or see [@ForSpr:2010]).
The product $F^{[p]}_{{{\sf S}}}\cdot F^{[q]}_{{{\sf T}}}$ has one term for each shuffle of $[p]$ with $\iota_p([q])$. The corresponding subset ${{\sf R}}\subseteq [p{+}q]$ is the image of $[p]$ in the shuffle (not just ${{\sf S}}$), together with the image of ${{\sf T}}$. For example, $$\begin{aligned}
\label{Eq:delsim_prod}
\raisebox{-3pt}{\includegraphics{figures/0.1.g.eps}}\cdot
\raisebox{-3pt}{\includegraphics{figures/1.3.r.eps}}&\ =\
\raisebox{-3pt}{\includegraphics{figures/2.4.gr.eps}}\ +\
\raisebox{-3pt}{\includegraphics{figures/2.4.rg.eps}}\ +\
\raisebox{-3pt}{\includegraphics{figures/13.4.eps}}\ +\ \vspace{-3pt}
\raisebox{-3pt}{\includegraphics{figures/14.4.eps}}\,,\\
\intertext{and}\vspace{-3pt}
\raisebox{-3pt}{\includegraphics{figures/1.3.g.eps}}\cdot
\raisebox{-3pt}{\includegraphics{figures/0.1.r.eps}}&\ =\
\raisebox{-3pt}{\includegraphics{figures/123.4.eps}}\ +\
\raisebox{-3pt}{\includegraphics{figures/124.4.eps}}\ +\
\raisebox{-3pt}{\includegraphics{figures/134.4.eps}}\ +\
\raisebox{-3pt}{\includegraphics{figures/234.4.eps}}\,. \nonumber\end{aligned}$$
Let $\alpha$ be the bijection between subsets ${{\sf S}}=\{
a,b,\dots,c,d\} \subseteq[n]$ and compositions $\alpha({{\sf S}})=(a,b{-}a,\dots,d-c,n{+}1{-}d)$ of $n{+}1$. Numbering the nodes of a composition tree $1,\dotsc,n$ from left to right, the subset of $[n]$ corresponding to the tree is comprised of the colored nodes. $$\begin{picture}(124,80)(0,-2)
\put( 0,-2){\includegraphics{figures/cc3214.big.eps}}
\put( 4,70){1} \put(15,70){2} \put(27,70){\Red{3}}
\put( 39,70){4} \put(51,70){\Red{5}} \put(65,70){\Red{6}}
\put( 79,70){7} \put(91,70){8} \put(104,70){9}
\put(112,70){10}
\end{picture}
\raisebox{36pt}{$\longleftrightarrow\quad \{3,5,6\}\ \longleftrightarrow\ (3,2,1,4)\,.$}$$
Applying this bijection to the indices of their fundamental bases gives a linear isomorphism $\balpha\colon\delsym\stackrel{\simeq}{\longrightarrow}\csym\circ\csym$. Comparing the definitions above, this is clearly and isomorphism of coalgebras. Compare and . If we use the second product on $\csym \circ \csym$ (induced by the connection $f_r$), then $\balpha$ is nearly an isomorphism of the algebra, which can be seen by comparing the examples and . In fact, from the definitions given above, it is an [[**]{}]{} ($\balpha(p\cdot
q)=\balpha(q)\cdot\balpha(p)$) of one-sided algebras. We may summarize this discussion as follows.
The map $\balpha \colon \delsym \to (\csym \circ \csym,f_r)^{\mathrm{op}}$ is an isomorphism of one-sided Hopf algebras (with left-sided unit and right-sided antipode).
\[thm: F-S is cofree\] The one-sided Hopf algebra of simplices introduced in [@ForSpr:2010] is cofree as a coalgebra.
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|
---
abstract: 'In this paper, we give a more physical proof of Liouville’s theorem for a class generalized harmonic functions by the method of parabolic equation.'
address:
- 'School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, P.R.China 100049'
- 'School of Mathematics, Putian University, Putian 351100, P.R. China'
author:
- Weihua Wang
- Qihua Ruan
title: |
Liouville’s theorem for the generalized harmonic function\
\[2ex\]
*This paper is dedicated to our advisors*
---
Liuville’s theorem; generalized harmonic functions; parabolic equation\
**2010 Mathematics Subject Classification:** [Primary 35J15; Secondary 35K40]{}
{#section .unnumbered}
Liouville’s theorem for harmonic functions is based on the mean value property ([@Ev]), which has a self-evident advantage that the theorem is derived via Harnack’s inequality under a weaker assumption that the function is bounded below ([@GT]). Miyazaki [@Mi] give a proof of Liouville’s theorem for harmonic functions by the method of heat kernels, which reflects the physical essence that there is no more heat exchange after the system of heat diffusion reaches thermal equilibrium. Inspired by Miyazaki’s method [@Mi], we present a proof of Liouville’s theorem for a class generalized harmonic function by the method of parabolic equation. In fact, we generalize the result of Theorem 1 in [@Mi].
\[def1\] We say $u(x)$ is a generalized harmonic function in $\Omega $ in $\mathbb{R}^3$ if there exists a constant vector $\boldsymbol{c}$ in $\mathbb{R}^d$ such that $u$ satisfy $-\triangle u + \boldsymbol{c}\cdot\nabla u=0$ in the distributional sense: $$\label{eq1}
\int_{\Omega}u(x)(\triangle \varphi(x) + \boldsymbol{c}\cdot\nabla \varphi(x))dx=0$$ for all $\varphi\in C_0^{\infty}(\Omega)$, where $\triangle= \sum_{j=1}^d \frac{\partial^2}{\partial x_j^2}$, $\nabla=(\frac{\partial}{\partial x_1},\cdots, \frac{\partial}{\partial x_d} )$.
In definition\[def1\], u is a classic harmonic function in the distributional sense when $\boldsymbol{c}\equiv0$.
\[thm1\] Let the generalized harmonic function $u$ is bounded continuous in $\mathbb{R}^d$, then $u$ must be a constant.
Let $${
K(x,t)=
\left\{
\begin{array}{ll}
(4\pi t)^{-\frac{d}{2}}\exp\left(-\frac{(x-\boldsymbol{c}t)^2}{4t}\right), &(s\in \mathbb{R}^d, t>0)\\
0 &(s\in\mathbb{R}^d, t<0),
\end{array}
\right.}$$ then for any non-zero $t\in\mathbb{R}$, $K(x,t)$ is a function in the Schwartz space $\mathscr{S}(\mathbb{R}^d)$ and satisfied the parabolic equation $$\partial_t K(x,t)-\triangle K(x,t)+\boldsymbol{c}\cdot\nabla K(x,t)=0$$ and $$\int_{\mathbb{R}^d}K(x,t)dx=1.$$ We define the function $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
v(x,t) &=& K(\cdot, t)\ast u(x)\\
&=& \int_{\mathbb{R}^d}K(x-y,t)u(y)dy,\end{aligned}$$ which is in $C^{\infty}(\mathbb{R}^d\times\mathbb{R}_+)$. We notice the equation also holds for all $\varphi$ in $\mathscr{S}(\mathbb{R}^d)$ by the bound of $u$, since $C_0^{\infty}(\mathbb{R}^d)$ is dense in $\mathscr{S}(\mathbb{R}^d)$. By the properties of $K(x,t)$ and assumption on $u$ we have
$$\begin{aligned}
% \nonumber to remove numbering (before each equation)
& &\partial_t v(x,t)\\
&=& \int_{\mathbb{R}^d}\partial_t K(x-y,t)u(y)dy \\
&=& \int_{\mathbb{R}^d}\{(\triangle_x -\boldsymbol{c}\cdot\nabla_x)K(x-y,t)\}u(y)dy \\
&=& \int_{\mathbb{R}^d}\{(\triangle_y +\boldsymbol{c}\cdot\nabla_y)K(x-y,t)\} u(y)dy\\
&=&0.\end{aligned}$$
Hence $v(x,t)$ is independent of $t$. Since $\lim_{t\to 0^+}v(x,t)=u(x)$, we have $$v(x,t)=u(x),$$ which implies that $u$ is a $C^{\infty}$ function. Differentiating in $x_j$ gives
$$\partial_{x_j}u(x) = \int_{\mathbb{R}^d}\partial_{x_j}K(x-y,t)u(y)dy,$$
hence, $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
& &|\partial_{x_j}u(x)| \\
&\leq& \|\partial_{x_j}K(x,t)\|_{L^1(\mathbb{R}^d)} \|u\|_{L^{\infty}(\mathbb{R}^d)}\\
&=&\int_{\mathbb{R}^d}\left|(4\pi t)^{-\frac{d}{2}}\exp\left(-\frac{(x-\boldsymbol{c}t)^2}{4t}\right)\frac{(x_i-\boldsymbol{c}_it)}{2t}\right|dx \|u\|_{L^{\infty}(\mathbb{R}^d)}\\
&=& t^{-\frac{1}{2}}\int_{\mathbb{R}^d}\left|(4\pi )^{-\frac{d}{2}}\exp(-y^2)y_i\right|dy\|u\|_{L^{\infty}(\mathbb{R}^d)}.
\end{aligned}$$ Here, the replacement $$y=\frac{x-\boldsymbol{c}t}{2t^{\frac{1}{2}}}$$ is used in the last equation of the above equations.\
Let $t\to +\infty$, and we obtain $\partial_{x_j}u(x) =0$ for all $x\in \mathbb{R}^d$ and $1\leq j\leq d$. Therefore $u(x)$ is a constant.
In theorem \[thm1\], we obtain immediately the result of theorem 1 in [@Mi] just by letting $\boldsymbol{c}=0$.
Acknowledgments {#acknowledgments .unnumbered}
===============
Two authors would like to thank the referees for their valuable suggestions.
References {#references .unnumbered}
==========
[10]{} L.C. Evans, *Partial Differential Equations, second ed.*, AMS, Providence, 2010. D. Gilbarg, N.S. Trudinger, *Elliptic Partial Differential Equations of Second Order*, Springer, Berlin/ Heidelberg, 1998. Y. Miyazaki, *Liouville’s theorem and heat kernels*, Expo. Math., 33 (2015), pp. 101-104.
|
---
abstract: 'We present a security proof for establishing private entanglement by means of recurrence-type entanglement distillation protocols over noisy quantum channels. We consider protocols where the local devices are imperfect, and show that nonetheless a confidential quantum channel can be established, and used to e.g. perform distributed quantum computation in a secure manner. While our results are not fully device independent (which we argue to be unachievable in settings with quantum outputs), our proof holds for arbitrary channel noise and noisy local operations, and even in the case where the eavesdropper learns the noise. Our approach relies on non-trivial properties of distillation protocols which are used in conjunction with de-Finetti and post-selection-type techniques to reduce a general quantum attack in a non-asymptotic scenario to an i.i.d. setting. As a side result, we also provide entanglement distillation protocols for non-i.i.d. input states.'
address: '${1}$ Institut für Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria ${2}$ Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany'
author:
- 'Alexander Pirker$^{1}$, Vedran Dunjko$^{1,2}$, Wolfgang Dür$^{1}$ and Hans J. Briegel$^{1}$'
title: Entanglement generation secure against general attacks
---
Introduction
============
Entanglement is a key resource in quantum information processing. Entanglement can be used to teleport quantum information [@BennettTele], to implement remote quantum gates [@BennettRemote], or for distributed quantum computation [@Cirac]. It allows one to perform tasks that are not possible by classical means, such as secret key expansion vital for secure classical communication. The latter is achieved through the famous and extensively studied quantum key distribution (QKD) protocols [@bib.lo; @bib.got; @bib.sho; @bib.bai; @bib.postselect; @bib.renner.diss; @bib.applyexpdef]. In these works, security was proven in a variety of ever more general scenarios, considering noisy channels, imperfect devices and device-independent (DI) settings, where even the local quantum devices are untrusted [@bib.acin; @bib.lim; @bib.umseh].
In contrast, the perhaps equally important task of establishing private entanglement, and the closely related problem of establishing secure quantum channels, has not been resolved in equal generality. The latter has, historically, received significantly less attention [@Barnum], until the very recent increase of interest [@Hayden; @SecChannelBroadbent; @SecChannelPortmann; @SecChannelGarg] in security under ideal settings. The task of establishing private entanglement has been considered in the context of noisy channels and both perfect [@bib.dejmps] operations, and operations with local depolarizing noise [@bib.aschauer; @AschauerPRA]. In these works, either initial states that are identical and independently distributed (i.i.d.), or asymptotic scenarios are assumed.
Here, we present a comprehensive treatment for the security of distillation protocols. To make our results broadly applicable, we generalize the security model (i.e. powers of the adversary) over standard settings for protocols with quantum outputs. Furthermore, we remove the need for asymptotic, or i.i.d. assumptions, allow for more general noise models, and formulate and prove security criteria which ensure composability – i.e. the security of the protocols when they are used in arbitrary contexts, e.g. as sub-routines of larger protocols.
More specifically, we consider arbitrary attacks employed by an adversary (Eve, the distributer of noisy or corrupt Bell-pairs) and assume noisy communication channels and noisy local operations – essentially arbitrary noise describing imperfect single- and two-qubit gates. We also extend adversarial powers beyond standard: the noisy apparatus may leak all the information about the noise processes which occurred in a run of the protocol to Eve.
Our scenario, by necessity, falls short from full DI, as security under such weakest assumptions is not attainable for protocols with a quantum output – any device used in any protocol with which a client can interact classically, perhaps to test its performance, but which eventually outputs a quantum system, can always deviate from honest behavior when the final quantum output is eventually demanded (independent of how elaborate the testing may have been). This raises the questions of how DI assumptions can be relaxed such that security becomes possible also for quantum output protocols, or how standard security models can be further extended.
DI assumptions can be understood as an extreme noisy scenario, where Eve has absolute control over the noise processes. Our model relaxes this: Eve’s control is not exact (deterministic), but rather probabilistic, however still perfectly heralded – while Eve may fail in her interventions, she still learns the noise realized. In this sense, generalizing the types of noise the protocol is provably secure under in our model, corresponds to scenarios which are ever closer to DI. Naturally, other generalizations of DI settings which make sense for protocols with quantum outputs may be possible [^1].
We proceed by first providing a security analysis for i.i.d. inputs, and then generalize to non-i.i.d. states. This is done by employing de-Finetti and post-selection symmetrization-based techniques. However, since we are interested in security in arbitrary contexts, we must go beyond standard scenarios considered in entanglement distillation works [@bib.dejmps; @bib.aschauer; @AschauerPRA] and explicitly consider the adversarial quantum systems (containing e.g. purifications of all quantum states) as well. Therefore the symmetrization-based techniques cannot be straightforwardly applied, but need to be adapted. We present and discuss the required additional steps of preprocessing, and provide entanglement distillation protocols that are not restricted to i.i.d. inputs, but are capable of dealing with general inputs. The latter is related to recent results in [@Brandao; @Buscemi; @Waeldchen].
Structure of the paper
======================
The paper is organized as follows. In Sec. \[sec:model\] we introduce the basic concepts, specify the overall setting and define the confidentiality of entanglement distillation protocols. Next, we summarize our main contribution in Sec. \[sec:contr\]. In Sec. \[sec:conf\] we show confidentiality of recurrence-type entanglement distillation protocols by proving confidentiality for i.i.d. inputs in Sec. \[sec:conf:iid\] and we extend this results to arbitrary initial states in Sec. \[sec:conf:arb\] and \[sec:conf:1\]. Finally we prove confidentiality whenever the noise transcripts leak to Eve in Sec. \[sec:conf:2\]. We summarize and discuss our results in Sec. \[sec:disc\].
The model and security guarantees {#sec:model}
=================================
Entanglement distillation is modelled by considering three players, Alice and Bob, who wish to generate a shared Bell pair, and Eve, who provides the initial pairs. Thus, Eve is connected to Alice and to Bob via a (generally noisy) quantum channel which may be completely under her control. Alice and Bob are connected by a classical authenticated, but not confidential, channel. In entanglement distillation protocols Alice and Bob apply local, in general noisy, quantum operations to their pairs. To model this noise, we extend the approach of [@bib.aschauer], where a noise register, referred to as the “lab demon” (L) register $L$ is used to store classical information about the local noise history, is appended to Alice and Bob’s pairs. In this work, the L register is a quantum register, attached to Alice and Bob. We represent the noisy maps of the entanglement distillation process as unitaries acting on an enlarged Hilbert space. L thereby coherently applies Pauli operators onto the registers of Alice and Bob. Due to the symmetry of Bell states $\ket{B_{00}} = 1/\sqrt{2}(\ket{00}+\ket{11}),$ it suffices to consider the case when the noise is applied on Alice’s register only. To model the setting where Eve acquires information about the noise transcript during the execution of the protocol, we assume that L informs Eve which noise operator was applied at each step. The setting is illustrated in Fig. \[fig:model\]. In the remainder of this paper we elaborate further on the full quantum treatment of L and Eve in terms of purifications, going beyond the setting of [@bib.aschauer].
The proposed overall protocol under i.i.d. assumption involves several steps. First, Eve distributes $n$ pairs (the *initial states*), to Alice and Bob who apply local “twirl” operations (random, correlated local operations). Next, Alice and Bob sacrifice some $m\approx\sqrt{n}$ pairs to check whether the fidelity, given with $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma \rho^{1/2}}$ for density operators $\rho$ and $\sigma$, of the pairs is sufficient for entanglement distillation, via local $\sigma_x$ and $\sigma_z$ measurements. If the fidelity $F$ relative to $\ket{B_{00}}$ is insufficient, they abort. Otherwise they proceed with a recurrence-type entanglement distillation to produce a high fidelity Bell-pair from the remaining initial states, which may also be aborted. Finally, Alice and Bob output their final state. For i.i.d. inputs, the twirl ensures that local $\sigma_z$ and $\sigma_x$ correlation measurements can be used to estimate the fidelity of individual pairs. This estimate is crucial for ensuring entanglement distillation via recurrence-type entanglement distillation protocols. Later, we will generalize to non-i.i.d. settings by prepending the protocol with symmetrization (permuting of the pairs) and tracing-out steps.
To formalize the security requirements, we define the ideal map ${\mathcal{F}}^{\alpha,l}$, mapping the initial states of Alice and Bob to a single Bell-pair, where $\alpha$ (abstractly) characterizes the noise levels in the channels connecting Eve to Alice and Bob, and also the noise of the local devices, and $l$ indicates that the noise transcripts leak to Eve. The ideal map can intuitively be thought of as a map which simulates a real protocol as follows. In the case of an abort, it replaces the final state with a fixed state $\sigma^{\perp}_{ABE}$. In the non-aborting case, however, it replaces the actual output with a special state $\sigma^{\alpha,\mathcal{P},l}_{ABE},$ which corresponds to the output of a real protocol where the noise transcripts leak to Eve, utilizing distillation protocol $\mathcal{P}$, that was successfully run with asymptotically many high-fidelity i.i.d. initial pairs. This is the best the noisy entanglement distillation protocol $\mathcal{P}$ could ever do. As we show later, $\sigma^{\alpha,\mathcal{P},l}_{ABE}$ is a well-defined state for the entanglement distillation protocols and noise parameters considered here. That is, it depends on the local noise parameters *only*, and not the initial states. Formally, we have for a given real map (that is, the map realized by the execution of a real protocol) $$\begin{aligned}
({\mathcal{E}}^{\alpha,l} \otimes {id}_E) \left(\dm{\psi}_{ABE} \right) = p_{\rho} \sigma_{ABE} \otimes \dm{ok}_{f} + (1-p_{\rho}) \sigma^{\perp}_{ABE} \otimes \dm{fail}_{f} \label{eq:realmap}\end{aligned}$$ a corresponding ideal map $$\begin{aligned}
({\mathcal{F}}^{\alpha,l} \otimes {id}_E) \left(\dm{\psi}_{ABE} \right) = p_{\rho} \sigma^{\alpha,\mathcal{P},l}_{ABE} \otimes \dm{ok}_{f} + (1-p_{\rho}) \sigma^{\perp}_{ABE} \otimes \dm{fail}_{f} \label{ideal}\end{aligned}$$ where $\ket{\psi}_{ABE}$ is a purification of the initial $n$-partite ensemble $\rho^{(n)}_{AB}$ provided by Eve, $p_{\rho}$ is the success probability depending on the initial state $\rho^{(n)}_{AB}$, and $\sigma^{\perp}_{ABE}$ is a fixed state output if the protocol is aborted. Observe that the corresponding success probabilities $p_{\rho}$, per definition, are identical for the real and ideal maps ${\mathcal{E}}^{\alpha,l}$ and ${\mathcal{F}}^{\alpha,l}$ in (\[eq:realmap\]) and (\[ideal\]) respectively. The two-level flag system $f$ distinguishes the accepting and aborting branches. The state $\sigma^{\alpha,\mathcal{P},l}_{ABE}$ is the asymptotic state of the entanglement distillation protocol $\mathcal{P}$ and is of the form $$\begin{aligned}
\sigma^{\alpha, \mathcal{P},l}_{ABE} = \left(\sum\limits^1_{i,j = 0} \omega_{ij}(\alpha,\mathcal{P}) \dm{B_{ij}}_{AB} \otimes \dm{\eta_{ij}}_{E} \right) \otimes \sigma_{E}
\label{eq:asympstate}\end{aligned}$$ where $\ket{\eta_{ij}}$ are the leaked noise transcripts of Eve, $\ket{B_{ij}} = ({id}\otimes {\sigma_x}^j {\sigma_z}^i) \ket{B_{00}}$ the Bell-basis states, and $\omega_{ij}(\alpha,\mathcal{P})$ are probabilities which depend on the noise level of the local devices and the entanglement distillation protocol $\mathcal{P}$. For instance, if the local devices are perfect, then $\omega_{ij} = 1$ if and only if $i=j=0,$ hence $AB$ contains a perfect Bell-pair. Finally, the states $\ket{\eta_{ij}}$ specify the sequences of noise operations, and are orthogonal for different $i,j$. If the noise transcripts are not leaked to Eve, we denote the ideal protocol by ${\mathcal{F}}^{\alpha}$. In that case, $\ket{\eta_{ij}}$ in (\[eq:asympstate\]) is not accessible to Eve, hence we replace $\sigma^{\alpha, \mathcal{P},l}_{ABE}$ by $\sigma^{\alpha, \mathcal{P}}_{ABE} = \left(\sum_{i,j} \omega_{ij}(\alpha,\mathcal{P}) \dm{B_{ij}}_{AB} \right) \otimes \sigma_{E}$ in (\[ideal\]). Observe that the ideal map ${\mathcal{F}}^{\alpha,l}$, which mathematically defines the type of process we wish to realize, is a global operation beyond LOCC (local operations and classical communication) which can be decomposed by concatenating the real protocol ${\mathcal{E}}^{\alpha,l}$ and a replacement map $\mathcal{S}$ (which replaces the final state only if the real protocol succeeds according to the system $f$ in (\[ideal\])), i.e. ${\mathcal{F}}^{\alpha,l} = \mathcal{S} \circ {\mathcal{E}}^{\alpha,l}$.
An entanglement distillation protocol (together with the noise maps), given as a CPTP map $\mathcal{E}^ {\alpha,(l)}$, is confidential if it is close to the ideal map:
\[def.confidentiality\] The protocol ${\mathcal{E}}^{\alpha,(l)}$ is $\varepsilon$-confidential, if $$\begin{aligned}
\label{eqn.confidentiality}
\| ({\mathcal{E}}^{\alpha,(l)} & \otimes id_{E} -{\mathcal{F}}^{\alpha,(l)} \otimes id_{E}) ({\left| \psi \right\rangle \left\langle \psi \right|_{ABE}}) \|_{1} \leq \varepsilon\end{aligned}$$ holds for all initial states $\ket{\psi}_{ABE},$ where $\| \rho \|_{1} = \tr{\sqrt{\rho \rho^\dagger }}$ is the operator 1-norm for a density operator $\rho$.
The system $E$ above may contain any purification of the initial states Eve provided. In this work, we use the term security in a generic sense, and the precise meaning depends on the context. For instance, in QKD applications, security means that Alice and Bob establish a perfectly random and secret key which the adversary has negligible information about [@Gottesman; @Lo; @bib.sho; @bib.got; @bib.renner.diss; @Koenig]. In recent times, composable security definitions have become commonplace, in which, roughly speaking, security is defined via an ideal process, and security level via the amount by which the process realized by the protocol deviates from the ideal process. In the context of QKD, this distance reduces to the distance on the generated final states of the ideal vs. realized protocol. The ideal protocol outputs a completely mixed state on Alice and Bobs system which is in tensor product with Eve. More formally, see also [@bib.renner.diss], a QKD protocol $\mathcal{Q}$ is said to be $\varepsilon-$secure for initial state $\rho_{ABE}$ if $$\begin{aligned}
\| \sigma_{S_A S_B C E} - \sigma_{SS} \otimes \sigma_{CE} \|_1 \leq \varepsilon \label{eq:def:qkd:renner}\end{aligned}$$ holds where $\sigma_{S_A S_B C E} = (\mathcal{Q} \otimes {id}_E)(\rho_{ABE})$, $S_A$ and $S_B$ denote the output systems of Alice and Bob (corresponding the generated key), $C$ denotes the classical communication and $\sigma_{SS} = 1/|S| \sum_{s \in S} \dm{s} \otimes \dm{s}$ for orthogonal states $s$. The state $\sigma_{SS} \otimes \sigma_{CE}$ corresponds to the output of the ideal protocol. The confidentiality criterion which we introduce here follows the distance-on-maps approach introduced in the context of QKD like in e.g. [@bib.postselect]. Observe that such an approach is especially tailored to compose different protocols, as the confidentiality definition concerns the distance of the real process with respect to an ideal process. Therefore the real and ideal maps ${\mathcal{E}}^{\alpha,(l)}$ and ${\mathcal{F}}^{\alpha,(l)}$ respectively are motivated by abstracting the protocol in terms of processes. It is straightforward to abstract and define the ideal map in terms of input and output relations, reflecting an ideal entanglement distillation process. As we discuss above, the ideal protocol has an $ok-$ and $fail-$branch. The $fail-$branch corresponds to the case whenever Alice and Bob abort the procedure, outputting the state $\sigma^\perp_{ABE}$. However, if the procedure succeeds then we might think of the ideal map as running the entanglement distillation protocol for infinitely many initial states, ending up in the fixed state $\sigma^{\alpha, \mathcal{P},l}_{ABE}$ of the entanglement distillation protocol $\mathcal{P}$ for noise level $\alpha$. We observe two important facts regarding that particular state: first, its the best the entanglement distillation protocol $\mathcal{P}$ can do in the presence of noise of level $\alpha$, and second, as Eve is disentangled from Alice and Bob, this state is useful for applications like quantum teleportation. Hence we refer to this state also as a private state, or equivalently, Alice and Bob share private entanglement. In contrast to (\[eq:def:qkd:renner\]), the target state $\sigma^{\alpha, \mathcal{P},l}_{ABE}$ in the $ok-$branch is only in tensor product with respect to Eve if the noise transcripts do not leak to the adversary. In that case a secure quantum channel is feasible in terms of quantum teleportation. Otherwise, that is if the noise transcripts $\ket{\eta_{ij}}$ leak to Eve, she is in a separable state with respect to Alice and Bob, but still enabling for confidential applications. By confidential we mean here that when the final state is used for quantum teleportation no information about the teleported state is leaked, but the final state does not guarantee that Eve cannot change the teleported state. This observation motivates the term confidentiality rather than security. The classical communication is not correlated to the output of the real protocol, thus it can be ignored, see \[app:sec:eppiid\] for details. The robustness of the protocol [^2] is considered in \[app:sec:robust\], which enables us to assume for the subsequent analysis that all basic distillation steps succeed.
Main contribution {#sec:contr}
=================
We summarize the main findings of our paper as follows: recurrence-type entanglement distillation protocols prepended by a symmetrization and a system discarding step enable confidentiality, provided that the noise transcripts do not leak to the adversary for all noise levels $\alpha$ for which distillation would be possible in the i.i.d. case. We also show that this alone implies that the final state in the accepting branch, is close to a tensor product state – Eve is factored out. The results regarding the BBPSSW protocol [@Bennett] are analytic whereas for the DEJMPS protocol [@bib.dejmps] the results rely on strong numerical evidence. For low noise rates, we achieve better results via the post-selection-based reduction. In that case, no system discarding step is necessary. Finally we find that if an entanglement distillation protocol is confidential when the noise transcripts do not leak, then it also confidential if they do leak to the adversary. In particular, even in the case that Eve picks up information about all the realized noise processes during the protocol, the final output system still enables confidential quantum applications like e.g. quantum teleportation. The paper proceeds as follows. We establish necessary conditions to guarantee confidentiality for recurrence-type entanglement distillation protocols restricted to i.i.d. inputs whenever the noise transcripts are not leaked to Eve. Then, we generalize this to arbitrary initial states via the de-Finetti theorem [@bib.oneandahalf]. Next, we use them to prove the confidentiality criterion (\[eqn.confidentiality\]) for entanglement distillation protocols where the noise transcripts are not leaked. Finally, this will be used to derive the confidentiality bound whenever the noise transcripts are leaked.
Confidentiality of entanglement distillation protocols {#sec:conf}
======================================================
Entanglement distillation for i.i.d inputs {#sec:conf:iid}
------------------------------------------
The basic step of a recurrence-type entanglement distillation protocol is summarized as follows: Alice and Bob share two noisy Bell-pairs, i.e. both have two qubits, each representing a “half” of a noisy Bell pair, and they first apply local operations to their respective parts of the Bell-pairs; next, they measure one Bell-pair and classically communicate their outcomes. Depending on the entanglement distillation protocol and the outcomes they either keep or discard the unmeasured pair. The basic step is applied to all pairs of the initial states, which comprises one distillation round. This distillation round is iterated where output states of the previous round are used as inputs for the next round. In the limit, a noiseless entanglement distillation protocol outputs a perfect Bell-pair (implying that Eve is factored out). Here, we allow for any type of noise acting (independently) on the single- and two-qubit gates appearing in the protocol [^3]. Using the results of [@DuerStd], by utilizing random basis changes and adding additional noise, any such general noise can be brought to a standard form: depolarizing noise for imperfect single- and two-qubit CNOT-type operations, see \[app:sec:eppiid\]. Thus, it is sufficient to address noise in such standard form.
For such noise, one can analytically show [@Duer] that for the BBPSSW protocol [@Bennett], there exists a unique attracting fixed point of the protocol which only depends on the noise parameters. That is, whenever the fidelity of the initial states is above some minimum fidelity $F_{\text{min}}$, depending on the noise parameters, the protocol converges towards that unique fixed point which we denote by $\sigma^{\alpha;\text{B}}_{AB}$. Observe that $\sigma^{\alpha;\text{B}}_{AB}$ is related to $\sigma^{\alpha, \mathcal{P},l}_{ABE}$ of (\[eq:asympstate\]) by letting $\mathcal{P} = \text{B}$ and tracing out Eves system, i.e. $\sigma^{\alpha;\text{B}}_{AB} = {\text{tr}_{E}\left[ \sigma^{\alpha, \text{B},l}_{ABE} \right]}$. In particular, we mean by $\mathcal{P} = \text{B}$ that the BBPSSW protocol is used for entanglement distillation. We find that the output state $\sigma^{N}_{AB}$, where $N = \log_2 n$ denotes the number of successfully completed distillation layers, satisfies $\| \sigma^{N}_{AB} - \sigma^{\alpha;\text{B}}_{AB} \|_{1} \leq \epsilon_{\text{B}}$, where $\epsilon_{\text{B}}$ is a function of $N$, and it holds that $\epsilon_{\text{B}} \leq F(n) \in O\left(n^{-b_{\text{B}}(\alpha)} \right)$ and $0 < b_{\text{B}}(\alpha) \leq \log_2 3 - 1$. For the entanglement distillation protocol of Deutsch et. al. [@bib.dejmps] (referred to as the DEJMPS protocol) the fixed point analysis is more complicated. In the noiseless case, DEJMPS was proven to have a unique attracting fixed point [@Macchiavello]. For the noisy case, we can only provide extensive numerical evidence that there exists a unique attracting fixed point, depending on the noise parameters only which we denote by $\sigma^{\alpha;\text{D}}_{AB}$, see \[sec.sup.dejmps\]. Again, observe that $\sigma^{\alpha;\text{D}}_{AB}$ is related to $\sigma^{\alpha, \mathcal{P},l}_{ABE}$ of (\[eq:asympstate\]) by setting $\mathcal{P} = \text{D}$ and tracing out Eves system, i.e. $\sigma^{\alpha;\text{D}}_{AB} = {\text{tr}_{E}\left[ \sigma^{\alpha, \text{D},l}_{ABE} \right]}$. We numerically find that for the state $\sigma^{N}_{AB}$ obtained after successfully completing $N = \log_2 n$ layers of distillation that $\| \sigma^{N}_{AB} - \sigma^{\alpha;\text{D}}_{AB} \|_{1} \leq \epsilon_{\text{D}}$ where $\epsilon_{\text{D}}$ is a function of $N$, and it holds that $\epsilon_{\text{D}} \leq F(n) \in O\left(n^{-b_{\text{D}}(\alpha)} \right)$. $b_{\text{D}}(\alpha)$ is a positive function. We note that a similar analysis, but also with analytic findings for the noiseless DEJMPS protocol was first performed in [@Macchiavello]. We reiterate that we assume for our analysis that all basic distillation steps succeed, since we deal with failures due to the entanglement distillation protocol with a quadratic overhead in terms of initial states, see \[app:sec:robust\]. The final state of the entanglement distillation protocol $\mathcal{P}$ in the $ok-$branch, $\sigma_{AB}$, depends on whether the parameter estimation on $\sqrt{n}$ initial states was accurate or not. The latter occurs with an exponentially small probability in terms of initial states, see the discussion of the robustness of the protocol in \[app:sec:robust\]. This in turn implies that the parameter estimation was accurate with probability exponentially close to unity. Therefore the results regarding $n$ i.i.d. initial states as input to the distillation protocol $\mathcal{P}$ above imply that $$\begin{aligned}
p_{\rho} \| \sigma_{AB} - \sigma^{\alpha;\mathcal{P}}_{AB}\|_1 \leq \epsilon_{\mathcal{P}}(n) + 2p_{\mathrm{PE}} \leq \epsilon'_{\mathcal{P}}(n) =: \varepsilon_{\mathcal{P}}(n+\sqrt{n}) \label{eq:scale:iid:dist}\end{aligned}$$ where $p_{\mathrm{PE}} \in O(\exp(-\sqrt{n}))$ for all i.i.d. inputs $\rho_{AB}^{\otimes n + \sqrt{n}}$. This equation attains exactly the same form for both protocols with the difference in the labels, so if we substitute $\mathcal{P}$ with $\mathrm{B}$ (by writing, for example $\epsilon_{\mathrm{B}}(n)$) we refer to the BBPSSW protocol, where substituting $\mathcal{P}$ with $\mathrm{D}$ refers to the DEJMPS protocol. In similar fashion we refer from now by $\epsilon_{\mathcal{P}}(n)$ to $\epsilon'_{\mathcal{P}}(n)$ for the sake of clarity. So to summarize, the distance for $n+\sqrt{n}$ i.i.d. initial states in the $ok-$branch of the protocol is bounded by $\varepsilon_{\mathcal{P}}(n+\sqrt{n})$. Since, in the abort case, the outputs of the overall protocol ${\mathcal{E}}^\alpha$ and the ideal protocol ${\mathcal{F}}^{\alpha}$ are identical we obtain that where the probability $p_{\rho}$ depends on the initial state $\rho$ for both protocols and corresponds to the probability of parameter estimation succeeding and completing $\log_{2}(n-\sqrt{n})$ distillation layers successfully for initial state $\rho$. Hence, in both cases, the final distance to the respective fixed points scales polynomial in terms of $n$. The functions $b_{\text{B}}(\alpha)$ and $b_{\text{D}}(\alpha)$ of the local noise level $\alpha$ govern the rate of convergence of the real protocol to the ideal protocol in the i.i.d case for entanglement distillation protocols. We numerically found that these functions monotonically increase as the local noise rate $\alpha$ tends to zero \[app:sec:eppiid\]. Thus, increasing the fidelity of local devices (through e.g. fault tolerance) directly influences the rate of convergence, which in turn governs the confidentiality level. In contrast to $b_{\text{B}}(\alpha)$, the function $b_{\text{D}}(\alpha)$ is not upper bounded, which implies that for certain noise parameters $\alpha$ the DEJMPS protocol needs to perform fewer distillation rounds than the BBPSSW protocol to achieve the required confidentiality levels. This fast convergence is crucial for the powerful post-selection technique [@bib.postselect] for non i.i.d. initial states, which is not applicable for the BBPSSW protocol. Now we use the established fixed point properties of entanglement distillation protocols for i.i.d. initial states to show that similar results hold for arbitrary initial states.
Entanglement distillation for arbitrary inputs {#sec:conf:arb}
----------------------------------------------
In generalizing the previous results to arbitrary initial states we make use of the de Finetti theorem [@bib.oneandahalf]. The basic de-Finetti results guarantee that the reduced state $\text{tr}_{n-k}\left(\rho^{(n)}_{AB}\right)$ of a permutation-invariant $n-$partite state $\rho^{(n)}_{AB}$ is close to an i.i.d state $\int \sigma_{AB}^{\otimes k}\ d\sigma$, with distance which scales as $O(k/n).$ This enables the following Lemma.
\[lem.localconvergence\] Let $n,k \in {\mathbb{N}}$ where $k \leq n$. Furthermore, let ${\mathcal{E}}^{s \& t}$ be the real protocol and ${\mathcal{F}}^{s \& t}$ the ideal protocol including symmetrization and the tracing out of $n-k$ pairs. Moreover, let $\rho_{AB}$ be a bipartite mixed state of $n$ systems shared by Alice and Bob and let ${\mathcal{E}}$ and ${\mathcal{F}}$ denote the real and ideal protocol after symmetrization and tracing out $n-k$ pairs. Then $$\begin{aligned}
\label{eq.basicdefin}
\| {\mathcal{E}}^{s \& t}(\rho_{AB}) - {\mathcal{F}}^{s \& t}(\rho_{AB}) \|_1 \leq \frac{64k}{n} + \max_{\mu_{AB}} \|{\mathcal{E}}(\mu^{\otimes k}_{AB}) - {\mathcal{F}}(\mu^{\otimes k}_{AB})\|_1\end{aligned}$$
Let $\rho_{AB}$ be a mixed state. After Alice and Bob apply a symmetrization they share a permutation invariant state $\tilde{\rho}_{AB}$. Thus we can apply Theorem II.7 of [@bib.oneandahalf] and have for $\xi^k_{AB} := {\text{tr}_{n-k}\left[ \tilde{\rho}_{AB} \right]}$ the inequality $\|\xi^k_{AB} - \int \mu^{\otimes k}_{AB} dm(\mu_{AB})\|_1 \leq 32k/n$ for some probability measure $m$ on the set of mixed states on $AB$. Moreover we note that ${\mathcal{E}}$ and ${\mathcal{F}}$ are CPTP maps. We define $\tau_k := \int \mu^{\otimes k}_{AB} dm(\mu_{AB})$. A straightforward computation shows $$\begin{aligned}
\|{\mathcal{E}}^{s \& t}(\rho_{AB}) - {\mathcal{F}}^{s \& t}(\rho_{AB})\|_1 & = \|{\mathcal{E}}(\xi^k_{AB}) - {\mathcal{F}}(\xi^k_{AB})\|_1 \leq \|{\mathcal{E}}(\xi^k_{AB}) - {\mathcal{E}}(\tau_k)\|_1 + \|{\mathcal{E}}(\tau_k) - {\mathcal{F}}(\xi^k_{AB})\|_1 \\
& \leq \|{\mathcal{E}}(\xi^k_{AB}) - {\mathcal{E}}(\tau_k)\|_1 + \|{\mathcal{E}}(\tau_k) - {\mathcal{F}}(\tau_k)\|_1 + \|{\mathcal{F}}(\tau_k) - {\mathcal{F}}(\xi^k_{AB})\|_1 \\
& \leq 2 \|\tau_k - \xi^k_{AB} \|_1 + \|{\mathcal{E}}(\tau_k) - {\mathcal{F}}(\tau_k)\|_1 \leq \frac{64k}{n} + \left\|({\mathcal{E}}-{\mathcal{F}})\left(\int \mu^{\otimes k}_{AB} dm(\mu_{AB}) \right) \right\|_1 \\
& \leq \frac{64k}{n} + \max_{\mu_{AB}} \left\|({\mathcal{E}}-{\mathcal{F}})\left(\mu^{\otimes k}_{AB} \right) \right\|_1\end{aligned}$$ which completes the proof.
Therefore the application of the de-Finetti theorem introduces an additive term $\frac{64k}{n}$ when reducing arbitrary initial states to i.i.d. initial states. As the right hand side of (\[eq.basicdefin\]) is independent of the initial state $\rho_{AB}$, (\[eq.basicdefin\]) holds for all initial states $\rho_{AB}$. In (\[eq.basicdefin\]) we have omitted the superscript $\alpha$ characterizing the noise level, and we will use it only if it is specifically needed. Inequality (\[eq.basicdefin\]) implies that the properties of the fixed point (uniqueness, attractivity, noise-dependence) also hold for arbitrary initial states, if the protocol is prepended by symmetrization and a trace-out step. This enables us to prove the confidentiality criterion of Definition \[def.confidentiality\] for entanglement distillation protocols, where the noise transcripts of L are not leaked, which will, in turn, imply the confidentiality criterion (\[eqn.confidentiality\]) whenever the noise transcripts are leaked.
Confidentiality of entanglement distillation protocols {#sec:conf:1}
------------------------------------------------------
The inequality in (\[eq:local-iid\]) establishes the local properties of the protocol, and is more-or-less typical for studies of the convergence of entanglement distillation protocols in the i.i.d. case. However, it falls short of the complete characterization captured by the confidentiality criterion (\[eqn.confidentiality\]) in two ways: first, the input states are restricted (i.i.d.); second, it fails to consider the purifying system of Eve [^4], vital in cryptographic contexts. While the prior issue is the subject of de-Finetti and post-selection-type reductions, the latter issue can be a problem in general, as small distance of corresponding subsystems does not imply a small distance of the total systems.
However, we can resolve this issue by using the fixed point properties of entanglement distillation protocols. More precisely, we relate the two distances by the following general Lemma, proven in \[app:sec:defin\].
\[thm.globalclosenessfactor\] Let $\rho$ be an arbitrary mixed state shared by Alice and Bob and let $\ket{\psi}_{ABE}$ be a purification thereof held by Eve. Furthermore, let $\mathcal{P}_1$ correspond to a (distillation-type) real protocol and $\mathcal{P}_2$ correspond to the associated (distillation-type) ideal protocol, i.e. $$\begin{aligned}
\mathcal{P}_1 (\rho) &= p_{\rho} \sigma_{AB} \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \dm{fail}, \\
\mathcal{P}_2(\rho) &= p_{\rho} \sigma^{\alpha}_{AB} \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \dm{fail}. \notag\end{aligned}$$ where $\alpha$ characterizes the level of the noise, $\sigma^{\alpha}_{AB}$, and $\sigma^{\perp}_{AB}$ are two fixed two qubit states. Furthermore, let $\mathcal{P}_1$ and $\mathcal{P}_2$ satisfy the following properties:
1. The noise transcripts do not leak to Eve.
2. The protocol $\mathcal{P}_1$ guarantees to converge towards some state $\sigma^{\alpha}_{AB}$ within the ok-branch of the protocol and $\max_{\mu_{AB}} \|(\mathcal{P}_1 - \mathcal{P}_2)(\mu_{AB})\|_1 \leq \varepsilon$.
Then it holds that $$\begin{aligned}
\| (\mathcal{P}_1 \otimes id_{E} -\mathcal{P}_2 \otimes id_{E} )({\left| \psi \right\rangle \left\langle \psi \right|_{ABE}})\|_1 \leq (34 \cdot 4^8 +1) \varepsilon. \label{eq.globalclosefactor}\end{aligned}$$
The factor $34 \cdot 4^8 +1$ arises as an upper bound on the distance of the given states from states in product form based on the notion of non-steerability we introduce (see \[app:sec:defin\] for details). In our computations we managed to prove the key lemma in a manner which is proportional to the dimension of the systems, more precisely, the overall size of the corresponding density matrix. It may be the case that the bound of Lemma \[thm.globalclosenessfactor\] could hold without the dependence on the system size (and indeed, with smaller constants), however this was not necessary for our purposes. Lemma \[thm.globalclosenessfactor\] is vital as it allows us to employ the de-Finetti theorem [@bib.oneandahalf]. Hence, for the protocols ${\mathcal{E}}^{s\&t}$ and ${\mathcal{F}}^{s\&t}$, by combining Lemma \[lem.localconvergence\] with Lemma \[thm.globalclosenessfactor\], we obtain the following Theorem.
Let ${\mathcal{E}}^{s \& t}$ be the real protocol and ${\mathcal{F}}^{s \& t}$ the ideal protocol including symmetrization and the tracing out of $n-k$ pairs, taking $n$ input pairs and $k \leq n$ and utilizing entanglement distillation protocol $\mathcal{P}$. Then we have $$\begin{aligned}
\max_{\ket{\psi}_{ABE}} \|({\mathcal{E}}^{s \& t} \otimes id_{E})({\left| \psi \right\rangle \left\langle \psi \right|_{}}) - ({\mathcal{F}}^{s \& t} \otimes id_{E})({\left| \psi \right\rangle \left\langle \psi \right|_{}})\|_1 \leq (34 \cdot 4^8 +1) \left(\frac{64k}{n} + \varepsilon_{\mathcal{P}}(k) \right) \label{eq:defin:tech}\end{aligned}$$ where $\varepsilon_{\mathcal{P}}(k)$ denotes the maximum distance of the real and ideal protocol without symmetrization and tracing out step using entanglement distillation protocol $\mathcal{P}$ in the $ok-$branch for $k$ i.i.d. initial states, i.e. Eq. (\[eq:local-iid\]).
Suppose Eve prepares a purification $\ket{\psi}_{ABE}$ of the state $\rho_{AB}$ shared by Alice and Bob. Recall that the real and ideal protocol including symmetrization and the tracing out of $n-k$ pairs applied to initial state $\rho_{AB}$ read as $$\begin{aligned}
{\mathcal{E}}^{s \& t} (\rho_{AB}) = p_{\rho} \sigma_{AB} \otimes \dm{ok} + (1-p_{\rho}) \sigma^{\perp}_{AB} \otimes \dm{fail}, \\
{\mathcal{F}}^{s \& t} (\rho_{AB}) = p_{\rho} \sigma^{\alpha,\mathcal{P}}_{AB} \otimes \dm{ok} + (1-p_{\rho}) \sigma^{\perp}_{AB} \otimes \dm{fail}\end{aligned}$$ and observe that we have for the initial state $\rho_{AB}$ by Lemma \[lem.localconvergence\] that $$\begin{aligned}
\| ({\mathcal{E}}^{s \& t} -{\mathcal{F}}^{s \& t})(\rho_{AB})\|_1 = p_{\rho} \| \sigma_{AB} - \sigma^{\alpha,\mathcal{P}}_{AB} \|_1 \leq \left(\frac{64k}{n} + \max_{\mu_{AB}} \left\|({\mathcal{E}}-{\mathcal{F}})\left(\mu^{\otimes k}_{AB} \right) \right\|_1 \right) \label{eq:defin:proof:2}\end{aligned}$$ where ${\mathcal{E}}$ and ${\mathcal{F}}$ denote the real and ideal protocol after symmetrization and tracing out $n-k$ pairs. Since the right-hand side of (\[eq:defin:proof:2\]) is independent of the initial state $\rho_{AB}$ it holds for all initial states of the protocol. Therefore, the properties of the fixed point (unique, attracting and depending on the noise parameters only) translate from i.i.d. initial states to arbitrary initial states. Hence the protocol guarantees that it converges towards the fixed point of the entanglement distillation protocol. Additionally, by inserting (\[eq:local-iid\]) in (\[eq:defin:proof:2\]) we find $$\begin{aligned}
\| ({\mathcal{E}}^{s \& t} -{\mathcal{F}}^{s \& t})(\rho_{AB})\|_1 \leq \left(\frac{64k}{n} + \varepsilon_{\mathcal{P}}(k) \right) \label{eq:defin:proof:3}.\end{aligned}$$ This implies that the real protocol indeed converges towards the fixed point, and, thus we can apply Lemma \[thm.globalclosenessfactor\] to the protocols ${\mathcal{E}}^{s \& t}$ and ${\mathcal{F}}^{s \& t}$ for the purification $\ket{\psi}_{ABE}$ of $\rho_{AB}$ and we find by using (\[eq:defin:proof:3\]) that $$\begin{aligned}
\|({\mathcal{E}}^{s \& t} \otimes id_{E})({\left| \psi \right\rangle \left\langle \psi \right|_{}}) - ({\mathcal{F}}^{s \& t} \otimes id_{E})({\left| \psi \right\rangle \left\langle \psi \right|_{}})\|_1 \leq (34 \cdot 4^8 +1) \left(\frac{64k}{n} + \varepsilon_{\mathcal{P}}(k) \right). \label{eq:defin:proof:4}\end{aligned}$$ Taking the maximum in (\[eq:defin:proof:4\]) completes the proof.
Thus, we can reach arbitrary confidentiality levels, however at the cost of wasting some pairs. The scaling of the confidentiality parameter, i.e. the right-hand side of (\[eq:defin:tech\]), is linear in the number of initial states $n$, due to the use of the “basic” de Finetti approach. If the local noise is low, we can do better in terms of scaling and efficiency, using the post-selection technique [@bib.postselect]. For that purpose, we first establish a result similar to (\[eq.globalclosefactor\]) by using the fact that the resulting state of the protocol, including L, is pure, see \[app:sec:eppiid\]. More precisely, we have the following Lemma, proven in \[app:sec:post\].
\[lem:forpost\] Let ${\mathcal{E}}$ be the real protocol which guarantees to converge towards a unique and attracting fixed point depending on the noise parameter only and let ${\mathcal{F}}$ be the ideal protocol. Furthermore let $\rho$ be a mixed state (consisting of $n$ systems) shared by Alice and Bob. If the extension of ${\mathcal{E}}$ and ${\mathcal{F}}$ to the system of L satisfies $\|{\mathcal{E}}_{\text{L}}(\rho) - {\mathcal{F}}_{\text{L}}(\rho)\|_1 \leq \varepsilon(n)$, then $$\begin{aligned}
\|({\mathcal{E}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}}) &- ({\mathcal{F}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}})\|_1 \leq 4 \sqrt{\varepsilon(n)}\end{aligned}$$ for all purifications $\ket{\psi}_{ABE'}$ of $\rho$.
This Lemma allows us to prove the closeness on any purification from the closeness of the reduced systems, and finally to derive confidentiality from the performance of the ideal protocol via the following Theorem.
Let ${\mathcal{E}}^{s}$ be the real protocol and ${\mathcal{F}}^{s}$ the ideal protocol preceded by a symmetrization step operating on $n$ input pairs. Furthermore let $\max_{\mu_{AB}} \|{\mathcal{E}}(\mu^{\otimes n}_{AB}) - {\mathcal{F}}(\mu^{\otimes n}_{AB})\|_1 \leq \varepsilon_{\mathcal{P}}(n)$, see (\[eq:local-iid\]), where ${\mathcal{E}}$ and ${\mathcal{F}}$ denote the sub-protocols after symmetrization (i.e. the protocols without the symmetrization step) and $\mathcal{P}$ the entanglement distillation protocol. Then we have $$\begin{aligned}
\max_{\ket{\psi}_{ABE'}} \|({\mathcal{E}}^{s} \otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{}}) - ({\mathcal{F}}^{s} \otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{}})\|_1 \leq 4 \sqrt{2} g_{n,d} \sqrt[4]{\varepsilon_{\mathcal{P}}(n)} \label{eq:post:redudction}\end{aligned}$$ where $g_{n,d} = {n+15 \choose n}$.
We observe that ${\mathcal{E}}^{s}$ and ${\mathcal{F}}^{s}$ are permutation invariant maps due to the symmetrizazion step. Thus we can apply the post-selection technique of [@bib.postselect] which implies $$\begin{aligned}
\label{inequ.post.1}
\max_{\ket{\psi}_{ABE'}} \|({\mathcal{E}}^{s} \otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{}}) - ({\mathcal{F}}^{s} \otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{}})\|_1 \leq g_{n,d} \|({\mathcal{E}}^{s} \otimes id_{E'})({\left| \tau \right\rangle \left\langle \tau \right|_{ABE'}}) - ({\mathcal{F}}^{s} \otimes id_{E'})({\left| \tau \right\rangle \left\langle \tau \right|_{ABE'}})\|_1\end{aligned}$$ where $\ket{\tau}_{ABE'}$ is a purification of the de-Finetti Hilbert-Schmidt state, hence ${\text{tr}_{E'}\left[ {\left| \tau \right\rangle \left\langle \tau \right|_{ABE'}} \right]} = \int \mu^{\otimes n}_{AB} d \eta(\mu) =: \tau'$ where $\eta$ is the measure induced by the Hilbert-Schmidt metric on $\text{End}({\mathbb{C}}^4)$. Furthermore, we note that we have for the extensions of ${\mathcal{E}}^{s}$ and ${\mathcal{F}}^{s}$ to L, i.e. the maps ${\mathcal{E}}^{s}_{\mathrm{L}}$ and ${\mathcal{F}}^{s}_{\mathrm{L}}$, that $$\begin{aligned}
\|{\mathcal{E}}^{s}_{\mathrm{L}}(\tau') - {\mathcal{F}}^{s}_{\mathrm{L}}(\tau')\|_1 = \left\|({\mathcal{E}}^{s}_{\mathrm{L}} - {\mathcal{F}}^{s}_{\mathrm{L}})\left(\int \mu^{\otimes n}_{AB} d \eta(\mu) \right) \right\|_1 \leq \max_{\mu_{AB}} \left\|({\mathcal{E}}_{\mathrm{L}}-{\mathcal{F}}_{\mathrm{L}})\left(\mu^{\otimes n}_{AB} \right) \right\|_1. \label{eq:post:proof:1}\end{aligned}$$ According to \[sec:det:l\], which implies that the distance including L scales as the square root of the $1-$norm induced distance without L, i.e. Alice and Bob only, we find for (\[eq:post:proof:1\]) by using the assumption $\max_{\mu_{AB}} \|{\mathcal{E}}(\mu^{\otimes n}_{AB}) - {\mathcal{F}}(\mu^{\otimes n}_{AB})\|_1 \leq \varepsilon_{\mathcal{P}}(n)$ that $$\begin{aligned}
\left\|({\mathcal{E}}_{\mathrm{L}}-{\mathcal{F}}_{\mathrm{L}})\left(\mu^{\otimes n}_{AB} \right) \right\|_1 \leq 2 \sqrt{\left\|({\mathcal{E}}-{\mathcal{F}})\left(\mu^{\otimes n}_{AB} \right) \right\|_1} \leq 2 \sqrt{\varepsilon_{\mathcal{P}}(n)}.\end{aligned}$$ As $\ket{\tau}_{ABE'}$ is a purification of $\tau'$ we can apply Lemma \[lem:forpost\] which gives, for (\[inequ.post.1\]), $$\begin{aligned}
\max_{\ket{\psi}_{ABE'}} \|({\mathcal{E}}^{s} \otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{}}) - ({\mathcal{F}}^{s} \otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{}})\|_1 & \leq 4 g_{n,d} \sqrt{\max_{\mu_{AB}} \left\|({\mathcal{E}}_{\mathrm{L}}-{\mathcal{F}}_{\mathrm{L}})\left(\mu^{\otimes n}_{AB} \right) \right\|_1} \\
& \leq 4 g_{n,d} \sqrt{2 \sqrt{\varepsilon_{\mathcal{P}}(n)}} \\
& = 4 \sqrt{2} g_{n,d} \sqrt[4]{\varepsilon_{\mathcal{P}}(n)}\end{aligned}$$ which completes the proof.
Observe that $\varepsilon_{\mathcal{P}}(n)$, which governs the rate of convergence of the overall protocol, relates to the rate of convergence of the entanglement distillation protocol $\mathcal{P}$ via $\varepsilon_{\mathcal{P}}(n) = \epsilon_{\mathcal{P}}(n - \sqrt{n})$, as $\sqrt{n}$ initial states are used for parameter estimation. We remind the reader that the preprocessing steps (symmetrization, tracing out) of the entanglement distillation protocol and the Lemmas of this section are non-trivial and crucial for the proof of the de-Finetti-based and post-selection-based reduction technique. Furthermore we point out that the proof regarding the BBPSSW protocol is analytic and necessarily relies on the de-Finetti-based reduction technique because of its slow convergence rate. The rate of convergence for the BBPSSW protocol can easily be derived, see \[app:sec:eppiid\] for details. For the DEJMPS protocol it turns out that we have polynomial scaling depending on the noise parameter $\alpha$, i.e. $\max_{\sigma_{AB}} \left\|({\mathcal{E}}-{\mathcal{F}})\left(\sigma^{\otimes n}_{AB} \right) \right\|_1 \leq \varepsilon_{\text{D}}(n) \leq O(n^{-b_{\mathrm{D}}(\alpha)})$, see (\[eq:local-iid\]). However, the protocol needs to converge sufficiently quickly, as the post-selection technique incurs a multiplicative increase in the effective distance between real and ideal protocols, which scales as a (15 degree) polynomial in $n$, see (\[eq:post:redudction\]). The resulting confidentiality level scales therefore as $O(n^{15-b_{\mathrm{D}}(\alpha)/4})$, which leads to an acceptable noise level that is rather low, e.g. about $10^{-19}$ for the DEJMPS protocol in the setting of binary pairs [^5], see \[sec:det:l\]. This very low rate is due to the polynomial factor introduced by applying the post-selection technique, i.e. $g_{n,d}$ in (\[eq:post:redudction\]) with $d=4$. Observe that these small rates are determined by properties of recurrence-type entanglement distillation protocols, i.e. $b(\alpha)$ for the recurrence-type entanglement distillation protocols studied here, and may be improved by either considering hashing-type protocols [@BennettHash] or through fault-tolerant constructions. Indeed, the noise threshold for fault-tolerant quantum computation also applies to this case, yielding a tolerable noise level of about $10^{-4}$. We reiterate that the post-selection technique is not applicable to the BBPSSW protocol, due to its slow convergence.
Confidentiality of entanglement distillation protocols when the noise transcripts leak {#sec:conf:2}
--------------------------------------------------------------------------------------
Finally, we provide confidentiality guarantees for entanglement distillation protocols when the noise transcripts are leaked to Eve. For that purpose, we relate the confidentiality criterion (\[eqn.confidentiality\]) for protocols where the noise transcripts are leaked to the earlier results. More formally, we have the following Theorem.
\[lem.leaklesstoleak\] Let ${\mathcal{E}}$ be the real protocol and ${\mathcal{F}}$ be the ideal protocol satisfying the assumptions of Lemma \[thm.globalclosenessfactor\]. Furthermore, let ${\mathcal{E}}^l$ denote the real and ${\mathcal{F}}^l$ the ideal protocol when the noise transcripts leak to Eve. Then $$\begin{aligned}
\|({\mathcal{E}}^l \otimes {id}_E - {\mathcal{F}}^l \otimes {id}_E) (\dm{\psi}) \|_1 \leq 2 \sqrt{\varepsilon(n)}\end{aligned}$$ for all purifications $\ket{\psi}_{ABE}$ of initial state $\rho_{AB}$ consisting of $n$ systems.
The proof, see \[app:sec:leak\], uses the unitary equivalence of purifications. Theorem \[lem.leaklesstoleak\] establishes via (\[eq.leakreduction\]) that if an entanglement distillation protocol is $\varepsilon-$confidential according to Definition \[def.confidentiality\] then the protocol is $2\sqrt{\varepsilon}-$confidential if the noisy apparatus leaks the noise transcripts.
Discussion {#sec:disc}
==========
We have shown that recurrence-type entanglement distillation protocols ensure private entanglement without referring to the asymptotic limit. This holds true even when the local devices are noisy, and when the potential eavesdropper is able to completely monitor the operation of these devices in run-time (i.e., the noisy apparatus leaks information about the realized noise processes). If the noise transcripts are not leaked, Eve is “factored out” – in tensor product with Alice and Bob, and only classically correlated otherwise. Our protocol can, for instance, be used to realize confidential quantum channels by means of teleportation - the only information that may leak to Eve after teleportation is which noise map was applied to the sent state, but nothing about the state itself (see \[app:sqc\] for details). More generally, our results imply the confidentiality of the protocols in arbitrary settings (beyond the application to quantum channels), thus opening the way for the confidential realization of various quantum tasks: from establishing quantum channels and quantum networks, to applications such as distributed quantum computation. Aside from cryptographic aspects, the proposed protocol can be used to generate high quality entanglement from non-iid sources.
**Acknowledgments:\
** We acknowledge the support by the Austrian Science Fund (FWF) through the SFB FoQuS F 4012 and project P28000-N27. AP and VD are grateful to Christopher Portmann for useful discussions, comments, and advice concerning technical aspects of this work.
Entanglement distillation for i.i.d. inputs {#app:sec:eppiid}
===========================================
The DEJMPS protocol {#sec.sup.dejmps}
-------------------
We first provide an overview of the DEJMPS protocol [@bib.dejmps] and then extend the description incrementally to our proposed setting (including L and Eve).
The DEJMPS protocol is a recurrence-type entanglement distillation protocol which combines several noisy copies of a mixed state $\rho$ to distill a state arbitrarily close to the maximally entangled state $\ket{B_{00}}{}$, where $\ket{B_{ij}} = ({id}\otimes {\sigma_x}^j {\sigma_z}^i) (\ket{00} + \ket{11})/\sqrt{2}$ for $i \in \lbrace 0,1 \rbrace$ and $j \in \lbrace 0,1 \rbrace$, provided that the fidelity $F=\bra{B_{00}}{} \rho \ket{B_{00}}{}$ satisfies $F > 1/2$ for the noiseless case. If the apparatus is noisy, then the minimal required fidelity $F$ needs to satisfy $F > F_{min}$ (where $F_{min}$ depends on the noise level of the apparatus) to achieve distillation. For more details on recurrence-type entanglement distillation protocols in general we refer the interested reader to [@bib.reviewepp]. A basic step of the DEJMPS protocol is as follows:
Input state of Alice and Bob: $\rho^{(a_1, b_1)} \otimes \rho^{(a_2, b_2)}$ \[enu.oxford.step1\] Alice and Bob apply the local basis change $U_x = e^{-i \pi/4 {\sigma_x}^{(a_1)}} \otimes e^{i \pi/4 {\sigma_x}^{(b_1)}} \otimes e^{-i \pi/4 {\sigma_x}^{(a_2)}} \otimes e^{i \pi/4 {\sigma_x}^{(b_2)}}$: $$\begin{aligned}
U_x \left(\rho^{(a_1, b_1)} \otimes \rho^{(a_2, b_2)}\right) U_x^\dagger.
\end{aligned}$$ \[enu.oxford.bcnot\] Alice and Bob apply a bilateral CNOT (BCNOT): $$\begin{aligned}
\left(\text{CNOT}_{a_1 \to a_2} \otimes \text{CNOT}_{b_1 \to b_2} \right) \rho^{(a_1, b_1)} \otimes \rho^{(a_2, b_2)} \left(\text{CNOT}_{a_1 \to a_2} \otimes \text{CNOT}_{b_1 \to b_2} \right)^\dagger.
\end{aligned}$$ \[enu.oxford.meas\] Alice and Bob apply a $ \sigma_z^{(a_2)} = \sigma_z \otimes id$ and a $\sigma_z^{(b_2)} = id \otimes \sigma_z$ measurement \[enu.oxford.steplast\] Alice and Bob communicate their measurement outcomes, $z_a$ and $z_b$ respectively, over a classical authentic channel Alice and Bob keep the subsystems $a_1$ and $b_1$ of step \[enu.oxford.bcnot\] Alice and Bob discard the measured subsystems $a_2$ and $b_2$ Alice and Bob discard both pairs
Hence, we can write one basic distillation step of the DEJMPS protocol as the linear map $O_{\text{2-EPP}} (\rho \otimes \rho) = O'_{\text{2-EPP}} (\rho \otimes \rho) O'^\dagger_{\text{2-EPP}}$ where
$$\begin{aligned}
O'_{\text{2-EPP}} = \left(id_{a_1,b_1} \otimes P^{(a_2)}_{z} \otimes P^{(b_2)}_{z} \right) \left(\text{CNOT}_{a_1 \to a_2} \otimes \text{CNOT}_{b_1 \to b_2} \right) U_x\end{aligned}$$
modulo a normalization factor and where $P_{z} = {\left| z \right\rangle \left\langle z \right|_{}}, \, z \in \lbrace 0,1 \rbrace$ denotes the respective outcome of step \[enu.oxford.meas\] of Protocol \[protocol.rd\].
The basic step is applied to all initial pairs, which comprises one distillation round. This distillation round is iterated where output states of the previous round are used as inputs for the next round. So we summarize the DEJMPS protocol as follows:
Input state of Alice and Bob: $\bigotimes^{2^n}_{i = 1} \rho^{(a_i,b_i)}$ where $F = \bra{B_{00}}{} \rho^{(a_i,b_i)} \ket{B_{00}}{} > 1/2$ for all $i \in \lbrace 1,..,2^n \rbrace$ Apply Protocol \[protocol.rd\] to all pairs Use the outputs of the previous step as input for the next distillation round
We remind the reader that the recurrence relations of the protocol (i.e. update functions of the coefficients of an ensemble) are central for the convergence analysis of the DEJMPS protocol. For Bell-diagonal states, i.e. states of the form $$\begin{aligned}
\rho = p_{00} {\left| B_{00} \right\rangle \left\langle B_{00} \right|_{}} + p_{11} {\left| B_{11} \right\rangle \left\langle B_{11} \right|_{}} + p_{01} {\left| B_{01} \right\rangle \left\langle B_{01} \right|_{}} + p_{10} {\left| B_{10} \right\rangle \left\langle B_{10} \right|_{}}\end{aligned}$$ where $\sum_{ij} p_{ij} = 1, \, p_{ij} \geq 0$, a straightforward computation yields the recurrence relations for the DEJMPS protocol to be $$\begin{aligned}
\label{eqn.recurrencedejmps}
\tilde{p}_{00} = \frac{p_{00}^2 + p_{11}^2}{N}, \qquad \tilde{p}_{11} = \frac{2 p_{01} p_{10}}{N}, \notag \\
\tilde{p}_{01} = \frac{p_{01}^2 + p_{10}^2}{N}, \qquad \tilde{p}_{10} = \frac{2 p_{00} p_{11}}{N}\end{aligned}$$ where $N = (p_{00} + p_{11})^2 + (p_{01} + p_{10})^2$, see e.g. [@bib.dejmps].
In [@Macchiavello] it has been shown analytically that the recurrence relations (\[eqn.recurrencedejmps\]) converge towards a unique and attracting fixed point provided the initial fidelity with $\ket{B_{00}}$, $p_{00}$, is above $1/2$.
The recurrence relations of the DEJMPS protocol taking independent single qubit white noise, i.e. noise of the form $N \rho = f \rho + (1-f)/4 (\rho + {\sigma_x}\rho {\sigma_x}+ {\sigma_y}\rho {\sigma_y}+ {\sigma_z}\rho {\sigma_z})$ acting on each qubit of Alice into account, read far more complex. In the presence of noise we have strong numerical evidence that the DEJMPS protocol converges towards a unique and attracting fixed point depending on the noise level $f$ only.
$n$
From figure \[fig:dejmps\] we suggest a linear relationship between $\log \|\rho_{\text{fix}} - \rho_{n} \|_{1}$ (where $\rho_{\text{fix}}$ and $\rho_n$ denote the fixed point and the state after successfully completing $n$ distillation rounds respectively) and the number of successful distillation rounds $n$. We immediately observe that the slope only depends on the noise parameter $f$, i.e. we have that
$$\begin{aligned}
\log \|\rho_{\text{fix}} - \rho_{n} \|_{1} = a(f) - n b(f).\end{aligned}$$
Using $\log_2 N = n$, where $N$ denotes the number of input pairs, this implies $\|\rho_{\text{fix}} - \rho_{n} \|_{1} = e^{a(f)} e^{-b(f) \log_2 N} = a'(f) N^{-b'(f)}$, i.e. $\|\rho_{\text{fix}} - \rho_{n} \|_{1}$ scales as $F(N) \in O(N^{-b'(f)})$ as mentioned in the main text. Furthermore we numerically find that the function $b'(f)$ monotonically grows for $f \to 1$.
For two qubit correlated noise, we refer the reader to the analysis including L, as the fixed point and the scaling can be recovered from that analysis by tracing out the system of L.
### Detailed analysis including L {#sec:det:l}
We outline the remainder of this section as follows: First we derive the recurrence relations of the DEJMPS protocol in the most general setting, taking the noise applied by L into account as well as assuming that Eve receives the leaked noise transcripts of L. We use those recurrence relations in the next subsection to provide analytical results regarding the fixed point of the recurrence relations, where the inputs are binary pairs and L only applies either $id$ or ${\sigma_x}$ operators. We close the section with numerical results for general i.i.d. Bell-diagonal pairs and the most general noise maps of L.
For i.i.d. input states the state of each system subject to distillation at an intermediate distillation round of the DEJMPS protocol is of the form $\ket{\Psi}_{ABEL} = \sum_{i,j,k,l} P_{ijkl} \ket{B_{ij}}_{AB} \ket{kl}_{L} \ket{ijkl}_{E}$, where $P_{ijkl}$ are probability amplitudes, if we assume the noise is leaked to Eve after every distillation round. The system $AB$ models the pair of Alice and Bob, $L$ the system of L (where the content of the register corresponds to the effective noise introduced to $AB$) and $E$ the system of Eve. L applies the noise processes before a basic protocol step to the systems of Alice. Moreover, L keeps track of the effective noise introduced using its system in a sense we clarify later.
In the following we use the notation
$$\begin{aligned}
\sigma_{0,0} = id, \quad \sigma_{0,1} = {\sigma_x}, \quad \sigma_{1,0} = {\sigma_z}, \quad \sigma_{1,1} = {\sigma_y}\end{aligned}$$
for the four Pauli-operators. Furthermore we denote by superscripts in brackets particle labels and by superscripts without brackets the power of an operator.
L introduces the noise maps $U_{\alpha_1,\beta_1, \alpha_2, \beta_2} = U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2,\beta_2}$ where $U^{(a_k)}_{\alpha,\beta} = \sigma^{(a_k)}_{\alpha,\beta} \otimes \left(({\sigma_x}^\alpha) \otimes ({\sigma_x}^\beta)\right)^{(L_k)}$. We observe that applying the noise map $U_{\alpha_1,\beta_1, \alpha_2, \beta_2}$ might flip the contents of the registers $L_1$ and $L_2$ depending on the values of $\alpha_1,\beta_1, \alpha_2$ and $\beta_2$. This enables L to keep track of the noise introduced to a pair.
There are two approaches how L can apply the noise maps $U_{\alpha_1,\beta_1, \alpha_2, \beta_2}$: stochastically in terms of CPTP maps, or coherently in terms of unitaries acting on an enlarged Hilbert space. Here we assume the latter approach, but provide the analysis of the noisy DEJMPS protocol in terms of CPTP maps and purifications.
To show that these are equivalent, first suppose that L owns a register $H$ set to the state $\sum_{\alpha_1, \beta_1, \alpha_2, \beta_2} \sqrt{\tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2}} \ket{\alpha_1 \beta_1 \alpha_2 \beta_2}_{H}$ where $\tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2}$ are the probabilities of applying the respective noise map $U_{\alpha_1,\beta_1, \alpha_2, \beta_2}$. L uses the register $H$ to apply the noise maps $U_{\alpha_1,\beta_1, \alpha_2, \beta_2}$ coherently controlled to the input state $\ket{\Psi}_{ABEL}$. We observe that tracing out $H$ after applying all the noise maps $U_{\alpha_1,\beta_1, \alpha_2, \beta_2}$ in a controlled fashion yields $$\begin{aligned}
\sum\limits_{\alpha_1, \beta_1, \alpha_2, \beta_2} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} U_{\alpha_1,\beta_1, \alpha_2, \beta_2} \left({\left| \Psi \right\rangle \left\langle \Psi \right|_{}} \otimes {\left| \Psi \right\rangle \left\langle \Psi \right|_{}}\right) U^\dagger_{\alpha_1,\beta_1, \alpha_2, \beta_2}.\end{aligned}$$ On the other hand, assume that L applies the noise process in terms of a CPTP map $N$, i.e. $$\begin{aligned}
N \rho = \sum\limits_{\alpha_1, \beta_1, \alpha_2, \beta_2} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} U_{\alpha_1,\beta_1, \alpha_2, \beta_2} \left({\left| \Psi \right\rangle \left\langle \Psi \right|_{}} \otimes {\left| \Psi \right\rangle \left\langle \Psi \right|_{}}\right) U^\dagger_{\alpha_1,\beta_1, \alpha_2, \beta_2}.\end{aligned}$$ We observe that $N \rho$ will be, in general, a mixed state, thus there exists a purification on a larger Hilbert space. As all purifications are unitarily equivalent, see e.g. [@Nielsen], we choose the purification $$\begin{aligned}
\ket{\Phi} = \sum\limits_{\alpha_1, \beta_1, \alpha_2, \beta_2} \sqrt{\tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2}} U_{\alpha_1,\beta_1, \alpha_2, \beta_2} \ket{\Psi} \otimes \ket{\Psi} \otimes \ket{\alpha_1 \beta_1 \alpha_2 \beta_2}_{H}.\end{aligned}$$ Hence ${\text{tr}_{H}\left[ \dm{\Phi} \right]} = N \rho$. Furthermore, we observe that the pure state $\ket{\Phi}$ can be generated by applying the unitaries $U_{\alpha_1,\beta_1, \alpha_2, \beta_2}$, coherently controlled by the register $H$, to $\ket{\Psi} \otimes \ket{\Psi} \otimes \left(\sum_{\alpha_1, \beta_1, \alpha_2, \beta_2} \sqrt{\tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2}} \ket{\alpha_1 \beta_1 \alpha_2 \beta_2}_{H} \right)$.
This equivalence allows us to assume that L introduces the noise as a CPTP map, applying $U_{\alpha_1,\beta_1, \alpha_2, \beta_2}$ with respective probabilities $f_{\alpha_1, \beta_1, \alpha_2, \beta_2}$ and purifying the state after the basic distillation step is executed by Alice and Bob.
Since the noise of L is applied before the basic distillation step is executed by Alice and Bob, the result of one noisy distillation step reads as $$\begin{aligned}
\label{eqn.overall}
\rho' & = \sum\limits_{\alpha_1, \beta_1, \alpha_2, \beta_2} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} U_u O'_{\text{2-EPP}} (U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2}) \left({\left| \Psi \right\rangle \left\langle \Psi \right|_{}} \otimes {\left| \Psi \right\rangle \left\langle \Psi \right|_{}}\right) (U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2})^\dagger O'^\dagger_{\text{2-EPP}} U_u^\dagger\end{aligned}$$ which needs finally to be purified.
In order to evaluate (\[eqn.overall\]), we proceed as follows:
- Step 1: We first compute $$\begin{aligned}
O'_{\text{2-EPP}} (U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2}) \ket{\Psi} \otimes \ket{\Psi}.\end{aligned}$$ which corresponds to the state after the noise map $U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2}$ is applied by L and the basic distillation step of the entanglement distillation protocol is executed by Alice and Bob.
- Step 2: We apply the unitary $U_u$, which acts only on L’s systems and whose purpose we clarify later, to the previous equality.
- Step 3: We have to determine the purification held by Eve if the noise is leaked to her. In doing so, we trace out Eve and then provide her with the purification of the resulting state (which corresponds to leaking the noise transcripts to Eve).
We observe that applying the noise map $U^{(a_1)}_{\alpha,\beta}$ to $\ket{\Psi}{}$ yields $$\begin{aligned}
\label{eqn.proofrec1}
U^{(a_1)}_{\alpha,\beta} \ket{\Psi}{} & = U^{(a_1)}_{\alpha,\beta} \sum\limits_{i,j,k,l} P_{ijkl} \ket{B_{ij}}_{AB} \ket{kl}_{L} \ket{ijkl}_{E} \\ \notag
& = \sum\limits_{i,j,k,l} P_{ijkl} \ket{B_{(i \oplus \alpha)(j \oplus \beta)}}_{AB} \ket{(k \oplus \alpha)(l \oplus \beta)}_{L} \ket{ijkl}_{E} \\ \notag
& = \sum\limits_{i,j,k,l} P_{(i \oplus \alpha)(j \oplus \beta)(k \oplus \alpha)(l \oplus \beta)} \ket{B_{ij}}_{AB} \ket{kl}_{L} \ket{(i \oplus \alpha)(j \oplus \beta)(k \oplus \alpha)(l \oplus \beta)}_{E}.\end{aligned}$$ This observation suggests the following notational simplifications: $$\begin{aligned}
P^{\alpha\beta}_{ijkl} = P_{(i \oplus \alpha)(j \oplus \beta)(k \oplus \alpha)(l \oplus \beta)} \quad \text{ and } \quad \ket{e^{\alpha\beta}_{ijkl}}_{E} = \ket{(i \oplus \alpha)(j \oplus \beta)(k \oplus \alpha)(l \oplus \beta)}_{E}.\end{aligned}$$ Using this notation we rewrite (\[eqn.proofrec1\]) as $U^{(a_1)}_{\alpha,\beta} \ket{\Psi}{} = \sum_{i,j,k,l} P^{\alpha\beta}_{ijkl} \ket{B_{ij}}_{AB} \ket{kl}_{L} \ket{e^{\alpha\beta}_{ijkl}}_{E}$. This is the state of Alice, Bob, L, and Eve after the noise map $U^{(a_1)}_{\alpha,\beta}$ is applied by L to the first pair. In order to compute (\[eqn.overall\]) we define $$\begin{aligned}
\ket{\Psi''_{\alpha_1, \beta_1, \alpha_2, \beta_2}}{} & = (U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2}) \ket{\Psi}{} \ket{\Psi}{} \\
& = \sum\limits_{i_1, j_1, i_2, j_2} \sum\limits_{k_1, l_1, k_2, l_2} A^{\alpha_1\beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 j_2 k_2 l_2} \ket{B_{i_1 j_1}}_{AB_1} \ket{B_{i_2 j_2}}_{AB_2} \ket{k_1 l_1}_{L_1} \ket{k_2 l_2}_{L_2} \\
& \hspace*{0.4cm} \otimes \ket{e^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1}}_{E_1} \ket{e^{\alpha_2 \beta_2}_{i_2 j_2 k_2 l_2}}_{E_2}\end{aligned}$$ which corresponds to the state after the noise map $U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2}$ is applied and $$\begin{aligned}
\label{eqn.proofrec00}
\ket{\Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2}}{} = U_u O_{\text{2-EPP}} \ket{\Psi''_{\alpha_1, \beta_1, \alpha_2, \beta_2}}{}\end{aligned}$$ which is the state after the noise map $U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2}$, one basic distillation step and the update of L’s noise register by $U_u$. Thus we rewrite (\[eqn.overall\]) as $$\begin{aligned}
\label{eqn.proofrec0}
\rho' = \sum\limits_{\alpha_1, \beta_1, \alpha_2, \beta_2} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} {\left| \Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right\rangle \left\langle \Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right|_{}}.\end{aligned}$$ According to (\[eqn.proofrec00\]) Alice and Bob apply one basic distillation step of the DEJMPS protocol to the state $\ket{\Psi''_{\alpha_1, \beta_1, \alpha_2, \beta_2}}{}$. Recall that step \[enu.oxford.step1\] of Protocol \[protocol.rd\] maps $\ket{B_{ij}}{}$ to $\ket{B_{i (i \oplus j)}}{}$ and that step \[enu.oxford.bcnot\] maps $\ket{B_{ij}}{} \ket{B_{i'j'}}{}$ to $\ket{B_{(i \oplus i')j}}{} \ket{B_{i'(j \oplus j')}}{}$. Thus we conclude that after step \[enu.oxford.step1\] and \[enu.oxford.bcnot\] of Protocol \[protocol.rd\] the state of Alice, Bob, L, and Eve is $$\begin{aligned}
\label{eqn.proofrec2}
\sum\limits_{i_1, j_1, i_2, j_2} \sum\limits_{k_1, l_1, k_2, l_2} P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 j_2 k_2 l_2} & \ket{B_{(i_1 \oplus i_2)(i_1 \oplus j_1)}}_{AB_1} \ket{B_{i_2 (i_1 \oplus j_1 \oplus i_2 \oplus j_2)}}_{AB_2} \ket{k_1 l_1}_{L_1} \ket{k_2 l_2}_{L_2} \notag \\
& \ket{e^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1}}_{E_1} \ket{e^{\alpha_2 \beta_2}_{i_2 j_2 k_2 l_2}}_{E_2}\end{aligned}$$ Following Protocol \[protocol.rd\], a ${\sigma_z}$-measurement of the target pair of the BCNOT, i.e. the subsystem $AB_2$, is applied to (\[eqn.proofrec2\]). Next Alice and Bob communicate their respective measurement outcomes over a classic authentic channel. If the measurement outcomes coincide, Alice and Bob keep the source pair, i.e. subsystem $AB_1$ of step \[enu.oxford.bcnot\], else they discard both subsystems $AB_1$ and $AB_2$. We assume that both measurements yield the outcome $1$. If both measurement outcomes yield $0$, no phase factor $(-1)^{i_2}$ would be required in the expression (\[eqn.proofrec3\]). The coinciding measurement outcomes imply $i_1 \oplus j_1 \oplus i_2 \oplus j_2 = 0$. To summarize, the state post-selected on the measurement outcomes $1$ of Alice and Bob is $$\begin{aligned}
\label{eqn.proofrec3}
& \sum\limits_{i_1, j_1, i_2, j_2} \sum\limits_{k_1, l_1, k_2, l_2} (-1)^{i_2} P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2} \ket{B_{(i_1 \oplus i_2)(i_1 \oplus j_1)}}_{AB_1} \ket{k_1 l_1}_{L_1} \ket{k_2 l_2}_{L_2} \ket{e^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1}}_{E_1} \ket{e^{\alpha_2 \beta_2}_{i_2(i_1 \oplus j_1 \oplus i_2) k_2 l_2}}_{E_2}.\end{aligned}$$ Recall that L stores in its register attached to the pair of Alice and Bob the effective noise introduced. For that purpose we introduce the unitary $U_u$ as well as an ancilla system $L_3$ set to the state $\ket{00}_{L_3}$. Applying $U_u$ to all three registers of L yields $U_u \ket{00}{}\ket{i}{}\ket{j}{} \ket{i'}{}\ket{j'}{} = \ket{u(i,j,i',j')}{}\ket{i}{}\ket{j}{} \ket{i'}{}\ket{j'}{}$ where $u$ is the so called flag update function defined in [@bib.aschauer]. The function $u$ returns the effective noise introduced on the source pair of step \[enu.oxford.bcnot\] of Protocol \[protocol.rd\]. Applying $U_u$ to (\[eqn.proofrec3\]) gives $$\begin{aligned}
\ket{\Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2}}{} = \sum\limits_{i_1, j_1, i_2, j_2} \sum\limits_{k_1, l_1, k_2, l_2} (-1)^{i_2} P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2} & \ket{B_{(i_1 \oplus i_2)(i_1 \oplus j_1)}}_{AB_1} \ket{k_1 l_1}_{L_1} \ket{k_2 l_2}_{L_2} \ket{u(k_1, l_1, k_2, l_2)}_{L_3} \\ \notag
& \otimes \ket{e^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1}}_{E_1} \ket{e^{\alpha_2 \beta_2}_{i_2(i_1 \oplus j_1 \oplus i_2) k_2 l_2}}_{E_2}.\end{aligned}$$ We remind the reader that $\ket{\Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2}}{}$ is the state after the application of i) the noise map $U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2}$, ii) a basic distillation step, and iii) the update of L’s noise register by $U_u$.
Since the noise transcripts - by assumption for this analysis - leak to Eve, we attribute the systems $L_1$ and $L_2$ to Eve. In order to treat the most general situation, we assume that Eve holds a purification of ${\text{tr}_{L_1,L_2,E_1,E_2}\left[ \rho' \right]}$. We determine this purification by computing $\rho'_{1} = {\text{tr}_{L_1,L_2}\left[ \rho' \right]}$ and $\rho'_{2} = {\text{tr}_{E_1,E_2}\left[ \rho'_{1} \right]}$ and attribute the purification of $\rho'_{2}$ to Eve.
By the linearity of the partial trace we have $$\begin{aligned}
\rho'_{1} &= {\text{tr}_{L_1,L_2}\left[ \rho' \right]} = \sum\limits_{\alpha_1, \beta_1, \alpha_2, \beta_2} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} {\text{tr}_{L_1,L_2}\left[ {\left| \Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right\rangle \left\langle \Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right|_{}} \right]}.\end{aligned}$$ It is useful to define $\rho'_{\alpha_1, \beta_1, \alpha_2, \beta_2} = {\text{tr}_{L_1,L_2}\left[ {\left| \Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right\rangle \left\langle \Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right|_{}} \right]}$ which evaluates to $$\begin{aligned}
\rho'_{\alpha_1, \beta_1, \alpha_2, \beta_2} &= {\text{tr}_{L_1,L_2}\left[ {\left| \Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right\rangle \left\langle \Psi'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right|_{}} \right]} \\
& = \sum (-1)^{i_2 \oplus i'_2} P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2} (P^{\alpha_1 \beta_1}_{i'_1 j'_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i'_2 (i'_1 \oplus j'_1 \oplus i'_2) k_2 l_2})^{*} {\left| B_{(i_1 \oplus i_2)(i_1 \oplus j_1)} \right\rangle \left\langle B_{(i'_1 \oplus i'_2)(i'_1 \oplus j'_1)} \right|_{}} \\
& \hspace*{0.4cm} \otimes {\left| u(k_1, l_1, k_2, l_2) \right\rangle \left\langle u(k_1, l_1, k_2, l_2) \right|_{}} \otimes {\left| e^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} \right\rangle \left\langle e^{\alpha_1 \beta_1}_{i'_1 j'_1 k_1 l_1} \right|_{}} \otimes {\left| e^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2} \right\rangle \left\langle e^{\alpha_2 \beta_2}_{i'_2 (i'_1 \oplus j'_1 \oplus i'_2) k_2 l_2} \right|_{}}.\end{aligned}$$ In the previous expression we neglected the indices appearing in the sum for simplicity, but it is understood that the sum ranges over all indices except $\alpha_1, \beta_1, \alpha_2$ and $\beta_2$.
In order to determine the state of Alice, Bob, and L which Eve finally purifies we have to compute $\rho'_{2} = {\text{tr}_{E_1,E_2}\left[ \rho'_{1} \right]}$. Again, the linearity of the partial trace yields $$\begin{aligned}
\label{eqn.proofrec6}
\rho'_{2} = {\text{tr}_{E_1,E_2}\left[ \rho'_{1} \right]} = \sum\limits_{\alpha_1, \beta_1, \alpha_2, \beta_2} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} {\text{tr}_{E_1,E_2}\left[ \rho'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right]}.\end{aligned}$$ We remind the reader that $\ket{e^{\alpha \beta}_{i j k l}}_{E_1} = \ket{(i \oplus \alpha)(j \oplus \beta)(k \oplus \alpha)(l \oplus \beta)}_{E_1}$. Hence, for fixed $\alpha_1$ and $\beta_1$, we have $\tr{{\left| e^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} \right\rangle \left\langle e^{\alpha_1 \beta_1}_{i'_1 j'_1 k_1 l_1} \right|_{}}} = \delta_{i_1 i'_1} \delta_{j_1 j'_1}$, which implies that $i'_1 = i_1$ and $j'_1 = j_1$. Thus, we also have $$\begin{aligned}
\tr{{\left| e^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2} \right\rangle \left\langle e^{\alpha_2 \beta_2}_{i'_2 (i'_1 \oplus j'_1 \oplus i'_2) k_2 l_2} \right|_{}}} = \tr{{\left| e^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2} \right\rangle \left\langle e^{\alpha_2 \beta_2}_{i'_2 (i_1 \oplus j_1 \oplus i'_2) k_2 l_2} \right|_{}}} = \delta_{i_2 i'_2}.\end{aligned}$$ Hence $$\begin{aligned}
\label{eqn.proofrec5}
{\text{tr}_{E_1,E_2}\left[ \rho'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right]} &= \notag \\[3ex]
&= \sum\limits_{i_1, i_2,j_1} \sum\limits_{k_1, l_1, k_2, l_2} P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2} (P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2})^{*} \notag \\
& \hspace*{0.4cm} {\left| B_{(i_1 \oplus i_2) (i_1 \oplus j_1)} \right\rangle \left\langle B_{(i_1 \oplus i_2) (i_1 \oplus j_1)} \right|_{}} \otimes {\left| u(k_1, l_1, k_2, l_2) \right\rangle \left\langle u(k_1, l_1, k_2, l_2) \right|_{}} \notag \\[3ex]
&= \sum\limits_{i_1, i_2,j_1} \sum\limits_{k_1, l_1, k_2, l_2} \left|P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2} \right|^2 \notag \\
& \hspace*{0.4cm} {\left| B_{(i_1 \oplus i_2)(i_1 \oplus j_1)} \right\rangle \left\langle B_{(i_1 \oplus i_2)(i_1 \oplus j_1)} \right|_{}} \otimes {\left| u(k_1, l_1, k_2, l_2) \right\rangle \left\langle u(k_1, l_1, k_2, l_2) \right|_{}}.\end{aligned}$$ By inserting (\[eqn.proofrec5\]) in (\[eqn.proofrec6\]) we get $$\begin{aligned}
\rho'_{2} & = {\text{tr}_{E_1,E_2}\left[ \rho'_{1} \right]} \\[3ex]
&= \sum\limits_{\alpha_1, \beta_1, \alpha_2, \beta_2} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} {\text{tr}_{E_1,E_2}\left[ \rho'_{\alpha_1, \beta_1, \alpha_2, \beta_2} \right]} \\[3ex]
& = \sum\limits_{\alpha_1, \beta_1, \alpha_2, \beta_2} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} \sum\limits_{i_1, i_2,j_1} \sum\limits_{k_1, l_1, k_2, l_2} \left|P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2} \right|^2 \\
& \hspace*{0.4cm} {\left| B_{(i_1 \oplus i_2)(i_1 \oplus j_1)} \right\rangle \left\langle B_{(i_1 \oplus i_2)(i_1 \oplus j_1)} \right|_{}} \otimes {\left| u(k_1, l_1, k_2, l_2) \right\rangle \left\langle u(k_1, l_1, k_2, l_2) \right|_{}} \\[3ex]
& = \sum_{i_1, i_2, j_1}{\left| B_{(i_1 \oplus i_2)(i_1 \oplus j_1)} \right\rangle \left\langle B_{(i_1 \oplus i_2)(i_1 \oplus j_1)} \right|_{}} \otimes \sum_{\gamma_0, \gamma_1} \left(\sum_{\stackrel{\alpha_1, \beta_1, \alpha_2, \beta_2, k_1, l_1, k_2, l_2}{u(k_1, l_1, k_2, l_2) = (\gamma_0, \gamma_1)}} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} \left|P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2}\right|^2 \right) \\
& \hspace*{0.4cm} {\left| \gamma_0 \gamma_1 \right\rangle \left\langle \gamma_0 \gamma_1 \right|_{}}.\end{aligned}$$ Rearranging the sum over $i_1, i_2$ and $j_1$ in the previous equation gives $$\begin{aligned}
\label{eqn.proofrec7}
\sum_{\delta_0, \delta_1}{\left| B_{\delta_0 \delta_1} \right\rangle \left\langle B_{\delta_0 \delta_1} \right|_{}} \otimes \sum_{\gamma_0, \gamma_1} & \left(\sum_{\stackrel{i_1, i_2, j_1}{i_1 \oplus i_2 = \delta_0, i_1 \oplus j_1 = \delta_1}}\sum_{\stackrel{\alpha_1, \beta_1, \alpha_2, \beta_2, k_1, l_1, k_2, l_2}{u(k_1, l_1, k_2, l_2) = (\gamma_0, \gamma_1)}} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} \left|P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2}\right|^2 \right) \\ \notag
& {\left| \gamma_0 \gamma_1 \right\rangle \left\langle \gamma_0 \gamma_1 \right|_{}}.\end{aligned}$$ Using the definition $$\begin{aligned}
\label{eq.proof.recurrencerelation}
|\tilde{P}_{\delta_0 \delta_1 \gamma_0 \gamma_1}|^2 = \sum_{\stackrel{i_1, i_2, j_1}{i_1 \oplus i_2 = \delta_0, i_1 \oplus j_1 = \delta_1}}\sum_{\stackrel{\alpha_1, \beta_1, \alpha_2, \beta_2, k_1, l_1, k_2, l_2}{u(k_1, l_1, k_2, l_2) = (\gamma_0, \gamma_1)}} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} \left| P^{\alpha_1 \beta_1}_{i_1 j_1 k_1 l_1} P^{\alpha_2 \beta_2}_{i_2 (i_1 \oplus j_1 \oplus i_2) k_2 l_2}\right|^2\end{aligned}$$ where $\delta_0, \delta_1, \gamma_0, \gamma_1 \in \lbrace 0,1 \rbrace$ and omitting the normalization factor for clarity, (\[eqn.proofrec7\]) simplifies to $$\begin{aligned}
\sum_{\delta_0, \delta_1}{\left| B_{\delta_0 \delta_1} \right\rangle \left\langle B_{\delta_0 \delta_1} \right|_{}} \otimes \sum_{\gamma_0, \gamma_1} |\tilde{P}_{\delta_0 \delta_1 \gamma_0 \gamma_1}|^2 {\left| \gamma_0 \gamma_1 \right\rangle \left\langle \gamma_0 \gamma_1 \right|_{}}\end{aligned}$$ which is the state of Alice, Bob, and L after one noisy distillation step. Since this final state is purified by Eve with the leaked noise transcripts and all purifications are unitarily equivalent, the state of Alice, Bob, L, and Eve after one noisy distillation step can be written without loss of generality as $$\begin{aligned}
\ket{\psi^{DEJMPS}} = \sum\limits_{\delta_0, \delta_1, \gamma_0, \gamma_1} \tilde{P}_{\delta_0, \delta_1, \gamma_0, \gamma_1} \ket{B_{\delta_0,\delta_1}}_{AB} \ket{\gamma_0 \gamma_1}_{L} \ket{\delta_0 \delta_1 \gamma_0 \gamma_1}_{E}.\end{aligned}$$ This also implies that (\[eq.proof.recurrencerelation\]) are the recurrence relations of the noisy DEJMPS protocol.
First we study the scaling of the systems of Alice, Bob, and L and extend those results then to the (possibly leaked) noise transcripts of Eve in terms of purifications.
Suppose that the initial i.i.d. pairs of Alice and Bob are mixtures of $\ket{B_{00}}$ and $\ket{B_{01}}$ and that L applies either the identity or a ${\sigma_x}$-operator with respective probabilities $\tilde{f}_0$ and $\tilde{f}_1 = 1-\tilde{f}_0$ independently to each pair. We remind the reader that Eve purifies the state of Alice, Bob, and L with the leaked noise transcripts, i.e. each individual state taking Eve into account at an intermediate round of the DEJMPS protocol reads as $\sum_{i,j} P_{ij} \ket{B_{0i}}_{AB} \otimes \ket{\eta_j}_{L} \otimes \ket{\eta_{ij}}_{E}$. Using $p_{ij} = |P_{ij}|^2$, the recurrence relations (\[eq.proof.recurrencerelation\]) for the setting we are concerned with here simplify to $$\begin{aligned}
\label{eq.recurrencerelation.bin1}
\tilde{p}_{00} & = 1/N(\tilde{f}^2_0 \left( p^2_{00} + 2 p_{00} p_{01} \right) + \tilde{f}^2_1 \left(p^2_{11} + 2 p_{10} p_{11} \right) + 2 \tilde{f}_0 \tilde{f}_1 \left(p_{11} p_{00} + p_{10} p_{00} + p_{11} p_{01} \right)), \\
\tilde{p}_{01} & = 1/N(\tilde{f}^2_0 p^2_{01} + 2 \tilde{f}_0 \tilde{f}_1 p_{10} p_{01} + \tilde{f}^2_1 p^2_{10}), \\
\tilde{p}_{10} & = 1/N(\tilde{f}^2_0 \left( p^2_{10} + 2 p_{10} p_{11} \right) + \tilde{f}^2_1 \left(p^2_{01} + 2 p_{00} p_{01} \right) + 2 \tilde{f}_0 \tilde{f}_1 \left(p_{01} p_{10} + p_{00} p_{10} + p_{01} p_{11} \right)), \\ \label{eq.recurrencerelation.bin4}
\tilde{p}_{11} & = 1/N(\tilde{f}^2_0 p^2_{11} + 2 \tilde{f}_0 \tilde{f}_1 p_{00} p_{11} + \tilde{f}^2_1 p^2_{00}).\end{aligned}$$ where $N = (\tilde{f}^2_0 + \tilde{f}^2_1)((p_{00} + p_{01})^2 + (p_{10} + p_{11})^2) + 4 \tilde{f}_0 \tilde{f}_1 (p_{00}+p_{01})(p_{10}+p_{11})$. In the following we denote the recurrence relations (\[eq.recurrencerelation.bin1\])–(\[eq.recurrencerelation.bin4\]) by the vector-valued mapping $\mathbf{f}$, i.e. $\mathbf{p} \stackrel{\mathbf{f}}{\to} \mathbf{\tilde{p}},$ where $\mathbf{p}=(p_{00}, p_{01}, p_{10}, p_{11})$. A simple computation yields the following fixed points of $\mathbf{f}$: $$\begin{aligned}
\label{eq.fixedpoint.bin}
& p^\infty_{00} = 1/2 + \sqrt{4 \tilde{f}_0 - 3}/(4\tilde{f}_0 -2) \quad p^\infty_{01} = p^\infty_{10} = 0 \quad p^\infty_{11} = 1-p^\infty_{00}, \\
& p^\infty_{00} = 1/2 - \sqrt{4 \tilde{f}_0 - 3}/(4\tilde{f}_0 -2) \quad p^\infty_{01} = p^\infty_{10} = 0 \quad p^\infty_{11} = 1-p^\infty_{00}, \\
& p^\infty_{00}= p^\infty_{11} = 1/2 \quad p^\infty_{01} = p^\infty_{10} = 0.\end{aligned}$$ The parameter estimation phase guarantees that the fidelity $F$ with $\ket{B_{00}}$ is sufficiently high for distillation. Hence the fixed point of interested is (\[eq.fixedpoint.bin\]), i.e. $$\begin{aligned}
\label{eqn.bin.fixedpoints}
\mathbf{p}^\infty = (1/2 + \sqrt{4 \tilde{f}_0 - 3}/(4\tilde{f}_0 -2), 0, 0, 1/2 - \sqrt{4 \tilde{f}_0 - 3}/(4\tilde{f}_0 -2)).\end{aligned}$$ From (\[eqn.bin.fixedpoints\]) we observe that in the limit the ‘cross-probabilities’ $p_{01}$ and $p_{10}$, vanish, hence $L$ is fully correlated to $AB$.
It is of central importance, regarding convergence that the fixed point $\mathbf{p}^\infty$ is an attractor, as only this ensures convergence towards that fixed point. Note that $\mathbf{p}^\infty$ is an attractor if and only if the largest eigenvalue $\lambda_{max}$ of $\mathbf{f'(p^\infty)}$ satisfies $\lambda_{max} < 1$. We easily find that $\lambda_{max} = (\tilde{f}_0 \sqrt{4 \tilde{f}_0 - 3} - \tilde{f}_0)/(2 \tilde{f}_0 - 1) < 1$ for $0.78 \leq \tilde{f}_0 \leq 1$. The fixed point $\mathbf{p}^\infty$ enables us to determine the rate of convergence. For that purpose, we expand $\mathbf{f}$ in terms of its Taylor series around the fixed point $\mathbf{p}^\infty$, i.e. $\mathbf{\tilde{p}} = \mathbf{f}(\mathbf{p}) \approx \mathbf{f}(\mathbf{p}^\infty) + \mathbf{f'}(\mathbf{p}^\infty)(\mathbf{p} - \mathbf{p}^\infty).$ Hence by defining $\mathbf{e} = \mathbf{p} - \mathbf{p^\infty}$ we find $\mathbf{\tilde{e}} = \mathbf{f'(p^\infty)} \mathbf{e}$, providing an estimate of the error propagation for one successful distillation round. The state of Alice, Bob, and L after $n$ successful distillation rounds and at the fixpoint read as $\rho_n = \sum_{ij} p^{(n)}_{ij} \dm{B_{0i}}_{AB} \otimes \dm{\eta_j}_{L}$ and $\rho_{\text{fix}} = \sum_{i} p^\infty_{ii} \dm{B_{0i}}_{AB} \otimes \dm{\eta_i}_{L}$ respectively, which implies for their distance induced by the $1$-norm $$\begin{aligned}
\label{eq:scaling:binary:local}
\epsilon_{n} = \| \rho_n - \rho_{\text{fix}} \|_{1} = \left\| \sum_{i,j} (p^{(n)}_{ij} - p^\infty_{ij}) \dm{B_{0i}}_{AB} \otimes \dm{\eta_j}_{L} \right\|_{1} = \underbrace{\sum_{i,j} |p^{(n)}_{ij} - p^\infty_{ij}|}_{\|\mathbf{e_{n}} \|_{1;\text{v}}} \leq \|\mathbf{f'}(\mathbf{p^\infty})^{n-1} \| \| \mathbf{e_{1}}\|_{1;\text{v}}.\end{aligned}$$ where $\| \mathbf{x} \|_{1;\text{v}} = \sum^k_{i=1} |x_i|$ denotes the $1$-norm of vectors in ${\mathbb{C}}^k$.
Eq. (\[eq:scaling:binary:local\]) only concerns the systems of Alice, Bob, and L. To complete the analysis we recall that Eve purifies $\rho_{n}$ and $\rho_{\text{fix}}$ with the leaked noise transcripts of L. If we take this purifying system, $E$, into account, i.e. consider $\|\dm{\psi^{n}}_{ABEL} - \dm{\psi^\alpha}_{ABEL}\|_{1}$ where $\rho_n = {\text{tr}_{E}\left[ \dm{\psi^{n}}_{ABEL} \right]}$, $\ket{\psi^\alpha}_{ABEL} = \sum_{i,j} P^{\infty}_{ij} \ket{B_{0i}}_{AB} \otimes \ket{\eta_j}_{L} \otimes \ket{\eta_{ij}}_{E}$ with $|P^{\infty}_{ij}|^2 = p^\infty_{ij}$ and $\rho_{\text{fix}} = {\text{tr}_{E}\left[ \dm{\psi^\alpha}_{ABEL} \right]}$, we find $$\begin{aligned}
\|\dm{\psi^{n}}_{ABEL} - \dm{\psi^\alpha}_{ABEL}\|_{1} \leq \sqrt{\epsilon_{n}} \label{eq:scaling:bin:1}\end{aligned}$$ since purifications scale with a square root.
In order to apply the post-selection-based reduction, we need to relate the previously obtained results for i.i.d. input pairs to general ensembles. As stated in the main text, we exclude the parameter estimation step on $\sqrt{n}$ initial states for simplicity. We remind the reader, as we have stated in the main text, that for all purifications $\ket{\psi}_{ABE'}$ of a $n$-partite input state $\rho_{AB}$ we have $$\begin{aligned}
\label{eq:postselection:bin}
\| ({\mathcal{E}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}}) &- ({\mathcal{F}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}})\|_1 \leq 4 g_{n,d} \sqrt{\max_{\sigma_{AB}} \left\|({\mathcal{E}}_{\text{L}}-{\mathcal{F}}_{\text{L}})\left(\sigma^{\otimes n}_{AB} \right) \right\|_1}\end{aligned}$$ where $g_{n,d} = {n + d^2 -1 \choose n}$. Thus, inserting the previous result for $2^n$ i.i.d. input states (necessary to achieve $n$ rounds of distillation) in (\[eq:postselection:bin\]) yields $$\begin{aligned}
\|({\mathcal{E}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}}) &- ({\mathcal{F}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}})\|_1 \leq 4 g_{2^n,d} \epsilon_{n}^{1/4}.\end{aligned}$$ One square root in the expression above arises from inequality (\[eq:scaling:bin:1\]) and the other square root appears from inequality $(\ref{eq:postselection:bin})$.
Hence, for confidentiality we necessarily need $g_{2^n,d} \epsilon_{n}^{1/4} \to 0$ for $n \to \infty$. Thus $\epsilon_{n}^{1/4}$ should decay faster than $g_{2^n,d}$ grows in $n$. Numerical simulations suggest that, for $\tilde{f}_0 = 1-10^{-19}$, this turns out to be true, i.e. the post-selection-based reduction is applicable (see Figure \[fig:binpairspostselect\]). As stated in the main text such rates are unlikely to be achievable on the physical level, but they are, at least in principle, possible through fault-tolerant constructions.
$n$
In the following we show that the previous established results also hold true for the general i.i.d. setting where L applies all four Pauli operators and each individual pair is arbitrary. We remind the reader that the recurrence relations for states $\sum_{i,j,k,l} P_{ijkl} \ket{B_{ij}}_{AB} \otimes \ket{\eta_{kl}}_{L} \otimes \ket{\eta_{ijkl}}_{E}$ (i.e. Eve purifies $\rho_n = \sum_{i,j,k,l} |P_{ijkl}|^2 {\left| B_{ij} \right\rangle \left\langle B_{ij} \right|_{AB}} \otimes {\left| \eta_{kl} \right\rangle \left\langle \eta_{kl} \right|_{L}}$ with the leaked noise transcripts) read (by denoting $|P_{ijkl}|^2 = p_{ijkl}$) as $$\begin{aligned}
\tilde{p}_{\delta_0 \delta_1 \gamma_0 \gamma_1} = \sum_{\stackrel{i_1, i_2, j_1}{i_1 \oplus i_2 = \delta_0, i_1 \oplus j_1 = \delta_1}}\sum_{\stackrel{\alpha_1, \beta_1, \alpha_2, \beta_2, k_1, l_1, k_2, l_2}{u(k_1, l_1, k_2, l_2) = (\gamma_0, \gamma_1)}} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} p_{(i_1 \oplus \alpha_1) (j_1\oplus \beta_1) (k_1 \oplus \alpha_1) (l_1 \oplus \beta_1)} p_{(i_2 \oplus \alpha_2) (i_1 \oplus j_1 \oplus i_2 \oplus \beta_2) (k_2 \oplus \alpha_2) (l_2 \oplus \beta_2)}\end{aligned}$$ modulo the normalization factor $\sum_{\delta_0 \delta_1 \gamma_0 \gamma_1} \tilde{p}_{\delta_0 \delta_1 \gamma_0 \gamma_1}$.
For simplicity we assume independent single qubit white noise, i.e. $\tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} = \tilde{f}_{\alpha_1, \beta_1} \tilde{f}_{\alpha_2, \beta_2}$ as well as $\tilde{f}_{\alpha_1, \beta_1} = f$ if $\alpha_1=\beta_1=0$ and $(1-f)/3$ otherwise. Furthermore, we assume that the initial fidelity $F$ with $\ket{B_{00}}$ is sufficiently high for distillation. Numerically iterating the recurrence relations (which we again denote by $\mathbf{p} \stackrel{\mathbf{f}}{\to} \mathbf{\tilde{p}}$) reveal that, for a sufficiently large number of iterations, the ‘cross-probabilities’ vanish, i.e. $p^{\infty}_{ijkl} = 0 \Leftrightarrow i \neq k$ or $j \neq l$. Hence, to obtain a fixed point $\mathbf{p^{\infty}} = (p^{\infty}_{ijkl})^1_{i,j,k,l = 0}$ of $\mathbf{f}$, it is reasonable to assume that $p^{\infty}_{ijkl} =0 \Leftrightarrow i \neq k$ or $j \neq l$.
Thus the fixed point $\mathbf{p^{\infty}}$ is determined by four equations in four unknowns, namely the equations
$$\begin{aligned}
p_{\delta_0 \delta_1 \delta_0 \delta_1} = \frac{1}{N} \sum_{\stackrel{i_1, i_2, j_1}{i_1 \oplus i_2 = \delta_0, i_1 \oplus j_1 = \delta_1}}\sum_{\stackrel{\alpha_1, \beta_1, \alpha_2, \beta_2}{u(i_1, j_1, i_2, i_1 \oplus j_1 \oplus i_2) = (\delta_0, \delta_1)}} \tilde{f}_{\alpha_1, \beta_1} \tilde{f}_{\alpha_2, \beta_2} & p_{(i_1 \oplus \alpha_1) (j_1 \oplus \beta_1) (i_1 \oplus \alpha_1) (j_1 \oplus \beta_1)} \\
& \cdot p_{(i_2 \oplus \alpha_2) (i_1 \oplus j_1 \oplus i_2 \oplus \beta_2) (i_2 \oplus \alpha_2) (i_1 \oplus j_1 \oplus i_2 \oplus \beta_2)}.\end{aligned}$$
where $\delta_0, \delta_1 \in \lbrace 0,1 \rbrace$ and $N = \sum_{\delta_0, \delta_1} p_{\delta_0 \delta_1 \delta_0 \delta_1 }$. Figure \[fig:p0000:fixed\] illustrates the numerical estimate of $p^{\infty}_{0000}$ as a function of $f$.
$f$
Similar to the case of binary pairs, we can write the recurrence relations $\mathbf{f}$ in terms of its Taylor series expansion around the fixed point $\mathbf{p}^\infty$, i.e. $\mathbf{\tilde{p}} = \mathbf{f}(\mathbf{p}) \approx \mathbf{f}(\mathbf{p}^\infty) + \mathbf{f'}(\mathbf{p}^\infty)(\mathbf{p} - \mathbf{p}^\infty).$ Hence by defining $\mathbf{e} = \mathbf{p} - \mathbf{p^\infty}$ we have $\mathbf{\tilde{e}} = \mathbf{f'(p^\infty)} \mathbf{e}$, i.e. as for binary pairs, the error induced by the $1-$norm of the state of Alice, Bob, and L after $n$ successful distillation rounds satisfies $$\begin{aligned}
\| \rho_n - \rho_{\text{fix}} \|_{1} = \left\| \sum_{i,j,k,l} \left(p^{(n)}_{ijkl} - p^\infty_{ijkl} \right) \dm{B_{ij}}_{AB} \otimes \dm{\eta_{kl}}_{L} \right\|_{1} \leq \sum_{i,j,k,l} \left|p^{(n)}_{ijkl} - p^\infty_{ijkl} \right| \leq \|\mathbf{f'}(\mathbf{p^\infty})^{n-1} \| \| \mathbf{e_{1}}\|_{1;\text{v}}. \label{eq:scaling:general}\end{aligned}$$
$n$
Figure \[fig:specnormjac\] suggests a linear relationship between the number of successful distillation rounds $n$ and $\log \|\mathbf{f'}(\mathbf{p^\infty})^{n-1} \|$ for each noise level $f$, i.e. $b(f) n + a(f) = \log \|\mathbf{f'}(\mathbf{p^\infty})^{n-1} \|$. As the number $N$ of pairs necessary to achieve $n$ distillation rounds is $N = 2^n$ ($\Leftrightarrow n = \log_2 N$) we have $b(f) \log_2 N + a(f) = \log \|\mathbf{f'}(\mathbf{p^\infty})^{n-1} \|$, which is equivalent to $$\begin{aligned}
\|\mathbf{f'}(\mathbf{p^\infty})^{n-1} \| = e^{a(f)} e^{b(f) \log_2 N} = a'(f) N^{b'(f)}.\end{aligned}$$ Hence, $\|\mathbf{f'}(\mathbf{p^\infty})^{n-1} \|$ scales as $F(N) \in O(N^{b'(f)})$ where $b'(f) < 0$ and $b'(f)$ decays for $f \to 1$.
What is left to show, is that the fixed point $\mathbf{p^\infty}$ is an attracting fixed point. For that purpose we numerically compute the largest eigenvalue of $\mathbf{f'}(\mathbf{p^\infty})$, see Fig. \[fig:lambdamaxjac\], and observe that, for noise below $10^{-1}$, i.e. $1-f < 10^{-1}$, the largest eigenvalue $\lambda_{max}$ of $\mathbf{f'}(\mathbf{p^\infty})$ fulfills $\lambda_{max} < 1$, proving that $\mathbf{p^\infty}$ is an attracting fixed point.
$1-f$
This implies that, if the initial fidelity $F$ with $\ket{B_{00}}$ is sufficiently large for distillation, the DEJMPS protocol necessarily converges towards the fixed point $\mathbf{p^\infty}$ where the ‘cross-probabilities’ vanish.
The analysis so far still lacks Eve’s system $E$ for the leaked noise transcripts. Suppose $\ket{\psi^{n}}_{ABEL}$ and $\ket{\psi^f}_{ABEL}$ are purifications of $\rho_n$ and $\rho_{\text{fix}}$, i.e. $\rho_n = {\text{tr}_{E}\left[ \dm{\psi^{n}} \right]}$ and $\rho_{\text{fix}} = {\text{tr}_{E}\left[ \dm{\psi^f} \right]}$ respectively. This implies $\epsilon_{n} = \|\dm{\psi^{n}} - \dm{\psi^f}\|_{1} \leq \sqrt{F(N)}$, i.e. $\epsilon_{n} \in O(N^{b'(f)/2})$ which we also confirmed with our numeric results.
It is straightforward to extend the analysis above to two-qubit correlated noise introduced by L on the system of Alice and Bob. For that purpose we assume that $\tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} = \tilde{f} + (1-\tilde{f})/16$ if $\alpha_1=\beta_1=\alpha_2=\beta_2=0$ and $(1-\tilde{f})/16$ otherwise. Also in that case we numerically observe that $p^{\infty}_{ijkl} = 0 \Leftrightarrow i \neq k$ or $j \neq l$. Hence it is reasonable to assume that $p^{\infty}_{ijkl} =0 \Leftrightarrow i \neq k$ or $j \neq l$ in order to obtain a fixed point $\mathbf{p^{\infty}} = (p^{\infty}_{ijkl})^1_{i,j,k,l = 0}$ of $\mathbf{f}$.
The fixed point $\mathbf{p^{\infty}}$ is determined by four equations in four unknowns, namely the equations
$$\begin{aligned}
p_{\delta_0 \delta_1 \delta_0 \delta_1} = \frac{1}{N} \sum_{\stackrel{i_1, i_2, j_1}{i_1 \oplus i_2 = \delta_0, i_1 \oplus j_1 = \delta_1}}\sum_{\stackrel{\alpha_1, \beta_1, \alpha_2, \beta_2}{u(i_1, j_1, i_2, i_1 \oplus j_1 \oplus i_2) = (\delta_0, \delta_1)}} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} & p_{(i_1 \oplus \alpha_1) (j_1 \oplus \beta_1) (i_1 \oplus \alpha_1) (j_1 \oplus \beta_1)} \\
& \cdot p_{(i_2 \oplus \alpha_2) (i_1 \oplus j_1 \oplus i_2 \oplus \beta_2) (i_2 \oplus \alpha_2) (i_1 \oplus j_1 \oplus i_2 \oplus \beta_2)}.\end{aligned}$$
where $\delta_0, \delta_1 \in \lbrace 0,1 \rbrace$ and $N = \sum_{\delta_0, \delta_1} p_{\delta_0 \delta_1 \delta_0 \delta_1 }$. Figure \[fig:p0000:fixed:2qu\] illustrates the numerical estimate of $p^{\infty}_{0000}$ as a function of $\tilde{f}$.
$\tilde{f}$
Furthermore we numerically compute the largest eigenvalue of $\mathbf{f'}(\mathbf{p^\infty})$ and observe that if $\tilde{f} > 0.8284$, the largest eigenvalue $\lambda_{max}$ of $\mathbf{f'}(\mathbf{p^\infty})$ fulfills $\lambda_{max} < 1$, hence $\mathbf{p^\infty}$ is an attracting fixed point, see Fig. \[fig:lambdamaxjac\_2qu\].
$1-\tilde{f}$
Finally, we obtain again a linear relationship between the number of successful distillation rounds $n$ and $\log \|\mathbf{f'}(\mathbf{p^\infty})^{n-1} \|$ for each noise level $\tilde{f}$, i.e. $b_2(\tilde{f}) n + a_2(\tilde{f}) = \log \|\mathbf{f'}(\mathbf{p^\infty})^{n-1} \|$, see Fig. \[fig:specnormjac:2qu\]. This implies, similar to the case of single qubit white noise, that the right-hand-side of (\[eq:scaling:general\]) converges polynomial fast towards zero in terms of initial states. The rate of convergence is governed by $\tilde{f}$, i.e. $\| \rho_n - \rho_{\text{fix}} \|_{1} \leq F_2(N)$ where $F_2(N) \in O(N^{b_2(\tilde{f})})$ and $b_2(\tilde{f}) < 0$ with $b_2(\tilde{f})$ decays for $\tilde{f} \to 1$.
$n$
Taking the system of leaking noise transcripts into account,this implies that $\epsilon_{n} = \left\|\dm{\psi^{n}} - \dm{\psi^{\tilde{f}}} \right\|_{1} \leq \sqrt{F_2(N)}$, i.e. $\epsilon_{n} \in O(N^{b_2(\tilde{f})/2})$.
To conclude the analysis, we now show that the noise model of two-qubit depolarizing noise is actually sufficient to cover [*any*]{} noise process for two-qubit operations. This is the case because for any CNOT-type gate (which we need to apply in the case of both recurrence-type entanglement distillation protocols we consider), one can depolarize these gates to a standard form [@DuerStd]. This is done by randomly applying single-qubit operations before and after the application of the gate, which allows one to reduce any noise characteristics to a specific form with 8 parameters without altering the fidelity of the gate. A further simplification is possible if the noise characteristic of the apparatus is known [@DuerStd], which could in some cases be achieved through quantum process tomography. In this case, one can add additional (local) noise by randomly choosing to apply the gate, or some other (separable) operation. This allows one to bring any CNOT-type gate (i.e. any two-qubit gate that is equivalent to a CNOT gate up to single qubit unitary operations that are applied before and after the gate) to the standard form $$\begin{aligned}
{\mathcal{E}}(\rho) = \tilde{f} U \rho U^{\dagger} + \frac{1-\tilde{f}}{16}\sum\limits^1_{\alpha_1, \beta_1, \alpha_2, \beta_2 = 0} \sigma_{\alpha_1, \beta_1} \sigma_{\alpha_2, \beta_2} \rho \sigma_{\alpha_1, \beta_1} \sigma_{\alpha_2, \beta_2} \label{eq:ox:gen:1}\end{aligned}$$ As outlined in [@DuerStd] this depolarization procedure causes a change in the gate fidelity of the utilized quantum gates. More precisely, if the fidelity of the quantum gate before the depolarization was $F_{g} = 1-x$ then the gate fidelity after the depolarization is $F'_{g} > 1- 17x$. Thus one reduces the quality of the gate by about an order of magnitude in the worst case by depolarizing to this standard form. We observe that (\[eq:ox:gen:1\]) can be rewritten as $$\begin{aligned}
{\mathcal{E}}(\rho) &= \tilde{f} U \rho U^{\dagger} + \frac{1-\tilde{f}}{16}\sum\limits^1_{\alpha_1, \beta_1, \alpha_2, \beta_2 = 0} \sigma_{\alpha_1, \beta_1} \sigma_{\alpha_2, \beta_2} \rho \sigma_{\alpha_1, \beta_1} \sigma_{\alpha_2, \beta_2} \notag \\
&= U\left(\tilde{f} \rho + \frac{1-\tilde{f}}{16}\sum\limits^1_{\alpha_1, \beta_1, \alpha_2, \beta_2 = 0} \sigma_{\alpha_1, \beta_1} \sigma_{\alpha_2, \beta_2} \rho \sigma_{\alpha_1, \beta_1} \sigma_{\alpha_2, \beta_2} \right) U^\dagger \notag \\
&= U \left(\sum\limits^1_{\alpha_1, \beta_1, \alpha_2, \beta_2 = 0} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} \sigma_{\alpha_1, \beta_1} \sigma_{\alpha_2, \beta_2} \rho \sigma_{\alpha_1, \beta_1} \sigma_{\alpha_2, \beta_2} \right) U^\dagger \label{eq:ox:gen:2}\end{aligned}$$ where $\tilde{f}_{0,0,0,0} = \tilde{f} + (1-\tilde{f})/16$ and $\tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} = (1-\tilde{f})/16$ otherwise. Recall, that one noisy distillation step of the DEJMPS protocol including L is given by (\[eqn.overall\]). By introducing $O_{\mathrm{D}} = U_{u} O'_{\mathrm{2-EPP}}$ we rewrite (\[eqn.overall\]) as $$\begin{aligned}
\rho' & = \sum\limits_{\alpha_1, \beta_1, \alpha_2, \beta_2} \tilde{f}_{\alpha_1, \beta_1, \alpha_2, \beta_2} O_{\mathrm{D}} (U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2}) \left({\left| \Psi \right\rangle \left\langle \Psi \right|_{}} \otimes {\left| \Psi \right\rangle \left\langle \Psi \right|_{}}\right) (U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2})^\dagger O^\dagger_{\mathrm{D}}. \label{eq:ox:gen:3}\end{aligned}$$ We observe that the noise maps $U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2}$ in (\[eq:ox:gen:3\]) act on Alice’s part of the systems only. But this is sufficient due to the symmetry of Bell-states - noise on Bobs side can be moved to the other side. Furthermore the additional ${\sigma_x}$-flips introduced on the system(s) of L by the unitaries $U^{(a_1)}_{\alpha_1, \beta_1} \otimes U^{(a_2)}_{\alpha_2, \beta_2}$ are used to keep track of the noise map applied. Because Alice and Bob apply the depolarization procedure as described in [@DuerStd] and L keeps track of the effective error introduced, we can safely assume that the additional ${\sigma_x}$-flips will be introduced *after* Alice and Bob complete the depolarization procedure, hence it is sufficient to consider two qubit correlated noise introduced at Alice’s part of the systems.
The BBPSSW protocol
-------------------
The protocol proposed in [@Bennett] (also referred to as BBPSSW protocol) is very similar to the DEJMPS protocol. Instead of step \[enu.oxford.step1\] of Protocol \[protocol.rd\] Alice and Bob apply a correlated depolarization procedure (twirl) to their input states which brings them to Werner form.
For the subsequent analysis, suppose that each pair of Alice and Bob is of the form $\rho(p) = p {\left| B_{00} \right\rangle \left\langle B_{00} \right|_{}} + (1-p)\frac{1}{4} id$. We assume that the apparatus applies independent and identical noise of the form $N \rho(p) = f \rho(p) + (1-f)/4 (\rho(p) + {\sigma_x}\rho(p) {\sigma_x}+ {\sigma_y}\rho(p) {\sigma_y}+ {\sigma_z}\rho(p) {\sigma_z})$ before each distillation step. In similar fashion to the DEJMPS protocol one easily obtains the recurrence relation for the noisy BBPSSW protocol: $$\begin{aligned}
\tilde{p} = \frac{4 p^2 f^2 + 2 p f}{3 p^2 f^2 + 3} = b(p).\end{aligned}$$ The fixed point $p^\infty$ of the protocol is obtained by solving the equation $b(p^\infty) = p^\infty$. A straightforward computation gives the fixed point $p^\infty = 2/3 + 1/3 \sqrt{4 - 9/f^2 + 6/f}$ (which depends on the noise parameter $f$). It was shown in [@Duer] that this fixed point is an attractor assuming sufficiently high initial fidelity with $\ket{B_{00}}$ per input pair. Expressing the recurrence relation $b$ in terms of its Taylor series around $p^\infty$ leads to $$\begin{aligned}
\label{eqn.bbpswtaylor}
\tilde{p} = b(p) \approx b(p^\infty) + b'(p^\infty)(p-p^\infty).\end{aligned}$$ Hence, (\[eqn.bbpswtaylor\]) provides an approximation of the error in terms of fidelity with $\ket{B_{00}}$ after $n+1$ successful distillation rounds, i.e. $\epsilon_{n+1} = \left(b'(p^\infty)\right)^{n} \epsilon_{1}$, see also the plots within Fig. \[fig:ibm\]. Moreover, we compute the first derivative of $b$ by $$\begin{aligned}
b'(p) = \frac{2 f (1 + 4 f p - f^2 p^2)}{3 (1 + f^2 p^2)^2}.\end{aligned}$$ Evaluating $b'$ at $p^\infty$ yields $$\begin{aligned}
\label{eqn.bbpsswderivativefix}
b'(p^\infty) = \frac{9 - 3 f}{f (3 + 2 (2 + \sqrt{4 - 9/f^2 + 6/f}) f)}.\end{aligned}$$
$n$
From this we conclude that, if the apparatus is perfect, i.e. $f=1$ in (\[eqn.bbpsswderivativefix\]), the error in terms of fidelity with $\ket{B_{00}}$ after $n+1$ successful distillation rounds scales as $\epsilon_{n+1} = (2/3)^{n} \epsilon_{1}$.
Using $\log_2 N = n$, where $N$ denotes the number of initial states, we infer for $\epsilon_{n+1}$ that $$\begin{aligned}
\epsilon_{n+1} = \epsilon_{1} b'(p^\infty)^{\log_2 N} = \epsilon_{1} \left(2^{\log_2 b'(p^\infty)} \right)^{\log_2 N} = \epsilon_{1} N^{\log_2 b'(p^\infty)}.\end{aligned}$$ This implies that $\epsilon_{n+1}$ scales as $F(N) \in O(N^{\log_2 b'(p^\infty)})$ and thus $\|\rho_{\text{fix}} - \rho_{n} \|_{1}$, where $\rho_{\text{fix}}$ and $\rho_{n}$ denote the fixed point and the state after $n$ successful distillation rounds respectively, scales also as $F(N) \in O(N^{\log_2 b'(p^\infty)})$ as mentioned in the main text. For the analysis of two qubit correlated noise we assume that the noisy operations used by the BBPSSW protocol are of the form $$\begin{aligned}
O_{12} \rho = \tilde{f} O^{\mathrm{ideal}}_{12} \rho + \frac{1-\tilde{f}}{4} {\text{tr}_{12}\left[ \rho \right]} \otimes {id}_{12} \label{eq:ibm:noisyop}\end{aligned}$$ where $\rho$ is a two qubit density operator and $O^{\mathrm{ideal}}_{12}$ denotes the ideal two qubit quantum gate. Observe that (\[eq:ibm:noisyop\]) coincides with the standard form of [@DuerStd]. If the noisy quantum gates are not of the form (\[eq:ibm:noisyop\]) we bring them to that standard form via the same depolarization procedure mentioned in the analysis of the DEJMPS protocol. Hence the following anaylsis is not restricted to this specific noise model, but actually applies to arbitrary noise processes describing noisy two qubit gates. It has been shown in [@Duer] that the BBPSSW protocol converges for noisy CNOT gates of the form (\[eq:ibm:noisyop\]) to a unique and attracting fixed point if $\tilde{f}$ is sufficiently high. The recurrence relation for the fidelity relative to $\ket{B_{00}}$ obtained in [@Duer] is given by the formula $$\begin{aligned}
F' = \frac{\tilde{f}^2 (F^2 + (\frac{1-F}{3})^2) + \frac{1-\tilde{f}^2}{8}}{\tilde{f}^2(F^2 + \frac{2F(1-F)}{3} + 5 (\frac{1-F}{3})^2) + \frac{1-\tilde{f}^2}{2}}. \label{eq:ibm:recgeneral}\end{aligned}$$ Hence one obtains as in [@Duer] the respective fixed points of (\[eq:ibm:recgeneral\]) to be $$\begin{aligned}
F_{\mathrm{min,max}} = \frac{3 \pm \sqrt{10 - 9/\tilde{f}^2}}{4}.\end{aligned}$$ For $F \in (F_{\mathrm{min}}, F_{\mathrm{max}})$ we have that $F' > F$ which shows that $F_{\mathrm{max}}$ is an attracting fixed point. By replacing $F'$ in (\[eq:ibm:recgeneral\]) with $\tilde{b}(F)$ we observe similar to (\[eqn.bbpswtaylor\]) that the error after $n+1$ successful distillation rounds scales for two qubit correlated noise as $F(N) \in O(N^{\log_2 \tilde{b}'(F_{\mathrm{max}})})$ where $N$ denotes the number of initial states. Finally we provide a worst case analysis of the BBPSSW protocol. For that purpose assume the following scenario: The noisy apparatus performs with probability $f_{\mathrm{I}}$ the ideal distillation step ${\mathcal{E}}_{\mathrm{I}}$ and introduces with probability $1-f_{\mathrm{I}}$ an arbitrary noise map ${\mathcal{E}}_{\perp}$. More precisely, we decompose the distillation step taken by Alice and Bob before the measurement of the target system as the CP map $$\begin{aligned}
{\mathcal{E}}(\rho) = f_{\mathrm{I}} {\mathcal{E}}_{\mathrm{I}}(\rho) + (1-f_{\mathrm{I}}) {\mathcal{E}}_{\perp} (\rho)\end{aligned}$$ where $\rho$ is a four qubit density operator. Notice that one can always decompose a noisy map in this form, where both maps are completely positive and trace preserving. We remark, however, that the map ${\mathcal{E}}_{\mathrm{I}}$ denotes the ideal protocol which includes an abort option, i.e. we only keep the first pair if the results of the measurements on the second pair coincide. The map ${\mathcal{E}}_{\perp}$ may similarly contain such an abort branch. The noise parameter $f_{\mathrm{I}}$ describes the quality of the overall map [^6], i.e. one can think of the process that with probability $f_{\mathrm{I}}$ the desired procedure (including gates and measurements) is performed, while with probability $(1-f_{\mathrm{I}})$ something else happens (described by the map ${\mathcal{E}}_{\perp}$). We will now consider the worst case for the map ${\mathcal{E}}_{\perp}$ w.r.t. entanglement distillation. The worst case for the BBPSSW protocol is that the apparatus introduces a state orthogonal to $\ket{B_{00}}$ on the source system and the state $\ket{B_{00}}$ on the target system as this will always contribute to the overall success probability of a distillation step of the BBPSSW protocol but lead at the same time to a lower fidelity relative to $\ket{B_{00}}$ after the measurement of the target system compared to the ideal distillation step. One example for such a map is given by ${\mathcal{E}}_{\perp} (\rho) = \dm{B_{01}} \otimes \dm{B_{00}}$. Any other map will lead to a larger fidelity after the distillation step followed by depolarization to Werner form. We thus have $$\begin{aligned}
F ' \geq \frac{f_{\mathrm{I}}(F^2 + (\frac{1-F}{3})^2)}{f_{\mathrm{I}}(F^2 + \frac{2F(1-F)}{3} + 5 (\frac{1-F}{3})^2) + 1 - f_{\mathrm{I}}} \label{eq:ibm:gen:rec}\end{aligned}$$ for the fidelity relative to $\ket{B_{00}}$. This formula can be understood as follows: The ideal protocol is applied with probability $f_{\mathrm{I}}$, and succeeds with probability $f_{\rm suc}$, thereby producing a fidelity $\tilde F$. The map ${\mathcal{E}}_{\perp}$ is applied with probability $(1-f_{\mathrm{I}})$, does never abort and does not contribute to the final fidelity (which is clearly the worst case). We thus have $F' \geq f_{\mathrm{I}} f_{\rm suc} \tilde F/[f_{\mathrm{I}} f_{\rm suc} + (1-f_{\mathrm{I}})]$
We now analyze the worst case scenario, i.e. assuming equality in (\[eq:ibm:gen:rec\]). Since we know that at each step the actual noise map produces an output density operator with a larger fidelity than the worst-case map, we can conclude that the resulting fidelity of any noise map will be larger than the fixed point which is achieved by the worst-case map. We remark, however, that this does not constitute a full confidentiality proof for arbitrary noise maps, as it is not evident from this analysis that for any fixed noise map a unique fixed point is reached. Assuming equality in (\[eq:ibm:gen:rec\]), one can compute that the fixed points of the noisy BBPSSW protocol are in this case given by the solutions of $$\begin{aligned}
-f_{\mathrm{I}} + (9-2f_{\mathrm{I}})F_{\infty} - 14f_{\mathrm{I}} F^2_{\infty} + 8f_{\mathrm{I}} F^3_{\infty} = 0 \label{eq:ibm:gen:fixeq}\end{aligned}$$ which only depend on the noise parameter $f_{\mathrm{I}}$. We define $g_{\mathrm{fix}}(x,f_{\mathrm{I}}) = -f_{\mathrm{I}} + (9-2f_{\mathrm{I}})x - 14f_{\mathrm{I}} x^2 + 8f_{\mathrm{I}} x^3$ which implies that (\[eq:ibm:gen:fixeq\]) reads as $g_{\mathrm{fix}}(F_{\infty},f_{\mathrm{I}}) = 0$. The question how many solutions of (\[eq:ibm:gen:fixeq\]) are real we easily answer by the discriminant of $g_{\mathrm{fix}}$. We obtain for the discriminant of $g_{\mathrm{fix}}$ $$\begin{aligned}
\Delta(f_{\mathrm{I}}) = -36 (648 f_{\mathrm{I}} - 873 f_{\mathrm{I}}^2 - 212 f_{\mathrm{I}}^3 + 436 f_{\mathrm{I}}^4). \label{eq:ibm:gen:disc}\end{aligned}$$ Hence if $\Delta(f_{\mathrm{I}}) > 0$ then all three solutions of (\[eq:ibm:gen:fixeq\]) are real. We numerically estimate that $\Delta(f_{\mathrm{I}_{\mathrm{crit}}}) = 0$ for $f_{{\mathrm{I}}_{\mathrm{crit}}} \approx 0.9641$, hence for $f_{\mathrm{I}} > f_{{\mathrm{I}}_{\mathrm{crit}}}$ there exist three real solutions of (\[eq:ibm:gen:fixeq\]) because $\Delta(f_{\mathrm{I}}) > 0$ for $f_{\mathrm{I}} > f_{{\mathrm{I}}_{\mathrm{crit}}}$, see Fig. \[fig:ibm:gen:disc\].
$f_{\mathrm{I}}$
Thus, for $f_{\mathrm{I}} > f_{{\mathrm{I}}_{\mathrm{crit}}}$, we compute the fixed points of the noisy BBPSSW protocol via solving (\[eq:ibm:gen:fixeq\]). Fig. \[fig:ibm:gen:fixpoint\] shows the function $g_{\mathrm{fix}}$ for different values of $f_{\mathrm{I}}$.
$F$
From Fig. \[fig:ibm:gen:disc\] and \[fig:ibm:gen:fixpoint\] we infer that we have three possible fixed points for $f_{\mathrm{I}} > f_{{\mathrm{I}}_{\mathrm{crit}}}$. Hence we need to show that the fixed point with the highest fidelity relative to $\ket{B_{00}}$ obtained via (\[eq:ibm:gen:fixeq\]) is an attracting fixed point. We solve this issue by showing that $F' > F$ for $F \in (F_\text{min}, F_{\max})$ (where $F_\text{min}$ denotes the second, and $F_{\max}$ the third fixed point in Fig. \[fig:ibm:gen:fixpoint\]). From Fig. \[fig:ibm:gen:attr\] we find that $F' - F > 0$ for $f_{\mathrm{I}} > f_{{\mathrm{I}}_{\mathrm{crit}}}$, hence $F' > F$ which shows that $F_{\max}$ is an attracting fixed point whenever starting with initial fidelity $F > F_\text{min}$.
$F$
Furthermore, by assuming equality in (\[eq:ibm:gen:rec\]) and replacing $F'$ with $b_{\perp}(F)$, we observe similar to (\[eqn.bbpswtaylor\]) that the error after $n+1$ successful distillation rounds scales in this worst case analysis as $F(N) \in O(N^{\log_2 b'_{\perp}(F_{\mathrm{max}})})$ where $N$ denotes the number of initial states.
Confidentiality of entanglement distillation protocols {#sec.supp.noleak}
======================================================
In this section we provide the proofs of Lemma \[thm.globalclosenessfactor\] and Lemma \[lem:forpost\] of the main text, crucial for the de-Finetti-based and post-selection-based reduction techniques. Both proofs require only one specific property of the real protocol ${\mathcal{E}}^\alpha$: after passing the parameter estimation phase the entanglement distillation protocol always converges to *one* fixed point, i.e. the fixed point is *unique*, an *attractor* for all the states which pass the parameter estimation and depends on the noise parameters *only*, as this implies that the distance with respect to the $1-$norm within the $ok-$branch of the protocol is bounded and converges towards zero.
Proof of Lemma \[thm.globalclosenessfactor\] {#app:sec:defin}
--------------------------------------------
We first state the following lemma which establishes a connection between measurements on one subsystem of a bipartite state and tensor product states.
\[lem.steering\]\[Steering of local states\] Let $\rho_{AB}$ be a bipartite (in general, mixed) state and let $\rho_A = {\text{tr}_{B}\left[ \rho_{AB} \right]}$ and $\rho_B = {\text{tr}_{A}\left[ \rho_{AB} \right]}$. Furthermore let $\rho^{\phi}_{B}$ be defined as $$\begin{aligned}
\rho^{\phi}_{B} = \frac{{\text{tr}_{A}\left[ (\dm{\phi} \otimes I) \rho_{AB} \right]}}{p_{A}(\phi)}\end{aligned}$$ where $\ket{\phi} \in {\mathcal{H}}_A$ and $p_{A}(\phi) = \tr{\left(\dm{\phi} \rho_{A} \right)}$. If $\|\rho^{\phi}_{B} - \rho_B \|_1 \leq \epsilon$ for all $\ket{\phi} \in {\mathcal{H}}_A$, then $$\begin{aligned}
\label{inequ.steering}
\|\rho_{AB} - \rho_A \otimes \rho_B \|_1 \leq 2 C \epsilon\end{aligned}$$ where $C$ only depends on the dimensions of $A$ and $B$. In particular, if we fix the number of qubits of $A$ and $B$ to $2$ respectively, then we have $C = 4^8$.
In the following we denote the four Pauli operators by $$\begin{aligned}
\sigma_{0} = id, \quad \sigma_{1} = {\sigma_x}, \quad \sigma_{2} = {\sigma_z}, \quad \sigma_{3} = {\sigma_y}.\end{aligned}$$ First we decompose $\rho_{AB}$ in the Pauli basis, i.e. we have $$\begin{aligned}
\label{equ.lem.meas.rhoab}
\rho_{AB} = \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i},\mathbf{j}} \alpha_{\mathbf{ij}} \sigma_{\mathbf{i}} \otimes \sigma_{\mathbf{j}}\end{aligned}$$ where $n$ and $m$ denote the number of qubits of $A$ and $B$ respectively and we use the notations $\mathbf{i} = (i_1,..,i_{n})$ and $\mathbf{j}=(j_1,..,j_{m})$ where each $i_k$ and $j_k$ are in $\lbrace 0,..,3 \rbrace$ as well as $\sigma_\mathbf{i} = \bigotimes^{n}_{k = 1} \sigma_{i_k}$ and $\sigma_\mathbf{j} = \bigotimes^{m}_{k = 1} \sigma_{j_k}$. Recall that $\tr{(\sigma_0)} = 2$ and $\tr{(\sigma_1)} = \tr{(\sigma_2)} = \tr{(\sigma_3)} = 0$. From this one easily computes $\rho_A$ and $\rho_B$ by $$\begin{aligned}
\rho_A &= {\text{tr}_{B}\left[ \rho_{AB} \right]} = \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i,j}} \alpha_{\mathbf{ij}} \sigma_{\mathbf{i}} \tr{(\sigma_{\mathbf{j}})} = \frac{1}{2^{n}} \sum\limits_{\mathbf{i}} \alpha_{\mathbf{i0}} \sigma_{\mathbf{i}}, \label{equ.lem.meas.partials.1} \\
\rho_B &= {\text{tr}_{A}\left[ \rho_{AB} \right]} = \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i,j}} \alpha_{\mathbf{ij}} \tr{(\sigma_{\mathbf{i}})} \sigma_{\mathbf{j}} = \frac{1}{2^{m}} \sum\limits_{\mathbf{j}} \alpha_{\mathbf{0j}} \sigma_{\mathbf{j}}. \label{equ.lem.meas.partials.2}\end{aligned}$$ Using (\[equ.lem.meas.rhoab\]), (\[equ.lem.meas.partials.1\]) and (\[equ.lem.meas.partials.2\]) we obtain for (\[inequ.steering\]) $$\begin{aligned}
\label{inequ.lem.meas.1}
\|\rho_{AB} - \rho_A \otimes \rho_B \|_1 & \leq \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i,j}} \|(\alpha_{\mathbf{ij}} - \alpha_{\mathbf{i0}} \alpha_{\mathbf{0j}}) \sigma_{\mathbf{i}} \otimes \sigma_{\mathbf{j}} \|_1 = \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i,j}} |\alpha_{\mathbf{ij}} - \alpha_{\mathbf{i0}} \alpha_{\mathbf{0j}}| \cdot \|\sigma_{\mathbf{i}} \otimes \sigma_{\mathbf{j}}\|_1 \notag \\
& = \frac{2^{n+m}}{2^{n+m}} \sum\limits_{\mathbf{i,j}} |\alpha_{\mathbf{ij}} - \alpha_{\mathbf{i0}} \alpha_{\mathbf{0j}}| = \sum\limits_{\mathbf{i,j}} |\alpha_{\mathbf{ij}} - \alpha_{\mathbf{i0}} \alpha_{\mathbf{0j}}| = \|\mathbf{a} - \mathbf{a'} \|_{1;{\mathbb{C}}^{4^{n+m}}}\end{aligned}$$ where $\mathbf{a} = (\alpha_{00},..,\alpha_{3^n 3^m})$, $\mathbf{a'} = (\alpha_{00} \alpha_{00},..,\alpha_{3^n 0} \alpha_{0 3^m})$ and $\| \cdot \|_{1;{\mathbb{C}}^{4^{n+m}}}$ denotes the $1-$norm of vectors in ${\mathbb{C}}^{4^{n+m}}$. Hence in order to prove (\[inequ.steering\]) it is sufficient to prove $\|\mathbf{a} - \mathbf{a'} \|_{1;{\mathbb{C}}^{4^{n+m}}} \leq 2C \epsilon$. By assumption we have for $\rho^{\phi}_{B}$ where $\ket{\phi} \in {\mathcal{H}}_A$ and $p_{A}(\phi) = \tr{\left((\dm{\phi} \otimes I) \rho_{AB}\right)}$ that $\|\rho^{\phi}_{B} - \rho_B \|_1 \leq \epsilon$ for all $\ket{\phi} \in {\mathcal{H}}_A$. Moreover, according to Theorem 9.1 in [@Nielsen] we have for all $\ket{\xi} \in {\mathcal{H}}_B$ $$\begin{aligned}
\label{inequ.lem.meas.nc}
\frac{1}{2} |p_B(\xi | \phi) - q_B(\xi)| = \frac{1}{2}\left|\tr{\left(\dm{\xi} \rho^{\phi}_{B} \right)} - \tr{\left(\dm{\xi} \rho_{B} \right)} \right| \leq \max_{E_m} \frac{1}{2} \sum\limits_{m} \left|\tr{\left(E_m \rho^{\phi}_{B} \right)} - \tr{\left(E_m \rho_{B} \right)} \right| = \|\rho^{\phi}_{B} - \rho_B \|_1 \leq \epsilon\end{aligned}$$ where $p_B(\xi | \phi)$ denotes the conditional probability of obtaining the outcome $\phi$ on system $A$ and the outcome $\xi$ on system $B$ and $\{E_m \}$ denotes a POVM on the subsystem of $B$. Suppose we perform a projective measurement on the systems of $A$ and $B$ denoted by $\{\ket{\psi_k}_{AB} \} = \{\ket{\phi_k}_{A} \otimes \ket{\xi_k}_{B} \}$ where $k \in \lbrace 1,.., 4^{n+m}\rbrace$ on $\rho_{AB}$ and $\rho_A \otimes \rho_B$. This yields for the respective probabilities $p_{AB}(\psi_k)$ and $q_{AB}(\psi_k)$ of observing outcome $k$ for $\rho_{AB}$ and $\rho_A \otimes \rho_B$ $$\begin{aligned}
p_{AB}(\psi_k) &= \tr{(\dm{\psi_k} \rho_{AB})} = \tr{\left(\dm{\phi_k}_A \otimes \dm{\xi_k}_B \rho_{AB} \right)} = \tr{\left(\dm{\xi_k}_B {\text{tr}_{A}\left[ (\dm{\phi_k}_A \otimes I) \rho_{AB} \right]}\right)} \\
& = \tr{\left(\dm{\xi_k}_B p_A(\phi_k) \rho^{\phi_k}_{B} \right)} = p_A(\phi_k) \tr{\left(\dm{\xi_k}_B \rho^{\phi_k}_{B} \right)} = p_A(\phi_k) p_B(\xi_k | \phi_k),\\
q_{AB}(\psi_k) &= \tr{(\dm{\psi_k} \rho_{A} \otimes \rho_{B})} = \tr{(\dm{\phi_k} \rho_{A})} \tr{(\dm{\xi_k} \rho_{B})} = q_A(\phi_k) q_B(\xi_k)\end{aligned}$$ where $p_B(\xi_k | \phi_k)$ denotes the conditional probability of obtaining outcome $\phi_k$ on system $A$ first and obtaining outcome $\xi_k$ on system $B$. We observe $p_A(\phi_k) = q_A(\phi_k)$. Thus we obtain $$\begin{aligned}
|p_{AB}(\psi_k) - q_{AB}(\psi_k)| = p_A(\phi_k) |p_B(\xi_k | \phi_k) - q_B(\xi_k)| \leq 2\epsilon p_A(\phi_k)\end{aligned}$$ using (\[inequ.lem.meas.nc\]). In order to compute a bound for (\[inequ.lem.meas.1\]) we use quantum state tomography, see e.g. [@tomo]. For that purpose we perform an informationally complete POVM induced by different separable bases on ${\mathcal{H}}_A \otimes {\mathcal{H}}_B$. More precisely, we choose that many POVMs such that we have in total $4^{n+m}$ different outcomes. We observe for $\ket{\psi_k}_{AB} = \ket{\phi_k}_{A} \otimes \ket{\xi_k}_{B}$ that $$\begin{aligned}
\label{equ.lem.meas.pandqtomo}
p_{AB}(\psi_k) = \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i,j}} \bra{\phi_k} \sigma_\mathbf{i} \ket{\phi_k} \bra{\xi_k} \sigma_\mathbf{j} \ket{\xi_k} \alpha_{\mathbf{ij}} \quad \text{and} \quad q_{AB}(\psi_k) = \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i,j}} \bra{\phi_k} \sigma_\mathbf{i} \ket{\phi_k} \bra{\xi_k} \sigma_\mathbf{j} \ket{\xi_k} \alpha_{\mathbf{i0}} \alpha_{\mathbf{0j}}.\end{aligned}$$ Enumerating (\[equ.lem.meas.pandqtomo\]) for $1 \leq k \leq 4^{n+m}$ yields $4^{n+m}$ equations for $\mathbf{a}$, i.e. $$\begin{aligned}
\label{inequ.lem.meas.20}
p_{AB}(\psi_1) &= \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i,j}} \bra{\phi_1} \sigma_\mathbf{i} \ket{\phi_1} \bra{\xi_1} \sigma_\mathbf{j} \ket{\xi_1} \alpha_{\mathbf{ij}},\\
&... \notag \\
p_{AB}(\psi_{4^{n+m}}) &= \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i,j}} \bra{\phi_{4^{n+m}}} \sigma_\mathbf{i} \ket{\phi_{4^{n+m}}} \bra{\xi_{4^{n+m}}} \sigma_\mathbf{j} \ket{\xi_{4^{n+m}}} \alpha_{\mathbf{ij}} \label{inequ.lem.meas.21}\end{aligned}$$ as well as $4^{n+m}$ equations for $\mathbf{a'}$ $$\begin{aligned}
\label{inequ.lem.meas.30}
q_{AB}(\psi_1) &= \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i,j}} \bra{\phi_1} \sigma_\mathbf{i} \ket{\phi_1} \bra{\xi_1} \sigma_\mathbf{j} \ket{\xi_1}\alpha_{\mathbf{i0}} \alpha_{\mathbf{0j}}, \\
& ... \notag \\
q_{AB}(\psi_{4^{n+m}}) &= \frac{1}{2^{n+m}} \sum\limits_{\mathbf{i,j}} \bra{\phi_{4^{n+m}}} \sigma_\mathbf{i} \ket{\phi_{4^{n+m}}} \bra{\xi_{4^{n+m}}} \sigma_\mathbf{j} \ket{\xi_{4^{n+m}}} \alpha_{\mathbf{i0}} \alpha_{\mathbf{0j}} \label{inequ.lem.meas.31}.\end{aligned}$$ We can rewrite the systems of equations (\[inequ.lem.meas.20\])-(\[inequ.lem.meas.21\]) and (\[inequ.lem.meas.30\])-(\[inequ.lem.meas.31\]) using $$\begin{aligned}
T = \begin{pmatrix}
\bra{\phi_1} \sigma_0 \ket{\phi_1} \bra{\xi_1} \sigma_0 \ket{\xi_1} & ... & \bra{\phi_1} \sigma_{3^n} \ket{\phi_1} \bra{\xi_1} \sigma_{3^m} \ket{\xi_1} \\
... & ... & ... \\
\bra{\phi_{4^{n+m}}} \sigma_0 \ket{\phi_{4^{n+m}}} \bra{\xi_{4^{n+m}}} \sigma_0 \ket{\xi_{4^{n+m}}} & ... & \bra{\phi_{4^{n+m}}} \sigma_{3^n} \ket{\phi_{4^{n+m}}} \bra{\xi_{4^{n+m}}} \sigma_{3^m} \ket{\xi_{4^{n+m}}}
\end{pmatrix}\end{aligned}$$ and $\mathbf{p} = (p_{AB}(\psi_1),..,p_{AB}(\psi_{4^{n+m}}))$ and $\mathbf{q} = (q_{AB}(\psi_1),..,q_{AB}(\psi_{4^{n+m}}))$ as $$\begin{aligned}
\mathbf{p} = \frac{1}{2^{n+m}} T \mathbf{a} \quad \text{and} \quad \mathbf{q} = \frac{1}{2^{n+m}} T \mathbf{a'}\end{aligned}$$ respectively. Hence $2^{n+m}(\mathbf{p} - \mathbf{q}) = T\left(\mathbf{a} - \mathbf{a'} \right)$. Moreover we observe that $T$ is invertible if the POVM is informationally complete, see [@tomo] for details. Thus, inverting $T$ and taking norms on both sides yields $$\begin{aligned}
\| \mathbf{a} - \mathbf{a'} \|_{1;{\mathbb{C}}^{4^{n+m}}} & \leq 2^{n+m} \|T^{-1}\| \| \mathbf{p} - \mathbf{q}\|_{1;{\mathbb{C}}^{4^{n+m}}} = 2^{n+m} \|T^{-1}\| \sum_{k} |p_{AB}(\psi_k) - q_{AB}(\psi_k)| \\
& \leq 2^{n+m} \|T^{-1}\| \sum_{k} 2 \epsilon p_A(\phi_k) \leq 2 \|T^{-1}\| 4^{n+m} 2^{n+m} \epsilon\end{aligned}$$ which completes the proof for the general case with $C = \|T^{-1}\| 4^{n+m} 2^{n+m}$. Before we complete the Lemma we need to determine $C$ for the case of $n=m=2$. We choose $\ket{\phi_{4^3(j_1-1) + 4^2(j_2-1) + 4(j_3-1) + j_4}} = \ket{\phi'_{j_1}} \otimes \ket{\phi'_{j_2}} \otimes \ket{\phi'_{j_4}} \otimes \ket{\phi'_{j_4}}$ where $j_1,j_2,j_3,j_4 \in \lbrace 1,2,3,4 \rbrace$ and $$\begin{aligned}
\ket{\phi'_{1}} &= (\ket{0} + \ket{1})/\sqrt{2} , \label{eq:tomoset:1} \\
\ket{\phi'_{2}} &= (\ket{0} + i\ket{1})/\sqrt{2}, \label{eq:tomoset:2} \\
\ket{\phi'_{3}} &= \ket{0}, \label{eq:tomoset:3} \\
\ket{\phi'_{4}} &= (\ket{0} - \ket{1})/\sqrt{2}, \label{eq:tomoset:4} \end{aligned}$$ which is informationally complete and thus a valid choice. This choice of $\ket{\phi'_{l}}$ corresponds to a Pauli tomography on a single qubit. We observe that the matrix $T$ is invertible and compute $\|T^{-1}\| = 16$. Thus $C = 4^8$ which completes the proof.
Roughly speaking Lemma \[lem.steering\] states that if all post-selected reduced states of a bipartite state, where each partition consists of two qubits, are $\eta-$close then the overall state is $2 \cdot 4^8 \eta$ close to a product state. We gave the lemma in a more general form as it may have utility beyond the scope of this paper. However for our purposes we need a stronger, but more specific result. In the following lemma we will show that we can achieve the same result even if the measurements must succeed above a threshold, which is important in the application of the lemma.
\[lem:steering:prob\] In the situation of Lemma \[lem.steering\] for $n=m=2$ it suffice to consider measurements on the subsystem $A$ which have a probability greater than or equal to $1/16$. More precisely, for every state $\rho_{AB}$ there exists a unitary $U$ acting on system $A$ and a state $\rho'_{AB} = (U \otimes I_{B}) \rho_{AB} (U \otimes I_{B})^\dagger$, such that if the state $\rho'_{AB}$ meets the conditions of Lemma \[lem.steering\], i.e. subsystem $B$ is $\epsilon-$non-steerable via measurements on subsystem $A$ for all measurements with probability greater than or equal to $1/16$, then $$\begin{aligned}
\| \rho_{AB} - \rho_{A} \otimes \rho_{B} \|_1 \leq 2C \epsilon.\end{aligned}$$
First we construct the state $\rho'_{AB}$ associated with $\rho_{AB}$ and show that it suffice to consider measurements of probability greater than or equal to $1/16$. Recall the situation of Lemma \[lem.steering\]. Let $\rho_{AB}$ be a bipartite (in general, mixed) state and let $\rho_A = {\text{tr}_{B}\left[ \rho_{AB} \right]}$ and $\rho_B = {\text{tr}_{A}\left[ \rho_{AB} \right]}$. Furthermore let $\rho^{\phi}_{B}$ be defined as $$\begin{aligned}
\rho^{\phi}_{B} = \frac{{\text{tr}_{A}\left[ (\dm{\phi} \otimes I) \rho_{AB} \right]}}{p_{A}(\phi)}\end{aligned}$$ where $\ket{\phi} \in {\mathcal{H}}_A$ and $p_{A}(\phi) = \tr{\left(\dm{\phi} \rho_{A} \right)}$. Then the claim of Lemma \[lem.steering\] was: If $\|\rho^{\phi}_{B} - \rho_B \|_1 \leq \epsilon$ for all $\ket{\phi} \in {\mathcal{H}}_A$, then $$\begin{aligned}
\|\rho_{AB} - \rho_A \otimes \rho_B \|_1 \leq 2 C \epsilon\end{aligned}$$ where $C$ only depends on the dimensions of $A$ and $B$. In particular, if we fix the number of qubits of $A$ and $B$ to $2$ respectively, then we have $C = 4^8$. Further recall that the set $\ket{\phi_{4^3(j_1-1) + 4^2(j_2-1) + 4(j_3-1) + j_4}} = \ket{\phi'_{j_1}} \otimes \ket{\phi'_{j_2}} \otimes \ket{\phi'_{j_4}} \otimes \ket{\phi'_{j_4}}$ where $j_1,j_2,j_3,j_4 \in \lbrace 1,2,3,4 \rbrace$ of Lemma \[lem.steering\], i.e. (\[eq:tomoset:1\])–(\[eq:tomoset:4\]), is informationally complete and thus suffice to reconstruct any $4$ qubit quantum state where $$\begin{aligned}
\ket{\phi'_{1}} &= (\ket{0} + \ket{1})/\sqrt{2} , \label{eq:lem:prob:tom:1}\\
\ket{\phi'_{2}} &= (\ket{0} + i\ket{1})/\sqrt{2}, \label{eq:lem:prob:tom:2}\\
\ket{\phi'_{3}} &= \ket{0}, \label{eq:lem:prob:tom:3}\\
\ket{\phi'_{4}} &= (\ket{0} - \ket{1})/\sqrt{2}. \label{eq:lem:prob:tom:4}\end{aligned}$$ In order to prove the claim, we use the following observation: The state $\rho_{A} = \text{tr}_{B}[\rho_{AB}]$ is a two qubit quantum state, so it can be written as $$\begin{aligned}
\rho_{A} = \sum^3_{j=0} \lambda_j \dm{\Psi_j} \label{eq:lem:prob:1}\end{aligned}$$ where the states $\ket{\Psi_j}$ correspond to the (orthogonal) eigenstates of $\rho_{A}$ for the real non-negative eigenvalues $\lambda_{j}$. Hence there exists at least one $j' \in \lbrace 0,1,2,3 \rbrace$ such that $\lambda_{j'} \geq 1/4 $, which corresponds to the maximum of the eigenvalues $\lambda_j$. Now we choose a local unitary $U$ such that $U \ket{\Psi_{j'}} = \ket{0} \otimes \ket{0}$. Applying this unitary to (\[eq:lem:prob:1\]) therefore leads to the state $$\begin{aligned}
\rho'_{A} = U \left(\sum^3_{j=0} \lambda_j \dm{\Psi_j} \right) U^\dagger = \lambda_{j'} \dm{00} + \sum^3_{j \neq j'} \lambda_j \dm{\varphi_{j}} \label{eq:lem:prob:2}\end{aligned}$$ where $\ket{\varphi_j} = U \ket{\psi_j}$. We compute the probability for any projector applied on $\rho'_{A}$ which is taken from the set (\[eq:lem:prob:tom:1\])–(\[eq:lem:prob:tom:4\]) and of the form $\dm{\phi'} = \dm{\phi'_{k}} \otimes \dm{\phi'_{l}}$ by $$\begin{aligned}
\text{tr} \left(\dm{\phi'} \rho'_{A} \right) &= \text{tr} \left(\dm{\phi'_{k}} \otimes \dm{\phi'_{l}} U \rho_{A} U^\dagger \right) = \sum^3_{j=0} \lambda_j \text{tr} \left(\dm{\phi'_{k}} \otimes \dm{\phi'_{l}} U \dm{\Psi_j} U^\dagger \right) \notag \\
& \geq \frac{1}{4} \text{tr} \left(\dm{\phi'_{k}} \otimes \dm{\phi'_{l}} U \dm{\Psi_{j'}} U^\dagger \right) = \frac{1}{4} \text{tr} \left(\dm{\phi'_{k}} \otimes \dm{\phi'_{l}} \dm{00} \right) \notag \\
& = \frac{1}{4} \text{tr} \left(\dm{\phi'_{k}} \dm{0} \right) \text{tr} \left(\dm{\phi'_{l}} \dm{0} \right) \geq \frac{1}{4} \frac{1}{2} \frac{1}{2} = \frac{1}{16} \label{eq:lem:prob:4}\end{aligned}$$ where we have used that $\text{tr}(AB) = \text{tr}(BA)$ for matrices $A$ and $B$ and that $\text{tr}\left(\dm{\phi'_{k}} \dm{0} \right) \geq 1/2$ for all $k \in \lbrace 0,1,2,3 \rbrace$. So we define the state $\rho'_{AB}$ as $\rho'_{AB} = (U \otimes I_{B}) \rho_{AB} (U \otimes I_{B})^\dagger$. Observe that the probabilities of all projectors within the tomographic set (\[eq:lem:prob:tom:1\])–(\[eq:lem:prob:tom:4\]) are greater than or equal to $1/16$ for the state $\rho'_{A}$. Now suppose we perform a measurement from the tomographic set (\[eq:lem:prob:tom:1\])–(\[eq:lem:prob:tom:4\]) on the subsystem $A$ of $\rho'_{AB}$ yielding outcome $\ket{\phi}$. The post-selected state conditioned on $\ket{\phi}$ reads as $$\begin{aligned}
\rho'^{\phi}_{B} = \frac{{\text{tr}_{A}\left[ (\dm{\phi} \otimes I) \rho'_{AB} \right]}}{p_{A}(\phi)}\end{aligned}$$ where $p_{A}(\phi) \geq 1/16$. Furthermore assume as in Lemma \[lem.steering\] that $\|\rho'^{\phi}_{B} - \rho_{B} \|_1 \leq \epsilon$ for all such $\ket{\phi} \in \mathcal{H}_A$. Then Lemma \[lem.steering\] implies that $$\begin{aligned}
\|\rho'_{AB} - \rho'_{A} \otimes \rho_{B} \|_1 \leq 2C \epsilon.\end{aligned}$$ The proof completes by observing that $\rho'_{AB}$ and $\rho'_{A} \otimes \rho_{B}$ are related by the local unitary $U$ to $\rho_{AB}$ and $\rho_{A} \otimes \rho_{B}$ and the unitary equivalence of the trace distance, i.e. $$\begin{aligned}
\|\rho_{AB} - \rho_{A} \otimes \rho_{B} \|_1 &= \| (U \otimes I_B)(\rho_{AB} - \rho_{A} \otimes \rho_{B})(U \otimes I_B)^\dagger \|_1 \\
&= \| \rho'_{AB} - \rho'_{A} \otimes \rho_{B} \|_1 \leq 2C \epsilon.\end{aligned}$$
We observe that, due to the proof of Lemma \[lem:steering:prob\], which relies on the informationally complete set (\[eq:lem:prob:tom:1\])–(\[eq:lem:prob:tom:4\]), it suffices to be non-steerable with respect to the measurements within that set for a probability of measurement above or equal to $1/16$. We actually have proven a stronger result, as the actual choice of measurements does not matter, provided the probability of success is above or equal to the threshold $1/16$.
Let $\rho$ be an arbitrary mixed state shared by Alice and Bob and let $\ket{\psi}_{ABE}$ be a purification thereof held by Eve. Furthermore, let $\mathcal{P}_1$ correspond to a (distillation-type) real protocol and $\mathcal{P}_2$ correspond to the associated (distillation-type) ideal protocol, i.e. $$\begin{aligned}
\mathcal{P}_1 (\rho) &= p_{\rho} \sigma_{AB} \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \dm{fail}, \\
\mathcal{P}_2(\rho) &= p_{\rho} \sigma^{\alpha}_{AB} \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \dm{fail}. \notag\end{aligned}$$ where $\alpha$ characterizes the level of the noise, $\sigma^{\alpha}_{AB}$, and $\sigma^{\perp}_{AB}$ are two fixed two qubit states. Furthermore, let $\mathcal{P}_1$ and $\mathcal{P}_2$ satisfy the following properties:
1. The noise transcripts do not leak to Eve.
2. The protocol $\mathcal{P}_1$ guarantees to converge towards some state $\sigma^{\alpha}_{AB}$ within the ok-branch of the protocol and $\max_{\mu_{AB}} \|(\mathcal{P}_1 - \mathcal{P}_2)(\mu_{AB})\|_1 \leq \varepsilon$.
Then it holds that $$\begin{aligned}
\| (\mathcal{P}_1 \otimes id_{E} -\mathcal{P}_2 \otimes id_{E} )({\left| \psi \right\rangle \left\langle \psi \right|_{ABE}})\|_1 \leq (34 \cdot 4^8 +1) \varepsilon. \label{eq.thm.localsteer}\end{aligned}$$
The proof relies on Lemma \[lem.steering\] and \[lem:steering:prob\]. Suppose Eve prepares the pure state $\ket{\psi}_{ABE}$ and let ${\text{tr}_{E}\left[ \dm{\psi} \right]} = \rho_{AB}$ be the state received by Alice and Bob. Then we have $$\begin{aligned}
(\mathcal{P}_1 \otimes {id}_E)(\dm{\psi}) &= p_{\rho} \sigma_{ABE} \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \sigma_E \otimes \dm{fail}, \label{eq.thm.localsteer.1} \\
(\mathcal{P}_2 \otimes {id}_E)(\dm{\psi}) &= p_{\rho} \sigma^{\alpha}_{AB} \otimes \sigma_E \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \sigma_E \otimes \dm{fail}. \notag\end{aligned}$$ If we post-select Eq. (\[eq.thm.localsteer.1\]) on the $ok-$branch we have after normalization $$\begin{aligned}
\label{eq.thm.localsteer.2}
\frac{1}{p_{\rho}}({id}_{ABE} \otimes \dm{ok}) (\mathcal{P}_1 \otimes {id}_E)(\dm{\psi}) = \sigma_{ABE} \otimes \dm{ok}.\end{aligned}$$ It is obvious from the fact that the protocol is performed by Alice and Bob per definition that any measurement of Eve in the $ok-$branch can be commuted to the beginning of the protocol $\mathcal{P}_1$ because Eve is not part of the protocol. Hence her measurement only changes the input of the protocol $\mathcal{P}_1$ and thus either cause an abort or not.
We call the final state of Alice and Bob $\eta-$Eve-non steerable if for all $\phi \in {\mathcal{H}}_E$ we have $\|\sigma_{AB}^{\phi} - \sigma_{AB} \|_1 \leq \eta$ where $\sigma_{AB}^{\phi} = {\text{tr}_{E}\left[ \frac{1}{p_E(\phi)}({id}_{AB} \otimes \dm{\phi}_E)\sigma_{ABE} \right]}$. We sketch the remainder of this proof as follows: We show, that the final state of Alice and Bob is Eve-non steerable in the sense of Lemma \[lem.steering\] by making use of the bounded distance of the protocols $\mathcal{P}_1$ and $\mathcal{P}_2$. Furthermore, Lemma \[lem:steering:prob\] implies that it suffice to consider measurements of Eve of having probability greater than or equal to $1/16$. Therefore Lemma \[lem.steering\] and Lemma \[lem:steering:prob\] completes the proof. Because the output of Alice and Bob are $2$ qubits the purifying system that Eve holds is without loss of generality also a two-qubit system. Hence, according to Lemma \[lem:steering:prob\], there exists a state $\sigma'_{ABE}$, which is unitarly related to $\sigma_{ABE}$ via an unitary $U$ on Eve’s system only (which is not part of the protocol) and for which it suffice to consider measurements of Eve having probability greater than or equal to $1/16$. Furthermore observe that this local unitary of Eve can not change the success probability of the overall protocol as unitaries are CPTP. In other words, the success probabilities associated with $\sigma_{ABE}$ and $\sigma'_{ABE}$ are identical. More formally, suppose Eve performs a projective measurement on this state $\sigma'_{ABE}$ (which stems from a purification $\ket{\psi'}_{ABE}$ of $\rho'_{AB}$ which is unitarly related to the purification $\ket{\psi}_{ABE}$ of $\rho_{AB}$ and both having the same success probability, see paragraph above) and observes outcome $\ket{\phi} \in {\mathcal{H}}_E$ having probability greater than or equal to $1/16$. Then the post-selected state of Alice, Bob, and Eve conditioned on that particular outcome $\phi$ reads as $$\begin{aligned}
& \frac{1}{p_E(\phi)}({id}_{AB} \otimes \dm{\phi}_E)(\sigma'_{ABE} \otimes \dm{ok}) = \frac{1}{p_E(\phi)}({id}_{AB} \otimes \dm{\phi}_E) \frac{1}{p_{\rho}}({id}_{ABE} \otimes \dm{ok}) (\mathcal{P}_1 \otimes {id}_E)(\dm{\psi'}_{ABE}) \\
& = \frac{1}{p_{\rho^\phi}} ({id}_{ABE} \otimes \dm{ok}) (\mathcal{P}_1 \otimes {id}_E)\underbrace{\left(\frac{{id}_{AB} \otimes \dm{\phi}_E}{p'_E(\phi)} \dm{\psi'}_{ABE}\right)}_{=: \rho^{\phi}_{ABE}} = \frac{1}{p_{\rho^\phi}} ({id}_{ABE} \otimes \dm{ok}) (\mathcal{P}_1 \otimes {id}_E) (\rho^{\phi}_{ABE}) \\
& = \sigma^{\phi}_{ABE} \otimes \dm{ok}.\end{aligned}$$ More importantly, we relate the probability of the protocol succeeding for initial state $\rho'$, $p_{\rho}$, and the probability of measuring $\phi$ after the protocol, $p_E(\phi)$, to the probability of the protocol succeeding for the initial state $\rho^{\phi}_{ABE}$ (measurement of Eve commuted to the beginning of the protocol), $p_{\rho^\phi}$, and the probability of measuring $\phi$ before the protocol has started, $p'_E(\phi)$, via $$\begin{aligned}
p_{\rho} p_E(\phi) = p_{\rho^\phi} p'_E(\phi). \label{eq:lemma:prod:1}\end{aligned}$$ Observe that (\[eq:lemma:prod:1\]) is equivalent to $$\begin{aligned}
\frac{p_{\rho} p_E(\phi)}{p'_E(\phi)} = p_{\rho^\phi} \label{eq:lemma:prod:2}\end{aligned}$$ We note that the state $\sigma^{\phi}_{ABE}$ is in the $ok-$branch of the protocol $\mathcal{P}_1$. The next step is to apply Lemma \[lem.steering\] which relates the distances $\| \sigma'_{ABE} - \sigma_{AB} \otimes \sigma'_E \|_1$ and $\|\sigma'_{AB} - \sigma^{\phi}_{AB}\|_1$. In particular we show that for all measurements of Eve with outcome $\ket{\phi} \in {\mathcal{H}}_E$ having a probability greater than or equal to $1/16$ we have that $\|\sigma'_{AB} - \sigma^{\phi}_{AB}\|_1 \leq 17 \varepsilon / p_{\rho}$. This then implies using Lemma \[lem:steering:prob\] that $\| \sigma_{ABE} - \sigma_{AB} \otimes \sigma_E \|_1 \leq 34C \varepsilon / p_{\rho}$. In detail, using the triangle inequality we compute for the distance between $\sigma'_{AB}$ and $\sigma^{\phi}_{AB}$ $$\begin{aligned}
\|\sigma'_{AB} - \sigma^{\phi}_{AB}\|_1 & \leq \|\sigma'_{AB} - \sigma^{\alpha}_{AB}\|_1 + \|\sigma^{\alpha}_{AB} - \sigma^{\phi}_{AB}\|_1 = \frac{1}{p_{\rho}} \|(\mathcal{P}_1 - \mathcal{P}_2)(\rho'_{AB})\|_1 + \frac{1}{p_{\rho^\phi}} \|(\mathcal{P}_1 - \mathcal{P}_2)(\rho^\phi_{AB})\|_1 \notag \\
& \leq \left(\frac{1}{p_{\rho}} + \frac{1}{p_{\rho^\phi}} \right) \max_{\mu_{AB}} \|(\mathcal{P}_1 - \mathcal{P}_2)(\mu_{AB})\|_1. \label{eq:lemma:prod:3}\end{aligned}$$ Now we employ (\[eq:lemma:prod:2\]) in (\[eq:lemma:prod:3\]) which yields $$\begin{aligned}
\|\sigma'_{AB} - \sigma^{\phi}_{AB}\|_1 & \leq \left(\frac{1}{p_{\rho}} + \frac{p'_E(\phi)}{p_{\rho} p_E(\phi)} \right) \max_{\mu_{AB}} \|(\mathcal{P}_1 - \mathcal{P}_2)(\mu_{AB})\|_1 \leq \left(\frac{1}{p_{\rho}} + \frac{1}{p_{\rho} p_E(\phi)} \right) \max_{\mu_{AB}} \|(\mathcal{P}_1 - \mathcal{P}_2)(\mu_{AB})\|_1 \notag \\
& = \frac{1}{p_{\rho}} \left(1 + \frac{1}{p_E(\phi)} \right) \max_{\mu_{AB}} \|(\mathcal{P}_1 - \mathcal{P}_2)(\mu_{AB})\|_1 \leq \frac{1}{p_{\rho}} \left(1 + 16 \right) \max_{\mu_{AB}} \|(\mathcal{P}_1 - \mathcal{P}_2)(\mu_{AB})\|_1 \notag \\
&\leq \frac{17}{p_{\rho}} \varepsilon \label{eq:lemma:prod:4}\end{aligned}$$ because $p_E(\phi) \geq 1/16$ and $\max_{\mu_{AB}} \|(\mathcal{P}_1 - \mathcal{P}_2)(\mu_{AB})\|_1$ is bounded by $\varepsilon$ by assumption. Hence we apply Lemma \[lem.steering\] to $\sigma'_{ABE}$ with $\epsilon = \frac{17}{p_{\rho}} \varepsilon$ which implies for the distance between $\sigma'_{ABE}$ and $\sigma_{AB} \otimes \sigma'_{E}$ that $$\begin{aligned}
\label{eq.thm.localsteer.4}
\|\sigma'_{ABE} - \sigma_{AB} \otimes \sigma'_{E} \|_1 \leq \frac{34 \cdot 4^8}{p_{\rho}} \varepsilon\end{aligned}$$ where the factor $4^8$ is the constant $C$ of Lemma \[lem.steering\] depending on the dimensions of the systems of Alice/Bob and Eve, for which we have $n=m=2$. Furthermore, this implies via Lemma \[lem:steering:prob\] that $$\begin{aligned}
\|\sigma_{ABE} - \sigma_{AB} \otimes \sigma_{E} \|_1 \leq \frac{34 \cdot 4^8}{p_{\rho}} \varepsilon \label{eq.thm.localsteer.5}\end{aligned}$$ because $\sigma_{ABE}$ and $\sigma_{AB} \otimes \sigma_{E}$ are unitarly related to $\sigma'_{ABE}$ and $\sigma_{AB} \otimes \sigma'_{E}$ via the unitary $U$ on Eve’s system. Finally, employing (\[eq.thm.localsteer.5\]) in (\[eq.thm.localsteer\]) yields $$\begin{aligned}
\|(\mathcal{P}_1 \otimes {id}_E)(\dm{\psi}) &- (\mathcal{P}_2 \otimes {id}_E)(\dm{\psi})\|_1 = p_{\rho} \|\sigma_{ABE} - \sigma^{\alpha}_{AB} \otimes \sigma_E \|_1 \\
& \leq p_{\rho} (\|\sigma_{ABE} - \sigma_{AB} \otimes \sigma_E \|_1 + \|\sigma_{AB} \otimes \sigma_E - \sigma^{\alpha}_{AB} \otimes \sigma_E \|_1) \\
& \leq 34 \cdot 4^8 \varepsilon + \varepsilon = (34 \cdot 4^8 +1) \varepsilon.\end{aligned}$$
Proof of Lemma \[lem:forpost\] {#app:sec:post}
------------------------------
Now we turn to the proof of Lemma \[lem:forpost\] of the main text. For that purpose we remind the reader that the final state after the distillation protocol including the system of L is pure. Thus, the following Lemma will turn out to be very useful.
\[lem.localstates\] Let $\rho_{AB}$ and $\varphi_{AB} = {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \otimes \mu_{B}$ be two mixed states. Furthermore, assume that $\rho_A = {\text{tr}_{B}\left[ \rho_{AB} \right]} $ satisfies $\|\rho_A - {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \|_1 \leq \varepsilon$ and $\rho_B = {\text{tr}_{A}\left[ \rho_{AB} \right]} = \mu_{B}$. Then $\|\rho_{AB} - \varphi_{AB} \|_{1} \leq 4 \sqrt{\varepsilon}$.
By assumption we have $\| \rho_{A} - {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \|_1 \leq \varepsilon$. Moreover, let $\ket{\psi}_{ABR}$ be a purification of $\rho_{AB}$. According to Lemma A.2.7 in [@bib.renner.diss] there exists a purification $\ket{\varphi}_{A} \otimes \ket{\xi}_{BR}$ of $\varphi_{AB}$ such that $\|\ket{\psi}_{ABR} - \ket{\varphi}_{A} \otimes \ket{\xi}_{BR}\|_{\text{vec}} \leq \sqrt{\| \rho_{A} - {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \|_1 } = \sqrt{\varepsilon}$ where $\|\ket{\psi}{}\|_{\text{vec}} = \sqrt{\langle \psi | \psi \rangle}$ and $_{ABR} \langle \psi | \varphi \rangle_{A} | \xi \rangle_{BR}$ is real and non-negative. Moreover, Lemma A.2.3 of [@bib.renner.diss] gives $$\begin{aligned}
\| {\left| \psi \right\rangle \left\langle \psi \right|_{ABR}} - {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \otimes {\left| \xi \right\rangle \left\langle \xi \right|_{BR}} \|_1 \leq 2 \|\ket{\psi}_{ABR} - \ket{\varphi}_{A} \otimes \ket{\xi}_{BR}\|_{\text{vec}} \leq 2 \sqrt{\epsilon}.\end{aligned}$$ We define $\xi_{B} = {\text{tr}_{R}\left[ {\left| \xi \right\rangle \left\langle \xi \right|_{BR}} \right]}$. As the $1$-norm does not increase under the partial trace we have $$\begin{aligned}
\|\rho_{B} - \xi_{B}\|_1 \leq \|\rho_{AB} - {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \otimes \xi_{B}\|_1 \leq \|{\left| \psi \right\rangle \left\langle \psi \right|_{ABR}} - {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \otimes {\left| \xi \right\rangle \left\langle \xi \right|_{BR}}\|_1 \leq 2 \sqrt{\epsilon}\end{aligned}$$ by construction. Moreover, the assumption $\rho_{B} = \mu_{B}$ implies $\|\mu_{B} - \xi_{B}\|_1 = \|\rho_{B} - \xi_{B}\|_1 \leq 2 \sqrt{\varepsilon}$. This gives us $\|{\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \otimes \mu_{B} - {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \otimes \xi_{B} \|_1 = \|\mu_{B} - \xi_{B}\|_1 \leq 2 \sqrt{\varepsilon}$. If we combine these results we obtain $$\begin{aligned}
\|\rho_{AB} - \varphi_{AB} \|_1 = \|\rho_{AB} - {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \otimes \mu_{B} \|_1 \leq \|\rho_{AB} - {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \otimes \xi_{B} \|_1 + \|{\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \otimes \xi_{B} - {\left| \varphi \right\rangle \left\langle \varphi \right|_{A}} \otimes \mu_{B} \|_1 \leq 4 \sqrt{\varepsilon}\end{aligned}$$ which proves the claim.
Lemma \[lem.localstates\] enables us to prove Lemma \[lem:forpost\] of the main text.
Let ${\mathcal{E}}$ be the real protocol which guarantees to converge towards a unique and attracting fixed point depending on the noise parameter only. Let ${\mathcal{F}}$ be the ideal protocol as defined in the main text. Furthermore let $\rho$ be a mixed state (consisting of $n$ systems) shared by Alice and Bob. If the extension of ${\mathcal{E}}$ and ${\mathcal{F}}$ to the system of L satisfies $\|{\mathcal{E}}_{\text{L}}(\rho) - {\mathcal{F}}_{\text{L}}(\rho)\|_1 \leq \varepsilon(n)$, then $$\begin{aligned}
\|({\mathcal{E}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}}) &- ({\mathcal{F}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}})\|_1 \leq 4 \sqrt{\varepsilon(n)}\end{aligned}$$ for all purifications $\ket{\psi}_{ABE'}$ of $\rho$.
As mentioned in the main text, we introduce a two-level flag system held by Alice which indicates whether they aborted the protocol or not. So we observe $$\begin{aligned}
{\mathcal{E}}_{\text{L}}(\rho) &= p_\rho \sigma_{ABEL} \otimes {\left| ok \right\rangle \left\langle ok \right|_{}} + (1-p_\rho) \sigma^{\perp}_{ABEL} \otimes {\left| fail \right\rangle \left\langle fail \right|_{}}, \\
{\mathcal{F}}_{\text{L}}(\rho) &= p_\rho \dm{\psi^{f}}_{ABEL} \otimes {\left| ok \right\rangle \left\langle ok \right|_{}} + (1-p_\rho) \sigma^{\perp}_{ABEL} \otimes {\left| fail \right\rangle \left\langle fail \right|_{}},\end{aligned}$$ where $E$ denotes the system of leaked noise transcripts to Eve. By assumption we have $\|{\mathcal{E}}_{\text{L}}(\rho) - {\mathcal{F}}_{\text{L}}(\rho)\|_1 \leq \varepsilon(n)$. This is equivalent to $p_{\rho} \|\sigma_{ABEL} - {\left| \psi_{f} \right\rangle \left\langle \psi_{f} \right|_{ABEL}}\|_1 \leq \varepsilon(n)$ since ${\mathcal{E}}_{\text{L}}(\rho)$ and ${\mathcal{F}}_{\text{L}}(\rho)$ are equal on the fail branch. This we can rewrite to $\|\sigma_{ABEL} - {\left| \psi_{f} \right\rangle \left\langle \psi_{f} \right|_{ABEL}}\|_1 \leq \varepsilon(n) / p_{\rho}$.
Moreover, applying the real and ideal protocol to the purification $\ket{\psi}_{ABE'}$ results in $$\begin{aligned}
({\mathcal{E}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}}) &= p_{\rho} \sigma_{ABEE'} \otimes {\left| ok \right\rangle \left\langle ok \right|_{}} + (1-p_{\rho}) \sigma^{\perp}_{ABEE'} \otimes {\left| fail \right\rangle \left\langle fail \right|_{}}, \\
({\mathcal{F}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}}) &= p_{\rho} \sigma^{f}_{ABE} \otimes \rho_{E'} \otimes {\left| ok \right\rangle \left\langle ok \right|_{}} + (1-p_{\rho}) \sigma^{\perp}_{ABEE'} \otimes {\left| fail \right\rangle \left\langle fail \right|_{}}.\end{aligned}$$ Again, both expression are equal in the fail branch, thus the $1$-norm simplifies to $$\begin{aligned}
\label{equ.global.1}
\|({\mathcal{E}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}}) - ({\mathcal{F}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}})\|_1 = p_\rho \|\sigma_{ABEE'} - \sigma^{f}_{ABE} \otimes \rho_{E'} \|_1.\end{aligned}$$ Hence it is sufficient to show $p_\rho \|\sigma_{ABEE'} - \sigma^{f}_{ABE} \otimes \rho_{E'}\|_1 \leq 4 \sqrt{\varepsilon(n)}$. We observe that by introducing the system $L$ held by L that $$\begin{aligned}
\label{inequ.global.extension}
p_\rho \|\sigma_{ABEE'} - \sigma^{f}_{ABE} \otimes \rho_{E'}\|_1 \leq p_\rho \|\sigma_{ABELE'} - \dm{\psi^\alpha}_{ABEL} \otimes \rho_{E'}\|_1.\end{aligned}$$ One easily verifies ${\text{tr}_{E'}\left[ \sigma_{ABELE'} \right]} = \sigma_{ABEL}$ and ${\text{tr}_{ABEL}\left[ \sigma_{ABELE'} \right]} = \rho_{E'}$ because the system $E'$ is not changed by the protocol ${\mathcal{E}}$. Moreover, by assumption we have $\|\sigma_{ABEL} - {\left| \psi_{f} \right\rangle \left\langle \psi_{f} \right|_{ABEL}}\|_1 \leq \varepsilon(n) / p_{\rho}$. Thus we apply Lemma \[lem.localstates\] to $\rho_{A'B'} := \sigma_{ABELE'}$ and $\varphi_{A'B'} = {\left| \psi_f \right\rangle \left\langle \psi_f \right|_{ABEL}} \otimes \rho_{E'}$ where $A' := ABEL$ and $B' := E'$ which implies $$\begin{aligned}
\label{inequ.global.2}
\|\sigma_{ABELE'} - {\left| \psi_{f} \right\rangle \left\langle \psi_{f} \right|_{ABEL}} \otimes \rho_{E'}\|_1 \leq 4 \sqrt{\varepsilon(n) / p_{\rho}}.\end{aligned}$$ Employing (\[inequ.global.extension\]) and (\[inequ.global.2\]) in (\[equ.global.1\]) yields $$\begin{aligned}
\|({\mathcal{E}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}}) - ({\mathcal{F}}\otimes id_{E'})({\left| \psi \right\rangle \left\langle \psi \right|_{ABE'}})\|_1 \leq p_\rho 4 \sqrt{\varepsilon(n) / p_{\rho}} = 4 \sqrt{p_{\rho} \varepsilon(n)} \leq 4 \sqrt{\varepsilon(n)}\end{aligned}$$ which completes the proof.
Confidentiality of entanglement distillation protocols whenever the noise transcripts leak {#app:sec:leak}
==========================================================================================
In this section we show how the confidentiality guarantees regarding an entanglement distillation protocol can be extended to the case whenever the noise transcripts leak to Eve. We remind the reader that it is not necessary to leak the noise transcripts to Eve after every single distillation round. It is sufficient to copy all noise transcripts at the very end to Eve’s register, as L is not accessible and Eve is not part of the protocol being executed by Alice and Bob.
Let ${\mathcal{E}}$ be the real protocol and ${\mathcal{F}}$ be the ideal protocol. Furthermore, let ${\mathcal{E}}^l$ be the real and ${\mathcal{F}}^l$ be the ideal protocol when the noise transcripts leak to Eve. Then $$\begin{aligned}
\|({\mathcal{E}}\otimes {id}_E) (\dm{\psi}_{ABE}) - ({\mathcal{F}}\otimes {id}_E) (\dm{\psi}_{ABE})\|_1 \leq \varepsilon(n)\end{aligned}$$ implies that $$\begin{aligned}
\label{eq.supp.leakreduction}
\|({\mathcal{E}}^l \otimes {id}_E) (\dm{\psi}_{ABE}) - ({\mathcal{F}}^l \otimes {id}_E) (\dm{\psi}_{ABE})\|_1 \leq 2 \sqrt{\varepsilon(n)}.\end{aligned}$$ for all purifications $\ket{\psi}_{ABE}$ of initial state $\rho_{AB}$ consisting of $n$ systems
We observe that $$\begin{aligned}
({\mathcal{E}}\otimes {id}_E)(\dm{\psi}) &= p_{\rho} \sigma_{ABE} \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \sigma_E \otimes \dm{fail}, \\
({\mathcal{F}}\otimes {id}_E)(\dm{\psi}) &= p_{\rho} \sigma^{\alpha}_{AB} \otimes \sigma_E \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \sigma_E \otimes \dm{fail}.\end{aligned}$$ So by assumption we have $$\begin{aligned}
\| ({\mathcal{E}}\otimes {id}_E)(\dm{\psi}) - ({\mathcal{F}}\otimes {id}_E)(\dm{\psi}) \|_1 = p_{\rho} \| \sigma_{ABE} - \sigma^{\alpha}_{AB} \otimes \sigma_E\|_1 \leq \varepsilon(n),\end{aligned}$$ i.e. $\| \sigma_{ABE} - \sigma^{\alpha}_{AB} \otimes \sigma_E\|_1 \leq \varepsilon(n) / p_{\rho}$. As outlined in the main text we model L in terms of purifications. Because purifications are unitarly equivalent we choose a particular purification of $\sigma^{\alpha}_{AB} \otimes \sigma_E$. Thus we fix $\ket{\psi_{{\mathcal{F}}}}_{ABL_1L_2E} = \ket{\psi'}_{ABL_1} \otimes \ket{\psi''}_{L_2E}$ where $\ket{\psi'}_{ABL_1} = \sum_{i,j} \omega_{ij}(\alpha)\ket{B_{ij}}_{AB} \ket{ij}_{L_1}$. The purifying systems $L_1$ and $L_2$ we attribute to the Lab Demon. Moreover, according to Lemma A.2.7 in [@bib.renner.diss] there exists a purification $\ket{\psi_{{\mathcal{E}}}}$ of $\sigma_{ABE}$ such that $\|\ket{\psi_{{\mathcal{F}}}}_{ABL_1L_2E} - \ket{\psi_{{\mathcal{E}}}}_{ABL_1L_2E} \|_{\text{vec}} \leq \sqrt{\varepsilon(n) / p_{\rho}}$ where $\|\ket{\psi}{}\|_{\text{vec}} = \sqrt{\langle \psi | \psi \rangle}$ and $_{ABL_1L_2E} \langle \psi_{{\mathcal{F}}} | \psi_{{\mathcal{E}}} \rangle_{ABL_1L_2E}$ is real and non-negative. Furthermore, Lemma A.2.3 of [@bib.renner.diss] gives $$\begin{aligned}
\label{eq.leakreduction.0}
\| \dm{\psi_{{\mathcal{E}}}}_{ABL_1L_2E} - \dm{\psi_{{\mathcal{F}}}}_{ABL_1L_2E} \|_1 \leq 2 \|\ket{\psi_{{\mathcal{E}}}}_{ABL_1L_2E} - \ket{\psi_{{\mathcal{F}}}}_{ABL_1L_2E} \|_{\text{vec}} \leq 2 \sqrt{\varepsilon(n) / p_{\rho}}.\end{aligned}$$ When the noise transcripts leak to Eve, L effectively copies the noise transcripts $\ket{ij}_{L_1}$ to Eve, resulting in the pure state $\ket{\phi}_{ABL_1L_2EE'} = \left(\sum_{i,j} \ket{B_{ij}}_{AB} \ket{ij}_{L_1} \ket{ij}_{E'}\right) \otimes \ket{\psi}_{L_2E}.$ Hence we can model the leakage of the noise transcripts to Eve by a unitary $U_{M}$ such that $U_{M} \ket{\psi_{{\mathcal{F}}}}_{ABL_1L_2E} \ket{0}_{E'} = \ket{\phi}_{ABL_1L_2EE'}$. For the protocol when the noise transcripts leak to Eve we have $$\begin{aligned}
({\mathcal{E}}^l \otimes {id}_E)(\dm{\psi}) &= p_{\rho} \sigma'_{ABE} \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \sigma_E \otimes \dm{fail} \\
& = p_{\rho} {\text{tr}_{L_1 L_2}\left[ U_{M} \dm{\psi_{{\mathcal{E}}}} U^\dagger_{M} \right]} \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \sigma_E \otimes \dm{fail} \\
({\mathcal{F}}^l \otimes {id}_E)(\dm{\psi}) &= p_{\rho} \sigma'^{\alpha}_{ABE} \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \sigma_E \otimes \dm{fail} \\
& = p_{\rho} {\text{tr}_{L_1 L_2}\left[ U_{M} \dm{\psi_{{\mathcal{F}}}} U^\dagger_{M} \right]} \otimes \dm{ok} + (1-p_\rho) \sigma^{\perp}_{AB} \otimes \sigma_E \otimes \dm{fail}.\end{aligned}$$ Because the real and the ideal protocol are equal in the fail-branch we obtain by using (\[eq.leakreduction.0\]) $$\begin{aligned}
\|({\mathcal{E}}^l \otimes {id}_E) (\dm{\psi}_{ABE}) &- ({\mathcal{F}}^l \otimes {id}_E) (\dm{\psi}_{ABE})\|_1 \\
& = p_{\rho} \left\| \sigma'_{ABE} \otimes \dm{ok} - \sigma'^\alpha_{ABE} \otimes \dm{ok} \right\|_1 \\
& = p_{\rho} \left\|{\text{tr}_{L_1 L_2}\left[ U_{M} \dm{\psi_{{\mathcal{E}}}} U^\dagger_{M} \right]} - {\text{tr}_{L_1 L_2}\left[ U_{M} \dm{\psi_{{\mathcal{F}}}} U^\dagger_{M} \right]} \right\|_1 \\
& \leq p_{\rho} \left\|U_{M} \dm{\psi_{{\mathcal{E}}}} U^\dagger_{M} - U_{M} \dm{\psi_{{\mathcal{F}}}} U^\dagger_{M} \right\|_1 \\
& = p_{\rho} \left\| \dm{\psi_{{\mathcal{E}}}} - \dm{\psi_{{\mathcal{F}}}} \right\|_1 \leq 2 \sqrt{\varepsilon(n) p_{\rho}} \leq 2 \sqrt{\varepsilon(n)}\end{aligned}$$ which proves (\[eq.supp.leakreduction\]).
Thus the confidentiality of a protocol where the noise transcripts leak to Eve is bounded by the confidentiality of the same protocol when they do not.
Quantum one-time padding after the real protocol {#sec.supp.decouple}
================================================
In this section we show that a final secret twirl applied to the pair of Alice and Bob decouples Eve completely from the remaining state. Keep in mind that for this Alice and Bob require two classical bits unknown to Eve.
Recall that the state of Alice, Bob, Eve, and L after $n$ distillation rounds is pure and of the form $\ket{\psi}{} = \sum_{i,j,k,l} P_{ijkl} \ket{B_{ij}}_{AB} \ket{\eta_{kl}}_{L} \ket{\eta_{ijkl}}_{E}$. Tracing over L yields the mixed state $$\begin{aligned}
\label{eqn.decouple.1}
\rho_{ABE} = \sum\limits_{i_1,i_2,j_1,j_2}\sum\limits_{k,l} P_{i_1 j_1 k l} P^{*}_{i_2 j_2 k l}{\left| B_{i_1 j_1} \right\rangle \left\langle B_{i_2 j_2} \right|_{}} \otimes {\left| \eta_{i_1 j_1 k l} \right\rangle \left\langle \eta_{i_2 j_2 k l} \right|_{}}.\end{aligned}$$ Suppose Alice and Bob apply a secret twirl ${\mathcal{T}}$ to (\[eqn.decouple.1\]), i.e. they apply stochastically the family of operators $\lbrace id, K_1, K_2, K_1 K_2 \rbrace$ where $K_1 = {\sigma_x}\otimes {\sigma_x}$ and $K_2 = {\sigma_z}\otimes {\sigma_z}$. These are two stabilizers of the Bell state, i.e., $$\begin{aligned}
K^{r_1}_1 \ket{B_{i_1 j_1}}{} &= (-1)^{i_1 r_1} \ket{B_{i_1 j_1}}{}, \\
K^{r_2}_2 \ket{B_{i_1 j_1}}{} &= (-1)^{j_1 r_2} \ket{B_{i_1 j_1}}{}.\end{aligned}$$ Hence, applying the secret twirl ${\mathcal{T}}$ to (\[eqn.decouple.1\]) gives $$\begin{aligned}
{\mathcal{T}}\rho_{ABE} &= \sum\limits_{\stackrel{r_1,r_2}{i_1,i_2,j_1,j_2,k,l}} \frac{1}{4} P_{i_1 j_1 k l} P^{*}_{i_2 j_2 k l} K^{r_1}_1 K^{r_2}_2 {\left| B_{i_1 j_1} \right\rangle \left\langle B_{i_2 j_2} \right|_{}} K^{r_1}_1 K^{r_2}_2 \otimes {\left| \eta_{i_1 j_1 k l} \right\rangle \left\langle \eta_{i_2 j_2 k l} \right|_{}} \\
& = \sum\limits_{\stackrel{r_1,r_2}{i_1,i_2,j_1,j_2,k,l}}(-1)^{i_1 r_1} (-1)^{j_1 r_2} (-1)^{i_2 r_1} (-1)^{j_2 r_2} \frac{1}{4} P_{i_1 j_1 k l} P^{*}_{i_2 j_2 k l} {\left| B_{i_1 j_1} \right\rangle \left\langle B_{i_2 j_2} \right|_{}} \otimes {\left| \eta_{i_1 j_1 k l} \right\rangle \left\langle \eta_{i_2 j_2 k l} \right|_{}} \\
& = \sum\limits_{i_1,i_2,j_1,j_2} {\left| B_{i_1 j_1} \right\rangle \left\langle B_{i_2 j_2} \right|_{}} \otimes \frac{1}{4} \sum\limits_{k,l} P_{i_1 j_1 k l} P^{*}_{i_2 j_2 k l} {\left| \eta_{i_1 j_1 k l} \right\rangle \left\langle \eta_{i_2 j_2 k l} \right|_{}} \sum\limits_{r_1,r_2} (-1)^{(i_1 + i_2) r_1} (-1)^{(j_1 + j_2) r_2} \\
& = \sum\limits_{i_1,j_1} {\left| B_{i_1 j_1} \right\rangle \left\langle B_{i_1 j_1} \right|_{}} \otimes \sum\limits_{k,l} |P_{i_1 j_1 k l}|^2 {\left| \eta_{i_1 j_1 k l} \right\rangle \left\langle \eta_{i_1 j_1 k l} \right|_{}}.\end{aligned}$$ Note that in the resulting state $\sum\limits_{i_1,j_1} {\left| B_{i_1 j_1} \right\rangle \left\langle B_{i_1 j_1} \right|_{}} \otimes \sum\limits_{k,l} |P_{i_1 j_1 k l}|^2 {\left| \eta_{i_1 j_1 k l} \right\rangle \left\langle \eta_{i_1 j_1 k l} \right|_{}}$ Eve decouples, i.e. Alice/Bob and Eve have a separable state. The obtained resource state can be used to establish a confidential quantum channel by means of quantum teleportation.
Robustness of recurrence-type entanglement distillation protocol {#app:sec:robust}
================================================================
To complete the security characterization of entanglement distillation protocols we also consider the robustness of an entanglement distillation protocol. To define this term precisely we first need the definition of a honest eavesdropper.
We call an eavesdropper honest, if the states sent by the eavesdropper are of the form $\ket{B_{00}}{}^{\otimes 2^n}$.
It is obvious that a honest eavesdropper is not entangled with the ensemble delivered to Alice and Bob via the noisy quantum channel. Moreover we formally define the robustness of a protocol by:
We call a protocol ${\mathcal{E}}^\alpha$ $\varepsilon_R$-robust, if for a honest eavesdropper the probability of aborting the protocol is at most $\varepsilon_R$.
Now we show that we can tune the robustness of a recurrence-type entanglement distillation protocol to be exponentially small in terms of necessary number of input pairs.
Let $M \in {\mathbb{N}}$ such that Alice and Bob achieve $\varepsilon$-confidentiality by succeeding $M$ rounds of a recurrence-type entanglement distillation protocol. Furthermore assume that Alice and Bob receive $n$ pairs from a honest eavesdropper over the quantum channel $\Phi^{\otimes n}$ (where $\Phi(\rho) = \beta \rho + (1-\beta)/4 \left(\sum_{i,j} \sigma_{i,j} \rho \sigma_{i,j} \right)$) such that, after the parameter estimation step of the proposed protocol, $k - \sqrt{k}$ pairs (where $k - \sqrt{k} = c 2^M$ and $c = \xi 2^{M+2}$) are left for entanglement distillation. Then, the robustness $\varepsilon_{R}$ of the protocol is bounded by $$\begin{aligned}
\varepsilon_{R} \leq \exp\left(- (3\beta - 4 F_{min}(\alpha) - 1)^2 \sqrt{k}/128 \right) + M \exp\left( - \xi \right).\end{aligned}$$
The basic idea of the proof is to request sufficiently many pairs from Eve such that the probabilities of abort during the protocol to be exponentially small while still having enough pairs left to achieve $M$ rounds of a recurrence-type entanglement distillation protocol. We divide the proof into two parts:
- Part 1: We prove that the probability of aborting the recurrence-type entanglement distillation protocol due to parameter estimation is exponentially small.
- Part 2: We prove the same holds true for aborting the protocol during entanglement distillation.
Suppose Eve sends the state $\ket{B_{00}}{}^{\otimes n}$ through the noisy quantum channel $\Phi^{\otimes n}$ to Alice and Bob. Applying $\Phi$ to ${\left| B_{00} \right\rangle \left\langle B_{00} \right|_{}}$ yields $$\begin{aligned}
\label{eqn.robust.qchannel}
\rho_{AB} = \Phi\left({\left| B_{00} \right\rangle \left\langle B_{00} \right|_{}} \right) = (3\beta + 1)/4 {\left| B_{00} \right\rangle \left\langle B_{00} \right|_{}} + (1-\beta)/4 \left({\left| B_{10} \right\rangle \left\langle B_{10} \right|_{}} + {\left| B_{01} \right\rangle \left\langle B_{01} \right|_{}} + {\left| B_{11} \right\rangle \left\langle B_{11} \right|_{}} \right).\end{aligned}$$ Thus the state Alice and Bob receive is $\rho_{AB}^{\otimes n}.$ According to the preceding protocols proposed in the main text, Alice and Bob apply a symmetrization to $\rho_{AB}^{\otimes n}$, and, depending on the noise level of the apparatus, they might have to trace out $n-k$ pairs or not. For the subsequent analysis we assume that this tracing out step is necessary, i.e. the de-Finetti-based reduction needs to be applied. Hence, Alice and Bob continue by applying a twirl to each remaining pair. Since $\rho_{AB}^{\otimes k}$ is invariant under permutations and $\rho_{AB}$ is Bell-diagonal, the remaining state after twirling is equal to $\rho_{AB}^{\otimes k}$.
Next, they apply to $\sqrt{k}$ of the remaining $k$ pairs the parameter estimation for estimating the fidelity of each pair. Necessary for convergence of all recurrence-type entanglement distillation protocols is that the fidelity $F$ of $\rho_{AB}$ with $\ket{B_{00}}{}$ satisfies $F > F_{min}(\alpha)$. Hence this step is crucial in order to guarantee successful distillation.
For that purpose, we measure $\lfloor \sqrt{k} \rfloor$ of $k$ pairs by applying two-qubit measurements. To be more precise, we apply a ${\sigma_x}\otimes {\sigma_x}$ to the first and ${\sigma_z}\otimes {\sigma_z}$ measurement to the second pair. We refer to this measurements by $M_1$ and $M_2$ respectively. We observe that the state $\ket{B_{00}}{}$ is a common eigenstate of $M_1$ and $M_2$ with eigenvalue $1$. We define to each pair of pairs a random variable $X_i$ for $i \in \lbrace 1,.., \lfloor \sqrt{k} \rfloor /2 \rbrace$ with $X_i = 1$ whenever both measurements $M_1$ and $M_2$ yield outcome $1$ and $X_i = 0$ else.
Furthermore we assume for the expected value $\mathbb{E}(X)$ of the fidelity with $\ket{B_{00}}$ that $\mathbb{E}(X) = F_{min}(\alpha) + \delta$, where $\delta > 0$ will be fixed below. The protocol will be aborted if the estimate is below $F_{min}(\alpha) + \delta$.
From (\[eqn.robust.qchannel\]) we observe that, whenever $(3\beta + 1)/4 \leq F_{min}(\alpha)$, the entanglement distillation protocol will not distill any entanglement. This implies for the quantum channel $\Phi$ that, if $\beta \leq (4F_{min}(\alpha) - 1)/3$ the parameter estimation step will abort, independent of the input provided by Eve. Thus we assume for the subsequent analysis that $\beta > (4F_{min}(\alpha) - 1)/3$.
Moreover we define $\eta = \delta/2$. Hence we get by the Hoeffdings inequality [@Hoeffding] for the probability of an error larger than $\eta$ in our measured estimate $\overline{X}$ for the fidelity the following expression: $$\begin{aligned}
\mathbb{P}(|\mathbb{E}(X) - \overline{X}| \geq \eta) \leq \exp\left( - \eta^2 \sqrt{k}/2 \right) =: p_{\text{pe-abort}}.\end{aligned}$$ Thus the probability of aborting the protocol due to an error in the parameter estimation is exponentially small in number of necessary input pairs. In order to fix $\delta$ we recognize that Alice and Bob abort the protocol whenever $(3\beta + 1)/4 < F_{min}(\alpha) + \delta$. This is equivalent to $\delta > (3\beta - 4 F_{min}(\alpha) - 1)/4$. Inserting the definition of $\eta$ yields $\eta > (3\beta - 4 F_{min}(\alpha) - 1)/8$ and thus $p_{\text{pe-abort}} < \exp\left(- (3\beta - 4 F_{min}(\alpha) - 1)^2 \sqrt{k}/128 \right)$.
What remains to be shown is that the probability of aborting the protocol in the distillation phase is also exponentially small in the number of input pairs. For that purpose, we assume that the noise level $\alpha$ of the apparatus is such that distillation is feasible. In the following we show that we can force the probability of abort due to entanglement distillation to be exponentially small in terms of requested input pairs.
We assume that Alice and Bob are left with $c 2^M$ pairs after parameter estimation. Recall that the Chernoff inequality for a sequence of independent Bernoulli random variables $X_1, ... , X_n$ where $\mathbb{P}\left(X_i = 1 \right) = p$ and $d \in [0,1]$ reads as $$\begin{aligned}
\mathbb{P}\left(\sum_i X_i \leq (1-d) p n \right) \leq \exp\left( - \frac{d^2}{2} p n \right).\end{aligned}$$ Moreover, we observe that a basic distillation step can be modelled by a Bernoulli random variable $X_i$ where $\mathbb{P}\left(X_i = 1 \right) = p$ is the probability of succeeding (measurement outcomes coincide).
Suppose we perform $m$ rounds of entanglement distillation. Let $N_m$ denote the number of input pairs to the $m$-th round and let $d \in [0,1]$.
Then the Chernoff inequality implies that the probability that less than $(1-d)p N_m$ basic distillation steps at round $m$ have succeeded is bounded by $\exp\left( - \frac{d^2}{2} p N_m \right)$, i.e. $$\begin{aligned}
\label{inequ.robustness.1}
p_{\text{abort},m} = \mathbb{P}\left(\sum_i X_i \leq (1-d) p N_m \right) \leq \exp\left( - \frac{d^2}{2} p N_m \right).\end{aligned}$$ But this also implies that, with probability $1-p_{\text{abort},m}$, at least $(1-d) p N_m + 1$ basic distillation steps have succeeded at round $m$. Thus we may safely assume that $N_{m+1} = (1-d) p N_m + 1$. The situation is summarized in Fig. \[app:fig:robustness\]. Furthermore we have $N_{1} = c 2^M$. Eliminating the recurrence relation yields $N_{m+1} = (1-d)^{m} p^{m} c 2^M + \sum_{i=0}^{m-1}(1-d)^i p^i$. This implies for (\[inequ.robustness.1\]) $$\begin{aligned}
p_{\text{abort},m} \leq \exp\left( - \frac{d^2}{2} p \left((1-d)^{m-1} p^{m-1} c 2^M + \underbrace{\sum_{i=0}^{m-2}(1-d)^i p^i}_{> 0} \right) \right) \leq \exp\left( - \frac{d^2}{2} (1-d)^{m-1} p^{m} c 2^M \right).\end{aligned}$$ Furthermore, we compute the probability of aborting the protocol at distillation round $m$ (assuming that the previous rounds $1,..,m-1$ succeeded) by $$\begin{aligned}
\label{inequ.robustness.2}
p_{\text{abort at round $m$}} = p_{\text{abort},m} \underbrace{\prod\limits^{m-1}_{k=1} p_{\text{succeed},k}}_{\leq 1} \leq p_{\text{abort},m} \leq \exp\left( - \frac{d^2}{2} (1-d)^{m-1} p^{m} c 2^M \right).\end{aligned}$$ The events of aborting the distillation protocol at two different rounds $i$ and $j$ are disjoint. Thus we have for the probability of aborting in any of $m$ rounds $p_{\text{abort in any of $m$ rounds}} = \sum_{k=1}^m p_{\text{abort at round $k$}}$. A simple consequence thereof is $$\begin{aligned}
\label{inequ.robustness.3}
p_{\text{abort in any of $M$ rounds}} &= \sum\limits^M_{k=1} p_{\text{abort at round $k$}} \leq \sum\limits^M_{k=1} \exp\left( - \frac{d^2}{2} (1-d)^{k-1} p^{k} c 2^M \right)\end{aligned}$$ where we have used (\[inequ.robustness.2\]). Inserting $p=1/2$ and $d =1/2$ in (\[inequ.robustness.3\]) yields $$\begin{aligned}
\notag
p_{\text{abort in any of $M$ rounds}} & \leq \sum\limits^M_{k=1} \exp\left( - \frac{1}{8} \frac{1}{2^{2k-1}} c 2^M \right) = \sum\limits^M_{k=1} \exp\left( - c 2^{M-2k-2} \right) \leq M \exp\left( - c 2^{M-2M-2} \right) \\ \label{inequ.robustness.4}
& = M \exp\left( - c 2^{-(M+2)} \right).\end{aligned}$$ By assumption we have $c = 2^{M+2} \xi$ which implies for (\[inequ.robustness.4\]) $$\begin{aligned}
p_{\text{abort in any of $M$ rounds}} & \leq M \exp\left( - \xi 2^{M+2} 2^{-(M+2)} \right) = M \exp\left( - \xi \right).\end{aligned}$$ Thus, the probability of aborting the protocol satisfies $$\begin{aligned}
\varepsilon_{R} & \leq p_{\text{pe-abort}} + (1-p_{\text{pe-abort}})p_{\text{abort in any of $M$ rounds}} \leq \exp\left(- (3\beta - 4 F_{min}(\alpha) - 1)^2 \sqrt{k}/128 \right) + M \exp\left( - \xi \right)\end{aligned}$$ which completes the proof.
Establishing a confidential quantum channel {#app:sqc}
===========================================
For illustration purposes, we show how confidential quantum channels can be realized using our proposal in conjunction with standard teleportation. By our results, the joint state of Alice, Bob, and Eve after the distillation protocol is $\epsilon$ close to the output of the ideal protocol. The latter, since the register of L is not accessible to any of the parties and thus is traced out, yields the state of the form (provided the protocol was not aborted) The teleportation of any state $\rho$ from Alice to Bob will yield the state Thus the only information Eve can obtain is what noise operator was applied on the teleported state, and nothing more – thus, the channel is confidential. Moreover, the probabilities for the different noise processes are not under Eve’s control, but depend on the local devices.
[——]{}
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[^1]: E.g., we assume very primitive, but trusted, quantum devices, such as a device which can either forward an input quantum system, or measure it in one basis. Already such a simple device invalidates our no-go observation.
[^2]: The robustness is quantified by the abort probability in the all-honest, but noisy setting.
[^3]: We assume that the noise characteristics of the quantum gates are constant throughout the protocol.
[^4]: Technically, inequality (\[eq:local-iid\]) is a statement about the operator norm-induced distance on maps, where expression of (\[eqn.confidentiality\]) is the completely bounded diamond norm, relevant for security statements.
[^5]: For this simplified analysis we assumed that no parameter estimation is necessary.
[^6]: We remark that a similar analysis can be performed by modelling local operations of Alice and Bob sepearetely in this way.
|
---
abstract: 'In this paper, we give a brief review of the Minimal Supersymmetric Standard Model (MSSM) and “$\mu$ from $\nu$” Supersymmetric Standard Model ($\mu \nu$SSM). Then we propose a generalization of $\mu \nu$SSM in order to explain the recent ATLAS, CMS and LHCb results. This “new" $\mu \nu$SSM generalizes the superpotential $W_{suppot}$ of $\mu \nu$SSM by including two terms that generate a mixing among leptons, gauginos and higgsinos while keeping the charginos and neutralinos masses unchanged. Also, it is potentially interesting for cosmological applications as it displays flat directions of the superpotential and a viable leptogenesis mechanism.'
author:
- |
M. C. Rodriguez and I. V. Vancea\
[^1] *[Grupo de F[í]{}sica Teórica e Matemática Física, Departamento de Física, ]{}*\
*[Universidade Federal Rural do Rio de Janeiro (UFRRJ),]{}*\
*[Cx. Postal 23851, BR 465 Km 7, 23890-000 Seropédica - RJ, Brasil ]{}*
date: 05 July 2016
title: '**Flat Directions and Leptogenesis in a “New" $\mu \nu$SSM**'
---
PACS number(s): 12.60. Jv
Keywords: Supersymmetric models
Introduction
============
Despite its successful predictions, the Standard Model (SM) suffers from a major drawback as it contains massless neutrinos to all orders in the perturbation theory. Even after including the non-perturbative effects, this problem persists. This is in contradiction with the experimental results that suggest that the neutrinos have non-zero masses and oscillations. The best-fit values at $1 \sigma$ error level for these neutrino oscillation parameters in the three-flavor framework are summarised as follows [@nu_best-fit] $$\begin{aligned}
\Delta |m_{atm}^{2}|&=& \Delta |m^{2}_{31}| = 2.40^{+0.12}_{-0.11} \times 10^{-3} {\mbox eV}^{2} \;,\quad
\sin^{2} \theta_{atm}= \sin^{2} \theta_{12} = 0.304^{+0.022}_{-0.016}\;, \nonumber \\
\Delta m_{solar}^{2}&=& \Delta m^{2}_{21} = 7.65^{+0.23}_{-0.20} \times 10^{-5} {\mbox eV}^{2} \;,\quad
\sin^{2} \theta_{solar}=\sin^{2} \theta_{23} = 0.50^{+0.07}_{-0.06}\;, \nonumber \\
\sin^{2} \theta_{13} &\leq& 0.01^{+0.016}_{-0.011}\;. \nonumber \\
\label{eq1:best-fit_mass}\end{aligned}$$ Thus, the oscillation experiments indicate that at least some neutrinos must be massive. However, the above relations do not provide the overall scale of masses which means that other methods must be employed to understand the neutrino mass spectrum. One way to obtain meaningful bounds on the absolute scale for the neutrinos is to look for kinematic effects that can be present as a consequence of their non-zero masses in the tritium $\beta$-decay (${}^{3}$H $\rightarrow {}^{3}{\mbox He} + \overline{\nu}_{e} + e^{-}$). Two groups from Mainz [@mainz] and from Troitsk [@troitsk], respectively, have reported on the bounds of $m_\nu < 2.3$ eV and $m_\nu < 2.5$ eV. Also, the upcoming KATRIN experiment [@katrin] is expected to produce results at a sensitivity of about 0.3 eV, which will further narrow down the scale of the neutrino spectrum. Another way to probe the neutrino mass scale is via studies of the lepton number ($L$) violating neutrinoless double $\beta$-decay ($^{A}_{Z}\left[\mbox{Nucl}\right]
\rightarrow \;^{\;\;\;\;A}_{Z+2}\left[\mbox{Nucl}^{\prime}\right] + 2 e^{-}$) [@bb_Maj_test]. Several groups such as Heidelberg-Moscow [@heidelberg-moscow] and IGEX [@IGEX] collaborations conducted experiments with $^{76}{\mbox Ge}$, while the more recent CUORICINO experiment [@CUORICINO] used $^{130}{\mbox Te}$ to test the lepton number conservation. The best upper bounds on the decay lifetimes are presently provided by CUORICINO (which is still running), whose results are translated to $$m_{\nu}< 0.19 -0.68 {\mbox eV } \;(90\% {\mbox C.L.}) \;,
\label{eq1:numass_CUOR}$$ for the neutrino mass. Note that the large range is due to the uncertainty in the nuclear matrix elements. Upcoming experiments like CUORE [@CUORE], GERDA [@GERDA] and MAJORANA [@MAJORANA] are expected to further improve these results with projected sensitivity of about 0.05 eV. Finally, it must be mentioned that some of the strongest bounds on the overall scale for neutrino masses come from cosmology. The studies of the data from the Wilkinson Microwave Anisotropy Probe (WMAP) and the Sloan Digital Sky Survey (SDSS) have deduced that the sum of neutrino masses (three species assumed) is constrained by $\sum_{i} \,
|m_{i}| \leq 0.6$ [@wmap_numass] and $1.6$ eV [@sdss_numass].
On the other hand, the LhCb reported recently a deviation of $2.6 \sigma$ of the measured ratios of the branching fractions $R_{K}$ in the individual lepton flavour model with respect to the SM in the low invariant mass region given by $1\ {\mbox GeV}^2 \leq M_{\ell\ell} \leq 6\ {\mbox GeV}^2$ [@Aaij:2014ora]. In this range, $R_{K}$ is defined as $$R_K =
\frac{\int^{q^{2}_{max}}_{q^{2}_{min}}\frac{d\Gamma(B^+ \rightarrow K^+ \mu^+ \mu^-)}{dq^2} dq^2}{\int^{q^{2}_{max}}_{q^{2}_{min}} \frac{d\Gamma(B^+ \rightarrow K^+ e^+ e^-)}{dq^2} dq^2}
\label{RK}$$ Thus, the experimental results from [@Aaij:2014ora] put new numerical constraints on the scalar and pseudoscalar couplings. As observed in [@Aaij:2014ora; @Aaij:2013pta], the low invariant mass range excludes the resonant regions $J/\psi \rightarrow \mu^+ \mu^- $ and $J/\psi \rightarrow e^+ e^-$ thus improving the theoretical predictions. After these interesting results, the CMS collaboration published an intriguing deviation from the SM in the $eejj$ channel, in the mass region $1.8$ [TeV]{}$< m_{eejj}<2.2 {\mbox TeV}$. No significant deviation was observed in the $\mu\mu jj$ channel [@Khachatryan:2014dka; @CMS1]. The ATLAS measured an excess[^2] with respect to SM predictions in the production of di-electroweak gauge bosons VV (where V= W;Z) that decay hadronically [@Aad:2015owa]. The ATLAS and CMS collaborations have recently presented the results of di-photon resonance searches[^3] in early Run II of $\sqrt s=13$ TeV data [@ATLAS-CONF-2015-081; @CMS:2015dxe; @atlas13; @CMS:2016owr].
Beside the neutrino masses and the results from Atlas, LhCb and CMS, there are other features of the SM that require an explanation from a more fundamental point of view:
1. The coupling constants do not meet at a single definite value [@amaldi].
2. The hierarchy problem [@hier].
3. The naturalness or fine tuning problem [@fine].
4. The large number of parameters [@dress; @Baer:2006rs; @Aitchison:2005cf].
Also, it is expected that in a fundamental theory, the gravity take a natural place alongside the other three fundamental interactions.
One promising class of theories that could solve the problems of the SM is formed by the supersymmetric extensions of the SM based on a postulated fundamental symmetry between the bosons and the fermions. The model from this class that contains a minimum number of physical states and interactions is the Minimal Supersymmetric Standard Model (MSSM) [@dress; @Baer:2006rs; @Aitchison:2005cf; @R; @ssm; @grav][^4]. The MSSM suffers from the $\mu$-problem which is the generation of a $\mu$ coupling in the $\mu \hat{H}_{1}\hat{H}_{2}$ term of the order of the electro-weak scale. The $\mu$-problem is solved by $\mu \nu$SSM proposed in [@LopezFogliani:2005yw][^5] which represents a modification of the MSSM by introducing new Yukawa interactions $Y_{\nu}^{ij} \, \hat{H}_{2}\, \hat{L}_{i} \, \hat{\nu}^{c}_{j}$ that generates light neutrino masses, as we will present at Sec.(\[sec:newsup\]).
The aim of the present paper is to propose a modification $\mu\nu$SSM that can explain the recent data from Atlas, CMS and LHCb by introducing new interactions among the leptons with gauginos and higgsinos while the masses of charginos and neutralinos are left unchanged. Since the new model has an explicit broken $R$-parity and lepton number, there are flat directions of the superpotential that can generate the cosmological inflation. Also, the matter anti-matter asymmetry could be obtained from the letogenesis mechanism.
This paper is organized as follows. In order to make the paper self-contained, we review in Section 2 the $\mu\nu$MSSM and establish our notations. In Section 3 we present a model that generalizes the $\mu\nu$MSSM . Next, we calculate all flat directions of this model and show that it can generate a viable leptogenesis mechanism [^6]. Also, we show how this model can explain the data from CMS and LHCb. The last section is devoted to conclusions.
Review of the $\mu \nu$SMM
===========================
The Minimal Supersymmetric Standard Model (MSSM) is the supersymmetric extension of the SM that contains a minimal number of states and interactions [@R; @ssm; @grav]. It aims at providing a general frame for the solving of the hierarchy problem, for the stabilization of the weak scale, for the unification of the coupling constants and for addressing the dark matter issues, among other things. The model has the gauge symmetry $SU(3)_{C} \otimes SU(2)_{L} \otimes U(1)_{Y}$ extended by the supersymmetry to include the supersymmetric partners of the SM fields which have spins that differ by $+1/2$ as required by the supersymmetric algebra [@dress; @Baer:2006rs; @Aitchison:2005cf; @Martin:1997ns]. Since the SM fermions are left-handed and right-handed and they transform differently under $SU(3)_{C}$, $SU(2)_{L}$ and $U(1)_{Y}$ groups, the particles of the MSSM must belong to chiral or gauge supermultiplets. The degrees of freedom are grouped in gauge superfields for gauge bosons and left-handed chiral superfields for spinors.
The chiral supermultiplet [@dress; @Baer:2006rs; @Aitchison:2005cf] contains three families of left-handed (right-handed) quarks $\hat{Q}_{i}\sim({\bf 3},{\bf2},1/3)$ and $(\hat{u}^{c}_{i}\sim({\bf \bar{3}},{\bf1},-4/3)$, $\hat{d}^{c}_{i}\sim({\bf \bar{3}},{\bf1},2/3))$. Here, the numbers in parenthesis refers to the $(SU(3)_{C}, SU(2)_{L}, U(1)_{Y}$) quantum numbers, respectively and $i=1,2,3$ refers to the generation index (or flavor indices) and we neglected the color indices. Also, we use the notation in the anti-right-chiral superfield $e^{-}_{R}=(e^{+}_{L})^{c}$ according to [@Baer:2006rs]. The model contains three families of leptons $\hat{L}_{i}\sim({\bf 1},{\bf2},-1)$ ($\hat{l}^{c}_{i}\sim({\bf 1},{\bf1},2)$), respectively. The Higgs boson has spin 0, therefore it must belong to a chiral supermultiplet. However, in this case a single Higgs boson cannot provide mass for all quarks that have different weak isospin charges $T_{3} = \pm (1/2)$. Therefore, the MSSM contains two left-handed chiral superfields for Higgs fields $\hat{H}_{1}\sim({\bf 1},{\bf2},-1),\hat{H}_{2}\sim({\bf 1},{\bf \bar{2}},1)$ [@dress; @Baer:2006rs; @Aitchison:2005cf; @Martin:1997ns; @Kuroda:1999ks]. The particle content of each chiral superfield given above is presented in the Tables (\[lfermionnmssm\]) and (\[rfermionnmssm\]) below
$\mbox{ Left-Chiral Superfield} $ $\mbox{ Fermion} $ $\mbox{ Scalar} $
----------------------------------- -------------------- -------------------
$\hat{L}_{i}$ $L_{i}$ $\tilde{L}_{i}$
$\hat{Q}_{i}$ $Q_{i}$ $\tilde{Q}_{i}$
$\hat{H}_{1}$ $\tilde{H}_{1}$ $H_{1}$
: Particle content in the left-chiral superfields in MSSM, $i$ is flavour index ($i=1,2,3$).[]{data-label="lfermionnmssm"}
$\mbox{ Anti-Right-Chiral Superfield} $ $\mbox{ Fermion} $ $\mbox{ Scalar} $
----------------------------------------- -------------------- ---------------------
$\hat{l}^{c}_{i}$ $l^{c}_{i}$ $\tilde{l}^{c}_{i}$
$\hat{u}^{c}_{i}$ $u^{c}_{i}$ $\tilde{u}^{c}_{i}$
$\hat{d}^{c}_{i}$ $d^{c}_{i}$ $\tilde{d}^{c}_{i}$
$\hat{H}_{2}$ $\tilde{H}_{2}$ $H_{2}$
: Particle content in the anti-right-chiral superfields in MSSM, $i$ is flavour index ($i=1,2,3$).[]{data-label="rfermionnmssm"}
The gauge supermultiplets are described by three vector superfilds $\hat{V}^{a}_{c}\sim({\bf 8},{\bf 1}, 0)$, where $a=1,2, \ldots ,8$, $\hat{V}^{i}\sim({\bf 1},{\bf 3}, 0)$ with $i=1,2,3$ and $\hat{V}^{\prime}\sim({\bf 1},{\bf 1}, 0)$. The particle content in each vector superfield is presented in the Table \[gaugemssm\].
${\rm{Vector \,\ Superfield}}$ ${\rm{Gauge \,\ Bosons}}$ ${\rm{Gaugino}}$ ${\rm Gauge \,\ constant}$
-------------------------------- --------------------------- ---------------------- ----------------------------
$\hat{V}^{a}_{c}$ $g^{a}$ $\tilde{g}^{a}$ $g_{s}$
$\hat{V}^{i}$ $V^{i}$ $\tilde{V}^{i}$ $g$
$\hat{V}^{\prime}$ $V^{\prime}$ $\tilde{V}^{\prime}$ $g^{\prime}$
: Particle content in the vector superfields in MSSM.[]{data-label="gaugemssm"}
The supersymetric Lagrangian of the MSSM is given by $$\mathcal{L}_{SUSY} = \mathcal{L}^{chiral}_{SUSY} + \mathcal{L}^{Gauge}_{SUSY} .
\label{SUSY-Lagrangian1}$$ The Lagrangian defined in the equation (\[SUSY-Lagrangian1\]) contains contributions from all sectors of the model $$\mathcal{L}^{chiral}_{SUSY} =
\mathcal{L}_{Quarks} + \mathcal{L}_{leptons} + \mathcal{L}_{Higgs} ,
\label{SUSY-Lagrangian}$$ and the terms have the following explicit form $$\begin{aligned}
{\cal L}_{Quarks}&=& \int d^{4}\theta\;\sum_{i=1}^{3}\left[\,
\hat{\bar{Q}}_{i}e^{2g_{s}\hat{V}_{c}+2g\hat{V}+g^{\prime} \left( \frac{1}{6} \right) \hat{V}^{\prime}} \hat{Q}_{i} +
{\hat{ \bar{u^c}}}_{i}e^{2g_{s}\hat{V}_{c}+ g^{\prime} \left( - \frac{2}{2}\right) \hat{V}^{\prime}}\hat{u}^{c}_{i} +
\hat{ \bar{d^{c}}}_{i}e^{2g_{s}\hat{V}_{c}+g^{\prime}\left( \frac{1}{3}\right) \hat{V}^{\prime}}\hat{d}^{c}_{i}
\,\right] \,\ . \end{aligned}$$ here, $\hat{V}_{c}=T^{a}\hat{V}^{a}_{c}$ and $T^{a}=\lambda^{a}/2$ (with $a=1,\cdots,8$) are the generators of $SU(3)_{C}$ and $\hat{V}=T^{i}\hat{V}^{i}$ where $T^{i}=\lambda^{i}/2$ (with $i=1,2,3$) are the generators of $SU(2)_{L}$. As usual, $g_{s}$, $g$ and $g^{\prime}$ are the gauge couplings for the $SU(3)$, $SU(2)$ and $U(1)$ groups, respectively, as shown in the Table \[gaugemssm\]. The action in the lepton and Higgs sectors are defined by the following Lagrangians $$\begin{aligned}
{\cal L}_{lepton}&=& \int d^{4}\theta\;\sum_{i=1}^{3}\left[\,
\hat{ \bar{L}}_{i}e^{2g\hat{V}+g^{\prime} \left( - \frac{1}{2}\right) \hat{V}^{\prime}} \hat{L}_{i} +
\hat{ \bar{l^{c}}}_{i}e^{g^{\prime}\hat{V}^{\prime}} \hat{l}^{c}_{i}\,\right] \,\ , \nonumber \\
{\cal L}_{Higgs}&=& \int d^{4}\theta\;\left[\,
\hat{ \bar{H}}_{1}e^{2g\hat{V}+g^{\prime}\left( - \frac{1}{2}\right) \hat{V}^{\prime}}\hat{H}_{1}+
\hat{ \bar{H}}_{2}e^{2g\hat{V}+g^{\prime}\left( \frac{1}{2}\right) \hat{V}^{\prime}}\hat{H}_{2}
+ W+ \bar{W} \right]\!.
\label{allsusyterms} \end{aligned}$$ The last two terms define the superpotential of the MSSM as $W=W_{H}+W_{Y}$ where $$\begin{aligned}
W_{H}&=& \mu\; \epsilon_{\alpha \beta}\hat{H}_{1}^{\alpha}\hat{H}_{2}^{\beta} \,\ ,
\label{mu-HH} \\
W_{Y}&=& \epsilon_{\alpha \beta}\sum_{i,j=1}^{3}\left[\,
f^{l}_{ij}\hat{H}^{\alpha}_{1}\hat{L}^{\beta}_{i}\hat{l}^{c}_{j}+
f^{d}_{ij}\hat{H}^{\alpha}_{1}\hat{Q}^{\beta}_{i}\hat{d}^{c}_{j}+
f^{u}_{ij}\hat{H}^{\alpha}_{2}\hat{Q}^{\beta}_{i}\hat{u}^{c}_{j}\,\right] \,\ .
\label{Y-H} \end{aligned}$$ The supersymmetric parameter $\mu$ is a complex numbers and the $f$ terms are elements of the complex $3 \times 3$ Yukawa coupling matrices in the family space. The color indices on the triplet (antitriplet) superfield $\hat{Q}$ $( \hat{u}^{c}, \hat{d}^{c})$ contract trivially, and have been suppressed. The second terms of the Lagrangian defined by the equation (\[SUSY-Lagrangian1\]) is given by the following equation $$\begin{aligned}
\mathcal{L}^{Gauge}_{SUSY}&=& \frac{1}{4} \int d^{2}\theta\;
\left[ \sum_{a=1}^{8} W^{a \alpha}_{s}W_{s \alpha}^{a}+ \sum_{i=1}^{3} W^{i \alpha}W_{ \alpha}^{i}+
W^{ \prime \alpha}W_{ \alpha}^{ \prime}\, + h.c. \right] \,\ . \nonumber \end{aligned}$$ The gauge superfields have the following explicit form $$\begin{aligned}
W^{a}_{s \alpha}&=&-\frac{1}{8g_{s}}\,\bar{D}\bar{D}e^{-2g_{s}\hat{V}^{a}_{c}}D_{\alpha}e^{2g_{s}\hat{V}^{a}_{c}}
\,\ , \nonumber \\
W^{i}_{\alpha}&=&-\frac{1}{8g}\,\bar{D}\bar{D}e^{-2g\hat{V}^{i}}D_{\alpha}e^{2g\hat{V}^{i}}
\,\ , \nonumber \\
W_{\alpha}^{\prime}&=&-\frac{1}{4}\,D D \bar{D}_{\alpha} \hat{V}^{\prime} \,\ ,
\label{W-a}\end{aligned}$$ where $\alpha = 1,2$ is a spinorial index.
In principle, one could add to the Lagrangian defined by the equation (\[SUSY-Lagrangian\]) other terms that, even if they break the baryon number and the lepton number conservation laws, are still allowed by the supersymmetry. However, no physical process with this property has been discovered so far. This phenomenological fact suggest imposing a symmetry that rules out such terms called $R$-parity which is defined in terms of the following operators $$P_{M} = (-1)^{3(B-L)}, \hspace{0,5cm}
P_{R} = P_{M}(-1)^{2s},$$ where $B$ and $L$ are the baryon and lepton numbers, respectively, and $s$ is the spin for a given state. The $R$-parity of the Lagrangian implies that the usual particles of the SM have $P_{M}=1$ while their superpartners have $P_{M}=-1$. The terms that break $R$-parity are $$\begin{aligned}
W_{2RV}&=&\epsilon_{\alpha \beta} \sum_{i=1}^{3}\mu_{0i} \hat{L}^{\alpha}_{i}\hat{H}^{\beta}_{2},\nonumber \\
W_{3RV}&=&\epsilon_{\alpha \beta} \sum_{i,j,k=1}^{3} \left( \lambda_{ijk}\hat{L}^{\alpha}_{i}\hat{L}^{\beta}_{j}\hat{l}^{c}_{k}+
\lambda^{\prime}_{ijk}\hat{L}^{\alpha}_{i}\hat{Q}^{\beta}_{j}\hat{d}^{c}_{k}+
\lambda^{\prime\prime}_{ijk}\hat{u}^{c}_{i}\hat{d}^{c}_{j}\hat{d}^{c}_{k} \right) .
\label{mssmrpv}\end{aligned}$$ Here, we have suppressed the $SU(2)$ indices and $\epsilon$ is the antisymmetric $SU(2)$ tensor. Some of the coupling constants in the equation (\[mssmrpv\]) should be set to zero in order to avoid a too fast proton decay and neutron-anti-neutron oscillation [@dress; @Baer:2006rs; @barbier; @moreau]. The choice of the $R$-parity violation couplings $\lambda^{\prime}_{11k}$ with $k = 2$ and 3, are constrained from various low energy observables such as: (i) charge-current universality, (ii) $e-\mu-\tau$ universality, (iii) atomic parity violation. The bounds on the product $|\lambda^{\prime}_{112}\lambda^{\prime}_{113}|$ can be obtained from the charged $B$-meson decay mixing $B^{\pm}_d \rightarrow \pi^{\pm} K^0$, $B_{s}-\bar{B}_{s}$ and the transition $B\rightarrow X_s\gamma$ as observed in [@Biswas:2014gga].
The experimental evidence suggests that the supersymmetry is not an exact symmetry. Therefore, supersymmetry breaking terms should be added to the Lagrangian defined by the equation (\[SUSY-Lagrangian\]). One possibility is by requiring that the divergences cancel at all orders of the perturbation theory. The most general soft supersymmetry breaking terms, which do not induce quadratic divergence, where described by Girardello and Grisaru [@10]. They found that the allowed terms can be categorized as follows: a scalar field $A$ with mass terms $${\cal L}_{SMT}=-m^{2} A^{\dagger}A,$$ a fermion field gaugino $\lambda$ with mass terms $${\cal L}_{GMT}=- \frac{1}{2} (M_{ \lambda} \lambda^{a} \lambda^{a}+h.c.)$$ and finally trilinear scalar interaction terms $${\cal L}_{INT}= \Xi_{ij}A_{i}A_{j}+ \Upsilon_{ij}A_{i}A_{j}+ \Omega_{ijk}A_{i}A_{j}A_{k}+h.c.$$ The terms in this case are similar with the terms allowed in the superpotential of the model we are going to consider next.
Taken all this information into account, we can add the following soft supersymmetry breaking terms to the MSSM $$\begin{aligned}
{\cal L}^{MSSM}_{Soft} &=& {\cal L}^{MSSM}_{SMT} + {\cal L}^{MSSM}_{GMT}+ {\cal L}^{MSSM}_{INT} \,\ ,
\label{The Soft SUSY-Breaking Term prop 2aaa}\end{aligned}$$ where the scalar mass term ${\cal L}_{SMT}$ is given by the following relation $$\begin{aligned}
{\cal L}^{MSSM}_{SMT} &=& - \sum_{i,j=1}^{3} \left[\,
\left( M_{L}^{2}\right)_{ij}\;\tilde{L}^{\dagger}_{i}\tilde{L}_{j}+
\left( M^{2}_{l}\right)_{ij} \tilde{l^{c}}^{\dagger}_{i}\tilde{l^{c}}_{j}+
\left( M_{Q}^{2}\right)_{ij}\;\tilde{Q}^{\dagger}_{i}\tilde{Q}_{j}
\right. \nonumber \\
\hspace{1.7cm} &+& \left.
\left( M^{2}_{u}\right)_{ij} \tilde{u^{c}}^{\dagger}_{i}\tilde{u^{c}}_{j}+
\left( M^{2}_{d}\right)_{ij} \tilde{d^{c}}^{\dagger}_{i}\tilde{d^{c}}_{j}
+ M_{1}^{2} H^{\dagger}_{1}H_{1} +
M_{2}^{2} H^{\dagger}_{2}H_{2}
\right] \,\ ,
\label{burro}\end{aligned}$$ The $3 \times 3$ matrices $M_{L}^{2},M^{2}_{l},M_{Q}^{2},M^{2}_{u}$ and $M^{2}_{d}$ are hermitian and $M_{1}^{2}$ and $M_{2}^{2}$ are real. The gaugino mass term is written as $$\begin{aligned}
{\cal L}^{MSSM}_{GMT} &=&- \frac{1}{2} \left[
\left(\,M_{3}\; \sum_{a=1}^{8} \lambda^{a}_{C} \lambda^{a}_{C}
+ M\; \sum_{i=1}^{3}\; \lambda^{i}_{A} \lambda^{i}_{A}
+ M^{\prime} \; \lambda_{B} \lambda_{B}\,\right)
+ h.c. \right] \,\ .
\label{The Soft SUSY-Breaking Term prop 3}\end{aligned}$$ Here, $M_{3},M$ and $M^{\prime}$ are complex. Finally, there is an interaction term ${\cal L}_{INT}$, see the equation (\[mssmrpv\]), of the form $$\begin{aligned}
{\cal L}^{MSSM}_{INT} =- M_{12}^{2}\epsilon H_{1}H_{2}
+ \epsilon \sum_{i,j,k=1}^{3} \left[
\left( A^{E}\right)_{ij} H_{1}\tilde{L}_{i}\tilde{l}^{c}_{j}+
\left( A^{D}\right)_{ij} H_{1}\tilde{Q}_{i}\tilde{d}^{c}_{j}+
\left( A^{U}\right)_{ij} H_{2}\tilde{Q}_{i}\tilde{u}^{c}_{j}\right] +h.c. \,\ .\end{aligned}$$ The $3 \times 3$ matrices $M_{12}^{2}$ and $A$ matrices are complex.
The total Lagrangian of the MSSM is obtained by adding all Lagrangians above $$\mathcal{L}^{MSSM} = \mathcal{L}_{SUSY} + \mathcal{L}^{MSSM}_{soft},
\label{L-total}$$ see the equations (\[SUSY-Lagrangian1\],\[The Soft SUSY-Breaking Term prop 2aaa\]). The MSSM contains 124 free parameters [@Baer:2006rs] and the symmetry breaking parameters are completely arbitrary [@dress]. The main goal in the SUSY phenomenology is to find some approximation about the way we can break SUSY in order to have a drastic reduction in the number of these parameters[^7]. Many phenomenological analyses adopt the universality hypothesis at the scale $Q \simeq M_{GUT}\simeq 2 \times 10^{16}$ GeV: $$\begin{aligned}
g_{s}&=&g=g^{\prime} \equiv g_{GUT}, \nonumber \\
M_{3}&=&M=M^{\prime}\equiv m_{1/2}, \nonumber \\
M_{L}^{2}&=&M^{2}_{l}=M_{Q}^{2}=M^{2}_{u}=M^{2}_{d}=M_{1}^{2}=M_{2}^{2}\equiv m^{2}_{0}, \nonumber \\
A^{E}&=&A^{D}=A^{U}\equiv A_{0}.
\label{msugra}\end{aligned}$$ The assumptions that the MSSM is valid between the weak scale and GUT scale, and that the “boundary conditions”, defined by the equation (\[msugra\]) hold, are often referred to as mSUGRA, or minimal supergravity model. The mSUGRA model is completely specified by the parameter set [@dress; @Baer:2006rs] $$\begin{aligned}
m_{0}, \,\ m_{1/2}, \,\ A_{0}, \,\ \tan \beta , \,\ {\mbox sign}( \mu ) .\end{aligned}$$ The new free parameter $\beta$ is defined in the following way $$\begin{aligned}
\tan \beta \equiv \frac{v_{2}}{v_{1}},
\label{defbetapar}\end{aligned}$$ where $v_{2}$ is the vev of $H_{2}$ while $v_{1}$ is the vev of the Higgs in the doublet representation of $SU(2)$ group. Due the fact that $v_{1}$ and $v_{2}$ are both positive, it imples that $0 \leq \beta \leq (\pi/2) \,\ {\mbox rad}$.
In the context of the MSSM, it is possible to give mass to all charged fermions. With this superpotential we can explain the mass hierarchy in the charged fermion masses as showed in [@cmmc; @cmmc1]. On the other hand, ${\cal L}_{Higgs}$ give mass to the gauge bosons: the charged ones ($W^{\pm}$) and the neutral ($Z^{0}$ and get a massless foton but the neutrinos remain massless. Due to this fact, it is generated a spectrum that contains five physical Higgs bosons, two neutral scalar ($H,h$), one neutral pseudoscalar ($A$), and a pair of charged Higgs particles ($H^{\pm}$). At the level of tree level, we can write the following relations hold in the Higgs sector [@dress; @Baer:2006rs]: $$\begin{aligned}
m^{2}_{H^{\pm}}&=&m^{2}_{A}+m^{2}_{W}, \nonumber \\
m^{2}_{h,H}&=&\frac{1}{2}\left[
(m^{2}_{A}+m^{2}_{Z}) \mp \sqrt{(m^{2}_{A}+m^{2}_{Z})^{2}-4m^{2}_{A}m^{2}_{Z}\cos^{2}\beta} \right], \nonumber \\
m^{2}_{h}+m^{2}_{H}&=&m^{2}_{A}+m^{2}_{Z}, \nonumber \\
\cos^{2}( \beta - \alpha )&=& \frac{m^{2}_{h}(m^{2}_{Z}-m^{2}_{h})}{m^{2}_{A}(m^{2}_{H}-m^{2}_{h})}. \end{aligned}$$ Therefore, the light scalar $h$ has a mass smaller than the $Z^{0}$ gauge boson at the tree level. This implies that one has to consider the one-loop corrections which lead to the following result [@Haber:1990aw] $$m_{h}\simeq m^{2}_{Z}+ \frac{3g^{2}m^{4}_{Z}}{16 \pi^{2}m^{2}_{W}}\left\{
\ln \left( \frac{m^{2}_{\tilde{t}}}{m^{2}_{t}}\right) \left[ \frac{2m^{4}_{t}-m^{2}_{t}m^{2}_{Z}}{m^{4}_{Z}}\right] +
\frac{m^{2}_{t}}{3m^{2}_{Z}}\right\}.$$ In the MSSM there are four neutralinos ($\tilde{\chi}^{0}_{i}$ with $i=1,2,3,4$) and two charginos ($\tilde{\chi}^{\pm}_{i}$ with $i=1,2$) [@dress; @Baer:2006rs].
The mass matrix of neutrinos from this model was studied in [@hall; @banks; @rv1; @fb]. The mass matrix has two zero eigenvalues. Thus, there are two neutrinos $\nu_{1,2}$, which are massless at the tree level. More realistic neutrino masses require radiative corrections [@rv1; @rnm; @rv2; @marta]. The neutrinos are Majorana particles, therefore the neutrinoless double beta decay must be observed. Neutrinoless double beta decay $0\nu\beta\beta$ is a sensitive probe of physics beyond the SM since it violates the conservation of the lepton number [@hdmo94; @baudis97; @heidel01; @arxivnote; @expbb]. The nucleon level process of $0\nu\beta\beta$ decay, $ n +n \to p+p + e^{-} + e^{-}$, can be obtained via the lepton number violating sub-process $d+d \to u +u +e+e$.
The supersymmetric mechanism of $0\nu\beta\beta$ decay was first suggested by Mohapatra [@rnm] and further studied in [@Vergados; @HKK1]. In [@HKK2], the $R$-parity violating Yukawa coupling of the first generation is strongly bounded by $\lambda^{\prime}_{111} \leq 3.9\cdot 10^{-4}$ due to the gluino exchange $0\nu\beta\beta$-decay. Babu and Mohapatra [@BM] have latter implemented another contribution comparable with that via the gluino exchange. This set stringent bounds on the products of $R$-parity violating Yukawa couplings $\lambda^{\prime}_{11i}\lambda^{\prime}_{1i1}$ of $i$th generation index [@hir96] $$\begin{aligned}
\lambda_{113}^{\prime}\lambda_{131}^{\prime} &\leq & 1.1 \cdot 10^{-7},\\
\lambda_{112}^{\prime}\lambda_{121}^{\prime}&\leq & 3.2 \cdot 10^{-6}. \end{aligned}$$
On the other hand, the confrontation of the experimental results with the predictions of the MSSM set a phenomenological constraint on the magnitude of $\mu \sim {\cal O}(M_{W})$. Indeed, the mass of Higgssino from the equation (\[mu-HH\]) and the terms from the ${\cal L}_{Soft}$, given by the equation (\[The Soft SUSY-Breaking Term prop 2aaa\]), are of order of the electro-weak scale of $246 \,\ GeV$ while the natural cut-off scale is the Planck scale $1.22 \times 10^{19}\,\ GeV$. The MSSM does not provide any mechanism to explain the difference between the two scales. This is know as the *$\mu$ problem*.
In order to address this problem, the Next-to-the-Minimal Supersymmetric Standard-Model (NMSSM) [@R; @nmssm] was developed within the framework of the Grand Unification Theory (GUTs) as well as the superstring theorie [@barr; @nilsredwy; @derendinger][^8]. The NMSSM is characterized by the a new singlet field introduced in the following chiral superfield[^9] [@dress] $$\begin{aligned}
\hat{N}(y, \theta )&=&n(y)+ \sqrt{2}\theta \tilde{n}(y)+ \theta \theta F_{n}(y), \end{aligned}$$ where $n$ is the scalar in the singlet and its vacuum expectation value is given by $\sqrt{2} \langle n \rangle =x$. Its superpartner $\tilde{n}$ is known as the singlino. The rest of the particle content of this model is the same as of the MSSM given above in the Tables (\[lfermionnmssm\]), (\[rfermionnmssm\] and \[gaugemssm\]). The superpotential of the NMSSM model has the following form $$\begin{aligned}
W_{NMSSM}&=&W_{Y}+ \epsilon_{\alpha \beta}\lambda \hat{H}_{1}^{\alpha} \hat{H}_{2}^{\beta} \hat{N}+
\frac{1}{3} \kappa \hat{N}\hat{N}\hat{N},\end{aligned}$$ where $W_{3RV}$ is defined by the equation (\[Y-H\]). The way in which the $\mu$-problem is solved in the NMSSM is by generating dynamically the $\mu$ term in the superpotential through $\mu = \lambda x$ with a dimensionless coupling $\lambda$ and the vacuum expectation value $x$ of the Higgs singlet. Another essential feature of the NMSSM is the fact that the mass bounds for the Higgs bosons and neutralinos are weakened. For more details about the scalar sector of this model see [@Drees:1988fc]. We summarize them in the Table \[tabvec\] below
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
${\rm{Symbol}}$ ${\rm{Decomposition}}$
-------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------
$H^{\pm}$ $\sin(\beta) h_{1}^{\pm}+ \cos(\beta) h_{2}^{\pm}$
$A_{1}$, $A_{2}$, $m_{A_{1}}\leq m_{A_{2}}$ $A_{1}= \cos( \alpha_{PS})a^{0}+ \sqrt{2}\sin( \alpha_{PS}) \mathfrak{Im} \left[ n \right]$
$A_{2}=- \sin( \alpha_{PS})a^{0}+ \sqrt{2}\cos( \alpha_{PS}) \mathfrak{Im} \left[ n \right]$
$h_{1}$, $h_{2}$, $h_{3}$, $m_{h_{1}}\leq m_{h_{2}}\leq m_{h_{3}}$ $h_{i}= \sqrt{2} \mathfrak{Re} \left[ {\cal O}_{i1}(h^{0}_{1}-v_{1})+ {\cal O}_{i2}(h^{0}_{2}-v_{2})
+ {\cal{O}}_{i3}(n-x) \right]$
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: The physical Higgs states of the NMSSM [@dress], the $\beta$ parameter is defined by the equation (\[defbetapar\]).[]{data-label="tabvec"}
Note that the neutralino sector is extended to a $5 \times 5$ mass matrix. If the following vector basis for fields is adopted (see, e. g. [@dress]) $$(\psi^{0})^{T}=(\lambda_{0},\lambda_{3},\tilde{h}^{1}_{1},\tilde{h}^{2}_{2},\tilde{n}) ,$$ the mass matrix takes the following form [^10] $$Y= \left(
\begin{array}{ccccc}
M_{1} & 0 & -m_{Z}\sin \theta_{W} \cos \beta & m_{Z}\sin \theta_{W}\sin \beta & 0 \\
0 & M_{2} & m_{Z}\cos \theta_{W}\cos \beta & -m_{Z}\cos \theta_{W}\sin \beta & 0 \\
-m_{Z}\sin \theta_{W}\cos \beta & m_{Z}\cos \theta_{W}\cos \beta & 0 & - \lambda \frac{x}{\sqrt{2}} &
- \lambda \frac{v_{1}}{\sqrt{2}} \\
m_{Z}\sin \theta_{W}\sin \beta & m_{Z}\cos \theta_{W}\sin \beta & - \lambda \frac{x}{\sqrt{2}} & 0 &
- \lambda \frac{v_{2}}{\sqrt{2}} \\
0 & 0 & - \lambda \frac{v_{1}}{\sqrt{2}} & - \lambda \frac{v_{2}}{\sqrt{2}} & \sqrt{2}\kappa x
\end{array}
\right) .$$ We note that the singlino $\tilde{n}$ does not mix directly with the gauginos $\lambda_{0},\lambda_{3}$ but it can mix with the neutral higgsinos [@dress]. The neutrinos are massless. However, the term $\hat{\nu}^{c}_{i} \hat{H}_{1}\hat{H}_{2}$ can produce an effective $\mu$ term when the sneutrinos get vev as we will show later on. This would allow us to solve the $\mu$ problem [@mupb], without having to introduce an extra singlet superfield as we have done in the NMSSM. This new model is called "$\mu$ from $\nu$” Supersymmetric Standard Model ($\mu \nu$SSM). The field content of the $\mu \nu$SSM is the same as MSSM supplemented by three neutrino superfields $\hat{\nu}^{c}_{i}$ [@LopezFogliani:2005yw] and is given by the following equation given by: $$\begin{aligned}
\hat{\nu}^{c}_{i}(y, \theta )= \tilde{\nu}^{c}_{i}(y)+ \sqrt{2}\theta \nu^{c}_{i}(y)+ \theta \theta
F_{\nu^{c}_{i}}(y).\end{aligned}$$ If the terms like $\hat{H}_{2}\hat{L}_{i}\hat{\nu}^{c}_{j}$ are considered, then the term $\mu_{0i}$ is induced when the right handed sneutrinos acquires a vev. By adding right handed neutrinos to the model, one can choose only the terms that break the lepton number conservation instead of the ones that break the baryon number conservation. The vev of these models are $$\begin{aligned}
\langle \tilde{\nu}^{c}_{i} \rangle &\equiv& \frac{v_{\nu^{c}_{i}}}{\sqrt{2}}, \nonumber \\
\langle \tilde{\nu}_{i} \rangle &\equiv& \frac{v_{\nu_{i}}}{\sqrt{2}}.
\label{vevemunussm}\end{aligned}$$ In this case, all the neutrinos of the model can get mass at the tree level. Then the double beta decay can occur and the nucleon is stabilized.
The ${\cal Z}_{3}$ symmetry generates the following transformation of each chiral superfield $$\Phi \rightarrow \exp \left( \frac{2 \pi \omega}{3} \right) \Phi ,$$ where $\omega$ is an entire number. The superpotential of this model can be obtained by requiring that it be ${\cal Z}_{3}$-symmetric invariant. As a consequence, it takes the following form $$\begin{aligned}
W_{\mu \nu suppot} &=&W_{Y}+ \sum_{i,j,k=1}^{3} \left(
f^{\nu}_{ij} \, \hat{H}_{2}\, \hat{L}_{i} \, \hat{\nu}^{c}_{j} +
h_{i}^{\nu}\hat{H}_{2}\hat{H}_{1}\hat{\nu}^{c}_{i}
+ \frac{1}{3}\kappa^{ijk} \hat{\nu}^{c}_{i}\hat{\nu}^{c}_{j}\hat{\nu}^{c}_{k} \right) \,.
\label{superpotential}\end{aligned}$$ where $W_{Y}$ is defined by the equation (\[Y-H\]). It turns out that ${\cal Z}_{3}$ symmetry forbids all the bilinear terms in the superpotential. The expression obtained in the equation (\[superpotential\]) is consistent with the phenomenological models derived from the superstring theory that generate only trilinear couplings.
In the present context the string theory is relevant to the unification of all interactions, including gravity. The term proportional to $\kappa$ gives an effective Majorana mass term to neutrinos, while the coupling $f^{\nu}$ generates Dirac mass term to neutrinos.
When the scalar components of the superfields $\hat{\nu}^{c}_{i}$, denoted by $\tilde{\nu}^{c}_{i}$, acquire vev’s of the order of the electroweak scale, an effective interaction $\mu \hat{H}_{1} \hat{H}_{2}$ is generated with the effective coupling $\mu$ given by $$\mu \equiv h_{i}^{\nu} \langle \tilde{\nu}^{c}_{i} \rangle .$$ In the same situation, the term $\mu_{0i}\hat{H}_{2}\hat{L}_{i}$ can be generated with $$\mu_{0i}\equiv \sum_{j=1}^{3} f^{\nu}_{ij}\langle \tilde{\nu}^{c}_{j} \rangle ,$$ the contribution of $f^{\nu}\leq 10^{-6}$ to the minimization conditions for the left-handed neutrinos $\mu_{0i}\ll \mu$. That provides an explanation for the neutrino’s masses in MSSM [@Montero:2001ch].
In this model the $R$-parity (and also the lepton number conservation) is broken explicitly. One of the candidates for the dark matter in NMSSM is the gravitino. Recently, some experimental bounds on gravitino masses were presented in see [@Catena:2014pca]. For an analysis of the gravitino as dark matter without $R$-parity see [@yamaguchi]. Other possibilities that LSP be the axino were presented in [@axino]. The mass spectrum of this model can be found in [@Escudero:2008jg] and ths spectrum is consistent with the experimental values obtained for both masses and mixing. The nice phenomenological aspects of this model were discussed in [@Munoz:2009an]. There are some works in $\mu\nu$SSM that consider gravitino as dark matter [@Choi:2009ng; @GomezVargas:2011ph; @Albert:2014hwa].
A “new" $\mu\nu$SSM
===================
In this section we propose a generalization of the $\mu\nu$SSM by adding new interaction terms that explicitly break the $R$-parity and the lepton number symmetries, respectively. Therefore, the new model has potentially interesting cosmological consequences such as flat directions that provide a mechanism for the cosmological inflation and leptogenesis which explains the asymmetry between the matter and the anti-matter. We determine the flat directions of the generalized $\mu\nu$SSM and explain the leptogenesis mechanism. Then we show how our proposal addresses the recent experimental results from CMS and LHCb obtained in [@Aaij:2014ora; @Aaij:2013pta] and discussed in [@Biswas:2014gga].
New terms in the Superpotential of $\mu \nu$SSM Model {#sec:newsup}
-----------------------------------------------------
The superpotential of the $\mu\nu$SSM model given by the equation (\[superpotential\]) can be generalized as follows $$\begin{aligned}
W &=& W_{\mu \nu suppot}+ \sum_{i,j,k=1}^{3} \left( \lambda^{\prime}_{ijk} \hat{L}_{i}\hat{L}_{j}\hat{l}^{c}_{k}+ \lambda^{\prime \prime}_{ijk}
\hat{L}_{i}\hat{Q}_{j}\hat{d}^{c}_{k} \right) \,.
\label{superpotential1}\end{aligned}$$ In the above equation, we have introduced two new terms that explicitly break the $R$-parity and the lepton number symmetry. There is a new parameter $\lambda^{\prime}$ that generates one more contribution to the mixing between the usual leptons with higgsinos. The usual techniques allow to determine from the superpotential $W$ the following mass matrix elements $$\begin{aligned}
&-&\left[ f^{l}_{ij} \left( H_{1}L_{i}l^{c}_{j}+ \tilde{H}_{1}\tilde{L}_{i}l^{c}_{j}\right) +
f^{\nu}_{ij}\left( \tilde{H}_{2}L_{i}\tilde{\nu}^{c}_{j}+H_{2}L_{i}\nu^{c}_{j}+ \tilde{H}_{2}\tilde{L}_{a}\nu^{c}_{b} \right) \right. \nonumber \\
&+& \left. h_{i}^{\nu}\left( \tilde{H}_{1}\tilde{H}_{2}\tilde{\nu}^{c}_{i}+H_{1}\tilde{H}_{2}\nu^{c}_{i}+ \tilde{H}_{1}H_{2}\nu^{c}_{i}\right) + \kappa_{ijk}\tilde{\nu}^{c}_{i}\nu^{c}_{j}\nu^{c}_{k}+2 \lambda^{\prime}_{ijk}\tilde{L}_{i}L_{j}l^{c}_{k} \right] \,\ . \nonumber \\\end{aligned}$$ The terms that describe the mixing between the usual leptons with the gauginos are the same as in the MSSM. In our notation, they are given by the following relations $$\begin{aligned}
\imath \sqrt{2}g\left[ \bar{L}_{i} \left( \frac{\sigma^{a}}{2} \right) \overline{\tilde{W}^{a}} \tilde{L}_{i} -
\overline{\tilde{L}}_{i}\left( \frac{\sigma^{a}}{2} \right) \tilde{W}^{a} L_{i} \right]
- \imath \sqrt{2} \left[ \bar{L}_{i} \left( - \frac{1}{2} \right) \tilde{L}_{i}
\overline{\tilde{V}^{\prime}}- \overline{\tilde{L}}_{i}\left( - \frac{1}{2} \right)L_{i}\tilde{V}^{\prime} \right]. \end{aligned}$$ The mixing between the usual leptons with the higgsinos is given by the equation $$\begin{aligned}
&-&f^{l}_{ij}\tilde{H}_{1}\tilde{L}_{i}l^{c}_{j} +
f^{\nu}_{ij}\left( \tilde{H}_{2}L_{i}\tilde{\nu}^{c}_{j}+ \tilde{H}_{2}\tilde{L}_{i}\nu^{c}_{j} \right) +
h_{i}^{\nu}\left( H_{1}\tilde{H}_{2}\nu^{c}_{i}+ \tilde{H}_{1}H_{2}\nu^{c}_{i}\right) .\end{aligned}$$ Beside the new mixing sectors given above, there are interactions between gauginos and higgsinos given by the same terms as in the MSSM. One can calculate the mass matrices of the charged leptons following the reference [@Escudero:2008jg]. The result in the basis $\Psi^{- \,\ T}=(-i \tilde{W}^{-},\tilde{H}^{-}_{1},l_{1},l_{2},l_{3})^{T}$ is given by the matrix $$M_{C}= \frac{1}{\sqrt{2}}
\left(
\begin{array}{ccccc}
\sqrt{2}M_{2} & gv_{2} & 0 & 0 & 0\\
gv_{1} & \lambda_{i} v_{\nu^{c}_{i}} & -f^{l}_{i1} v_{\nu_{i}} &
-f^{l}_{i1} v_{\nu_{i}} & -f^{l}_{i1} v_{\nu_{i}} \\
gv_{\nu_{1}} & -f^{\nu}_{1i}v_{\nu^{c}_{1}} & a_{11} & a_{12} & a_{13} \\
gv_{\nu_{2}} & -f^{\nu}_{2i}v_{\nu^{c}_{2}} & a_{21} & a_{22} & a_{23} \\
gv_{\nu_{3}} & -f^{\nu}_{3i}v_{\nu^{c}_{3}}& a_{31} & a_{32} & a_{33}
\end{array}
\right),
\label{charginosmasses}$$ where $$a_{ij}= f^{l}_{ij}v_{1}-\sum_{k=1}^{3}\lambda^{\prime}_{kij} v_{\nu_{k}}\sim f^{l}_{ij}v_{1}.$$ Here, the winos $\tilde{W}^{\pm}$ (superpartners of the $W$-boson) defined as $$\sqrt{2}\tilde{W}^{\pm}\equiv \tilde{V}^{1}\mp \tilde{V}^{2} \, ,
\label{Winos-def}$$ See also the equation (\[vevemunussm\]). In the neutralino sector we use the basis $$\Psi^{0 \,\ T}=(-i \tilde{V}^{\prime},-i \tilde{V}^{3},\tilde{H}^{0}_{1},\tilde{H}^{0}_{2},\nu^{c}_{1},\nu^{c}_{2},\nu^{c}_{3},\nu_{1},\nu_{2},\nu_{3})^{T}.
\label{neutralino-basis}$$ Then the mass matrices take the following form $$M_{N}=
\left(
\begin{array}{cc}
M_{7 \times 7} & m_{3 \times 7} \\
(m_{3 \times 7})^{T} & 0_{3 \times 3}
\end{array}
\right)_{10 \times 10},
\label{neutralinosmasses}$$ where $M_{N}$ is the neutralino mass matrix presented in [@Escudero:2008jg]. However, one should emphasize that the neutrinos are Majorana particles. Therefore, the double beta decay without neutrinos can occur in this model. This process is permitted by the new first term from the equation (\[superpotential1\]). The second term alone will not induce neither the fast proton decay, nor the neutron anti-neutron oscillation [@dress].
It is important to note that the last term in the superpotential will modify the down quarks masses. Some algebra shows that they are given by the following relation $$\begin{aligned}
m_{d}= \frac{1}{\sqrt{2}} \left(f^{d}v_{1}+\lambda^{\prime}_{ijk} v_{\nu_{i}} \right) \sim
\frac{1}{\sqrt{2}}f^{d}v_{1}.
\label{massa do lepton}\end{aligned}$$ The new superpotential given by the equation (\[superpotential1\]) has all properties required in [@LopezFogliani:2005yw; @Escudero:2008jg; @Munoz:2009an]. It has the advantage that it induces the double beta decay while maintaining the nucleon stability [@dress] as was discussed in the previous section.
Flat direction of $\mu \nu$SSM Model {#sec:cosmological}
------------------------------------
One of the most remarkable aspects of the supersymmetric gauge theories is that they have a vacuum degeneracy at the classical level. It is a well established fact that the renormalizable scalar potential is a sum of squares of $F$-terms and $D$-terms which implies that it can vanish identically along certain *flat directions* [^11] in the space of fields [^12] . The properties of the space of flat directions of a supersymmetric model are crucial for making realistic considerations in cosmology and whenever the behaviour of the theory at large field strengths is an issue.
In the MSSM the flat directions can give rise to a host of cosmologically interesting dynamics (for a review, see [@Enqvist:2003gh]). These include Affleck-Dine baryogenesis [@Affleck:1984fy; @Dine:1995uk; @Dine:1995kz], the cosmological formation and fragmentation of the MSSM flat direction condensate and subsequent $Q$-ball formation [@Kusenko:1997zq; @Enqvist:1997si; @Jokinen:2002xw; @Kasuya:1999wu], reheating the Universe with $Q$-ball evaporation [@Enqvist:2002rj], generation of baryon isocurvature density perturbations [@Enqvist:1998pf], as well as curvaton scenarios where MSSM flat directions reheat the Universe and generate adiabatic density perturbations [@Enqvist:2002rf]. Adiabatic density perturbations induced by fluctuating inflaton-MSSM flat direction coupling has also been discussed in [@Enqvist:2003uk].
We can calculate all flat directions in the MSSM using the prescription given in [@Dine:1995kz; @Gherghetta:1995dv]. It turns out that a MSSM flat direction is some linear combination of the MSSM scalars and can be thought of as a trajectory in the moduli space described by a single scalar degree of freedom.
Flat direction $(B-L)$
------------------------------------------- ---------
$\hat{H}_{2}\hat{Q}\hat{u}^{c}$ 0
$\hat{H}_{1}\hat{Q}\hat{d}^{c}$ 0
$\hat{H}_{1}\hat{L}\hat{l}^{c}$ 0
$\hat{H}_{2}\hat{L}\hat{\nu}^{c}$ 0
$\hat{H}_{1}\hat{H}_{2}\hat{\nu}^{c}$ 1
$\hat{\nu}^{c}\hat{\nu}^{c}\hat{\nu}^{c}$ 3
$\hat{L}\hat{L}\hat{e}^{c}$ -1
$\hat{L}\hat{Q}\hat{d}^{c}$ -1
: Flat direction of the model $\mu \nu$SSM.[]{data-label="tab:flatdir"}
Using the same technique from [@Dine:1995kz; @Gherghetta:1995dv] we can calculate the flat directions in $\mu \nu$SSM model. The Table (\[tab:flatdir\]) presents the computed flat directions when the only terms taken into account are the renormalizable terms. In this model $\hat{H}_{2}\hat{L}_{i}\hat{\nu}^{c}_{j}$ gives a particular flat direction. It follows that the field $$\phi_1 = \frac{H_{2}+ \tilde{L}+ \tilde{\nu}^{c}}{\sqrt{3}},
\label{flat1}$$ generates a similar inflanton scenario as the one discussed in [@Allahverdi:2007wt; @Allahverdi:2006cx; @Allahverdi:2009kr]. Another flat direction is given by the interaction term $\hat{\nu}^{c}_{i}\,\hat{H}_{1} \hat{H}_{2}$ from which is generated the following field $$\phi_2 = \frac{H_{1}+H_{2}+ \tilde{\nu}^{c}}{\sqrt{3}}.
\label{flat2}$$ It will be interesting to compare the cosmological consequences of the fields $\phi_1$ and $\phi_2$ given above.
Leptogenesis in $\mu \nu$SSM Model {#sec:leptom1}
----------------------------------
The mechanisms to create a baryon asymmetry from an initially symmetric state must in general satisfy the three basic conditions for baryogenesis as pointed out by Sakharov [@sakharov]:
1. Violate baryon number, $B$, conservation.
2. Violate $C$ and $CP$ conservation.
3. To be out of thermal equilibrium.
It is found that the $CP$ violation observed in the quark sector [@KM_CPviolation], e.g. in $K^0$-$\bar{K}^0$ or $B^0$-$\bar{B}^0$ mesons system, is far too small to give rise to the observed baryon asymmetry [@CP_in_SM_tooSmall]. Therefore, these conditions should be extended to include the lepton number $L$ violation processes.
The classic leptogenesis scenario of Fukugita and Yanagida [@fukugita_yanagida_86] described in [@LopezFogliani:2005yw; @Escudero:2008jg] can occur in the $\mu\nu$SSM model [@Chung:2010cd]. The Yukawa coupling can then induce heavy right handed neutrino $N$ decays via the following two channels: $$\label{eq1:lepto_std_decay_channels}
N_k \rightarrow
\left\{
\begin{array}{r}
\;\;\; l_{j} + \overline{\phi} \;,\\
\;\;\; \overline{l}_{j} + \phi \;,
\end{array}
\right.
,$$ that violate the lepton number by one unit. In the new $\mu\nu$SSM model given by the superpotential (\[superpotential1\]), it is one of the heavies neutralinos that is responsible for the right handed neutrion decay according to the equation (\[neutralinosmasses\]). All Sakharov’s conditions for leptogenesis are satisfied if these decays violate $CP$ and go out of equilibrium at some stage during the evolution of the early universe. The requirement for $CP$ violation means that the coupling matrix $Y$ must be complex and the mass of $N_k$ must be greater than the combined mass of $l_{j}$ and $\phi$, so that the interferences between the tree-level processes and the one-loop corrections with on-shell intermediate states will be non-zero [@Law:2009vh; @Law:2010zz]. Since $\phi$ is the scalar field of the SM, the usual Higgs can suffer the following decays $$\begin{aligned}
H^{0}_{1} & \rightarrow & l_{a}l^{c}_{b},
\\
H^{0}_{2} & \rightarrow & \nu_{a}\nu^{c}_{b},
\\
H^{-}_{1} & \rightarrow & \nu_{a}l^{c}_{b} ,
\\
H^{+}_{2} & \rightarrow & l_{a}\nu^{c}_{b}.\end{aligned}$$ Note that none of these decays violate the lepton number conservation. Nevertheless, in this model the fields $\tilde{\nu}$ have both chiralities. Therefore, they will induce the followings decays $$\begin{aligned}
\tilde{\nu}^{c}_{c} & \rightarrow & \nu^{c}_{a}\nu^{c}_{b},
\\
\tilde{\nu}_{a} & \rightarrow & l_{b}l^{c}_{c}.\end{aligned}$$ Thus, both violate the lepton number conservation. On the other hand, we note that there are scattering processes that can alter the abundance of the neutrino flavour $N_K$ in the $s$-channel $N \ell \leftrightarrow q_L \bar{t}_R$ and $t$-channel $N t_R \leftrightarrow q_L \bar{\ell}, N q_L \leftrightarrow t_R \ell$ besides the tree-level interaction ($N\leftrightarrow \ell \bar{\phi}$). In addition to these, there are also $\Delta L = \pm 2$ scattering processes mediated by $N_k$ which can be important for the evolution of $(B-L)$. Also, if we consider the couplings $Y_{\nu}^{ij}$ and $\lambda^{i}$ to be complex, we can generate the leptogenesis in this model as shown in [@Law:2009vh] by inducing decays as $\tilde{\chi}^{0}l \rightarrow d\bar{u}$.
It is interesting to note that the superpotential from the equation (\[superpotential1\]) induces the following processes [@dress; @Baer:2006rs; @barbier; @moreau]
1. New contributions to the neutrals $K\bar{K}$ and $B\bar{B}$ Systems.
2. New contributions to the muon decay.
3. Leptonic Decays of Heavy Quarks Hadrons such as $D^{+} \rightarrow \overline{K^{0}}l^{+}_{i}\nu_{i}$.
4. Rare Leptonic Decays of Mesons like $K^{+} \rightarrow \pi^{+}\nu \bar{\nu}$.
5. Hadronic $B$ Meson Decay Asymmetries.
Also, ir gives the following direct decays of the lightest neutralinos $$\begin{aligned}
\tilde{\chi}^{0}_{1} &\rightarrow& l^{+}_{i}\bar{u}_{j}d_{k}, \,\
\tilde{\chi}^{0}_{1} \rightarrow l^{-}_{i}u_{j}\bar{d}_{k},
\nonumber \\
\tilde{\chi}^{0}_{1} &\rightarrow& \bar{\nu}_{i}\bar{d}_{j}d_{k},
\,\ \tilde{\chi}^{0}_{1} \rightarrow \nu_{i}d_{j}\bar{d}_{k},
\label{lepto1}\end{aligned}$$ and for the lightest charginos $$\begin{aligned}
\tilde{\chi}^{+}_{1} &\rightarrow& l^{+}_{i}\bar{d}_{j}d_{k}, \,\
\tilde{\chi}^{+}_{1} \rightarrow l^{+}_{i}\bar{u}_{j}u_{k},
\nonumber \\
\tilde{\chi}^{+}_{1} &\rightarrow& \bar{\nu}_{i}\bar{d}_{j}u_{k},
\,\ \tilde{\chi}^{+}_{1} \rightarrow \nu_{i}u_{j}\bar{d}_{k}.
\label{lepto2}\end{aligned}$$ These decays are similar to the ones from the MSSM when $R$-Parity violating scenarios are taken into account. Therefore, we expect that the missing energy plus jets be the main experimental signal in the “new” $\mu \nu$SSM as is in the MSSM. These decays violate only the lepton number conservation but they conserve the baryon number.
As we have seen above, all necessary conditions to generate a viable leptogenesis mechanism from the $\mu \nu$SSM model are present in the “new” $\mu \nu$SSM model [@Kajiyama:2009ae] as well as the CP violation processes. Also, this model could contain an invisible axion. These properties deserve a deeper study. Another interesting phenomenological avenue is to analyse the total cross section of the Dark Matter-Nucleon (DM-N) elastic scattering process.
Explanation of the data from ATLAS, CMS and LHCb in $\mu \nu$SSM Model. {#sec:data}
=======================================================================
One possible explanation to the excess of electrons is given if the following processes are considered [@Biswas:2014gga; @Allanach:2014lca] $$\begin{aligned}
pp & \rightarrow & \tilde{e} \rightarrow e^{-} \tilde{\chi}_{1}^{0} \rightarrow e^+e^-jj, \nonumber \\
pp & \rightarrow & \tilde{\nu_{e}} \rightarrow e^{-} \tilde{\chi}_{1}^{+} \rightarrow e^+e^-jj.
\label{cmsexplanation}\end{aligned}$$ Neglecting finite width effects, the color and spin-averaged parton total cross section of a single slepton production is [@Dimopoulos:1988jw; @Allanach:2009iv] $$\begin{aligned}
\hat{\sigma} &=& \frac{\pi}{12\hat{s}}|\lambda^{\prime}_{111}|^2\delta \left( 1-\frac{m^{2}_{\tilde{l}}}{\hat{s}} \right),\end{aligned}$$ where $\hat{s}$ is the partonic center of mass energy, and $m_{\tilde{l}}$ is the mass of the resonant slepton. Including the effects of the parton distribution functions, we find the total cross section $$\begin{aligned}
\sigma (pp \rightarrow \tilde{l}) \propto |\lambda^{\prime}_{111}|^{2} / m_{\tilde{l}}^{3}, \end{aligned}$$ to a good approximation in the parameter region of interest.
As was discussed in [@Allanach:2014lca], these processes represent one of the possibilities to explain the data of CMS [@Khachatryan:2014dka; @CMS1] if the selectron mass is fixed to $2.1 {\mbox TeV}$ and the lightest neutralino mass is taken to be in the range from 400 GeV up to 1 TeV. The $R_{K}$ measurement can be consistent with the new physics arising from the electron or muon sector of the SM and it was shown in [@Biswas:2014gga] that if we consider the muon sector in the MSSM with $R$-parity violation scenarios, the $R_{K}$ can also account for both data arising from CMS and LHCb. In the “new” $\mu\nu$SSM model we have both terms present. With respect with the di-boson data, there is a similar explanation. Indeed, in the case of $V=W,Z$ there is the single production of smuons [@Allanach:2015blv], while in the case of di-photons the stau is produced [@Allanach:2015ixl]. Due this fact, we expect that our model fit the new data coming from ATLAS [@atlas13], CMS [@CMS1] and from LHCb [@Aaij:2014ora; @Aaij:2013pta]. To confirm that this is the true mechanism employed, the double beta decay must be detected in experiments like CUORE [@CUORE], GERDA [@GERDA] and MAJORANA [@MAJORANA] and no proton decay must occur in the neutron anti-neutron oscillation.
Conclusions {#sec:conclusion}
===========
In this article we have reviewed some of the basic properties of the MSSM, NMSSM and $\mu \nu$SSM essential to the cosmological applications. Also, in order to incorporate the recent data from the CMS and LHCb into this class of models, we have proposed a “new” $\mu \nu$SSM model characterized by the superpotential given in the equation (\[superpotential1\]). The terms added to the $W_{superpot}$ of the $\mu\nu$SSM in order to obtain the modified model, explicitly break the $R$-parity and the lepton number conservation. This makes the model attractive for cosmological applications as it presents flat directions that represent a possibility to generate inflation and a viable leptogenesis mechanism that is necessary to generate the matter anti-matter asymmetry. These properties make the model interesting for further investigations on which we hope to report in the near future.
[**Acknowledgments**]{}
M. C. R would like to thanks to Laboratório de Física Experimental at Centro Brasileiro de Pesquisas Físicas (LAFEX-CBPF) for their nice hospitality and special thanks to Professores J.A. Helayël-Neto and A. J. Accioly. Both authors acknowledge R. Rosenfeld for hospitality at ICTP-SAIFR where part of this work was accomplished. We acknowledge P. S. Bhupal Dev for information on the latest results from ATLAS and CMS in di-bosons, and D. E. Lopez-Fogliani for useful correspondence on the gravitino in the $\mu \nu$SSM.
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[^1]: marcoscrodriguez@ufrrj.br, ionvancea@ufrrj.br
[^2]: The excess was at a di-boson invariant masses in the range from 1.3 to 3.0 TeV.
[^3]: The ressonance appear at around 750 GeV in the di-photon invariant mass.
[^4]: About the history of MSSM, see e. g. [@Fayet:2001xk; @Rodriguez:2009cd]
[^5]: The term $h^{i}_{\nu}\hat{H}_{2}\hat{H}_{1}\hat{\nu}^{c}_{i}$ generate the $\mu$ term when the sneutrino get its values expectation values.
[^6]: We recall that the flat directions provide a viable mechanism to generate the cosmological inflation and the leptogenesis is important to explaining the asymmetry between the matter and the anti-matter.
[^7]: Different assumptions result in different version of the Constrained Minimal Supersymmetric Model (CMSSM).
[^8]: References to the original work on the NMSSM may be found in the reviews [@Ananthanarayan:1996zv; @Ellwanger:1998jk]
[^9]: $y^{m}\equiv x^{m}-i \theta \sigma^{m}\bar{\theta}$, where $\sigma^{m}$ are the three Pauli matrices plus the $I_{2 \times 2}$ the identity matrix.
[^10]: Where we have defined $e=g \sin \theta_{W}=g^{\prime}\cos \theta_{W}$.
[^11]: The flat directions are noncompact lines and surfaces in the space of scalars fields along which the scalar potential vanishes. The present flat direction is an accidental feature of the classical potential and gets removed by quantum corrections.
[^12]: In quantum field theories, the possible vacua are usually labelled by the vacuum expectation values of scalar fields, as Lorentz invariance forces the vacuum expectation values of any higher spin fields to vanish. These vacuum expectation values can take any value for which the potential function is a minimum. Consequently, when the potential function has continuous families of global minima, the space of vacua for the quantum field theory is a manifold (or orbifold), usually called the *vacuum manifold*. This manifold is often called the moduli space of vacua, or just the moduli space.
|
---
abstract: 'Originating in Wuhan, the novel coronavirus, severe acute respiratory syndrome 2 (SARS-CoV-2), has astonished health-care systems across globe due to its rapid and simultaneous spread to the neighboring and distantly located countries. To gain the systems level understanding of the role of global transmission routes in the COVID-19 spread, in this study, we have developed the first, empirical, global, index-case transmission network of SARS-CoV-2 termed as C19-TraNet. We manually curated the travel history of country wise index-cases using government press releases, their official social media handles and online news reports to construct this C19-TraNet that is a spatio-temporal, sparse, growing network comprising of $187$ nodes and $199$ edges and follows a power-law degree distribution. To model the growing C19-TraNet, a novel stochastic scale free (SSF) algorithm is proposed that accounts for stochastic addition of both nodes as well as edges at each time step. A peculiar connectivity pattern in C19-TraNet is observed, characterized by a fourth degree polynomial growth curve, that significantly diverges from the average random connectivity pattern obtained from an ensemble of its $1,000$ SSF realizations. Partitioning the C19-TraNet, using edge betweenness, it is found that most of the large communities are comprised of a heterogeneous mixture of countries belonging to different world regions suggesting that there are no spatial constraints on the spread of disease. This work characterizes the superspreaders that have very quickly transported the virus, through multiple transmission routes, to long range geographical locations alongwith their local neighborhoods.'
author:
- '$\text{Vikram Singh}^{\dagger}$'
- Vikram Singh
bibliography:
- 'CoV\_TrNet.bib'
title: 'C19-TraNet: an empirical, global index-case transmission network of SARS-CoV-2'
---
Introduction
============
Recent outbreak of COVID-19 has been emerged due to a new species of human infecting coronaviruses (HCovs), namely, SARS-CoV-2 [@of2020species] that was isolated from a cluster of pneumonia cases in Wuhan, capital city of Hubei Province, China [@zhu2020novel] in December 2019. It rapidly spread across the globe and was declared as a Public Health Emergency of International Concern (PHEIC) on [@world2005statement], and was later declared a pandemic on by the World Health Organization (WHO) [@world2020director]. As of , more than $9.1$ million confirmed cases, and more than $0.47$ million deaths have been reported due to COVID-19 in $216$ countries/territories [@WHOSR109]. The outbreak is attributed to a zoonotic transmission of SARS-CoV-2 from primordial or intermediate host to humans, it could have further propagated through human-to-human contacts [@wang2020clinical; @bai2020presumed]. Globalization of transport systems *via* air, sea and land routes has resulted in drastic decrease in journey time alongwith an unprecedented increase in transport volume of passengers. Although global transport has played a pivotal role in world’s economic growth, it has also facilitated the disease causing agents to move to remote and otherwise unreachable places very quickly [@tatem2006global], and therefore has enhanced the potential of any endemic to become a pandemic [@fidler1996globalization].
In the past five centuries the world has witnessed several major pandemics, like, plague [@keeling2000metapopulation] (caused by *Yersinia pestis*), cholera [@sack2004sack] (*Vibrio cholera*), influenza [@ferguson2003ecological] (influenza virus), HIV/AIDS [@perrin2003travel] (HIV), SARS [@peiris2004severe] (SARS-CoV) etc., that had badly affected most of the world population as well as economy. All these pandemics had been originated in localized regions and had different means of human-to-human transmission, however, they were transported to distant geographical locations by human movements *via* air, land or sea transport [@zinsser1935rats; @reidl2002vibrio; @palese2004influenza; @heymann2004international]. There are many other emerging, reemerging infections, with varying consequences in terms of severity, morbidity, mortality, which the world continues to confront [@bloom2019infectious]. Different epidemiological strategies, like, modeling disease dynamics, spread, mitigation and other epidemiological parameters of different outbreaks have been applied successfully in the past [@epstein2009modelling; @gates2018innovation; @weiss2003hiv]. Network theoretic approaches have also been successfully utilized to gain understanding about the topology of different epidemiological disease transmission networks [@salathe2010high; @enright2018epidemics; @keeling2005networks; @craft2015infectious]. Topological understanding of infection transmission networks provides us critical insights about infection growth and distribution. Furthermore, knowledge of potential transmission routes allows us to devise effective control strategies to contain infection [@bell1999centrality].
Due to lack of specific medication for SARS-CoV-2, non-pharmaceutial containment based intervention methods have become indispensable tools to grapple with the infection. A large number of studies are being published to investigate important metrics, like, CFR, basic reproduction rate ($R_0$), spread kinetics etc., however, a comprehensive study investigating the transmission routes of the virus across countries is still lacking. The actual structure (topology) underlying the global spread of the SARS-CoV-2 is also less explored. To prevent virus re-invasion in the countries/territories/regions (from where the virus has been eradicated) considering large incubation period of SARS-CoV-2, at this stage of outbreak, it is crucial to gain insights about the short range human-to-human contacts as well as long range geographically separated country-to-country global transmission pathways. It may have implications in making SARS-CoV-2 prevention policies as well as better prepare ourselves to fight against future outbreaks. In this study, we have constructed an empirical global transmission network of SARS-CoV-2 and explored its topological space to explore the mechanism of global spread of infection. Increased topological knowledge will provide us the understanding about the epidemiological dynamics of the virus invading remote geographical locations.
Methods
=======
Network construction
--------------------
In this work, we have constructed the index-case empirical global transmission network of SARS-CoV-2 by manually going through the daily situation reports published by WHO, government press releases of various countries, their official social media handles (facebook, twitter), and other online news reports providing relevant information regarding the travel history of the patients for a period of $84$ days ranging from to . The transmission network is a dynamic network having the potential to grow on daily basis. Countries comprise the nodes of this network, while infected index-case individuals carrying the infection from a given country $X$ to some other country $Y$ connect these two countries with a directed link with $X$ as the source node and $Y$ as the target node. The direction of an edge indicate the flow of virus infection across borders. The connection between two countries have been placed on the basis of travel history of the patient. Any foreign national reported to be corona positive in any country has been considered as a carrier of virus to that country irrespective of his home nation. For the countries reporting more than one patient as their index-cases, if the reported patients had different travel histories we first observed the timeline to find the countries in which virus had been already spread out and if the the virus had already reached in the source countries then we placed a link between all these source countries with the target country. On the other hand for the source countries where the virus had yet not reached, we considered that the patient has contracted the virus *enroute* and did not place link between them.
Stochastic scale free (SSF) random network models
-------------------------------------------------
SARS-CoV-2 or any other outbreak that spread through human-to-human contacts will follow a peculiar spread pattern where the infection starts from a source area and then gradually spreads across other countries/areas/regions. To model this specific spread pattern, we are proposing a novel algorithm, namely, stochastic scale free (SSF). SSF is developed as an extension of the basic Barabási-Alberts (BA) model of scale free network. In the proposed SSF algorithm, network grows stochastically over time with number of nodes and their associated edges both being drawn from an appropriate distribution. The algorithm follows three basic steps: (1) The infection spread is unidirectional *i.e.* from source to target, new nodes (targets) attach preferentially with already existing nodes (source) and have linkage probabilities proportional to source degree. (2) At every time step, any number of nodes ($0, 1, 2, \dots,\id{maximum-nodes}$) can be incorporated into the network. (3) Every node can bring one or more number of links ($1, 2, 3, \dots, \id{maximum-nodes}$) with it. The key purpose of stochastically adding both nodes and edges is to model the index-case of SARS-CoV-2 infection across various countries as a growing transmission network. In this network, the index-case import in various countries may have two different scenarios, namely, (i) more than one country can import their index-case on any given day *i.e.* more than one nodes may be added at a single time step. (ii) a country can receive their index-case from more than one countries *i.e.* an incoming node may have more than one in-degree.
The main characteristic of the growth of C19-TraNet is that initially very few new countries get infection so the probability of infection transmission to new countries at any given day is very low, however, this probability increases very rapidly with increase in number of infected countries. Therefore, while generating an SSF model of C19-TraNet, to obtain the number of nodes and links associated with them at a unit time interval $t$ (one day), we have used beta distribution that is given by $f(x;\alpha,\beta)
= \frac{\Gamma (\alpha + \beta) }{\Gamma (\alpha) \Gamma(\beta)} x^{\alpha -
1}(1 - x)^{\beta-1}$ where $\Gamma$ is a gamma function, and $\alpha$, $\beta$ are two non-zero parameters that control the shape of this distribution. The scaling parameters at any time interval $t$ were computed as $\beta_t =
\frac{\id{Max-time} - \id{Current-time} }{\id{Max-time}}$ and $\alpha_t = 1 -
\beta_t$, where $\id{Max-time}$ is the number of days for which the infection is to be modeled ($84$ in the current study) and $\id{Current-time}$ is the $t^{th}$ time step. On the other hand, scaling parameters $\alpha = 0.05$ and $\beta = 0.95$ were qualitatively selected to draw the number of edges asociated with the incoming nodes. The small value for $\alpha$ and a high value for $\beta$ are deliberately selected considering a very low average connectivity ($\approx 2.1$) of C19-TraNet. The maximum number of nodes ($\id{maximum-nodes}$) that can be incorporated into the network at a discrete time step is given by $\id{maximum-nodes} = \langle k_{in + out} \rangle + 1
\sigma $, while maximum number of edges that a node can bring along with it was computed as $\id{maximum-nodes} = \langle k_{out} \rangle + 1 \sigma $. Values obtained were truncated to retain only the integer part. Corresponding to the manualy curated C19-TraNet, we constructed an ensemble of $1,000$ stochastic scale free random networks having same number of nodes, and approximately same number of edges.
Global network metrics and community detection
----------------------------------------------
Four global network metrics, namely, (i) degree ($k$) distribution, (ii) network density $(\rho)$, (iii) average clustering coefficient $(C)$, and (iv) average path length $(L)$ were estimated for C19-TraNet and compared with the corresponding $1,000$ SSF random network models generated *via* proposed SSF algorithm. The degree of a node is the number of edges connected with it. In a directed network, the number of edges incident on a particular node constitute its indegree ($k_{in}$), while the number of edges extending outward constitute its oudegree ($k_{out}$) [@Newman2003]. The in and out degree distributions of a network are defined as fraction of nodes having $k$ incoming or outgoing links. Furthermore, we computed the density of network that is defined as a fraction of existing edges in the network to the all possible edges *i.e.* $\rho = \frac{m}{ \binom{n}{2}}$. We also enumerated clustering coefficient for every node that is defined as the number of edges existing between first neighbors of the node $i$ to the total possible edges existing between the first neighbors, averaged over all the nodes $i$ [@watts1998collective]. If $N_i$ is the set of nodes directly connected to node $i$ then its clustering coefficient for a directed network is defined as $$C_i = \frac{|\{e_{jk}: v_j, v_ \in N_i, e_{jk} \in E\}|}{|N_i| (|N_i| - 1)
}$$, where $E$ is the edge set of network $G(V,E)$. Average path length of a network is the average of geodesic paths between all possible pairs of nodes in the network [@watts1998collective] and is defined as $$L =
\frac{2}{n(n-1)} \sum \limits_{a = 1}^{n-1} \sum \limits_{b = a + 1}^{n}
\delta_{ab}$$. We identified communities based on edge betweenness, where the network is partitioned into node sets by removing high betweenness edges [@newman2004finding]. We generated $1,000$ random realizations of SSF models and Mann-Whitney U test was performed on distributions of average outdegree obtained at different time steps for both real and random networks [@mann1947test].
Results and Discussion
======================
The dynamic effects of heterogeneous communication routes has been central to the propagation of many diseases [@Mangili2005], hence the interplay between them and the underlying transmission network of SARS-CoV-2 across different countries/territories/areas is the focus of this study. We catalogued the first person diagnosed to have contracted SARS-CoV-2 virus for $187$ countries. By investigating the travel history of index-cases from government press releases, their official social media handles and other online news reports, we inferred a probable source (country where the patient contracted the virus) and the target (country confirming COVID-19 infection in the diseased) of each edge to be incorporated in the empirical COVID-19 global transmission network (C19-TraNet). The constructed C19-TraNet is comprised of $187$ nodes and $199$ directed edges specifying the route of infection transmission from the source country to the target country (\[fig:Net\]). The final network is a combined snapshot representing SARS-CoV-2 transmission across the countries that has grown over a period of $84$ days from to .
SARS-CoV-2 global transmission routes
-------------------------------------
The outbreak originated from China (Western Pacific region) and within two months (by ) the virus had been spread to at least one country of all the six WHO regions. By the end of January, the contagion had been reported in five WHO regions, namely, Western Pacific Region ($45 \%$), South-East Asia Region ($40 \%$), European Region ($13.33$), Region of Americas ($5.56 \%$) and Eastern Mediterranean ($4.54 \%$). The numbers in parentheses indicate percentage of infected countries per region. Except Malaysia and Spain, which imported virus from Singapore and Germany, respectively, all the other $23$ countries imported it from China. Global transmission of SARS-CoV-2 can broadly be attributed to two tidal waves, namely, (1) a Chinese wave, from the date of origin of outbreak till the end of January, 2020 and (2) a European wave that took off from mid February and continued until almost all the countries got infection. The major reason for this quick spread is, (1) emergence of China as an economic globaliser in past few decades that has increased international business and trade, and (2) expansion of highly connected local and global travel systems [@trindade2018globaliser].
Travel data suggests that approximately $3$ billion people traveled through trains or flights during the Chunyun (Lunar New Year) holidays [@daszak2020strategy]. Also, a large volume of passengers have been estimated to travel within [@zhao2020association] and outside China [@bogoch2020pneumonia] through land and air routes from previous travel data. These studies suggest a significant relation between transport volume and likelihood of imported cases of SARS-CoV-2. Although ban was implemented in Wuhan city on , which was followed by travel restrictions to $14$ of its neighboring cities of Hubei Province on , however, by this time the virus had been transmitted to other Chinese cities as well as to $9$ different countries belonging to four WHO regions [@WHOSR5]. From these sources, the virus kept on spreading locally as well as outside China till the end of January when the results of containment measures became visible with overall decrease in cases exported from China [@chinazzi2020effect] (diminishing the effects of Chinese wave). Then from mid of February, second tidal wave took off from European Region and circulated the infection to most parts of the globe. Italy, Iran and Spain were among the super spreaders which aggressively spread the contagion. Owing to large incubation period of SARS-CoV-2 along with lack of necessary infrastructure, sophisticated testing facilities, herd immunity, a large number of undocumented asymptomatic or presymptomatic cases passed undetected into susceptible populations of different countries/territories/areas, thus provided unprecedented routes to virus transmission [@li2020substantial; @rothe2020transmission]. These large number of possible routes also increased the strength of second wave by providing opportunities to circulate the virus to most of the uninfected countries belonging to different regions. Therefore, we may infer that a large incubation period, high movement and multiple routes helped global circulation of the disease.
Global topological metrics of C19-TraNet
----------------------------------------
The final network is sparse with a very low density ($0.01$). This is expected, as we constructed only the index-case transmission network. Average connectivity of the network is found to be $2.128$, computed by ignoring the edge directions. Average or characteristic path length (APL) of a network represents closeness in the network, so lesser the magnitude of APL closer the nodes are. The characteristic path length of C19-TraNet is found to be $1.56$, while its diameter and radius are found to be $4$ and $1$, respectively. As per APL, most of the countries are infected by one or two transmission events. Network diameter suggests that most distant country in the network is infected by a maximum of four transmission events. These short path related metrics explain the quick global spread of virus, *i.e.* the virus got access to those countries which have a wider reach on the globe that helped in propelling the infection across the world in a very short span of time. Short APL and a very small network diameter also indicate that there are some long range connections connecting distant geographical locations bringing an element of small worldness to the network. There are some hub nodes (very few) with a large number of edges in C19-TraNet. Plotting the degree distribution on log-log scale, we obtained a straight line (with scaling exponent $-1.63$) indicating that C19-TraNet follows power law degree distribution and possess the property of scale freeness (Fig. \[fig:DDist\]). The power law pattern of degree distribution has a simple interpretation that only few countries in the network are responsible for spreading the virus to most other countries. It has been reported that for any scale free disease transmission networks with scaling exponent less than $3$, the virus reproduction number ($R_0$) is greater than $1$ *i.e.* very few nodes (super spreaders) are sufficient to maintain virus load in the susceptible populations [@schneeberger2004scale]. Its main implications lie in designing control strategies, like, targeting the super spreaders by implementing the basic containment strategies *e.g.* prohibit transport, sanitization etc..
Global infection exportation potential (GIEP)
---------------------------------------------
Emergence of a large number of countries in C19-TraNet as leaf nodes (having very low average clustering coefficient) implies that the infection was not equally spread by all the countries, rather only a few countries were responsible for its spread. We observed that only $38 (\approx 20 \%)$ countries among $187$ have at least one edge projecting outward. Among them, only seven countries are found to have an outdegree of $10$ or more, with Italy at the top of the tally contributing a total of $47$ exportation events followed by china ($29$) and Iran ($16$) (see Table \[tab:Prop\]). Furthermore, infection import to every target node in C19-TraNet would have occurred in two possible ways, either a domicile of the target country has brought the virus from a source country, or a foreign national visiting a source country has spread the infection to the target country. To investigate the proportion of these two cases in the global spread of infection, we enlisted the nationalities of SARS-CoV-2 carriers and found that most of the index-cases are local nationals who contracted the infection in some source country and then brought it to their home nations.
A total of $131 (\approx 65 \%)$ out of $202$ index-cases were found to be local nationals. Here we mentioned $202$ index-cases because some countries, like, Kuwait, Aruba, Kazakhstan had reported more than one cases on the same day (Supplementary Table 1). Remaining $71$ ($45 \%$) index-cases were foreign nationals belonging to $22$ countries. Among these foreign nationals, $19$ are found to be Chinese followed by $14$ Italians (see Table \[tab:ISP\]). To estimate the global infection exportation potential (GIEP) per country, we further normalized the number foreign nationals belongiing to each of the source country by the total number of exportation events ($71$) carried out by foreign nationals (Fig. \[fig:Rgiep\]). Above data clearly suggests that although most of the infections are imported to the target countries by their domiciles, however, a significant proportion of countries are infected by foreign nationals. As an example, the number of virus transmission events ($47$) that took place from Italy is very high, however, there were only $14$ Italians involved in these transmission events. Thus, in the light of above data, it can be inferred that highly robust and efficient transportation system has played a huge role in spreading the infection across international borders. Therefore, more caution is required specially from aviation departments regarding individual’s health at international transportation junctions.
Connectivity dynamics modeling and analysis of C19-TraNet
---------------------------------------------------------
Since the pandemic evolves with time, so exploring the connectivity of growing C19-TraNet at different instances of time ($t$) can provide critical insights about the network topology and growth mechanisms. Keeping that in mind we computed average out-degree of C19-TraNet for each of the $84$ days. Further, $1,000$ random networks using SSF model corresponding to C19-TraNet were generated and average out-degree vectors for all these random networks were computed which were further averaged to yield a single vector. We then performed Mann-Whitney U test to verify the null hypothesis ($H_0$) that the two outdegree distributions (of real network and average of $1,000$ random networks) are equal, which is refuted by a very low $p-value = 3.041e-09$. Distributions of average outdegree per unit time (days) for both real and average of SSF models are shown in Fig \[fig:Deg\]. We fitted four distributions *viz.* linear, logarithmic, exponential and, polynomial (up to order four), among which $4^{th}$ order polynomial curve is found to have the highest adjusted coefficient of determination (Adj. $R^2$) for both the real and SSF distributions (Table \[tab:Trend\]).
In the initial growth phase (*i.e.* till the end of January, 2020), both the curves followed same trend, however, from February onwards a peculiar phase of no growth (for about a fortnight) is observed in C19-TraNet which is absent in the random networks (Fig \[fig:Deg\]). This delay emerges due to implementation of Wuhan ban and can be attributed to the border control measures including local and international travel ban, quarantine, symptom screening at airports etc. simultaneously adopted by China and other countries. As stated earlier, most of the infections in Chinese wave are disseminated from China, therefore the two consecutive travel bans implemented by China substantially reduced the rapid local and international flow of virus [@kraemer2020effect; @wells2020impact]. The second surge in average connectivity (starting from the end of February, 2020) marks the onset of European wave which can be attributed to passage of undocumented pre- or asymptomatic infected people from countries other than China where restrictions were not that much strict [@wells2020impact; @li2020substantial]. Though the contagion invaded most of the world, however, our model suggests that these containment measures have delayed the global transmission of virus by $51$ days. If the growth had continued to follow the initial trends, the virus would have invaded South Sudan ($187^{th}$ country to be infected) on . We obtained similar trend, in reverse order, characterized by a phase of no growth for approximately same period when density is plotted against time (see Fig \[fig:Den\]). Network density peaked when the network was very small. With the passage of time, addition of new countries brought small increase in network size in comparison to the network order resulting in an overall decrease in network density.
Community structure of C19-TraNet
---------------------------------
We explored the structure of C19-TraNet by leveraging edge betweenness, in which high betweenness nodes were recursively removed to identify sparsely connected ‘node sets’ (communities) whose members are densely connected among themselves [@newman2004finding]. Communities provide an insight about the local connectivity patterns in C19-TraNet by partitioning the network into densely connected, small sub-graphs. A total of $23$ different communities (Fig. \[fig:CNet\]) were obtained with the largest set consisting of $39$ countries followed by $16$ countries in the second community and $14$ in the third set (Table. \[ Tab:ComCount\]). Most of the communities were small sized, while very few consisted of large number of countries (hubs), indicating that few countries (having high volume of international traffic) have acted as super spreaders. We further grouped the countries into six geographical regions as per WHO, namely, European Region (ER), Western Pacific Region (WPR), South-East Asia Region (SEAR), Eastern Mediterranean Region (EMR), Region of the Americas (RA), African Region (AR) and enumerated the country share of each region per community.
As shown in Fig. \[fig:CHmap\], most of communities are composed of a heterogeneous mix of countries from different regions. Most of the communities are comprised of countries from more than one region among which larger regions (in terms of number of countries) including ER, RA and AR are the major contributors. Approximately $62\%$ of the countries in Community$\_0$ belongs to ER followed by a relatively few countries of RA ($15\%$) and AR ($\approx
13\%$) regions. Similarly, Community$\_1$ is also comprised mostly of countries from two regions, ER ($42\%$) and EM ($\approx 53\%$) (Fig. \[fig:CHmap\]). These results indicate that there is not any kind of positional constraint imposed on the spread of infection, *i.e.*, infected countries from a given region have spread the infection to their neighboring countries (short range interactions) as well as across non neighboring countries or territories which are geographically far away from them (long range interactions).
Summary and Conclusion
======================
In this study, we have constructed the an empirical index-case global transmission network of infection caused by SARS-CoV-2. Our main objective here is to develop a systems level understanding of emergence and growth of COVID-19 from a local outbreak into a global pandemic. To achieve this goal, we relied on the travel history of index-case patients obtained from various reliable sources. It allowed us to map multiple transmission routes through which the virus intruded into different geographical locations.
We observe that, after its origin in China, the global transmission of SARS-CoV-2 can be attributed to two waves, namely, the Chinese wave and the European wave. Initially, during Chinese wave, the virus invaded to a large number of neighboring as well as distantly located countries from China due to a large volume of international traffic. By the end of January, 2020, the effect of Chinese wave started to diminish due to implementation of various border control measures within and outside China. However, soon after a small stagnation phase, the virus started to spread from other infected countries mainly from European Regions. The main reason of this European wave was that a large number of asymptomatic patients (local or foreign nationals) passed undetected to different countries and by the time we responded, it had already victimized the whole world.
Topological structure of C19-TraNet is found to be sparse, however, its degree distribution follows a power law distribution, *i.e.*, it possess scale free property suggesting that few countries have played the role of super spreaders in virus transmission. Considering the C19-TraNet as an evolving network, we further modeled the temporal growth of its average outdegree and found a delay of $51$ days in virus transmission that can be attributed to different border control measures. Data suggests that large incubation period, free flow of international traffic and lack of coordinated action at international level in implementing various control strategies have played a vital role in global transmission of the virus. Exploration of community structure of the network revealed that large communities are not comprised of countries from a single region, however, they are heterogeneous composition of countries from different regions suggesting the presence of local transmission, to neighboring countries of the same region, as well as long range transmission to countries belonging to different regions. These multiple transmission paths allowed the virus to rapidly invade susceptible subpopulations located in distant geographical regions. This conclusion emphasizes the imposition of travel restrictions which although implemented, however, lack coordinated actions from different countries. It suggests that a coordinated action plan is needed to put in places to fight with any future outbreak.
The stochastic scale free (SSF) algorithm, proposed in this work to model the first case global transmission network of SARS-CoV-2, can be implemented very easily to model other human-to-human transmission disease networks with minor changes. The main advantage of the proposed algorithm is that the probability of infecting new nodes changes with change in number of infected nodes, that increases continuously until the network order approaches saturation.
We thank Central University of Himachal Pradesh for providing the required infrastructure and computational facilities. $\text{VS}^{\dagger}$ thanks Council of Scientific and Industrial Research (CSIR), India for providing Junior Research Fellowship (JRF).
Author Contributions
====================
$\text{VS}^*$ conceptualized and designed the research framework. $\text{VS}^{\dagger}$ performed the computational experiments. $\text{VS}^{\dagger}$ and $\text{VS}^*$ analyzed the data and interpreted results. $\text{VS}^{\dagger}$ and $\text{VS}^*$ wrote and finalized the manuscript.
Conflict of Interests
=====================
The authors declare that they have no conflict of interests.
[1]{} {width="\textwidth"}
[0.43]{} {width="\textwidth"}
[0.55]{} ![ Outdegree histogram and global infection exportation potential (GIEP) of $22$ source countries: (a) Total number of index-case exportation events from a source country (blue), and the number of such events carried out by its domiciles (red). (b) Global infection exportation potential (GIEP) that is defined as the ratio of index-case exportation events from the domiciles of a source country to the total number of such events ($71$). It may be noted that China has the largest GIEP followed by Italy and USA.[]{data-label="fig:Rgiep"}](GIEP_bar "fig:"){width="\textwidth"}
[0.55]{} ![ Outdegree histogram and global infection exportation potential (GIEP) of $22$ source countries: (a) Total number of index-case exportation events from a source country (blue), and the number of such events carried out by its domiciles (red). (b) Global infection exportation potential (GIEP) that is defined as the ratio of index-case exportation events from the domiciles of a source country to the total number of such events ($71$). It may be noted that China has the largest GIEP followed by Italy and USA.[]{data-label="fig:Rgiep"}](GIEP "fig:"){width="\textwidth"}
[0.55]{} {width="\textwidth"}
[0.55]{} {width="\textwidth"}
[0.70]{} {width="\textwidth"}
[0.65]{} {width="\textwidth"}
--- ---- --- --- ---- ---- ---
1 24 0 2 2 6 5
2 8 1 0 10 0 0
3 2 5 2 2 3 0
4 0 1 2 0 10 3
5 3 1 0 1 1 8
6 1 0 0 0 7 6
7 5 1 0 1 3 3
--- ---- --- --- ---- ---- ---
: Number of countries per region present in individual community.
\[tab:ComCount\]
---------------------------------- --- ---- --------
Italy 1 47 1.2034
Spain 1 11 1.1538
Iran 1 16 1.1111
French Republic 1 15 1.1176
United States of America 1 16 1.0000
United Kingdom 1 14 1.0667
Germany 1 4 1.8824
Switzerland 1 4 1.0000
Netherlands 1 3 1.0000
Portugal 2 1 1.0000
Japan 1 3 1.0000
United Arab Emirates 1 3 1.0000
India 1 3 1.2500
Belgium 1 3 1.2500
Singapore 1 2 1.3333
Ecuador 1 1 1.0000
Norway 1 2 1.3333
Ghana 2 1 1.0000
Greece 1 2 1.0000
Malaysia 1 1 1.0000
Dominican Republic 1 1 1.0000
Morocco 1 1 1.0000
Saudi Arabia 1 1 1.0000
Hungary 1 1 1.0000
Cameroon 1 1 1.0000
Thailand 1 2 1.0000
Côte d’Ivoire 1 1 1.0000
Australia 1 1 1.0000
Philippines 1 1 1.0000
Russian Federation 1 1 1.0000
Democratic Republic of the Congo 1 1 1.0000
Rwanda 1 1 1.0000
Mauritius 1 1 1.0000
Togo 1 1 1.0000
Burkina Faso 1 1 1.0000
China 0 29 1.9836
Turkey 0 1 1.5000
Panama 0 1 1.0000
---------------------------------- --- ---- --------
: Network metrics of countries with at least one outdegree.
\[tab:Prop\]
---------------------------------- ---- --------
China 19 0.4043
Italy 14 0.2979
United States of America 5 0.1064
United Kingdom 4 0.0851
Germany 3 0.0638
Iran 3 0.0638
Spain 3 0.0638
French Republic 2 0.0426
India 2 0.0426
Japan 2 0.0426
Netherlands 2 0.0426
Russian Federation 2 0.0426
Australia 1 0.0213
Belgium 1 0.0213
Democratic Republic of the Congo 1 0.0213
Ecuador 1 0.0213
Morocco 1 0.0213
Nigeria 1 0.0213
Norway 1 0.0213
Philippines 1 0.0213
Turkey 1 0.0213
United Arab Emirates 1 0.0213
---------------------------------- ---- --------
: Number of domiciles of source country spreading infection to target countries and global infection exportation potential (GIEP) obtained by normalizing individual number by total out degree.
\[tab:ISP\]
----------------- ------- ---------- ------- ---------- ------- ---------- ------- ----------
****
****
****
**Linear** 0.623 2.80E-19 0.599 3.58E-18 0.441 3.54E-12 0.567 9.09E-17
**Quadratic** 0.762 1.01E-24 0.906 2.24E-37 0.725 5.95E-19 0.889 5.14E-34
**Cubic** 0.887 6.54E-35 0.987 8.93E-70 0.882 1.96E-29 0.984 1.02E-64
**Quartic** 0.97 1.44E-55 0.995 2.14E-84 0.957 2.12E-44 0.994 1.72E-81
**Quintic** 0.977 1.02E-59 0.995 1.19E-84 0.972 5.75E-51 0.994 3.29E-81
**Log** 0.906 4.97E-44 0.911 5.16E-45 0.822 1.20E-32 0.896 3.00E-42
**Exponential** 0.536 1.50E-15 0.54 1.11E-15 0.906 4.57E-44 0.929 5.17E-49
----------------- ------- ---------- ------- ---------- ------- ---------- ------- ----------
: R adjusted ($R_{adj}$) and *p-values* obtained by fitting different models on outdegree and density obtained from both C19-TraNet and corresponding SSF models (for comparison, average of $1,000$ SSF models is used.)
\[tab:Trend\]
|
---
abstract: 'We define and investigate the properties of the jaggedness of path integral trajectories. The new quantity is shown to be scale invariant and to satisfy a self-averaging property. Jaggedness allows for a classification of path integral trajectories according to their relevance. We show that in the continuum limit the only paths that are not of measure zero are those with jaggedness 1/2, i.e. belonging to the same equivalence class as random walks. The set of relevant trajectories is thus narrowed down to a specific subset of non-differentiable paths. For numerical calculations, we show that jaggedness represents an important practical criterion for assessing the quality of trajectory generating algorithms. We illustrate the obtained results with Monte Carlo simulations of several different models.'
address: |
Scientific Computing Laboratory, Institute of Physics\
P. O. Box 57, 11001 Belgrade, Serbia and Montenegro
author:
- 'A. Bogojevi'' c'
- 'A. Balaž'
- 'A. Beli'' c'
title: Jaggedness of path integral trajectories
---
,
,
Path integral ,Quantum theory ,Self-similarity ,Monte Carlo 03.65.-w ,03.65.Db ,05.45.Df ,05.10.Ln
Introduction {#sec:intro}
============
The set of all the trajectories that one integrates over in a path integral is huge (cardinality $\aleph_2$). In order to have successful numerical simulations it is important to generate a finite number of trajectories that are as representative of the whole set as possible. For this reason it is necessary to broaden the amount of analytical information regarding the subset of trajectories that give dominant contributions to path integrals. This subset is much smaller than the set of all trajectories. Investigation of its properties is of great importance. Ultimately we would like to find ways to generate only these relevant trajectories.
Classification of trajectories according to their relevance is just as important for analytical calculations of continuum limit amplitudes. As a result of the continuum limit many trajectories are in fact of measure zero, i.e. give no contribution to path integrals. For example, it is well known that differentiable trajectories are of measure zero. This has been a severe impediment to the development of path integration theory as most of mathematics (and practically all of our intuition) is firmly based on smooth, differentiable functions. In fact, the principle reason that we have made any analytical progress with path integrals lies in the fact that, although of measure zero, some differentiable functions (e.g. classical solutions) turn out to be important markers of “nearby" non-differentiable paths that do give important contributions.
We are in a precarious position in which we know very little about the paths we need to work with. In addition, most of our knowledge is negative, i.e. tells us which trajectories are not important rather than which are. This is a substantial problem in the analytical approach. In numerical simulations, on the other hand, we work with discrete trajectories about whose classification we know even less. Therefore, the implementing of a systematic classification of the relevance of path integral trajectories (both in the discretized and continuum theories) is a crucial starting point in the quest for more efficient calculation schemes.
One of the few positive statements concerning path integral trajectories is that they are stochastically self-similar [@feynmanhibbs; @feynman]. An exhaustive review of various aspects of the path integral method can be found in [@kleinert]. A direct consequence of self-similarity is that relevant path integral trajectories have [@kroeger] fractal (Hausdorff) dimension $d_H=2$. In this paper we introduce a new classification property – the jaggedness of a trajectory. An analytical and numerical analysis of the properties of jaggedness leads to a new classification of trajectories according to their relevance to path integrals.
Self-similarity of trajectories {#sec:selfsim}
===============================
The property of stochastic self-similarity of path integral trajectories has important repercussions both on the dynamics as well as on the construction of efficient numerical algorithms for generating paths. In a previous set of papers [@prl05a; @prb05a; @pla05a] we have used self-similarity to obtain analytical relations between discretizations of different coarseness for the case of a general theory. The newly developed analytical method systematically improves the convergence of path integrals of a generic $N$-fold discretized theory through the explicit construction of a set of effective actions $S^{(p)}$ for $p=1,2,3,\ldots$. These effective actions are equivalent to the starting action (in the sense that they lead to the same continuum amplitudes), however, the path integrals calculated using them converge to the continuum limit ever faster. Discretized amplitudes calculated using the $p$ level effective action tend to the continuum limit as $1/N^p$. Using the general procedure we obtained explicit effective actions up to $p=9$. Self-similarity played a crucial role in this procedure allowing us to derive, and asymptotically solve, an integral equation relating discretized theories viewed at different coarseness. It was shown [@pla05a] that this approach is in fact equivalent to the derivation of a generalization of Euler’s summation formula to path integrals.
Self-similarity has also been successfully utilized in constructing efficient numerical algorithms for generating paths for Monte Carlo simulations [@pollockceperley; @ceperley]. In particular the Lévy construction discussed in these references generates self-similar paths through a simple iterative procedure. One begins with the fixed end points $a$ at time $t=0$ and $b$ at time $t=T$ and samples a bisecting point at time $t=T/2$ from a Gaussian centered in the middle between $a$ and $b$ and with width $\sigma=\sqrt{T/2}$. Having sampled the point at $t=T/2$ one now bisects the two new intervals $[0,T/2]$ and $[T/2,T]$ in the same way generating new bisection points at $t=T/4$ and $t=3T/4$ with appropriately centered Gaussians of width $\sigma=\sqrt{T/4}$. The procedure is continued recursively doubling the number of sampled points at each level and using the width $\sigma=\sqrt{\epsilon/2}$, where $\epsilon$ is the current time step. This method exactly samples free particle trajectories, but also works very well for interacting theories. The principle benefit of the method is that computational effort scales as $O(N)$, where $N$ is the coarseness of the discretization.
It is important to strengthen the relation between these two types of approaches. For a deeper understanding of the role of self-similarity it is necessary to classify trajectories according to their relative contribution to the path integral. As we have already mentioned, the first and most natural such classification of paths was with respect to their fractal dimension. The fractal dimension $d_H$ is calculated from the formula $$\label{fractaldimension}
\langle L\rangle\propto N^{\frac{d_H-1}{d_H}}\ ,$$ where $\langle L\rangle$ is the expectation value of $L=\sum_{i=0}^{N-1}|q_{i+1}-q_i|$, the total trajectory length. It was shown by Kr" oger [@kroeger] that for theories with (Euclidean) action of the form $$\label{action}
S=\int dt\,\left(\frac{1}{2}\dot q^2+V(q)\right)\ ,$$ the only trajectories that contribute to the path integral are those with fractal dimension $d_H=2$. The reason for this is that for short times of propagation the kinetic term dominates over the potential and each model looks like a random walk (which has $d_H=2$). This is illustrated in Fig. 1 on the example of an anharmonic oscillator with quartic coupling $V(q)=\frac{1}{2}q^2+\frac{1}{4!}gq^4$. Kr" oger has also shown that the addition of velocity dependent interactions changes the fractal dimension of relevant paths.
{height="4.8cm"} {height="4.8cm"}
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Let us now introduce another quantity (complementary to the fractal dimension) which will serve to further classify path integral trajectories according to their relevance.
The $N$-fold discretized path from $q_0=a$ to $q_N=b$ is determined by the $N-1$ intermediate positions $q_1, q_2, \ldots, q_{N-1}$. We define the jaggedness of a path as $$\label{jaggedness}
J=\frac{1}{N-1}\sum_{i=0}^{N-2}\frac{1}{2}\Big(1-\mbox{sgn}(\delta_i\delta_{i+1})\Big)\ ,$$ where $\delta_i=q_{i+1}-q_i$. From the above definition we see that $J$ in fact counts the number of peaks (both minima and maxima) divided by $N-1$ (the maximal number peaks for an $N$-fold discretized trajectory). Therefore, for all values of $N$, we have $J\in[0,1]$. Fig. 2 illustrates a typical path with $N=14$ and jaggedness $J=\frac{10}{13}$ (intermediate points that are peaks are depicted by black circles).
An important property of the jaggedness is that it is scale invariant up to $1/N$ terms. Namely, if we scale the intermediate points $q_i\to\lambda q_i$ the only thing that can change are the first and last term in the above sum (since the end points need to remain fixed). This brings about a change in $J$ that is $O(1/N)$. Therefore, we see that in the continuum limit $J$ is in fact scale invariant. $J$ is also scale invariant for finite $N$ when $a=b=0$. This behavior is quite different from that of fractal dimension. From eq. (\[fractaldimension\]) we see that the fractal dimension is not invariant under scaling with an $N$-dependent $\lambda$. For this reason the classification of trajectories with respect to $J$ is independent of a classification with respect to $d_H$.
Analytical and numerical analysis of jaggedness {#sec:jag}
===============================================
In the continuum limit all smooth and differentiable trajectories have $J=0$. This is not only true of monotonic trajectories but also of those having a finite number of extrema. Paths with $J=0$ are of measure zero. A classification of paths with non-vanishing jaggedness is therefore a path integral motivated classification of non-differentiable paths.
From eq. (\[jaggedness\]) we see that (up to $1/N$ terms) the jaggedness satisfies an averaging property, i.e. if we split a trajectory with $2N$ discrete time steps into two equal halves (with jaggedness $J_1$ and $J_2$), the total jaggedness equals $$J=\frac{1}{2}(J_1+J_2)-\frac{1}{2N-1}\left[\frac{1-\mbox{sgn}(\delta_{N-1}\delta_N)}{2}-\frac{J_1+J_2}{2}\right]\ .$$ As we can see, for finite $N$ the averaging is broken by a $1/N$ term. In the continuum limit, however, the averaging of jaggedness is exact. Cutting up a trajectory into $k$ equal pieces we get $J=\frac{1}{k}\sum_{i=1}^kJ_i$. Stochastic self-similarity now implies that all the $J_i$’s are equal to each other, and in fact that they are equal to the jaggedness of the whole path. We denote this property of jaggedness as self-averaging. The jaggedness of the whole trajectory (corresponding to a motion for time $T$) is thus equal to the jaggedness of even the smallest piece of that trajectory (corresponding to the propagation for a much shorter time $\Delta t$).
It is well known that for short times of propagation the dynamics of any model is well approximated by that of a random walk, i.e. does not depend on the potential. Therefore, self-similarity of trajectories implies that the jaggedness of the trajectories of a general model with action of the form given in eq. (\[action\]) is equal to the jaggedness of trajectories for a random walk.
The expectation value of the jaggedness for a random walk is quite easily calculated. For a random walk $\delta_i$ and $\delta_{i+1}$ are not correlated and so $\langle J\rangle =\frac{1}{2}$. We therefore conclude that for all the models with action of the form of eq. (\[action\]) we have $\langle J\rangle=\frac{1}{2}$. Similarly, for finite $N$ we find that $\langle J\rangle$ deviates from the continuum as $O(1/N)$.
{width="10cm"}
{width="10cm"}
Fig. 3 gives the relative contributions to the path integral of trajectories of different jaggedness for the case of $N=512$. As is shown in the figure, the distribution is practically indistinguishable for different models and widely varying parameters. Detailed numerical investigations show that for all models studied deviations from a Gaussian distribution (e.g. skew and kurtosis) go to zero as $N\to\infty$. In agreement with our previous analytical argument based on self-similarity the center of the distribution is $\langle J\rangle = \frac{1}{2}+ O(1/N)$. The width of the distribution $\sigma_J$ vanishes as $\sigma_J\sim 1/\sqrt{N}$ as illustrated in Fig. 4. The conclusion is that for models with action of the form of eq. (\[action\]) the only paths that are not of measure zero are those with $J=\frac{1}{2}$.
There are several important consequences of this that we need to mention. The analytical consequence of the above results is that they give us a much more detailed understanding of which trajectories are relevant and which are not. Not only are smooth, differentiable trajectories of measure zero (i.e. those with $J=0$), but in fact most of the non-differentiable trajectories also do not contribute to path integrals. To understand path integrals better we need to focus on a much narrower class of non-differentiable trajectories – those with $J=\frac{1}{2}$, i.e. those belonging to the same jaggedness equivalence class as the random walk.
The numerical consequence of the above results is that for finite $N$ it is important to have algorithms that generate trajectories that are distributed in a way that mimics, as much as possible, the physical distribution of relative contributions to the path integral such as the one shown in Fig. 3. The most important thing to focus on is the center of the distributions.
{width="10cm"}
Fig. 5 compares the physical average $\langle J\rangle$, which is algorithm independent, with $\bar J$ the average jaggedness of the trajectories generated by the algorithm (in this case the Le' vy construction). Note that for the Le' vy construction $\bar J$ is completely independent of the dynamics. The reason why this method represents a good algorithm (at least for large $N$) is that $\bar J\to\frac{1}{2}$ in the continuum limit. This figure also indicates that a useful measure of the quality $Q$ of an algorithm for generating trajectories is roughly the inverse of $|\langle J\rangle - \bar J|$. We will use this measure of quality to compare the Le' vy construction with the diagonalization algorithm in which trajectories are generated by a Gaussian distribution function using a semi-classical expansion. The computing time of this algorithm scales as $O(N^2)$ since it is necessary to diagonalize the quadratic form in the exponential of the distribution function.
In Fig. 6 we plot $|\langle J\rangle - \bar J|$ for these two algorithms. The top curve is for the Le' vy construction, the bottom for the diagonalization method. The latter algorithm outperforms the former by an order of magnitude for all values of $N$. As a result of this the Monte Carlo errors calculated using the two methods differ by an order of magnitude. However, one should not forget that quality needs to be balanced by an assessment of cost (in computing time). Although it gives a relatively lower quality, the computing time for the Le' vy method scales as $N$. The diagonalization method outperforms the Le' vy method for a given $N$, however its computation time scales as $N^2$. Both algorithms are good (since $|\langle J\rangle - \bar J|$ vanishes in the continuum limit). A simple cost-benefit analysis tells us that Le' vy construction is better for large $N$, while diagonalization is better for smaller $N$.
{width="10cm"}
Not all algorithms for generating paths are good, however. For example, if we modify the Le' vy construction using uniform distributions (of appropriate widths) instead of Gaussians we get an algorithm for which $\bar J$ does not tend to $\frac{1}{2}$. A signature of a bad algorithm is that there exists a maximum quality $Q_{max}$ that can’t be passed irrespective of computational cost.
As we have mentioned, it is important to strengthen the connection between the dynamical (calculation of path integrals) and kinematical (generation of trajectories) aspects of self-similarity. The effective action approach [@prl05a; @prb05a; @pla05a] has brought about an immense speedup in path integral calculations. The speedup is a direct consequence of the fact that effective actions allow us to work with much smaller values of $N$ to obtain the same precision. Using standard algorithms (which have all been derived to be optimal for large $N$) we have obtained a speedup of many orders of magnitude. The important step that next needs to be taken is to develop a path generating algorithm that is tailored for small $N$’s, i.e. for which $\bar J$ is near to $\langle J\rangle$ for coarse discretizations. Note that the diagonalization algorithm is one such method, but that it is not computationally optimized. We are currently working on developing these kinds of algorithms and tying them in to the effective action approach.
At the very end we wish to make further contact between jaggedness and random walks. In order to be near the random walk limit, the potential term in the discretized action $\epsilon V$, where $\epsilon=T/N$, must be smaller than the kinetic term $\delta^2/2\epsilon$. One should expect that when we are near the random walk regime all quantities depend on the ratio of these two. On the other hand, for a random walk we have $\delta^2/\epsilon\sim 1$, so that the ratio is in fact $\epsilon V(q_c)$ where $q_c$ is a characteristic length. A rough value for the characteristic length follows from $q_c\, p\sim 1$ (essentially Heisenberg’s uncertainty relation in $\hbar=1$ units). We are using units in which $m=1$, so $p=\delta/\epsilon\sim1/\sqrt{\epsilon}$. The last step follows from the basic random walk relation $\delta^2/\epsilon\sim 1$. Finally we find that everything should be expressed in terms of the ratio $\epsilon V(\sqrt{\epsilon})$. For example, for the anharmonic oscillator with quartic coupling this ratio is $\epsilon^3g$. We have shown that $\langle J\rangle$ differs from $\frac{1}{2}$ by a term proportional to $1/N$. From this it follows that for the oscillator with quartic anharmonicity $|\langle J\rangle -\frac{1}{2}|$ should be proportional to $g^{1/3}/N$. A similar back of the envelope calculation for a particle moving in a modified Pöschl-Teller potential gives that $|\langle J\rangle -\frac{1}{2}|$ should be proportional to $\frac{\alpha^2\beta(\beta-1)}{N}$. Fig. 7 illustrates that these simple calculations do in fact hold.
{height="4.8cm"} {height="4.8cm"}
To conclude, we have identified and investigated the properties of a quantity that we call the jaggedness and that is useful for obtaining a more detailed classification of relevant path integral trajectories. For discrete calculation, i.e. numerical simulations, the properties of the jaggedness are useful for obtaining more efficient algorithms for generating representative paths. Furthermore, we have shown that jaggedness can be used as an important practical criterion of the quality of trajectory generating algorithms. In the continuum limit (analytical calculations) we found that only trajectories with jaggedness equal to $\frac{1}{2}$ contribute to the path integral. In this way we greatly narrow the set of trajectories that are not of measure zero to those belonging to the equivalence class of the random walk. Classification of paths with respect to jaggedness is thus a classification of relevant non-differentiable paths. In the continuum limit, jaggedness is shown to be scale invariant as well as self-averaging.
[00]{}
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R. P. Feynman, *Statistical Mechanics* (W. A. Benjamin, New York, 1972).
H. Kleinert, *Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets* (World Scientific, 2004).
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|
---
abstract: |
We report our spectroscopic monitoring of the detached, grazing, and slightly eccentric 12-day double-lined eclipsing binary [EPIC 219568666]{} in the old nearby open cluster Ruprecht147. This is the second eclipsing system to be analyzed in this cluster, following our earlier study of [EPIC 219394517]{}. Our analysis of the radial velocities combined with the light curve from the [[*K2*]{}]{} mission yield absolute masses and radii for [EPIC 219568666]{} of $M_1 = 1.121 \pm 0.013~M_{\sun}$ and $R_1 = 1.1779
\pm 0.0070~R_{\sun}$ for the F8 primary, and $M_2 = 0.7334 \pm
0.0050~M_{\sun}$ and $R_2 = 0.640 \pm 0.017~R_{\sun}$ for the faint secondary. Comparison with current stellar evolution models calculated for the known metallicity of the cluster points to a primary star that is oversized, as is often seen in active M dwarfs, but this seems rather unlikely for a star of its mass and with a low level of activity. Instead, we suspect a subtle bias in the radius ratio inferred from the photometry, despite our best efforts to avoid it, which may be related to the presence of spots on one or both stars. The radius sum for the binary, which bypasses this possible problem, indicates an age of $2.76 \pm 0.61$ Gyr that is in good agreement with a similar estimate from the binary in our earlier study.
author:
- 'Guillermo Torres, Andrew Vanderburg, Jason L. Curtis, David Ciardi, Adam L. Kraus, Aaron C. Rizzuto, Michael J. Ireland, Michael B. Lund, Jessie L. Christiansen, and Charles A. Beichman'
title: 'Eclipsing binaries in the open cluster Ruprecht147. II: [EPIC 219568666]{}'
---
Introduction {#sec:introduction}
============
Detached, double-lined eclipsing binaries are the primary source of precise empirical masses and radii for normal stars [see, e.g., @Andersen:1991; @Torresetal:2010]. When coupled with stellar evolution models they can also provide accurate ages. If the binary happens to be located in a cluster, the age determination serves as an important check against more traditional estimates obtained by isochrone fitting or gyrochronology, or with other techniques of dating single member stars such as asteroseismology.
Ruprecht147 (NGC 6774) is a nearby, middle-aged open cluster with a slightly supersolar chemical composition [distance $\sim$300 pc, age $\sim$3 Gyr, ${\rm [Fe/H]} =
+0.10$; @Curtis:2013]. It is endowed with no fewer than 5 detached eclipsing binaries identified by [@Curtis:2016] that are relatively bright and amenable to detailed investigation. In an earlier study we reported results for the first of these, [EPIC 219394517]{} [@Torres:2018 Paper I], a pair of very similar early G-type stars in a 6.5-day orbit that yielded highly accurate masses and radii good to better than 0.2% and 1%, respectively. These properties gave an excellent fit to current stellar evolution models, and an improved age for the cluster between about 2.5 and 2.6 Gyr, depending on the model, with a formal uncertainty of about 0.3 Gyr.
The present paper reports on our follow-up observations and analysis of a second eclipsing binary system in Ruprecht147, [EPIC 219568666]{} (TYC 6296-2012-1, 2MASS J19165992$-$1625176, Gaia DR2 4183920989590558720, $V = 11.86$), which has the potential to add significantly to the characterization of the cluster. By virtue of the location of Ruprecht147 on the ecliptic, [EPIC 219568666]{} was observed along with the other eclipsing systems by NASA’s [[*K2*]{}]{} mission in the final months of 2015 (Campaign 7). Initial follow-up by [@Curtis:2016] determined it to be a 12-day period, slightly eccentric single-lined spectroscopic binary, which we find here to be double-lined. However, unlike the previous binary we reported on, the mass ratio in this case is very different from unity, which makes the secondary very faint. Membership in the cluster was established by [@Curtis:2016], and confirmed more recently based on parallax and proper motion information from the [*Gaia*]{} mission by [@Cantat-Gaudin:2018].
As we describe below, the seemingly straightforward analysis of the high-precision [[*K2*]{}]{} light curve of [EPIC 219568666]{} turns out to present difficulties that compromise the accuracy of the results. Although this does not allow us to take full advantage of the precise individual mass and radius measurements to test models of stellar evolution, the system still yields a useful estimate of the age of the cluster. This case serves as an interesting illustration of the pernicious effects of systematic errors that can easily go unnoticed, particularly those in the photometry that can bias the determination of the stellar radii in a significant way.
The layout of the paper is as follows. Section \[sec:observations\] describes our reduction and treatment of the raw [[*K2*]{}]{} photometry of [EPIC 219568666]{}, our new high-resolution spectroscopic observations, and archival and new imaging observations to explore the field surrounding the binary. In Section \[sec:analysis\] we present the joint spectroscopic and light curve analysis, leading to the physical properties of the system discussed in Section \[sec:dimensions\]. We compare these properties with current stellar evolution models as well as with the properties of the binary in our earlier study in Section \[sec:models\]. While this reveals discrepancies that could be due to a bias in the radius ratio preventing us from relying on the individual radii to infer an age, we are still able to make use of the radius sum, which appears to be well determined. Section \[sec:discussion\] then discusses multiple tests to investigate the source of the discrepancies. Closing remarks are given in Section \[sec:remarks\].
Observations {#sec:observations}
============
Photometry {#sec:photometry}
----------
[EPIC 219568666]{} was observed by [[*K2*]{}]{} during its 7th observing campaign for 81 days between October and December of 2015, with a cadence of 29.4 minutes (3654 measurements). The target fell within a large super-aperture designed to observe many members of the Ruprecht147 cluster together. Following downlink from the spacecraft and calibration by the NASA Ames [[*K2*]{}]{} pipeline, we proceeded to download the pixel time series from the Ruprecht147 super-aperture from the Mikulski Archive for Space Telescopes (MAST).[^1] Because [EPIC 219568666]{} is located in a dense region of stars, we extracted the light curve by calculating the flux within a concentric set of 10 circular moving apertures, to ensure that third-light contamination from nearby stars was kept constant as the telescope’s pointing drifted. We performed a first-pass correction for [[*K2*]{}]{}’s spacecraft systematics using the methods described by [@Vanderburg:2014] and [@Vanderburg:2016], and selected the circular moving aperture that yielded the highest photometric precision after the systematics correction. The aperture we selected had a radius of 3.97 pixels, or 1580. We then refined the systematics correction by simultaneously fitting for the eclipse shapes using [@Mandel:2002] transit models, [[*K2*]{}]{} roll systematics using splines as a function of [[*Kepler*]{}]{}’s pointing position, and long-term variability using splines in time, as described by [@Vanderburg:2016]. The resulting light curve has a scatter of about 100 ppm per 30 minute exposure, and contains 7 primary eclipses and 6 secondary eclipses. We removed low-frequency variability by dividing away the best-fit spline from our simultaneous light curve solution. The data processed in this way (3654 measurements) are given in Table \[tab:photometry\].
[cc]{} 57301.4866 & 0.99987691\
57301.5070 & 0.99997035\
57301.5275 & 1.00011420\
57301.5479 & 1.00008800\
57301.5683 & 1.00009520
Imaging {#sec:imaging}
-------
Several fainter stars near [EPIC 219568666]{} fall within the 1580 aperture we used to extract the photometry, and can potentially affect the parameters derived from the light curve. In Figure \[fig:CFHT\] we show an image of the target in a passband similar to Sloan $r$ taken in 2008 by [@Curtis:2013] with the MegaCam instrument [@Hora:1994] on the Canada-France-Hawaii Telescope (CFHT). We measured the position relative to the target and the brightness in the CFHT $gri$ filters[^2] of all numbered companions within the aperture, and list them in Table \[tab:CFHT\]. This information is used later in Section \[sec:analysis\] to make a quantitative estimate of the flux contamination. The table includes $J$- and $K$-band brightness measurements based on UKIRT/WFCAM imaging [@Curtis:2016]. We also report the properties of several of these companions that are listed in the [*Gaia*]{}/DR2 catalog [@Gaia:2018], though none appear to be members of the cluster, based on their parallax.
[rccccccccccccc]{} 1 & 19:17:00.172 & $-$16:25:20.98 & 131.8 & 4.9 & & & & & 18.38 & 17.58 & 0.05 & &\
2 & 19:16:59.577 & $-$16:25:13.18 & 312.7 & 7.0 & & & & & 17.03 & 16.62 & 0.02 & 18.59 & $0.398 \pm 0.344$\
3 & 19:16:59.832 & $-$16:25:25.20 & 190.4 & 7.5 & & & & & 17.79 & 16.95 & 0.03 & 20.62 &\
4 & 19:17:00.023 & $-$16:25:25.56 & 169.8 & 7.9 & & & & & 18.78 & 17.71 & 0.05 & &\
5 & 19:16:59.731 & $-$16:25:09.50 & 341.3 & 8.8 & 16.65 & 16.12 & 15.93 & 0.02 & 14.99 & 14.55 & 0.02 & 16.36 & $0.348 \pm 0.077$\
6 & 19:16:59.323 & $-$16:25:18.97 & 262.4 & 9.1 & 17.67 & 17.26 & 17.07 & 0.02 & 16.68 & 16.25 & 0.02 & 17.98 & $0.071 \pm 0.223$\
7 & 19:16:59.686 & $-$16:25:26.91 & 200.9 & 9.8 & 21.09 & 20.64 & 20.20 & 0.11 & 18.45 & 17.84 & 0.06 & 20.27 & $0.850 \pm 1.229$\
8 & 19:16:59.313 & $-$16:25:22.11 & 244.1 & 10.2 & 18.43 & 17.82 & 17.54 & 0.02 & 16.23 & 15.60 & 0.02 & 17.97 & $0.401 \pm 0.184$\
9 & 19:17:00.361 & $-$16:25:08.31 & 33.4 & 11.5 & 18.72 & 18.38 & 18.29 & 0.02 & 17.54 & 17.07 & 0.03 & 18.89 & $0.021 \pm 0.311$\
10 & 19:17:00.724 & $-$16:25:22.56 & 112.4 & 12.9 & 20.06 & 19.43 & 19.17 & 0.05 & 17.77 & 17.31 & 0.04 & 19.31 & $1.280 \pm 0.455$\
11 & 19:16:58.939 & $-$16:25:16.02 & 277.2 & 14.9 & 19.60 & 18.88 & 18.63 & 0.04 & 17.25 & 16.70 & 0.02 & 18.82 & $0.293 \pm 0.317$\
12 & 19:17:00.952 & $-$16:25:15.29 & 80.3 & 15.6 & 25.82 & 24.04 & 22.62 & 0.81 & 18.61 & 17.65 & 0.05 & &
In order to search for blended stellar companions to [EPIC 219568666]{} inside the inner working angle of the CFHT imaging, we used the 10m KeckII telescope with the NIRC2 facility adaptive optics (AO) imager to obtain natural guide star adaptive optics imaging and non-redundant aperture mask interferometry (NRM) in the $K^\prime$ filter ($\lambda = 2.124\thinspace \mu$m) on 2016 June 16 UT. These observations followed the standard observing strategy described by [@Kraus:2016] and previously reported for Ruprecht147 targets by [@Torres:2018]. For [EPIC 219568666]{}, we obtained a short sequence of 6 images and 12 interferograms in vertical angle mode. In both cases, calibrators were drawn from the other Ruprecht147 members observed on the same night.
The images were analyzed following the methods described by [@Kraus:2016] To briefly recap, the primary star point spread function (PSF) was subtracted using both an azimuthal median profile and an independent calibrator that most closely matches the speckle pattern. Within each image, the residual fluxes as a function of position were measured in apertures of radius 40mas, centered on each pixel, and the noise was estimated from the RMS of fluxes within concentric rings around the primary star. Finally, the detections and detection limits were estimated from the flux-weighted sum of the detection significances in the stack of all images, and any location with a total significance greater than 6$\sigma$ was visually inspected to determine if it was a residual speckle or cosmic ray. No candidates remained after this visual inspection.
The interferograms were analyzed following the methods described by [@Kraus:2008] and [@Ireland:2013]. We Fourier-transformed the interferograms to extract the complex visibilities, and from those we computed the corresponding closure phases for each triplet of baselines. We calibrated the closure phases with other observations of targets nearby on the sky and in time, and then fit the calibrated closure phases with binary source models to search for significant evidence of a companion, but did not find any. We determined the detection limits using a Monte Carlo process that randomizes the phase errors and determines the distribution of possible binary fits, indicating the 99.9% upper limit on companions in bins of projected separation.
Within the observations taken with KeckII/NIRC2 on 2016 June 16, some targets showed evidence of variable AO correction, possibly tied to variable seeing over the course of the night. In particular, some observations of [EPIC 219568666]{} displayed elongation of the PSF core that results when the gain in the AO system does not adapt quickly enough to changes in seeing. This resulted in little impact on the imaging limits on wide separations, but did appear to impact the sensitivity of the NRM observations.
To verify that there were no companions, we therefore made additional near-infrared high-resolution observations on 2019 June 14 UT with the PHARO instrument on the Palomar Observatory 5m telescope, behind the natural guide star AO system. We used a 5-point quincunx dither pattern that is standard in this type of observation. The dither pattern step size was 5 and was repeated three times, with each dither offset from the previous dither by 05.
The images were made in the narrow-band Br-$\gamma$ filter ($\lambda =
2.1686\thinspace \mu$m; $\Delta\lambda = 0.0326\thinspace \mu$m) with an integration time of 20 seconds and one coadd per frame for a total of 300 seconds on target. The camera was in the narrow-angle mode with a full field of view of $\sim$25 and a pixel scale of approximately $0\farcs025\thinspace {\rm pixel}^{-1}$. We detected no additional stellar companions to within a resolution of $\sim$01 FWHM (see Figure \[fig:ao\]).
The sensitivity of the final combined Palomar AO image was determined by injecting simulated sources azimuthally around the target every 45 at separations of integer multiples of the central source’s FWHM [@Furlan:2017]. The brightness of each injected source was scaled until standard aperture photometry detected it with 5$\sigma$ significance. The resulting brightness of the injected sources relative to the target set the contrast limits at that injection location. The final 5$\sigma$ limit at each separation was determined from the average of all of the determined limits at that separation and the uncertainty on the limit was set by the RMS dispersion of the azimuthal slices at a given radial distance.
The sensitivity curves from the KeckII/NIRC2 and Palomar/PHARO observations are displayed in Figure \[fig:ao\] along with an inset image zoomed in on the target, showing no other companion stars.
Spectroscopy {#sec:spectroscopyy}
------------
Spectroscopic monitoring of [EPIC 219568666]{} was carried out between 2016 May and 2018 June with the Tillinghast Reflector Echelle Spectrograph [TRES; @Szentgyorgyi:2007; @Furesz:2008] on the 1.5m Tillinghast reflector at the Fred L. Whipple Observatory on Mount Hopkins (Arizona, USA). We gathered a total of 25 spectra at a resolving power of $R \approx 44,000$ covering the wavelength region 3800–9100 Å in 51 orders. For the order centered at $\sim$5187 Å containing the b triplet the signal-to-noise ratios range from 17 to 42 per resolution element of 6.8 [kms$^{-1}$]{}.
Radial velocities were measured with TODCOR, a two-dimensional cross-correlation technique introduced by [@Zucker:1994]. A template matching each component was selected from a pre-computed library of synthetic spectra based on model atmospheres by R. L. Kurucz, and a line list tuned to better match the spectra of real stars [see @Nordstrom:1994; @Latham:2002]. These templates cover a limited wavelength region of $\sim$300 Å centered at 5187 Å.
For the primary star we estimated the effective temperature ($T_{\rm
eff}$) and projected rotational velocity ($v \sin i$) by running grids of one-dimensional cross-correlations of the observed spectra against synthetic spectra, following [@Torres:2002]. The grids were run over a broad range in those parameters (4500–7500 K, 0–30 [kms$^{-1}$]{}) at fixed values of the surface gravity ($\log g$) and metallicity (\[Fe/H\]), ignoring the presence of the secondary star because it is very faint (only about 4% of the brightness of the primary; see below). We then chose the combination giving the highest value of the cross-correlation coefficient averaged over all 25 spectra, weighted by the strength of each exposure. We repeated this for $\log g$ values of 4.0 and 4.5, bracketing our best estimate reported later, and \[Fe/H\] values of 0.0 and +0.5 on either side of the known cluster abundance. By interpolation we obtained estimates of $T_{\rm eff} = 6140$ K and $v \sin i = 6~{\ifmmode{\rm km\thinspace s^{-1}}\else km\thinspace s$^{-1}$\fi}$, with estimated uncertainties of 100 K and 2 [kms$^{-1}$]{}, respectively. These uncertainties are based on the scatter from the individual spectra, conservatively increased to account for possible systematic errors. The temperature corresponds approximately to a spectral type of F8. For the velocity determinations we adopted primary template parameters of 6250 K and 6 [kms$^{-1}$]{}, the nearest in our grid, along with $\log g = 4.5$ and ${\rm
[Fe/H]} = 0.0$.
While the presence of the secondary does not affect the above determinations, its faintness does prevent us from obtaining direct estimates of its temperature and $v \sin i$ from our spectra. We therefore adopted a value of $T_{\rm eff} = 4500$ K appropriate for its mass (typical of a mid-K star), and $v \sin i = 4~{\ifmmode{\rm km\thinspace s^{-1}}\else km\thinspace s$^{-1}$\fi}$. The surface gravity and metallicity for the secondary template were taken to be the same as for the primary.
The heliocentric radial velocities measured with TODCOR are presented in Table \[tab:rvs\] along with their uncertainties. We also determined the average flux ratio at the mean wavelength of our observations (5187 Å), which is $\ell_2/\ell_1 = 0.044 \pm 0.003$. A graphical representation of the velocities is shown in Figure \[fig:rvs\], along with our final model described below. The orbit is clearly eccentric. We note that two observations near phase 0.46 (the first and third in Table \[tab:rvs\]) are safely outside of eclipse, and are thus not affected by the Rossiter-McLaughlin effect.
[lccccc]{} 57528.9254 & 42.17 & 0.15 & 39.43 & 1.39 & 0.4623\
57550.9028 & $-$4.96 & 0.17 & 108.99 & 1.48 & 0.2951\
57552.8726 & 41.13 & 0.14 & 39.26 & 1.31 & 0.4594\
57553.8662 & 64.90 & 0.14 & 5.10 & 1.29 & 0.5422\
57554.9623 & 79.50 & 0.14 & $-$19.80 & 1.28 & 0.6336\
57555.9456 & 81.95 & 0.15 & $-$22.28 & 1.39 & 0.7157\
57556.9458 & 75.93 & 0.14 & $-$15.55 & 1.36 & 0.7991\
57557.9371 & 63.21 & 0.17 & 8.08 & 1.53 & 0.8817\
57558.9318 & 45.53 & 0.17 & 33.42 & 1.52 & 0.9647\
57566.8695 & 79.32 & 0.18 & $-$17.51 & 1.58 & 0.6266\
57583.8587 & 26.07 & 0.23 & 64.89 & 2.08 & 0.0434\
57584.7819 & 7.08 & 0.15 & 92.09 & 1.38 & 0.1204\
57700.5724 & 78.30 & 0.15 & $-$18.06 & 1.39 & 0.7766\
57705.5734 & $-$5.50 & 0.17 & 112.75 & 1.51 & 0.1937\
57706.5771 & $-$7.17 & 0.18 & 117.29 & 1.57 & 0.2774\
57710.5737 & 77.23 & 0.17 & $-$12.98 & 1.50 & 0.6107\
57854.9921 & 81.17 & 0.10 & $-$21.17 & 0.93 & 0.6542\
57878.9487 & 81.09 & 0.17 & $-$18.60 & 1.51 & 0.6521\
57908.9128 & 1.21 & 0.18 & 99.40 & 1.58 & 0.1509\
57910.8838 & $-$1.82 & 0.25 & 106.00 & 2.35 & 0.3153\
57939.9190 & 81.46 & 0.14 & $-$20.99 & 1.32 & 0.7366\
58034.6680 & 80.25 & 0.18 & $-$19.28 & 1.62 & 0.6381\
58274.8628 & 81.86 & 0.19 & $-$20.29 & 1.74 & 0.6688\
58277.9134 & 55.22 & 0.17 & 19.99 & 1.54 & 0.9232\
58294.8240 & 2.07 & 0.14 & 101.25 & 1.35 & 0.3335
Analysis {#sec:analysis}
========
The light curve analysis of [EPIC 219568666]{} was carried out using the [eb]{} code of [@Irwin:2011], which is based on the Nelson-Davis-Etzel binary model [@Etzel:1981; @Popper:1981] implemented in the popular EBOP program and its descendants. Further details may be found in our earlier work [@Torres:2018]. The main adjustable parameters are the orbital period ($P$) and reference epoch of primary eclipse ($T_0$, which is strictly the time of inferior conjunction in this code), the central surface brightness ratio in the [[*Kepler*]{}]{} bandpass ($J \equiv J_2/J_1$), the sum of the relative radii normalized by the semimajor axis ($r_1+r_2$) and their ratio ($k
\equiv r_2/r_1$), the cosine of the inclination angle ($\cos i$), and the eccentricity parameters $e \cos\omega$ and $e \sin\omega$, with $e$ being the eccentricity and $\omega$ the longitude of periastron. We adopted a quadratic limb-darkening law for this work, with coefficients $u_1$ and $u_2$ for each star.
Our detrending procedure is designed to eliminate the obvious modulation in the light curve due to spots, as well as any long-term trends (deemed to be of instrumental origin), and at the same time it also removes other out-of-eclipse variability including tidal distortions (ellipsoidal variability) and reflection, which for this long-period and well-detached system are very small in any case. Therefore, our modeling was run with the calculation of ellipsoidal variability and reflection disabled, and we restricted the analysis to data within 0.02 in phase from the center of each eclipse (about 2.5 times the total duration of the eclipses). We accounted for the finite time of integration of the [[*K2*]{}]{} long-cadence observations by oversampling the model light curve and then integrating over the 29.4-minute duration of each cadence prior to the comparison with the observations [see @Gilliland:2010; @Kipping:2010].
Our method of solution used the [emcee]{}[^3] code of [@Foreman-Mackey:2013], which is a Python implementation of the affine-invariant Markov chain Monte Carlo (MCMC) ensemble sampler proposed by [@Goodman:2010]. We used 100 walkers with chain lengths of 10,000 each, after discarding the burn-in. Uniform (non-informative) or log-uniform priors over suitable ranges were adopted for all parameters (see below), and convergence of the chains was checked visually, requiring also a Gelman-Rubin statistic of 1.05 or smaller for each parameter [@Gelman:1992].
Flux contamination from neighboring stars was accounted for by including a third light parameter in our model ($\ell_3$), defined such that $\ell_1 + \ell_2 + \ell_3 = 1$, in which the values of $\ell_1$ and $\ell_2$ for this normalization correspond to the light at first quadrature (phase 0.2235 in this system). An estimate of $\ell_3$ was obtained from the brightness measurements in Table \[tab:CFHT\] of all companions within the photometric aperture, using the magnitudes in the CFHT $r$ band, which is close to the [[*Kepler*]{}]{} bandpass. To guard against possible errors arising from the slight bandpass difference, we conservatively inflated the companion magnitude uncertainties to be no less than 0.2 mag. In this way we obtained $\ell_3 = 0.026 \pm 0.006$, which we used as a Gaussian prior in our MCMC analysis. All companions near the edge of the circular aperture are very faint, so the result is insensitive to the treatment of partial pixels for those stars.
In eccentric orbits such as that of [EPIC 219568666]{} it is usually the case that $e \cos\omega$ is tightly constrained by the light curve (from the location of the secondary eclipse), but $e \sin\omega$ is not. The reverse is true of the radial-velocity curves, making it beneficial to combine the two types of observations into a joint solution. We took this approach, solving for three additional adjustable parameters: the center-of-mass velocity ($\gamma$), and the velocity semiamplitudes of the primary and secondary ($K_1$ and $K_2$). We handled the relative weighting between the photometry and the primary and secondary velocities by including additional adjustable multiplicative parameters ($f_{{{\it K2\/}}}$, $f_1$, and $f_2$, respectively) to rescale the observational errors. These scale factors were solved for self-consistently and simultaneously with the other orbital quantities [see @Gregory:2005]. The initial error assumed for the photometric measurements is 200 parts per million (ppm), and the initial errors for the velocities are those listed in Table \[tab:rvs\].
For completeness we chose to account for light travel time across the binary, which can contribute to the displacement of the secondary eclipse from phase 0.5, although the effect is negligible in this case: a delay of 26 s, more than three orders of magnitude smaller than the measured displacement corresponding to about 54,800 s. Strictly speaking, then, our $T_0$ is referred to the barycenter of the binary.
As shown below the eclipses of [EPIC 219568666]{} are partial and quite shallow ($\sim$11% and $\sim$4%). With the expectation that the radius ratio $k$ would be poorly constrained, as it often is in such cases, we chose to take advantage of our spectroscopic measurement of the average light ratio to help constrain $k$ indirectly, given that the two quantities are strongly correlated ($\ell_2/\ell_1 \propto
k^2$). Our measured value of $\ell_2/\ell_1 = 0.044 \pm 0.003$ at a mean wavelength of 5187 Å was transformed to the [[*Kepler*]{}]{} band by using synthetic spectra based on PHOENIX models taken from the grid of [@Husser:2013]. For this we adopted temperatures of 6100 K and 4500 K, close to those of the components, and a preliminary estimate of the radius ratio ($k = 0.60$) with which we are able to reproduce the measured light ratio at 5187 Å. The result, $\ell_2/\ell_1 =
0.084 \pm 0.005$, was then applied as a prior in our analysis.
Initial solutions revealed that the second-order limb-darkening coefficients $u_2$ were poorly constrained for both stars. This is to be expected given the grazing orientation of the system (see below). Therefore, for the remainder of this work they were held fixed at the theoretical values tabulated by [@Claret:2011], interpolated according to the adopted temperatures, the metallicity, and our final surface gravities reported below. The values of $u_2$ adopted for the [[*Kepler*]{}]{} band are 0.295 and 0.075 for the primary and secondary, respectively, and are based on ATLAS model atmospheres and the least-squares procedure favored by those authors for calculating the coefficients.
We report the results of our analysis in Table \[tab:LCfit\], where the values given correspond to the mode of the posterior distributions. Posterior distributions of the derived quantities listed in the bottom section of the table were constructed directly from the MCMC chains of the adjustable parameters involved. Our adopted model along with the [[*K2*]{}]{} observations can be seen in Figure \[fig:LCfit\], together with an illustration of the fairly grazing configuration of the system. The residuals, with an overall scatter of about 120 ppm, appear slightly larger in the secondary eclipse. This may be due to spottedness, which would not be unexpected for a mid-K star such as the secondary.
[lcc]{} $P$ (days)& $11.991313^{+0.000013}_{-0.000013}$ & \[11, 13\]\
\[1ex\] $T_0$ (HJD$-$2,400,000)& $57727.23362^{+0.00042}_{-0.00042}$ & \[57725, 57729\]\
\[1ex\] $J$& $0.2707^{+0.0051}_{-0.0051}$ & \[0.02, 1.0\]\
\[1ex\] $r_1+r_2$& $0.06715^{+0.00045}_{-0.00045}$ & \[0.01, 0.20\]\
\[1ex\] $k$& $0.544^{+0.016}_{-0.016}$ & \[0.1, 1.0\]\
\[1ex\] $\cos i$& $0.04128^{+0.00064}_{-0.00064}$ & \[0, 1\]\
\[1ex\] $e \cos\omega$& $-0.083091^{+0.000013}_{-0.000013}$ & \[$-$1, 1\]\
\[1ex\] $e \sin\omega$& $-0.0747^{+0.0010}_{-0.0010}$ & \[$-$1, 1\]\
\[1ex\] $\ell_3$& $0.0261^{+0.0060}_{-0.0061}$ & \[0.0, 0.3\]\
\[1ex\] $\gamma$ ([kms$^{-1}$]{})& $+40.732^{+0.033}_{-0.033}$ & \[30, 50\]\
\[1ex\] $K_1$ ([kms$^{-1}$]{})& $45.440^{+0.044}_{-0.044}$ & \[20, 80\]\
\[1ex\] $K_2$ ([kms$^{-1}$]{})& $69.46^{+0.37}_{-0.37}$ & \[20, 80\]\
\[1ex\] $\ln f_{{{\it K2\/}}}$& $-0.479^{+0.047}_{-0.044}$ & \[$-5$, 0.9\]\
\[1ex\] $\ln f_1$& $+0.09^{+0.17}_{-0.14}$ & \[$-5$, 5\]\
\[1ex\] $\ln f_2$& $-0.04^{+0.15}_{-0.14}$ & \[$-5$, 5\]\
\[1ex\] Primary $u_1$& $0.291^{+0.064}_{-0.064}$ & \[0, 1\]\
\[1ex\] Secondary $u_1$& $0.455^{+0.063}_{-0.064}$ & \[0, 1\]\
\[1ex\]\
\[-1.5ex\]\
\[1ex\]\
\[-1.5ex\] $r_1$& $0.04349^{+0.00021}_{-0.00022}$ &\
\[1ex\] $r_2$& $0.02364^{+0.00061}_{-0.00059}$ &\
\[1ex\] $i$ (degree)& $87.634^{+0.037}_{-0.036}$ &\
\[1ex\] $\ell_2/\ell_1$& $0.0781^{+0.0039}_{-0.0037}$ &\
\[1ex\] $e$& $0.11176^{+0.00069}_{-0.00069}$ &\
\[1ex\] $\omega$ (degree)& $221.98^{+0.38}_{-0.41}$ &\
\[1ex\] $f_{{{\it K2\/}}}$& $0.620^{+0.029}_{-0.027}$ &\
\[1ex\] $f_1$& $1.09^{+0.20}_{-0.14}$ &\
\[1ex\] $f_2$& $0.95^{+0.16}_{-0.12}$ &\
\[1ex\] Prim. duration (days)& $0.20800^{+0.00047}_{-0.00047}$ &\
\[1ex\] Sec. duration (days)& $0.19582^{+0.00062}_{-0.00062}$ &\
\[1ex\] Prim. impact param.& $1.012^{+0.020}_{-0.019}$ &\
\[1ex\] Sec. impact param.& $0.872^{+0.017}_{-0.017}$ &
-------------------------------------------------
{width="8.0cm"}
\[0.5ex\] {width="8.0cm"}
\[1ex\] {width="8.0cm"}
-------------------------------------------------
To account for the possibility that the increased secondary residuals (or the fact that we held the second-order limb-darkening coefficients fixed) may be causing our parameter uncertainties to be underestimated, we performed a residual permutation exercise as follows. We shifted the residuals from our adopted model by an arbitrary number of time indices, we then added them back into the model curve at each time of observation (with wrap-around), and we carried out a new MCMC analysis on this synthetic data set. We repeated this 50 times. The residual permutation was done separately for the in-eclipse and out-of-eclipse regions [see @Hartman:2018]. For each new MCMC analysis we simultaneously perturbed the theoretical quadratic limb-darkening coefficients by adding Gaussian noise with a standard deviation of 0.10. The scatter (standard deviation) of the resulting distributions for each parameter was added in quadrature to the internal errors from our original MCMC analysis to arrive at the final uncertainties reported in Table \[tab:LCfit\]. The simulation errors are larger than the internal errors for $P$, $T_0$, and the linear limb-darkening coefficients.
We note, finally, that our analysis reveals strong correlations among several of the fitted elements, which is not unexpected for a system of this configuration. The strongest correlations are between $k$ and $r_1+r_2$ (correlation coefficient +0.921), between $k$ and $\cos i$ (+0.938), between $r_1+r_2$ and $\cos i$ (+0.992), and between $P$ and $T_0$ (+0.997).
Absolute dimensions {#sec:dimensions}
===================
The physical properties of [EPIC 219568666]{} derived from our MCMC analysis are collected in Table \[tab:dimensions\]. The relative uncertainties for the masses are $\sim$1% or smaller. The radii are formally good to 0.6% and 2.5% for the primary and secondary, although we argue below that systematic errors may be larger.
As described earlier, the primary effective temperature was determined directly from our spectra, while the secondary value is adopted. These temperatures along with broadband photometry from the literature allow us to obtain an estimate of the interstellar reddening along the line of sight. We proceeded as follows. With the values in Table \[tab:dimensions\] the luminosity-weighted spectroscopic mean temperature of the system is $\langle T_{\rm eff} \rangle_{\rm sp} =
5877 \pm 84$ K. Color indices from published photometry along with color/temperature calibrations allow a determination of a mean photometric temperature, $\langle T_{\rm eff} \rangle_{\rm ph}$, which depends on the overall reddening, $E(B-V)$. We used photometry in the Johnson, Tycho-2, 2MASS, and Sloan systems [@Zacharias:2013; @Hog:2000; @Skrutskie:2006; @Henden:2015] to construct 13 non-independent color indices, and derived a temperature for each one using the calibrations by [@Casagrande:2010] and [@Huang:2015]. Following [@Torres:2018], we adjusted the temperatures from the [@Casagrande:2010] calibrations by $-130$ K to remove a systematic difference compared to those of [@Huang:2015]. The results were then averaged. The metallicity terms in these calibrations were calculated using ${\rm [Fe/H]} =
+0.10 \pm 0.04$ [@Curtis:2013], and the reddening corrections appropriate for each color index were made as prescribed by [@Cardelli:1989]. We determined the optimal reddening $E(B-V)$ by varying it until $\langle T_{\rm eff} \rangle_{\rm ph} = \langle
T_{\rm eff} \rangle_{\rm sp}$. In this way we obtained $E(B-V) = 0.126
\pm 0.023$ mag, corresponding to $A_V = 0.391 \pm 0.071$ mag for a ratio of total to selective extinction of $R_V = 3.1$. This is in good agreement with the value of $E(B-V) = 0.112 \pm 0.029$ mag derived in our earlier study of the Ruprecht147 eclipsing system [EPIC 219394517]{}, and also with independent estimates for the cluster by [@Bragaglia:2018] and [@Olivares:2019].
The distance to [EPIC 219568666]{} was estimated from the luminosities, bolometric corrections from [@Flower:1996] [see also @Torres:2010], our extinction estimate, and an adopted visual magnitude out of eclipse of $V = 11.859 \pm 0.050$ [@Zacharias:2013]. The result, $d = 300^{+21}_{-20}$ pc, corresponds to a parallax in good accord with the entry in the [*Gaia*]{}/DR2 catalog [@Gaia:2018 see Table \[tab:dimensions\]]. Finally, given the $\sim$3 Gyr age of the cluster, the non-zero eccentricity of the binary ($e \approx 0.112$) is consistent with the expectation from tidal theory, which predicts an orbital circularization timescale [formally 630 Gyr; e.g., @Hilditch:2001] that is longer than the age of the Universe.
[lcc]{} $M$ ($\mathcal{M}_{\sun}^{\rm N}$)& $1.121^{+0.013}_{-0.013}$ & $0.7334^{+0.0050}_{-0.0049}$\
\[1ex\] $R$ ($\mathcal{R}_{\sun}^{\rm N}$)& $1.1779^{+0.0070}_{-0.0070}$ & $0.640^{+0.017}_{-0.016}$\
\[1ex\] $q \equiv M_2/M_1$&\
\[1ex\] $a$ ($\mathcal{R}_{\sun}^{\rm N}$)&\
\[1ex\] $\log g$ (dex)& $4.3457^{+0.0049}_{-0.0048}$ & $4.690^{+0.022}_{-0.022}$\
\[1ex\] $T_{\rm eff}$ (K)& 6140 $\pm$ 100 & 4500 $\pm$ 100\
\[1ex\] $L$ ($L_{\sun}$)& $1.77^{+0.12}_{-0.11}$ & $0.152^{+0.016}_{-0.015}$\
\[1ex\] $M_{\rm bol}$ (mag)& $4.109^{+0.072}_{-0.072}$ & $6.78^{+0.11}_{-0.11}$\
\[1ex\] $BC_V$ (mag)& $-0.027 \pm 0.100$ & $-0.624 \pm 0.100$\
\[1ex\] $M_V$ (mag)& $4.14^{+0.12}_{-0.12}$ & $7.40^{+0.15}_{-0.15}$\
\[1ex\] $v_{\rm sync} \sin i$ ([kms$^{-1}$]{})& $4.966^{+0.029}_{-0.029}$ & $2.699^{+0.070}_{-0.067}$\
\[1ex\] $v \sin i$ ([kms$^{-1}$]{})& $6 \pm 2$ & 4 (adopted)\
\[1ex\] $E(B-V)$ (mag)&\
\[1ex\] $A_V$ (mag)&\
\[1ex\] Dist. modulus (mag)&\
\[1ex\] Distance (pc)&\
\[1ex\] $\pi$ (mas)&\
\[1ex\] $\pi_{Gaia/{\rm DR2}}$ (mas)&
Rotation and activity {#sec:rotation}
---------------------
The light curve of [EPIC 219568666]{} presents obvious variations out of eclipse with a peak-to-peak amplitude of about 2 milli-magnitudes. We attribute this to rotational modulation by spots, although it is unclear which star is responsible. While it may well be the brighter primary, the secondary is likely to be spotted as well, as suggested by the somewhat larger photometric residuals during secondary eclipse. In that case, the dilution effect would imply an intrinsic amplitude from spots on the secondary of about 2.5% in the [[*Kepler*]{}]{} band.
The observations after removal of the eclipses and a long-term drift are presented in Figure \[fig:rotation\], along with the corresponding Lomb-Scargle periodogram. The peak location at $P_{\rm
rot} = 12.2^{+1.1}_{-0.9}$ days (uncertainties estimated from the half width at half maximum) is consistent with the orbital period, suggesting synchronous rotation. Our spectroscopically measured projected rotational velocity of the primary star ($v \sin i = 6 \pm
2~{\ifmmode{\rm km\thinspace s^{-1}}\else km\thinspace s$^{-1}$\fi}$) agrees with the predicted value of $v_{\rm sync} \sin i$ shown in Table \[tab:dimensions\]. Both of these indications are consistent with the expected synchronization timescale of $\sim$300 Myr [e.g., @Hilditch:2001], given the $\sim$3 Gyr age of the parent Ruprecht147 cluster.
Aside from the small photometric modulation described above, the activity level of the binary appears to be low, as we see no indicators of such activity in our spectra, nor does the object appear to have been detected in X rays.
Comparison with theory {#sec:models}
======================
Our precise determinations of the masses and radii of [EPIC 219568666]{}, and the increased leverage afforded by a mass ratio significantly different from unity, offer an opportunity for a valuable comparison with current stellar evolution models. They also permit an independent estimate of the age of the Ruprecht147 cluster to supplement the one from our previous study of [EPIC 219394517]{} [@Torres:2018]. Figure \[fig:MR\] displays the measured radius and temperature of the components as a function of their measured mass, together with model isochrones from the PARSEC series [@Chen:2014] computed for the cluster metallicity of ${\rm
[Fe/H]} = +0.10$. For reference we include the results for [EPIC 219394517]{} from Paper I. The model isochrones are the same as shown in Figure 7 of that work, and span ages between 2 and 3 Gyr. The heavy dashed line represents the 2.65 Gyr isochrone that provided the best fit to [EPIC 219394517]{} in our earlier study.
While the isochrones match the effective temperature of the primary of [EPIC 219568666]{} to within its uncertainty (the secondary $T_{\rm eff}$ is adopted from models), the radius of the primary appears $\sim$4% larger than expected for its mass (6.7$\sigma$), if we take the best-fitting model from [EPIC 219394517]{} as the reference. Furthermore, the radius of the secondary is about 6% smaller than the models predict (2.2$\sigma$), assuming the mass is accurate. These differences are somewhat surprising, particularly for the primary, given the high precision of the observations and the relatively straightforward analysis, and the expectation of fewer complications from the longer orbital period compared to our previous study of the more active 6.5-day system [EPIC 219394517]{}. Repeating the comparison with the MIST models [@Choi:2016] gives essentially the same result.
A casual look at Figure \[fig:MR\] may give the impression that relatively small shifts in the locations of the primary and secondary in just the right directions would bring satisfactory agreement with the reference model from [EPIC 219394517]{}. For instance, an increase in $M_1$ by about 2$\sigma$ together with an increase in $R_2$ also by about 2$\sigma$ would be sufficient to obtain a reasonably good fit. However, this ignores the strong correlations that exist among the inferred masses and radii, which restrict the shifts one may apply in each direction if they are to remain within the multi-dimensional confidence region mapped out by our MCMC analysis. As an example, $M_1$ and $M_2$ have a correlation coefficient of +0.981, indicating they cannot be varied independently.
A more quantitative assessment of how well the measurements agree with the reference isochrone that implicitly accounts for all correlations may be obtained using the chains from our [emcee]{} analysis, with a combined length of $10^6$ links. For each link we determined the normalized distance in four-dimensional parameter space between the values of \[$M_1$, $M_2$, $R_1$, $R_2$\] and the nearest pair of points on the isochrone representing the location of the primary and secondary. We normalized the separation along each axis by the standard deviation of the corresponding variable as determined from the chains. The masses and radii on the isochrone that are closest to the link giving the smallest distance are represented with square symbols in Figure \[fig:corner\]. This figure displays a “corner” plot showing the correlations among the four quantities. The contours correspond to the 1-$\sigma$, 2-$\sigma$, and 3-$\sigma$ confidence levels, and the panels for \[$M_1$, $R_1$\] and \[$M_2$, $R_2$\] also show the reference isochrone from Figure \[fig:MR\]. The square symbols in the latter two panels deviate from the values reported in Table \[tab:LCfit\] by more than 2$\sigma$ for the primary and more than 3$\sigma$ for the secondary. Similar offsets are seen in the other panels. The overall discrepancy with the model is at about the 3-$\sigma$ level, which we consider significant.
If we assume for a moment that the masses of both stars are unbiased, the positive deviation in radius for the primary star may be speculated to be due to stellar activity, which is believed to be the underlying cause of the phenomenon of “radius inflation” in convective stars [see, e.g., @Mullan:2001; @Chabrier:2007; @Torres:2013; @Feiden:2013; @Feiden:2014; @Somers:2015]. We note, though, that the activity level of the primary seems relatively low, based on the photometric amplitude of the rotational modulation (provided it comes from this star and not the secondary) and on the lack of spectroscopic activity indicators (H$\alpha$ or H and K emission) or X-ray emission. Furthermore, radius inflation is most often seen in K and M dwarfs, and much more rarely among higher-mass convective stars near a solar mass, though a few examples do exist [e.g., the secondary components of CVBoo, FLLyr, V1061Cyg, and V636Cen, with masses of 0.97, 0.96, 0.93 and 0.85 $M_{\sun}$, respectively; see @Torres:2008; @Helminiak:2019; @Torres:2006; @Clausen:2009]. None are as massive as the F8 primary of [EPIC 219568666]{}, however, and all rotate more rapidly.
On the other hand, we can think of no plausible physical explanation for the small radius of the secondary at its nominal mass, although the deviation is admittedly less significant than for the primary. Instead, we postulate that the discrepancies for both stars are more likely to be due to systematic errors stemming from our analysis and/or from the observations themselves. In the next section we discuss possible sources of these errors, and the tests we carried out to investigate them.
We note here that light curve solutions for systems with shallow eclipses such as [EPIC 219568666]{} can often lead to a biased measure of the radius ratio (especially, but not necessarily, when the components are similar, which is not the case here) because of degeneracies among several orbital elements [see, e.g., @Andersen:1991; @Torresetal:2010; @Kraus:2015]. Such degeneracies are present in our own analysis, as pointed out at the end of Section \[sec:analysis\]. On the other hand, the sum of the radii is typically more robust [e.g., @Andersen:1983]. Interestingly, we estimate that an increase from the value of $k \approx 0.54$ we determine to about 0.60 would yield very good agreement with the models shown in Figure \[fig:MR\]. This may be taken as circumstantial evidence of a possible bias in the radius ratio, although the situation is likely more complex given the correlations between $k$ and other elements.
If we set aside for now the individual radii and rely only on the radius sum ($R_1 + R_2 = 1.819 \pm 0.014~R_{\sun}$), which we expect to be more accurate, it is still possible to estimate an age for the system using the same PARSEC models as above. This is illustrated in Figure \[fig:age\], in which the solid curve gives the age predicted from theory (for the adopted cluster metallicity of ${\rm [Fe/H]} =
+0.10$) as a function of the radius sum computed at the measured masses for [EPIC 219568666]{}. The corresponding 1$\sigma$ error interval that comes from the uncertainty in the measured masses is shown with dashed lines. The intersection of this curve with the measured radius sum for [EPIC 219568666]{} (horizontal shaded area) leads to an age estimate of $2.76 \pm
0.61$ Gyr (dot and error bars at the bottom). Although the uncertainty is more than twice that of the estimate for [EPIC 219394517]{} from Paper I ($2.65 \pm 0.25$ Gyr at fixed metallicity), the determinations are consistent within the smallest of their errors. In both cases we used the same models. Together the two binaries therefore support an age for the cluster slightly under 3 Gyr.
Discussion {#sec:discussion}
==========
In this section we examine a number of possible explanations for the unexpected disagreement illustrated in Figure \[fig:MR\] between models and the mass and radius measurements for [EPIC 219568666]{}, which appears statistically significant, as discussed above. Barring an unusual instance of (activity-related?) radius inflation for a star as massive as the primary, or severely underestimated mass and/or radius uncertainties that we think is unlikely, we proceed on the assumption that there may be a subtle bias somewhere in our analysis that is causing the primary to appear to be too large and the secondary too small compared to predictions, or perhaps the primary mass to be too small. We address potential errors in the spectroscopy, errors in the priors for the light ratio and third light that we applied in our MCMC analysis, the possibility of biases in our detrending procedures for the [[*K2*]{}]{} photometry, and the impact of the cadence of the photometric observations.
Mass errors
-----------
Earlier we proposed that an underestimate of the radius ratio is one plausible explanation for the measured slope in the mass-radius diagram being too steep (Figure \[fig:MR\]). In principle a bias in the mass ratio may contribute as well. This could result from a mismatch between the templates used in our TODCOR analysis (involving mainly the $T_{\rm eff}$ parameter, which affects the velocities the most) and the spectra of the real components. While the primary temperature is well determined directly from our spectra, the secondary temperature (4500 K) was adopted from models because that star is too faint for an independent determination. We repeated the radial-velocity measurements for a range of temperatures for the secondary template between 4000 and 5000 K in steps of 250 K, but found that the residuals from a spectroscopic orbital solution were worse, the masses changed by less than 1% compared to the adopted values, and the mass ratio changed by even less. We conclude from this that the masses appear to be robust, and are therefore not likely to be the main source of the discrepancy, though they may still have some influence. This then shifts the focus to the radii, which depend more critically on the photometry.
Alternate solutions
-------------------
A separate MCMC analysis without the radial velocities gave essentially the same results for both the radius sum and radius ratio, again suggesting that the problem may lie in the photometry. Splitting the photometric data set in two produced similar parameters in each half, from which we conclude any bias is not time-dependent.
Priors
------
Our adopted solution in Table \[tab:LCfit\] used Gaussian priors for both the light ratio and third light, each based on empirical constraints. To investigate the effect of these priors, we repeated the analysis without them as well as using one prior but not the other. All three solutions resulted in values of $k$ and $r_1+r_2$ (and also of $\ell_2/\ell_1$ and $\ell_3$) that were very close to those obtained when applying both priors together, suggesting little or no tension between the photometry and the external constraints. Nevertheless, we examined each of the priors more closely.
### Spectroscopic light ratio
The prior on the light ratio can potentially have the strongest impact on the inferred value of the radius ratio because of the direct correlation between them ($\ell_2/\ell_1 \propto k^2$). For example, grids of MCMC solutions imposing Gaussian light ratio priors ranging from 0.030 to 0.150 (straddling our adopted value of $0.084 \pm
0.005$) and the same prior on $\ell_3$ as in our original analysis show indeed that the derived $k$ values change significantly from 0.50 to 0.73 (a 46% change), while the values of $r_1+r_2$ change in tandem but less, from about 0.066 to 0.071 (8%), supporting the notion that the radius sum is more robust. Removing the $\ell_3$ prior in this exercise causes the radius sum to change in the opposite direction, from 0.071 to 0.066. However, in both cases the quality of the solution as measured by the $f_{{{\it K2\/}}}$ scale factor for the photometric errors degrades considerably toward the upper end of the $\ell_2/\ell_1$ range we explored.
Our adopted light ratio prior in the [[*Kepler*]{}]{} band is extrapolated from our measured value at $\sim$5187 Å, as described in Section \[sec:analysis\], and as such it is subject to error. The extrapolation used synthetic spectra by [@Husser:2013] appropriate for each component in order to calculate the wavelength dependence of the flux ratio. The normalization was performed using $k = 0.60$, which is the value of the radius ratio we find is needed in order to reproduce the measured $\ell_2/\ell_1$ value at 5187 Å ($0.044 \pm
0.003$). Interestingly, this value of $k$ (which is independent of our MCMC analysis) is also the one that seems to provide a match between the PARSEC evolutionary models and the individual masses and radii of [EPIC 219568666]{}, assuming $r_1+r_2$ is accurate. As noted above, however, our MCMC analysis returns a lower $k$ value close to 0.54, not 0.60.
As a check on our extrapolation we repeated the light ratio determination with TODCOR in other echelle orders available in our spectra sampling the entire [[*Kepler*]{}]{} bandpass. Because our standard template library [@Nordstrom:1994; @Latham:2002] only spans about 300 Å around the region of the b triplet, we used synthetic spectra based on PHOENIX models from the same [@Husser:2013] library as above that cover the entire optical range. We note that these synthetic spectra were not used for our original radial-velocity determinations because at high resolution they do not match real stars as well as the templates from our own library, which are based on a line list tuned for that purpose. Nevertheless, the [@Husser:2013] library suffices for a determination of the light ratio, which does not require high resolution.
We measured $\ell_2/\ell_1$ in 31 spectral orders between 4300 Å and 8900 Å that have high enough flux and are sufficiently free from telluric lines. The results are displayed in Figure \[fig:fluxratio\] as points with error bars. Also shown is the predicted flux ratio from the synthetic spectra for two different values of the radius ratio. The top curve is for $k = 0.60$, the value required to reproduce the measured light ratio at 5187 Å, and the bottom curve is for $k = 0.54$, which is our result from the light curve analysis. The open squares on the top, $k = 0.60$ curve mark the predicted values at 5187 Å (matching the spectroscopic $0.044 \pm
0.003$ measurement by construction) and at the [[*Kepler*]{}]{} band ($0.084
\pm 0.005$). They were computed by integrating the theoretical flux ratio over the corresponding response functions (assumed to have a top-hat shape for the b order). The [[*Kepler*]{}]{} band prediction for the $k = 0.54$ curve is marked with an open circle. For reference the shaded area represents the response function for [[*Kepler*]{}]{}, with arbitrary normalization.
Our order-by-order light ratio measurements over the [[*Kepler*]{}]{} band clearly agree best with the upper ($k = 0.60$) curve in Figure \[fig:fluxratio\], and thus support the accuracy of the $\ell_2/\ell_1$ prior used in our analysis, which was based on that value of the radius ratio. At the same time, this purely spectroscopic approach conflicts with the result for $k$ returned by our MCMC analysis, suggesting the latter value is underestimated. We note that this conclusion is independent of the stellar evolution models.
### Third light
Our Gaussian prior for $\ell_3$ ($0.026 \pm 0.006$) relies on brightness measurements for all known stars within the photometric aperture, based on a seeing-limited image from the CFHT (Figure \[fig:CFHT\] and Section \[sec:analysis\]). While our higher-resolution AO imaging in Section \[sec:imaging\] revealed no additional companions outside of 01, it is still possible we are missing flux due to even closer companions. To explore this possibility and its effect on the analysis, we ran a grid of MCMC solutions in which we varied the $\ell_3$ prior between 0.03 and 0.12, maintaining our light ratio prior as in our original analysis. We found only a very small increase in $k$ (0.539–0.544) coupled with a reduction in $r_1+r_2$ (0.0670–0.0654), which together result in an even smaller value for the secondary radius than indicated in Table \[tab:dimensions\]. Furthermore, if the missing third light were due to a physically associated single star, a value as high as $\ell_3 = 0.12$ would imply a brightness for that star relative to the primary of $\ell_3/\ell_1
\approx 0.13$ in the [[*Kepler*]{}]{} band, which is even brighter than the secondary star ($\ell_2/\ell_1 = 0.084$). There is no sign of such a tertiary in our spectra, nor do we see any trend in the radial-velocity residuals from our spectroscopic orbit that might indicate an outer companion bound to the binary.
Limb darkening
--------------
For the present analysis we adopted the quadratic limb-darkening law for both stars, which is commonly used when dealing with space-based photometry. We allowed the first-order coefficients to vary and held the second-order coefficients fixed from theory because they are poorly constrained by the data given that the eclipses are grazing. The first-order coefficient we obtain for the primary, $0.291 \pm
0.064$, is quite consistent with the theoretical value of 0.343 from [@Claret:2011], while the one for the secondary star, $0.455 \pm
0.064$, is lower than expected (0.674). In principle the use of a different or possibly more sophisticated limb-darkening prescription could affect the resulting geometric parameters to some extent, particularly the radius sum and radius ratio. Unfortunately we are not able to test alternate formulations here because the public [eb]{} code we use restricts our choices to either the linear or quadratic laws. Nevertheless, as an experiment we repeated the analysis with the linear law, and found that neither $k$ nor $r_1+r_2$ changed significantly. The linear coefficients we derived in this case for the primary and secondary, $0.464 \pm 0.012$ and $0.428 \pm 0.034$, may be compared with the theoretical values of 0.564 and 0.730, respectively. Using the linear law for one star and the quadratic law for the other also did not affect the geometric parameters in any meaningful way. While these experiments do not completely rule out the possibility that a different or higher-order limb-darkening formula might have a more significant impact, the fact that even reverting to the simpler linear law did not change the results makes that seem less likely. In retrospect this is not all that surprising, as the grazing configuration means that the photometry typically has very little discriminating power on the details of limb darkening.
Detrending
----------
Given the above evidence that neither our spectroscopy nor our Gaussian priors on $\ell_2/\ell_1$ or $\ell_3$ seem to be causing a bias in our lightcurve analysis, suspicion falls on the photometry itself, and in particular on the processing (detrending) to which it was subjected prior to analysis.
We therefore tested different ways of treating the photometry, focusing both on the systematics-removal stage and on the removal of low-frequency variability, to see if residual systematic errors or subtle biases from our low-frequency removal might explain the discrepancy between our best-fit radius ratio and the radii predicted by evolutionary models.
First, we tried removing low-frequency variability from the [[*K2*]{}]{} light curve in a different way. Instead of dividing away the best-fit spline from the light curve solution, we fit for a new spline after the removal of systematics while manually masking the eclipses. We then tweaked our systematics removal process to see if subtle biases from our simultaneous fit to the eclipse shapes, the [[*K2*]{}]{} systematics, and low-frequency variability were causing a bias in the radius ratio. We tested the light curve with only a first-pass systematics correction, and found that the best-fit radius ratio from this light curve was essentially unchanged compared to our original processing.
We also tested different light curves produced by simultaneously fitting the systematics and low-frequency variability along with more sophisticated models for the eclipses. The [@Mandel:2002] code we used in our detrending was designed for modeling the light curves of transiting planets, and assumes that each set of transits or eclipses has the same limb-darkening profile, and furthermore, it does not include a third-light contribution. In our tests we relaxed both of these assumptions by including a third-light contribution in the eclipse model and by allowing the flexibility for the code to fit each set of eclipses with their own quadratic limb darkening coefficients. Interestingly, none of these changes significantly affected the best-fit radius ratio $k$ or the radius sum $r_1+r_2$ derived from the light curve.
Impact of the radius ratio on the light curve
---------------------------------------------
In order to quantify the impact that a change in the radius ratio has on the light curve itself, we compared our adopted model with another in which we forced $k$ to be approximately 0.60. We did this by applying a tight prior on $k$ and removing the prior on $\ell_2/\ell_1$ so as to avoid conflict. All other parameters were solved for in the same way as before. The difference between the synthetic light curves from these two models has a maximum amplitude of about 100 ppm during the ingress/egress of the secondary eclipse, and less than half of that during the primary eclipse. We note that this is of the same order as the overall scatter of the [[*K2*]{}]{} photometry from our adopted model ($\sim$120 ppm). We speculate that this difference may be too small for our analysis to detect given the photometric noise.
Cadence of the photometric observations
---------------------------------------
Finally, we explored the possibility that the relatively long 29.4 min sampling of the [[*K2*]{}]{} light curve may be negatively impacting our results, given that only a handful of observations were recorded near the points of first and last contact (see Figure \[fig:LCfit\]), which are critical for defining the radii. For this we used the same $k = 0.60$ solution mentioned in the previous section, and oversampled the corresponding model to emulate the 1 min short cadence (SC) observations that [[*K2*]{}]{} has obtained for many targets (but not for [EPIC 219568666]{}). As a check, we also sampled it at the same 29.4 min rate (long cadence, LC) as the real observations. We then added Gaussian noise appropriate to each cadence, and subjected the two synthetic data sets to identical MCMC analyses as with the real data, integrating the model over 29.4 min for the LC time series but not for the SC time series. We repeated this several times, each with different noise, and found that within the uncertainties both synthetic data sets gave the same results for $k$ and $r_1+r_2$, on average, and that those results were also similar to the input values of those parameters for this experiment. We conclude that the cadence of the photometric observations has little influence in this particular case.
Summary
-------
From the tests described above we conclude that the apparent bias in the radius ratio suggested by Figure \[fig:MR\] and supported by Figure \[fig:fluxratio\] does not seem to be caused by errors in the radial velocity measurements, by incorrect priors on $\ell_2/\ell_1$ or $\ell_3$, by the subtleties of our detrending procedures, or by the finite cadence of the observations. The change in $k$ required to match the slope in the mass-radius diagram (from $\sim$0.54 to $\sim$0.60) is about 11% of the value we measure. While this change may seem significant at face value (about 3.7 times the formal uncertainty in $k$), the impact on the shape of the light curve is actually very small, as indicated earlier, and it may be that the data available are insufficient to discern this difference. Another possibility that cannot be ruled out has to do with the presence of spots on one or both stars, especially given the grazing configuration of the [EPIC 219568666]{} system. Spots can cause systematic errors in the measured eclipse depths (changing the surface brightness ratio), which in turn can lead to biases in other geometric parameters such as the radius ratio, the inclination angle, and even the radius sum. An example with a much more detailed discussion of spot effects on eclipsing binary parameters may be found in the work of [@Irwin:2011].
Concluding remarks {#sec:remarks}
==================
[EPIC 219568666]{} is the second eclipsing binary we have analyzed in the Ruprecht147 cluster after [EPIC 219394517]{} [@Torres:2018], based on [[*K2*]{}]{} photometry and follow-up spectroscopic observations. While the formal precision of our mass and radius determinations is quite high ($\sim$1% relative errors in the masses, and about 0.6% and 2.5% errors in the radii), the agreement with stellar evolution models is less satisfactory for [EPIC 219568666]{} than it was for the system we studied earlier. The primary star seems too large for its mass (a 6.7$\sigma$ discrepancy) while the secondary is a bit too small (though only by 2.2$\sigma$). Activity-related radius inflation seems unlikely for the primary for the reasons indicted earlier, and there is no evidence of any strong systematic errors in the masses, which leads us to suspect a bias in the radius ratio we inferred from the [[*K2*]{}]{} light curve.
A weakly constrained radius ratio is not an uncommon occurrence in light curve analyses, though it often goes unnoticed. The usual cure for this problem is the use of an external constraint on the light ratio derived, e.g., from spectroscopy, as we have done here. For [EPIC 219568666]{} this does not seem to have completely eliminated biases, despite our best attempts. Careful examination of the critical ingredients for the mass and radius determinations, as detailed in the preceding section, has given no clues as to what could be causing $k$ to be underestimated in our analysis. A remaining possibility is a bias in the geometric parameters caused by spots on one or both stars. Simulations by [@Morales:2010] have shown that these effects appear to be maximized when the spots are concentrated at the poles. Although we have no information on the latitudinal distribution of spots in this system, it is possible that acquiring multi-color observations might help, as the different sensitivity to spots as a function of wavelength could help break degeneracies with other effects such as limb darkening, particularly in a grazing configuration such as that of the present system.
The example of [EPIC 219568666]{} serves as a cautionary tale about the need to be aware of the potential for biases in the solution of light curves, especially regarding properties that often poorly constrained such as the radius ratio. This can even affect photometric measurements from space-based missions, which may well be internally highly precise but could still suffer from subtle systematic errors not reflected in the formal uncertainties. Such errors could result, e.g., from the complex processing space photometry is typically subjected to, prior to use, or from physical effects such as the presence of spots, as mentioned above. Independent high-precision photometric observations of [EPIC 219568666]{} such as may be obtained in the future with NASA’s Transiting Exoplanet Survey Satellite (TESS), or at other wavelengths from the ground, may shed some light on this issue.
We have shown that it is still possible to infer a useful estimate of the age of [EPIC 219568666]{} by relying only on the sum of the radii, bypassing the use of the individual radii involving $k$. The result, $2.76 \pm
0.61$ Gyr at a metallicity fixed to that of the cluster, is consistent with the independent estimate obtained in our earlier study of [EPIC 219394517]{} ($2.65 \pm 0.25$ Gyr), if much less precise. Together these two age estimates therefore point to an age for Ruprecht147 near 2.7 Gyr. We expect to strengthen this determination even further as studies are completed for three additional eclipsing binaries in the cluster that are underway.
The spectroscopic observations of [EPIC 219568666]{} were gathered with the help of P. Berlind, M. Calkins, G. Esquerdo, and D. Latham. J. Mink is thanked for maintaining the CfA echelle database. We are also grateful to J. Irwin for implementing changes in the [eb]{} program that facilitated the present analysis, and to the anonymous referee for helpful comments and suggestions. G.T. acknowledges partial support from NASA’s Astrophysics Data Analysis Program through grant 80NSSC18K0413, and to the National Science Foundation (NSF) through grant AST-1509375. J.L.C. is supported by the NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1602662, and by NASA under grant NNX16AE64G issued through the [[*K2*]{}]{} Guest Observer Program (GO 7035). This work was performed in part under contract with the California Institute of Technology (Caltech)/Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. The research has made use of the SIMBAD and VizieR databases, operated at the CDS, Strasbourg, France, and of NASA’s Astrophysics Data System Abstract Service. The research was made possible through the use of the AAVSO Photometric All-Sky Survey (APASS), funded by the Robert Martin Ayers Sciences Fund. Data products were also used from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded NASA and the NSF. The work has also made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<https://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement. The computational resources used for this research include the Smithsonian Institution’s “Hydra” High Performance Cluster.
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[^1]: <http://archive.stsci.edu/>
[^2]: <http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/megapipe/docs/filtold.html>
[^3]: http://dan.iel.fm/emcee .
|
---
abstract: 'We calculate the coefficient of bulk viscosity by considering the non-leptonic weak interactions in the cores of hybrid stars with both hyperons and quarks. We first determine the dependence of the production rate of neutrons on the reaction rate of quarks in the non-leptonic processes, that is $\Gamma_{n}=K_{s}\Gamma_{s}+\Gamma_{\Lambda}+2\Gamma_{\Sigma^{-}}$. The conversion rate, $K_{s}$ in our scenario is a complicated function of baryon number density. We also consider medium effect of quark matter on bulk viscosity. Using these results, we estimate the limiting rotation of the hybrid stars, which may suppress the r-mode instability more effectively. Hybrid stars should be the candidates for the extremely rapid rotators .'
author:
- |
Na-Na Pan$^{1}$, Xiao-Ping Zheng$^{1}$[^1] and Jia-Rong Li$^{2}$\
1.Institute of Astrophysics, Huazhong Normal University, Wuhan 430079, China\
2.Institute of Particle Physics, Huazhong Normal University, Wuhan 430079, China
date: 'Accepted 0000, Received 0000'
title: 'Bulk viscosity of Mixed nucleon-hyperon-quark Matter in Neutron stars'
---
\[firstpage\]
dense matter — gravitation —stars: neutron — stars: rotation — stars: oscillations
Introduction
============
It has been recognized that weak interaction processes in neutron star matter contribute to the bulk viscosity(Sawyer & Soui[@1]; Sawyer [@2]; Haensel & Schaeffer [@3]). When only nucleons and leptons exist in the interior of neutron stars, the bulk viscosity results from Urca processes. Some exotic particles, such as hyperons and quarks, can be produced as the density inside neutron stars increases (Weber [@4]; Glendenning et al. [@5]). The non-leptonic processes involving both hyperons and quarks lead to larger coefficient of bulk viscosity of dense nuclear matter(Jones [@6]; Wang & Lu [@8]). Therefore, the calculations of the bulk viscosity induced by non-leptonic reactions have received considerable attention in the past few years(Sawyer [@9]; Madsen [@10]; Goyal et al. [@11]; Haensel & Levenfish [@12]; Haensel et al. [@34];Lindblom & Owen [@13]; Zheng et al.[@14]; Zheng et al.[@15]; Zheng et al.[@16])since the discovery by Andersson, Friedman and Morsink of r-mode instability in neutron stars(Andersson [@17]; Friedman & Morsink [@18]). Actually, neutron stars can be constituted by a larger variety of particles other than just neutrons and protons or pure quark matter. The mixed phases, with nucleon-hyperon, nucleon-quark or nucleon-hyperon-quark matter, may be contained in the interior of neutron stars(Glendenning [@19]; Glendenning [@20]). The bulk viscosities of mixed nucleon-hyperon matter and nucleon-quark matter have been discussed by some authors so far(Lindblom & Owen [@13]; Drago et al. [@21]; Nayyar & Owen [@22]). However, the calculations are reduced to single reaction case since only one type of reaction is involved, either hadron or quark non-leptonic process. The purpose of this paper is to investigate the general case thoroughly. We focus on mixed nucleon-hyperon-quark matter to evaluate its bulk viscosity by considering the reactions involving both hyperons and quarks. According to Lindblom & Owen (LO) formalism (Lindblom & Owen [@13]; Nayyar & Owen [@22]), the bulk viscosity depends on the relaxation time governed by the hyperon and quark reactions. Following the LO approach, we first convert reaction rates for hyperons and quarks into the production rate of neutrons per unit volume in order to give the relaxation time of the system, and then simulate the bulk viscosity of the mixed phase using the rates. In our study, the conversion rate from quarks to neutrons varies with a variable baryon number density, and we will see below that the numerical solution is necessary. Our study is under the neutron stars models based on $Ghosh-Phatak-Sahu$ $(GPS)$ equations of state(Ghosh et al.[@23]) (EOS) for hadron matter and $SGT$ model (Schertler, Greiner & Thoma [@24]) for quark matter in medium. At the high density, $GPS$ EOSs are favorable for the equilibrium state to include a sequence of hyperons. Our model also concerns a $Baym-Pethick-Sutherland $ $(BPS)$(Baym et al.[@25]) crust. Our analysis shows that there exist two instability windows for a neutron star with strangeness. The low-temperature window indicates the possibility of existence of submillisecond pulsars.
The organization of the rest of this paper is as follows. In section 2, we provide details of structures in stars which we are studying by taking into account a first-order deconfinement phase transition into quarks from hadrons in the cores of the stars. Then in section 3,we employ LO approach to derive the bulk viscosity of mixed nucleon-hyperon-quark matter, including numerical aspects of the evaluations of the viscosity coefficient. Finally in section 4, we work out r-mode instability window based on the competition between increasing of gravitational radiation and damping of viscosities.
Equation of state
=================
In this paper, we consider the compact stars inside which the deconfinement transition occurs at high density. Concerning the hadronic phase (HP), we use the $BPS$ EOS at subnuclear densities for the crust of the star, and the $GPS$ EOS for supranuclear density in the framework of the relativistic self-consistent mean field theory (Glendenning [@19]). In this theory, $\Sigma^{-}$ and $\Lambda$ hyperons are included for nuclear densities, where nucleons interact through the nuclear force mediated by the exchange of isoscalar and isovector mesons $(\sigma,\omega,\rho)$. The model parameters used in this paper are arranged in table 1. These two EOSs are matched at $\epsilon\approx10^{14}g/cm^{3}\approx0.4\epsilon_{0}$, here $\epsilon_{0}=140Mev/fm^{3}\approx2.5\times10^{14}g/cm^{3}$ is the saturation density of nucleon, above which we allow the HP to undergo a first order phase transition to a deconfined quark matter phase (QP) that is described with the effective mass bag model considering medium effect represent by the coupling constant $g$ for strong action (Schertler, Greiner & Thoma [@24]). This phase transition makes it possible that the occurrence of a mixed hadron-quark phase (MP) in a finite density range inside compact stars. The relative constraints of various species in the MP are determined at each density by imposing $\beta$-equilibrium, Gibbs condition and global charge neutrality. At the high density of interest to us, these equilibrium constraints are $$\mu_{n}=\mu_{u}+2\mu_{d},$$ $$\mu_{p}=2\mu_{u}+\mu_{d},$$ $$\mu_{\mu}=\mu_{e},$$ $$\mu_{\Sigma^{-}}=\mu_{n}+\mu_{e},$$ $$\mu_{\Lambda}=\mu_{n},$$ $$\mu_{s}=\mu_{d},$$ $$\mu_{u}=\mu_{d}-\mu_{e},$$ $$P_{HP}(\mu_{n},\mu_{e})=P_{QP}(\mu_{n},\mu_{e}),$$ $$(1-\chi)(n_{p}-n_{\Sigma^{-}}-n_{HP,
e}-n_{\mu})=\chi(-\frac{2}{3}n_{u}+\frac{1}{3}n_{d}+\frac{1}{3}n_{s}+n_{QP,
e}).$$
Here $\mu_{i},n_{i}$ are the chemical potential and the number density of the $i^{th}$ species respectively, $P_{i}$ is the pressure of the $i^{th}$ phase, and $\chi$ is the fraction of the quark matter in MP. Solving these constraints, we can give the compositions of the equilibrium matter in hybrid stars. Figure \[GPS fixed population\] and \[g fixed population\] show some solutions for different parameters. Evidently, they contain a significant number of hyperons and strange quark matter in the highest density portion of their cores, and the additional hyperons appear as the coupling among quarks increases. Using the Tolman-Oppenheimer-Volkoff (TOV) equation, we obtain the structures of some hybrid stars with maximum masses for a few EOSs shown in Figure \[structure\], where we also plot the structures of hyperon stars for the same $GPS$ EOSs (Figure \[structure\](c)) as comparison. The EOSs of the stars are softened when deconfinement phase transition happens, but the structures of the stars are not sensitive to the changes of $GPS$ EOSs unlike hyperon stars, while medium effect in quark matter does significantly change the stellar masses and radii. As is known, the Hulse-Taylor pulsar has the measured mass of 1.44 $M_{sun}$ (Taylor & Weisberg [@35]). Strictly speaking, the EOSs models producing the maximum mass smaller than the measured value should be rejected. We, however, take the lower values around 1.44 $M_{sun}$ to compare with $GPS$ models (hyperon stars), since the small difference of the masses can’t change our results significantly below.
Bulk viscosity
==============
Bulk viscosity is the dissipative process in which the macroscopic compression or expansion of a fluid element is converted to heat(Landau & Lifshitz [@26]). The dissipation due to bulk viscosity is carried out via microscopic reactions which come from weak interactions in compact star matter. In general, there are two types of reactions in dense nuclear matter. One produces bulk viscosity because its timescales are comparable with the period of the perturbation, the other puts constraints on the variations of different particles’ densities. As in most cases, the volume per unit mass variation and the chemical potential imbalance due to the oscillation in the star are so small that it is the linear part that contributes most to the viscosity, so we will use the relaxation time approximation method(Lindblom & Owen [@13]; Drago et al.[@21]; Landau & Lifshitz [@26]) to compute the bulk viscosity of dense nuclear matter.
Here we are interested in the case that hyperons and quarks are all present in the mixed phase. It is thought that the non-leptonic processes are dominating over the bulk viscosity of nuclear matter with strangeness (Jones [@6]; Lindblom & Owen [@13]; Drago et al.[@21]). In the initial works(Drago et al.[@21]), one estimated the results arising from a single reaction, assuming either an absence of hyperon or quark pairing in mixed hadron-quark phase. That treatment was for simplification in the calculations. In fact, the processes that produce bulk viscosity of the mixed system are $$\label{en21}
u+d\longleftrightarrow u+s$$ for strange quark matter, and $$\label{en220}
n+n\longleftrightarrow p+\Sigma^{-},$$ $$\label{en22}
n+p\longleftrightarrow p+\Lambda$$ for hyperon matter.
Accordingly, the processes which put constraints on the variations of the densities of different particles come from the melting reactions of nucleons $$\label{en23}
p\longleftrightarrow 2u+d,$$ $$\label{en24}
n\longleftrightarrow u+2d,$$ $$\label{en240}
n+\Lambda\longleftrightarrow p+\Sigma^{-},$$ and hyperons $$\label{en25}
\Lambda\longleftrightarrow u+d+s$$ into free quarks.
In the mixed phase, since the degrees of freedom, for example, the number densities of various baryons, are related to each other even out of thermodynamic equilibrium by constraints such as conservation of baryon number, we could express all of the perturbed quantities in terms of a single one. We here choose the number density of neutrons $n_{n}$ as our primary variable. Since the three reactions (\[en21\])(\[en220\])(\[en22\]) all contribute to the change of neutron numbers, we express the production rate of neutrons per unit volume as $$\label{en160000}
\Gamma_{n}=K_{s}\Gamma_{s}+K_{\Lambda}\Gamma_{\Lambda}
+K_{\Sigma^{-}}\Gamma_{\Sigma^{-}},$$ where $\Gamma_{s}$, $\Gamma_{\Lambda}$, $\Gamma_{\Sigma^{-}}$ indicate the reaction rates of processes (\[en21\]), (\[en220\]) and (\[en22\]) respectively, and $K_{s}$, $K_{\Lambda}$, $K_{\Sigma^{-}}$ are the rates converting reaction rates of each exotic particles, s quark, $\Lambda$ and $\Sigma^{-}$ hyperon, into the production rate of neutrons. For fast processes, we know that their contributions to the relaxation time of the system could be omitted compared with the slow processes, so we may easily know that $K_{\Lambda}=1$ and $K_{\Sigma^{-}}=2$ by the reactions (\[en220\]) and (\[en22\]). The determination of $K_{s}$ is a slightly complicated matter. Its treatment will be given later. Obviously, if the system is perturbed, the fraction of the quark matter in the mixed phase $\chi$ would be changed through reactions (\[en23\]), (\[en24\]) and (\[en25\]), and the pressure of the system would be equilibrated again after the perturbation in order to satisfy the mechanical equilibrium. Therefore we have following constraints:
$$\begin{aligned}
\label{en26}
0=(1-\chi)(\delta n_{n}+\delta n_{p}+\delta n_{\Sigma^{-}}+\delta
n_{\Lambda})+\chi(\delta n_{u}+\delta n_{d}+\delta
n_{s})/3\nonumber\\
+\delta \chi((n_{u}+n_{d}+n_{s})/3
-n_{n}-n_{p}-n_{\Sigma^{-}}-n_{\Lambda}),\end{aligned}$$
$$\begin{aligned}
\label{en27}
0=(1-\chi)(\delta n_{p}-\delta n_{\Sigma^{-}})+\chi(2\delta
n_{u}-\delta n_{d}-\delta n_{s})/3\nonumber\\
+\delta \chi((2n_{u}-n_{d}-n_{s})/3-n_{p}+n_{\Sigma^{-}}),\end{aligned}$$
$$\label{en28}
0=\sum_{{H}}p_{H}\delta n_{H}-\sum_{{Q}}p_{Q}\delta n_{Q},$$
$$\label{en29}
0=\alpha_{pn}\delta n_{n}+\alpha_{pp}\delta
n_{p}+\alpha_{p\Sigma^{-}}\delta
n_{\Sigma^{-}}+\alpha_{p\Lambda}\delta
n_{\Lambda}-2\alpha_{uu}\delta n_{u}-\alpha_{dd}\delta n_{d},$$
$$\label{en30}
0=\alpha_{nn}\delta n_{n}+\alpha_{np}\delta
n_{p}+\alpha_{n\Sigma^{-}}\delta
n_{\Sigma^{-}}+\alpha_{n\Lambda}\delta
n_{\Lambda}-\alpha_{uu}\delta n_{u}-2\alpha_{dd}\delta n_{d},$$
$$\label{en300}
0=\beta_{n}\delta n_{n}+\beta_{p}\delta
n_{p}+\beta_{\Sigma^{-}}\delta
n_{\Sigma^{-}}+\beta_{\Lambda}\delta n_{\Lambda},$$
$$\label{en31}
0=\alpha_{\Lambda n}\delta n_{n}+\alpha_{\Lambda p}\delta
n_{p}+\alpha_{\Lambda \Sigma^{-}}\delta
n_{\Sigma^{-}}+\alpha_{\Lambda\Lambda}\delta
n_{\Lambda}-\alpha_{uu}\delta n_{u}-\alpha_{dd}\delta
n_{d}-\alpha_{ss}\delta n_{s}.$$
The first constraint (\[en26\]) is the baryon number conservation of the system. The second one (\[en27\]) imposes the conservation of electric charge assuming all leptonic reaction rates are much smaller and neglected. The third constraint (\[en28\]) is related to the mechanical equilibrium. The last four (\[en29\])-(\[en31\]) describe the equilibrium with respect to reactions (\[en23\])-(\[en25\]). Noteworthily, when the $\Sigma^{-}$ density vanishes in the system, $\delta
n_{\Sigma^{-}}=0$, and the melting process (\[en240\]) doesn’t exist, nor does the process (\[en220\]).
Here we use $$\alpha_{ij}=(\partial \mu_{i}/\partial n_{j})_{n_{k}, k\neq j},$$ $$\beta_{i}=\alpha_{ni}+\alpha_{\Lambda
i}-\alpha_{pi}-\alpha_{\Sigma^{-}i},$$ $$p_{i}=\partial P/\partial n_{i}$$ for shortening. Then we could obtain the variations of various particles’ number densities $\delta n_{i}$ and the quark fraction $\delta \chi$ in the form of $\delta n_{n}$.
In order to determine $K_{s}$, we need to describe the variation of neutrons $\delta n_{n}$ following (\[en160000\]) as
$$\label{en16000}
-\delta n_{n}=K_{s}\delta n_{s}+\delta n_{\Lambda} +2\delta
n_{\Sigma^{-}}.$$
Thus we have $$\label{en1600000}
K_{s}=\frac{-\delta n_{n}-\delta n_{\Lambda}-2\delta
n_{\Sigma^{-}}}{\delta n_{s}},$$ which is a function of baryon number density for given equation of state under thermodynamic equilibrium in the mixed phase system. Combining the constraints conditions to equation (\[en1600000\]) with the linear dependence of $\delta
n_{\Lambda}$, $\delta n_{\Sigma^{-}}$ and $\delta n_{s}$ on $\delta n_{n}$, the $K_{s}$ can be determined. An example of the numerical result is showed in the Figure \[Ks coefficient\]. In the extreme, if the system doesn’t undergo the deconfinement phase transition ($\delta n_{s}=0$) or is in the absence of hyperons ($\delta n_{\Lambda}=\delta n_{\Sigma^{-}}=0$), the calculation is reduced to the situation of a single reaction(Lindblom & Owen [@13]; Drago et al.[@21]).
The relaxation time $\tau$ of the system due to these reactions must be: $$\label{en1600}
\frac{1}{\tau}=(\frac{K_{s}\Gamma_{s}}{\delta\mu}+\frac{\Gamma_{\Lambda}}{\delta\mu}
+\frac{2\Gamma_{\Sigma^{-}}}{\delta\mu})\frac{\delta\mu}{\delta
n_{n}}.$$ Here $\Gamma_{s}$, $\Gamma_{\Lambda}$ and $\Gamma_{\Sigma^{-}}$ have already been calculated by $Dai$ under the high temperature limit $(2\pi k T\gg \delta\mu)$ (Dai & Lu [@27]) and $Lindblom$ $et$ $al.$ (Lindblom & Owen [@13]; Nayyar & Owen [@22]) separately. Considering the reactions (\[en23\])(\[en24\])(\[en240\])(\[en25\]), the overall chemical potential imbalance $\delta\mu$ is obtained through the equilibrium of the system : $$\delta \mu\equiv \delta \mu_{d}-\delta \mu_{s}= \delta
\mu_{n}-\delta \mu_{\Lambda}=2 \delta \mu_{n}-\delta
\mu_{p}-\delta \mu_{\Sigma^{-}},$$ which could also be described in the form of $\delta n_{n}$.
Now we use the LO formula $$\label{en16000}
Re\zeta=\frac{p(\gamma_{\infty}-\gamma_{0})\tau}{1+(\hat{\omega}\tau)^{2}}$$ to evaluate the bulk viscosity. Thereinto, $$\gamma_{\infty}-\gamma_{0}\equiv
-\frac{n_{B}^{2}}{p}\frac{\partial p}{\partial
n_{n}}\frac{d\tilde{x}_{n}}{dn_{B}}.$$ Here $\tilde{x}_{n}=n_{n}/n_{B}$ is the relative population of neutrons in the equilibrium state of the mixed phase calculated in the part 2. $\hat{\omega}$ is the perturbation frequency in compact stars, which is typically $10^{3}\sim10^{4}s^{-1}$. We have chosen $\hat{\omega}=10^{4}s^{-1}$ for calculation . By the equation (\[en16000\]), we know that for a perturbation of fixed frequency , the bulk viscosity $\zeta$ is in proportion to the reciprocal of the relaxation time $\tau$ under the low temperature limit $1\ll \hat{\omega}\tau$ , that is $\zeta\propto\tau^{-1}$ ; on the other hand $\zeta$ is in proportion to $\tau$ under condition of the high temperature limit $1\gg \hat{\omega}\tau$, that is $\zeta\propto\tau$. Thereby, $\zeta(n_{B})$ could be divided into two different behaviors, in another word, $\zeta(n_{B})$ will increase with the temperature for low temperature regime; and the circs is just opposite for the high temperature regime. The transition temperature is the function of baryon density and perturbation frequency (Figure \[trantemperature\] for $GPS2, g=2.0$ EOS).
Figure \[GPS fixed bulk viscosity\] and Figure \[g fixed bulk viscosity\] display the bulk viscosity of mixed phase as a function of baryon density for various temperatures and given EOSs. The numerical results show that i) the bulk viscosity of mixed phase, either nucleon-hyperon phase or nucleon-quark phase or nucleon-hyperon-quark phase, is of almost the same order of the bulk viscosity of pure strange quark matter, ii) the differences between EOSs only change the compositions and sizes of the mixed phase in the interior of the stars, iii) the transition temperature between low and high temperature regime is about $10^{8}\sim 10^{9} K$ in our models.
R-mode instability windows of hybrid stars
==========================================
The r-modes in a perfect fluid star succumbing to gravitational radiation (GR) drive the Chandrasekhar-Friedman-Schutz instability for all rates of stellar rotation, and arise due to the action of Coriolis force with positive feedback of the increasing of GR(Andersson & Kokkotas [@28]; Stergioulas [@29]). Actually, the viscosity of stellar matter can hold back the growth of the modes effectively. Based on the competition between the destabilizing effect of GR and the damping effect of viscosity, a r-mode instability window can be defined. Now, we could apply our results of the bulk viscosity in the previous section together with those of the HP and QP to considering the r-mode instability windows of rotating compact stars using the standard formulae given in literatures(Lindblom et al.[@30]; Lindblom et al.[@31]). We first integrate the viscosity on the structure of stars, which has been showed in Figure \[structure\] for a wide range of EOSs, to obtain the bulk viscosity timescale $\tau_{B}$ $$\frac{1}{\tau_{B}}=-\frac{1}{2\widetilde{E}}(\frac{d\widetilde{E}}{dt})_{B},$$ where $$\widetilde{E}=\frac{1}{2}\alpha^{2}\Omega^{2}R^{-2}\int_{0}^{R}\epsilon(r)r^{6}dr,$$ $$(\frac{d\widetilde{E}}{dt})_{B}=-4\pi\int_{0}^{R}\zeta(\epsilon(r))<|\overrightarrow{\nabla}\cdot\delta\overrightarrow{\upsilon}|^{2}>r^{2}dr.$$ Here $\widetilde{E}$ is the mode energy in a frame rotating with the star, $(\frac{d\widetilde{E}}{dt})_{B}$ is the rate at which energy is drained from the mode by viscosity, $\alpha$ is a dimensionless amplitude coefficient, $\Omega$ is the angular frequency of a spinning star, $\epsilon(r)$ is the energy density of star at radius r, and $\delta\overrightarrow{\upsilon}$ is the Eulerian velocity perturbation. In general the expansion of the mode $\overrightarrow{\nabla}\cdot\delta\overrightarrow{\upsilon}$ is a complicated function of radius and angle. Although the function can only been determined numerically, we adopt an excellent analytical fit to those numerical solutions $$<|\overrightarrow{\nabla}\cdot\delta\overrightarrow{\upsilon}|^{2}>={\frac{\alpha^{2}\Omega^{2}}{690}(\frac{r}{R})^{6}[1+0.86(\frac{r}{R})^{2}](\frac{\Omega^{2}}{\pi
G\bar{\epsilon}})^{2}},$$ which is given by Lindblom (Lindblom & Owen [@13]; Nayyar & Owen [@22]). Finally, we could obtain the critical rotation frequency for the star with a solid crust as a function of temperature following from $$-\frac{1}{\tau_{GR}}+\frac{1}{\tau_{B}}+\frac{1}{\tau_{SR}}=0$$ for low temperature limit($T< 10^{9}K$), and $$-\frac{1}{\tau_{GR}}+\frac{1}{\tau_{B}}+\frac{1}{\tau_{mU}}=0$$ for high temperature limit($T> 10^{9}K$). Here $\tau_{GR}$ is the timescale for gravitational radiation to effect the r-mode(Andersson & Kokkotas [@28]), while $\tau_{SR}$ is the timescale for surface rubbing due to the presence of the viscous boundary layer if a solid crust exists(Bildsten & Ushomirsky [@32]; Andersson et al.[@33]), and $\tau_{mU}$ is the viscosity timescale of modified URCA reactions for low density hadronic matter(Lindblom et al.[@31]). Our results for the critical rotation frequency are shown in the Figure \[rmode\].
We may know from it that there are two instability windows for such hybrid stars(Andersson & Kokkotas [@28]). We find that the r-mode instability is completely suppressed for the hybrid stars from the low temperature windows. We realize that hadronic EOSs, either stiffer of softer, hardly influence the instability windows. It is because that the structure of the star including quark phase hardly changes for different $GPS$ EOSs. The increasing medium effect of quarks enlarges the window at low $g$ values but reduces the window at high $g$ values. The properties may imply that extremely rapid rotating or submillisecond pulsars would exist, and just be the hybrid stars.
Conclusion
==========
Hyperons and quarks exist in cores of neutron stars at high density. To evaluate accurately the bulk viscosity (due to strangeness-changing four-baryon weak interactions involving both hyperons and quarks non-leptonic processes), it is necessary to compute detailed and accurate models for the compositions and structures of the neutron-star cores. We use $GPS$ EOSs for hadrons and $SGT$ model with medium effect for quarks to evaluate composition of the nuclear matter inside the stellar core and solve the TOV equation to obtain the stellar structure, and particularly study the effect of differences between EOSs on the structures of stars.
We numerically calculate the bulk viscosity of mixed phase matter of various EOSs, and especially give the treatment of the case that involves both hyperons and quarks reactions. It is a generalization of single reaction case in previous work (Drago et al.[@21]). Our results show that there is high coefficient of bulk viscosity as long as exotic particle appears in high density nuclear matter, either strange quark or hyperon.
We have considered medium effect of quark matter. The bulk viscosity of hybrid star increases as the coupling among quarks becomes stronger. The coefficient of bulk viscosity at large coupling constant is higher than assumed in reference (Drago et al.[@21]), and the mixed phase regime is expanded. Just so, the hybrid stars have extremely high limiting rotation frequencies from the low temperature windows even if the stellar masses are large. The submillisecond pulsars may exist.
We must emphasize that we are concerned about the bulk viscosity of mixed normal nucleon-hyperon-quark phase. Nucleon, hyperon superfluidity and quark superconductivity may occur at $T < 10^{8}
K$ and given critical density. These effects would significantly influence the dissipation of hybrid star matter. These considerations, which are made in hyperon star model (Lindblom & Owen [@13]) and hybrid star model (Drago et al.[@21]), are our future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by NFSC under Grant Nos.90303007 and 10373007, and the Ministry of Education of China with project No.704035. We are especially grateful to Professor D.F. Hou for reading and checking the text.
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$Name$ $\rho_{0}(fm^{-3})$ $B/A(Mev)$ $a_{sym}(Mev)$ $K(Mev)$ $m^{*}/m_{N}$ $x_{\sigma}=x_{\rho}=x_{\omega}$
-------- --------------------- ------------ ---------------- ---------- --------------- ----------------------------------
$GPS1$ 0.150 -16.0 32.5 250 0.83 $\sqrt{\frac{2}{3}}$
$GPS2$ 0.150 -16.0 32.5 300 0.83 $\sqrt{\frac{2}{3}}$
$GPS3$ 0.150 -16.0 32.5 350 0.83 $\sqrt{\frac{2}{3}}$
. \[parameter\]
![Influences of medium effect on the relative particles populations as functions of baryon number densities in the hybrid stars. Note that under the condition $g=1.0$, $\Lambda$ does also exist in the MP, but its population is small compared with others’, so we don’t show it here. Here they are the relative populations of baryon number densities for u,d,s quark only. For all QP EOSs, we choose the bag constant $B^{1/4}=170.0 Mev$ and the mass of the s quark $m_{s}=150.0 Mev$ in calculations.[]{data-label="GPS fixed population"}](GPSfixedpopulation.eps){width="90.00000%"}
![Influences of hadron EOSs on relative particles populations as functions of baryon number densities in the hybrid stars. Under the condition $GPS1$, $\Lambda$ also exists in the MP, but we don’t show it here as its population is small compared with others’. Parameters of QP EOSs are the same with the Figure 1 in calculations.[]{data-label="g fixed population"}](gfixedpopulation.eps){width="60.00000%"}
![Structures of maximum masses hybrid stars (a) (b) using different EOSs discussed in the Figure 1 and 2, and the corresponding hyperon stars (c) for same hadron EOSs with case (b) : the relations between total energy densities(in units of $10^{15}g/cm^{3}$) and the distances from the center of the stars .[]{data-label="structure"}](rourrelation.eps){width="80.00000%"}
![Converting rate $K_{s}$ as a function of baryon number density in the mixed phase under the conditions $GPS2$ and $g=3.0$. []{data-label="Ks coefficient"}](Kscoefficient.eps){width="60.00000%"}
![Influences of medium effect on bulk viscosities as functions of baryon number densities for a range of temperatures. Parameters are the same with Figure 1.[]{data-label="GPS fixed bulk viscosity"}](GPSfixedbulkviscosity.eps){width="90.00000%"}
![Influences of hadron EOSs on bulk viscosities as functions of baryon number densities for a range of temperatures. Parameters are the same with Figure 2.[]{data-label="g fixed bulk viscosity"}](gfixedbulkviscosity.eps){width="60.00000%"}
![Critical transition temperatures as functions of baryon number densities and perturbation frequencies under the conditions $GPS2$ and $g=2.0$.[]{data-label="trantemperature"}](trantemperatureok.eps){width="60.00000%"}
 (b) respectively. Here $\nu$ is the critical rotation frequency of the stars.[]{data-label="rmode"}](GPSgfixedrmode.eps){width="60.00000%"}
[^1]: E-mail:zhxp@phy.ccnu.edu.cn
|
---
abstract: |
We present a new sample of distant ultraluminous infrared galaxies. The sample was selected from a positional cross–correlation of the $IRAS$ Faint Source Catalog with the FIRST database. Objects from this set were selected for spectroscopy by virtue of following the well-known star-forming galaxy correlation between 1.4 GHz and 60 $\mu$m flux, and by being optically faint on the POSS. Optical identification and spectroscopy were obtained for 108 targets at the Lick Observatory 3 m telescope. Most objects show spectra typical of starburst galaxies, and do not show the high ionization lines of active galactic nuclei. The redshift distribution covers $0.1 < z <
0.9$, with 13 objects at $z > 0.5$ and an average redshift of $\bar{z}
= 0.31$. $K$-band images were obtained at the IRTF, Lick, and Keck observatories in sub-arcsec seeing of all optically identified targets. About 2/3 of the objects appear to be interacting galaxies, while the other 1/3 appear to be normal. Nearly all the identified objects have far-IR luminosities greater than $10^{11} L_\odot$, and $\sim$25 % have $L_{FIR} > 10^{12} L_\odot$.
author:
- 'S.A. Stanford'
- Daniel Stern
- Wil van Breugel
- Carlos De Breuck
title: 'The FIRST Sample of Ultraluminous Infrared Galaxies at High Redshift I. Sample and Near-IR Morphologies'
---
Introduction
============
Observations by the Infrared Astronomy Satellite ($IRAS$) led to the discovery of a class of galaxies with enormous far-IR luminosities. Subsequent observations over a large range of wavelengths have shown that these objects, called ULIG for ultraluminous infrared galaxies, have 1) bolometric luminosities and space densities comparable to those of optical quasars (Sanders et al. 1988); 2) a broad range in host galaxy spectral type, including starburst galaxies, Seyfert I and II, radio galaxies, and quasars; 3) morphologies often suggestive of recent interactions or merging (Carico et al. 1990; Leech et al.1994; Rigopoulou et al. 1999); and 4) large amounts of molecular gas concentrated in small ($<$1 kpc) central regions (e.g. Scoville et al. 1989; Solomon et al. 1997). Understanding the nature of the prime energy source in ULIG has proven difficult (e.g. Smith, Lonsdale, & Lonsdale 1998). Many of the observed characteristics indicate that very strong starbursts could be the culprit. Alternatively, an active galactic nucleus (AGN) may power the ULIG (e.g. Lonsdale, Smith, & Lonsdale 1993). The very high luminosities suggest an evolutionary connection between ULIG and quasars, wherein a dust-enshrouded central massive black hole is gradually revealed as the appearance of the object changes from ULIG to quasar (Sanders et al. 1988).
Much effort has been expended in trying to determine the primary source of energy—starbursts or AGN—driving the large FIR luminosities. The recent studies using ISO indicate that the vast majority of the power comes from starbursts in $\sim80\%$ of the observed systems (Genzel et al. 1998; Lutz et al. 1998). Rigopoulou et al. (1999) present the results of an expanded version of the mid-IR spectroscopic survey first reported by Genzel et al. (1998). Using ISO to observe 62 ULIG at $z < 0.3$, they measured the line to continuum ratio of the 7.7 $\mu$m polycyclic aromatic hydrocarbon (PAH) feature to differentiate between starburst and AGN as the dominant source of the large FIR luminosity. PAH features have been shown to be strong in starburst galaxies and weak in AGN (Moorwood 1986; Roche et al. 1991). Rigopoulou et al. confirmed the results of Genzel et al. (1998), and also found, based on near-IR imaging, that approximately 2/3 of their sample have double nuclei and nearly all the objects show signs of interactions. For a recent review of ULIG see Sanders & Mirabel (1996).
ULIG are also of great interest for studies of early star formation in the building of galaxies. Recent sub-mm observations suggest that objects similar to ULIG may contain a significant fraction of the star formation at high redshifts (e.g. Lilly et al. 1999). But so far most studies have found ULIG only in the nearby universe. Sanders et al. (1988) initially studied a group of 10 objects at $z < 0.1$. Previously published systematic surveys have found objects mostly at $z < 0.4$ (Leech et al. 1994; Clements et al. 1996a, 1996b). A few high redshifts objects have been found, all of which turn out to contain hidden AGN. These include FSC 15307+3252 at $z=0.926$ (Cutri et al. 1994) and FSC 10214+4724 at $z=2.286$ (Rowan-Robinson et al.1991). The former object was found to exhibit a highly polarized continuum, indicating the presence of a buried quasar (Hines et al.1995) while the latter was found to be lensed (Eisenhardt et al.1996) and also shows signs of containing a hidden AGN (Lawrence et al. 1993; Elston et al. 1994; Goodrich et al. 1996). Further progress in this field has been hampered by the lack of identified ULIG at moderately high redshifts.
No new deep far-IR survey will become available prior to the launch of [*SIRTF*]{}, which will be capable of studying ULIG in detail at high redshifts. So, the $IRAS$ database remains the primary source of targets for finding high redshift ULIG. Radio observations provide a relatively unbiased method for extracting FIR galaxies from the $IRAS$ Faint Source Catalog (FSC; Moshir et al. 1992) because radio continuum emission is relatively unaffected by extinction in dense gas and dust. Such FIR/radio samples are ideal for detailed investigations of the complex relationships between the interstellar media, starbursts, and possible AGN in ULIG. For example, a sample of radio-loud objects was constructed by cross-correlating the $IRAS$ FSC with the Texas 365 MHz radio catalog (TXFS; Dey & van Breugel 1990). Subsequent optical identifications and spectroscopy showed that the TXFS objects tend to be distant AGN. So a radio-quiet sample, extracted from the FSC, should be an excellent means of finding ULIG without AGN—i.e. powered by starbursts—at interesting cosmological distances. In this paper, we report on such a sample: we describe the sample selection process and discuss the near-IR imaging. We defer a detailed analysis of the radio properties and optical spectroscopy to future papers.
The FIRST/FSC Sample
====================
We have used two large area surveys in the radio and far-IR, which we briefly describe here, to select ULIG candidates. In the radio, we have used the FIRST (Faint Images of the Radio Sky at Twenty cm; Becker, White, & Helfand 1995). Using the VLA, this project is surveying $\pi$ steradians down to a 5$\sigma$ limit of 1 mJy with 5 arcsec resolution and subarcsec positional accuracy. One of the problems with finding distant ULIG using $IRAS$ is that there are many faint galaxies visible in a deep optical image within the relatively large error ellipse of an FIR source. The high resolution and good positional information of FIRST offer an excellent means of choosing the best of the many optical candidates on which to spend valuable large telescope time getting redshifts. We used the second version of the catalog (released 1995 October 16), which samples 2925 degrees$^2$ in two regions of sky in the North ($7^h20^m < $ RA(J2000) $<
17^h20^m$, $22\fdg2 < $ Dec(J2000) $< 42\fdg5$) and South ($21^h20^m <
$ RA(J2000) $< 3^h20^m$, $-2\fdg5 < $ Dec(J2000) $< 1\fdg6$) Galactic Caps. In the far-IR we have used the $IRAS$ FSC (Moshir et al. 1992) which resulted from the Faint Source Survey (FSS). Relative to the $IRAS$ Point Source Catalog, the FSS achieved better sensitivity by point-source filtering the detector data streams and then coadding those data before finding sources. At 60 $\mu$m (the band used for defining our candidates), the FSC covers the sky at $|b|
\gtrsim 10\fdg0$ and has a reliability (integrated over all signal-to-noise ratios) of $\gtrsim 94\%$. The limiting 60 $\mu$m flux density of the FSC is approximately 0.2 Jy, where the signal-to-noise ratio (SNR) is $\sim$5. The FSS also resulted in the Faint Source Reject file which contains extracted sources not in the FSC with an SNR above 3.0. We used the FSR, in addition to the FSC, with part of FIRST to increase the number of targets in the fall sky.
The $IRAS$ FSC was positionally cross-correlated with the second version of the FIRST catalog, with the requirements that an FSC source must have a real 60 $\mu$m detection ($f_{qual} \geq 1$) and that it be within 60 arcsec of the FIRST source. The 60 $\mu$m band was chosen because it is more reliable than the 100 $\mu$m band and samples close to the wavelength peak of the ULIG power. The resulting FIRST-FSC (FF) catalog contains 2328 matches. To increase the available objects in the fall sky, we also performed a positional match of the FSR with the South Galactic Cap portion of FIRST, which yielded an additional 176 matches. The 20 cm and 60 $\mu$m flux densities for this sample of 2504 sources are plotted in Figure \[f20v60\]. The majority of the FF sources fall along the well-known radio-FIR correlation (Condon et al. 1991), extending from nearby starburst galaxies to much fainter FIR/radio flux levels. The surface density of such objects is approximately 1 degree$^{-2}$ down to the $\sim 5 \sigma$ limits of 1 mJy at 20 cm and $\sim$0.2 Jy at 60 $\mu$m.
We generated optical finding charts using the Digitized Sky Surveys, available from the Space Telescope Science Institute, for all 2504 matches. The radio source position and the FSC error ellipse were overlaid on these charts. Visual inspection of these finding charts was carried out to select optically faint targets for further study, with the expectation that such targets would be distant ULIG. Approximately 150 targets, which will carry the designation FF along with the usual coordinate naming scheme, were selected in this manner. A strict cutoff in optical magnitude was not employed, and we make no attempt to construct a sample which has a well-defined limiting magnitude in the optical. In practice, the magnitude of the targets selected for optical imaging and spectroscopy depended on the observing conditions, i.e. some targets which are not visually faint on the DSS image were observed during cloudy conditions. While the FIRST and FSC catalogs do have well-defined flux limits, our sample was not constructed in order to be complete to a chosen flux level in either the radio nor the far-IR bands. The main goal of the survey is simply to find high-redshift ULIG. It is worth noting that our target list would include objects with observed characteristics in the radio, optical, and far-IR similar to those of FSC10214+4724 (which itself lies outside of the FIRST area that we used and so cannot fall into our catalog). We have not found any ULIG at redshifts as great as that of FSC10214+4724 in the $\sim$3000 degree$^2$ surveyed.
Observations
============
During several runs from March 1996 to April 1999, the Kast spectrograph (Miller & Stone 1994) at the Shane 3 m telescope of Lick Observatory was used to obtain optical images and spectroscopy of the candidate ULIG from our FF catalog. The observing procedure typically consisted of taking two 300 s images in the $r_S$ band, identifying the optical counterpart of the FF source in these data, and immediately following up with slit spectroscopy of the optical object. Because the resolution and positional accuracy of FIRST are high, it was usually clear which optical object coincided with the radio source. The FWHM of the seeing in the images was usually in the range 1.5–2.0 arcsec. Standard stars were observed in imaging mode when conditions were photometric. However, because much of the data were obtained during non-photometric conditions, $r_S$ magnitudes will not be presented here for the sample. Unless the source morphology demanded a particular value, the position angle of the slit was set to the parallactic angle. The object was dithered along the slit by $\sim 10$ arcsec between two exposures to aid in fringe subtraction. Optical spectra of 1200-6000s duration were obtained of the optical source using the 300 line mm$^{-1}$ grating in the red-side spectrograph, which provides $\sim$4.6 Å pixel$^{-1}$ resolution from 5070–10590 Å, and a 452/3306 grism in the blue-side spectrograph which provides $\sim$2.5 Å pixel$^{-1}$ resolution from 3000–5900 Å. The slit width was set at 2 arcsec. The images and spectra were reduced using standard techniques.
Near-infrared images were obtained of the targets for which redshifts had been determined in order to better ascertain the morphologies of the galaxies. $K'$ images were obtained for nearly all identified targets with NSFCAM at the IRTF 3 m telescope in 1998 August and 1999 February. Additional observations of 3 targets were obtained in service mode in September 1999. NSFCAM was used in its 0.3 arcsec pixel$^{-1}$ mode which provides a 77$\times$77 arcsec field. Typical total exposure times per object were 960s; more distant objects were observed for twice this period. Conditions were photometric with seeing averaging 0.9 arcsec. Observations of standard stars from the Persson et al. (1998) list were obtained and used to calibrate the images onto the California Institute of Technology (CIT) system, which is defined in Elias et al. (1982). The data were reduced using standard techniques.
Five targets were observed in the $K$ band using Gemini (McLean et al. 1993) at the Shane 3 m telescope on 1998 October 7. Gemini has 0.68 arcsec pixels which give it a 174 arcsec field. Objects were observed for 1080 s each in photometric conditions with seeing of $\sim$1.2 arcsec. The data were reduced using standard techniques and calibrated onto the CIT system using observations of UKIRT faint source standards (Casali & Hawarden 1992).
Two distant targets were imaged at the Keck I telescope with NIRC (Matthews & Soifer 1994) in 1998 April. FF1106+3201 was observed in the $K$ band for 16 minutes and FF1614+3234 was observed for 32 minutes in the $K_s$ band. Both objects were observed in clear conditions with $\sim$0.5 arcsec seeing. These data were reduced using standard techniques and calibrated onto the CIT system using observations of UKIRT faint source standards (Casali & Hawarden 1992).
Results
=======
Optical
-------
We attempted spectroscopic observations of approximately 150 IRAS/FIRST candidates, of which 116 yielded redshift information. The 108 with infrared imaging are listed in Table 1; the 8 sources with redshifts but lacking infrared images are not considered further. The sources which did not provide useful spectra were usually observed in poor conditions; the reasons for their lack of redshifts were not because of having intrinsically challenging spectra. The object names in Table 1 are based on the FIRST radio position. The source in the FIRST catalog would have the name given by the object’s coordinates shown in our Table 1, in the format FIRST J[*hhmmss.s+ddmmss*]{} where the coordinates are truncated, not rounded. In the $IRAS$ FSC, the FIR source name is different from that implied by our FF name, so we have included the FSC source name as a column in Table 1. The Z designation in the FSC name means that the FIR source is from the FSR catalog.
The typical resolution of the spectroscopy was $\approx 15$ Å (FWHM) at $\lambda > 5500$ Å, implying typical uncertainties of $\lesssim 0.002$ in redshift. Redshifts were determined from the spectra after identifying probable emission lines and continuum features. In practice the features most often used were the \[O II\]$\lambda$3727, \[O III\]$\lambda$4959,5007, and H$\alpha$ lines, and the D4000 break. The vast majority of the spectra have the emission lines characteristic of star formation; very few show any signs, such as high ionization lines, of an AGN. Four sample spectra, covering a range in redshift, signal to noise, and spectral type, are shown in Figure \[spex\]. A more detailed analysis of the optical spectra is deferred to a later paper.
Near Infrared
-------------
The $K'$ images are displayed for each object, along with the optical finding chart from the DSS, in Figure \[optir\]. Photometry of the FF objects was obtained from the $K'$ images. In Table 1, the $K$ magnitudes within 3 arcsec diameter apertures, centered on the peak of the near-IR emission, are given for each object. The 5 $\sigma$ detection limit in most of the images is $K \sim 19$ so the limiting factor in the uncertainty of the photometry is not the signal to noise, since most objects have magnitudes some 3–4 magnitudes brighter than the detection limit, but rather systematics in the zeropoint. We estimate that the uncertainty in the zeropoint is $0.03$ mag. For most objects the 3 arcsec diameter aperture contains $\sim 70\%$ of the total light. The morphologies of the objects tend to show signs of galaxy interactions, including tidal tails, multiple nuclei, and disturbed outer envelopes. Approximately 2/3 of the sample show such features, while 1/3 of the sample appear to be normal galaxies. A brief description of the near-IR morphology for each FF is included in Table 1.
Radio
-----
One of the major advantages of using FIRST in our survey is the high accuracy of its positional information. The coordinates listed in Table 1 are those of the radio source as given by the FIRST catalog, which has an absolute astrometric uncertainty of $\sim$1 arcsec. The 20 cm VLA images of all objects listed in Table 1 were extracted from the FIRST database. The radio morphologies were classified by visual inspection of these cutout images, and by consulting the deconvolved sizes listed in the FIRST catalog. The 20 cm morphological information is given for each FF source in Table 1. The 20 cm flux densities listed in Table 1 have typical uncertainties of 10% at the 2 mJy level.
Far Infrared
------------
Improved $IRAS$ flux densities were obtained for all objects in Table 1 with the ADDSCAN utility at IPAC. In addition to the 60 $\mu$m band used to construct our FF catalog, data at 12 $\mu$m, 25 $\mu$m, and 100 $\mu$m was searched for detections. Almost none of the objects in Table 1 were detected at either of the shorter two wavelengths, so no information is included from these wavebands in Table 1. Many detections were obtained in the 100 $\mu$m data; these are included where available in Table 1, and one $\sigma$ upper limits are indicated in parentheses. The uncertainty in the typical 60 $\mu$m measurement in the sample is $\sim 10\%$, and $\sim 15\%$ in the 100 $\mu$m band where detected. The 60 $\mu$m and 100 $\mu$m flux densities were used to calculate the far-infrared luminosity, as defined by Sanders & Mirabel (1996): L(40–500 $\mu$m) $= 4 \pi ~ D^2_L ~ C ~ F_{FIR}
[L_\odot]$, where $D^2_L$ is the luminosity distance in Mpc, $F_{FIR}
= 1.26 \times 10^{-14} (2.58 \times f_{60} + f_{100}) [W m^{-2}]$, and $C = 1.6$. Throughout this paper we use H$_0 = 65$ km s$^{-1}$ Mpc$^{-1}$ and $\Omega_M = 0.3$ with $\Lambda = 0$. When 100 $\mu$m detections could not be obtained with ADDSCAN, the 1$\sigma$ limiting flux densities were used in the calculation of the L$_{FIR}$. The L$_{FIR}$ are given in Table 1, and are plotted in Figure \[p20vlfir\]. The uncertainty in the L$_{FIR}$ is dominated by a combination of the typical $\sim 10-15\%$ flux measurement errors and the $\sim 15\%$ uncertainty in the scaling factor C which accounts for the extrapolated flux longward of the 100 $\mu$m band.
Discussion
==========
The reliability of the optical identification with the radio source for the objects in Table 1 is very high. The optical/radio source association with the FSC far-infrared source is less certain, because of the relatively large positional uncertainty of the $IRAS$ detections. But there are at least four reasons to believe that the identified optical/radio sources are indeed the FSC sources as well. First, in all cases, the FIRST position is within twice the 1 $\sigma$ error ellipse of the FSC source. Second, the optical spectra show emission lines typical of star-forming galaxies, as expected for most far-IR luminous objects. Third, in the cases where both 60 and 100 $\mu$m detections were obtained, the $IRAS$ flux ratios are typical of FIR luminous galaxies (Soifer et al. 1987). Finally, in nearly all cases the FIR/radio flux ratio lies on the well-established correlation as seen in Figure \[f20v60\].
There are at least two possible ways that the wrong association is being made. First, a galaxy optically brighter than the identified FF source could lie just outside the $IRAS$ error ellipse and still be the source of the IRAS detection. But in our sample there are no such galaxies which are also detected by FIRST, as would be expected if such objects were the true source of the FIR emission. Second, a faint radio source could be missed by the FIRST survey that would coincide with the $FSC$ source. Using the rms given in the FIRST catalog for each object, lower limits to the resulting $S_{60\mu{\rm m}}/S_{20cm}$ ratios for objects beneath the $FIRST$ detection limits are found to be $\sim$500-700, far greater than the standard ratio for star-forming galaxies. Finally its worth noting that on average one expects to find only 0.035 FIRST sources within the typical IRAS error ellipse area, so the probability of a random radio source being associated with an FSC source is low. In our sample of 108 identified FF objects, approximately 4 could be due to radio sources unassociated with the far-IR source.
Although we defer the scientific analysis of the new sample to future contributions, we will briefly compare some basic global properties of our FF sample to those of other similarly large samples of high redshift ULIG which have been previously published, e.g. Leech et al. (1994) and Clements et al. (1996). Our new sample has ULIG at higher average redshift ($\bar{z} \sim 0.3$) than that of Leech et al. ($\bar{z} \sim 0.17$) and that of Clements et al. ($\bar{z} \sim
0.21$). As for the interaction rate, on which there has been some disagreement in the literature (Sanders & Mirabel 1996), our result that 2/3 of the sample shows signs of galaxy interactions is in agreement with Leech et al. but not Clements et al., who found that $\sim 90\%$ of their sample were interacting systems. Finally, the rate of AGN-type optical spectra in our sample, which is only $10\%$, is somewhat less than the $\sim 25\%$ found by Clements et al. We do see a higher incidence of AGN-type spectra at the highest L$_{FIR}$ as has been noted previously by several studies; for a review of this topic see Sanders & Mirabel (1996).
Summary
=======
We have constructed a new survey of ULIG using a match of the $IRAS$ FSC with the second version of the FIRST catalog, which covered nearly 3000 degrees$^2$. By choosing for further study only optically faint matches from the DSS which also fall on the radio-FIR flux correlation, we have attempted to find high redshift ULIG which are powered primarily by starbursts. Optical images and spectra were obtained of 108 such targets, which were found to lie in the redshift range $0.1 < z < 0.9$; a redshift histogram is shown in Figure \[zhist\]. Nearly all of these targets have $L_{FIR}$ greater than $10^{11}
L_\odot$, and have a higher average redshift of $\bar{z}=0.31$ than in other recent ULIG surveys. Near-IR imaging shows that while more than the majority of objects show clear signs of galaxy interactions, nearly 1/3 appear to be normal at arcsec resolution in the $K$ band. With this sample, we intend to examine the nature of ULIG evolution in future contributions.
The authors thank the staff of Lick Observatory for their help in obtaining the optical data, and Bill Vacca and Dave Griep at the IRTF for their assistance in obtaining the near-IR data, and for conducting servicing observing for three of the sample objects. We have made use of the online facilities provided by IPAC. We also thank Bob Becker for providing us with the FIRST catalog and for help in using its contents. Finally we thank Rob Kennicutt for a speedy referee report. The Digitized Sky Surveys were produced at the Space Telescope Science Institute under U.S. Government grant NAG-W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the compressed digital form with permission of these institutions. The work by SAS, WvB, and CDB at IGPP/LLNL was performed under the auspices of the U.S. Department of Energy under contract W-7405-ENG-48 to the University of California. The work by DS was supported by IGPP grants 98-AP017 and 99-AP026.
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abstract: 'Generalization of the Itô-Wentzell formula for the generalized Itô’s SDE (Itô’s GSDE) system with a non-centered measure is constructed on the basis of the stochastic kernel of integral transformation. The Itô’s GSDE system for the kernel the solution of which is the kernel of the integral invariant, is formed. This invariant is connected with the solution of the Itô’s GSDE system non-centered measure. The concept of a stochastic first integral of the Itô’s GSDE system with non-centered measure is introduced and conditions that when being performed the random function is the first integral of the set Itô’s GSDE system are defined.'
author:
- |
Elena V. Karachanskaya\
[Pacific National University, Russia, Khabarovsk]{}
title: 'On a generalization of the Itô-Wentzell formula for system of generalized Itô’s SDE and the stochastic first integral'
---
Introduction {#introduction .unnumbered}
============
The specific approach to the stochastic integration based on the fact that not the point but its neighborhoods is reflected according to some quite different from the traditional rule, based on Malliavin’s calculus, was presented in [[[@D_78; @D_89; @D_02]]{}]{}.
The transfer to the stochastic equations for definite realizations in based on comparing the dynamics of the space structures element marked by an integer to the dynamics of the trajectory’s realization for which this integer is the initial value. This possibility is stipulated by the proximity of the trajectories, in the sense of probability, beginning the domain of the initial neighborhood for each of the temporary intervals.
The investigation of open systems that are characterized by the presence of random disturbances in them both continuous described in Wiener’s processes and jump-like ones, presented in Poisson’s processes is urgent nowadays. At the same time in case of random disturbances some characteristics of the system must be presented. And the problems of constructing the program control in the open systems arise. When a disturbance is caused by Wiener processes only there are some types of algorythms for constructing program control. The Poisson component of the random disturbance still not let to do program control for such systems. The present generalization of the Itô-Wentzell formula solves this problem [@11_KchUpr].
The stochastic volume
======================
Let ${\bf x}(t)$ – is the random dynamic process and this process is solution of the next SDE system: $$\label{Ayd01.1}
\begin{array}{c}
dx_{i}(t)= \displaystyle a_{i}\Bigl(t;{\bf x}(t) \Bigr)\, dt
+\sum\limits_{k=1}^{m}
b_{i,k}\Bigl(t;{\bf x}(t)\Bigr)\, dw_{k}(t) +
\int\limits_{\mathbb{R}(\gamma)}g_{i}\Bigl(t;{\bf x}(t);\gamma\Bigr)\nu(dt;d\gamma), \\
{\bf x}(t)={\bf x}\Bigl(t;{\bf x}(0) \Bigr) \Bigr|_{t=0}={\bf
x}(0), \ \ \ \ \ \ i=\overline{1,n}, \ \ \ \ \ \ t\geq 0,
\end{array}$$ where ${\bf w} (t)$ is $m$–dimensional Wiener process, $\nu(t;\Delta\gamma)$ – is homogeneous on $t$ non centered Poisson measure.
We suppose that the coefficients ${ a}(t;{\bf x})$, ${ b}(t;{\bf x})$, and ${g}(t;{\bf x};\gamma)$ are bounded and continuous together with own derivations $
\displaystyle \frac{\partial a_{i}(t;{\bf x})}{\partial x_{j}}$, $\displaystyle \frac{\partial b_{i,k}(t;{\bf x})}{\partial
x_{j}}$, $\displaystyle \frac{\partial g_{i}(t;{\bf
x};\gamma)}{\partial x_{j}}$, $i,j=\overline{1,n}
$ respect to $(t;{\bf x};\gamma)$.
These conditions are sufficient for existing random variables $$\textit{J}_{i,j}(t)= \mathop {\operatorname{l}
.i.m.}\limits_{\left|\Delta \right|= \Delta_{j}\rightarrow
0}\displaystyle \frac{x_{i}\Bigl(t; {\bf x}(0)+\Delta
\Bigr)-x_{i}\Bigl(t; {\bf x}(0)\Bigr)}{\Delta_{j}},$$ that can be defined by solving the system (\[Ayd01.1\]) and the system of stochastic equation [[[@GS_68]]{}]{}: $$\label{Ayd2.2}
\begin{array}{c}
d\textit{J}_{i,j}(t)=\textit{J}_{l,j}(t)\, \biggl[\displaystyle
\frac{\partial a_{i}(t;{\bf x}(t))}{\partial x_{l}(0)}\, dt +
\beta \,\displaystyle \frac{\partial b_{i,k}(t;{\bf
x}(t))}{\partial
x_{l}(0)}\, dw_{k}(t) + \\
+\displaystyle \int\limits_{\mathbb{R}(\gamma)}
\frac{\partial g_{i}(t;{\bf x}(t))}{\partial x_{l}(0)}\nu(dt;d\gamma) \biggr], \\
\textit{J}_{i,j}(t)=\textit{J}_{i,j}\Bigl(t; {\bf x}(0) \Bigr)
\Bigr|_{t=0}=\delta_{i,j}=\left\{
\begin{array}{ll}
0, & i\neq j, \\
1, & i=j.
\end{array}
\right.
\end{array}$$
Let $\textit{J}(t)= \textit{J}\Bigl(t; {\bf x}(0)\Bigr)=\det\Bigl(\textit{J}_{ij}\Bigl(t; {\bf x}(0)\Bigr)\Bigr)$, $i,j=\overline{1,n}$. Then the random variable $\textit{J}(t)$ can be considerer a Jacobian of transition from ${\bf x}(0)$ to ${\bf x}(t;{\bf x}(0))$.
By means of random variables $\textit{J}_{i,j}(t)$ it is possible to construct different random manifolds ${\cal M}\Bigl(t; {\bf x}(t;\lambda ) \Bigr)$ ($\lambda\in {\cal
D} \subset \mathbb{R}^{r}, \ {\bf x}(t
;\lambda )\in\mathbb{R}^{n}, \ r \leq n $), having compared them to configuration manifolds of the initial manifold ${\cal M}\Bigl(0; {\bf x}(0;\lambda ) \Bigr)$ ($\lambda\in {\cal
D} \subset \mathbb{R}^{r}, \ {\bf x}(0;\lambda )={\bf
x}(\lambda )\in\mathbb{R}^{n}, \ r \leq n $).
\[Aydo2\][[[@D_89]]{}]{} Let’s call the structure $
d\Gamma(t)=\textit{J}(t) \,d\Gamma\Bigl({\bf x}(0)\Bigr)
$, where $
d\Gamma({\bf x})= \displaystyle \prod \limits_{i=1}^{n} dx_{i}$, an element of stochastic volume that it generated by some random process ${\bf x}\Bigl(t; {\bf x}(0;
\lambda)\Bigr)$, where $\textit{J}(t)$ is a Jacobian built of elements $\textit{J}_{ij}\Bigl(t; {\bf x}(0)\Bigr)$.
According to definition \[Aydo2\] a random structure is introduced: $$\label{Ayd2.3}
\begin{array}{c}
{\cal I}_{\Gamma(t)}(t)=\underbrace{\int \cdots
\int}\limits_{\Gamma (t)} f(t;{\bf z}) \, d \Gamma ({\bf z})
=\underbrace{\int \cdots \int} \limits_{\Gamma (0)}f\Bigl(t;{\bf
x}(t;{\bf y})\Bigr) \, \textit{J}(t)\, d \Gamma ({\bf y}),
\end{array}$$ where ${\bf x}(t;{\bf y}) $ is a stochastic process with random initial condition ${\bf y}$.
in so doing the integration over a stochastic volume $\Gamma(t)$ is carried out Riemann’s integration sums, mean square limit concept for any continuous functions by ${\bf z}$, and in general, random functions $f(t;{\bf z})$.
Generically, the structure is a random variable. The statement of the classic integral calculus can be associated with random structure . In this sense the equality is an analogue for substituting variables in the classic calculus. Here, a Jacobian of transformation $\textit{J}_{i,j}(t)$ is a random value comparising the solutions of GSDE system (\[Ayd2.2\]).
Let us determine the conditions for the invariance (or conservation) of the stochastic volume.
Stochastic kernel of the stochastic integral invariant
======================================================
Let’s consider the Itô’s GSDE system of the following kind (\[Ayd01.1\]). Let’s assume that $\rho(t;{\bf x};\omega)$ – is a random function which is measured with respect to $\sigma$-algebras flow $\Bigl\{\mathcal{F}\Bigr\}_{0}^{T}$, $\mathcal{F}_{t_{1}}\subset\mathcal{F}_{t_{2}}$, $t_{1}<t_{2}$. This flow is adapted with the processes ${\bf w} (t)$ and $\nu(t;\Delta\gamma)$. Each function $f(t;{\bf x})$ from the set of local bounded functions which have second bounded derivatives with respect to ${\bf x}$ the next relations hold: $$\label{Aydyad1}
\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(t;{\bf x})f(t;{\bf
x})d\Gamma({\bf x})=\int\limits_{\mathbb{R}^{n}}\rho(0;{\bf
y})f(t;{\bf x}(t;{\bf y}))d\Gamma({\bf y})$$ $$\label{Aydyad2}
\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(0;{\bf
x})d\Gamma({\bf x})=1,$$ $$\label{Aydyad3}
\begin{array}{c}
\displaystyle \lim\limits_{|{\bf x}|\to \infty}\rho(0;{\bf x})=0,
\ \ \ d\Gamma({\bf x})=\prod\limits_{i=1}^{n}dx_{i}.
\end{array}$$ Here ${\bf x}(t;{\bf y})$ is the solution of (\[Ayd01.1\]).
If we have $f(t;{\bf x})=1$, then taking into consideration the conditions (\[Aydyad1\]) and (\[Aydyad2\]) we’ll have the equality $$\label{usl-inv}
\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(t;{\bf
x})d\Gamma({\bf x})=\int\limits_{\mathbb{R}^{n}}\rho(0;{\bf
y})d\Gamma({\bf y})=1.$$ It means that there exists a random functional, which retains the constant value for the random function $\rho(t;{\bf x})=\rho(t;{\bf
x};\omega)$: $$\label{ro1}
\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(t;{\bf
x})d\Gamma({\bf x})=1.$$ We consider this functional a stochastic volume. Then the equation under conditions and is a stochastic integral invariant for function $f(t;{\bf x})$.
Let’s call the non-negative function $\rho(t;{\bf x})$ a stochastic kernel or stochastic density of the stochastic integral invariant if equation , and are kept.
But the essential distinction that let to consider the invariance of the random volume on the basis of integral invariant kernel in [[@D_89; @D_02]]{} to that there is a functional factor in . Thus, the notion of integral invariant kernel for the system of the ordinary differential equations can be considered a particular case of the introduced one in [[@D_89; @D_02]]{} if we take $f(t;{\bf x})=1$ and examine the integration by determined volume, having excluded the randomness preset in Wiener and Poisson processes.
Let arbitrary function $f(t;{\bf x})$ has the first derivation with respect to $t$ and the second derivations wish respect to ${\bf x}$. We determine relations for function $\rho(t;{\bf x})$ which implies that $\rho(t;{\bf x})$ is integral invariant kernel for $f(t;{\bf x})$.
Let ${\bf x}(t)$ – is solution of (\[Ayd01.1\]) and $f(t;{\bf x}(t))$ is random function. We apply the generalized Itô’s formula [[[@GS_68]]{}]{}: $$\label{Ayd2}
\begin{array}{c}
\displaystyle d_{t}f(t;{\bf x}(t))=\Bigl[ \frac{\partial f(t;{\bf x}(t)) }{\partial t}+
\sum\limits_{i=1}^{n}a_{i}(t;{\bf x}(t))\frac{\partial f(t;{\bf x}(t)) }{\partial x_{i}}+\Bigr.\\
\Bigl.+ \displaystyle \frac{1}{2} \sum\limits_{i=1}^{n}
\sum\limits_{j=1}^{n}
\sum\limits_{k=1}^{m}b_{i\,k}(t;{\bf x}(t))b_{j\,k}(t;{\bf x}(t))
\frac{\partial^{\,2} f(t;{\bf x}(t)) }{\partial x_{i}x_{j}}\Bigr]dt +\\
+\displaystyle \sum\limits_{i=1}^{n}
\sum\limits_{k=1}^{m}b_{i\,k}(t;{\bf x}(t))\frac{\partial f(t;{\bf x}(t)) }{\partial x_{i}} dw_{k}(t)+\\
+\displaystyle \int\limits_{\mathbb{R}(\gamma)}
\Bigl[ f\left(t;{\bf x}(t)+g(t;{\bf x}(t);\gamma)\right)- f(t;{\bf x}(t))
\Bigr]\nu(dt;d\gamma).
\end{array}$$
Later on for notational simplicity we’ll not use the summation symbol if we have double occurred indexes.
Let’s differentiate both parts of equation wish respect to $t$ taking into account that $f(t;{\bf x})$ of the left part is a determined function while $f(t;{\bf x}(t;{\bf y}))$ of the right part is a random process. $$\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}\Bigl(f(t;{\bf
x})d_{t}\rho(t;{\bf x})+\rho(t;{\bf x})\frac{\partial f(t;{\bf
x})}{\partial t}dt\Bigr)d\Gamma({\bf
x})= \int\limits_{\mathbb{R}^{n}}\rho(0;{\bf y})d_{t}f(t;{\bf x}(t;{\bf
y}))d\Gamma({\bf y}).
\end{array}$$ Then, taking into account both (\[Aydyad1\]) and (\[Ayd2\]) we get: $$\label{Aydyad5}
\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}\Bigl(f(t;{\bf
x})d_{t}\rho(t;{\bf x})+\rho(t;{\bf x})\frac{\partial f(t;{\bf
x})}{\partial t}dt\Bigr)d\Gamma({\bf
x})=\\
=\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(0;{\bf y})d_{t}
f(t;{\bf x}(t;{\bf y}))d\Gamma({\bf y})
=\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(t;{\bf
x})d_{t}f(t;{\bf
x})d\Gamma({\bf x})= \\
=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})\cdot \biggl[ \Bigl[\frac{\partial f(t;{\bf
x}(t)) }{\partial t}+
a_{i}(t;{\bf x})\frac{\partial f(t;{\bf x}) }{\partial x_{i}}+\Bigr.\biggr.\\
\Bigl.+ \displaystyle \frac{1}{2} b_{i\,k}(t;{\bf
x})b_{j\,k}(t;{\bf x})
\frac{\partial^{\,2} f(t;{\bf x}) }{\partial x_{i} \partial x_{j}}\Bigr]dt
+\displaystyle
b_{i\,k}(t;{\bf x})\frac{\partial f(t;{\bf x}) }{\partial x_{i}} dw_{k}(t)+\\
\biggl. +\displaystyle \int\limits_{\mathbb{R}(\gamma)}
\Bigl[ f\left(t;{\bf x}+g(t;{\bf x};\gamma)\right)- f(t;{\bf x})
\Bigr]\nu(dt;d\gamma)\biggr].
\end{array}$$ Let’s consider the integral $$\label{Aydy1}
I=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})
f\left(t;{\bf x}+g(t;{\bf x};\gamma)\right).$$ We change variables: $$\label{Aydx1y}
{\bf x}+g(t;{\bf x};\gamma)={\bf y},$$ Having taken into account the transition from ${\bf x}$ to ${\bf y}$, we get $
{\bf x}={\bf y}-g(t;{\bf x};\gamma)={\bf y}-g(t;{\bf
x}^{-1}(t;{\bf y};\gamma);\gamma)
$. And the integral $I$ becomes: $$\label{Aydy2}
\begin{array}{c}
I=\displaystyle \int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
y})\rho\left(t;{\bf y}-g(t;{\bf x}^{-1}(t;{\bf
y};\gamma);\gamma)\right)
\cdot f(t;{\bf y})\cdot D\left( {\bf x}^{-1}(t;{\bf y};\gamma) \right),
\end{array}$$ where $D\left( {\bf x}^{-1}(t;{\bf y};\gamma)\right) $ is a Jacobian of transition from ${\bf x}$ to ${\bf y}$.
Taking into accout (\[Aydy2\]) and integration over the $\mathbb{R}^{n}$ we have: $$\label{Aydy3}
\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})
\displaystyle \int\limits_{\mathbb{R}(\gamma)}
\Bigl[ f\left(t;{\bf x}+g(t;{\bf x};\gamma)\right)- f(t;{\bf x})
\Bigr]\nu(dt;d\gamma)= \\
=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})
\displaystyle \int\limits_{\mathbb{R}(\gamma)}
f\left(t;{\bf x}+g(t;{\bf
x};\gamma)\right)\nu(dt;d\gamma)-\\
-\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})
\displaystyle \int\limits_{\mathbb{R}(\gamma)} f(t;{\bf
x})
\nu(dt;d\gamma)= \\
=\displaystyle \int\limits_{\mathbb{R}^{n}}d\Gamma({\bf x})
\int\limits_{\mathbb{R}(\gamma)} \Bigl(\Bigr.\rho\left(t;{\bf x}-g(t;{\bf x}^{-1}(t;{\bf
x};\gamma);\gamma)\right) \cdot\\
\cdot
f(t;{\bf x})\cdot D\left( {\bf x}^{-1}(t;{\bf x};\gamma) \right)\Bigl.\Bigr) \nu(dt;d\gamma)-\\
-\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})
\displaystyle \int\limits_{\mathbb{R}(\gamma)} f(t;{\bf x}) \nu(dt;d\gamma)=\\
=\displaystyle \int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})f(t;{\bf x})\int\limits_{\mathbb{R}(\gamma)}
\Bigl[\rho\left(t;{\bf x}-g(t;{\bf x}^{-1}(t;{\bf
x};\gamma);\gamma)\right)
\cdot \Bigr.\\
\cdot\Bigl. D\left( {\bf x}^{-1}(t;{\bf x};\gamma) \right)-\rho(t;{\bf
x})\Bigr]\nu(dt;d\gamma).
\end{array}$$ With due regard for (\[Aydyad2\]) we calculate the following integrals using integration by parts: $$\label{Aydi01}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})a_{i}(t;{\bf x})\frac{\partial f(t;{\bf x})
}{\partial x_{i}},$$ $$\label{Aydi02}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})b_{i\,k}(t;{\bf x})\frac{\partial f(t;{\bf x})
}{\partial x_{i}},$$ $$\label{Aydi03}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})
\frac{\partial^{\,2} f(t;{\bf x}) }{\partial x_{i} \partial x_{j}}.$$ For equation (\[Aydi01\]) we get: $$\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})a_{i}(t;{\bf x})\frac{\partial f(t;{\bf x})
}{\partial x_{i}}
=\displaystyle\int\limits_{-\infty}^{+\infty}\prod\limits_{j=1}^{n}dx_{j}\rho(t;{\bf
x})a_{i}(t;{\bf x})\frac{\partial f(t;{\bf x}) }{\partial x_{i}}= \\
=\displaystyle\int\limits_{-\infty}^{+\infty}\prod\limits_{j=1}^{n-1}dx_{j}
\int\limits_{-\infty}^{+\infty}\rho(t;{\bf
x})a_{i}(t;{\bf x})\frac{\partial f(t;{\bf x}) }{\partial
x_{i}}dx_{i}.
\end{array}$$ Then we calculate the inner integral using integration in parts, having taken into account (\[Aydyad3\]): $$\begin{array}{c}
\displaystyle\int\limits_{-\infty}^{+\infty}\rho(t;{\bf
x})a_{i}(t;{\bf x})\frac{\partial f(t;{\bf x}) }{\partial
x_{i}}dx_{i}=\\
=\rho(t;{\bf x})a_{i}(t;{\bf x})f(t;{\bf
x})\biggl|_{-\infty}^{+\infty}-\displaystyle\int\limits_{-\infty}^{+\infty}f(t;{\bf
x})\displaystyle\frac{\partial \left( \rho(t;{\bf x})a_{i}(t;{\bf
x})\right) }{\partial
x_{i}}dx_{i} =\\
=-\displaystyle\int\limits_{-\infty}^{+\infty}f(t;{\bf
x})\displaystyle\frac{\partial \left( \rho(t;{\bf x})a_{i}(t;{\bf
x})\right) }{\partial x_{i}}dx_{i} .
\end{array}$$ Therefore, we get $$\label{Aydi10}
\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})a_{i}(t;{\bf x})\frac{\partial f(t;{\bf x})
}{\partial x_{i}}
=\displaystyle\int\limits_{-\infty}^{+\infty}\prod\limits_{j=1}^{n}dx_{j}\rho(t;{\bf
x})a_{i}(t;{\bf x})\frac{\partial f(t;{\bf x}) }{\partial x_{i}}= \\
=\displaystyle\int\limits_{-\infty}^{+\infty}\prod\limits_{j=1}^{n-1}dx_{j}
\int\limits_{-\infty}^{+\infty}\rho(t;{\bf
x})a_{i}(t;{\bf x})\frac{\partial f(t;{\bf x}) }{\partial
x_{i}}dx_{i}=\\
=-\displaystyle\int\limits_{-\infty}^{+\infty}\prod\limits_{j=1}^{n}dx_{j}
\displaystyle\int\limits_{-\infty}^{+\infty}f(t;{\bf
x})\displaystyle\frac{\partial \left( \rho(t;{\bf x})a_{i}(t;{\bf
x})\right) }{\partial
x_{i}}dx_{i} =\\
=
-\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})f(t;{\bf x})\displaystyle\frac{\partial \left( \rho(t;{\bf
x})a_{i}(t;{\bf x})\right) }{\partial x_{i}}
\end{array}$$
Similarly we calculate the second integral (\[Aydi02\]) and the third integral (\[Aydi03\]) by using twice the integration in parts: $$\label{Aydi20}
\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})b_{i\,k}(t;{\bf x})\frac{\partial f(t;{\bf x})
}{\partial x_{i}}
=-\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})f(t;{\bf x})\frac{\partial\rho(t;{\bf x})b_{i\,k}(t;{\bf x})
}{\partial x_{i}},
\end{array}$$ $$\label{Aydi30}
\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})
\frac{\partial^{\,2} f(t;{\bf x}) }{\partial x_{i} \partial x_{j}}=\\
=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf x})f(t;{\bf
x})
\frac{\partial^{\,2} \left(\rho(t;{\bf x})b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})\right) }{\partial x_{i} \partial
x_{j}}.
\end{array}$$
In equation (\[Aydyad5\]) we move all terms to the right part and taking into account (\[Aydy3\]), (\[Aydi10\]), (\[Aydi20\]) and (\[Aydi30\]) we obtain the following equation: $$\label{Aydi}
\begin{array}{c}
0= \displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf x})f(t;{\bf
x})\cdot \biggl[-d_{t}\left(\rho(t;{\bf x})\right)-\frac{\partial
f(t;{\bf x}(t)) }{\partial t}+\biggr.\\
+ \displaystyle\frac{\partial f(t;{\bf x}(t)) }{\partial
t}-\frac{\partial\rho(t;{\bf x})b_{i\,k}(t;{\bf x}) }{\partial
x_{i}}dw_{k}(t)+\\
+\Bigl(\Bigr.-\displaystyle\frac{\partial \left( \rho(t;{\bf
x})a_{i}(t;{\bf x}(t))\right) }{\partial x_{i}}
+\displaystyle\frac{1}{2}\frac{\partial^{\,2} \left(\rho(t;{\bf
x})b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})\right) }{\partial x_{i} \partial x_{j}}\Bigl.\Bigr)dt + \\
\biggl.+\displaystyle\int\limits_{\mathbb{R}(\gamma)} \Bigl[\rho\left(t;{\bf
x}-g(t;{\bf x}^{-1}(t;{\bf x};\gamma);\gamma)\right)
\cdot D\left( {\bf x}^{-1}(t;{\bf x};\gamma) \right)-\rho(t;{\bf
x})\Bigr]\nu(dt;d\gamma)\biggr].
\end{array}$$ To make the equation (\[Aydi\]) true for any locally bounded function $f(t;{\bf x})$ obtaining the limited second-order derivatives the integral invariant $\rho(t;{\bf x})$ must be the solution for the following stochastic equation: $$\label{Aydii}
\begin{array}{c}
d_{t}\rho(t;{\bf x})=
-\displaystyle\frac{\partial\rho(t;{\bf x})b_{i\,k}(t;{\bf x})
}{\partial x_{i}}dw_{k}(t) +\Bigl(-\displaystyle\frac{\partial
\left( \rho(t;{\bf x})a_{i}(t;{\bf x})\right) }{\partial
x_{i}}+\\
+\displaystyle\frac{1}{2}\frac{\partial^{\,2} \left(\rho(t;{\bf
x})b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})\right) }{\partial x_{i} \partial x_{j}}\Bigr)dt + \\
+\displaystyle\int\limits_{\mathbb{R}(\gamma)} \Bigl[\rho\left(t;{\bf
x}-g(t;{\bf x}^{-1}(t;{\bf x};\gamma);\gamma)\right)
\cdot D\left( {\bf x}^{-1}(t;{\bf x};\gamma) \right)-\rho(t;{\bf
x})\Bigr]\nu(dt;d\gamma).
\end{array}$$ The following terms must be kept: $$\label{Aydii1}
\begin{array}{c}
\rho(t;{\bf x})\Bigl|_{t=0}=\rho(0;{\bf x})\in C_{0}^{2}, \\
\lim\limits_{|{\bf x}|\to \infty}\rho(0;{\bf x})=0, \ \ \ \
\displaystyle\lim\limits_{|{\bf x}|\to
\infty}\frac{\partial\rho(0;{\bf x})}{\partial x_{i}}=0.
\end{array}$$ Thus, we get the invariance conditions of a stochastic volume and the following theorem is proved.
\[thro1\] Let ${\bf x}(t)$, ${\bf x}\in\mathbb{R}^{n}$ – is the solution of Itô’s GSDE system: $$\begin{array}{c}
dx_{i}(t)= \displaystyle a_{i}\Bigl(t;{\bf x}(t) \Bigr)\, dt
+\sum\limits_{k=1}^{m}
b_{i,k}\Bigl(t;{\bf x}(t)\Bigr)\, dw_{k}(t) +
\int\limits_{\mathbb{R}(\gamma)}g_{i}\Bigl(t;{\bf x}(t);\gamma\Bigr)\nu(dt;d\gamma), \\
{\bf x}(t)={\bf x}\Bigl(t;{\bf x}(0) \Bigr) \Bigr|_{t=0}={\bf
x}(0), \ \ \ \ \ \ i=\overline{1,n}, \ \ \ \ \ \ t\geq 0,
\end{array}$$ where ${\bf w} (t)$ – is $m$-dimentional Wiener process, $\nu(t;\Delta\gamma)$ – is homogeneous on $t$ non centered Poisson measure. Assume that $\rho(t;{\bf x})$ is a random function which is measurable function with respect to $\sigma$-algebras flow $\Bigl\{\mathcal{F}\Bigr\}_{0}^{T}$, $\mathcal{F}_{t_{1}}\subset\mathcal{F}_{t_{2}}$, $t_{1}<t_{2}$, which adapting ${\bf w} (t)$ and $\nu(t;\Delta\gamma)$ processes and with respect to any function $f(t;{\bf x})\in\mathfrak{S}$ from the set of locally bounded functions having the second bounded derivatives in ${\bf x}$. The function $\rho(t;{\bf x})$ is the stochastic kernel of the stochastic integral invariant for arbitrary locally bounded function $f(t;{\bf x})\in\mathfrak{S}$ if it’s a solution of Itô’s GSDE system meeting the initial conditions .
The generalization of Itô-Wentzell formula
==========================================
The equation (\[Aydyad1\]) under conditions (\[Aydyad2\]), and equation (\[Aydii\]) for the integral invariant kernel conformed to a determinate function $f(t;{\bf x})$. Let’s assume, that analogical equation for random function ${\bf
z}(t;{\bf x})$ holds: $$\label{Aydz3p}
\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(t;{\bf x}){\bf z}(t;{\bf
x})d\Gamma({\bf x})=
\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(0;{\bf y}){\bf
z}(t;{\bf x}(t;{\bf y}))d\Gamma({\bf y}),
\end{array}$$ here ${\bf x}(t;{\bf y})$ is a solution of (\[Ayd01.1\]).
Let’s consider a compound stochastic process ${\bf z}\left(t;{\bf x}(t;{\bf
y})\right)\in \mathbb{R}^{n_{o}}$, where ${\bf x}(t;{\bf y})$ – is a solution of (\[Ayd01.1\]) where the process ${\bf z}(t;{\bf
x})$ – is a solution of the system: $$\label{Aydz1}
\begin{array}{c}
d_{t}{\bf z}(t;{\bf x})=\Pi(t;{\bf x})dt+D_{k}(t;{\bf x})dw_{k}(t)
+
\displaystyle\int\limits_{R(\gamma)}\nu(dt;d\gamma)G(t;{\bf x};\gamma).
\end{array}$$
As far as the random functions $\Pi(t;{\bf x})$, $D_{k}(t;{\bf
x})$, $G(t;{\bf x};\gamma)$ defined on $\mathbb{R}^{n_{o}}$ are concerned we can assume that they are continuous and bounded together with their own derivatives with respect to all variables. These functions are measurable with respect to $\sigma$-algebras flow $\mathcal{F}_{t}$, which adapted with processes ${\bf w}(t)$ and $\nu(t;\Delta\gamma)$ from equation (\[Ayd01.1\]).
Relying on the equation for the stochastic integral invariant. We can construct the differentiation law for the compound stochastic process ${\bf z}\left(t;{\bf x}(t;{\bf
y})\right)$.
Let us consider the integral $
\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(t;{\bf x}){\bf
z}(t;{\bf x})d\Gamma({\bf x})
$. As we do integration over the $\mathbb{R}^{n}$ and ${\bf x}\in \mathbb{R}^{n}$, we use equation (\[Aydz3p\]) and having changed the places of the equation, we get : $$\label{Aydz3}
\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(0;{\bf y}){\bf
z}(t;{\bf x}(t;{\bf y}))d\Gamma({\bf y})=
\int\limits_{\mathbb{R}^{n}}\rho(t;{\bf x}){\bf z}(t;{\bf
x})d\Gamma({\bf x}).
\end{array}$$ Lrt’s differentiate both parts of (\[Aydz3\]) with respect to $t$: $$\label{Aydz4}
\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(0;{\bf y})d_{t}{\bf
z}(t;{\bf x}(t;{\bf y}))d\Gamma({\bf
y})= \\
\displaystyle\int\limits_{\mathbb{R}^{n}}\Bigl(\Bigr.\rho(t;{\bf x})d_{t}{\bf
z}(t;{\bf x})+{\bf z}(t;{\bf x})d_{t}\rho(t;{\bf x}) -
D_{k}(t;{\bf x}) \displaystyle\frac{\partial\rho(t;{\bf
x})b_{i\,k}(t;{\bf x}) }{\partial x_{i}}dt\Bigl.\Bigr)d\Gamma({\bf
x}) ,
\end{array}$$
Taking into account that integration is carried out over the $\mathbb{R}^{n}$ and the variable is changed we should keep in ind this substitution (\[Aydx1y\]) when integration over the curve $R(\gamma)$.
As the function $\rho(t;{\bf x})$ – is the integral invariant kernel we apply the theorem \[thro1\]. Substitute (\[Aydii\]) and (\[Aydz1\]) to the right part of (\[Aydz4\]): $$I_{1}= \displaystyle \int\limits_{\mathbb{R}^{n}}\Bigl(\rho(t;{\bf
x})d_{t}{\bf z}(t;{\bf x})+{\bf z}(t;{\bf x})d_{t}\rho(t;{\bf x})
-D_{k}(t;{\bf x}) \displaystyle\frac{\partial\rho(t;{\bf
x})b_{i\,k}(t;{\bf x})
}{\partial x_{i}}dt\Bigr)d\Gamma({\bf x})=$$ $$= \displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf x})\biggl\{\rho(t;{\bf x})
\Bigl[\Bigr.\Pi(t;{\bf x})dt+D_{k}(t;{\bf x})dw_{k}(t)
+$$ $$+\displaystyle\int\limits_{R(\gamma)}G(t;{\bf x}+g(t;{\bf
x}^{-1}(t;{\bf
x};\gamma);\gamma)\nu(dt;d\gamma)\Bigl.\Bigr]+$$ $$+{\bf z}(t;{\bf x})\biggl[-\displaystyle\frac{\partial\rho(t;{\bf
x})b_{i\,k}(t;{\bf x}) }{\partial
x_{i}}dw_{k}(t)+$$ $$+\Bigr.\displaystyle\Bigl(\Bigr.-\displaystyle\frac{\partial
\left( \rho(t;{\bf x})a_{i}(t;{\bf x})\right) }{\partial
x_{i}}+
\displaystyle\frac{1}{2}\frac{\partial^{\,2} \left(\rho(t;{\bf
x})b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})\right) }{\partial x_{i} \partial x_{j}}\Bigl.\Bigr)dt +$$ $$+\displaystyle\int\limits_{\mathbb{R}(\gamma)} \Bigl[\Bigr.\rho\left(t;{\bf
x}-g(t;{\bf x}^{-1}(t;{\bf x};\gamma);\gamma)\right)
\cdot D\left( {\bf x}^{-1}(t;{\bf x};\gamma) \right)-\rho(t;{\bf
x})\Bigl.\Bigr]\nu(dt;d\gamma)\Bigl.\Bigl.\biggr]-$$ $$-D_{k}(t;{\bf
x}) \displaystyle\frac{\partial\rho(t;{\bf x})b_{i\,k}(t;{\bf x})
}{\partial x_{i}}dt\biggr\}.$$ Collect similar terms: $$\label{Aydz5}
\begin{array}{c}
I_{1}=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\biggl(\rho(t;{\bf x})\Pi(t;{\bf x}) -{\bf z}(t;{\bf
x})\displaystyle\frac{\partial \left( \rho(t;{\bf x})a_{i}(t;{\bf
x})\right) }{\partial x_{i}}+\biggr.\\
-D_{k}(t;{\bf x}) \displaystyle\frac{\partial\rho(t;{\bf
x})b_{i\,k}(t;{\bf x}) }{\partial x_{i}}
+\displaystyle\frac{1}{2}{\bf z}(t;{\bf x})\frac{\partial^{\,2}
\left(\rho(t;{\bf x})b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})\right)
}{\partial x_{i}
\partial x_{j}}\biggl.\biggr)dt+\\
+\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\biggl(\rho(t;{\bf x})D_{k}(t;{\bf x}) -{\bf z}(t;{\bf
x})\displaystyle\frac{\partial \left( \rho(t;{\bf
x})b_{i\,k}(t;{\bf x})\right) }{\partial
x_{i}}\biggr)dw_{k}(t)+\\
+\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\biggl[\rho(t;{\bf x}) \cdot\int\limits_{R(\gamma)}G(t;{\bf
x}+g(t;{\bf x}^{-1}(t;{\bf
x};\gamma);\gamma))\nu(dt;d\gamma) +\Bigr.\\
+{\bf z}(t;{\bf
x})\displaystyle\int\limits_{R(\gamma)}\Bigl[\rho\left(t;{\bf
x}-g(t;{\bf x}^{-1}(t;{\bf x};\gamma);\gamma)\right)
\cdot D\left( {\bf x}^{-1}(t;{\bf x};\gamma) \right)-\rho(t;{\bf
x})\Bigr]\nu(dt;d\gamma)\Bigl.\biggr].
\end{array}$$ And because of (\[Aydi10\]), (\[Aydi20\]), (\[Aydi30\]) we have: $$\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf x}){\bf
z}(t;{\bf x})\frac{\partial \left(\rho(t;{\bf x})a_{i}(t;{\bf
x})\right) }{\partial x_{i}} =
-\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})a_{i}(t;{\bf x})\displaystyle\frac{\partial
{\bf z}(t;{\bf x})
}{\partial x_{i}},
\end{array}$$ $$\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})D_{k}(t;{\bf x})\frac{\partial\left(\rho(t;{\bf
x})b_{i\,k}(t;{\bf x})\right) }{\partial x_{i}}=
-\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})b_{i\,k}(t;{\bf x})\frac{\partial D_{k}(t;{\bf
x}) }{\partial x_{i}},
\end{array}$$ $$\begin{array}{c}
\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf x}){\bf
z}(t;{\bf x})
\frac{\partial^{\,2} \left(\rho(t;{\bf x})b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})\right) }{\partial x_{i} \partial
x_{j}}=
\\
= \displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})
\frac{\partial^{\,2} {\bf z}(t;{\bf x}) }{\partial x_{i} \partial
x_{j}}.
\end{array}$$ Calculate the last integral in (\[Aydz5\]): $$\label{Ayde6}
\begin{array}{c}
I_{2}=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x}){\bf z}(t;{\bf x})
\cdot\rho\Bigl(t;{\bf x}-g(t;{\bf x}^{-1}(t;{\bf
x};\gamma);\gamma)\Bigr)D({\bf x}^{-1}(t;{\bf x};\gamma)).
\end{array}$$ Let’s introduce the variables’ substitution: $$\begin{array}{c}
{\bf x}-g(t;{\bf x}^{-1}(t;{\bf x};\gamma);\gamma)={\bf y}, \\
{\bf x}={\bf y}+g(t;{\bf x}^{-1}(t;{\bf x};\gamma);\gamma)={\bf y}+g(t;{\bf
y};\gamma).
\end{array}$$ Let $D_{o}({\bf x}^{-1}(t;{\bf y};\gamma))$ be a Jacobian of transformation from the vector ${\bf x}$ to the vector ${\bf
y}$. Then, by virtue of (\[Aydy1\]) and further formal substitution of the integration variable’s notation, we get: $$\label{Ayde7}
\begin{array}{c}
I_{2}=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf y}){\bf
z}\Bigl(t;{\bf y}+g(t;{\bf
y};\gamma)\Bigr)
\cdot\rho(t;{\bf y})D_{o}({\bf x}^{-1}(t;{\bf y};\gamma))D({\bf
x}^{-1}(t;{\bf x};\gamma))= \\
=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf y}){\bf
z}\Bigl(t;{\bf y}+g(t;{\bf
y};\gamma)\Bigr)\rho(t;{\bf y})
=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf x}){\bf
z}\Bigl(t;{\bf x}+g(t;{\bf
x};\gamma)\Bigr)\rho(t;{\bf x}).
\end{array}$$ As a result, the right part of (\[Aydz4\]) becomes: $$\label{Ayde8}
\begin{array}{c}
I_{1}=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
x})\rho(t;{\bf x})\biggl\{\Bigl( D_{k}(t;{\bf x}) +b_{i\,k}(t;{\bf
x})\frac{\partial {\bf z}(t;{\bf x}) }{\partial x_{i}}
\Bigr)dw_{k}(t)+\biggr.\\+
\Bigl(\Pi(t;{\bf x})+a_{i}(t;{\bf
x})\displaystyle\frac{\partial
{\bf z}(t;{\bf x})
}{\partial x_{i}}+ b_{i\,k}(t;{\bf
x})\frac{\partial
D_{k}(t;{\bf x})
}{\partial x_{i}}+\\
+\displaystyle\frac{1}{2}b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})
\frac{\partial^{\,2} {\bf z}(t;{\bf x}) }{\partial x_{i} \partial
x_{j}}\Bigr)dt+\\
+\displaystyle\int\limits_{R(\gamma)}G(t;{\bf x}+g(t;{\bf
x}^{-1}(t;{\bf x};\gamma);\gamma)\nu(dt;d\gamma)+\\
+
\displaystyle\int\limits_{R(\gamma)}\Bigl[{\bf z}\Bigl(t;{\bf x}+g(t;{\bf
x};\gamma)\Bigr)
-{\bf z}(t;{\bf
x})\Bigr]\nu(dt;d\gamma)\biggl.\biggr\}.
\end{array}$$ In equation (\[Aydz4\]) we transfer all the terms to the right part. And having taken (\[Aydyad1\]) into account, we have: $$\begin{array}{c}
0=\displaystyle\int\limits_{\mathbb{R}^{n}}d\Gamma({\bf
y})\rho(0;{\bf y})\biggl\{-d_{t}{\bf z}(t;{\bf x}(t;{\bf
y}))+\\
+\biggr.\Bigl( D_{k}(t;{\bf x}(t;{\bf y}))+b_{i\,k}(t;{\bf x}(t;{\bf
y}))\displaystyle\frac{\partial {\bf z}(t;{\bf
x}) }{\partial x_{i}} \Bigr)dw_{k}(t)+\\
+ \Bigl(\Pi(t;{\bf x}(t;{\bf y}))+a_{i}(t;{\bf
x})\displaystyle\frac{\partial
{\bf z}(t;{\bf x})
}{\partial x_{i}}
+b_{i\,k}(t;{\bf
x})\frac{\partial
D_{k}(t;{\bf x})
}{\partial x_{i}}+\Bigr.\\
+\Bigl.\displaystyle\frac{1}{2}b_{i\,k}(t;{\bf x}(t;{\bf y}))b_{j\,k}(t;{\bf x}(t;{\bf y}))
\frac{\partial^{\,2} {\bf z}(t;{\bf x}(t;{\bf y})) }{\partial x_{i} \partial
x_{j}}\Bigr)dt+\\
+\displaystyle\int\limits_{R(\gamma)}G(t;{\bf x}(t;{\bf y})+g(t;{\bf
x}(t;{\bf y});\gamma)\nu(dt;d\gamma)+\\
+
\displaystyle\int\limits_{R(\gamma)}\Bigl[{\bf z}\Bigl(t;{\bf x}(t;{\bf y})+g(t;{\bf
x}(t;{\bf y});\gamma)\Bigr)-
{\bf z}(t;{\bf
x}(t;{\bf y}))\Bigr]\nu(dt;d\gamma)\biggl.\biggr\}.
\end{array}$$ Therefore, we’ll have the differentiation law for the compound stochastic process as the theorem.
\[thro2\] Let ${\bf x}(t;{\bf y})\in\mathbb{R}^{n} $ is the solution of SDE system , ${\bf z}(t;{\bf
x})$ is the solution of SDE system and ${\bf z}\left(t;{\bf x}(t;{\bf
y})\right)\in \mathbb{R}^{n_{o}}$ is the random process. The random functions that are coefficients of equations and , which are defined on $\mathbb{R}^{n_{o}}$ and $\mathbb{R}^{n}$ respectively. This functions are continuous and bounded together with their own derivatives respect to all variables. Also these functions are measurable respect to $\sigma$-algebras flow $\mathcal{F}_{t}$, which is adapted with processes ${\bf w}(t)$ and $\nu(t;\Delta\gamma)$ from equation . Then the compound stochastic process ${\bf z}\left(t;{\bf x}(t;{\bf
y})\right)$ is the solution of SDE system: $$\label{Ayde9}
\begin{array}{c}
d_{t}{\bf z}(t;{\bf x}(t;{\bf y}))=\displaystyle\Bigl(
D_{k}(t;{\bf x}(t;{\bf y})) +b_{i\,k}(t;{\bf x}(t;{\bf
y}))\frac{\partial {\bf z}(t;{\bf
x}(t;{\bf y})) }{\partial x_{i}} \Bigr)dw_{k}(t)+\\
+\Bigl(\Pi(t;{\bf x}(t;{\bf y}))+a_{i}(t;{\bf
x}(t;{\bf y}))\displaystyle\frac{\partial
{\bf z}(t;{\bf x}(t;{\bf y}))
}{\partial x_{i}}
+b_{i\,k}(t;{\bf
x}(t;{\bf y}))\displaystyle\frac{\partial
D_{k}(t;{\bf x}(t;{\bf y}))
}{\partial x_{i}}+\Bigr.\\
+\Bigl.\displaystyle\frac{1}{2}b_{i\,k}(t;{\bf x}(t;{\bf y}))b_{j\,k}(t;{\bf x}(t;{\bf y}))
\frac{\partial^{\,2} {\bf z}(t;{\bf x}(t;{\bf y})) }{\partial x_{i} \partial
x_{j}}\Bigr)dt+\\
+\displaystyle\int\limits_{R(\gamma)}G(t;{\bf x}(t;{\bf y})+g(t;{\bf
x}(t;{\bf y});\gamma)\nu(dt;d\gamma)+\\
+
\displaystyle\int\limits_{R(\gamma)}\Bigl[{\bf z}\Bigl(t;{\bf x}(t;{\bf y})+g(t;{\bf
x}(t;{\bf y});\gamma)\Bigr)
-{\bf z}(t;{\bf
x}(t;{\bf y}))\Bigr]\nu(dt;d\gamma).
\end{array}$$
On the analogy of the well-known terminology let’s call the formula (\[Ayde9\]) as generalized Itô-Wentzell formula for the generalized SDE Itô of non-centered Poisson measure. In addition we can settle one more statement.
\[thro3\] If ${\bf z}(t;{\bf x}(t;{\bf y}))$ – is the solution for equation meeting the initial condition $${\bf z}(t;{\bf x}(t;{\bf y}))\Bigr|_{t=0}={\bf z}(0;{\bf y}), \ \
\ {\bf z}(0;{\bf y})\in C_{o}^{2},$$ then the random function $\rho(t;{\bf x})$, which is a function that measurable with respect to $\sigma$-algebras flow $\Bigl\{\mathcal{F}\Bigr\}_{0}^{T}$, $\mathcal{F}_{t_{1}}\subset\mathcal{F}_{t_{2}}$, $t_{1}<t_{2}$, that is adapted with processes ${\bf w} (t)$ and $\nu(t;\Delta\gamma)$, is the stochastic kernel of the stochastic integral invariant .
The stochastic first integral {#PintSt}
=============================
In the article [@D_78] the concept of the first integral for Itô’s SDE system (without Poisson part) has been introduced. In the article [@D_02] the concept of the stochastic first integral for Itô’s GSDE system with centered measure has been given. Let’s introduce the similar concept for the non-centered Poisson measure.
\[Ayddf1\] Let random function $u(t;{\bf x};\omega)$ and solution ${\bf x}(t)$ of the Itô’s GSDE system are defined on the same probability space. The function $u(t;{\bf x};\omega)$ is called the stochastic first integral for Itô’s GSDE system with non centered Poissin measure, if condition $$u\Bigl(t;{\bf x}(t; {\bf x}(0));\omega\Bigr)=u\Bigl(0;{\bf
x}(0);\omega\Bigr)$$ is held with probability is equaled to 1 for each solution ${\bf x}(t;{\bf x}(0);\omega)$ of the system .
We set conditions for the function $u(t;{\bf x};\omega)$ which imply that $u(t;{\bf x};\omega)$ is the stochastic first integral for Itô’s GSDE system .
\[l1\] If the function $\rho(t;{\bf x})$ is the stochastic kernel of the integral invariant of order $n$ of the stochastic process ${\bf x}(t)$ outgoing from a random point ${\bf x}(0)={\bf y}$, then for any $t$ it complies with the equation $\rho\Bigl(t;{\bf x}(t;{\bf y})\Bigr)\textit{J}(t;{\bf y})=\rho(0;{\bf y})$, where $\textit{J}(t;{\bf y})=\textit{J}(t;{\bf x}(0))$ is a transition Jacobian from ${\bf x}(t)$ to ${\bf x}(0)={\bf y}$.
Let’s proceed to the equation . If we substitute the variables of the integral we’ll get the integration is fulfilled according one and the same random volume: $$1=\int\limits_{\mathbb{R}^{n}}\rho(0;{\bf
y})d\Gamma({\bf y})=\displaystyle\int\limits_{\mathbb{R}^{n}}\rho(t;{\bf
x})d\Gamma({\bf x})=\displaystyle\int\limits_{\mathbb{R}^{n}}\rho\Bigl(t;{\bf x}(t;{\bf y})\Bigr)\textit{J}(t;{\bf
y})d\Gamma({\bf y}).$$
As the random process ${\bf x}(t)$ is defined within the extended phase space of general dimension $\mathbb{R}^{n+1}$, then to determine the uniqueness of the trajectory the system of differential equations for stochastic kernel should consists of at least $n+1$ equations: $$\label{sys-yad}
\left\{
\begin{array}{l}
d_{t}\rho_{l}(t;{\bf x})=
-\displaystyle\frac{\partial\rho_{l}(t;{\bf x})b_{i\,k}(t;{\bf x})
}{\partial x_{i}}dw_{k}(t) +\Bigl(-\displaystyle\frac{\partial
\left( \rho_{l}(t;{\bf x})a_{i}(t;{\bf x})\right) }{\partial
x_{i}}+\\
+\displaystyle\frac{1}{2}\frac{\partial^{\,2} \left(\rho_{l}(t;{\bf
x})b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})\right) }{\partial x_{i} \partial x_{j}}\Bigr)dt + \\
+\displaystyle\int\limits_{\mathbb{R}(\gamma)} \Bigl[\rho_{l}\left(t;{\bf
x}-g(t;{\bf x}^{-1}(t;{\bf x};\gamma);\gamma)\right)
\cdot D\left( {\bf x}^{-1}(t;{\bf x};\gamma) \right)-\rho_{l}(t;{\bf
x})\Bigr]\nu(dt;d\gamma), \\
\rho_{l}(t;{\bf x}(t))\Bigl.\Bigr|_{t=0}=\rho_{l}(0;{\bf x}(0))=\rho_{l}(0;{\bf y}), \ \ \ \ \ l=\overline{1, n+1}.
\end{array}
\right.$$
As you know the collection of kernels is called complete if any other function that is the integral invariant kernel of the $n$-order can be presented as the function of this collection’ elements.
\[th-yad\] The GSDE system possesses the complete collection of kernels, which consists of $(n+1)$ functions and each of these functions is the solution for equation .
Let $\rho_{l}(t;{\bf x})$, $l=\overline{1,m}$, $m\geq n+1$ – are the integral invariant kernels of . As follows from lemma \[l1\] that for any $l\neq n+1$ the ratio $\dfrac{\rho_{l}(t;{\bf x}(t;{\bf y}))}
{\rho_{n+1}(t;{\bf x}(t;{\bf y}))}$ is a constant depending only on ${\bf x}(0)={\bf y}$ for any solution ${\bf x}(t)$ of system : $
\dfrac{\rho_{l}(t;{\bf x}(t;{\bf y}))}
{\rho_{n+1}(t;{\bf x}(t;{\bf y}))}=\dfrac{\rho_{l}(0;{\bf y})}
{\rho_{n+1}(0;{\bf y})}.
$ Let’s construct the functions $\theta_{l}(t;{\bf x})= \rho_{l}(t;{\bf x})\,\rho_{n+1}^{-1}(t;{\bf
x})$, $l=\overline{1,n}$, taking into account the linear independence of the functions $\rho_{l}(0;{\bf y})$ and $\rho_{n+1}(0;{\bf y})$.
Depending on the specific approach, described in the Introduction and the obtained independence of the functions $\theta_{l}(t;{\bf x})$ with $t=0$ it was possible to construct a one-to-one correspondence: $$\label{inv-1}
x_{i}=q_{i}(\theta_{l},\theta_{2},\ldots,\theta_{n}).$$
Let’s introduce the following notation just to shorten the entries: $$\overrightarrow{\theta}=
\left\{\theta_{l},\theta_{2},\ldots,\theta_{n} \right\},$$ $$\overrightarrow{q}=\left\{q_{l},q_{2},\ldots,q_{n} \right\}
\in {\mathbb R}^{n},$$ $$\overrightarrow{\rho}(t;{\bf x})=\left\{\rho_{1}(t;{\bf
x}),\rho_{2}(t;{\bf x}),\ldots, \rho_{3}(t;{\bf x}) \right\}
\in {\mathbb R}^{n}.$$ Because of the condition for some $s\geq1$ and any other function $
\chi(t;{\bf x})=\rho_{n+s}(t;{\bf x})\,\rho_{n+1}^{-1}(t;{\bf x})
$ at the time moment $t=0$ can be represented like this: $$\chi(0;{\bf y})=\rho_{n+s}(0;{\bf y})\,\rho_{n+1}^{-1}(0;{\bf y})=\overline{\psi} \Bigl(
\overrightarrow{q}(\overrightarrow{\theta})\Bigr).$$ Then, consequently, for any function $\rho_{n+s}(t;{\bf x})$ being a kernel of the integral invariant and because the lemma \[l1\] conclusion, we get: $\theta_{l}(t;{\bf x})$: $$\chi(t;{\bf x}(t;{\bf y}))=\rho_{n+s}(t;{\bf x}(t;{\bf
y}))\,\rho_{n+1}^{-1}(t;{\bf x}(t;{\bf y}))=$$ $$=\rho_{n+s}(0;{\bf y})\,\rho_{n+1}^{-1}(0;{\bf y})=\chi(0;{\bf y})=
\overline{\psi} \biggl(
\overrightarrow{q}\Bigl(\rho_{n+1}^{-1}(t;{\bf x}(t;{\bf y}))\cdot\overrightarrow{\rho}(t;{\bf x}(t;{\bf
y})) \Bigr)\biggr).$$ It follows from this relation the for any $ t\geq 0 $ and each $s\geq 1$: $$\rho_{n+s}(t;{\bf x}(t;{\bf y}))=\rho_{n+1}(t;{\bf x}(t;{\bf
y}))\, \overline{\psi} \biggl(
\overrightarrow{q}\Bigl(\rho_{n+1}^{-1}(t;{\bf x}(t;{\bf y}))\cdot\overrightarrow{\rho}(t;{\bf x}(t;{\bf
y})) \Bigr)\biggr).$$ Then the complete collection of the kernels consists of $(n+1)$ linearly independent functions.
\[c1\] The complete collection of linear independent stochastic first integrals of system consists of $n$ functions.
The function $\widetilde{u}_{l}(t;{\bf x})=\rho_{l}(t;{\bf x})\rho_{n+1}^{-1}(t;{\bf x})$, possesses the properties described in the definition \[Ayddf1\] and presents the stochastic first integral of the system . There is $n$ number of such functions.
Let the random process ${\bf x}(t)$ is the solution of the generalized Itô’s SDE system that is presented like this: $${d}_{t}{\bf x}(t)=a(t;{\bf x}(t))dt + b(t;{\bf
x}(t))d{\bf w}(t)+ dP(t,\Delta\gamma)=\widetilde{d}_{t}{\bf x}(t)+dP(t,\Delta\gamma).$$ Then the generalized Itô’s formula can be taken down as an equation: $$\label{AydIt2}
d_{t}f(t;{\bf x}(t))= \widetilde{d}_{t}f(t;{\bf
x}(t))+\widetilde{d}_{t}P(t,\Delta\gamma),$$ where $\widetilde{d}_{t}f(t;{\bf x}(t))$ is the Itô’s differential for the part $\widetilde{d}_{t}{\bf x}(t)$ and $\widetilde{d}_{t}P(t,\Delta\gamma)$ is the differential for the Poisson’s component $dP(t,\Delta\gamma)$.
Let’s construct the equation for $u(t;{\bf x})$ using the corollary \[c1\] and the formula: $$\label{Ayd2.11}
\ln u_{s}(t;{\bf x})=\ln \rho_{s}(t;{\bf x})-\ln\rho_{l}(t;{\bf
x}).$$ By using , the Itô’s generalized formula and (\[AydIt2\]) we’ll do the differentiation of $\ln \rho(t;{\bf x})$: $$\label{Aydln}
\begin{array}{c}
\displaystyle d_{t}\ln\rho(t;{\bf x})= \frac{1}{\rho(t;{\bf
x})}\widetilde{d}_{t}\rho(t;{\bf x})-\frac{1}{2\rho^{2}(t;{\bf
x})}\left(-\displaystyle\frac{\partial(\rho(t;{\bf
x})b_{i\,k}(t;{\bf
x})) }{\partial x_{i}}\right)^{2}dt + \\
+\displaystyle\int\limits_{R(\gamma)}\Bigl[\Bigr.\ln\left\{\right.
\rho_{s}\left( t;{\bf x}-g(t;{\bf
x}(t;{\bf y});\gamma);\gamma\right)
\cdot D\left( {\bf x}^{-1}(t;{\bf x};\gamma) \right)
\left.\right\}-\\
-\ln\rho_{s}(t;{\bf x})\Bigl.\Bigr]\nu(dt;d\gamma).
\end{array}$$ Now let us examine the sum of the first two summands, dropping the functions’ arguments: $$\label{Aydlnp}
\begin{array}{c}
S_{1}=\displaystyle \frac{1}{\rho}\, \widetilde{d}_{t}\rho-\frac{1}{2\rho^{2}}\,
\left(-\displaystyle\frac{\partial(\rho b_{i\,k})}{\partial x_{i}}\right)^{2} =\\
= \displaystyle\frac{1}{\rho} \biggl[\Bigl( -\frac{\partial (\rho a_{i})}{\partial x_{i}}+
\frac{1}{2}\frac{\partial ^{\,2}(\rho b_{i\,k}b_{j\,k})}{\partial x_{i}\partial x_{j}}
\Bigr)dt + \Bigl( -\displaystyle\frac{\partial(\rho b_{i\,k}) }{\partial
x_{i}}\Bigr)dw_{k}(t)\biggr]
-\\
-\displaystyle\frac{1}{2\rho^{2}}\,\left(-\displaystyle\frac{\partial(\rho b_{i\,k})}{\partial
x_{i}}\right)^{2}dt=\\
= \displaystyle\biggl[\biggr. - \frac{\partial a_{i}}{\partial x_{i}}-a_{i}\frac{\partial \ln\rho}{\partial x_{i}} +
\frac{1}{2\rho}\frac{\partial}{\partial x_{i} }\left(\rho\,\frac{\partial b_{i\,k}b_{j\,k}}{\partial
x_{j}}+ b_{i\,k}b_{j\,k}\,\frac{\partial \rho}{\partial x_{i}} \right)-
\\
-\displaystyle\frac{1}{2\rho^{2}}\,\left(\displaystyle\rho\,\frac{\partial b_{i\,k}}{\partial
x_{i}}+b_{i\,k}\, \frac{\partial \rho}{\partial
x_{i}}\right)^{2}\biggl.\biggr]dt
-\left( b_{i\,k}\displaystyle\frac{\partial\ln\rho }{\partial
x_{i}}+\frac{\partial b_{i\,k}}{\partial
x_{i}}\right)dw_{k}(t)=\\
=S_{2}dt-\left( b_{i\,k}\displaystyle\frac{\partial\ln\rho }{\partial
x_{i}}+\frac{\partial b_{i\,k}}{\partial
x_{i}}\right)dw_{k}(t).
\end{array}$$ We transform the part $S_{2}$: $$\label{Aydlnp1}
\begin{array}{c}
S_{2}= - \displaystyle \frac{\partial a_{i}}{\partial
x_{i}}-a_{i}\frac{\partial \ln\rho}{\partial x_{i}} +
\frac{\partial b_{i\,k}b_{j\,k}}{\partial
x_{j}}\frac{\partial \ln\rho}{\partial x_{i}} +\\
+
\displaystyle\frac{1}{2}\,\frac{\partial^{\,2}( b_{i\,k}b_{j\,k})}{\partial x_{i}\partial x_{j}}
+\frac{1}{2}b_{i\,k}b_{j\,k}\,\frac{1}{\rho}\,\frac{\partial^{\,2} \rho}{\partial x_{i}\partial x_{j}}
-\\
-
\displaystyle\frac{1}{2}\,\left(\displaystyle\frac{\partial b_{i\,k}}{\partial
x_{i}}+b_{i\,k}\, \frac{\partial \ln\rho}{\partial
x_{i}}\right)\left(\displaystyle\frac{\partial b_{j\,k}}{\partial
x_{j}}+b_{j\,k}\, \frac{\partial \ln\rho}{\partial
x_{j}}\right)=\\
=- \displaystyle \frac{\partial a_{i}}{\partial
x_{i}}-a_{i}\frac{\partial \ln\rho}{\partial x_{i}} +
\frac{\partial b_{i\,k}b_{j\,k}}{\partial
x_{j}}\frac{\partial \ln\rho}{\partial x_{i}} +\\
+
\displaystyle\frac{1}{2}\,\frac{\partial^{\,2}( b_{i\,k}b_{j\,k})}{\partial x_{i}\partial
x_{j}}+\frac{1}{2}\,b_{i\,k}b_{j\,k}\frac{\partial^{\,2} \ln\rho}{\partial x_{i}\partial
x_{j}}+
+\displaystyle\frac{1}{2}\,b_{i\,k}b_{j\,k}\frac{\partial\ln\rho}{\partial x_{i}}\,\frac{\partial\ln\rho}{\partial
x_{j}}-\\
-\displaystyle\frac{1}{2}\left[\frac{\partial b_{i\,k}}{\partial x_{i}}\,
\frac{\partial b_{j\,k}}{\partial x_{j}} +2\, b_{i\,k}\frac{\partial b_{j\,k}}{\partial
x_{j}}\,\frac{\partial \ln\rho}{\partial x_{i}} +
b_{i\,k}b_{j\,k}\,\frac{\partial\ln\rho}{\partial x_{i}}\,\frac{\partial\ln\rho}{\partial
x_{j}} \right]=\\
=- \displaystyle \frac{\partial a_{i}}{\partial
x_{i}}-a_{i}\frac{\partial \ln\rho}{\partial x_{i}} +
\frac{\partial b_{i\,k}b_{j\,k}}{\partial
x_{j}}\,\frac{\partial \ln\rho}{\partial x_{i}} +\\
+\displaystyle
\frac{1}{2}\,\frac{\partial^{\,2}( b_{i\,k}b_{j\,k})}{\partial x_{i}\partial
x_{j}}+\frac{1}{2}\,b_{i\,k}b_{j\,k}\frac{\partial^{\,2} \ln\rho}{\partial x_{i}\partial
x_{j}}
-\displaystyle\frac{1}{2} \frac{\partial b_{i\,k}}{\partial x_{i}}\,
\frac{\partial b_{j\,k}}{\partial x_{j}} - b_{i\,k}\frac{\partial b_{j\,k}}{\partial
x_{j}}\,\frac{\partial \ln\rho}{\partial x_{i}}.
\end{array}$$ Hence, by substituting (\[Aydlnp1\]) into (\[Aydlnp\]) and then into (\[Aydln\]). We get the following equation for $\ln
\rho(t;{\bf x})$: $$\label{Ayd2.11a}
\begin{array}{c}
d_{t}\ln\rho(t;{\bf x})=\biggl[\biggr.- \displaystyle
\frac{\partial a_{i}(t;{\bf x})}{\partial x_{i}}-
a_{i}(t;{\bf x})\frac{\partial \ln\rho(t;{\bf x})}{\partial x_{i}}-\\
-\displaystyle \frac{1}{2}\,\frac{\partial b_{i\,k}(t;{\bf x})}
{\partial x_{i}}\,\frac{\partial b_{j\,k}(t;{\bf x})}
{\partial x_{j}}
+
\displaystyle \frac{1}{2}b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})
\frac{\partial^{2} \ln\rho(t;{\bf x})}{\partial x_{i}\partial x_{j}}+\\
+\displaystyle\frac{\partial \left(b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})\right)}
{\partial x_{i}}\,
\frac{\partial \ln\rho(t;{\bf x})}{\partial x_{j}}+\\
+\displaystyle\frac{1}{2}\frac{\partial^{2}\left( b_{i\,k}(t;{\bf x})b_{j\,k}(t;{\bf x})\right)}
{\partial x_{i}\partial x_{j}}
- b_{j\,k}(t;{\bf x})\displaystyle\frac{\partial b_{i\,k}(t;{\bf x})}
{\partial x_{i}}\,\frac{\partial \ln\rho(t;{\bf x})}{\partial x_{j}}\biggl.\biggr] dt +\\
+\displaystyle\int\limits_{R(\gamma)}\Bigl[\Bigr.\ln\left\{\right.
\rho\left( t;{\bf x}-g(t;{\bf
x}(t;{\bf y});\gamma);\gamma\right)
\cdot D\left( {\bf x}^{-1}(t;{\bf x};\gamma) \right)
\left.\right\}-\\
-\ln\rho(t;{\bf x})\Bigl.\Bigr]\nu(dt;d\gamma)
-\biggl[\displaystyle \frac{\partial b_{i\,k}(t;{\bf x})}{\partial x_{i}}+
b_{j\,k}(t;{\bf x})\frac{\partial \ln\rho(t;{\bf x})}{\partial
x_{j}}
\biggr]dw_{k}(t).
\end{array}$$
Relaying upon the obtained formula (\[Ayd2.11a\]) we can construct the equations for $\rho_{s}(t;{\bf x})$ and $\rho_{l}(t;{\bf x})$, and having taken (\[Ayd2.11\]) into consideration we make a conclusion that the stochastic first integral $u(t;{\bf x};\omega)$ of the generalized Itô’s equation is the solution for this equation: $$\label{Ayd2.12}
\begin{array}{c}
d_{t}u(t;{\bf x})=\biggl[\biggr. - a_{i}(t;{\bf x})\displaystyle\frac{\partial u(t;{\bf x})}{\partial
x_{i}} +\displaystyle \frac{1}{2}\,b_{i\,k}(t;{\bf
x})b_{j\,k}(t;{\bf x})
\frac{\partial^{2} u(t;{\bf x})}{\partial x_{i} \partial x_{j}} -\\
- b_{i\,k}(t;{\bf x})\displaystyle\frac{\partial}{\partial
x_{i}}\left(b_{j\,k}(t;{\bf x})\frac{\partial u(t;{\bf
x})}{\partial x_{j}}\right)\biggl.\biggr]dt
- b_{i\,k}(t;{\bf x})\displaystyle\frac{\partial u(t;{\bf x})}{\partial
x_{i}}\,dw_{k}(t)+ \\
+ \displaystyle\int\limits_{R(\gamma)}\Bigl[u\Bigl(t; {\bf
x}-g(t;{\bf
x}^{-1}(t;{\bf x};\gamma));\gamma\Bigr)
-
u(t;{\bf x})\Bigr]\nu(dt;d\gamma).
\end{array}$$
The concept of the first integral for Itô’s SDE system as the non random function over arbitrary random perturbation realizations has been introduce by the article [[[@D_78]]{}]{}: $$u\Bigl(0;{\bf x}(0)\Bigr)=u\Bigl(t;{\bf x}(t; {\bf x}(0))\Bigr).$$ We can also speak however about a stochastic first integral for Itô’s SDE system. It should be observed that we can also speak about stochastic (but not only about determinate) first integral when Poisson’s perturbations are missing.
In case of just Wiener’s perturbations (Itô classical equation) the investigation of the properties of the first integral as a determined function was associated with stating the independence of such a function from realization ${\bf w}(t)$ [[[@D_78]]{}]{}. In the case under consideration from definition \[Ayddf1\] the function $u\Bigl(t;{\bf x}\Bigr)$ with its equation (\[Ayd2.12\]), depends over Poisson’s perturbation are added (Itô generalized equation) such a requirement, as it follows from equation (\[Ayd2.12\]), leads to the following conditions.
\[th-usl\] The random function $u\left(t;{\bf x}(t)\right) \in \mathcal{C}_{t,x}^{1,2}$ defined over the same probability space as the random process ${\bf x}(t)$ that is the solution for system is the first integral of the system iff the function $u\left(t;{\bf x}(t)\right)$ satisfies the terms $\left.\mathcal{L}\right)$\[uslL1\]:
1. $b_{i\,k}(t;{\bf x})\displaystyle\frac{\partial u(t;{\bf x})}{\partial
x_{i}}=0$, for all $k=\overline{1,m}$ (compensation of the Wiener’s perturbations);
2. $\displaystyle\frac{\partial u(t;{\bf x})}{\partial
t}+\displaystyle\frac{\partial u(t;{\bf x})}{\partial
x_{i}}\Bigl[a_{i}(t;{\bf x})-
\displaystyle\frac{1}{2}\,b_{j\,k}(t;{\bf x})\frac{\partial
b_{i\,k}(t;{\bf x})}{\partial x_{j}}\Bigr]=0$ (independente of the time);
3. $u(t;{\bf x})-u\Bigl(t;{\bf x}+g(t;{\bf
x};\gamma)\Bigr)=0$ for any $\gamma\in R(\gamma)$ in the whole field of the process definition (compensation of the Poisson’s jumps).
Let’s examine the equation , the solution of which is the function $u\left(t;{\bf x}(t)\right)$. First let’s carry everything to one side from the equals sing. Having taken into account that $d_{t}u(t;{\bf x})=\dfrac{\partial u(t;{\bf x})}{\partial t}\,dt$ at we get for any $t$: $$\begin{array}{c}
\biggl[\biggr.\dfrac{\partial u(t;{\bf x})}{\partial t}+ a_{i}(t;{\bf x})\displaystyle\frac{\partial u(t;{\bf x})}{\partial
x_{i}} -\displaystyle \frac{1}{2}\,b_{i\,k}(t;{\bf
x})b_{j\,k}(t;{\bf x})
\frac{\partial^{2} u(t;{\bf x})}{\partial x_{i} \partial x_{j}} +\\
+ b_{i\,k}(t;{\bf x})\displaystyle\frac{\partial}{\partial
x_{i}}\left(b_{j\,k}(t;{\bf x})\frac{\partial u(t;{\bf
x})}{\partial x_{j}}\right)\biggl.\biggr]dt
+ b_{i\,k}(t;{\bf x})\displaystyle\frac{\partial u(t;{\bf x})}{\partial
x_{i}}\,dw_{k}(t)- \\
- \displaystyle\int\limits_{R(\gamma)}\Bigl[u\Bigl(t; {\bf
x}-g(t;{\bf
x}^{-1}(t;{\bf x};\gamma));\gamma\Bigr)
+
u(t;{\bf x})\Bigr]\nu(dt;d\gamma)=0.
\end{array}$$ Hence, the multipliers must be equal to zero with $dt$, $dw(t)$ and $\nu(dt;d\gamma)$. The Wiener’s summand equaled zero: For the Wiener part we have: $$\label{usl-win}
b_{i\,k}(t;{\bf x})\displaystyle\frac{\partial u(t;{\bf x})}{\partial
x_{i}}=0 \ \ \ \ \textrm{for all}\ \ k=\overline{1,m}.$$ For Poisson’s part we’ll have: $$\label{usl-puass}
u\Bigl(t; {\bf x}-g(t;{\bf x}^{-1}(t;{\bf x};\gamma));\gamma\Bigr)-
u(t;{\bf x})=0.$$ Now we re-arrange the equality and proceed to variables: $${\bf y}={\bf x}-g(t;{\bf x}^{-1}(t;{\bf x};\gamma);\gamma).$$ But taking into account that ${\bf x}^{-1}(t;{\bf x};\gamma)$ is the notation an inverse function for $f({\bf x})={\bf x}+g(t;{\bf x};\gamma)$, we get convinced that this term is equivalent to the following: $$\label{usl-puass-1}
u(t;{\bf x})-u\Bigl(t;{\bf x}+g(t;{\bf
x};\gamma)\Bigr)=0 \ \ \ \textrm{for any} \ \ \ \gamma\in R(\gamma).$$ Then, using the rule of differentiating the product and the term , we obtain: $$\begin{array}{c}
\dfrac{\partial u(t;{\bf x})}{\partial t}+
a_{i}(t;{\bf x})\dfrac{\partial u(t;{\bf x})}{\partial
x_{i}} + \dfrac{1}{2}\,b_{i\,k}(t;{\bf
x})b_{j\,k}(t;{\bf x})
\dfrac{\partial^{2} u(t;{\bf x})}{\partial x_{i} \partial x_{j}}\, -\\
- b_{i\,k}(t;{\bf x})\dfrac{\partial}{\partial
x_{i}}\left(b_{j\,k}(t;{\bf x})\dfrac{\partial u(t;{\bf
x})}{\partial x_{j}}\right)= \\
= \dfrac{\partial u(t;{\bf x})}{\partial t}+
a_{i}(t;{\bf x})\cfrac{\partial u(t;{\bf x})}{\partial
x_{i}}\, +\\
+ \dfrac{1}{2}\,b_{i\,k}(t;{\bf
x})\left[\dfrac{\partial}{\partial x_{i}}\Bigl(b_{j\,k}(t;{\bf x})
\dfrac{\partial u(t;{\bf x})}{\partial x_{j}}\Bigr)-
\dfrac{\partial u(t;{\bf x})}{\partial x_{j}}
\dfrac{\partial b_{jk}(t;{\bf x})}{\partial x_{i}}\right]=\\
= \dfrac{\partial u(t;{\bf x})}{\partial t}+
a_{i}(t;{\bf x})\cfrac{\partial u(t;{\bf x})}{\partial
x_{i}}\, - \dfrac{1}{2}\,b_{i\,k}(t;{\bf
x})
\dfrac{\partial u(t;{\bf x})}{\partial x_{j}}
\dfrac{\partial b_{jk}(t;{\bf x})}{\partial x_{i}}.
\end{array}$$ Consequently, $$\displaystyle\frac{\partial u(t;{\bf x})}{\partial
t}+\displaystyle\frac{\partial u(t;{\bf x})}{\partial
x_{i}}\Bigl[a_{i}(t;{\bf x})-
\displaystyle\frac{1}{2}\,b_{j\,k}(t;{\bf x})\frac{\partial
b_{i\,k}(t;{\bf x})}{\partial x_{j}}\Bigr]=0.$$ Thus, we get all the terms of $\left.\mathcal{L}\right)$.
If we analyzed the concrete realization then the non random function $u(t;{\bf x})$ is the determinate first integral of the stochastic system.
The concept of the stochastic first integral for a centered Poisson measure was introduced in [[@D_02]]{}. The obtained conditions for its realization take the necessity of determining the density of intensive Poisson distribution. Thus, it makes no difference what is the probability distribution of intensities of Poisson jumps. This case is very important for constructing program controls [[[@11_KchUpr]]{}]{}.
The presented generalization of the Itô-Wentzell formula and the stochastic first integral concept [[[@D_02]]{}]{} allow to construct of the program control with the probability is equaled to 1 for dynamic systems which being subjected to the Wiener perturbations and the Poisson jumps [[[@11_KchUpr]]{}]{}.
[99]{}
Doobko, V. The first integral of the stochastic differential equations system : preprint. 1978. (in Russian)
Doobko, V. The questions about theory and applications of the stochastic diffirential equation. 1989. (in Russian)
Doobko, V. Open evolutioning systems. // The First International Scientific- Practical conference “Open evolutioning systems” devoted to theoretical, metodological and practical problems of evolution, stabilization and selforganization of open systems of arbitrary nature.(2002, Kiyv, Ukraina) 2002. – pp. 14-31. (in Russian)
Gihman, I. I. and Skorohod, A. V. Stochastic differential equations. 1972.
Karachanskaya, E. Construction of program control with probability one for a dynamical system with Poisson perturbations // Bulletin of PNU. – No 2 (21) 2011, pp. 51–60. (in Russian)
|
---
abstract: 'The Peierls instability in multi-channel metal nanowires is investigated. Hyperscaling relations are established for the finite-size-, temperature-, and wavevector-scaling of the electronic free energy. It is shown that the softening of surface modes at wavevector $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ leads to critical fluctuations of the wire’s radius at zero temperature, where ${k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ is the Fermi wavevector of the highest occupied channel. This Peierls charge density wave emerges as the system size becomes comparable to the channel correlation length. Although the Peierls instability is weak in metal nanowires, in the sense that the correlation length is exponentially long, we predict that nanowires fabricated by current techniques can be driven into the charge-density-wave regime under strain.'
author:
- 'D. F. Urban$^1$, C. A. Stafford$^2$, and Hermann Grabert$^1$'
bibliography:
- 'Urban06b.bib'
title: 'Scaling Theory of the Peierls-CDW in Metal Nanowires'
---
Introduction
============
Already long ago, Fröhlich [@Froehlich54] and Peierls [@Peierls-book] pointed out that a one-dimensional metal coupled to the underlying lattice is not stable at low temperatures. Electron-phonon interactions lead to a novel type of ground state with a charge-density wave (CDW) of wavevector $2k_F$ (see Ref. for a review). This state is characterized by a gap in the single-particle excitation spectrum, and by a collective mode with an associated charge density , where $\rho_0$ is the unperturbed electron density of the metal. Of particular interest are [*incommensurate*]{} systems, where the period of the CDW is not simply related to that of the unperturbed atomic structure. In that case, no long-range order is expected even at zero temperature due to quantum fluctuations.
In contrast to the usual Peierls systems, metallic nanowires[@Agrait03] are open systems with several inequivalent channels, for which the theory has not yet been developed. Interest in the Peierls transition in metal nanowires has been stimulated by recent experiments[@Yeom99; @Ahn03; @Ahn04; @Ahn05; @Snijders06] on nanowire arrays on stepped surfaces. In these systems, interactions between nanowires, mediated by the substrate, render the system quasi-two-dimensional at low temperatures, similar to the dimensional crossovers commonly observed in highly-anisotropic organic conductors.[@Gruener88] However, individual free-standing metal nanowires[@Agrait03] represent true (quasi)one-dimensional systems, in which the intrinsic behavior of the Peierls-CDW can be studied.
Due to quantization perpendicular to the wire axis, electron states in metal nanowires are divided into distinct channels, which are only weakly coupled. Each channel has a quadratic dispersion relation, and starts to contribute at a certain threshold energy, i.e., the eigenenergy $E_n$ of the corresponding transverse mode. This results in a sequence of quasi-one-dimensional systems with different Fermi wavevectors $$\begin{aligned}
{k_{{{\mbox{\tiny F}}}\!,\,n}}&=&\sqrt{\frac{2m_e}{\hbar^2}\left(E_F-E_n\right)}\,.\end{aligned}$$ The channel Fermi wavevectors are generically [*not commensurate*]{} with the underlying atomic structure in nanowires with more than one open channel.
The Peierls instability is weak in metal nanowires,[@Urban03] so that the system is close to the quantum critical point at which the transition from Fermi liquid behavior to a CDW state occurs. In the vicinity of the quantum critical point, the system exhibits an additional length scale, the correlation length $$\begin{aligned}
\xi_{\nu}&=&\frac{\hbar{v_{{{\mbox{\tiny F}}}\!,\,\nu}}}{2\Delta_{\nu}}.\end{aligned}$$ Here, the greek index $\nu$ labels the highest occupied channel, in which an energy gap $2\Delta_{\nu}$ opens and ${v_{{{\mbox{\tiny F}}}\!,\,\nu}}=({\hbar}/{m_e}){k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ is the Fermi velocity of this subband. Within a hyperscaling ansatz,[@Hertz76] the singular part of the energy is expected to scale like $E_{{{\rm sing}}}/L\sim\xi_{\nu}^{-1-z}$, where $L$ is the wire length, and the dynamic critical exponent takes the value $z=1$. Thus, $E_{{{\rm sing}}}/L \sim \Delta_\nu^2/\hbar{v_{{{\mbox{\tiny F}}}\!,\,\nu}}$. Near the singular point, we indeed find that the electronic energy is given by $$\begin{aligned}
\label{eq:scalingrelation:Intro}
\frac{E_{{{\rm sing}}}}{L}&\approx&\kappa_{\nu}\;
\frac{\Delta_{\nu}^2}{\hbar\,{v_{{{\mbox{\tiny F}}}\!,\,\nu}}}
\;\;\mbox{Y}\left(\xi_{\nu} \delta q,\frac{\xi_{\nu}}{L},
\frac{\xi_{\nu}}{L_T}\right),\end{aligned}$$ where $\mbox{Y}(x,y,z)\approx\ln(\mbox{max}\{x,y,z\})$ is a universal and dimensionless scaling function, $\delta
q=(q-2{k_{{{\mbox{\tiny F}}}\!,\,\nu}})$ is the detuning of the perturbation wavevector from its critical value $2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$, $L_T=\hbar\,{v_{{{\mbox{\tiny F}}}\!,\,\nu}}/k_BT$ is the thermal length at temperature $T$, and $\kappa_{\nu}=1\mbox{ or }
2$ is the degeneracy of the highest open channel $\nu$. This universal scaling behavior is quite different from that of a closed system with periodic boundary conditions,[@Nathanson92; @Montambaux98] where radically different behavior was found for odd or even numbers of fermions. Nonetheless, the correlation length $\xi$ was also found to control the finite-size scaling of the Peierls transition in mesoscopic rings.[@Montambaux98]
In this article, the quantum and thermal fluctuations of the nanowire surface are calculated in a continuum model,[@Zhang03] where the ionic background is treated as an incompressible, irrotational fluid. In contrast to the semiclassical theory of Ref. , which exhibited critical surface fluctuations only at finite temperature—due to the classical Rayleigh instability—the present fully quantum-mechanical theory exhibits critical zero-temperature surface fluctuations at wavevector $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$. In a finite system, these CDW correlations are found to grow in amplitude as the wire length $L$ approaches a critical length $L_c$ of order the correlation length $\xi_\nu$. A similar scaling is observed as the temperature is lowered, so that $L_T$ exceeds $\xi_\nu$. Although $\xi_\nu$ is typically very large for fully-equilibrated structures, consistent with the fact that Peierls distortions have not yet been observed in multi-channel nanowires, it is predicted that nanowires of dimensions currently produced in the laboratory can be driven into the CDW regime by applying strain.
This paper is organized as follows. Sec.\[sec:PeierlsInstability\] summarizes the (standard) Peierls theory for a one-dimensional metal with a half-filled band, and extends it to multi-channel wires. A description of the deformation of a nanowire through surface phonons is given in Sec. \[sec:SurfacePhonons\]. The correlation length $\xi$ is introduced in Sec. \[sec:ScalingRelations\], and the scaling relation (\[eq:scalingrelation:Intro\]) is established. Sec.\[sec:Correlations\] examines the critical surface fluctuations, and consequences for different materials are discussed. Finally, a summary and discussion are given in Sec.\[sec:Summary:Ch4\].
The Peierls instability {#sec:PeierlsInstability}
=======================
Consider the ground state of a one-dimensional linear chain of atoms, with lattice constant $a$ and periodic boundary conditions: In the presence of electron-phonon interactions, it is energetically favorable to introduce a periodic lattice distortion with period ${\lambda}=\pi/k_F$; this effect is known as the Peierls instability.[@Peierls-book] The lattice distortion opens up an energy gap $2\Delta$ in the electronic dispersion relation at the Fermi level $E_F$, so that the total electronic energy is lowered. If $\Delta\ll E_F$, then the gain in energy is given by [@Peierls-book] $$\begin{aligned}
\label{eq:Egain:gap:approx}
\frac{E_{{{\rm gain}}}}{L}&\approx&
\frac{1}{\pi}\,
\frac{\Delta^2}{2E_F/k_F}\left[
\log\!\left(\frac{\Delta}{4E_F}\right)-\frac{1}{2}
\right]
+{\mathcal{O}}\!
\left(\!\frac{\Delta}{E_F}\!\right)^{\!4}\!\!\!.\quad\end{aligned}$$ The size of the gap can be extracted from an energy balance: Let the increase of the elastic energy due to the deformation be given by $E_{{{\rm cost}}}/L=\alpha\,b^2$, where $b$ is the amplitude of the distortion. On the other hand, $\Delta$ is defined through the matrix element of the perturbation coupling states with longitudinal wave vector $\pm k_F$; it is linear in $b$ and we can set $\Delta=A\,b$. By finding the minimum of $\delta E(b) = E_{{{\rm gain}}}+E_{{{\rm cost}}}$, we can derive the optimal value of the distortion amplitude $b$ and from this derive the size $\Delta$ of the gap, $$\begin{aligned}
\label{eq:GapSize}
\Delta&=&4E_F\;\exp\!\left(-2\pi\alpha E_F/A^2k_F\right).\end{aligned}$$
The analysis of the Peierls instability in a one-dimensional metal can be extended to the case of a multi-channel system. While other instabilities of the Fermi liquid—induced by electron-electron interactions—compete with the Peierls instability in purely one-dimensional systems,[@Gruener88] their importance decreases as the number of channels increases,[@Matveev93] so that electron-electron interactions can in a first approximation be neglected in multi-channel nanowires. Moreover, electron-electron interactions are strongly screened in s-orbital metal nanowires with three or more conducting channels.[@Kassubek99; @Zhang05] Hence, including electron-electron interactions in the calculation will not lead to any qualitative change (except in the limit of very few conduction channels) whereas electron-phonon interactions do.[@footnoteEE] We therefore consider only electron-phonon coupling in this article.
For any given channel $n$, a perturbation of wavevector $q=2{k_{{{\mbox{\tiny F}}}\!,\,n}}$ will open a gap $2\Delta_n$ at the Fermi surface in the energy dispersion of this channel. The energy gain $E_{{{\rm gain}},n}$ and gap size $2\Delta_n$ are then given by Eqs.(\[eq:Egain:gap:approx\]) and (\[eq:GapSize\]), respectively, with $k_F$ replaced by ${k_{{{\mbox{\tiny F}}}\!,\,n}}$ and $E_F$ replaced by $\hbar^2{k_{{{\mbox{\tiny F}}}\!,\,n}}^2/2m_e$. The greatest effect will be seen close to the opening of the highest occupied channel (i.e. $n=\nu$), where the channel Fermi wavevector ${k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ is small. Note that the same perturbation (with $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$) will also modify the dispersion relations of lower-energy channels $n'$ with $E_{n'}<E_\nu$, but due to the finite spacing of the threshold energies, this modification will occur within the Fermi sea and there will be little net effect.[@Urban03]
The standard Peierls theory uses periodic boundary conditions for the wave functions. Its extension to the multi-channel case cannot be directly applied to a metallic nanowire of finite length $L$. The nanowire is part of a much larger system including the leads, and the longitudinal wavevectors $k_n$ in the subbands are not restricted to multiples of $\frac{2\pi}{L}$, as in the case of an isolated system with periodic boundary conditions. Therefore a perturbation of the nanowire with wavevector $q$ does not only couple states $k_n$ and $k_n'$ that exactly obey $k_n=k_n'+q$. Instead the state $k_n$ is coupled to a range of $k_n'$-states proportional to $1/L$. The dispersion relation remains smooth while, with increasing wire length, it develops a smeared-out *quasi-gap*, and only in the limit $L \rightarrow \infty$ do we recover the Peierls result with a jump at ${k_{{{\mbox{\tiny F}}}\!,\,n}}$.[@Urban03]
Surface phonons {#sec:SurfacePhonons}
===============
We describe the wire in terms of the Nanoscale Free Electron Model (NFEM),[@Stafford97a; @Buerki05b] treating the electrons as a Fermi gas confined within the wire by a hard-wall potential. The ionic structure is replaced by a uniform (jellium) background of positive charge, and we assume that this ionic medium is irrotational and incompressible. The approximations[@Kassubek99; @Zhang03; @Buerki05b] of the NFEM require strong delocalization of the valence electrons, good charge screening, and a spherical Fermi surface, conditions met in alkali metals and, to a lesser extent, noble metals such as gold.
In this continuum model, the ionic degrees of freedom are completely determined by the surface coordinates of the wire. Let us consider an initially uniform wire of radius $R_0$ and length $L$ which is axisymmetrically distorted. Its surface is given by the radius function $$\begin{aligned}
\label{eq:geometry}
R(z,t)&=&R_0\biggl(1+\sum_q b_q(t) e^{iqz}\biggr),\end{aligned}$$ where the time-dependent perturbation is written as a Fourier series with coefficients $b_q(t)$. Since $R(z,t)$ is real, we have $b_q=b_{-q}^*$ and we require that the volume of the wire is unchanged by the deformation. Other physically reasonable constraints are possible and will be discussed in detail in Sec. \[sec:Correlations\].
The kinetic energy of the ionic medium is given by[@Zhang03] $$\begin{aligned}
\label{eq:Ekin:phonons}
E_{kin}&=&
L\sum_{q>0}m(q,{R_0})\left|{R_0}\frac{\partial b_q(t)}{\partial t}\right|^2.\end{aligned}$$ Here, the *mode inertia* $m$ is a function of wire radius and phonon wavevector, and reads $$\begin{aligned}
m(q,R_0)&=&\rho_{ion}\cdot\frac{2\pi R_0 I_0(qR_0)}{q I_1(qR_0)}\;,
\label{eq:mode_inertia}\end{aligned}$$ where $\rho_{ion}$ is the ionic mass density, and $I_0$ and $I_1$ are the modified Bessel functions of zeroth and first order, respectively.[@footnote1] Considered as a function of $q$, the mode inertia has a singularity $\sim 1/q^2$ at $q=0$, and is monotonically decreasing, with $m\sim 1/q$ for large $q$.
Within the Born-Oppenheimer approximation, the potential energy of the ions is given by the grand canonical potential $\Omega$ of the confined electron gas. A linear stability analysis of cylindrical wires determining the leading-order change in $\Omega$ due to a small perturbation was recently presented in Ref. : $\delta\Omega$ is quadratic in the Fourier coefficients $b_q$ of the deformation, and can be written as $$\begin{aligned}
\label{eq:Epot:phonons}
\frac{\delta\Omega}{L}&=&\sum_{q\geq
0}|b_q|^2\alpha(q,R_0,L,T)+{\cal
O}\left(\frac{\lambda_F}{L}\right),\end{aligned}$$ where the terms of order $\lambda_F/L$ include nondiagonal contributions which can be neglected if the wire is long enough.[@footnote4] The explicit analytical expression for the *mode stiffness* $\alpha$ is given in App.\[app:alpha\]. Here, we are interested in the general behavior of the mode stiffness as a function of the perturbation wavevector $q$ for a given radius ${R_0}$, length $L$, and temperature $T$: $\alpha(q)$ is a smoothly increasing function, with a $L$- and $T$-dependent dip at $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$, where $\nu$ is the index of the highest open channel (see App.\[app:alpha\]). Therefore, we formally split $\alpha$ into a smoothly varying term and one containing the singular contribution of channel $\nu$, $$\begin{aligned}
\label{eq:modestiffness}
\alpha=\alpha_{{\rm smooth}}+ \alpha_{{\rm sing}}^{(\nu)}\;.\end{aligned}$$ The smooth part can be thought of as the sum of a Weyl contribution,[@Zhang03] which describes the effects of surface tension $\sigma_s$ and curvature energy $\gamma_s$, plus an electron-shell correction[@Zhang03] $$\alpha_{{\rm smooth}}= -2\pi R_0 \sigma_s + 2\pi R_0^2 (\sigma_s R_0 - \gamma_s)q^2 + \alpha_{{\rm shell}},
\label{eq:alpha:smooth}$$ whereas the singular part describes the onset of the Peierls instability in channel $\nu$. At zero temperature, it is given by[@Urban03] $$\begin{aligned}
\label{eq:alpha:sing}
\lefteqn{\alpha_{{\rm sing}}^{(\nu)}(q,{R_0},L)=
-\frac{2m_e}{\hbar^2}\;\frac{4\kappa_\nu E_{\nu}^2}{\pi
q}}\\
&&
\times\bigg[\ln\!\left|\frac{2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}\!\!+\!q}{2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}\!\!-\!q}\right|
\!-\! \mbox{F}\Big(\!(2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}\!\!+\!q)L\!\Big)
\!+\!
\mbox{F}\Big(\!|2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}\!\!-\!q|L\!\Big)\bigg],\nonumber\end{aligned}$$ where $\mbox{F}(x)=\mbox{Ci}(x)-\sin(x)/x$ and $\mbox{Ci}(x)$ is the cosine integral function.[@footnote2] The finite temperature mode stiffness is evaluated numerically by computing the integral $$\begin{aligned}
\label{eq:DefAlphaT}
\alpha(T)&=&\int dE
\left(-\frac{\partial f}{\partial E}\right)\alpha(E)\;,\end{aligned}$$ where $f(E)$ is the Fermi function and $\alpha(E)$ is obtained from the zero temperature results by replacing ${k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ by ${k_{{{\mbox{\tiny E}}},\nu}}\equiv[\frac{2m_e}{\hbar^2}(E-E_\nu)]^{-1/2}$.
(85,52) ( 0,0)[ Surface phonon dispersion relation $\omega(q)$ for a wire of length $L=L_c$, where $L_c$ is the critical length (see text). Different wire lengths are compared in the inset showing the vicinity of $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$. The curves are offset horizontally by $0.05$ units for clarity. (b) Temperature dependence of the minimum in $\omega(q)$ for a wire of length $L=L_c$ (curves are offset horizontally by $0.1$ units). The wire radius is ${R_0}=4.5k_F^{-1}$ for all curves, thus $L_c\sim6500k_F^{-1}$. ](fig1a.eps "fig:"){width="5.1cm"}]{} (52,0)[ Surface phonon dispersion relation $\omega(q)$ for a wire of length $L=L_c$, where $L_c$ is the critical length (see text). Different wire lengths are compared in the inset showing the vicinity of $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$. The curves are offset horizontally by $0.05$ units for clarity. (b) Temperature dependence of the minimum in $\omega(q)$ for a wire of length $L=L_c$ (curves are offset horizontally by $0.1$ units). The wire radius is ${R_0}=4.5k_F^{-1}$ for all curves, thus $L_c\sim6500k_F^{-1}$. ](fig1b.eps "fig:"){width="3.2cm"}]{} (0,49)[(a)]{} (51,49)[(b)]{}
Combining the kinetic energy (\[eq:Ekin:phonons\]) and potential energy (\[eq:Epot:phonons\]) yields a Hamiltonian for surface phonons, $H_{\mbox{\tiny
ph}}=\sum_q\hbar\omega_q\left(\hat{a}_q^{\dagger}\hat{a}_q+\frac{1}{2}\right)$, with frequencies $$\begin{aligned}
\label{eq:omegaPhonon}
\omega(q,R_0,L,T)&=&\sqrt{\frac{\alpha(q,R_0,L,T)}{R_0^2\,m(q,R_0)}}.\end{aligned}$$ These axisymmetric surface phonons correspond to the longitudinal acoustic mode of the nanowire. From Eqs. (\[eq:mode\_inertia\]), (\[eq:modestiffness\]) and (\[eq:omegaPhonon\]), we infer that $\omega(q)$ is a smoothly increasing function of $q$, with a dip at $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$. As expected for acoustic phonons, it is linear for small $q$. The $L$-dependent softening of the phonon modes with wavevector $2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ defines a *critical length* $L_c$ for which $\omega(2{k_{{{\mbox{\tiny F}}}\!,\,\nu}})=0$.
A plot of $\omega(q)$ for $L=L_c$ at zero temperature is shown in Fig. \[Fig:omegaQ\] (a). The inset shows a close-up of the minimum and compares different wire lengths, where the curves are horizontally offset for clarity. The temperature dependence of $\omega$ is illustrated in Fig. \[Fig:omegaQ\](b), concentrating on the vicinity of $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ (again, the curves are horizontally offset). With increasing temperature, the dip in $\omega$ disappears, and the curve becomes smoother.
Scaling relations {#sec:ScalingRelations}
=================
The opening of the Peierls gap in subband $\nu$ introduces a new energy scale, given by the gap $2\Delta_\nu$ for an infinitely long wire. $\Delta_\nu$ is determined by the matrix element of the perturbation coupling states with longitudinal wave vector $\pm
{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ and is linear in the distortion amplitude $b_q$. The perturbation potential matrix element is calculated in App.\[app:matelement\], and we find that $\Delta_\nu=2E_\nu b_q$.
On the other hand, the energy cost for creating a surface modulation with wavevector $q$ is determined by the smooth part of the mode stiffness, so that $\delta
E_{{\rm cost}}/L=\alpha_{{\rm smooth}}(q)|b_q|^2$. Following the arguments of Sec. \[sec:PeierlsInstability\] we can now calculate the length scale $\xi_\nu$, obtaining $$\begin{aligned}
\label{eq:xi}
\xi_{\nu}&=&\frac{\hbar{v_{{{\mbox{\tiny F}}}\!,\,\nu}}}{2\Delta_{\nu}}=\frac{1}{4{k_{{{\mbox{\tiny F}}}\!,\,\nu}}}
\;\exp\!\left[\left(\!\frac{\hbar^2}{2m_e}\!\right)
\frac{\pi{k_{{{\mbox{\tiny F}}}\!,\,\nu}}\alpha_{{{\rm smooth}}}}{2\kappa_{\nu}E_{\nu}^2}\right]\!,\quad\end{aligned}$$ where we have used Eq. (\[eq:GapSize\]). Introducing the correlation length $\xi_\nu$ allows us to derive the finite-size-, and wavevector-scaling of the electronic free energy near the critical point of the Peierls instability, given by $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$, $L\rightarrow\infty$, $T=0$, and weak electron–phonon coupling.
*Finite-size scaling*—First we examine the mode stiffness at zero temperature and for $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ as a function of wire length. Starting from Eqs. (\[eq:modestiffness\]) and (\[eq:alpha:sing\]), we take the limit $q\rightarrow 2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ and get $$\begin{aligned}
\alpha(L)\bigg|_{{q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}}\atop{T=0\;\quad}}\!\!\!
&=&\alpha_{{\rm smooth}}-\frac{4\kappa_\nu E_{\nu}^2}{\pi\hbar{v_{{{\mbox{\tiny F}}}\!,\,\nu}}}\,\left[
\ln\!\left(4{k_{{{\mbox{\tiny F}}}\!,\,\nu}}L\right)-c_1\right],\qquad\end{aligned}$$ where $c_1=1-\gamma_E+{\rm F}(4{k_{{{\mbox{\tiny F}}}\!,\,\nu}}L)\approx0.42$. Here $\gamma_E\simeq0.577$ is the Euler-Mascheroni constant, and we have used the fact that F$(x)\ll1$ for $x\gg1$. This expression for $\alpha(L)$ can further be simplified by the use of Eq. (\[eq:xi\]), so that $$\begin{aligned}
\alpha(L)\bigg|_{{q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}}\atop{T=0\quad\;}}
\label{eq:Lscaling:alpha}
&=&\frac{4\kappa_\nu E_{\nu}^2}{\pi\hbar{v_{{{\mbox{\tiny F}}}\!,\,\nu}}}\,\left[
\ln\!\left(\frac{\xi_\nu}{L}\right)+c_1\right].\end{aligned}$$ This defines the critical length $L_c$ for which $\alpha$ takes the value zero, $$\begin{aligned}
\label{eq:Lcrit}
L_c&\equiv&e^{c_1}\,\xi_\nu\;\approx\;1.52\,\xi_\nu\;.\end{aligned}$$ Note that the critical length is of the same order of magnitude as the correlation length.
*Wavevector scaling*—Now let ${\delta q}\equiv(q-2{k_{{{\mbox{\tiny F}}}\!,\,\nu}})$ be the detuning of the perturbation wavevector from its critical value $2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$. At zero temperature and in the limit $L\rightarrow\infty$, we expand the mode stiffness as a function of $\delta q$ and obtain $$\begin{aligned}
\label{eq:Qscaling:alpha}
\alpha({\delta q})\bigg|_{{L=L_c}\atop{T=0\;\,}}&\simeq&
\frac{4\kappa_\nu E_{\nu}^2}{\pi\hbar{v_{{{\mbox{\tiny F}}}\!,\,\nu}}}\,
\ln\!\left|\xi_\nu{\delta q}\right|\,+\,{\cal{O}}({\delta q})\;.\end{aligned}$$ Again, the result was written in a compact form by the use of Eq. (\[eq:xi\]).
*Temperature scaling*—Finally, we examine the effect of finite temperature for a wire of infinite length and a perturbation of wavevector $2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$. The main effect of finite temperature is to smear out the Fermi surface, so that the critical wavevector $q$ is detuned by ${\delta q_{{\mbox{\tiny E}}}}=2({k_{{{\mbox{\tiny F}}}\!,\,\nu}}-{k_{{{\mbox{\tiny E}}},\nu}})$ with ${k_{{{\mbox{\tiny E}}},\nu}}\equiv[\frac{2m_e}{\hbar^2}\left(E-E_\nu\right)]^{-1/2}$. Starting from the scaling behavior for finite ${\delta q}$-detuning, Eq. (\[eq:Qscaling:alpha\]), we calculate $\alpha(T)$ for small $T$ from Eq. (\[eq:DefAlphaT\]) by linearizing $E-E_F\approx\hbar {v_{{{\mbox{\tiny F}}}\!,\,\nu}}({k_{{{\mbox{\tiny E}}},\nu}}-{k_{{{\mbox{\tiny F}}}\!,\,\nu}})$ and find $$\begin{aligned}
\label{eq:Tscaling:alpha}
\alpha(T)\bigg|_{{L=L_c\quad}\atop{q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}}}&\simeq&
\frac{4\kappa_\nu E_{\nu}^2}{\pi\hbar{v_{{{\mbox{\tiny F}}}\!,\,\nu}}}\;\left[
\ln\!\left|\frac{\xi_\nu}{\hbar{v_{{{\mbox{\tiny F}}}\!,\,\nu}}\beta}\right|+c_2\right],\end{aligned}$$ with a numerical constant $c_2=-\gamma_E+\ln\pi\approx0.57$.
Combining the three scaling relations for length, perturbation wavevector, and temperature, Eqs. (\[eq:Lscaling:alpha\]), (\[eq:Qscaling:alpha\]), and (\[eq:Tscaling:alpha\]), respectively, we prove Eq. (\[eq:scalingrelation:Intro\]) for the singular part of the electronic energy.[@footnote7]
![ \[Fig:Yscaling:CrossOver\] (color online) Cross-over from $T$-scaling to $L$-scaling: The scaling function $\mbox{Y}(x,y,z)$ is plotted as a function of the ratio $y/z=L_T/L$ for fixed values of $y=\xi_\nu/L$ (red curves) and $z=\xi_\nu/L_T$ (green curves) at $x={\delta q}\xi_\nu=0$. For all curves $R_0=8k_F^{-1}$. ](fig2.eps){height="5cm"}
Finite temperature introduces a thermal length $L_T=\beta\hbar{v_{{{\mbox{\tiny F}}}\!,\,\nu}}$. Changing $L_T$ has an equivalent influence on the system properties as changing the length of the wire $L$. Near the critical point, we observe a crossover between finite-$T$ scaling and finite-$L$ scaling, depending on the ratio $L_T/L$ (see Fig. \[Fig:Yscaling:CrossOver\]). The smaller of the two lengths dominates the behavior of the singular part of the electronic energy. As long as $L_T<L$, a change of $L$, even by orders of magnitude, has only a small effect on Y, whereas the system is sensitive to small changes in $L_T$ and shows $T$-scaling. The situation is reversed for $L_T>L$: In this case, a change in temperature results in only small changes of Y, whereas the singular part of the energy depends strongly on $L$ and shows finite-length scaling. These two different cases are illustrated in Fig.\[Fig:Yscaling:CrossOver\] by the two sets of curves for different fixed values of $L$ and $L_T$, respectively.
So far we have considered ideal nanowires without disorder. Disordered structures exhibit an additional length scale, the electron elastic mean free path $\ell$. The scaling theory we have derived allows us to predict that the effect of disorder is to cut off the logarithmic scaling of the Peierls-CDW instability, exactly like the thermal length or wire length. We thus infer that the length-dependent criterion for the emergence of the Peierls-CDW in metal nanowires is given by $$\label{eq:CompareLscales}
L_c\lesssim L,\,L_T\,,\ell\;,$$ where the critical length $L_c$ (Eq. \[eq:Lcrit\]) is of order the correlation length $\xi_\nu$. Increasing temperature leads to a decreasing thermal length, and therefore destroys the phenomenon at sufficiently high $T$. Increasing disorder has a corresponding effect.
 CDW correlations for various values of $L/L_c$ at $T=0$ for a wire with $R=4.42k_F^{-1}$. The critical length is $L_c\sim 1560k_F^{-1}$. Curves offset vertically for clarity.](fig3.eps){height="5cm"}
Surface fluctuations {#sec:Correlations}
====================
The softening of the surface phonon modes with wavevector $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ leads to critical surface fluctuations. Given the mode stiffness $\alpha$ and phonon frequency $\omega(q)$, the fluctuations about the cylindrical shape are given by[@footnote5] $$\label{eq:correlation}
\frac{\left\langle R(z)R(0)\right\rangle}{R_0^2}=\frac{1}{2\pi}\int_{-\infty}^{\infty}dq
\frac{\hbar\omega(q)}{\alpha(q)}
\left[\frac{1}{e^{\beta\hbar\omega(q)}-1}+\frac{1}{2}\right]e^{iqz}.$$ Figure \[Fig:correlations\] shows the correlations for different wire lengths at zero temperature. Fluctuations with $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ increase with increasing wire length. Note that these CDW correlations may be pinned by disorder, or at the wire ends.[@Gruener88] The correlations shown in Fig.\[Fig:correlations\] are representative of regions far from an impurity or wire end.
For large $z$, we can use a saddle point approximation to estimate the integral in Eq. (\[eq:correlation\]), and find $$\label{eq:saddlepointapprox}
\frac{\left\langle R(z)R(0)\right\rangle}{R_0^2} \propto
\cos(2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}z)\;K_0\!\left(\sqrt{12\log\!\left({L_c}/{L}\right)}\;\frac{z}{L}\right),$$ where $K_0$ is the modified Bessel function of the second kind[@footnote3] of order 0. Its asymptotic behavior is given by[@Abramowitz-book] $$\begin{aligned}
K_0(x)&\sim&\left\{
\begin{array}{cl}
-\gamma_E-\log{x/2} & \quad\mbox{for}\;x\ll1,\\
\sqrt{\frac{\pi}{2x}}e^{-x} & \quad\mbox{for}\;x\gg1.
\end{array}
\right.\end{aligned}$$ Depending on the ratio of the wire length $L$ to the critical length $L_c$, we can distinguish two regimes: for $L\ll L_c$, the prefactor of ${z}/{L}$ in the argument of $K_0$ in Eq.(\[eq:saddlepointapprox\]) is large and the correlations decay exponentially for sufficiently large $z$. On the other hand, if $L\sim L_c$, our theory predicts a logarithmic decay of the correlations. Note that the ratio $L/L_c$ determines the crossover from a regime where the harmonic approximation about a Fermi liquid is valid ($L<L_c$) to that of a fully-developed CDW ($L>L_c$). The harmonic approximation, which we have used in our calculation, breaks down at $L=L_c$, where the wire can no longer be treated as a cylinder with small perturbations.
The correlation length $\xi$ is a material-specific quantity, since it depends exponentially on the smooth contribution (\[eq:alpha:smooth\]) to the mode stiffness \[cf. Eq.(\[eq:xi\])\]. As discussed in detail in Ref., the material-specific surface tension and curvature energy can be included in the NFEM through a generalized constraint on the allowed deformations of the wire $$\label{eq:constraint}
{\cal N}=k_F^3{\cal V}-\eta_s\,k_F^2\,{\cal S}+\eta_c\,k_F{\cal C}\;=\;\mbox{const}.\;$$ Here ${\cal V}$ is the volume of the wire, ${\cal S}$ its surface area, and ${\cal C}$ its integrated mean curvature. The constraint $\partial{\cal N}=0$ restricts the number of independent Fourier coefficients in Eq.(\[eq:geometry\]), and allows $b_0$ to be expressed in terms of the other Fourier coefficients. This results in a modification of the smooth part of the mode stiffness (see App. \[app:alpha\]). Through an appropriate choice of the dimensionless parameters $\eta_s$ and $\eta_c$, the surface tension and curvature energy can thus be set to the appropriate values for any given material.[@Urban06]
The upper panel of Fig. \[Fig:xi\] shows the correlation length calculated using the material parameters[@Tyson77] for Na and Au. For clarity, the plot is restricted to the cylindrical wires of so-called *magic radii* that prove to be remarkably stable even when allowing symmetry-breaking deformations.[@Urban04; @Urban06] These wires have conductance values $G/G_0=1,3,6,12,17,23,34,42,51,...$, where $G_0=2e^2/h$ is the conductance quantum.[@footnote6] Arrows point to the positions of the most stable wires, defined as those with the longest estimated lifetimes,[@Buerki05] which are near the minima of the shell potential for straight cylindrical wires, shown in the lower panel of Fig. \[Fig:xi\].
{width="13cm"}
The maximum length of free-standing metallic nanowires observed so far in experiments is that of gold wires produced by electron beam irradiation of thin gold films in ultra high vacuum,[@Kondo00] for which lengths of $L\sim3$–15 nm are reported. In those experiments, no sign of an onset of the Peierls-CDW was seen, presumably indicating that $L<\xi$. However, since the correlation length depends exponentially on the wire radius, [*we predict that nanowires of currently available dimensions can be driven into the CDW regime by applying strain*]{}. A tensile force of order 1 nN can change $\xi$ by orders of magnitude, and thereby drive the system into the CDW regime as soon as the condition (\[eq:CompareLscales\]) is met. Note that since $k_F$ is of order 1 Å$^{-1}$ for Na and Au, the thermal length $L_T$ at room temperature is of order 100 for Na and twice as large for gold, comparable to the lengths of the longest free-standing wires currently produced. By contrast, the elastic mean-free path can be several times as long[@Agrait03] and electron microscope images of gold nanowires[@Kondo00] show perfectly regular and disorder-free atomic arrangements. Surface roughness does not play a role in the parameter regime we consider, where electron shell effects dominate over ionic ordering. Thus CDW behavior should be observable at room temperature in free-standing metal nanowires under strain, in contrast to the behavior of quasi-one-dimensional organic conductors,[@Gruener88] where CDW behavior is observed only at cryogenic temperatures.
Summary and Discussion {#sec:Summary:Ch4}
======================
In conclusion, we have presented a scaling theory of the Peierls-CDW in multi-channel metal nanowires. Near the critical point, scaling relations for the $L$-, $q$- and $T$-dependence of the singular part of the free energy, which drives the Peierls instability, were established. A hyperscaling ansatz was verified and the universal scaling function was analyzed, which was found to be logarithmic. The crossover from a regime where the harmonic approximation about a Fermi liquid is valid ($L<L_c$) to that of a fully-developed CDW ($L>L_c$) occurs at a critical length of order the correlation length $\xi_\nu$, which is material dependent. We predict that the Peierls-CDW should be observable at room temperature in currently available metal nanowires under an applied strain.
The critical length is shortest in materials whose surface tension is small in natural units (i.e., in units of $E_F
k_F^2$). Notable in this respect[@Tyson77] is Al with $\sigma_s = 0.0018 E_F k_F^2$, some five times smaller than the value for Au. Although Al is a multivalent metal, it has a very free-electron-like band structure in an extended-zone scheme, and thus may be treated within the NFEM, although the continuum approximation is more severe. We thus predict that Al should be an ideal candidate for the observation of the Peierls-CDW.
Our findings on the finite-size scaling of the Peierls-CDW in metal nanowires are in contrast with previous theoretical studies[@Nathanson92; @Montambaux98] of mesoscopic rings: For spinless fermions in a one-dimensional ring, the Peierls transition was found to be suppressed for small systems when the number of fermions is odd, but enhanced when the number is even. Obviously, no such parity effect occurs in an open system, such as a metal nanowire suspended between two metal electrodes; we find that the CDW is always suppressed in nanowires with $L\ll
\xi_\nu$. Nonetheless, the correlation length $\xi$ was also found to control the (very different) finite-size scaling in mesoscopic rings.[@Montambaux98]
The finite-size scaling of the Peierls-CDW in metal nanowires is similar to that of the metal-insulator transition in the one-dimensional Hubbard model.[@Stafford93] In both cases, there is no phase transition in an infinite system, because the critical electron-phonon coupling and on-site electron-electron repulsion is $0^+$ in each case. However, for parameters such that the gap $2\Delta$ in an infinite system is sufficiently small, the system is close to a quantum critical point,[@Hertz76; @Stafford93] and the crossover from Fermi-liquid behavior to a Peierls-CDW or Mott insulator, respectively, can be described within hyperscaling theory.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge the Aspen Center of Physics, where the final stage of this project was carried out. This work was supported by the DFG and the EU Training Network DIENOW (D.F.U. and H.G.) and by NSF Grant No. 0312028 (C.A.S.).
Linear stability analysis {#app:alpha}
=========================
This appendix gives details on the linear stability analysis (cf. Ref. ) which determines the leading-order change in the grand canonical potential $\Omega$ of the electron gas due to a small deformation of a cylindrical nanowire. Since the nanowire is an open system connected to macroscopic metallic electrodes at each end, it is naturally described within a scattering matrix approach. The Schrödinger equation can be expanded as a series in the perturbation, and we solve for the energy-dependent scattering matrix $S(E)$ up to second order. The electronic density of states can then be calculated from $$\begin{aligned}
D(E) &=&\frac{1}{2\pi i}\;\mbox{Tr}\left\{
S^{\dagger}(E)\frac{\partial S}{\partial E} -
\frac{\partial S^{\dagger}}{\partial E}S(E)\right\},\end{aligned}$$ where a factor of 2 for spin degeneracy has been included. Finally, the grand canonical potential $\Omega$ is related to the density of states $D(E)$ by $$\label{eq:OmegaVonD}
\Omega=-k_{B} T \int \!dE\; D(E) \;
\ln\!\left[1+e^{-\frac{(E-\mu)}{k_{\mbox{\tiny B}}\,T}}
\right],$$ where $k_B$ is the Boltzmann constant, $T$ is the temperature, and $\mu$ is the chemical potential specified by the macroscopic electrodes. We find that the change $\delta\Omega$ due to the deformation of an initially axisymmetric geometry in leading order is quadratic in the Fourier coefficients $b_q$ of the deformation (cf. Eq. \[eq:geometry\]) and can be written as stated in Eq. (\[eq:Epot:phonons\]), defining the mode stiffness $\alpha(q,R_0, L, T)$. It is convenient to decompose $\alpha$ into three contributions, $\alpha=\alpha_{{\rm diag}}+\alpha_{{\rm nond}}+\alpha_{{\rm con}}.$ The coupling between channels mediated by the surface phonons determines $\alpha_{{\rm diag}}$, coming from scattering into the same channel (Eq. \[eq:alpha:diag\]), and $\alpha_{{\rm nond}}$, coming from scattering between different channels (Eq. \[eq:alpha:nond\]):
$$\begin{aligned}
\label{eq:alpha:diag} \alpha_{{\rm diag}}(q,R_0,L)&=&
\frac{1}{\pi}\sum_{n}\Theta(E_F\!-\!E_{n})\left[
12E_{n}{k_{{{\mbox{\tiny F}}}\!,\,n}}-\frac{4E_{n}^2}{q}\left(
\ln\left|\frac{2{k_{{{\mbox{\tiny F}}}\!,\,n}}+q}{2{k_{{{\mbox{\tiny F}}}\!,\,n}}-q}\right| - F((2{k_{{{\mbox{\tiny F}}}\!,\,n}}+q)L) + F(|2{k_{{{\mbox{\tiny F}}}\!,\,n}}-q|L)\right)
\right]\qquad
\\
\label{eq:alpha:nond} \alpha_{{\rm nond}}(q,R_0,L)
&=&-\frac{1}{\pi}\sum_{{n,n'}\atop{n\neq n'}}f_{n,n'}\Theta(E_F\!-\!E_{n})\;
\left[16{k_{{{\mbox{\tiny F}}}\!,\,n}}\frac{E_{n}E_{n'}}{E_{n}-E_{n'}}+\frac{4E_{n}E_{n'}}{q}
\ln\!\left|\frac{q^2+E_{n'}-E_{n}+2q{k_{{{\mbox{\tiny F}}}\!,\,n}}}{q^2+E_{n'}-E_{n}-2q{k_{{{\mbox{\tiny F}}}\!,\,n}}}\right|\;+
\right.\nonumber\\
&&\qquad\qquad\qquad
+\Theta(E_F\!-\!E_{n'})\frac{4E_{n}E_{n'}}{q}\Big(
F(|q-{k_{{{\mbox{\tiny F}}}\!,\,n}}-{k_{{{\mbox{\tiny F}}}\!,\,n'}}|L)-F((q+{k_{{{\mbox{\tiny F}}}\!,\,n}}+{k_{{{\mbox{\tiny F}}}\!,\,n'}})L)\Big)
\bigg]
\\
\label{eq:alpha:con}
\alpha_{{\rm con}}(q,R_0)&=&\frac{1}{\pi}\sum_{n}\Theta(E_F\!-\!E_{n})4E_{n}{k_{{{\mbox{\tiny F}}}\!,\,n}}\frac{1+\left(\eta_c-\eta_s\,{R_0k_F}\right)
\left({q}/{k_F}\right)^{\!2}}{1-{\eta_s}/({R_0k_F)}}.\end{aligned}$$
Here $f_{n,n'}=1$ for two channels having the same azimuthal symmetry and $f_{n,n'}=0$ otherwise. The function $F(x)=\mbox{Ci}(x)-\sin(x)/x$ smoothens the logarithmic divergences so that $\alpha_{{\rm diag}}$ and $\alpha_{{\rm nond}}$ are continuous functions having minima of length-dependent depth at $q=2{k_{{{\mbox{\tiny F}}}\!,\,n}}$ and $q={k_{{{\mbox{\tiny F}}}\!,\,n}}+{k_{{{\mbox{\tiny F}}}\!,\,n'}}$, respectively. The third contribution (Eq. \[eq:alpha:con\]) arises due to enforcing the constraint (\[eq:constraint\]) on allowed deformations.
The contribution to Eq. (\[eq:alpha:nond\]) from evanescent modes with $E_{n'}>E_F$ gives rise to the leading-order $q$-dependence of $\alpha_{{\rm nond}}$, which is quadratic. This term therefore essentially captures the change of surface and curvature energy \[cf. Eq. (\[eq:alpha:smooth\])\]. Figure \[fig:alpha\_compare\] compares the mode stiffness $\alpha$, computed from Eqs. (\[eq:alpha:diag\])–(\[eq:alpha:con\]), to the Weyl approximation \[first two terms on the r.h.s. of Eq.(\[eq:alpha:smooth\])\]. Here $R_0=5.75 k_F^{-1}$ and $L=1000
k_F^{-1}$. Both the results for Au ($\eta_s=0.76$, $\eta_c=-0.11$) and for a pure constant-volume constraint ($\eta_s=\eta_c=0$) are shown. In both cases, the overall $q^2$-dependence of $\alpha$ for large $q$ is evident, and consequently, the minimum at $q=2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}\approx 0.56 k_F$ is deeper than the other minima.
![(Color online) The mode stiffness $\alpha$ computed from Eqs. (\[eq:alpha:diag\])–(\[eq:alpha:con\]) \[solid curves\]. The dashed curves show the Weyl approximation to $\alpha$ \[first two terms on the r.h.s. of Eq. (\[eq:alpha:smooth\])\]. The results for gold and for an idealized free-electron metal are shown; in each case, the radius is $R_0=5.75 k_F^{-1}$ and the length is $L=1000k_F^{-1}$. []{data-label="fig:alpha_compare"}](fig5.eps){width="8.5cm"}
When examining the softening of phonon modes, the overall increase of $\alpha(q)$ with $q$ allows us to concentrate on the $q_0\equiv 2{k_{{{\mbox{\tiny F}}}\!,\,\nu}}$ mode, where $\nu$ labels the highest open channel. This mode will always be the dominant one. In general, the mode stiffness also shows less-pronounced minima at other values $q>q_0$, due to the lower-lying channels, but these contributions are less singular than the $q_0$ mode. It is therefore possible to approximate $\alpha$ by a smooth contribution $\propto q^2$ plus the explicit terms of Eq. (\[eq:alpha:diag\]) for the highest open channel.
Perturbation potential matrix elements {#app:matelement}
======================================
Consider free electrons confined within a cylindrical nanowire of radius $R_0$ and length $L$ by a step potential $V(r)=V_0\theta(r-R_0)$, where $\theta$ is the step function. The transverse eigenfunctions in polar coordinates are given by $\psi^\perp_{mn}(r,\varphi)=(2\pi)^{-\frac{1}{2}}e^{im\varphi}\chi_{mn}(r)$, where the radial function $\chi_{mn}(r)$ reads $$\chi_{mn}
=N_{mn}\!\left\{\!
\begin{array}{ll}
{ J_{m}}\!\big(\sqrt{E_{mn}}\,r\big), &r<R_0,\\
\frac{{ J_{m}}\!\big(\!\sqrt{\!E_{mn}}\,R_0\big){ K_{m}}\!\big(\sqrt{V_0\!-\!E_{mn}}\,r\big)}{{ K_{m}}\!\big(\!\sqrt{V_0-E_{mn}}\,R_0\big)}\,
, &r>R_0.
\end{array}\right.$$ Here $N_{mn}$ is a normalization factor, ${ J_{m}}$ is the Bessel function of order $m$, ${ K_{m}}$ is the modified Bessel function of the second kind of order $m$, and for simplicity of notation, we use the convention ${\hbar^2}/{2m_e}=1$. The transverse eigenenergies $E_{mn}$ are determined by the continuity of $\partial_r\chi_{mn}/\chi_{mn}$ at $r=R_0$. An axisymmetric perturbation of the wire changes the confinement potential by $${\delta V}(r,z)=V_0\big[\theta(r-R_0-\delta
R(z))-\theta(r-R_0)\big],$$ where the variation in radius is given by $\delta
R(z)/R_0=\sum_qb_qe^{iqz}$ and the perturbation wavevectors $q$ are restricted to integer multiples of $2\pi/L$. We expand the matrix elements of $\delta V$ with respect to the unperturbed eigenfunctions $\Psi_{mn k}(r\!,\varphi,z)=
L^{-1/2}e^{ikL}\psi^\perp_{mn}(r,\varphi)$ to first order in $b_q$ and get $$\begin{aligned}
\langle mn
k\vert\,{\delta V}\vert\,\bar{m}\bar{n}\bar{k}\rangle&\simeq&
-\delta_{m\bar{m}}V_0R_0\,{\chi}_{mn}(R_0){\chi}_{\bar{m}\bar{n}}(R_0)\nonumber\\
&&\times\sum_qb_q\frac{1-e^{i(\bar{k}-k+q)L}}{i(\bar{k}-k+q)L}\,,\end{aligned}$$ Taking the limit $V_0\rightarrow\infty$ we get $$\begin{aligned}
\lim_{V_0\rightarrow\infty}\langle mn
k\vert\,{\delta V}\vert\,m\bar{n}\bar{k}\rangle&\simeq&
2\sqrt{E_{mn}E_{m\bar{n}}}\nonumber\\
&&\times\sum_qb_q\frac{1-e^{i(\bar{k}-k+q)L}}{i(\bar{k}\!-\!k\!+\!q)L}.\qquad\end{aligned}$$ In the limit of $L\rightarrow\infty$ we have $$\lim_{{V_0\rightarrow\infty}\atop{L\rightarrow\infty}}\langle mn
k\vert\,{\delta V}\vert\,m\bar{n}\bar{k}\rangle\simeq-
2\sqrt{E_{mn}E_{m\bar{n}}}\;b_q\,\delta_{\bar{k},k-q}$$ and recover a coupling of states with $\bar{k}=k+q$, only.
|
---
author:
- 'Daniel J. Lennon'
- 'Roeland P. van der Marel'
- Mercedes Ramos Lerate
- 'William O’Mullane'
- Johannes Sahlmann
date: 'Received ; accepted'
subtitle: 'Candidate hypervelocity star discovery, and the nature of R71'
title: Gaia TGAS search for Large Magellanic Cloud runaway supergiant stars
---
[To search for runaway stars in the Large Magellanic Cloud (LMC) among the bright [*Hipparcos*]{} supergiant stars included in the [*Gaia*]{} DR1 TGAS catalog.]{} [We compute the space velocities of the visually brightest stars in the Large Magellanic Cloud that are included in the Gaia TGAS proper motion catalog. This sample of 31 stars contains a Luminous Blue Variable (LBV), emission line stars, blue and yellow supergiants and a SgB\[e\] star. We combine these results with published radial velocities to derive their space velocities, and by comparing with predictions from stellar dynamical models we obtain their (peculiar) velocities relative to their local stellar environment.]{} [Two of the 31 stars have unusually high proper motions. Of the remaining 29 stars we find that most objects in this sample have velocities that are inconsistent with a runaway nature, being in very good agreement with model predictions of a circularly rotating disk model. Indeed the excellent fit to the model implies that the TGAS uncertainty estimates are likely overestimated. The fastest outliers in this subsample contain the LBV R71 and a few other well known emission line objects though in no case do we derive velocities consistent with fast ($\sim$100km/s) runaways. On the contrary our results imply that R71 in particular has a moderate deviation from the local stellar velocity field (40km/s) lending support to the proposition that this object cannot have evolved as a normal single star since it lies too far from massive star forming complexes to have arrived at its current position during its lifetime. Our findings therefore strengthen the case for this LBV being the result of binary evolution. Of the two stars with unusually high proper motions we find that one, the isolated B1.5 Ia$^+$ supergiant Sk-672 (HIP22237), is a candidate hypervelocity star, the TGAS proper motion implying a very large peculiar transverse velocity ($\sim$360km/s) directed radially away from the LMC centre. If confirmed, for example by [*Gaia*]{} Data Release 2, it would imply that this massive supergiant, on the periphery of the LMC, is leaving the galaxy where it will explode as a supernova. ]{}
Introduction
============
A new model of the stellar dynamics of the Large Magellanic Cloud (LMC) was presented by [@vdM16] (from now on vdMS) based on the proper motions of 29 Hipparcos stars that are LMC members and that have suitably precise proper motions published in the Tycho-Gaia (hereafter TGAS) catalog of the first [*Gaia*]{} Data Release (DR1; [@gaiaa; @gaiab], [@lindegren]). Besides their use as test particles of the LMC velocity field, these stars are of great intrinsic interest due to their representing a sample of the visually brightest stars in a nearby star forming galaxy. This selection of blue and yellow supergiant stars (see Table 1 and Figure 1) includes a number of well known peculiar and interesting objects such as the Luminous Blue Variable (LBV) R71, a SgB\[e\] star (Sk-6946), a candidate LBV (Sk-6975; [@prinja]) and some peculiar emission line objects (such as HD37836; [@shore]).
The dynamical properties of these massive stars are of interest. Those field stars that are relatively isolated, being far from young clusters, are of particular significance since there are competing hypotheses to explain their solitude. Possible explanations include [*in situ*]{} formation of single massive stars ([@parker]), or ejection of runaway stars (with peculiar velocities of order 100 km/s) via disruption of a close binary by a SN explosion ([@blaauw]) or dynamical interaction in a massive cluster ([@poveda]). Indeed [@demink2012; @demink2014] point out that binary evolution channels enable a continuum of runaway velocities, even including slow runaways, the so-called walkaways. At the other extreme, even higher peculiar velocities (hundreds of km/s) are thought to be generated through interaction with the SMBH at the centre of our Milky Way ([@brown]) producing hyper-velocity stars, although alternative mechanisms are suggested with the discovery of hypervelocity B-stars that do not fit this paradigm ([@przybilla]). Clearly the dynamical signature of these processes is a crucial discriminant in disentangling these scenarios.
Within the Magellanic Clouds the investigation of runaway stars (in general) has been limited mainly to circumstantial evidence such as apparent isolation, peculiar radial velocity, or the presence of an infrared excess that might be attributed to the presence of a bow shock ([@evans], [@gvaramadze]). As a specific example we point to the current debate as to whether some, or all, LBVs are runaway stars. [@st15] suggests that LBVs are the ‘kicked mass gainers’ in binary evolution arguing that their relative isolation is inconsistent with single star evolution. Although this hypothesis is heavily debated (see [@hum16], [@smith] and [@davidson]) it is clear that the evidence is circumstantial since proper motions for stars in the LMC cannot yet discriminate between various propositions (but see [@platais] for a first attempt to measure precise relative proper motions of massive stars around 30 Doradus). The [*Gaia*]{} mission is set to change this picture and fortunately, as noted above, our sample contains the most isolated LBV in the LMC, R71, as well as a number of emission line objects potentially related to LBVs.
In this paper we use the proper motions from the [*Gaia*]{} TGAS catalog, along with published radial velocities to estimate stellar velocities relative to their local mean stellar velocity field in order to search for runaway stars. In Section 2 we describe our methods and discuss the results for the sample as whole, while in Section 3 we focus on a few individual objects that are of special interest.
![Open symbols represent positions of the stars in our LMC sample in the HR diagram. Effective temperatures and luminosities are taken from the literature where available, or estimated using spectral types and published photometry. As many of these stars are photometric and spectroscopic variables their positions in the HRD are illustrative. As an extreme example the positions of the LBV R71 are indicated (data from [@mehner]) joined by dotted lines. For convenience we indicate the position of the candidate hypervelocity star Sk-672. We also indicate the position of the zero age main sequence (left, labeled with initial masses, tracks from [@brott] and [@kohler]) and the approximate location of the Humphreys-Davidson limit.[]{data-label="Fig1"}](fig1.pdf)
![Black arrows indicate magnitudes and directions of proper motions of the 29 stars from vdMS (the LBV R71 is labelled), while blue arrows denote positions and proper motions of Sk-672 (HIP22237) and Sk-7142 (HIP25815) as indicated by the labels. The background image illustrates the stellar density map and contours for the LMC derived from the $Gaia$ catalogue.[]{data-label="Fig2"}](LMC_Fig2_color.pdf)
Stellar relative velocities
===========================
Cross-matching the TGAS catalogue with the master catalogue of LMC massive stars from [@bonanos], supplemented by the full Sanduleak catalogue ([@sanduleak]) of bright LMC members, reveals 311 sources in common. Examination of the uncertainties in the measured proper motions reveals two clear groupings; 31 sources, all [*Hipparcos*]{} sources, with proper motion uncertainties less than approximately 0.25 mas/yr, and all others, the [*TYCHO*]{} sources, with uncertainties in excess of approximately 0.6 mas/yr. In vdMS the TGAS proper motions of 29 of these 31 [*Hipparcos*]{} sources were used to improve the three dimensional model of the rotation field of that galaxy (see also [@kroupa]). This work built upon the previous work of [@vdM14] that modelled this field as a rotating flattened disk, constrained by 6790 line-of-sight (LOS) stellar velocities and the average proper motions of 22 fields in the LMC as derived from [*Hubble Space Telescope*]{} observations. Implicit in the use of these 29 stars to constrain this model is the expectation that their velocities are a reasonable reflection of the mean stellar motions at their respective positions. It follows that, say, any runaway star present in this sample, with a peculiar relative velocity in excess of around 50km/s (about 0.2 mas/yr in the LMC), would have larger residuals in the model fit than stars that are not runaway stars. In this context we note that the formal errors in the proper motions of these stars presented in TGAS have a mean value of $\sim$ 0.16 mas/yr in both right ascension and declination. While this is $\sim$37 km/s in velocity terms (see Table 1), as we discuss below we have reason to believe that the actual errors are substantially smaller than this. Regardless of this latter point, the precision of the TGAS proper motions is even now sufficient to search for (fast) runaway stars among our sample.
We therefore compute the residuals between each star’s proper motion, and the vdMS model prediction, as a measure of its velocity relative to the mean motion of stars in the LMC. We also compute the peculiar LOS velocity by comparing their radial velocities with the model estimates as constrained by [@vdM14] (see their Fig.4). These results are listed in Table 1 where we have converted proper motion to velocity assuming a distance to the LMC of $50.1\pm2.5$ kpc, corresponding to a distance modulus of m-M=$18.50\pm0.1$ ([@freedman]). We note that the distance uncertainty also implies a $\sim$5% systematic uncertainty in predicted and derived velocities. In Tabel 1 we also include two stars, HIP22237 (Sk-67 2) and HIP25815 (Sk-71 42), that were excluded by vdMS on the basis that their proper motions are unusual and their excess\_astrometric\_noise parameter is large compared to other stars in the sample. We have included these two objects in our analysis and will return to them in the next section. For now we point our that their resultant peculiar velocities are very large, more akin to hypervelocity stars, and we exclude them from the following discussion of the statistical properties of the sample of 29 stars. However we will return to these two objects in the next section, while the measured proper motions of all 31 stars are illustrated in Fig.2, that also serves as context for insight into each star’s local environment within the LMC.
In Fig.3 we illustrate our results as velocity dispersion histograms in each of the three directions of motion. One of the striking aspects of the overall results is that the goodness of fit to the model is excellent, there are no outliers beyond 50 km/s in right ascension or declination (see also Table 1). In fact the standard deviations of the residuals in right ascension and declination are each approximately 19 km/s, substantially less than the value that might be expected given that the formal proper motion errors predict a mean of 37km/s in velocity as mentioned above (well outside the 5% uncertainty in the distance to the LMC). Fig.4 shows the residual vectors in a polar plot indicating that there is no preferred direction for the peculiar velocities. We therefore interpret the proper motions of the present sample of stars as implying that they do not harbour fast runaways (with peculiar velocities in excess of 50km/s).
The LOS velocity information is more difficult to interpret since measured radial velocities are subject to a number of effects that introduce off-sets from the true LOS velocity. All of these are very luminous (in the approximate range $10^5 -10^6$ solar luminosities) and therefore many have strong winds that will effect the centroids of strong absorption lines and emission lines. The impact of the former is small, typically introducing a small blue-shift of $\sim$10km/s ([@arp]), while pure emission lines may be even more blue-shifted since they form in the outflowing wind (see the discussion of the forbidden \[Fe[[ii]{}]{}\] lines of R71 in the next section, and the discussion of the radial velocity of the suspected runaway or walkaway VFTS682 near 30 Doradus by [@vftsIII]). The details of course will depend crucially on the parameters of the stellar wind (stellar radius, mass-loss rate, terminal velocity and velocity law). Multiplicity can also play a role (it is known that Sk-6883 is an eclipsing binary with a semi-amplitude of $\sim$35km/s) and will in general lead to an overestimate of the velocity dispersion if ignored ([@vftsVII]). However the multiplicity characteristics of these types of stars is poorly understood at present, and many stars of our sample have only single epoch measurements. Given these caveats, and also noting the heterogeneous nature of the sources for the radial velocities, it is perhaps not too surprising to note that the velocity dispersion derived here of $\sim$20 km/s is larger than the 11.6 km/s derived by [@vdM14] for young red supergiants using the same model.
Summarising the results of this section, with the exception of the two anomalous stars Sk-67 2 and Sk-71 42, the proper motions and LOS velocities of this sample are consistent with the hypothesis that none are fast runaways, though there are some outliers that may be slower runaway stars with peculiar [*space*]{} velocities around 50km/s. A subset of the 5 fastest slow runaways includes the isolated B-hypergiant Sk-688, the LBV R71, the candidate LBV Sk-6975, and the SgB\[e\] star R66.
![Illustrated here are the velocity residuals in right ascension (v$_W$, red), declination (v$_N$, blue) and along the LOS (v$_{LOS}$, green). Histograms have a bin size of 10km/s and the mean peculiar velocities in W, N and LOS directions are 2.8, 1.5 and $-0.9$ km/s respectively, with standard deviations of 18.9, 18.8 and 20.3 km/s. Sk-672 (HIP22237) and Sk-7142 (HIP25815) are not included in these figures or estimates.[]{data-label="Fig3"}](fig3.pdf)
![Left panel is a polar plot of v$_W$ and v$_N$ (from Table 1) while the panel on the right illustrates the cumulative distribution function (CDF) of the positions angles of these vectors as a histogram. The straight dashed line represents the CDF of a random distribution. A formal comparison of the two distributions using the Kuiper test indicates that the observations are consistent with this random distribution at a significance level of 85%. []{data-label="Fig4"}](fig4.pdf)
Discussion: Individual objects of special interest
==================================================
In this section we discuss in more detail some implications of our results for a few objects of special interest.
R71 (= Sk-71 3, HDE269006)
--------------------------
R71 is a well studied LBV that has in recent years been undergoing an outburst, reaching a maximum visual magnitude $\sim$8.5 during the year 2012. It is particularly interesting as it is the most isolated LBV in the LMC, and is also one of the least luminous of this class, but note the influence of assumptions concerning extinction as pointed out my [@mehner], where more details on this object’s recent activity may be found.
There has been a recent suggestion by [@st15] that LBVs, such as R71, are the rejuvenated mass gainer products of massive star binary evolution. Central to this idea is their proposition that LBVs tend to be found in the field and well away from massive star clusters, a consequence of a more massive star donating material to the initially lower mass companion, before exploding as a SN and kicking the now more massive star out of its parent cluster. This runaway star subsequently evolves into its LBV phase though now as a relatively isolated star, before finally exploding as a SN far from its parent cluster. This is disputed by [@hum16] who argue that LBVs are predominantly the products of single star evolution arguing that the locations of massive LBVs are well correlated with O-type stars, while the locations of less luminous LBVs (such as R71) correlate well with locations of red supergiants. They further argue that none of their confirmed LBVs are runaway stars. In a counter argument [@smith] points out that runaway velocities predicted for the mass-gainer in the binary evolution scenario are not necessarily high, in many cases ranging from slow (walkaway) velocities of a few km/s to a few tens of km/s ([@demink2014]). The rejuvenated star also has a lifetime longer than would be expected for one of its luminosity if it were a single star, aiding and abetting the appearance of unusual isolation. Interestingly they attribute a LOS peculiar velocity of $-71$km/s to R71 arguing that it is consistent with it being a potential fast runaway, but see our discussion below, and the argument in [@davidson] concerning this star’s LOS velocity. While we cannot shed light on the LBV population of the LMC as a whole we can address the nature of R71. As discussed by [@st15] it is the most isolated of the confirmed LBVs in the LMC. They estimate that there is not any O-star within 300pc, the closest being almost half a degree distant (or 450pc) in projection.
Concerning R71’s LOS velocity, we adopt a value of 204km/s (see Table 1), compared to the value of 192km/s used by [@mehner] that is commonly also used by other authors. As they report however this latter estimate is based on the \[Fe[ii]{}\] lines. However [@wobig] reports a discrepancy between these lines and the absorption lines of neutral helium, metals and higher series Balmer lines that imply a systemic radial velocity of 204$\pm 5$kms, a result also obtained by [@wolf81]. Since the \[Fe[ii]{}\] lines form in the expanding wind (as also noted by both [@mehner] and [@wolf81]), we adopt the higher radial velocity of 204km/s that is weighted more towards the weaker photospheric absorption features in its spectrum that should be less influenced by its wind. We caution however that even these lines may well be slightly impacted and the actual systemic LOS velocity may be a little higher. Comparing with the model predicted LOS velocity at this position of 238km/s gives a peculiar LOS velocity of only $-34$km/s. (Comparing to the average red and yellow supergiant velocities from [@neugent] produces a similar result, implying a mean velocity at this position of $\sim$235 km/s and a very slightly lower LOS peculiar velocity of $-31$ km/s).
Our results for R71 indicate that both its proper motion and LOS velocity are most likely inconsistent with it being a fast runaway star (with peculiar velocity in the range 50–100km/s), having a transverse velocity of around 38km/s (see Table 1) though with a formal uncertainty of $\pm 34$km/s (but note our assessment that more realistic errors may be roughly half this value). At this velocity it would take almost 12Myr to travel the 450pc from the nearest grouping of OB stars, or about 6Myr assuming the uncertainty adds to the velocity (or 9Myr with our more optimistic error estimate). These numbers are difficult to reconcile with single-star main-sequence lifetimes of approximately 8, 5, 4 and 3 Myrs for stars with initial masses of 20, 30, 40 and 60M$_{\odot}$ ([@brott]).
Sk-672 (= HDE270754 = HIP22237)
-------------------------------
Due to its moderately high extinction of $E(B-V)$$\sim$ $0.2$, Sk-672 has been the subject of many observations with various facilities as a means of studying the interstellar medium of the LMC (see [@welty] for an example). The star is also classified by [@fitz] as belonging to a those B-type supergiants, denoted as [*N weak*]{} or [*BC*]{}, that appear nitrogen deficient with respect to the morphologically normal nitrogen rich supergiants. We find that this isolated star (it lies at the north-west edge of the galaxy, see Fig.2) has a very large peculiar velocity of 359km/s (Table 1) in a direction away from the LMC. This star was omitted from the vdMS sample, in part because its peculiar proper motion implies that it cannot help constrain the overall LMC rotation field. However, they also noted that this star has the second-highest astrometric\_excess\_noise parameter $a$ in the sample, after Sk-71 42 (see Section 3.3). However, while the latter star ($a$ = 2.58) is truly a strong outlier, this is not the case for Sk-67 2 ($a$ = 1.11). The remaining 29 sample stars have a mean $\langle a \rangle
= 0.51$ with an RMS of $0.18$. There are two other stars in the sample with $a \sim 0.9$, HIP 22849 and 27868, and these do not have unusual proper motions (see Table 1). Therefore, the astrometric\_excess\_noise for Sk-67 2 does not necessarily imply that its measured TGAS proper motion must suffer from unidentified systematics. It is consistent with being an $\sim$3$\sigma$ random Gaussian outlier in the overall distribution. We also note that Sk-67 2 has a strong mid- and far-infrared excess ([@bonanos]; see also Fig.5) that would be also be consistent with a high peculiar velocity if that motion were to drive a bow shock as the star ploughs through the interstellar medium.
We have a high resolution [feros]{} echelle spectrum of this object obtained in 2004, and we have extracted high resolution VLT/UVES echelle spectra from the ESO archive obtained in 2001 and 2012. Radial velocities measured from the weak metal lines in these data lead to a radial velocity of $320\pm1$km/s for all three epochs. Comparing our measured radial velocity with the predicted mean LOS velocity from [@vdM14] at this position of 277km/s leads to a peculiar velocity of $+43$km/s (comparing to the average red and yellow supergiant velocities from [@neugent] lead to a similar peculiar LOS velocity of $+49$km/s). [@vdM14] also cite a velocity dispersion for the young massive stars in the LMC of 11.6km/s implying that Sk-672 is a clear $\sim$4$\sigma$ outlier, though perhaps not such an obvious outlier in LOS as in proper motion. (Using instead $\sigma$=20 km/s from Fig.3 implies it is a 2.3$\sigma$ outlier.)
Based on the above evidence we argue that this star is a true LMC member and not some peculiar foreground Halo object and, since its velocity vector is almost parallel to the disk plane (see below), it is highly unlikely that it is a more distant star of unexplained origin or nature. To understand whether or not Sk-67 2 is bound we must compare its velocity with the escape velocity of the LMC. While this is currently uncertain, we can obtain an approximate value by comparison with a Milky Way value of 533 km/s as derived from the RAVE survey ([@piffl]). The ratio of these galaxies’ total dark halo masses of 0.061 ([@guo]) then implies an escape velocity for the LMC of $\sim131$ km/s. Alternatively if we directly compare circular velocities for the LMC (92 km/s, [@vdM14]) and Milky Way (239 km/s, [@mcmillan]) we obtain an escape velocity of around 205 km/s. Therefore Sk-67 2 is clearly unbound with respect to the LMC and, as we argue below, a possible hypervelocity star.
While Sk-67 2 is unbound its velocity is still significantly below the surface escape velocity of main sequence star of its approximate mass and therefore not an unambiguous candidate for a hypervelocity star since there are theoretical scenarios in which fast runaway stars, or hyper-runaway stars, might be created by classical runaway star scenarios. However, $N$-body simulations of dynamical ejection from massive clusters ([@perets]) predict very low rates of production of hyper-runaway stars, while binary evolution models have difficulty producing any runaway stars with velocities in excess of $\sim200$ km/s ([@eldridge]). Based on current models therefore it would appear that the hyper-runaway scenario is not a promising explanation of the nature of Sk-67 2.
One can calculate the 3D position and 3D velocity vectors of Sk-672 in an LMC-centric frame using the approach used in [@vdM2002] (this is a one-parameter family depending on the star’s actual distance). The minimum angle between its position and velocity vectors in this frame is 5.3$\pm$5.2 degrees, attained for a distance that is 50.6 kpc (i.e., 0.5 kpc larger than the distance of the LMC, but 0.5 kpc in front of the inclined LMC disk at this location). The uncertainty in this angle is dominated by the uncertainty in the position of the LMC center ($\sim 0.3$ deg per coordinate), which was taken from the joint fit to the HST and TGAS PM data in vdMS. In other words, its velocity vector is almost aligned with the LMC centre, as would be expected for a hypervelocity star ejected by a central massive black hole. We note here however that the star is $\sim 3.5$ kpc from the centre of the LMC implying that if the star distance is such that it is $\pm$3.5 kpc from the LMC disk plane, then the angle between the velocity vector and the vector to the LMC centre would be $\sim$45 degrees.
While no such black hole is known to exist, the presence of a black hole of mass $\leq 10^{7}$M$_{\odot}$ cannot be ruled out by the observed velocity field (Boyce et al.(2017), in preparation). Interestingly [@boubert] have suggested that the presence of a central massive black hole in the LMC might naturally explain the clustering of known hypervelocity stars in the constellations Leo and Sextens, provided that there is an as yet undiscovered southern population loosely focused on the LMC.
Given that its lifetime (below) clearly rules out a Milky Way origin, it is tempting to attribute membership of this population of hypervelocity stars to Sk-67 2. However there are still some issues with this explanation. For example, its distance from the centre of the LMC is $\sim$3.5 kpc that would take $\sim$10 Myr to traverse at its current velocity. Even allowing for the fact that we have ignored the effect of the LMC gravitational potential in estimating its flight time, implying this is an upper limit, it is still more than a factor of two longer than its likely main sequence lifetime. If, like R71, this star is a product of binary evolution, having been a hypervelocity binary ejected from a massive black hole at the centre of the LMC, then indeed there might be evolutionary channels that could be consistent with its current position. On the other hand, this discrepancy between lifetime and flight-time may provide some support for the hypothesis that some hypervelocity stars originate via interaction with an intermediate mass black hole (IMBH) as proposed by [@przybilla] for the star HE0437-5439, thought to have originated from the LMC. However the LMC is not known to host a confirmed IMBH either.
Stars in the LMC with unusual line-of-sight kinematics were previously reported by [@olsen]. These stars have velocity residuals up to $\sim 100$ km/s, and lower metallicities than typical for the LMC. They were interpreted as a population of accreted SMC stars. The three-dimensional velocity difference between the LMC and SMC is $128
\pm 32$ km/s ([@kallivayalil]), consistent with the scale of the observed residual velocities for these stars. By contrast, Sk-67 2 has a velocity residual of $\sim 360$ km/s compared to the LMC. Moreover, direct comparison to the known three-dimensional velocity of the SMC implies a residual velocity of $\sim 336$ km/s with respect to that galaxy. Therefore, Sk-67 2 cannot have an SMC origin (which would also be hard to reconcile with its short main sequence lifetime).
Finally in this subsection we note that the TGAS proper motion is essentially derived from the difference between the [*Hipparcos*]{} and [*Gaia*]{} positions of this object (since its parallax is negligible). One might suppose that some perturbation of the point-spread-function, such as the presence of a close visual companion, might cause an error in the estimated position. However Sk-672 is in a sparsely populated part of the LMC, and our high resolution [feros]{} spectrum shows no evidence for a bright companion. Further, to cause a spurious proper motion of $\sim$1mas/yr in TGAS, one would require an offset of $\sim$20mas in the Gaia astrometry. This is of order one-fifth of an AL/AC-average pixel and in addition this offset would have to be consistent for the different scan angles, that seems rather unlikely. In this context we also note that very good agreement is found between proper motions derived from Hubble and TGAS for the Magellanic Clouds (vdMS) and Globular Clusters in the Milky Way ([@wat]). We conclude that [*Sk-672 is a candidate hypervelocity star*]{} that if confirmed, by for example [*Gaia*]{} Data Release 2 (DR2), would warrant further investigation as to the nature of its origin.
![Spectral Energy Distribution of the candidate hypervelocity star Sk-672 showing its infrared excess from about 10 to 100 $\mu$m that is a signature of dust emission and a possible bow-shock (based on [*Spitzer*]{}, [*WISE*]{} and $IRAS$ data).[]{data-label="Fig5"}](fig5_sed.pdf)
Sk-7142 (=HDE269660 = HIP25815)
-------------------------------
The TGAS proper motions imply a peculiar velocity of $\sim$150 km/s for Sk-7142 but in contrast to Sk-672, it is in a very crowded region of the LMC and lies (in projection) within the environment of the nebula and supernova remnant LHA 120-N206 that hosts many massive O-type stars. Also, by contrast, this star has astrometric\_excess\_noise=2.58, substantially above the values of the rest of the present sample. Also, comparing our measured radial velocity of 238km/s with the mean LOS velocity predicted by the model of [@vdM14] at this position of 239km/s gives a peculiar LOS velocity of only $-1$km/s, while from the monitoring of [@morrell] we know that this star has constant radial velocity. (Comparing to the average red and yellow supergiant velocities of [@neugent] yields a peculiar velocity of $-12$km/s.) The lack of a peculiar LOS velocity, together with its unusually large astrometric\_excess\_noise parameter suggests that we must await further data on this object from [*Gaia*]{} DR2 before speculating further on its nature.
Summary
=======
We have shown that the [*Gaia*]{} TGAS catalogue is even now, with DR1, able to provide important dynamical constraints on a subset of the visually brightest massive stars in the LMC that can be used to address their nature as potential walkaway, runaway or hypervelocity stars. Specific conclusions concerning this sample include;
- Most of these very luminous stars are not runaways, the outliers mostly have peculiar velocities of less than 50km/s.
- R71 is rather unique in its isolation and we have shown that it’s peculiar space velocity is only moderately discrepant from its environment, that is difficult to reconcile with a single-star evolutionary scenario. A potential solution to this dilemma might be that this particular LBV is indeed the result of binary evolution but is the evolved product of a slow runaway binary (either a rejuvenated mass gainer or a stellar merger).
- The isolated B1.5Ia$^+$ supergiant Sk-672 is found to be a candidate hypervelocity star with a peculiar velocity of $\sim$360 km/s directed radially away from the LMC centre, suggesting possible ejection by an as yet undiscovered central black hole. However its main sequence lifetime is difficult to reconcile with the likely flight-time suggesting alternative hypotheses for the origin of its velocity will need to be explored.
This work has made use of data from the ESA space mission Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. DJL thanks Hassan Siddiqui, Uwe Lammers and Jose Hernandez of the ESAC [*Gaia*]{} Science Operations Centre for useful discussions. This research made use of Simbad and Vizier provided by CDS, Strasbourg; ESASky, developed by the ESAC Science Data Centre (ESDC); the ESO archive at Garching; and TOPCAT. J.Sahlmann was supported by an ESA Research Fellowship in Space Science.
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[llrlrlrllrlll]{} Name & HIP & v$_W$ & e$_W$ & v$_N$ & e$_N$ & v$_{\rm t}$ & e$_{\rm t}$ & v$_{\rm rad}$ & v$_{LOS}$ & Common aliases & Sp.Type & $Gaia$ source Id.\
Sk-67 2 & 22237 & 251.4 & 27.8 & 256.7 & 24.6 & 359.3 & 26.2 & 320 & 43.0 & R51, HDE270754 & B1.5 Ia$^+$ & 4662413602288962560\
Sk-69 7 & 22392 & -8.5 & 35.8 & -3.2 & 35.8 & 9.1 & 35.8 & 263 & 4.3 & R52, HDE268654 & B9 Ia & 4655349652394811136\
Sk-68 8 & 22758 & 30.3 & 37.9 & -40.6 & 40.6 & 50.7 & 39.7 & 271 & -1.3 & R58, HDE268729 & B5 Ia$^+$ & 4655510043652327552\
Sk-69 30 & 22794 & 11.2 & 20.8 & 0.5 & 20.9 & 11.3 & 20.8 & 258 & -8.1 & R59, HDE268757 & G7 Ia$^+$ & 4655460771785226880\
Sk-66 12 & 22849 & -9.0 & 28.6 & 8.6 & 30.8 & 12.5 & 29.7 & 284 & -4.4 & R61, HDE268675 & B8 Ia & 4661769941306044416\
Sk-67 19 & 22885 & 3.0 & 34.5 & 12.2 & 35.0 & 12.6 & 35.0 & 304 & 17.6 & HDE268719 & A2 Ia & 4661720532007512320\
Sk-69 37 & 22900 & 13.4 & 55.1 & 2.7 & 52.5 & 13.6 & 55.0 & 252 & -3.0 & HDE268819 & F5 Ia & 4655136518933846784\
R66 & 22989 & 29.3 & 55.7 & 25.1 & 53.9 & 38.6 & 54.9 & 264 & 6.4 & Sk-69 46, HDE268835, S73 & SgB\[e\] & 4655158131209278464\
Sk-65 8 & 23177 & -2.5 & 38.4 & 6.2 & 37.7 & 6.7 & 37.8 & 311 & 15.3 & HDE270920 & G0 Ia & 4662293892954562048\
R71 & 23428 & 33.7 & 34.7 & -16.8 & 33.4 & 37.7 & 34.4 & 204 & -33.5 & Sk-713, HDE269006, S155 & LBV & 4654621500815442816\
Sk-70 48 & 23527 & 26.5 & 57.1 & 16.6 & 54.7 & 31.3 & 56.4 & 216 & -28.3 & HDE269018, GV183 & B2.5 Ia & 4655036841335115392\
Sk-66 58 & 23665 & 1.0 & 24.8 & -20.6 & 27.2 & 20.7 & 27.2 & 300 & -0.2 & R75, HDE268946 & A0 Ia & 4661920986713556352\
Sk-67 44 & 23718 & 0.0 & 17.1 & 7.5 & 18.8 & 7.5 & 18.8 & 255 & -37.4 & R76, HD33579, S171 &A2 Ia$^+$ & 4661472145451256576\
HDE271018 & 23820 & 12.2 & 50.4 & 25.0 & 51.3 & 27.8 & 51.2 & 291 & -11.5 & GV538 & F8 Ia & 4662061311885050624\
Sk-71 14 & 24006 & -32.2 & 24.9 & -7.1 & 23.3 & 33.0 & 24.9 & 243 & -7.9 & R80, HDE269172 & A0 Ia & 4651629489160555392\
Sk-68 63 & 24080 & -10.5 & 21.9 & 4.0 & 22.4 & 11.3 & 21.9 & 253 & -24.9 & R81, HDE269128, S86 & B2.5 Ia$^+$ & 4658269336800428672\
Sk-69 75 & 24347 & -40.7 & 46.7 & 14.0 & 42.0 & 43.0 & 46.2 & 240 & -25.2 & HDE269216, S88 & B8 I & 4658204053297963392\
Sk-69 91 & 24694 & 10.4 & 31.1 & -17.7 & 29.9 & 20.5 & 30.2 & 230 & -30.2 & HDE269327, S95 & B2 Iae & 4658137739001073280\
Sk-65 48 & 24988 & 3.9 & 18.7 & 6.0 & 21.1 & 7.1 & 20.4 & 322 & 12.7 & R92, HDE271182 & F8 Ia & 4660601607121368704\
Sk-66 72 & 25097 & 10.4 & 41.1 & -9.8 & 45.5 & 14.4 & 43.2 & 305 & -3.8 & HDE269408 & A0 Ia & 4660444926713007872\
Sk-68 81 & 25448 & 19.8 & 32.8 & 41.3 & 31.7 & 45.8 & 31.9 & 276 & -1.0 & HDE269541, W9-26 & A2 Ia & 4658486455992620416\
Sk-67 125 & 25615 & 16.5 & 48.6 & 4.3 & 49.3 & 17.1 & 48.6 & 320 & 18.2 & HDE269594 & G0 Ia & 4660175580731856128\
Sk-71 42 & 25815 & -110.3 & 28.5 & 103.1 & 30.2 & 150.9 & 29.3 & 238 & -1.0 & R112, HDE269660 & B2 Ia & 4651835303997699712\
Sk-67 162 & 25892 & 9.4 & 43.3 & 37.0 & 44.2 & 38.2 & 44.2 & 311 & 10.4 & HDE269697, W24-21 & F6 Ia & 4660124762671796096\
Sk-67 201 & 26135 & -15.9 & 22.7 & -31.4 & 26.9 & 35.2 & 26.1 & 342 & 38.7 & R118, HDE269781 & A0 Iae & 4660246224352015232\
Sk-69 201 & 26222 & 25.8 & 44.5 & -9.3 & 44.6 & 27.4 & 44.5 & 267 & 4.1 & R123, HD37836, S124,MWC121 & B0 Iae & 4657280635327480832\
Sk-68 131 & 26338 & -30.3 & 44.0 & 12.5 & 47.5 & 32.7 & 44.5 & 276 & 0.0 & HDE269857, W27-61 & A9 Ia & 4657700408260606592\
Sk-69 267 & 26745 & -4.6 & 54.8 & -26.8 & 57.4 & 27.2 & 57.4 & 250 & -19.6 & R151W, HDE269982, W8-37 & A8 Ia & 4657627943562907520\
Sk-69 299 & 27142 & -5.8 & 33.7 & 0.9 & 32.5 & 5.9 & 33.7 & 275 & 0.6 & R153, HDE270086, S143 & A2 Ia$^+$ & 4657722879521554176\
Sk-68 178 & 27819 & 2.1 & 36.4 & 0.4 & 25.2 & 2.2 & 36.1 & 313 & 23.4 & HDE270296 & B6 Ia & 4659188769038018816\
Sk-68 179 & 27868 & -17.9 & 36.7 & 0.9 & 26.3 & 17.9 & 36.7 & 333 & 45.7 & HDE270305 & B3 Ia & 4659091084305723392\
\
Mean && 2.8 & 37.0 & 1.5 & 36.6 & 22.7 & & & -0.9 &&\
$\sigma$ && 18.9 & & 18.8 & & 13.7 & & & 20.3 &&\
|
---
abstract: 'We have observed emission lines of \[\] 10.51, H(7–6) 12.37, \[\] 12.81, \[\] 15.56, and \[\] 18.71 $\mu$m in a number of extragalactic regions with the [*Spitzer Space Telescope*]{}. A previous paper presented our data and analysis for the substantially face-on spiral galaxy M83. Here we report our results for the local group spiral galaxy M33. The nebulae selected cover a wide range of galactocentric radii (R$_G$). The observations were made with the Infrared Spectrograph with the short wavelength, high resolution module. The above set of five lines is observed cospatially, thus permitting a reliable comparison of the fluxes. From the measured fluxes, we determine the ionic abundance ratios including Ne$^{++}$/Ne$^+$, S$^{3+}$/S$^{++}$, and S$^{++}$/Ne$^+$ and find that there is a correlation of increasingly higher ionization with larger R$_G$. By sampling the dominant ionization states of Ne (Ne$^+$, Ne$^{++}$) and S (S$^{++}$, S$^{3+}$) for regions, we can estimate the Ne/H, S/H, and Ne/S ratios. We find from linear least-squares fits that there is a decrease in metallicity with increasing R$_G$: d log (Ne/H)/dR$_G$ = $-0.058\pm0.014$ and d log (S/H)/dR$_G$ = $-0.052\pm0.021$ dex kpc$^{-1}$. There is no apparent variation in the Ne/S ratio with R$_G$. Unlike our previous similar study of M83, where we conjectured that this ratio was an upper limit, for M33 the derived ratios are likely a robust indication of Ne/S. This occurs because the regions have lower metallicity and higher ionization than those in M83. Both Ne and S are primary elements produced in $\alpha$-chain reactions, following C and O burning in stars, making their yields depend very little on the stellar metallicity. Thus, it is expected that Ne/S remains relatively constant throughout a galaxy. The median (average) Ne/S ratio derived for regions in M33 is 16.3 (16.9), just slightly higher than the Orion Nebula value of 14.3. The same methodology is applied to [*Spitzer*]{} observations recently published for three massive regions: NGC 3603 (Milky Way), 30 Dor (LMC), and N 66 (SMC) as well as for a group of blue compact dwarf galaxies. We find median Ne/S values of 14.6, 11.4, 10.1, and 14.0, respectively. All of these values are in sharp contrast with the much lower “canonical", but controversial, solar value of $\sim$5. A recent nucleosynthesis, galactic chemical evolution (GCE) model predicts a Ne/S abundance of $\sim$9. Our observations may also be used to test the predicted ionizing spectral energy distribution of various stellar atmosphere models. We compare the ratio of fractional ionizations $<$Ne$^{++}$$>$/$<$S$^{++}$$>$, $<$Ne$^{++}$$>$/$<$S$^{3+}$$>$, and $<$Ne$^{++}$$>$/$<$Ne$^+$$>$ vs. $<$S$^{3+}$$>$/$<$S$^{++}$$>$ with predictions made from our photoionization models using several of the state-of-the-art stellar atmosphere model grids. The trends of the ionic ratios established from the prior M83 study are remarkably similar, but continued to higher ionization with the present M33 objects.'
author:
- 'Robert H. Rubin, Janet P. Simpson, Sean W.J. Colgan, Reginald J. Dufour, Gregory Brunner, Ian A. McNabb, Adalbert W.A. Pauldrach, Edwin F. Erickson, Michael R. Haas, and Robert I. Citron'
title: '[*Spitzer*]{} Observations of M33 and the Hot Star, Region Connection'
---
\#1 \#2 \#3/
Introduction
============
This work is a continuation of a similar previous study of 24 regions in the substantially face-on spiral galaxy M83 that we observed with the [*Spitzer Space Telescope*]{} (Rubin et al. 2007, hereafter R07). Here we present our more recent [*Spitzer*]{} observations of 25 regions in the local group spiral galaxy M33. The analyses of the new set, in conjunction with the M83 set, further elucidate topics investigated in R07. Since this paper should stand alone, that is, not require the reader to fully read R07 first, much of the material presented there is repeated here. At times, this may be verbatim.
Most observational studies of the chemical evolution of the universe rest on emission line objects, which define the mix of elemental abundances at advanced stages of evolution as well as the current state of the interstellar medium (ISM). Gaseous nebulae are laboratories for understanding physical processes in all emission-line sources and probes for stellar, galactic, and primordial nucleosynthesis. [*Spitzer*]{} has a unique ability to address the abundances of the elements neon and sulfur. This is particularly true in the case of regions, where one can observe simultaneously four emission lines that probe the dominant ionization states of Ne (Ne$^+$ and Ne$^{++}$) and S (S$^{++}$ and S$^{3+}$). The four lines, \[\] 12.81, \[\] 15.56, \[\] 18.71, and \[\] 10.51 $\mu$m can be observed cospatially with the Infrared Spectrograph (IRS) on the [*SST*]{}. Because of the sensitivity of [*SST*]{}, the special niche, relative to previous (and near-term foreseeable) instruments, is for studies of extragalactic regions. Toward this end, we have used [*SST*]{} to observe $\sim$25 regions each in galaxies with various metallicities and other properties. To the extent that all the major forms of Ne and S are observed, the true Ne/S abundance ratio could be inferred. For Ne, this is a safe assumption, but for S, there is the possibility of non-negligible contributions due to S$^+$ as well as what could be tied up in molecules and dust. Due to this, we surmised that the large values derived for Ne/S for the M83 regions, which are fairly low ionization, were in fact [*upper limits*]{}. With a portion of our Cycle 2 [*SST*]{} observations of M33 regions in hand, we concluded that the preliminary Ne/S results (from $\sim$12 to 21) were likely a reliable estimate because these nebulae had lower metallicity and much higher ionization than those in M83 (R07, Rubin et al. 2006).
The preliminary M33 Ne/S ratios and the Orion Nebula value of 14.3 (Simpson et al. 2004) were in sharp contrast with the much lower “canonical" solar value ($\sim$5) (Lodders 2003; Asplund, Grevesse, & Sauval 2005). There was an even larger difference compared with the Ne/S ratio predicted by GCE models. According to calculations based on the theoretical nucleosynthesis, galactic chemical evolution models of Timmes, Woosley, & Weaver (1995), the Ne/S ratio in the solar neighborhood would change little, from 3.80 to 3.75, between solar birth and the present time (apropos for the Orion Nebula). These calculations were provided by Frank Timmes (private communication). Timmes noted that although massive stars are expected to dominate the Ne and S production (and are all that were included in their non-rotating models with no wind losses), there is likely to be some re-distribution of Ne and S from rotation or from Wolf-Rayet phases of evolution, along with contributions of Ne from novae or even heavier intermediate mass stars. Hence, the Ne/S of 3.8 should be considered a lower bound because some potential sources of Ne are missing. With [*SST*]{} observations such as we have undertaken for M83 and M33, there is important feedback to the GCE field.
The presence of radial (metal/H) abundance gradients in the plane of the Milky Way is well established in both gaseous nebulae and stars (e.g., Henry & Worthey 1999; Rolleston et al. 2000). Radial abundance gradients seem to be ubiquitous in spiral galaxies, though the degree varies depending on a given spiral’s morphology and luminosity class. The gradients are generally attributed to the radial dependence of star formation history and ISM mixing processes (e.g., Shields 2002). Thus, the observed gradients are another tool for understanding galactic evolution (e.g., Hou et al. 2000; Chiappini et al. 2001; Chiappini et al. 2003). The premise is that star formation and chemical enrichment begins in the nuclear bulges of the galaxies and subsequently progresses outward into the disk, which has remained gas-rich. The higher molecular gas density in the inner regions produces a higher star formation rate, which results in a relatively greater return to the ISM of both “primary” $\alpha$-elements (including O, Ne, and S) from massive star supernovae, and “secondary” elements like N. Secondary nitrogen is produced by CNO burning of already existing carbon and oxygen in intermediate-mass stars and is subsequently returned to the ISM through mass loss. However, because chemical evolution models have uncertain input parameters, and because details of the abundance variations of each element are uncertain, current understanding of the formation and evolution of galaxies suffers (e.g., Pagel 2001).
Studies of regions in the Milky Way are hampered by interstellar extinction. For the most part, optical studies (e.g., Shaver et al. 1983) have been limited to those regions at galactocentric radius R$_G \gtrsim 6$ kpc (predicated on $R_\odot = 8$ kpc) because regions are very concentrated to the Galactic plane. Here extinction becomes severe with increasing distance from Earth. Observations using far-infrared (FIR) emission lines have penetrated the R$_G \lesssim 6$ kpc barrier. Surveys with the [*Kuiper Airborne Observatory (KAO)*]{} by Simpson et al. (1995), Afflerbach et al. (1997), and Rudolph et al. (2006) have observed 16 inner Galaxy regions. With the [*Infrared Space Observatory (ISO)*]{}, Martín-Hernández et al. (2002a) observed 13 inner Galaxy regions covering FIR and also mid-IR lines. A major finding of these studies is that inner Galaxy regions generally have lower excitation (ionization) compared to those at larger R$_G$. This holds for both heavy element ionic ratios O$^{++}$/S$^{++}$ (Simpson et al. 1995) and Ne$^{++}$/Ne$^{+}$ (Simpson & Rubin 1990; Giveon et al. 2002), and also He$^+$/H$^+$ measured from radio recombination lines (Churchwell et al. 1978; Thum, Mezger, & Pankonin 1980). Whether the observed increase in excitation with increasing R$_G$ comes entirely from heavy element opacity effects in the regions and stellar atmospheres, or also from a gradient in the maximum stellar effective temperature, [$T_{\rm eff}$]{}, of the exciting stars is still a point of controversy (e.g., Giveon et al. 2002; Martín-Hernández et al. 2002b; Smith, Norris, & Crowther 2002; Morisset et al. 2004).
It has become clear that nebular plasma simulations with photoionization modeling codes are enormously sensitive to the ionizing spectral energy distribution (SED) that is input (e.g., R07 and Simpson et al. 2004, and references in each). These SEDs need to come from stellar atmosphere models. In R07, we developed new observational tests of and constraints on the ionizing SEDs that are predicted from various stellar atmosphere models. We compared our [*SST*]{} observations of regions in M83 with the various tracks predicted from photoionization models that changed [*only the ionizing SEDs*]{}. M83 provided data for high metallicity (at least twice solar, e.g., Dufour et al. 1980; Bresolin & Kennicutt 2002) and lower-ionization regions. The best overall fit to the nebular models was obtained using the supergiant stellar atmosphere models computed with the WM-BASIC code (Pauldrach, Hoffmann, & Lennon 2001; Sternberg, Hoffmann, & Pauldrach 2003).
We discuss the M33 [*SST*]{}/IRS observations in section 2. In section 3, the data are used to test for a variation in the degree of ionization of the regions with R$_G$. We examine the Ne/S abundance ratio for our M33 region sample in section 4. In section 5, we test for a variation in the Ne/H and S/H ratio with R$_G$. Section 6 describes how these [*Spitzer*]{} data are used to constrain and test the ionizing SEDs predicted by stellar atmosphere models. In section 7, there is additional discussion pertaining to the Ne/S ratio. Last, we provide a summary and conclusions in section 8.
[*Spitzer Space Telescope*]{} Observations
==========================================
In the substantially face-on (tilt 56$^{\rm o}$) local group, spiral galaxy M33, we observed 25 regions, covering a wide range of deprojected galactocentric radii (R$_G$) from 0.71 to 6.73 kpc. We used the [*SST*]{}/IRS in the short wavelength, high dispersion (spectral resolution $\sim$ 600) configuration, called the short-high (SH) module (e.g., Houck et al. 2004). This covers the wavelength range from 9.9 – 19.6 $\mu$m permitting cospatial observations of all five of our programme emission lines: \[\] 10.51, hydrogen H(7–6) (Hu$\alpha$) 12.37, \[\] 12.81, \[\] 15.56, and \[\] 18.71 .
The data were collected under the auspices of Spitzer programme identification 20057. Most of the observations were made in 2006, January 15 to February 1 (UT) during [*SST*]{}/IRS campaign 28. The last set of observations, one Astronomical Observing Request (AOR) out of a total of 8, was made on 2007, February 11 during [*SST*]{}/IRS campaign 38. Figure 1 shows the regions and apertures observed, while Table 1 lists the region positions and the aperture grid configuration used to observe each. The nebulae are designated by their BCLMP number (Boulesteix et al. 1974). The regions observed in the remaining AOR in 2007 were \#62, 302, 691, 651, and 740W. The size of the SH aperture is 11.3$''$$\times$4.7$''$. Maps were arranged with the apertures overlapping along the direction of the long slit axis (the “parallel" direction). The purpose of overlapping is that most spatial positions will be covered in at least two locations on the array, minimizing the effects of bad detectors. In the direction of the short slit axis (the “perpendicular" direction), the apertures were arranged immediately abutting each other; that is, with no overlap or space between them. In all cases, we chose the mapping mode with aperture grid patterns varying from a 1$\times$2 grid to as large as a 2$\times$4 grid in order to cover the bulk of the expected emission.
Our terminology for the aperture grid pattern is intended to give roughly the map size in integer multiples of the parallel $\times$ perpendicular direction aperture size. For instance the 1$\times$2 and 1$\times$3 patterns have a shift of 3.45$''$ in the parallel direction and a 4.7$''$ shift in the perpendicular direction. The resulting map has a full size in the parallel direction of 14.75$''$ (11.3 + 3.45) which is closest to the integer one in the parallel direction. For the two other aperture grid patterns used, the 2$\times$3 and 2$\times$4 patterns, the shift (or step) is 3.77$''$ (1/3 the aperture length) in the parallel direction. Thus, the resulting maps have a full size that covers an area of 2$\times$3 or 2$\times$4 apertures, respectively. In order to save overhead time, we clustered the objects into AORs with the same aperture grid pattern. We used a ramp (exposure) time of 30 s and 12 cycles at each spatial position. This permits up to effectively 720 s (24 cycles) integration time for some spatial positions in the 1$\times$2 and 1$\times$3 patterns and up to 1080 s (36 cycles) for some positions in the 2$\times$3 and 2$\times$4 patterns.
Our data were processed and calibrated with version S15.3.0 of the standard IRS pipeline at the [*Spitzer*]{} Science Center. We use CUBISM, the CUbe Builder for IRS Spectral Mapping, (version 1.50) to build our post-BCD (basic calibrated data) data products. CUBISM is described in Smith et al. (2007a) and references therein, as well as in a manual detailing its use (Smith et al. 2007b). CUBISM was used to build maps, including accounting for aperture overlaps, and to deal effectively with bad pixels. From the IRS mapping observations, it can combine these data into a single 3-dimensional cube with two spatial and one spectral dimension. For each of our programme regions, we constructed a data cube. Global bad pixels (those occurring at the same pixel in every BCD) were removed manually. Record level bad pixels (those occurring only within individual BCDs) that deviated by 5 $\sigma$ from the median pixel value and occurred within at least 10 per cent of the BCDs were removed automatically in CUBISM with the “Auto Bad Pixels" function. In reducing our data, we were careful to monitor that the “Auto Bad Pixels" function did not incorrectly flag any of the pixels on our programme spectral lines as bad. Data cubes for each region were built without applying the slit-loss correction factors (SLCFs). This is discussed below. We varied our spatial extraction aperture size \[always a 2-D integer pixel grid\] due to the differences in the size of the observing grid over the particular nebula. Our further analysis of these spectra uses the line-fitting routines in the IRS Spectroscopy Modeling Analysis and Reduction Tool (SMART, Higdon et al. 2004).
The emission lines were measured with SMART using a Gaussian line fit. The continuum baseline was fit with a linear or quadratic function. Figures 2 (a)–(e) show the fits for each of the five lines in BCLMP 45 (object \#6 in Fig. 1). A line is deemed to be detected if the flux is at least as large as the 3 $\sigma$ uncertainty. We measure the uncertainty by the product of the full-width-half-maximum (FWHM) and the root-mean-square variations in the adjacent, line-free continuum; it does not include systematic effects.
In section 2 of our M83 paper (R07), we discussed and estimated systematic uncertainties as they affected the line fluxes. Here we recap (or repeat) the major points. Most likely the largest uncertainty is due to slit (aperture) loss correction factors (SLCFs). The pipeline flux calibration assumes that objects are point sources. Our nebulae are extended and that is why we mapped each with a grid that covers more than a single aperture. We did not make a correction for this effect. Thus we have implicitly assumed that the regions are close to the point-source limit within the SH 11.3$''$$\times$4.7$''$ aperture. If the region were uniformly extended within the SH aperture, SLCFs would need to be applied to our fluxes. These are: 0.697, 0.671, 0.663, 0.601, and 0.543 for the 10.5, 12.4, 12.8, 15.6, and 18.7 $\mu$m lines, respectively. These factors were obtained by interpolating in numbers provided from the $`$$b1\_slitloss\_convert.tbl$$'$ file from the [*Spitzer*]{} IRS Custom Extraction tool (SPICE) for the SH module. For the uniformly filled aperture, the maximum uncertainty in the flux due to this effect would be $\sim$46 per cent for the \[\] 18.7 line. The SLCFs would need to multiply our listed fluxes. We note that with regard to this effect, the fluxes listed in Table 2 are upper limits and that the uncertainty would be only in the direction to lower them. No correction factor was applied because we are likely closer to the point-source limit than the uniform-brightness limit. Because our science depends on line flux ratios, for our purposes, the possible uncertainty due to this effect would be lower, e.g., $\sim$22 per cent when we deal with the line flux ratio \[\] 10.5/\[\] 18.7. The possible uncertainty in the absolute flux calibration of the spectroscopic products delivered by the pipeline is likely confined to between $\pm$5 per cent and $\pm$10 per cent. Any uncertainty in the flux due to a pointing error is probably small and in the worst case should not exceed 10 per cent.
For the brighter lines, that is, most of the 10.5, 12.8, 15.6, and 18.7 $\mu$m lines, the systematic uncertainty far exceeds the measured (statistical) uncertainty. Even for the fainter lines, we estimate that the systematic uncertainty exceeds the measured uncertainty. In addition to the line flux, the measured FWHM and heliocentric radial velocities (V$_{helio}$) are listed in Table 2. Both the FWHM and V$_{helio}$ are useful in judging the reliability of the line measurements. The FWHM is expected to be the instrumental width for all our lines. With a resolving power for the SH module of $\sim$600, our lines should have a FWHM of roughly 500 km s$^{-1}$. The values for V$_{helio}$ should straddle the heliocentric systemic radial velocity for M33 of $-179$ km s$^{-1}$ (Corbelli & Schneider 1997). Most of our measurements are in agreement with these expectations.
Variation in the degree of ionization of the regions with R$_G$
===============================================================
We chose our sample of nebulae in order to cover a wide range in R$_G$ (in the plane of M33). To derive these deprojected galactocentric distances, we used a distance D of 840 kpc (Freedman, Wilson, & Madore 1991) an inclination angle ($i$ = 56$\pm1^{\rm o}$), and a position angle of the line of nodes ($\theta = 23\pm1^{\rm o}$) (Zaritsky, Elston, & Hill 1989). We assumed the centre of the galaxy is at $\alpha$, $\delta$ = $1^{\rm h}33^{\rm m}51\fs02$, $30^{\rm o}$39367 (J2000) (Cotton, Condon, & Arbizzani 1999). Table 3 lists R$_G$ for the centre of each object. These range from from 0.71 to 6.73 kpc.
From the measured fluxes, we estimate ionic abundance ratios, including Ne$^{++}$/Ne$^+$, S$^{3+}$/S$^{++}$, and S$^{++}$/Ne$^+$, for each of the regions. Important advantages compared with prior optical studies of various other ionic ratios are: (1) the IR lines have a weak and similar electron temperature ($T_e$) dependence while the collisionally-excited optical lines vary exponentially with $T_e$, and (2) the IR lines suffer far less from interstellar extinction. Indeed for our purposes, the differential extinction correction is negligible as the lines are relatively close in wavelength. In our analysis, we deal with ionic abundance ratios and therefore line flux ratios. In order to derive the ionic abundance ratios, we perform the usual semiempirical analysis assuming a constant $T_e$ and electron density ($N_e$) to obtain the volume emissivities for the five pertinent transitions. For the ions Ne$^+$, Ne$^{++}$, S$^{++}$, and S$^{3+}$, we use the atomic data described in Simpson et al. (2004) and Simpson et al. (2007). For H$^+$ from the H(7–6) line, we use Storey & Hummer (1995). There is a bit of a complication here because at Spitzer’s spectral resolution, the H(7-6) line is blended with the H(11-8) line. Their respective $\lambda$(vac) = 12.371898 and 12.387168 $\mu$m. In order to correct for the contribution of the H(11-8) line, we use the relative intensity of H(11-8)/H(7-6) from recombination theory (Storey & Hummer 1995) assuming case B and $N_e$ = 100 cm$^{-3}$. The ratio H(11-8)/H(7-6) = 0.122 and holds over a fairly wide range in $N_e$ and $T_e$ \[including 8000 and 10000 K, see below\] appropriate for our objects, and indeed for case A also.
For the entries in Table 3, we adopt a value for all the M33 regions of $T_e$ = 8000 K and $N_e$ = 100 cm$^{-3}$. In a recent paper, Magrini et al. (2007) were able to derive $T_e$ for 14 regions in M33 from the diagnostic flux ratio \[\] 4363/(5007 + 4959). Only one of their objects BCLMP 45 can be clearly identified with one of our regions. They found $T_e$\[\] = 8600$\pm$200 K. Furthermore, this object was one of only three where they also determined $T_e$\[\] from the flux ratio 5755/(6584 + 6548); the result was 8200$\pm$1000 K. There are two more of our programme sources with available $T_e$\[\]. These were derived by Crockett et al. (2006) from their optical observations of BCLMP 691 yielding 10000$\pm$200 K and recomputing $T_e$ with the measurements for BCLMP 280 (NGC 588) from Vílchez et al. (1988) resulting in 9300$^{+600}_{-400}$. While these values for $T_e$ are somewhat higher than the 8000 K we adopt, we point out a well-known bias. That is, both $T_e$\[\] and $T_e$\[\] derived from the ratio of fluxes of “auroral" to “nebular" lines are systematically higher than the so-called “$T_0$", which is the ($N_e$$\times$$N_i$$\times$$T_e$)–weighted average, where $N_i$ is the ion density of interest. The amount of this bias depends on the degree of $T_e$ variations in the observed volume (see Peimbert 1967 and many forward references). In our analysis, using the set of IR lines, it is more appropriate to be using a $T_e$ that is similar to $T_0$. Because of the insensitivity of the volume emissivities to $T_e$, particularly when working with ratios for these IR lines, our results depend very little on this $T_e$ choice. The effects on our analysis due to a change in the assumed $N_e$ are also small as will be discussed later.
We present the variation of Ne$^{++}$/Ne$^+$ with R$_G$ in Figure 3 using the values from Table 3. The error values here, as well as for all others in Table 3 and in Figures 4–8, represent the propagated flux measurement uncertainties and [*do not include the systematic uncertainties*]{}. Our assumed $N_e$ of 100 cm$^{-3}$ appears reasonable in view of previous observations that address the density. For instance, Magrini et al. (2007) derived $N_e$ from the familiar diagnostic line flux ratio \[\] 6717/6731. For most of their regions, they found that this ratio was consistent with the low-$N_e$ asymptotic limit, that is, $N_e$ $<$ 100 cm$^{-3}$. There is extremely little change in any of our derived Ne$^{++}$/Ne$^+$ ratios even when using an $N_e$ of 1000 cm$^{-3}$, which is likely an upper limit for these regions. A linear least-squares fit indicates a positive correlation with R$_G$ (in kpc),
Ne$^{++}$/Ne$^+$ = $-$0.44$\pm$0.22 + (0.46$\pm$0.067) R$_G$,
with miniscule change to this equation for $N_e$ = 1000 cm$^{-3}$. For all the least-squares line fits in this paper, each point is given equal weight because systematic uncertainties exceed the flux measurement uncertainties, as discussed earlier. The positive correlation of Ne$^{++}$/Ne$^+$ with R$_G$ as measured by the slope may be judged to be significant following the criterion that it exceeds the 3 $\sigma$ uncertainty. We also did a linear least-squares fit to log(Ne$^{++}$/Ne$^+$) vs. R$_G$ with the result
log(Ne$^{++}$/Ne$^+$) = $-$0.88$\pm$0.11 + (0.20$\pm$0.035) R$_G$.
This relation also produces a statistically significant slope. The transformation of this function is shown as the dashed line in Figure 3. Three objects (230, 702, and 740W) have been excluded in this Figure as well as in Figure 4 because of poor S/N and/or extreme deviancy from the trend of the 22 other sources. The slope in the linear plot here 0.46$\pm$0.067 is much steeper than the analogous slope for our M83 regions of 0.011$\pm$0.0035 (see Figure 3 in R07). Furthermore, the comparison with Figure 3 in the M83 paper shows dramatically the higher ionization of the M33 nebulae.
A similar fit to the S$^{3+}$/S$^{++}$ vs. R$_G$ data yields
S$^{3+}$/S$^{++}$ = $-$0.032$\pm$0.018 + (0.046$\pm$0.0057) R$_G$.
The slope exceeds the 1 $\sigma$ uncertainty by a factor of 8. Thus the increase in degree of ionization with increasing R$_G$ here too is significant. A linear least-squares fit to log(S$^{3+}$/S$^{++}$) vs. R$_G$ results in
log(S$^{3+}$/S$^{++}$) = $-$1.7$\pm$0.090 + (0.18$\pm$0.028) R$_G$.
Again there is a statistically significant slope. We map this relation onto Figure 3 as the dashed line. While there is no fundamental reason to expect either functional form shown in Figures 3 and 4, we note that the dashed line fits do have the advantage of not extrapolating to negative ionic ratios at small values of R$_G$.
Figure 5 plots the fractional ionic abundance ratio $<$S$^{++}$$>$/$<$Ne$^+$$>$ vs. R$_G$ for 23 regions (sources 230 and 702 are excluded). This ratio is obtained from the S$^{++}$/Ne$^+$ ratio by multiplying by an assumed Ne/S value (see below). The last three columns of Table 3 list this and other fractional ionic abundance ratios used in this paper. We show the linear least-squares fit for an assumed $N_e$ of 100 cm$^{-3}$. Here, the fit indicates a significant positive correlation with R$_G$,
$<$S$^{++}$$>$/$<$Ne$^+$$>$ = 0.56$\pm$0.23 + (0.31$\pm$0.069) R$_G$,
where angular brackets denote fractional ionization. In this figure and in the linear fit, we assume an Orion Nebula Ne/S abundance ratio of 14.3 (Simpson et al. 2004). Because Ne and S are “primary" elements, their production is expected to vary in lockstep and Ne/S would not be expected to show a radial gradient within a galaxy (Pagel & Edmunds 1981). There is a clear correlation of increasingly higher ionization with increasing R$_G$. One reason may be due to the lower metallicity at larger R$_G$ (see section 5) causing the exciting stars to have a harder ionizing spectrum.
Neon to Sulfur abundance ratio
==============================
For regions, we may approximate the Ne/S ratio with (Ne$^+$ + Ne$^{++}$)/(S$^{++}$ + S$^{3+}$). This includes the dominant ionization states of these two elements. However this relation does not account for S$^+$, which should be present at some level. We may safely ignore the negligible contributions of neutral Ne and S in the ionized region. Figure 6 shows our approximation for Ne/S vs. R$_G$. The linear least-squares fit to 23 objects (again omitting for cause sources 230 and 702) is
Ne/S = 18.0$\pm$1.3 $-$ (0.39$\pm$0.40) R$_G$,
plotted as the dotted line in Figure 6. We also show the fit, the solid line, after removing the remaining deviant large value (source 32). The relation becomes
Ne/S = 16.8$\pm$0.97 $-$ (0.14$\pm$0.29) R$_G$.
Both of these indicate that there is no significant slope. Our data also indicate that the lower envelope to Ne/S is well fit by a constant value equal to the Orion Nebula ratio of 14.3 (Simpson et al. 2004).
In our previous M83 results, there appeared to be a drop in the Ne/S ratio with increasing R$_G$. We argued that this was not a true gradient in Ne/S. Instead, it is most likely due to not accounting for the presence of sulfur in other forms – S$^+$, molecules, and dust. Because our observations of the regions in M83 showed an increasing degree of ionization with increasing R$_G$, the expected increasing fraction of S$^+$ towards the inner galaxy regions would lead to a flatter gradient. Another factor that could flatten the slope is the higher dust content (with S, but not Ne, entering grains) expected in the inner regions due to higher metallicity as is the case for the Milky Way. The refractory carbonaceous and silicate grains are not distributed uniformly throughout the Galaxy but instead increase in density toward the centre. A simple model suggests the dust density is $\sim$5 – 35 times higher in the inner parts of the Galaxy than in the local ISM (Sandford, Pendleton, & Allamandola 1995). The Ne/S abundance ratios that we derived for 24 regions in M83 varied from 41.9 to 24.4 (see Figure 5 in R07). All are considerably higher than the Orion Nebula value of 14.3 and, as an ensemble, significantly higher than what we find here for the nebulae in M33. Thus the evidence is strong that the derived Ne/S estimates for the M83 objects are upper limits and furthermore, those that are further from the centre will likely need less of a downward correction to obtain a true Ne/S ratio.
Because the M33 regions have a lower metallicity and because almost all have a significantly higher ionization than those we observed in M83, the amount of any correction needed for S in forms other than S$^{++}$ and S$^{3+}$ is minimized. Hence while our derived Ne/S ratios should still be considered as upper limits, these M33 values are a much more robust estimate of a true Ne/S ratio than those for M83. The scenario above is in excellent accord with the recent results of Wu et al. (2008), hereafter W08. They obtained an average Ne/S = 12.5$\pm$3.1 from [*Spitzer*]{} observations of 13 blue compact dwarf galaxies and found no correlation between their Ne/S and Ne/H ratios. The median (average) Ne/S ratio derived for 23 regions in M33 is 16.3 (16.9).
Variation in the Ne/H and S/H ratios with R$_G$
===============================================
There was a significant detection of the H(7-6) flux for 16 of the regions (see Table 2) which permits a determination of the heavy element abundances Ne/H and S/H (see Table 3). We present the results for Ne/H in Figure 7. A linear least-squares fit of log (Ne/H) vs. R$_G$ results in
log (Ne/H) = $-$4.07$\pm$0.04 $-$ (0.058$\pm$0.014) R$_G$.
This fit indicates a significant slope (4.1 $\sigma$). A similar plot for S/H is shown in Figure 8. Here the linear least-squares fit to log (S/H) vs. R$_G$ yields
log (S/H) = $-$5.31$\pm$0.06 $-$ (0.052$\pm$0.021) R$_G$.
It is interesting that the slope is nearly identical to the Ne/H relationship. However, the slope signal-to-noise ratio (2.5 $\sigma$) is less than 3 $\sigma$ and thus would be deemed only marginally significant. While [*Spitzer*]{} is an admirable machine for measuring both Ne and S abundances in regions, the neon abundances are determined more reliably. As previously mentioned, with the [*Spitzer*]{} observations alone, we are neither accounting for S$^+$ nor S that may be tied up in dust or molecules. In this sense, the S/H ratios in Figure 8 are lower limits. Both of the above measurements of a heavy element abundance gradient are in remarkable agreement with the recent value for the log (O/H) gradient of $-$0.054$\pm$0.011 dex kpc$^{-1}$ (Magrini et al. 2007). They derived this gradient from optical observations of 14 regions in M33 where the \[\] (5007, 4959, 4363 Å) and \[\] (7320, 7330 Å) emission line fluxes were all measured and a value for $T_e$ determined. The sources for their linear least-squares fit covered a range in R$_G$ from $\sim$2 to 7.2 kpc and resulted in log (O/H) = $-$3.47$\pm$0.05 $-$ (0.054$\pm$0.011) R$_G$. Because the slope for this is practically the same as ours above for log (Ne/H), we may use the respective y-intercepts to infer a Ne/O ratio of 0.28. This is in good agreement with the Ne/O = 0.25 value for the Orion Nebula (Simpson et al. 2004). Using [*ISO*]{}, Willner & Nelson-Patel (2002) measured the neon lines in M33. If the two outermost objects in their sample are neglected, they found a neon gradient of $-$0.05$\pm$0.02 dex kpc$^{-1}$, which agrees with the slope we find.
Constraints on the ionizing SED for the stars exciting the regions
==================================================================
Various fractional ionic abundances are highly sensitive to the stellar ionizing SED that apply to regions. The present [*Spitzer*]{} data probe the Ne$^+$ and Ne$^{++}$ fractional ionic abundances, as well as those of S$^{++}$ and S$^{3+}$. They may be used to provide further constraints and tests on the ionizing SED for the stars exciting these M33 nebulae, similar to what we had done in the M83 paper (R07). We use the ratio of fractional ionizations $<$Ne$^{++}$$>$/$<$S$^{++}$$>$ vs. $<$S$^{3+}$$>$/$<$S$^{++}$$>$ (Figure 9a), $<$Ne$^{++}$$>$/$<$S$^{3+}$$>$ vs. $<$S$^{3+}$$>$/$<$S$^{++}$$>$ (Figure 9b), and Ne$^{++}$/Ne$^+$ vs.$<$S$^{3+}$$>$/$<$S$^{++}$$>$ (Figure 9c). These ionic ratios are computed using our photoionization code NEBULA (e.g., Simpson et al. 2004; Rodríguez & Rubin 2005). The lines connect the results of the nebular models calculated using the ionizing SEDs predicted from various stellar atmosphere models. There are to the input parameters, just the SED. The stellar atmospheres used are representative of several non-LTE models that apply for O-stars. We also display the results from one set of LTE models (Kurucz 1992). His LTE atmospheres have been extensively used in the past as input for region models. Hence the comparison with the other non-LTE results reinforces the fact that more reliable SEDs for O-stars require a non-LTE treatment. Figures 9a–c dramatically illustrate how sensitive region model predictions of these ionic abundance ratios are to the ionizing SED input to nebular plasma simulations.
We list for each model the ([$T_{\rm eff}$]{} in kK, log $g$)-pair to identify it. There are other parameters for a stellar atmosphere model such as the elemental abundance mix and those that describe stellar winds, which are treated in each of these non-LTE codes. Basically, all the atmospheres use solar abundances. For the region models calculated with Pauldrach et al. (2001) atmospheres, the solid line connects models with “dwarf" atmospheres and the dashed line connects models with “supergiant" atmospheres. Proceeding from the hot to the cool end, the Pauldrach et al. dwarf set has (50, 4), (45, 3.9), (40, 3.75), (35, 3.8), and (30, 3.85) while the supergiant set has (50, 3.9), (45, 3.8), (40, 3.6), (35, 3.3), and (30, 3). In several instances, the loci are cut off at the edges of the plot at the cool end as they track toward the point computed with the coolest atmosphere. The Sternberg et al. (2003) paper also uses Pauldrach’s WM-BASIC code. At a given [$T_{\rm eff}$]{} we have used their model with the smallest log $g$ in order to be closest to the supergiant case. Because the locus using these Sternberg et al. atmosphere models is for the most part similar to the Pauldrach et al. supergiant locus, we do not show it in Figures 9 to avoid clutter.
The violet lines with inverted triangles are for region models calculated with Lanz & Hubeny (2003) atmospheres (TLUSTY code). The solid line connects models using atmospheres with (45, 4), (40, 4), (35, 4), and (30, 4) while the dotted line connects models using atmospheres with (45, 3.75), (40, 3.5), (37.5, 3.5), (35, 3.25), (32.5, 3.25), and (30, 3). The orange squares are the results of our nebular models with the atmospheres in Martins et al. (2005) that use Hillier’s CMFGEN code. The dotted line connects models using atmospheres with (48.53, 4.01), (42.56, 3.71), (40, 3.5), (37.5, 3.5), (35, 3.25), (35, 3.5), and (32.5, 3.5). The brown line with triangles result from the models using Kurucz (1992) atmospheres with (45, 4.5), (40, 4.5), (37, 4), (35, 4), and (33, 4).
To compare our data with the models in Figures 9a,b, we need to divide the observed Ne$^{++}$/S$^{++}$ and Ne$^{++}$/S$^{3+}$ ratios by an assumed Ne/S abundance ratio. For this purpose, we adopt a constant Ne/S = 14.3, the Orion Nebula value (Simpson et al. 2004). The open red circles are our prior results for the M83 regions. The green stars are the M33 results (adjusted by the assumed Ne/S) derived from our observed line fluxes using $N_e$ of 100 cm$^{-3}$. While the $<$Ne$^{++}$$>$/$<$Ne$^+$$>$ ratio in Figure 9c has the advantage of being independent of elemental abundance ratios, it appears to be more sensitive to the [*nebular*]{} parameters than the others, as will be discussed later. The trends of the ionic ratios established from the prior M83 study are remarkably similar, but continued to higher ionization with the present M33 objects. There are two sources, one in M33 and one in M83, that are deviantly low compared with the theoretical tracks and the other data. The point that is low in all three panels for M33 is BCLMP 702, which is the faintest of the M33 regions in the 10.5 and 12.8 $\mu$m lines. Also the FWHM of 903 km s$^{-1}$ measured for the 15.6 $\mu$m line is the largest (see Table 2) and possibly suspect. The deviant object in M83 (in all three panels) is source RK 268, which is the faintest of the M83 regions in the 12.8, 15.6, and 18.7 $\mu$m lines. For the 10.5 $\mu$m line, it is the second broadest line, FWHM of 809 km s$^{-1}$, of any line we measured in M83 (see Table 2 in R07). Thus we consider the position of both of these sources to be duly suspect in Figures 9. On the whole, the data for both galaxies in panels a and b lie closest to the theoretical loci obtained with the Pauldrach et al. supergiant SEDs. This is particularly notable in Figure 9b, where the other model loci are nearly perpendicular to the data point trend in the vicinity of where they intersect the data points. On the other hand, the data in panel c, for the most part, appear to lie closer to Martins et al., Lanz & Hubeny et al., and Pauldrach et al. [*dwarf*]{} loci.
The nebular models used to generate Figures 9 are all constant density, ionization-bounded, spherical models. We used a constant total nucleon density (DENS) of 1000 cm$^{-3}$ that begins at the star. Each model used a total number of Lyman continuum photons s$^{-1}$ ($N_{Lyc}$) = 10$^{49}$. The same [*nebular*]{} elemental abundance set was used for all nebular models. We use the same “reference" set as in Simpson et al. (2004) because in that paper we were studying the effects of various SEDs on other ionic ratios and other data sets. Ten elements are included with their abundance by number relative to H as follows: (He, C, N, O, Ne, Si, S, Ar, Fe) with (0.100, 3.75E$-$4, 1.02E$-$4, 6.00E$-$4, 1.50E$-$4, 2.25E$-$5, 1.05E$-$5, 3.75E$-$6, 4.05E$-$6), respectively. We continue with the same set of abundances as in the M83 paper (R07). As we mentioned there, the set of abundances used is roughly a factor of 1.5 higher than for Orion and not drastically different from solar. These heavy element abundances are too high for M33 and we investigate below how the theoretical loci will change when these abundances are scaled back accordingly.
We have investigated the effects of changing DENS, $N_{Lyc}$, and allowing for a central evacuated cavity, characterized by an initial radius (R$_{init}$) before the stellar radiation encounters nebular material. We term these shell models. In Figures 10a–c, the resulting changes to Figures 9a–c are shown for 12 nebular models run using the Pauldrach et al. (2001) supergiant atmospheres with [$T_{\rm eff}$]{} 35, 40, and 45 kK. These models are listed along with the symbol in Table 4. The original Pauldrach et al. (2001) supergiant locus (the dashed line connecting the filled circles) and points derived from the [*Spitzer*]{} M33 and M83 data are shown again.
The points for models 3, 7, and 11 are nearly identical to the original points for the Pauldrach supergiant models at the same respective [$T_{\rm eff}$]{}. This can be understood in terms of the ionization parameter ($U$), which is very useful for gauging ionization structure. An increase in $U$ corresponds to higher ionization (for a given $T_{\rm eff}$). For an ionization-bounded, constant density case, $$U = [N_e N_{Lyc} (\alpha - \alpha _1)^2/(36 \pi c^3)]^{1/3}~~,$$ where ($\alpha$ - $\alpha$$_1$) is the recombination rate coefficient to excited levels of hydrogen, and $c$ is the velocity of light (see Rubin et al. 1994, eq. 1 and adjoining discussion). Because($\alpha$ - $\alpha$$_1$) $\simeq$ 4.10$\times$10$^{-10}~T_e^{-0.8}$ cm$^{3}$ s$^{-1}$ (Rubin 1968 fit to Seaton 1959), there is only a weak dependence of $U$ on $T_e$ ($\sim$ $T_e^{-0.5}$). When $U$ is similar, as is the case here with the product of $N_e\times$$N_{Lyc}$, the ionization structure is similar. With regard to the three shell models in Figures 10, the Strömgren radius is $\sim$0.74 pc. Thus the radial thickness of the shell is slightly less than half the radius of the central cavity. From the visual appearance of our target regions, it is unlikely that the theoretical loci need to be tracked to higher dilutions.
Another [*nebular*]{} parameter that can alter the theoretical tracks is the set of elemental abundances used. To investigate this effect and to have a set more representative of the lower metallicity of M33, we have calculated three models (numbers 4, 8, and 12 in Table 4) reducing all the heavy element abundances by a factor of three from the “reference" set. As expected, lower metallicity results in a shift to higher ionization. In Figures 10a,b,c the point moves to the upper right. For [$T_{\rm eff}$]{} 35kK, the factors are $<$Ne$^{++}$$>$/$<$S$^{++}$$>$ = 1.85, $<$Ne$^{++}$$>$/$<$S$^{3+}$$>$ = 1.44, $<$S$^{3+}$$>$/$<$S$^{++}$$>$ = 1.29, and $<$Ne$^{++}$$>$/$<$Ne$^+$$>$ = 1.91. For [$T_{\rm eff}$]{} 40kK, the respective factors are 1.39, 1.23, 1.14, and 1.65. For [$T_{\rm eff}$]{} 45kK, the respective factors are 1.14, 1.10, 1.03, and 1.45. Likewise, higher metallicity shifts the point to the lower left (R07). The above factors show that there is a progression to a smaller influence on these ionic ratios with metallicity as [$T_{\rm eff}$]{} increases from 35 to 45kK. Additionally, the largest change is always in the $<$Ne$^{++}$$>$/$<$Ne$^+$$>$ ratio and the smallest is in the $<$S$^{3+}$$>$/$<$S$^{++}$$>$ ratio. While the $<$Ne$^{++}$$>$/$<$Ne$^+$$>$ ratio has the advantage of being independent of elemental abundance ratios, it appears to be more sensitive to the [*nebular*]{} parameters than does the other ratios. Note that the y-axis scales are different in Figures 10 a, b, and c and thus a simple visual guide from the length of the various line segments cannot be used as an accurate measure of the effect of changing a nebular parameter between the three panels. It is interesting that there appears to be a small, upward shift of the M33 star points with respect to the trend of the M83 points (circles), particularly noticeable in Fig. 10c (and in the electronic colour version). This is qualitatively what is expected for the lower metallicity regions of M33. The dashed line theoretical locus would move up, connecting the square points. Nevertheless, for both the case of the M33 and M83 nebulae, we may conclude that the predicted spread in Figures 10 due to a reasonable uncertainty in nebular metallicity is far less than that due to the SEDs of the various stellar atmosphere models.
The magnitude/direction of changes in Figures 10 due to varying the [*nebular*]{} parameters per Table 4 should be roughly similar for the other models shown in Figures 9. It is also noted that we have not considered matter-bounded nebular models. These would be higher ionization than the corresponding ionization-bounded model. There is also the effect of a change in the abundances used to compute the stellar atmosphere models. This will change the emergent stellar SED (e.g., Mokiem et al. 2004). As is the case for a change in the nebular model abundance set, such a modification in the stellar model will alter the shape of the SED in the same sense; that is, a higher metallicity will cause more opacity and soften the SED, and a lower metallicity will do the opposite. Mokiem et al. (2004) examined this using CMFGEN stellar models matching both the nebular and stellar metallicities. Their Figure 11 tracks the predicted variation in the \[\] 15.6/\[\] 12.8 flux ratio over a range of 0.1 – 2 Z$_{\sun}$. Although beyond the scope of this paper, it would be interesting to compare models using different stellar atmosphere metallicities, especially if the environment indicates significant departures from solar. However, the proper abundances are not accurately known and comparisons like those in this paper will help decipher the proper values. With the present paper, however, we have been investigating which of the models best fits the observations and such a comparison works most effectively with a fixed set of abundances [*common to all models*]{}, which are the solar ones.
There appears to be a remarkable “convergence" of the nebular models close to the hot (right) side of Figures 9. Depending on the set of stellar atmosphere models, there is a significant spread in [$T_{\rm eff}$]{} ($\sim$42–50kK) (as well as the log $g$ parameter) in the vicinity where this occurs. For instance, the results in all three panels are essentially identical using either the TLUSTY (45, 3.75) or the WM-BASIC (50, 3.9) SED. A likely explanation for this convergence behaviour in Figures 9 is that at these high [$T_{\rm eff}$]{} values, the ionization balance in the atmospheres shifts to higher ionization stages which have fewer lines that can influence the model calculations. Thus, blocking and blanketing effects no longer dominate the models as strongly, and the line radiation pressure is also less significant. This means that the influence of the expanding atmosphere decreases. As a result of all this, at high [$T_{\rm eff}$]{} the model calculation are no longer as critically dependent on the correct treatment of the complex details of the physics involved.
Discussion
==========
With the recent papers by W08 and Lebouteiller et al. (2008), hereafter L08, we are in a position to further examine the Ne/S ratio discussed in §4. Fundamental observational data are vital to test and constrain theories of nucleosynthesis and GCE. A very valuable adjunct would be to find how much the Ne/S ratio can vary or whether or not there is a fairly “universal" value. W08 obtained an average Ne/S = 12.5$\pm$3.1 from [*Spitzer*]{} observations of 13 blue compact dwarf galaxies. They found no correlation between their Ne/S ratios with metallicity (the Ne/H ratios). With their Table 3 of line fluxes, we apply the same methods used here for the M33 objects to derive the various ionic abundance ratios. For the same blue compact dwarf galaxy, we obtain somewhat higher Ne/S ratios. We are particularly interested in combining their observations with ours to address the Ne/S ratio and whether there is any correlation with degree of ionization as measured by the Ne$^{++}$/Ne$^+$ and S$^{3+}$/S$^{++}$ ratios. Nine of the galaxies in W08 had all four of the necessary lines detected to derive both of these ratios (i.e., we excluded their 4 objects which had at least one upper limit on a line flux). In Table 5, we list in columns 2–4, Ne$^{++}$/Ne$^+$, S$^{3+}$/S$^{++}$, and Ne/S derived by us using the individual $T_e$– and $N_e$–values in Wu et al.’s Table 2. In columns 5–7, we list the same set derived here assuming $T_e$ = 10000 K and $N_e$ = 100 cm$^{-3}$. It is apparent that there is very little difference between these due to what ($T_e$, $N_e$) values are used. The median (average) Ne/S for the 9 galaxies in column 4 is 14.0 (14.9), while it is 13.9 (14.9) from column 7. However, for these 9 objects, the median Ne/S is 12.4 and the average 12.7 (from Table 4 in W08).
L08 reported their Spitzer observations of three very massive region: the Galactic source NGC 3603; the extremely massive 30 Dor in the Large Magellanic Cloud; and N 66 (NGC 346), the largest in the Small Magellanic Cloud. With their Table 2 of line fluxes, we apply the same methods used here for the M33 objects to derive the various ionic abundance ratios. In their Table 6, they list derived abundance ratios for several position observed in each object: 3 for NGC 3603, 15 for 30 Dor, and 11 for N 66. Again, we find somewhat higher Ne/S ratios at all positions than the corresponding value obtained from their Table 6. In Table 6 here, we list in columns 2–4, Ne$^{++}$/Ne$^+$, S$^{3+}$/S$^{++}$, and Ne/S derived by us using the same $T_e$– and $N_e$–values as they had used. These are 10000 K and 1000 cm$^{-3}$ for NGC 3603; 10000 K and 100 cm$^{-3}$ for 30 Dor; and 12500 K and 100 cm$^{-3}$ for N 66. The median Ne/S ratios we derive for NGC 3603, 30 Dor, and N 66. are 14.6, 11.4, and 10.1, respectively.
Some variation is traceable to the adoption of different \[\] effective collision strengths. We use the set by Tayal & Gupta (1999); W08 and L08 stated that they used values from the IRON Project, which for this ion would be “paper X" (Galavís et al. 1995). There is a substantial difference between the values in these papers. For instance the values between the ground-state, fine structure levels (where the \[\] IR lines arise) at $T_e$ = 10000 K are: 3.98, 1.31, and 7.87 (Tayal & Gupta) and 2.331, 1.110,and 5.411 (Galavís et al.) for transitions $^3P_0$–$^3P_1$, $^3P_0$–$^3P_2$, and $^3P_1$–$^3P_2$ respectively. Furthermore, these three values show a change in [*opposite directions*]{} with $T_e$ decreasing from 10000 K. The above three collision strengths increase in the Tayal & Gupta work but decrease in the other study. In regions, S$^{++}$ is almost always more abundant than S$^{3+}$ as is the case for every point here except 1 of the 9 W08 galaxies addressed here. Because we use the larger \[\] effective collision strengths, the S$^{++}$ abundance we infer will be lower than what was derived in the other two papers. This will then result in the total S abundance (S$^{++}$ + S$^{3+}$) being smaller and hence the Ne/S abundance larger than what W08 and L08 found. Following Simpson et al. (2007), we repeat here the references for the other effective collision strengths we use: \[\] (Saraph & Storey 1999), \[\] (Saraph & Tully 1994), and \[\] (McLaughlin & Bell 2000). Apparently W08 and L08 did use the same sets for \[\] and \[\] but probably used “paper V" (Butler & Zeippen 1994) from the IRON Project for \[\]. We note that near $T_e$ = 10000 K, the values for \[\] are quite similar. Even without passing judgment on the relative merits of one set of effective collision strengths vs. another, it is crucial for the purposes of this paper that we continue the analysis with a homogeneous treatment. For this we will use the adjusted ionic ratios from columns 2–4 in Tables 5 and 6.
In Figure 11, we plot Ne/S vs. Ne$^{++}$/Ne$^+$ for our M33 results (star symbol) using the same 22 sources mentioned in Fig. 6. The linear least-squares fit to the M33 points is,
Ne/S = 17.02$\pm$0.63 $-$ (0.78$\pm$0.50) Ne$^{++}$/Ne$^+$,
which is shown as the solid line. The negative gradient is not statistically significant. The results from our prior M83 study (R07) are shown as circles. The wide spread in the M83 points dramatically demonstrates the limitations of our method to infer Ne/S \[except as an upper limit\] when Ne$^{++}$/Ne$^+$ is low (i.e., for low ionization objects). The squares show the W08 data for the 9 blue compact dwarf galaxies, as discussed above. The linear least-squares fit to those 9 points is,
Ne/S = 18.17$\pm$1.07 $-$ (0.93$\pm$0.24) Ne$^{++}$/Ne$^+$.
While our M33 data indicate no statistically significant slope, the fit to the 9 blue compact dwarf galaxies shows a statistically significant decrease with higher Ne$^{++}$/Ne$^+$ (the dotted line in Fig. 11). We note that the median (average) Ne/S for the 9 galaxies is 14.0 (14.9), very close to the Orion value of 14.3 shown as the dashed line. If we exclude the four lowest ionization galaxies and fit just the five with the highest ionization, then
Ne/S = 15.09$\pm$1.15 $-$ (0.49$\pm$0.20) Ne$^{++}$/Ne$^+$,
and this flatter negative slope (2.5 $\sigma$) is no longer statistically significant.
In Fig. 11, we also show the reanalyzed results of L08 (see Table 6). There appears to be a trend with the three positions in NGC 3603 (red asterisks) having the highest median Ne/S, followed by the 15 positions in 30 Dor (green triangles), and the 9 positions in N 66 (blue diamonds) having the lowest median ratio. This trend is correlated with the metallicity with NGC 3603 approximately solar, 30 Dor $\sim$0.7 solar, and N 66 $\sim$0.2 solar (L08 and references therein). No error bars are shown on these points, but they may be expected to be roughly 20 percent according to L08. A picture that is consistent with the data in Figure 11 is that at the lowest ionizations there remains considerable S still present at these positions in the form of S$^+$, molecular gas, and dust. The three 30 Dor positions with the lowest ionization, as measured by both Ne$^{++}$/Ne$^+$ and S$^{3+}$/S$^{++}$, are \#10 and 17, followed by \#8 in Table 6. An inspection of Fig. 2 in L08 shows that positions \#10 and 17 are near the periphery of their 30 Dor image and thus might be expected to be characterized by lower ionization than interior positions. The linear least-squares fit to the data in Table 6 excludes these three lowest ionization 30 Dor points as well as the NGC 3603 points. The dash-dot line fits 23 positions, those in N 66 and the 12 remaining in 30 Dor, with the highest ionization. We find,
Ne/S = 12.08$\pm$0.41 $-$ (0.37$\pm$0.12) Ne$^{++}$/Ne$^+$,
with a slope that is marginally significant (3.1 $\sigma$). We note that there is no basis to extrapolate the line to the edges of the graph as shown, and indeed no theoretical basis for a straight line. What we would expect once all other S species become negligible with respect to S$^{++}$ and S$^{3+}$ is that an asymptotic Ne/S will be approached at the higher ionizations.
Similar fits to Ne/S vs. S$^{3+}$/S$^{++}$ are shown in Fig 12 and result in,
Ne/S = 16.67$\pm$0.71 $-$ (3.11$\pm$5.41) S$^{3+}$/S$^{++}$,
for 22 sources in M33. The slope is not significant statistically. For the 9 blue compact dwarf galaxies,
Ne/S = 17.70$\pm$1.14 $-$ (6.39$\pm$2.00) S$^{3+}$/S$^{++}$.
This indicates a marginal, statistically significant (3.2 $\sigma$) negative slope as S$^{3+}$/S$^{++}$ increases. As with Fig. 11, if we exclude the four lowest ionization galaxies and fit just the five with the highest ionization, then,
Ne/S = 14.42$\pm$1.30 $-$ (2.84$\pm$1.72) S$^{3+}$/S$^{++}$.
This flatter negative slope (1.6 $\sigma$) is no longer statistically significant. The linear least-squares fit to the reanalyzed L08 data in Table 6 for the same 23 positions as done in Fig. 11 results in,
Ne/S = 12.14$\pm$0.44 $-$ (5.69$\pm$1.88) S$^{3+}$/S$^{++}$,
with a slope that is borderline significant (3.0 $\sigma$).
It is tantalizing to conjecture that for high ionization objects, all the dominant states of Ne and S are measured, resulting in a robust estimate of the true Ne/S abundance ratio. For objects as diverse as the Orion Nebula and NGC 3603, the M33 regions, 30 Dor, N 66, and the blue compact dwarf galaxies, there is remarkably little variation in the Ne/S derived. We note that if we were to adopt the smaller \[\] effective collision strengths (Galavís et al. 1995) that W08 and L08 used, all values for Ne/S that we have derived would be lower, including that for Orion. A rough estimate of how much lower is $\sim$2 (that is, $\sim$15–25 percent), which can be surmised from the comparison with the W08 and L08 Ne/S values and the recalculated values here in Tables 5 and 6. All of these Ne/S values point to a much larger ratio than the “canonical" solar value of $\sim$5 (see next section) and what had previously been predicted by the GCE models mentioned in the Introduction. The question, of how universal the Ne/S ratio may be, will require further study. Two of the blue compact dwarf galaxies (UM461 and IIZw40) with the highest degree of ionization strongly affect the line fits in Figures 11 and 12 and the conclusion that Ne/S continues to decrease, reaching a value of 10. It is possible that it does take such a high degree of ionization to percolate off any substantial amount of S that may still be tied up in grains, molecules, and S$^+$. Further [*Spitzer*]{} observations of high-ionization regions will be useful for this purpose. We note that the type of analysis done here will not hold for even higher ionization objects where substantial amounts of Ne exist as Ne$^{3+}$ or higher and/or S as S$^{4+}$ or higher.
Summary and conclusions
=======================
We have observed emission lines of H(7–6) 12.37, \[\] 12.81, \[\] 15.56, \[\] 18.71, and \[\] 10.51 $\mu$m cospatially with the [*Spitzer Space Telescope*]{} using the Infrared Spectrograph (IRS) in short-high mode (SH). From the measured fluxes, we estimate the ionic abundance ratios Ne$^{++}$/Ne$^+$, S$^{3+}$/S$^{++}$, and S$^{++}$/Ne$^+$ in 25 regions in the substantially face-on spiral galaxy M33. These nebulae cover a range from 0.71 to 6.73 kpc in deprojected galactocentric distance R$_G$. We find a correlation of increasingly higher ionization with increasing R$_G$. This is seen in the variation of Ne$^{++}$/Ne$^+$, S$^{3+}$/S$^{++}$, and $<$S$^{++}$$>$/$<$Ne$^+$$>$ with R$_G$ (see Figures 3–5). A possible reason may be due to the lower metallicity at larger R$_G$ causing the exciting stars to have a harder ionizing spectrum. As mentioned in the introduction and discussed in considerable detail (e.g., Morisset et al. 2004), there are other effects that could mimic this. We find a decrease in the metallicity, as measured by the Ne/H and S/H ratio, with increasing R$_G$ (see Figures 7–8). The linear-log gradients that we derive are in remarkable agreement with the recent value for the log (O/H) gradient of $-$0.054$\pm$0.011 dex kpc$^{-1}$ (Magrini et al. 2007) derived from optical observations of 14 regions in M33. Because their slope is practically identical to ours for log (Ne/H), we infer a Ne/O ratio of 0.28.
By sampling the dominant ionization states of Ne and S for regions, we can approximate the Ne/S ratio by (Ne$^+$ + Ne$^{++}$)/(S$^{++}$ + S$^{3+}$). For M33, we find no significant variation in the Ne/S ratio with R$_G$. Both Ne and S are the products of $\alpha$-chain reactions following carbon and oxygen burning in stars, with large production factors from core-collapse supernovae. Both are primary elements, making their yields depend very little on the stellar metallicity. Thus, at least to “first order", it is expected that Ne/S remains relatively constant throughout a galaxy. As discussed in §4 and §7, our estimate for Ne/S has accounted for neither the presence of S$^+$ nor S that may be tied up in grains or molecular gas.
The data presented here for M33, combined with other [*Spitzer*]{} data (see Figures 11 and 12), are consistent with our view that there are now reliable estimates for the [*total*]{} Ne/S ratio. As long as the degree of ionization is sufficiently high such that the amount of sulfur in forms other than S$^{++}$ and S$^{3+}$ is small, the methodology used here will provide a robust total Ne/S estimate. The median Ne/S value we derive for the M33 objects is 16.3, but this includes many with relatively low ionization. Although there is no statistically significant gradient with Ne$^{++}$/Ne$^+$, two of the sources, BCLMP 638 and 280, with the highest ionization are among those with the lowest Ne/S at 13.9 and 13.3, respectively. For the W08 blue compact dwarf galaxies, we find a median Ne/S for the 9 galaxies of 14.0. When we consider just the 5 with the highest ionization, the median drops to 13.2. From Table 6 with the recomputed Ne/S from the L08 observations, we find a median Ne/S of 10.1 for N 66 and 11.4 for 30 Dor. This median for 30 Dor drops to 11.3 if the three lowest ionization values are not included. Although the data are limited, we note the possibility that the true Ne/S ratio may be less for lower metallicity galaxies. N 66 has the lowest metallicity ($\sim$0.2 solar) while 30 Dor in the LMC has an intermediate value compared with the Milky Way. On the other hand, W08 found no correlation of Ne/S with metallicity for the blue compact dwarf galaxies.
The solar abundance, particularly of Ne, remains the subject of much controversy (e.g., Drake & Testa 2005; Bahcall, Serenelli, & Basu 2006; and references in each of these). While we cannot directly address the solar abundance with our observations of extragalactic regions, it is important to have reliable benchmarks for the Ne abundance. There appears to be a growing body of evidence that the Ne abundance \[its fractional number abundance relative to H log H = 12, by definition and termed $A$(H)\] is substantially higher in the solar neighborhood, and even in the Sun itself, than the “canonical" solar values given in two often-referenced papers. These papers have for the Sun: $A$(Ne) = 7.87, $A$(S) = 7.19 (Lodders 2003) and $A$(Ne) = 7.84, $A$(S) = 7.14 (Asplund, Grevesse, & Sauval 2005). Thus according to both, Ne/S $\sim$5. It is now generally accepted that Ne has the least well determined solar abundance among the most abundant elements. One of the proponents for a higher neon abundance pointed out that an $A$(Ne) = 8.29 would reconcile solar models with the helioseismological measurements (Bahcall, Basu, & Serenelli 2005). Using this value together with the $A$(S) values above, we obtain Ne/S of 12.6 and 14.1, respectively, close to the Orion Nebula ratio 14.3 (Simpson et al. 2004) and the various median ratios that range from 10.1 to 16.3 derived earlier.
As discussed in the last section, there is a surprisingly large difference between the effective collision strengths calculated for \[\]. It would be very worthwhile to resolve the differences with a new calculation or possibly experimental work. This is a very important matter because the dominant sulfur ionization state in the preponderance of regions is S$^{++}$. Another desirable avenue for future work would be ground-based optical observations that could cover the S$^+$ ion, which must be present at some level, and which cannot be done with [*Spitzer*]{}. A programme designed to cover \[\] 6716,31 Åas well as the \[\] 9069,9531 Å lines, particularly for the regions we deem high ionization here, would confirm whether or not the S$^+$ abundance is negligible and the reliability of our Ne/S values.
In our earlier M83 paper (R07), we discussed what the Ne/S ratio was predicted to be according to calculations based on the theoretical nucleosynthesis, galactic chemical evolution models of Timmes, Woosley, & Weaver (1995). The ratio was about 3.8. Since then there have been improved models (Woosley & Heger 2007). Their calculation considers massive stars from 12 to 120 M$_\odot$ starting with the solar abundance set of Lodders (2003). One difference from the Timmes et al. work is the addition now of a treatment for mass loss. This includes mass loss on the main sequence as well as red giant, and Wolf-Rayet phases. It also includes improvements in the explosion physics and nuclear cross sections. Stellar rotation is not included and according to Stan Woosley (private communication), will have an uncertain effect. Based on Fig. 7 in Woosley & Heger (2007), the Ne/S ratio is predicted to be $\sim$8.6. These models are based on starting with the solar abundances. Since several of the objects observed by [*Spitzer*]{} are lower than solar metallicity, it would be very interesting to compute the Ne/S nucleosynthesis yields starting with lower metallicities.
The data set here, combined with our previous M83 results (R07), may be used as constraints on the ionizing SEDs for the stars exciting these nebulae by comparing the ratio of fractional ionizations $<$Ne$^{++}$$>$/$<$S$^{++}$$>$, $<$Ne$^{++}$$>$/$<$S$^{3+}$$>$, and $<$Ne$^{++}$$>$/$<$Ne$^+$$>$ vs. $<$S$^{3+}$$>$/$<$S$^{++}$$>$ with predictions made from our photoionization models using stellar atmosphere models from several different sources. In Figures 9a,b we show the comparison assuming that the Ne/S ratio does not vary and equals the Orion Nebula value. Generally, the best fit is to the nebular models using the supergiant stellar atmosphere models (Pauldrach et al. 2001) computed with the WM-BASIC code. The comparison shown in Figure 9c is independent of the Ne/S ratio. For the most part, the Ne$^{++}$/Ne$^+$ values appear to lie closer to the theoretical loci that use the Martins et al., Lanz & Hubeny et al., and Pauldrach et al. [*dwarf*]{} SEDs. While the Ne$^{++}$/Ne$^+$ ratio has the advantage of being independent of elemental abundance ratios, it appears to be more sensitive to the [*nebular*]{} parameters than does the other ionic ratios used in Figures 9 and 10. This fact tends to make it less unique in its ability to discriminate between the stellar SEDs we present in this paper. We note that these comparisons are mainly qualitative since these ionic ratios depend not only on the SED, but also on the nebular parameters discussed as well as the effects of the stellar metallicity on the SED.
This work is based on observations made with the [*Spitzer Space Telescope*]{}, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under NASA contract 1407. Support for this work was provided by NASA for this [*Spitzer*]{} programme identification 20057. We thank Stan Woosley for providing information on the Ne/S ratio from a nucleosynthesis, galactic chemical evolution perspective. The referee Christophe Morisset provided comments that helped improve the paper. Our computer support at NASA Ames was handled very well by David Goorvitch. We thank Danny Key, Erik Krasner-Karpen, and Matt Lattanzi for assistance with the data reduction. The nebular models were run on JPL computers, initially a Cray and more recently, a Dell Intel Xeon cluster. Funding for computer use in this investigation was provided by the JPL Office of the Chief Information Officer. We are grateful to Heidi Lorenz-Wirzba for excellent computer support.
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[*[**(b)**]{}*]{} The same as panel (a) except the ordinate is $<$Ne$^{++}$$>$/$<$S$^{3+}$$>$. Both panels dramatically illustrate the sensitivity of the H [ii]{} region model predictions of these ionic abundance ratios to the ionizing SED that is input to nebular plasma simulations. Thesedata, for the most part, appear to track the Pauldrach et al. supergiant locus. Similar to panels (a) and (b) except the ordinate is $<$Ne$^{++}$$>$/$<$Ne$^+$$>$. This plot also illustrates the sensitivity of the H [ii]{} region model predictions of this ionic abundance ratio to the ionizing SED that is input to nebular plasma simulations. These data, for the most part, appear to lie closer to Martins et al., Lanz & Hubeny et al., and Pauldrach et al. [*dwarf*]{} loci.
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abstract: 'We compute the luminosity function (LF) and the formation rate of long gamma ray bursts (GRBs) by fitting the observed differential peak flux distribution obtained by the [*BATSE*]{} satellite in three different scenarios: i) GRBs follow the cosmic star formation and their LF is constant in time; ii) GRBs follow the cosmic star formation but the LF varies with redshift; iii) GRBs form preferentially in low–metallicity environments. We find that the differential peak flux number counts obtained by [*BATSE*]{} and by [*Swift*]{} can be reproduced using the same LF and GRB formation rate, indicating that the two satellites are observing the same GRB population. We then check the resulting redshift distributions in the light of [*Swift*]{} 2–year data, focusing in particular on the relatively large sample of GRBs detected at $z>2.5$. We show that models in which GRBs trace the cosmic star formation and are described by a constant LF are ruled out by the number of high–$z$ [*Swift*]{} detections. This conclusion does not depend on the redshift distribution of bursts that lack of optical identification, nor on the existence of a decline in star formation rate at $z>2$, nor on the adopted faint–end of the GRB LF. [*Swift*]{} observations can be explained by assuming that the LF varies with redshift and/or that GRB formation is limited to low–metallicity environments.'
author:
- 'R. Salvaterra, G. Chincarini,'
title: 'The Gamma Ray Burst Luminosity Function in the Light of the [*Swift*]{} 2–year Data'
---
Introduction
============
Gamma Ray Bursts (GRBs) are powerful flashes of high–energy photons occurring at an average rate of a few per day throughout the universe. Even though they are highly transient events very hard to localize, they are so bright that they can be detected up to very high redshift (the current record is $z=6.29$). The energy source of a GRB is believed to be associated to the collapse of the core of a massive star in the case of long–duration GRBs, and due to merger– or accretion–induced collapse for the short–hard class of GRBs (see Mészáros 2006 for a recent review). In this paper, we limit our analysis to the class of long–duration GRBs.
One of the main goals of the [*Swift*]{} satellite (Gehrels et al. 2004) is to trackle the key issue of the GRB luminosity function (LF). Unfortunately, although the number of GRBs with good redshift determination has been largely increased by [*Swift*]{}, the sample is still too poor (and bias dominated) to allow a direct measurement of the LF. Many studies (e.g. Lamb & Reichart 2000; Porciani & Madau 2001 (PM01); Schmidt 2001, Choudhury & Srianand 2002; Firmani et al. 2004; Guetta, Piran & Waxman 2005; Natarajan et al.2005; Daigne, Rossi & Mochkovitch 2006) tried to constrain the GRB LF under the assumption that GRBs trace the observed star formation rate, as suggested by the association of long GRBs to the death of massive stars. Following these works and assuming the most recent star formation rate determination, we derive the LF and formation rate of GRBs by fitting the observed [*BATSE*]{} differential peak flux distribution in three different scenarios: i) GRBs follow the cosmic star formation and have a constant LF; ii) the GRB LF varies with redshift; iii) GRBs form in low–metallicity environments. We check the results against the 2–year [*Swift*]{} data, focusing in particular on the large sample of high redshift ($z>2.5$) GRBs detected by this instrument.
Basic Equations
===============
The observed photon flux, $P$, in the energy band $E_{\rm min}<E<E_{\rm max}$, emitted by an isotropically radiating source at redshift $z$ is
$$P=\frac{(1+z)\int^{(1+z)E_{\rm max}}_{(1+z)E_{\rm min}} S(E) dE}{4\pi d_L^2(z)},$$
where $S(E)$ is the differential rest–frame photon luminosity of the source, and $d_L(z)$ is the luminosity distance. To describe the typical burst spectrum we adopt the functional form proposed by Band et al. (1993), i.e. a broken power–law with a low–energy spectral index $\alpha$, a high–energy spectral index $\beta$, and a break energy $E_b$. In this work, we take $\alpha=-1$ and $\beta=-2.25$ (Preece et al. 2000), and $E_b=511$ keV (PM01). Moreover, it is customary to define an isotropic equivalent intrinsic burst luminosity in the energy band 30-2000 keV as $L=\int^{2000\rm{keV}}_{30\rm{keV}} E S(E)dE$. Given a normalized GRB LF, $\phi(L)$, and the detector efficiency, $\epsilon(P)$, the observed rate of bursts with peak flux between $P_1$ and $P_2$ is
$$\begin{aligned}
\frac{dN}{dt}(P_1<P<P_2)&=&\int_0^{\infty} dz \frac{dV(z)}{dz}
\frac{\Delta \Omega_s}{4\pi} \frac{\Psi_{\rm GRB}(z)}{1+z} \nonumber \\
& \times & \int^{L(P_2,z)}_{L(P_1,z)} dL^\prime \phi(L^\prime)\epsilon(P),\end{aligned}$$
where $dV(z)/dz=4\pi c d_L^2(z)/[H(z)(1+z)^2]$ is the comoving volume element[^1], and $H(z)=H_0 [\Omega_M (1+z)^3+\Omega_\Lambda+(1-\Omega_M-\Omega_\Lambda)(1+z)^2]^{1/2}$. $\Delta \Omega_s$ is the solid angle covered on the sky by the survey, and the factor $(1+z)^{-1}$ accounts for cosmological time dilation. Finally, $\Psi_{\rm GRB}(z)$ is the comoving burst formation rate. In this work, we assume that the GRB LF is described by
$$\phi(L) \propto \left(\frac{L}{L_{\rm cut}}\right)^{-\xi} \exp \left(-\frac{L_{\rm cut}}{L}\right).$$
Models
======
We consider three different scenarios. In the first one, the GRB formation rate is proportional to the cosmic star formation rate (SFR), $\Psi_\star(z)$, i.e. $\Psi_{\rm GRB}(z)=k_{\rm GRB} \Psi_\star (z)$, and the LF does not evolve with redshift, i.e. $L_{\rm cut}={\rm const}=L_0$. The factor $k_{GRB}$ gives the number of GRBs formed per solar mass in stars and has units of $\Msun^{-1}$. $\Psi_\star(z)$ (in units of $\Msun$ Mpc$^{-3}$ yr$^{-1}$) is commonly parameterized with the form proposed by Cole et al. (2001) as
$$\Psi_\star(z)=\frac{(a_1+a_2z)h}{1+(z/a_3)^{a_4}}.$$
Recently, Hopkins & Beacom (2006) have provided the values of the coefficients $a$ by fitting the available UV and far–infrared measurements for $z<6$, corrected for dust obscuration. In this paper, we adopt their best fit parameters: $a_1=0.017$, $a_2=0.13$, and $a_3=3.3$ (Hopkins & Beacom 2006). The value of $a_4=4.3$ is taken to be slightly lower than the original one in order to match the decline of the SFR with $(1+z)^{-3.3}$ at $z\gsim 5$ suggested by recent deep–field data (see Stark et al. 2006 and references therein).
In the second scenario, while the GRB formation rate is still proportional to the observed SFR, the cut–off luminosity in the GRB LF increases with redshift as $L_{\rm cut}=L_0 (1+z)^\delta$. Lloyd–Ronning, Fryer & Ramirez–Ruiz (2002), using GRB redshifts and luminosities derived from the luminosity–variability relationship, found that the data imply $\delta\simeq 1.4\pm0.5$, and we adopt this as fiducial value.
Finally, we consider a case in which GRBs form only in environments with metallicity below a given threshold (no evolution in the LF is considered). In fact, some theoretical models (see Mészáros 2006 and reference therein) require that GRB progenitors should have metallicity $\lsim 0.1\;\Zsun$. Observations of GRB host galaxies (see Savaglio 2006 and reference therein) seems in agreement with this prescription, showing that GRB preferentially originates in low–metallicity regions. Langer & Norman (2006) have quantified the amount of star formation at a given metallicity, using a recent determination of the stellar mass function (Panter et al. 2004) and the observed mass–metallicity correlation (Savaglio et al. 2005). Adopting the metallicity redshift evolution derived from emission line studies (Kewley & Kobulnicky 2005), the fractional mass density belonging to metallicity below a given threshold, $Z_{th}$, can be computed as
$$\Sigma(z)=\frac{\hat{\Gamma}(0.84,(Z_{th}/\Zsun)^2 10^{0.3z})}{\Gamma (0.84)},$$
where $\hat{\Gamma}$ ($\Gamma$) are the incomplete (complete) gamma function, and $\Gamma(0.84)\simeq 1.122$. The GRB formation rate is then given by $\Psi_{\rm GRB}(z)= k_{\rm GRB}\Sigma(z)\Psi_\star(z)$. The main effect of this convolution is that the GRB formation rate peaks at higher redshift with respect to the cosmic SFR. We adopt $Z_{th}=0.1\;\Zsun$ as fiducial value, and, in this case, the GRB formation peaks at $z\sim 3.5$.
GRB number counts
=================
The free parameters in our model are the GRB formation efficiency $k_{\rm GRB}$, the cut–off luminosity at $z=0$, $L_0$, and the power index, $\xi$, of the GRB LF function. Following PM01, we optimized the value of these parameters by $\chi^2$ minimization over the observed differential number counts in the 50–300 keV band of [*BATSE*]{}. We use the off–line [*BATSE*]{} sample of Kommers et al. (2000), which includes 1998 archival (“triggered” plus “non–triggered”) bursts, and for which the detector efficiency is well described by the function $\epsilon(P)=0.5[(1+{\rm erf}(-4.801+29.868 P)]$ (Kommers et al. 2000). We report the best–fit parameters for our fiducial models in Table \[tab:fit\]. In the last column, we give the reduced $\chi^2$ for the best–fitting model, showing that it is always possible to find a good agreement with the data[^2]. Note that for the metallicity evolution scenario a higher GRB formation efficiency is required, since GRBs form only in a (small) fraction of star forming galaxies.
We can now use the best–fit parameters to compute the expected differential peak flux distribution of GRBs in the 15–150 keV band of the Burst Alert Telescope (BAT) instrument onboard of [*Swift*]{}. The results are plotted in Figure \[fig:swift\] and compared with the observed [*Swift*]{}/BAT data points. All models show a good agreement with the data without the need of any change of the GRB LF and formation efficiency, indicating that [*BATSE*]{} and [*Swift*]{} are observing essentially the same population of GRBs. This conclusion is rather insensitive to $20$% variations of the adopted GRB spectrum parameters, i.e. for the large majority of burst spectra (Kaneko et al. 2006).
GRB redshift distribution
=========================
Our model allows us to compute the expected redshift distribution of GRBs detected by [*Swift*]{}. We decide to avoid the comparison between model results and the overall observed distribution of bursts with known redshift, since this procedure implicitly assumes that the observed sample of GRBs with redshift determination is representative of all detected sources. Moreover, important information are missed by this kind of analysis: for example, that many bright GRBs are identified at high redshift. So, we try to answer this simple question: [*is the redshift distribution consistent with the number of [*Swift*]{} detections at $z>2.5$ and $z>3.5$?*]{}
The cumulative number of GRBs, identified during the two years of the [*Swift*]{} mission at $z>2.5$ (left panel) and $z>3.5$ (right panel), is plotted in Figure \[fig:zgt\] toghether with model predictions. Note that [*Swift*]{} detections are to be considered as a strong lower limit, since many high–$z$ bursts can be missed by optical follow–up searches.
The model with no LF evolution clearly underestimates the number of high redshift GRB detections at any photon flux and no bright GRBs are predicted for $z>3.5$. We checked that variations of the shape of the SFR do not affect this result: even assuming a constant SFR at $z\gsim 2$, the model predictions do not change significantly. In fact, for relatively bright GRBs, the rapid decline in the LF strongly hampers the detection of high redshift bursts. Furthermore, our analysis does not depend on the faint–end of the LF: increasing the population of faint GRBs would decrease the number of high–$z$ detections, strengthening our conclusion. So, models in which GRBs trace the cosmic SFR and are described by a constant LF, are ruled out by the large sample of high–$z$ [*Swift*]{} GRBs.
The number of high–$z$ [*Swift*]{} identifications can be justified assuming that the LF varies with redshift. In this case, high–$z$ GRBs are typically brighter than low–$z$ ones, so that are much easely detected. Assuming that the luminosity increases as $(1+z)^{1.4}$, we find many sources at $z>2.5$, but the model is barely consistent with the number of bright GRBs at $z>3.5$. Since some high–$z$ sources can be missed by optical follow–up searches, an even stronger evolution might be required to explain the data.
Finally, we consider the possibility that GRB formation is restricted to low–metallicity environments. In this case, the peak of the GRB formation is shifted towards higher redshift, so that the probability of high–$z$ detections increases. Assuming $Z_{th}=0.1\;\Zsun$, [*Swift*]{} identification are exceeded both at $z>2.5$ and $z>3.5$ without requiring any evolution in the LF. Thus, the model is consistent with a fraction of high redshift bursts missed by optical follow–up searches. Increasing the threshold metallicity will decrease the number of sources at high–$z$: for $Z_{th}\sim 0.4\;\Zsun$ the model becomes inconsistent with the number of observed GRBs at $z>3.5$. Higher threshold values would require evolution of the GRB luminosity and/or a more gentle decline of the SFR at high redshift.
In conclusion, the existence of a large sample of bursts at $z>2.5$ in the [*Swift*]{} 2-year data imply that GRBs have experienced some kind of evolution, being more luminous or more common in the past.
GRB rate at redshift larger than six
====================================
The discovery of GRB050904 at $z=6.29$ (Antonelli et al. 2005; Tagliaferri et al. 2005; Kawai et al. 2006) during the first year of the [*Swift*]{} mission has strengthened the idea that many GRBs should be observed out to very high redshift (e.g. Natarajan et al. 2005; Bromm & Loeb 2006; Daigne et al. 2006). Unfortunately, no other source at $z\gsim 6$ has been detected in the second year of observations.
In Figure \[fig:zgt6\], we plot the [*Swift*]{} detection rate expected for the three scenarios here considered. Models without evolution predict almost no sources to be detected at very high redshift. If luminosity evolution ($\delta=1.4$) is allowed, $\sim 2$ bursts/yr should lie above $z\sim 6$ for $P>0.2$ ph cm$^{-2}$ s$^{-1}$, whereas, in the metallicity evolution scenario ($Z_{th}=0.1\;\Zsun$), we expect $\sim 8$ GRBs/yr, one or two being at $z\gsim 8$.
The detection rate are found to decrease rapidly with increasing peak fluxes. Indeed, it is interesting to note that GRB050904 was relatively bright, being its observed photon flux $P=0.658$ ph cm$^{-2}$ s$^{-1}$. At this limit, only $\sim 1$ (2) bursts/yr would be at $z\gsim 6$, if luminosity (metallicity) evolution is assumed. Thus, the lack of very high redshift identification in the 2nd year of the [*Swift*]{} mission might be due to practical difficulties in the optical follow–up of faint GRBs. In fact, no GRB with observed photon fluxes below $0.5$ ph cm$^{-2}$ s$^{-1}$ has UVOT detection and only in a couple of cases a reliable redshift determination was possible. So, the identification of just one burst at $z\gsim6$ in two years of [*Swift*]{} mission is not very surprising. On the contrary, the discovery of GRB050904 may suggest that the [*Swift*]{} follow–up procedure is working very well, at least for relatively bright bursts.
Conclusions
===========
We have computed the luminosity function and the formation rate of long GRBs by fitting the [*BATSE*]{} differential peak flux number counts in three different scenarios: i) GRBs follow the cosmic star formation and have a redshift–independent LF; ii) the GRB LF varies with redshift; iii) GRBs are associated with star formation in low–metallicity environments. In all cases, it is possible to obtain a good fit to the data by adjusting the model free parameters. Moreover, using the same LF and formation rate, it is possible to reproduce both [*BATSE*]{} and [*Swift*]{} differential counts, showing that the two satellites are observing the same GRB population.
We have then computed the expected burst redshift distribution, testing the results against the number of high redshift GRBs, detected during the two years of the [*Swift*]{} mission. We find that models where GRBs trace the SFR and are described by a constant LF largely underestimate the number high–$z$ GRBs detected by [*Swift*]{}. This conclusion does not depend on the redshift distribution of burst lacking of optical identification, nor on the existence of a decline in the SFR at $z>2$, nor on the adopted faint–end of the LF. Alternatively, we find that the observed number of high–$z$ detection can be justified by assuming that the GRB luminosity increases with redshift and/or that GRBs preferentially form in low–metallicity environments.
Finally, we have estimated the detection rate of bursts at very high redshift. We find that $\sim 2$ (8) GRBs/yr should be observed at $z\gsim 6$, if luminosity (metallicity) evolution is assumed. The majority of these sources is faint and may be missed in optical follow–up searches, but $\sim 1$ (3) GRB/yr should be relatively bright, with an observed photon flux in excees to 0.5 ph cm$^{-2}$ s$^{-1}$.
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[lcccc]{} no evolution & $1.14\pm0.07$ & $9.54\pm4.55$ & $3.54\pm0.78$ & 0.83\
luminosity evolution ($\delta=1.4$) & $1.05\pm0.05$ & $0.77\pm0.13$ & $2.19\pm0.95$ & 0.80\
metallicity evolution ($Z_{th}=0.1\;\Zsun$) & $10.0\pm0.5$ & $16.7\pm5.7$ & $2.94\pm0.34$ & 0.84\
\[tab:fit\]
[^1]: We adopted the ’concordance’ model values for the cosmological parameters: $h=0.7$, $\Omega_m=0.3$, and $\Omega_\Lambda=0.7$.
[^2]: Note that strong covariance on $L_0$ and $\xi$ is observed in the parameter space surrounding the best–fit parameters (see also PM01)
|
---
abstract: 'Let $L=L(a,b)$ be a free Lie ring on two letters $a,b.$ We investigate the kernel $I$ of the map $L\oplus L \to L $ given by $(A,B)\mapsto [A,a]+[B,b].$ Any homogeneous element of $L$ of degree $\geq 2$ can be presented as $[A,a]+[B,b].$ Then $I$ measures how far such a presentation from being unique. Elements of $I$ can be interpreted as identities $[A(a,b),a]=[B(a,b),b]$ in Lie rings. The kernel $I$ can be decomposed into a direct sum $I=\bigoplus_{n,m} I_{n,m},$ where elements of $I_{n,m}$ correspond to identities on commutators of weight $n+m,$ where the letter $a$ occurs $n$ times and the letter $b$ occurs $m$ times. We give a full description of $I_{2,m};$ describe the rank of $I_{3,m};$ and present a concrete non-trivial element in $I_{3,3n}$ for $n\geq 1.$'
address:
- 'Laboratory of Continuous Mathematical Education (School 564 of St. Petersburg), nab. Obvodnogo kanala 143, Saint Petersburg, Russia.'
- 'Laboratory of Modern algebra and Applications, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia.'
- 'Laboratory of Continuous Mathematical Education (School 564 of St. Petersburg), nab. Obvodnogo kanala 143, Saint Petersburg, Russia.'
author:
- Boris Baranov
- 'Sergei O. Ivanov'
- Savelii Novikov
title: On two letter identities in Lie rings
---
Introduction {#introduction .unnumbered}
============
It is easy to check that the following identity is satisfied in any Lie ring (=Lie algebra over $\mathbb Z$) $$\label{1e}
[a,b,b,a]=[a,b,a,b],$$ where $[x_1,\dots,x_n]$ is the left-normed bracket of elements $x_1,\dots, x_n$ defined by recursion $[x_1,\dots,x_{n}]:=[[x_1,\dots, x_{n-1}],x_n].$ We denote by $[a,_i b]$ the Engel brackets of $a,b:$ $$[a,_0b]=a, \hspace{1cm} [a,_{i+1}b]=[[a,_ib],b].$$ For example, $[a,_3b]=[a,b,b,b].$ In [@Qbad] the second author together with Roman Mikhailov generalized the identity as follows $$\label{eq_Qbad_identities}
[[a,_{2n}b],a]= \left[\ \sum_{i=0}^{n-1}(-1)^i [[a,_{2n-1-i} b],[a,_{i}b]]\ ,\ b \ \right],$$ where $n\geq 1$. This identity is crucial in their proof that the wedge of two circles $S^1\vee S^1$ is a $\mathbb Q$-bad space in sense of Bousfield-Kan. Note that the letter $a$ occurs twice in each commutator of this identity and the letter $b$ occurs $2n$ times. Moreover, the identity has the form $[A,a]=[B,b].$
We are interested in identities of the form: $$\label{2}
[A(a,b),a]=[B(a,b),b],$$ where $A$ and $B$ are some expressions on letters $a$ and $b$. These identities can be interpreted as an equalities in the free Lie ring $L=L(a,b).$ Note that a description of all identities of such kind would give a full description of the intersection $[L,a] \cap [L,b].$ Consider a $\mathbb Z$-linear map $$\Theta:L \oplus L \to L,$$ $$\Theta(A,B)=[A,a]+[B,b].$$ Then the problem of describing identities of type can be formalised as the problem of describing $$I:={\rm Ker}(\Theta).$$ Any homogeneous element of $L$ of degree $\geq 2$ can be presented as $[A,a]+[B,b].$ So $I$ measures how far this presentation from being unique. The problem of describing of $I$ is different from the problem formulated on the formal language of identities, because $A,B$ here are not just formal expressions but they are elements of the free Lie ring. For example, the identity $[[b,b],a]=[[a,a],b]$ is not interesting for us because $([b,b],-[a,a])=(0,0)$ in $I.$ This work is devoted to the study of $I.$
The Lie ring $L$ has a natural grading by the weight of a commutator: $L=\bigoplus_{n\geq 1} L_n.$ Moreover, $L_n=\bigoplus_{k+m=n}L_{k,m}$, where $L_{k,m}\subseteq L_{k+m}$ is an abelian group generated by multiple commutators with $k$ letters $a$ and $m$ letters $b.$ We can consider the following restrictions of the map $\Theta$ $$\Theta_n:L_{n-1}\oplus L_{n-1} \to L_n, \hspace{1cm}\Theta_{k,l}:L_{k-1,l}\oplus L_{k,l-1} \to L_{k,l}$$ and set $I_n={\rm Ker}(\Theta_n)$ and $I_{k,l}={\rm Ker}(\Theta_{k,l}).$ It is easy to check that $$I=\bigoplus_{n\geq 1} I_n, \hspace{1cm} I_n=\bigoplus_{k+l=n} I_{k,l}.$$ The main results of the paper are the full description of $I_{2,n};$ the description of the rank of the free abelian group $I_{3,n};$ and the description of a concrete series of elements from $I_{3,3n}$ for any $n\geq 1.$
The rank of a free abelian group $X$ is called “dimension of $X$” in this paper and it is denoted by ${\rm dim}\, X.$ It well known that the dimension of $L_n$ can be computed by the Necklace polynomial $${\rm dim}\, L_n=\frac{1}{n} \sum_{d\mid n} \mu \left(\frac{n}{d}\right)2^d,$$ where $\mu$ is the Mobius function. $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
n &1 &2 &3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline
{\rm dim}\, L_n & 2 & 1 & 2 & 3 & 6 & 9 & 18 & 30 & 56 & 99 & 186 & 335 & 630 \\ \hline
\end{array}$$ Since the map $\Theta_n$ is an epimorphsm, we obtain $${\rm dim}\, I_n=2 \cdot {\rm dim}\ L_{n-1}-{\rm dim}\ L_n.$$ We prove the following $$\dim\, I_{2,m}=
\begin{cases}
0,\, \mbox{if } m \mbox{ is odd} \\
1,\, \mbox{if } m \mbox{ is even}
\end{cases}, \hspace{1cm} \dim\, I_{3,m}=\left\lfloor \frac{m+1}{2} \right\rfloor - \left\lfloor \frac{m-1}{3} \right\rfloor - 1.$$ (Proposition \[dimI2n\], Proposition \[dimI3m\]). For $n\leq 13$ we obtain the following table for dimensions. $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
n &2 &3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline
{\rm dim}\, I_{2,n-2} & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ \hline
{\rm dim}\, I_{3,n-3} & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 2 & 1 \\ \hline
{\rm dim}\, I_n & 3 & 0 & 1 & 0 & 3 & 0 & 6 & 4 & 13 & 12 & 37 & 40 \\ \hline
\end{array}$$
The computation of ${\rm dim}\, I_{2,m}$ shows that there are no any other elements in $I_{2,m}$ except those that come from the identities . In particular, all non-trivial identities corresponding to elements of $I_{2,n}$ have even weight.
Note that $I_n=0$ for odd $n<9.$ This can be interpreted as the fact that there is no a non-trivial identity of the type $[A,a]=[B,b]$ on two letters of odd weight lesser than $9.$ However, we have found a non-trivial identity of this type of weight $9$ (Theorem \[th\_I36\]). If we set $$C_n=[a,_nb],$$ then the following identity of weight $9$ holds in any Lie ring $$\label{eq_36}
\big[2[C_5,C_1]+5[C_4,C_2],\ a\ \big]=\big[ 2[C_4,C_1,C_0]+3[C_3,C_2,C_0]-2[C_3,C_1,C_1]+[C_2,C_1,C_2], \ b \ \big].$$
The main result of this paper is a concrete series of identities that correspond to non-trivial elements in $I_{3,3n}$ that generalise the identity (Theorem \[3aId\]). Namely, for any $n\geq 1$, the following identity is satisfied in $L_{3,3n}$. $$\left[\sum_{k=0}^{\left\lfloor\frac{n+1}{2}\right\rfloor}(-1)^{n+1}\alpha_{n+1-k, k}[C_{2n+1-k}, C_{n+k-1}]\ ,\ a\right] = \left[\sum_{i=0}^{n}\sum_{j=0}^{\left\lfloor\frac{i}{2}\right\rfloor}(-1)^{i+1}\alpha_{i-j, j}[C_{n+i-j}, C_{n+j-1},C_{n-i}]\ ,\ b\right],$$ where $\alpha_{0,0}=1$ and $\alpha_{i,j}=2\binom{i+j-1}{j}+\binom{i+j-2}{j-1}-\binom{i+j-2}{j-2}-2\binom{i+j-1}{j-2}$ for $i,j\geq 0$ and $(i,j)\ne (0,0)$. For $n=2$ we obtain the identity . For $n=1$ we obtain the identity $$[\ 3[a,b,b, [a,b]]+ 2[a,b,b,a]\ ,\ a]\ =\ [\ -[a,b,a,[a,b]] + 2[a,b,b,a,a]\ ,\ b]$$ that holds in any Lie ring.
Identities corresponding to elements in $I_{2,m}$
=================================================
If $w$ is a Lyndon word, we denote by $[w]$ the corresponding element of the Lyndon-Shirshov basis of the free Lie algebra $L$ (see [@Reutenauer]). If $w$ is a letter, then $[w]=w.$ If $w$ is not a letter then $w$ has a standard factorisation $w=uv$ and $[w]$ is defined by recursion $[w]=[[u],[v]].$ For example, $[a]=a$ and $[ab^n]=[[ab^{n-1}],b]=C_n$.
\[basL2n\] The following set is a basis of $L_{2,n}$ with $n\in\mathbb{N}$. $$\left\{ \,[C_k, C_l] \;|\; k>l,\: k+l=n \;\;k,l,m\in\mathbb{N} \,\right\}.$$
The intersection of the Lyndon-Shirshov basis with $L_{k,m}$ is a basis of $L_{k,m}.$ The basis of $L_{2,m}$ consists of commutators of Lyndon words with $2$ letters “$a$” and $m$ letters “$b$”. $$[ab^lab^k]=[[ab^l],[ab^k]]=-[[ab^l],[ab^k]]=-[C_k,C_l].$$ Word $ab^lab^k$ is a Lyndon word only when $k>l$. The assertion follows.
\[dim2n\] For any $n\in\mathbb{N}$ the following is satisfied: $$\dim L_{2,n}=\left\lceil \frac{n}{2} \right\rceil = \left\lfloor \frac{n+1}{2}\right\rfloor.$$
Consider the basis from lemma \[basL2n\]. Hence $L_{2,n}=
\langle \,\{ [C_{n_1},C_{n_2}] \, | \, n_1,n_2\in\mathbb{N}_0, n_1 > n_2 \mbox{ and } {n_1}+{n_2}=n \} \, \rangle$. Total number of words with $2$ letters $a$ and $n$ letters $b$ starting with $a$ is $n+1$. However, in our case $n_1>n_2$. Hence for odd $n$ number of such commutators is $\frac{n+1}{2}$ and for even $n$ it is $\frac{n}{2}$.
\[dimI2n\] For any $m\in\mathbb{N}$ we have $$\dim\, I_{2,m}=
\begin{cases}
0,\, \mbox{if } m \mbox{ is odd} \\
1,\, \mbox{if } m \mbox{ is even}.
\end{cases}$$
By definition, $I_{2,m}={\rm Ker}\, \Theta_{2,m}$. Then $\dim I_{2,m}=\dim (\ker\Theta_{2,m})=\dim (L_{1,m} \oplus L_{2,m-1})-\dim (\mathrm{Im} \, \Theta_{2,m})= \dim (L_{1,m} \oplus L_{2,m-1})-\dim L_{2,m}=\dim L_{1,m}+\dim L_{2,m-1}-\dim L_{2,m}=1+\dim L_{2,m-1}-\dim L_{2,m}=1+\left\lfloor \frac{m}{2}\right\rfloor -
\left\lfloor \frac{m+1}{2}\right\rfloor$ (see lemma \[dim2n\]). Let $m$ be even, then $\dim I_{2,m}=1+ \frac{m}{2} - \left\lfloor \frac{m}{2} -\frac{1}{2}\right\rfloor=1+\frac{m}{2}-\frac{m}{2}=1$. Consider the case of odd $m$. Then $\dim I_{2,m} = 1 + \left\lceil \frac{m-1}{2} \right\rceil - \frac{m+1}{2}=1+\frac{m}{2} -\frac{1}{2}-\frac{m}{2}-\frac{1}{2}=0$.
For any $m\in\mathbb{N}$ the following is satisfied $$I_{2,m}=
\begin{cases}
0,\, \mbox{if } m \mbox{ is odd} \\
\langle\, (\, C_m ,\, \sum_{i=1}^{\frac{m}{2}}(-1)^i [C_{m-i}, C_{i-1}]\, ) \, \rangle,\, \mbox{if } m \mbox{ is even}.
\end{cases}$$
Triviality of the kernel for odd $m$ can be easily proven using lemma \[dim2n\]. Consider the case when $m$ is even. Then basis of $L_{1,m}$ consists of one element $[ab^m]$, i.e. $L_{1,m}=\left\{ \alpha[ab^m] \; | \; \alpha\in\mathbb{Z} \right\}$. Basis of $L_{2,m-1}$ consists of $\left\lfloor \frac{m}{2}\right\rfloor$ elements (according to lemma \[dim2n\]). Because $m$ is even $\left\lfloor \frac{m}{2}\right\rfloor=\frac{m}{2}$. Hence the following equality is true $$L_{2,m-1}=\left\{ \alpha_1[aab^{m-1}]+\alpha_2[abab^{m-2}]+\dots+\alpha_{\frac{m}{2}}[ab^{\frac{m}{2}-1}ab^{m-\frac{m}{2}}] \; | \; \alpha_1,\alpha_2,\ldots,\alpha_{\frac{m}{2}}\in\mathbb{Z} \right\}.$$ By definition, $I_{2,m}=\ker \Theta_{2,m}$. We can apply map $\Theta_{2,m}$ to arbitrary element of $L_{1,m}\oplus L_{2,m-1}$ that is expressed as basis elements and equate the obtained to zero. Jacobi identity implies the following $$[[ab^nab^m], b]=[[ab^{n+1}],[ab^m]]+[ab^nab^{m+1}].$$ We can use this equality to transform an image of an element from $L_{2,m-1}$. $$\Theta_{2,m}\left(\alpha[ab^m], \sum_{i=1}^{\frac{m}{2}}\alpha_i [ab^{i-1}ab^{m-i}]\right)=\alpha[[ab^m],a] + \sum_{i=1}^{\frac{m}{2}}\alpha_i [[ab^{i-1}ab^{m-i}], b]=$$ $$=-\alpha[a,[ab^m]]+\sum_{i=1}^{\frac{m}{2}}\alpha_i \left([[ab^{i}],[ab^{m-i}]]+[ab^{i-1}ab^{m-i+1}]\right)=$$ For all $i\not = \frac{m}{2}$ commutator $[[ab^{i}],[ab^{m-i}]]=[ab^{i}ab^{m-i}]$. Then the sum can be rewritten as follows $$= -\alpha[aab^m]+\alpha_1[abab^{m-1}]+\alpha_1[aab^m]+\alpha_2[ab^2ab^{m-2}]+\alpha_2[abab^{m-1}]+\dots+\alpha_{i-1}[ab^{i-1}ab^{m-i+1}]+$$ $$+\alpha_{i-1}[[ab^{i-2}ab^{m-i+2}]]+\alpha_i[ab^{i}ab^{m-i}]+\alpha_i[ab^{i-1}ab^{m-i+1}]+\alpha_{i+1}[ab^{i+1}ab^{m-i+1}]+\alpha_{i+1}[ab^{i}ab^{m-i}]+\dots+$$ $$+\alpha_{\frac{m}{2}}[[ab^{\frac{m}{2}}],[ab^{m-\frac{m}{2}}]]+\alpha_{\frac{m}{2}}[ab^{\frac{m}{2}-1}ab^{m-\frac{m}{2}+1}]=0.$$ Last but one element of sum is equals to $0$ because $\alpha_{\frac{m}{2}}[[ab^{\frac{m}{2}}],[ab^{m-\frac{m}{2}}]]=
\alpha_{\frac{m}{2}}[[ab^{\frac{m}{2}}],[ab^{\frac{m}{2}}]]=0$. It is easy to see that for equality we need such coefficients $\alpha$ and $\alpha_i$ that terms of the sum will be reduced. Let $\alpha=1$, hence $\alpha_1=1$ because we need $-\alpha[aab^m]$ and $\alpha_1[aab^m]$ to be reduced. Other coefficients can be obtained similarly. Commutators with coefficients $\alpha_i$ and $\alpha_{i+1}$ will be reduced. Hence $\ker \Theta_{2,m}$ is generated by element $[ab^m]=C_m$ and sum $[aab^{m-1}]-[abab^{m-2}]+\dots\mp[ab^{\frac{m}{2}-1}ab^{m-\frac{m}{2}+1}]\pm[ab^{\frac{m}{2}}ab^{m-\frac{m}{2}}]=\sum_{i=1}^{\frac{m}{2}}(-1)^{i+1}[ab^{i-1}ab^{m-i}]=$ $=\sum_{i=1}^{\frac{m}{2}}(-1)^{i+1}[[ab^{i-1}],[ab^{m-i}]]=\sum_{i=1}^{\frac{m}{2}}(-1)^{i}[[ab^{m-i}],[ab^{i-1}]]=\sum_{i=1}^{\frac{m}{2}}(-1)^{i}[C_{m-i},C_{i-1}] $
Identities corresponding to elements in $I_{3,m}$
=================================================
\[basisL3n\] For any $n\in\mathbb{N}$ the following set is a basis of $L_{3,n}$. $$\left\{ \,[C_k, C_l, C_m] \;|\; k>l,\,k\geq m, \, k+l+m=n \;\;k,l,m\in\mathbb{N}_0 \,\right\}.$$
Lyndon words commutators of length $n+3$ with $3$ letters “a” and $n$ letters “b” construct the basis of $L_{3, n}$. It is easy to prove that $ab^{i}ab^{j}ab^{t}$ is a Lyndon word if and only if $i\leq j$ and $i < t$, where $i,j,t\in\mathbb{N}_0$. Consider two cases:
1. $j < t$ then $[ab^iab^jab^t]=[[ab^i],[ab^jab^t]]=[[ab^i],[[ab^j],[ab^t]]]=[[ab^t],[ab^j],[ab^i]]=[C_t,C_j,C_i]$. Take $t=k, \: j=l, \: i=m$ then $k > l, \: k>m$ and $l\geq m$.
2. $j\geq t$ then $[ab^iab^jab^t]=[[ab^iab^j],[ab^t]]=[[ab^i],[ab^j],[ab^t]]=[C_i ,C_j,C_t]=-[C_j ,C_i,C_t]$. Take $j=k, \: u=l, \: t=m$ then $k \geq m, \: l\leq k$ and $m>l$. Hence $k\geq m>l$, so $k>l$.
If we unite conditions of both cases, we get $k>l$ and $k\geq m$ for arbitrary $k,l,m\in\mathbb{N}_0$.
Generalized identity with three letters “a”
-------------------------------------------
\[recur\] For expression $\alpha_{i,j}=2\binom{i+j-1}{j}+\binom{i+j-2}{j-1}-\binom{i+j-2}{j-2}-2\binom{i+j-1}{j-2}$, where $i,j \in \mathbb{N}_0$ the following conditions are satisfied:
1. $\alpha_{i-1,j}+\alpha_{i,j-1}=\alpha_{i,j}$, when $i \ne j$ and $j\ne 0$
2. $\alpha_{i,i-1}=\alpha_{i,i}$, when $i\ge2$
3. $\alpha_{i,o}=2$, when $i\geq1$
We can use the recurrence relation for binomial coefficient to prove the first condition: $$\alpha_{i-1,j}+\alpha_{i,j-1}= 2\binom{i+j-2}{j}+\binom{i+j-3}{j-1}-\binom{i+j-3}{j-2}-2\binom{i+j-2}{j-2} +$$ $$+\: 2\binom{i+j-2}{j-1}+\binom{i+j-3}{j-2}-\binom{i+j-3}{j-3}-2\binom{i+j-2}{j-3}
= 2 \left(\binom{i+j-2}{j} + \binom{i+j-2}{j-1}\right) +$$ $$+\left(\binom{i+j-3}{j-1} + \binom{i+j-3}{j-2}\right) - \left(\binom{i+j-3}{j-2}+ \binom{i+j-3}{j-3}\right) - 2\left(\binom{i+j-2}{j-2} + \binom{i+j-2}{j-3}\right) =$$ $$=2\binom{i+j-1}{j}+\binom{i+j-2}{j-1}-\binom{i+j-2}{j-2}-2\binom{i+j-1}{j-2}=\alpha_{i,j}.$$ To prove the second condition we can substitute $j=i$ into $\alpha_{i, j}$ and express each term using recurrence relation for binomial coefficient: $$\alpha_{i,i}=2\binom{2i-1}{i}+\binom{2i-2}{i-1}-\binom{2i-2}{i-2}-2\binom{2i-1}{i-2}=$$ $$=\underline{2\binom{2i-2}{i-1}}+2\binom{2i-2}{i}+\underline{\binom{2i-3}{i-2}}+\binom{2i-3}{i-1}-\underline{\binom{2i-3}{i-3}}-\binom{2i-3}{i-2}-\underline{2\binom{2i-2}{i-3}}-2\binom{2i-2}{i-2}=$$ $$=\alpha_{i,i-1}+2\binom{2i-2}{i}+\binom{2i-3}{i-1}-\binom{2i-3}{i-2}-2\binom{2i-2}{i-2}.
$$ All we need to prove now is that $2\binom{2i-2}{i}+\binom{2i-3}{i-1}-\binom{2i-3}{i-2}-2\binom{2i-2}{i-2}=0$. Using symmetric property of binomial coefficient, i.e. $\binom{n}{k}=\binom{n}{n-k}$, all terms will be reduced: $$2\binom{2i-2}{i}+\binom{2i-3}{i-1}-\binom{2i-3}{i-2}-2\binom{2i-2}{i-2}=\binom{2i-3}{i-2}+2\binom{2i-2}{i-2}-\binom{2i-3}{i-2}-2\binom{2i-2}{i-2}=0.$$ Consider the case, when $j=0$. We need to mention that for $k<0$ binomial coefficient $\binom{n}{k}=0$. Then the expression will be as follows. $$\alpha_{i,0}= 2\binom{i-1}{0}+\binom{i-2}{-1}-\binom{i-2}{-2}-2\binom{i-1}{-2}=2\binom{i-1}{0}=2.$$
\[3aId\] For any $n\in\mathbb{N}$, the following identity is satisfied in $L_{3,3n}$. $$\left[\sum_{k=0}^{\left\lfloor\frac{n+1}{2}\right\rfloor}(-1)^{n+1}\alpha_{n+1-k, k}[C_{2n+1-k}, C_{n+k-1}], a\right] = \left[\sum_{i=0}^{n}\sum_{j=0}^{\left\lfloor\frac{i}{2}\right\rfloor}(-1)^{i+1}\alpha_{i-j, j}[C_{n+i-j}, C_{n+j-1},C_{n-i}], b\right],$$ where $\alpha_{0,0}=1$ and $\alpha_{i,j}=2\binom{i+j-1}{j}+\binom{i+j-2}{j-1}-\binom{i+j-2}{j-2}-2\binom{i+j-1}{j-2}$, where $i,j \in\mathbb{N}$.
Consider $n, k\in\mathbb{N}$. Lets prove the following identity for any $k\leq n$: $$\left[\sum_{i=0}^{k}\sum_{j=0}^{\left\lfloor\frac{i}{2}\right\rfloor}(-1)^{i+1}\alpha_{i-j, j}[C_{n+i-j}, C_{n+j-1},C_{n-i}], b\right]=\sum_{t=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+1}\alpha_{k+1-t, t}[C_{n+k+1-t}, C_{n-1+t}, C_{n-k}].$$ Denote the left part as $\omega_k$ and the right part as $\theta_k$. We will prove this equality using mathematical induction with variable $k$.\
We can expand the sum and use lemma \[recur\] $$\omega_{1}=\left[-\alpha_{0,0}[C_{n},C_{n-1},C_{n}]+\alpha_{1,0}[C_{n+1},C_{n-1},C_{n-1}],b\right]=-1[C_{n},C_{n-1},C_{n},b]+2[C_{n+1},C_{n-1},C_{n-1}, b]=$$ $$=-1[C_{n+1},C_{n-1},C_{n}]+1[C_{n+1},C_{n},C_{n-1}]+2[C_{n+1},C_{n-1},C_{n}]+2[C_{n+2},C_{n-1},C_{n-1}]+2[C_{n+1},C_{n-1},C_{n-1}]=$$ $$=2[C_{n+2},C_{n-1},C_{n-1}]+3[C_{n},C_{n-1},C_{n}]=\alpha_{2,0}[C_{n+2},C_{n-1},C_{n-1}]+\alpha_{1,1}[C_{n},C_{n-1},C_{n}]=\theta_1$$
We need to prove that $\omega_{k}=\theta_{k}$ implies $\omega_{k+1}=\theta_{k+1}$. We can expand $\omega_{k+1}$ as a sum of $\omega_{k}$ and the last element of the first sum in $\omega_{k+1}$: $$\omega_{k+1}=\underbrace{\omega_{k}}_{\theta_{k}}+\left[\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+1-j, j}[C_{n+k+1-j}, C_{n+j-1},C_{n-k-1}],b\right]$$ By expanding the commutator as follows $[C_{n+k+1-j}, C_{n+j-1},C_{n-k-1},b]=[C_{n+k+2-j}, C_{n+j-1},C_{n-k-1},b]+[C_{n+k+1-j}, C_{n+j},C_{n-k-1},b]+[C_{n+k+1-j}, C_{n+j-1},C_{n-k},b] $, we can express the second term as three different sums. One of them will be reduced by $\theta_{k}$. $$\omega_{k+1}=\sum_{t=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+1}\alpha_{k+1-t, t}[C_{n+k+1-t}, C_{n-1+t}, C_{n-k}]+\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+1-j, j}[C_{n+k+2-j}, C_{n+j-1},C_{n-k-1}]+$$ $$+\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+1-j, j}[C_{n+k+1-j}, C_{n+j},C_{n-k-1}]+\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+1-j, j}[C_{n+k+1-j}, C_{n+j-1},C_{n-k}]=$$ $$=\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+1-j, j}[C_{n+k+2-j}, C_{n+j-1},C_{n-k-1}]+\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+1-j, j}[C_{n+k+1-j}, C_{n+j},C_{n-k-1}]$$ Denote commutators in sums as $a_j$ and $b_j$ correspondingly. We can show that for any $j\geq 0$ it is satisfied that $a_{j+1}=b_j$. $$a_{j+1}=[C_{n+k+2-j-1}, C_{n+j+1-1},C_{n-k-1}]=[C_{n+k+1-j}, C_{n+j},C_{n-k-1}]=b_j.$$ Because of that, the expression can be written as a sum of $a_0$, $b_{\left\lfloor\frac{k+1}{2}\right\rfloor}$ with corresponding coefficients and one sum on index $j$ with summed coefficients. $$\omega_{k+1}=\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor-1}(-1)^{k+2}(\alpha_{k-j, j+1} + \alpha_{k+1-j, j})[C_{n+k+1-j}, C_{n+j},C_{n-k-1}] +(-1)^{k+2}\alpha_{k+1,0}[C_{n+k+2}, C_{n-1}, C_{n-k-1}]-$$ $$+(-1)^{k+2}\alpha_{k+1-\left\lfloor\frac{k+1}{2}\right\rfloor, \left\lfloor\frac{k+1}{2}\right\rfloor}[C_{n+k+1-\left\lfloor\frac{k+1}{2}\right\rfloor},C_{n+\left\lfloor\frac{k+1}{2}\right\rfloor},C_{n-k-1}].$$ Coefficients of commutators, in obtained sum on index $j$, can be transformed using the first case of lemma \[recur\]. Also, we can change index of sum by subtracting $1$ from it. Coefficient of $a_0$ can be rewritten using the third case of lemma \[recur\]. Then $b_{\left\lfloor\frac{k+1}{2}\right\rfloor}$ will become a zero element of sum on index $j$. $$\omega_{k+1}=\sum_{j=1}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+2-j, j}[C_{n+k+2-j}, C_{n-1+j},C_{n-k-1}] +(-1)^{k+2}\alpha_{k+2,0}[C_{n+k+2}, C_{n-1}, C_{n-k-1}]+$$ $$+(-1)^{k+2}\alpha_{k+1-\left\lfloor\frac{k+1}{2}\right\rfloor, \left\lfloor\frac{k+1}{2}\right\rfloor}[C_{n+k+1-\left\lfloor\frac{k+1}{2}\right\rfloor},C_{n+\left\lfloor\frac{k+1}{2}\right\rfloor},C_{n-k-1}]=$$ $$=\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+2-j, j}[C_{n+k+2-j}, C_{n-1+j},C_{n-k-1}] + (-1)^{k+2}\alpha_{k+1-\left\lfloor\frac{k+1}{2}\right\rfloor, \left\lfloor\frac{k+1}{2}\right\rfloor}b_{\left\lfloor\frac{k+1}{2}\right\rfloor}.$$ Consider two cases:
**1)** $k$ is odd. Then $\left\lfloor\frac{k+2}{2}\right\rfloor=\left\lfloor\frac{k+1}{2} + \frac{1}{2}\right\rfloor=\frac{k+1}{2}+\left\lfloor\frac{1}{2}\right\rfloor=\frac{k+1}{2}=\left\lfloor\frac{k+1}{2}\right\rfloor$, hence $b_{\left\lfloor\frac{k+1}{2}\right\rfloor}=0$. It is true because $b_{\left\lfloor\frac{k+1}{2}\right\rfloor}=[C_{n+k+1-\frac{k+1}{2}},C_{n+\frac{k+1}{2}},C_{n-k-1}]=[0,C_{n-k-1}]=0$. Consequently $\omega_{k+1}$ can be expressed as follows. $$\omega_{k+1}=\sum_{j=0}^{\left\lfloor\frac{k+2}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+2-j, j}[C_{n+k+2-j}, C_{n-1+j},C_{n-k-1}]= \theta_{k+1}.$$
**2)** $k$ is even. Then $\left\lfloor\frac{k+2}{2}\right\rfloor=\frac{k+2}{2}=\left\lfloor\frac{k}{2}\right\rfloor + 1=\left\lfloor\frac{k+1}{2}\right\rfloor + 1$, hence $\alpha_{k+2-\left\lfloor\frac{k+2}{2}\right\rfloor, \left\lfloor\frac{k+2}{2}\right\rfloor}=\alpha_{\left\lfloor\frac{k+1}{2}\right\rfloor + 1, \left\lfloor\frac{k+1}{2}\right\rfloor}$ because of the second case of lemma \[recur\]. It is important to mention that $\left\lfloor\frac{k+1}{2}\right\rfloor = \left\lfloor\frac{k}{2} + \frac{1}{2}\right\rfloor=\frac{k}{2}$. Hence $\alpha_{k+1-\left\lfloor\frac{k+1}{2}\right\rfloor, \left\lfloor\frac{k+1}{2}\right\rfloor} = \alpha_{\frac{k}{2}+1, \left\lfloor\frac{k+1}{2}\right\rfloor}=\alpha_{\left\lfloor\frac{k+1}{2}\right\rfloor + 1, \left\lfloor\frac{k+1}{2}\right\rfloor}$. We can express $\theta_{k+1}$ as a sum up to $\left\lfloor\frac{k+2}{2}\right\rfloor - 1 = \left\lfloor\frac{k+1}{2}\right\rfloor$ and the last addendum with $j=\left\lfloor\frac{k+2}{2}\right\rfloor=\left\lfloor\frac{k+1}{2}\right\rfloor + 1$: $$\theta_{k+1}=\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+2-j, j}[C_{n+k+2-j}, C_{n-1+j},C_{n-k-1}] +$$ $$+ (-1)^{k+2}\alpha_{k+2-\left\lfloor\frac{k+2}{2}\right\rfloor, \left\lfloor\frac{k+2}{2}\right\rfloor}[C_{n+k+2-\left\lfloor\frac{k+1}{2}\right\rfloor- 1},C_{n-1+\left\lfloor\frac{k+1}{2}\right\rfloor + 1},C_{n-k-1}]=$$ $$=\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{k+2}\alpha_{k+2-j, j}[C_{n+k+2-j}, C_{n-1+j},C_{n-k-1}] + (-1)^{k+2}\alpha_{k+1-\left\lfloor\frac{k+1}{2}\right\rfloor, \left\lfloor\frac{k+1}{2}\right\rfloor}b_{\left\lfloor\frac{k+1}{2}\right\rfloor}=\omega_{k+1}$$
\
\
Consequently, the identity above is satisfied for any $n,k\in\mathbb{N}$, such that $k\leq n$. If we substitute $k=n$, we will get the original identity.
Additional results
------------------
\[razn3n\] For any $n\in\mathbb{N}$ the following is satisfied: $$\dim L_{3,n}-\dim L_{3,n-1}=\left\lfloor \frac{n-1}{3} \right\rfloor + 1.$$
To calculate this expression, we need to count all Lyndon words of form $ab^{n_1}ab^{n_2}ab^{n_3}$, where $n_1,n_2,n_3\in\mathbb{N}_0 \mbox{ and } {n_1}+{n_2}+{n_3}=n$. Let $n_1=i$ and $n_2=j$, hence $n_3=n-i-j$. As it was mentioned before, $ab^{n_1}ab^{n_2}ab^{n_3}$ is a Lyndon word if and only if $n_1\leq n_2$ and $n_1 < n_3$, where $n_1,n_2,n_3\in\mathbb{N}$. We can portray integer points that satisfy these conditions on coordinate plane by drawing plots of functions $y=x$ and $y=n-2x$.
Abscissa of functions intersection point is $\frac{n}{3}$. $ab^{i}ab^{j}ab^{n-i-j}$ is a Lyndon word if point $(i,j)$ belongs to $ \bigtriangleup ABC$ (without point on the line $y=n-2x$). Then $\dim L_{3,n}$ equals to number of integer points in $ \bigtriangleup DEC$. Hence $\dim L_{3,n}-\dim L_{3,n-1}$ equals to number of integer points on segment $DE$, i.e. $\left\lfloor \frac{n-1}{3} \right\rfloor + 1$.
\[dimI3m\]
For any $m\in\mathbb{N}$ the following is satisfied: $$\dim I_{3,m}=\left\lceil \frac{m}{2} \right\rceil - \left\lfloor \frac{m-1}{3} \right\rfloor - 1.$$
By definition, $I_{3,m}=\ker\Theta_{3,m}$. Hence, according to lemmas \[dimI2n\] and \[razn3n\], $\dim I_{3,m}=\dim \ker\Theta_{3,m}=\dim (L_{2,m} \oplus L_{3,m-1})-\dim L_{3,m} = \dim L_{2,m} - (\dim L_{3,m} - \dim L_{3,m-1})=\left\lceil \frac{m}{2} \right\rceil - \left\lfloor \frac{m-1}{3} \right\rfloor - 1$.
\[razl\] For $k>l$, $k\geq m$ the following is satisfied: $$[C_k, C_l, C_m, b]=
\begin{cases}
[C_{k+1}, C_l, C_m] + [C_k, C_{l+1}, C_m] + [C_k, C_l, C_{m+1}], \mbox{if } k>l+1,\: k\geq m+1 \\
[C_{k+1}, C_l, C_m] + [C_k, C_l, C_{m+1}], \mbox{if } k=l+1,\: k\geq m+1 \\
2[C_{k+1}, C_l, C_m] - [C_{k+1}, C_{l+1}, C_{m-1}], \mbox{if } k=l+1,\: k=m \\
2[C_{k+1}, C_l, C_m] + [C_k, C_{l+1}, C_m] - [C_{k+1}, C_k, C_l], \mbox{if } k>l+1,\: k=m.
\end{cases}$$
It is easy to rewrite the expression in the first case using Jacobi identity: $$[C_k, C_l, C_m, b]=[C_k,C_l,b,C_m]+[C_k,C_l,[C_m,b]]
=[C_{k+1}, C_l, C_m] + [C_k, C_{l+1}, C_m] + [C_k, C_l, C_{m+1}].$$ Second case: $$[C_{k}, C_l, C_m, b]=[C_{l+2}, C_l, C_m] + [C_{l+1}, C_{l+1}, C_m] + [C_{l+1}, C_l, C_{m+1}]=[C_{k+1}, C_l, C_m] + [C_{k}, C_l, C_{m+1}].$$ Third case: $$[C_k, C_l, C_m, b]=[C_{k+1}, C_l, C_{m}] + [C_{k}, C_l, C_{m+1}]=[C_{k+1}, C_l, C_m]+[C_k,C_{m+1},C_l]+[C_{m+1},C_l,C_k]=$$ $$=2[C_{k+1},C_l,C_m]-[C_{k+1},C_{l+1},C_{m-1}].$$ Fourth case: $$[C_k, C_l, C_m, b]
=[C_{k+1}, C_l, C_m] + [C_k, C_{l+1}, C_m] + [C_k, C_l, C_{m+1}]= [C_{k+1}, C_l, C_m] + [C_k, C_{l+1}, C_m] +$$ $$+[C_k,C_{m+1},C_l]-[C_l,C_{m+1},k]=[C_{k+1}, C_l, C_m] + [C_k, C_{l+1}, C_m] + [C_k,C_{m+1},C_l]-[C_l,C_{m+1},k]=$$ $$=2[C_{k+1}, C_l, C_m] + [C_k, C_{l+1}, C_m] - [C_{k+1}, C_k, C_l].$$
\[I33\] The kernel of $\Theta_{3,3}$ is generated by the following element: $$(3[C_2, C_1]+ 2[C_3,C_0], [C_1,C_0,C_1] - 2[C_2,C_0,C_0] ).$$
According to lemma \[dimI3m\] $\dim I_{3,3}=\left\lceil \frac{3}{2} \right\rceil - \left\lfloor \frac{2}{3} \right\rfloor - 1=1$. Consequently, we have to provide only one identity to describe the whole $I_{3,3}$. Substitute $n=1$ into identity from theorem \[3aId\]: $$[\alpha_{2,0}[C_3,C_0] + \alpha_{1,1}[C_2,C_1],a]=[-\alpha_{0,0}[C_1,C_0,C_1]+\alpha_{1,0}[C_2,C_0,C_0],b].$$ By definition of $\alpha_{i,j}$, $\alpha_{2,0}=2, \: \alpha_{1,1}=3,\: \alpha_{0,0}=1$ and $\alpha_{1,0}=2$. We can move right part of the equality to the left side and it will become an image of the element from $L_{2,3} \oplus L_{3,2}$: $$\hspace{-0.5cm}
[2[C_3,C_0] + 3[C_2, C_1],a] + [[C_1,C_0,C_1] - 2[C_2,C_0,C_0],b] = \Theta_{3,3}(2[C_3,C_0]+3[C_2, C_1] , [C_1,C_0,C_1] - 2[C_2,C_0,C_0])= 0$$ As a result, we obtained the element that generates all identities in $L_{3,3}$ that is equivalent to description of $I_{3,3}$.
\[th\_I36\] The abelian group $I_{3,6}$ is generated by the following element $$(-2[C_5,C_1]-5[C_4,C_2]\ ,\ 2[C_4,C_1,C_0]+3[C_3,C_2,C_0]-2[C_3,C_1,C_1]+[C_2,C_1,C_2] )$$
Similarly to proof of the theorem \[I33\]. $\dim I_{3,6}=\left\lceil \frac{6}{2} \right\rceil - \left\lfloor \frac{5}{3} \right\rfloor - 1=1$. Substitute $n=2$ into identity from theorem \[3aId\]: $$[-\alpha_{3,0}[C_5,C_1]-\alpha_{2,1}[C_4,C_2], a]=[-\alpha_{0,0}[C_2,C_1,C_2]+\alpha_{1,0}[C_3,C_1,C_1]-\alpha_{2,0}[C_4,C_1,C_0]-\alpha_{1,1}[C_3,C_2,C_0],b].$$ Coefficients will be $\alpha_{3,0}=2, \: \alpha_{2,1}=5, \: \alpha_{0,0}=1, \: \alpha_{1,0}=2, \: \alpha_{2,0}=2$ and $\alpha_{1,1}=3$. Again, we’ve found an element of $L_{2,6} \oplus L_{3,5}$ that generates all possible identities. Coefficients in the right part will be multiplied by $-1$ because of moving to the left side.
[99]{} Sergei O. Ivanov, Roman Mikhailov: A finite $\mathbb{Q}$–bad space. arXiv:1708.00282 C. Reutenauer: Free Lie algebras, Oxford University Press, 1993
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abstract: 'The Hilbert Huang Transform is a new technique for the analysis of non–stationary signals. It comprises two distinct parts: [*[Empirical Mode Decomposition]{}*]{} (EMD) and the [*[Hilbert Transform]{}*]{} of each of the modes found from the first step to produce a Hilbert Spectrum. The EMD is an adaptive decomposition of the data, which results in the extraction of Intrinsic Mode Functions (IMFs). We discuss the application of the EMD to the calibration of two optical scintillometers that have been used to measure $C_n^2$ over horizontal paths on a building rooftop, and discuss the advantage of using the Marginal Hilbert Spectrum over the traditional Fourier Power Spectrum.'
author:
- |
Mark P. J. L. Chang, Erick A. Roura, Carlos O. Font, Charmaine Gilbreath and Eun Oh Physics Department, University of Puerto Rico, Mayagüez, Puerto Rico 00680\
U.S. Naval Research Laboratory, Washington D.C. 20375
bibliography:
- 'bib.bib'
title: 'Applying the Hilbert–Huang Decomposition to Horizontal Light Propagation $C_n^2$ data'
---
INTRODUCTION {#sect:intro}
============
The common practice when studying time series data is to invoke the tools of Fourier spectral analysis. Although extremely versatile and simple, the technique suffers from some stiff contraints that limit its usefulness when attempting to examine the effects of optical turbulence in the frequency domain. Namely, the system must be linear and the data must be strictly periodic or stationary. Strict stationarity is a constraint that is impossible to satisfy simply on practical grounds, since no detector can cover all possible points in phase space. The linearity requirement is also not generally fulfilled, since turbulent processes are by definition non–linear.
Fortunately a new technique that has come to be known as the Hilbert Huang Transform (HHT) has been developed[@Huang1998], patented by NASA. This allows for the frequency space analysis of non–stationary, non–linear signals. The HHT is composed of two main algorithms for filtering and analyzing such data series. Firstly it employs an adaptive technique to decompose the signal into a number of Intrinsic Mode Functions (IMFs) that have well prescribed instantaneous frequencies, defined as the first derivative of the phase of an analytic signal. The second step is to convert these IMFs into an energy–time–frequency relationship, by means of the Hilbert Transform.
Asides from overcoming the problems associated with more traditional Fourier methods, the HHT makes it possible to visualize the energy spread between available frequencies locally in time, rather like wavelet transform methods. The advantage the HHT has over wavelet transforms is that it is of much higher resolution, since it does not [*[a priori]{}*]{} assume a basis; rather it “lets the data do the talking”.
INSTANTANEOUS FREQUENCY {#sect:instfreq}
=======================
Key to the HHT is the idea of instantaneous frequency, which we will sometimes refer to as simply “the frequency”. The ideal instantaneous frequency is quite simply the frequency of the signal at a single time point. No knowledge is required of the signal at other times. Naturally such a statement leads to difficulties in definition; Huang et al[@Huang1999] take it to be the derivative of the phase of the analytic signal, found from the real and imaginary parts of the signal’s Hilbert Transform, which we follow.
The immediate problem in dealing with a phase so defined is that, for the most part, the Hilbert Transforms of the direct signals are not well behaved resulting in negative instantaneous frequencies which do not represent physical effects. The method by which this is circumvented is to ensure that the input to the Hilbert Transform obeys the following conditions:
- The number of local extrema of the input and the number of its zero crossings must be either equal or differ at most by one.
- At any point in time $t$, the mean value of the upper envelope (determined by the local maxima) and the lower envelope (determined by the local minima) is zero.
The functions that obey these are considered the IMFs.
EMPIRICAL MODE DECOMPOSITION {#sect:emd}
============================
We have implemented an IMF filtering algorithm, known as Empirical Mode Decomposition (EMD), following Huang et al[@Huang1998; @Huang2003]. The IMFs and the residual trend line thus obtained are verified to be complete by simply summing them to recreate the signal. The maximum relative error we have found is of the order $10^{-9}$ %.
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![\[fig:imfexample\] The IMFs found from a typical input signal (taken on 9-March-2006), with the signal itself shown at the top. For convenience, we refer to the lowest order IMF as one with the fastest oscillation. The bottom–most graph is the residual after removing all the IMFs and represents the overall trend.](pics/imf03092006-geology.eps "fig:"){height="60.00000%"}
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The IMFs show that in a very real sense the EMD method is acting as a filter bank, separating the more rapid oscillations from the slower oscillations. It seems that a subset of the individual IMFs may be added to determine the effect of physical variables, as suggested in Figure \[fig:solarfit\].
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![\[fig:solarfit\] (a) Stepwise summation of the 9-March-2006 IMFs and trend line to recreate the original input signal. (b) The sum of the trendline and the slowest 3 IMFs superimposed on the input signal. The overshoot into negative values of $C_n^2$ is unphysical, and serves to demonstrate that the information content of the subset is incomplete. Nevertheless, the fit does suggest that the IMFs represent an underlying physical process (primarily solar insolation).](pics/c2f03092006-geology.eps "fig:"){height="50.00000%"}
![\[fig:solarfit\] (a) Stepwise summation of the 9-March-2006 IMFs and trend line to recreate the original input signal. (b) The sum of the trendline and the slowest 3 IMFs superimposed on the input signal. The overshoot into negative values of $C_n^2$ is unphysical, and serves to demonstrate that the information content of the subset is incomplete. Nevertheless, the fit does suggest that the IMFs represent an underlying physical process (primarily solar insolation).](pics/solarfit03092006-geology.eps "fig:"){height="50.00000%"}
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In the absence of the major effects of solar insolation, the HHT technique reveals that the majority contribution to the $C_n^2$ signal lies in the highest order (slowest oscillation) IMFs, as can be seen from the extremely faithful fit to the data composed of the trend line and the 3 highest order IMFs shown in Figure \[fig:evening\].
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![\[fig:evening\] The 20-Feb-2006 $C_n^2$ and fit composed of the trend line and 3 highest order IMFs for both instruments (A on the left and B on the right). Sunset was at 18:31 local time.](pics/fit02202006-geology.eps "fig:"){height="35.00000%"} ![\[fig:evening\] The 20-Feb-2006 $C_n^2$ and fit composed of the trend line and 3 highest order IMFs for both instruments (A on the left and B on the right). Sunset was at 18:31 local time.](pics/fit02202006-stairs.eps "fig:"){height="35.00000%"}
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As a guide to the significance of the various modes, we examined the energy in the IMFs and compared the energy in each mode to the energy distribution of red noise. As an initial naïve estimator, we take IMF1 to be representative of the noise contained in the data. This is unlikely to be completely correct; probably a better noise estimator would be IMF1 - $<$IMF1$>$, where the angle brackets signify the mean value over the same temporal epoch (e.g. month or season). We do not do this simply because we do not have sufficient data.
We define the red noise (random) time series to be an AR1 process $$r(t_n) = \sigma E(t_n) + \rho r(t_{n-1})$$ The terms are: $$\begin{aligned}
\sigma & := & \textrm{standard deviation of IMF1} \\ \nonumber
E & := & \textrm{uniform distribution of random numbers between 1 and -1} \\ \nonumber
t_n & := & \textrm{the } n\textrm{th timestep} \\ \nonumber
r & := & \textrm{the random time series} \nonumber\end{aligned}$$ The IMFs of an ensemble of AR1 time series are generated and then this Monte Carlo is used to simulate the power distribution of the noise. The power of each $C_n^2$ derived IMF is then compared to the noise power distribution to determine the mode’s significance.
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![\[fig:montecarlo\] Monte Carlo simulations show the significance of the $C_n^2$ (solid line) derived IMFs for a single instrument on the afternoon/evening of 20-Feb-2006, compared to the equivalent red noise power (open circles).](pics/rednoisepower02202006-geology.eps "fig:"){height="50.00000%"}
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Figure \[fig:montecarlo\] shows that of the IMFs, the first three lie at or below the median red noise power. IMF4 to IMF9 lie above the median noise power, with the higher order IMFs being most significant. We interpret this simulation to mean that IMF6-IMF9 are highly physically significant.
HILBERT TRANSFORM OF IMFS {#sect:hht}
=========================
Following the decomposition into IMFs of the original signal, the derived components can be Hilbert transformed to produce a time–frequency map or spectrum.
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![\[fig:hhspectrum\] The Hilbert spectrum of the 9-March-2006 IMFs shown in Figure \[fig:imfexample\] plotted as a series of contour lines.](pics/hhspectrum03092006-geology.eps "fig:"){height="50.00000%"}
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Figure \[fig:hhspectrum\] shows that during the hours when there is no solar insolation the $C_n^2$ energy is distributed in the lowest frequencies. The highest frequencies sampled are only reached after the Sun is contributing energy into the lower atmosphere. The discontinuous, filamentary aspect of the plot indicates a large number of phase dropouts which shows that the data are non–stationary.
We are also able to find a Marginal Spectrum by integrating the Hilbert spectrum across time. The Marginal Spectrum shown in Figure \[fig:margfft\] clearly suffers from less leakage into the high frequencies than the Power Spectrum. The interpretation of both spectra are quite different: the Fourier Power Spectrum indicates that certain frequencies exist throughout the entire signal with a given squared amplitude. The Marginal Spectrum, on the other hand, describes the probability that a frequency exists at some local time point in the signal.
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![\[fig:margfft\] The Hilbert Marginal spectrum (solid line) of the 9-March-2006 IMFs compared to the Fourier Power Spectrum (dotted line) of the same data. The Power Spectrum has been shifted so that its maximum frequency coincides with that of the Marginal Spectrum. Note also that the units of the ordinate axis are arbitrary for the Fourier Power Spectrum.](pics/margfft03092006-geology.eps "fig:"){height="50.00000%"}
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It is clear that the two spectra have a difference in the standard deviations: the logarithm of the Marginal Spectrum has a standard deviation of 1.88 compared to the logarithm of the Fourier Power Spectrum, whose standard deviation is 2.24.
Applying a Kolmogorov–Smirnov test to the two spectra returns a P–value of 0 and a cumulative distribution function distance of 1, indicating that the data sets represent different distributions, as we might guess from their different gradients. We may therefore state that the spectra are unrelated and we argue on the basis of non–stationarity that the Fourier Power Spectrum has little, if any, physical meaning.
INSTRUMENTS AND DATA REDUCTION {#sect:instruments}
==============================
The $C_n^2$ data used in this study was collected during 2006 at the University of Puerto Rico, Mayagüez Campus, on the rooftop of the Physics Department.
The data were obtained with two commercially available scintillometers (model LOA-004) from Optical Scientific Inc, co–located such that the transmitter of system 1 was next to the receiver of system 2.
Each LOA-004 instrument comprises of a single modulated infrared transmitter whose output is detected by two single pixel detectors. For these data, the separation between transmitter and receiver was just under 100-m. The sample rate was set to 10 seconds, so that each $C_n^2$ point was found from a 10 second time average. The path integrated $C_n^2$ measurements are determined by the LOA instruments by computation from the log–amplitude scintillation ($C_\chi(r)$) of the two receiving signals[@Ochs1979; @Wang2002]. The algorithm for relating $C_\chi(r)$ to $C_n^2$ is based on an equation for the log–amplitude covariance function in Kolmogorov turbulence by Clifford [*[et al.]{}*]{}[@Clifford1974]
The data was collected by dedicated PCs, one per instrument. During analysis, the data were smoothed by a 120 point (10 minute) boxcar rolling average. This value was chosen for future ease of comparison with local weather station data, sampled at one reading per 10 minutes. Figure \[fig:comparisonimfs\] compares the extracted IMFs from a single day, from midnight to midnight. There are no data dropouts in the time signal for instrument A, while instrument B is 99.81% valid. A visual examination reveals that the measured $C_n^2$ functions are very similar and both instruments have 11 IMFs. Differences are to be found in the IMFs themselves.
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![\[fig:comparisonimfs\] The IMFs extracted from each instrument for the same day (3-March-2006), beginning and ending at midnight (A on the left and B on the right).](pics/imf03032006-geology.eps "fig:"){height="35.00000%"} ![\[fig:comparisonimfs\] The IMFs extracted from each instrument for the same day (3-March-2006), beginning and ending at midnight (A on the left and B on the right).](pics/imf03032006-stairs.eps "fig:"){height="35.00000%"}
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In Figure \[fig:comparisonmarginals\] we show the Hilbert Marginal Spectra derived from the IMFs together with a Kolmogorov Power Spectrum trend scaled to start coincident with the Marginal Spectra. Both Marginal Spectra follow each other fairly well, with similar frequency probabilities.
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![\[fig:comparisonmarginals\] The 3-March-2006 Marginal Spectra of both instruments with a comparison frequency$^{-5/3}$ (Kolmogorov Spectrum) trend line.](pics/marginals03032006.eps "fig:"){height="50.00000%"}
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Applying a Kolmogorov–Smirnov test to the two Marginal Spectra data sets gives the same means and standard deviations. This is indicative of a P–value of 1 and a cumulative distribution function distance of 0, so we may conclude that the instrument outputs have come from exactly the same distribution and are statistically identical. A further confirmation can be found by calculating the Hilbert phase difference between the two Marginal Spectra. Such a test displays phase synchronization, or lack thereof. In this case we find a phase difference of zero, so that the spectra are perfectly in phase.
CONCLUSIONS
===========
We have presented the results of applying the Hilbert Huang Transform to $C_n^2$ time series data. When used to compare the outputs of two of the same model of commercial scintillometer, we have been able to demonstrate that they provide identical output in terms of their Hilbert Marginal Spectra. It is clear that the HHT technique is a very useful tool in the analysis of non–stationary turbulence data and promises much in terms of understanding the nature of optical turbulence.
MPJLC would like to thank Norden Huang for introducing him to the Hilbert Huang Transform. Thanks also are due to Sergio Restaino and Christopher Wilcox for making available the scintillometers and providing data processing software.
|
---
abstract: 'We present a fourth-order accurate finite volume method for the solution of ideal magnetohydrodynamics (MHD). The numerical method combines high-order quadrature rules in the solution of semi-discrete formulations of hyperbolic conservation laws with the upwind constrained transport (UCT) framework to ensure that the divergence-free constraint of the magnetic field is satisfied. A novel implementation of UCT that uses the piecewise parabolic method (PPM) for the reconstruction of magnetic fields at cell corners in 2D is introduced. The resulting scheme can be expressed as the extension of the second-order accurate constrained transport (CT) Godunov-type scheme that is currently used in the Athena astrophysics code. After validating the base algorithm on a series of hydrodynamics test problems, we present the results of multidimensional MHD test problems which demonstrate formal fourth-order convergence for smooth problems, robustness for discontinuous problems, and improved accuracy relative to the second-order scheme.'
address:
- 'Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, United States'
- 'Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, United States'
author:
- Kyle Gerard Felker
- James Stone
bibliography:
- 'FelkerStone2017.bib'
title: 'A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport'
---
magnetohydrodynamics ,numerical methods ,high-order finite volume method ,constrained transport
Introduction {#sec:intro}
============
Numerical solutions to the equations of ideal compressible magnetohydrodynamics (MHD) are widely used to study astrophysical phenomena including jets, accretion disks, winds, solar flares, and magnetospheres. Solutions to these problems may possess both complex smooth features and strong shocks. It is a challenge for numerical methods to resolve the complex turbulent dynamics and capture discontinuous features in a robust and computationally cost effective manner.
Among methods for gas dynamics, Godunov-type schemes based on the local solutions to Riemann problems are well-suited to both capturing shocks and resolving nonlinear waves; therefore, they have been extended to MHD in a variety of approaches [@GardinerStone2005; @GardinerStone2008; @Crockett2005; @Fromang2006; @Dedner2002; @Toth2000; @RyuMinatiJonesFrank1998; @DaiWoodward1998b; @BalsaraSpicer1999]. Second-order spatially and temporally accurate Godunov-type methods are among the most popular upwind schemes in computational astrophysics. However, higher-order ($\mathcal{O}(\Delta x^r, \Delta t^p)$ where $ r, p \geq 3 $ here) schemes can improve smooth solutions at a much faster rate than second-order methods as the discrete resolution is improved. Further, high-order methods typically improve the spatial locality and reuse of memory references of the algorithm, and thus they reduce data transfers and operate more efficiently than low-order schemes [@Loffeld2017; @Guzik2015]. The on-node performance benefit increases for larger local domain box sizes, especially for higher dimensional problems on high-performance manycore architectures [@Loffeld2017; @Olschanowsky2014].
We are unaware of any finite volume methods for MHD of fourth-order or greater accuracy in active use in the computational astrophysics community. Discontinuous Galerkin (DG) methods for MHD have been formulated at arbitrarily high-order accuracy [@Mocz2014; @Luo2008; @LiShu2005], however such schemes are rarely used at or beyond $\mathcal{O}(\Delta x^4)$ accuracy, as it remains a challenge to prevent unphysical oscillations at shocks and discontinuities. Finite difference (FD) methods have been applied to MHD with third and fifth-order accuracy [@DelZanna2007; @Mignone2010a]. High-order finite volume (FV) methods offer the accuracy of DG and FD methods with robust shock capturing. Several high-order FV methods, typically employing variants of weighted essentially non-oscillatory (WENO) reconstruction, have been coupled to techniques such as GLM-MHD to correct errors that arise when evolving the magnetic fields [@Susanto2013; @NunezMunz2016-I]. In addition to the challenges of nonlinearity inherent in the Euler equations of compressible hydrodynamics, numerical integration of MHD must also contend with the non-convexivity and non-strict hyperbolicity of the system of equations. Moreover, the equations of MHD are intrinsically multidimensional and thus cannot be consistently expressed in a dimensionally split approach, a technique commonly used to reduce the complexity of a system of equations to a series of 1D problems. The ideal MHD system does not trivially map to a conservation law formulation, since the $$\nabla \cdot \mathbf{B} = 0 \, .\label{eq:div-b}$$ It is well known that the divergence-free constraint must be satisfied to numerical precision to omit nonphysical solutions generated by the presence of numerical magnetic monopoles [@BrackbillBarnes1980].
McCorquodale and Colella [@McCorquodaleColella2011] formulated a fourth-order method for hydrodynamics that is extensible to mapped, multiblock grids with adaptive mesh refinement (AMR) [@ColellaDorrHittingerMartin2009; @ColellaDorrHittingerMartin2011; @Guzik2015]. High-order accuracy is achieved for smooth problems using a combination of quadrature rules evaluated with finite difference approximations and $\mathcal{O}(\Delta t^4)$ Runge-Kutta (RK) temporal integration. The piecewise parabolic method (PPM) is used to guarantee the robustness of the method for problems with strong discontinuities. We use this approach as the basis of the algorithm we present here.
The constrained transport (CT) technique was introduced by Evans and Hawley [@EvansHawley1988] and addresses the need to enforce the divergence-free constraint. Unlike alternative MHD algorithms such as divergence cleaning [@BalsaraKim2004], Hodge projection, 8-wave schemes [@Powell1999], or GLM-MHD [@Dedner2002; @Mignone2010a], the CT discretization evolves the magnetic field using the induction equation directly. While this approach can strictly enforce the solenoidal condition in Equation to round-off error, there are difficulties in applying the technique in practice. Most crucially, the original CT formulation is limited to second-order accuracy [@Matsumoto2016]. When combined with upwinding of hydrodynamic fluxes, the CT discretization may lead to a scheme with dual independent representations of magnetic field quantities. The consistency of $\mathbf{B}$ on a staggered mesh relative to the conserved variable mesh quantity $\mathbf{B}$ is often guaranteed on a case-by-case basis for the particular underlying finite volume scheme [@GardinerStone2005; @GardinerStone2008].
The upwind constrained transport (UCT) framework of Londrillo and Del Zanna [@Londrillo2000; @Londrillo2004] extends CT formalism to a class of hybrid methods for MHD that inherently 1) maintain the divergence-free condition and 2) are consistent with the underlying Godunov scheme. The authors detail and test several high-order examples of their framework; a third-order accurate scheme based on WENO reconstruction produces the best results for their MHD test problems [@Londrillo2004].
In this paper, we combine the fourth-order finite volume method of McCorquodale and Colella for compressible hydrodynamics with a novel implementation of UCT that uses PPM for reconstructing field quantities across cell-faces [@McCorquodaleColella2011]. The paper is organized as follows. Section \[sec:fv-hydro\] reviews the high-order finite volume framework and specifies the details of the particular implementation used here for the hydrodynamics subsystem. Then, Section \[sec:hydro-tests\] validates the high-order hydrodynamics algorithm via three classes of test problems. The upwind constrained transport framework of Londrillo and Del Zanna is reviewed in Section \[sec:uct\], and the novel implementation using PPM coupled to the fourth-order finite volume hydrodynamics subsystem is introduced. Section \[sec:mhd-tests\] evaluates the overall algorithm using MHD test problems and compares the high-order scheme to a second-order scheme. Section \[sec:conclusion\] gives concluding remarks and discusses future work.
Hydrodynamics subsystem: fourth-order finite volume methods for hyperbolic conservation laws {#sec:fv-hydro}
============================================================================================
The equations governing compressible hydrodynamics can be formulated as a system of hyperbolic conservation laws $$\frac{\partial \mathbf{U}}{\partial t} + \vec{\nabla} \cdot \vec{\bm{\mathrm{F}}}(\bm{\mathrm{U}}) = 0 \, , \label{eq:conservation-law}$$ with conserved variable vector $\mathbf{U}= [\rho, \rho\mathbf{v}, E] $ and nonlinear flux function $ \vec{\bm{\mathrm{F}}}(\bm{\mathrm{U}}) $, where $\rho$ is the density, $\mathbf{v}$ is the gas velocity, and $E$ is the total energy per unit mass. In this section and the next, we consider the solution of the system using Godunov-type methods based on the solution of Riemann problems. The resulting finite volume method for the hydrodynamics subsystem will form the basis of the overall MHD scheme when combined with the constrained transport discretization of Section \[sec:uct\]. This terminology is equivalent to the “underlying scheme” in broader constrained transport literature and the “base scheme” in the nomenclature of Tóth [@Toth2000].
Colella, et al. [@ColellaDorrHittingerMartin2009; @ColellaDorrHittingerMartin2011] devised an approach to constructing high-order finite volume methods for the solution of Equation on mapped grids. An essential component of this class of methods is the use of high-order quadrature rules evaluated on cell interfaces, in particular for the numerical fluxes. It was first introduced by Barad and Colella [@BaradColella2005] in the context of a finite volume solver for the Poisson equation. The framework was extended from scalar hyperbolic PDEs to systems of nonlinear hyperbolic conservation laws by McCorquodale and Colella in [@McCorquodaleColella2011]. Guzik et al. in [@Guzik2015] combine the advances of [@ColellaDorrHittingerMartin2011] and [@McCorquodaleColella2011] to introduce a fourth-order accurate finite volume method for nonlinear systems of equations on mapped, adaptively refined grids. The method’s capabilities are demonstrated on canonical hydrodynamics problems involving strong shocks.
We focus our analysis on the algorithmic components that are essential to the high-order finite volume methods regardless of the coordinate system and mesh. Therefore, throughout this work we assume a uniform Cartesian mesh composed of cubic control volumes with grid spacing $h $, and we consider a physical domain equivalent to this abstract *computational space* in the mapped grid literature. An advantage of the mapped grid formalism is that the particular finite volume operators are simply described in the computational space. The complexities of nonuniform, refined, and smoothly curved meshes are encapsulated in the handling of the discrete metric terms of the non-analytic or analytic mapping [@Guzik2015]. While such grids are not presently considered in this study, the general applicability of the numerical methods to Cartesian and mapped, single-block and multiblock, refined and unrefined, structured and unstructured grids is important for the effective simulation of demanding relativistic and multiscale physics. In all cases, these high-order methods satisfy local conservation, guarantee freestream preservation (i.e., the solution to a uniform flow is unaffected by a smooth mapping or discretization), and are compatible with many underlying finite volume methods for computing fluxes in 1D such as WENO and PPM.
We adopt the notation used by [@ColellaDorrHittingerMartin2009; @ColellaDorrHittingerMartin2011; @McCorquodaleColella2011; @Guzik2015] and summarized in Section 2 of [@Guzik2015], but we restrict the notation to $D=2$ dimensions, without loss of generality. Subscripts with the letters $ i,j $ and integer offsets are used to index cell-centered quantities along the first and second dimensions, respectively, in the lab-frame coordinate system. Half-integer offsets are used to index the cell faces and corners along these directions. General 1D operators involving cell interfaces will be centered on the lower $ (i-\frac{1}{2}, j) $ interface, for example. Operators involving 2D cell corners will be centered on the lower left corner, for example, $ (i-\frac{1}{2}, j-\frac{1}{2}) $.
Beyond second-order accuracy, the midpoint approximation traditionally used in finite volume methods must be abandoned. The algorithm must distinguish all cell-/face-/edge-averaged quantities from pointwise cell-/face-/edge-centered approximations, and products of averages are only equal to averages of products to within a second order approximation. Angled brackets are used to denote the spatial averaging of a quantity defined on the mesh. For example, $ \langle Q \rangle_{i,j} $ indicates the cell-averaged value and $ \langle Q \rangle_{i-\frac{1}{2},j}, \langle Q \rangle_{i,j-\frac{1}{2}} $ are face-averaged quantities on faces in the first and second direction, respectively. In contrast, $ Q_{i,j} $ indicates the cell-centered value and $ Q_{i-\frac{1}{2},j}, Q _{i,j-\frac{1}{2}} $ are face-centered quantities.
Conversions between these averaged and pointwise values can be performed at fourth-order accuracy using stencils that approximate multidimensional data locally on the mesh. When integrating a quantity defined on a 2D cell volume, for example, the integrand can be replaced with a Taylor-series expansion about the cell-center. The odd-powered coordinate terms cancel out when integrating across the cell, so a finite difference Laplacian operator (of at least second-order accuracy) can be combined with the cell-centered value to get a fourth-order approximation to the cell-averaged quantity [@Guzik2015]. A similar argument holds for the inverse transformation. We use the following approximation to the Laplacian when converting between cell-centered and cell-averaged quantities in 2D: $$\Delta Q_{i,j} = \frac{1}{h^2} (Q_{i-1,j} - 2Q_{i,j} + Q_{i+1,j}) + \frac{1}{h^2} (Q_{i,j-1} - 2Q_{i,j} + Q_{i,j+1}) \, .\label{eq:laplacian}$$
For face-centered and face-averaged conversions, we require Laplacian operators $\Delta^{\perp,d} $ for each direction $d$ that only include derivative terms that are transverse to the surface normal vector $\mathbf{e}^d$:
$$\begin{aligned}
\Delta^{\perp,1} Q_{i-\frac{1}{2},j} =& \frac{1}{h^2} (Q_{i-\frac{1}{2},j-1} - 2Q_{i-\frac{1}{2},j} + Q_{i-\frac{1}{2},j+1}) \, , \label{eq:laplacian-transverse-x1} \\
\Delta^{\perp,2} Q_{i,j-\frac{1}{2}} =& \frac{1}{h^2} (Q_{i-1,j-\frac{1}{2}} - 2Q_{i,j-\frac{1}{2}} + Q_{i+1,j-\frac{1}{2}}) \, . \label{eq:laplacian-transverse-x2 }\end{aligned}$$
Temporal integration {#subsec:fv-temporal}
--------------------
While alternative techniques like ADER may be used to resolve flux gradients in time using compact stencils with local timesteps [@DumbserZanotti2013], we follow [@ColellaDorrHittingerMartin2009; @ColellaDorrHittingerMartin2011; @McCorquodaleColella2011; @Guzik2015] and use a simple semi-discrete/method of lines formulation of Equation $$\frac{\mathrm{d}}{\mathrm{d}t} \langle \mathbf{U} \rangle_{i,j} + \frac{1}{h} (\langle \mathbf{F}_1 \rangle_{i+\frac{1}{2},j} - \langle \mathbf{F}_1 \rangle_{i-\frac{1}{2},j}) + \frac{1}{h} (\langle \mathbf{F}_2 \rangle_{i,j+\frac{1}{2}} - \langle \mathbf{F}_2 \rangle_{i,j-\frac{1}{2}}) = 0 \, . \label{eq:fv-div}$$ One advantage of the method of lines is the decoupling of the temporal and spatial orders of accuracy of the overall scheme. This separation encapsulates the (often complicated) specifics of the given equations in the treatment of the spatial terms at a single stage. The initial discretization of all terms exclusively in space results in a time-dependent system of autonomous ODEs. The ODE system can be integrated with many general ODE integrators of different orders of accuracy and computational demands. For the purposes of this study, any explicit, multistage one-step integrator that is absolutely stable for the fourth-order central difference operator may be used. See of [@ColellaDorrHittingerMartin2011] for the corresponding linear stability analysis that computes the CFL restriction for the purely imaginary eigenvalues of the constant coefficient advection problem.
For the majority of the results presented here, we use a strong-stability preserving (SSP), low-storage variant of the fourth-order accurate Runge-Kutta method. See Gottlieb, Ketcheson, and Shu [@GottliebKetchesonShu2009] for the precise implementation details, and refer to [@Ketcheson2008; @Ketcheson2010; @SpiteriRuuth2002; @GottliebShu1998; @GottliebShuTadmor2001; @GottliebKetchesonShu2009] for the theory of low-storage, SSP RK integrators. This $\mathcal{O}(\Delta t^4)$ accurate RK variant uses five substages of flux updates and requires the storage of three intermediate solutions of the entire domain (registers) per time step. We will refer to the timestepper as RK4 throughout the paper. For smooth error convergence studies, we also employ the RK3 variant defined by [@ShuOsher1988] Equation 2.19. This method requires three substages and two registers per timestep. Future work will focus on comparing integrators with optimal effective SSP coefficients [@Ketcheson2008] in the context of challenging MHD applications.
Spatial discretization and numerical fluxes {#fv-spatial}
-------------------------------------------
In a finite volume method, the spatial operator must ultimately provide a consistent approximation to the average fluxes on all cell faces for the discrete divergence law in Equation . When solving nonlinear systems of hyperbolic PDEs at high-resolution, solution properties such as smoothness, monotonicity, positivity, and stability may become important considerations. These concerns may motivate the augmentation of the numerical flux function with intermediate steps, such as nonlinear variable transformations or projections, the enforcement of variable floors, and application of slope limiters. All of the steps in the algorithm must compute any intermediate approximations at $\mathcal{O}(\Delta x^4)$ in order to preserve the spatial accuracy of the overall scheme.
For completeness, we reproduce the outline detailed by Guzik, et al. in 3.2 of [@Guzik2015] for the procedure of computing the hydrodynamic fluxes while describing the differences of our particular implementation.
### Equation of state and variable inversion {#subsubsec:eos}
In finite volume methods for hydrodynamics, piecewise polynomial reconstruction of fluid profiles is frequently performed on the set primitive variables $ \mathbf{W} = [\rho, \mathbf{v}, P ]$ instead of the conserved variables $ \mathbf{U} $. Application of slope limiters to primitive reconstructions typically results in less oscillatory solutions, and positivity of density and fluid pressure can be explicitly enforced in the primitive variable space. The sets of variables are related by a nonlinear, invertible transformation $ \mathbf{W}(\mathbf{U}) $. Because the pointwise transformation is exact, the variable inversion at fourth-order accuracy is performed using approximations to the pointwise cell-centered values. The steps are as follows:
1. Convert from cell-averaged to cell-centered conserved variables $$\mathbf{U}_{i,j} = \langle \mathbf{U}\rangle_{i,j} - \frac{h^2}{24} \Delta \langle \mathbf{U} \rangle_{i,j} \, , \label{eq:U-cell-center}$$ via the finite difference Laplacian operator in Equation
2. Apply the variable inversion to the $\mathcal{O}(\Delta x^4)$ approximation to cell-centered conserved variables, resulting in a pointwise $\mathcal{O}(\Delta x^4)$ primitive variable approximation $$\mathbf{W}_{i,j} = \mathbf{W}(\mathbf{U}_{i,j}) \, . \label{eq:W-cell-center}$$
3. Apply the variable inversion to cell-averaged conserved variables, resulting in a $\mathcal{O}(\Delta x^2)$ approximation to cell-averaged primitive variables $$\overline{\mathbf{W}}_{i,j} = \mathbf{W}(\langle \mathbf{U} \rangle_{i,j}) \, .\label{eq:W-bar-cell-ave}$$
4. Combine both primitive variable approximations to compute a fourth-order approximation to the cell-averaged primitive variables $$\langle \mathbf{W} \rangle_{i,j} = \mathbf{W}_{i,j} + \frac{h^2}{24} \Delta \overline{\mathbf{W}}_{i,j} \, . \label{eq:W-cell-ave}$$
In addition to applying primitive variable floors to the density and pressure during the variable inversion steps in Equations and , the floors are reapplied after Equation .
### Primitive variable reconstruction and limiters {#subsubsec:fv-reconstruct}
From the $\mathcal{O}(\Delta x^4)$ cell-averaged primitive variables, the face-averaged primitive states are reconstructed using the piecewise parabolic method. The averages on each interface are initialized using a four-point stencil along the longitudinal direction. In the $x_1$ direction, for example, this is computed as $$\langle \mathbf{W}\rangle_{i-\frac{1}{2},j} = \frac{7}{12} (\langle \mathbf{W}\rangle_{i-1,j} + \langle \mathbf{W} \rangle_{i,j}) - \frac{1}{12} (\langle \mathbf{W} \rangle_{i+1,j} + \langle \mathbf{W} \rangle_{i-2,j} ) \, . \label{eq:PPM4}$$ While the interface approximation is single valued and fourth-order accurate when applied to smooth data, the presence of discontinuities and nonlinear dynamics demands the use of limiters to suppress spurious oscillations. Limiting may introduce multivalued face-averaged interface L/R Riemann states. We have tested more than five limiters when designing the overall scheme, including:
1. The original PPM limiter of Colella & Woodward [@ColellaWoodward1984]
2. The variant of Mignone [@Mignone2014] which has formulations for spherical and cylindrical coordinate systems
3. The smooth extremum preserving limiter of Colella & Sekora [@ColellaSekora2008]
4. A modification of [@ColellaSekora2008] presented by Colella, et al. in 4.3 of [@ColellaDorrHittingerMartin2011]
5. The improved version of [@ColellaDorrHittingerMartin2011] by McCorquodale & Colella in 2.4 of [@McCorquodaleColella2011]
We refer the reader to the original references for the complete implementation details. We have found that the final three limiters, which avoid the clipping of smooth extrema, all produce similar results in the test problems below with only minor differences in numerical dissipation. However, the modifications to the original smooth extremum preserving PPM limiter of [@ColellaSekora2008] made in [@ColellaDorrHittingerMartin2011] and [@McCorquodaleColella2011] cause the algorithm to lose the property of strict monotonicity-preservation. For example, Figure 4 of [@McCorquodaleColella2011] shows a non-monotonic solution for the 1D square wave advection problem. In addition, the derivative approximations in the McCorquodale variant lead to a 7-cell stencil, which is prohibitively expensive for the MHD applications under consideration.
Therefore, we implement PPM with fourth-order interface approximation (summarized as PPM4 in [@PetersonHammett2013]) and a smooth extremum preserving limiter variant based on the version in 4.3 [@ColellaDorrHittingerMartin2011]. Our implementation eschews the check for monotonicity of the derivative estimates when limiting the initial interface states (Equation 86 of [@ColellaDorrHittingerMartin2011]). We find that the additional dissipation relative to the more advanced limiter in [@McCorquodaleColella2011] is acceptable, especially in problems with strong discontinuities.
There are several typos in the above PPM limiter literature that we wish to identify here for clarity:
- In Colella & Sekora [@ColellaSekora2008], the final term in Equation 19 should have a factor of $\frac{1}{6}$, not $\frac{1}{3}$. This was identified in 4.3.1 of [@ColellaDorrHittingerMartin2011].
- In Colella & Sekora [@ColellaSekora2008], Equation 20 is missing an “or” conditional when checking two conditions for detecting local extrema.
- In Colella, et al [@ColellaDorrHittingerMartin2011], Equations 85a, 85c, 95b, 95c, 95d should not have the factors of $\frac{1}{2}$. These second-derivative cell-averaged stencils are not consistent with McCorquodale 2.4.1 nor Equation 21 of [@ColellaDorrHittingerMartin2011].
- In the original PPM reference [@ColellaWoodward1984], Equation 1.8 for van Leer limiting of the initial slopes, one of the terms in the $\min()$ function should be indexed with $j+1$, not $j-1$.
In certain MHD shock tube tests, we have found that reconstructing characteristic variable profiles instead of primitive variable profiles was necessary to suppress spurious oscillations. The eigenvectors of Appendix A of Stone, et al. [@Stone2008] are used for the characteristic projections, but the projection algorithm is fundamentally different from steps 1-5 in [@Stone2008] Section 4.2.2. The reconstruction procedure in characteristic variable space for the two $\langle \mathbf{W}^{R_1} \rangle _{i-\frac{1}{2},j}, \langle \mathbf{W}^{L_1} \rangle_{i+\frac{1}{2},j}$ primitive Riemann states local to each cell is:
1. The left and right eigenvectors of the linearized system are computed using the local cell-averaged primitive variables. The eigenmatrices are
$$\begin{aligned}
\vec{\bm{\mathrm{L}}}_{i,j} =& \vec{\bm{\mathrm{L}}}(\langle \mathbf{W} \rangle_{i,j} ) \, , \label{eq:L-eigenmatrix} \\
\vec{\bm{\mathrm{R}}}_{i,j} =& \vec{\bm{\mathrm{R}}}(\langle \mathbf{W} \rangle_{i,j} ) \, . \label{eq:R-eigenmatrix}\end{aligned}$$
2. The left eigenvectors are applied to all cell-averaged primitive variables within the stencil along the direction of reconstruction $$\langle \mathbf{Q} \rangle _{i+l,j} = \vec{\bm{\mathrm{L}}}_{i,j} \langle \mathbf{W} \rangle _{i+l,j} \, , \label{eq:char-projection}$$ where $ -2\leq l \leq 2 $ for PPM4.
3. Then, the same interpolation and limiting procedure as in the primitive variable case is followed using these projected quantities, resulting in characteristic Riemann states on the $x_1$ interfaces
$$\begin{aligned}
\langle \mathbf{Q}^{R_1} \rangle _{i-\frac{1}{2},j} \, , \label{eq:char-states-R1} \\
\langle \mathbf{Q}^{L_1} \rangle_{i+\frac{1}{2},j} \, . \label{eq:char-states-L1}\end{aligned}$$
4. Finally, the limited reconstructed characteristic variable interface states are converted to primitive interface states
$$\begin{aligned}
\langle \mathbf{W}^{R_1} \rangle _{i-\frac{1}{2},j} =& \vec{\bm{\mathrm{R}}}_{i,j} \langle \mathbf{Q}^{R_1} \rangle _{i-\frac{1}{2},j} \, , \label{eq:primitive-projection-R1} \\
\langle \mathbf{W}^{L_1} \rangle_{i+\frac{1}{2},j} =& \vec{\bm{\mathrm{R}}}_{i,j} \langle \mathbf{Q}^{L_1} \rangle_{i+\frac{1}{2},j} \, , \label{eq:primitive-projection-L1}\end{aligned}$$
using the right eigenvectors.
This procedure is summarized in the context of MP5 reconstruction by Equations 30-36 in Section 3.3 of Matsumoto, et al. [@Matsumoto2016]. Examples in the context of WENO reconstruction procedures are given in [@BalsaraShu2000] for monotonicity preserving MPWENO and [@JiangShu1996] for a reduced pressure and entropy projection scheme, WENO-LF-5-PS. In our particular formulation, the characteristic reconstruction process is computationally expensive because the projection step produces local stencils for each cell. The expense grows for high-order and multidimensional schemes [@Londrillo2000]. Furthermore, decomposing the wave along characteristics may not be feasible in relativistic simulations or in the presence of complex physics [@Matsumoto2016]. When presented with a problem for which primitive reconstruction produces oscillatory results and characteristic decomposition is impractical, alternative techniques such as artificial viscosity, slope flattening, or more aggressive limiting may be pursued [@McCorquodaleColella2011].
### Approximate flux calculation using Riemann solvers {#subsubsec:fv-flux}
Finally, fourth-order accurate interface-averaged fluxes $\langle \mathbf{F}_1\rangle _{i\pm\frac{1}{2}, j}, \langle \mathbf{F}_2\rangle _{i,j\pm\frac{1}{2}}$ are computed through the approximation of the face-centered fluxes. The process is compatible with any approximate or exact Riemann solver $ \mathscr{F}(\mathbf{W}^{L}, \mathbf{W}^{R}) $ that only depends on the L/R primitive states at an interface. In this publication, we use approximate HLL-type Riemann solvers, including HLLE [@Einfeldt1988], HLLC [@Toro1994], and HLLD [@MiyoshiKusano2005] (for MHD). The linearized Roe solver [@Roe1981] and the Local Lax-Friedrichs one-speed flux approximation are also implemented. The steps for $x_1$ faces are:
1. Convert face-averaged primitive L/R Riemann states to pointwise face-centered states
$$\begin{aligned}
\mathbf{W}^{L_1}_{i-\frac{1}{2}, j} =& \langle \mathbf{W}^{L_1} \rangle _{i-\frac{1}{2}, j} - \frac{h^2}{24} \Delta^{\perp,1} \langle \mathbf{W}^{L_1}\rangle_{i-\frac{1}{2}, j} \, , \label{eq:WL1-face-center} \\
\mathbf{W}^{R_1}_{i-\frac{1}{2}, j} =& \langle \mathbf{W}^{R_1} \rangle _{i-\frac{1}{2}, j} - \frac{h^2}{24} \Delta^{\perp,1} \langle \mathbf{W}^{R_1}\rangle_{i-\frac{1}{2}, j} \, . \label{eq:WR1-face-center}\end{aligned}$$
2. Using a Riemann solver $ \mathscr{F}() $, compute pointwise interface-centered fluxes $$\mathbf{F}_{1,i-\frac{1}{2},j} = \mathscr{F}(\mathbf{W}^{L_1}_{i-\frac{1}{2},j},\mathbf{W}^{ R_1}_{i-\frac{1}{2},j}) \, ,\label{eq:flux-face-center}$$ from interface-centered primitive states.
3. Using a Riemann solver $ \mathscr{F}() $, compute fluxes from the fourth-order accurate interface-averaged primitive states. This results in a $\mathcal{O}(\Delta x^2) $ accurate approximation to the interface-averaged fluxes $$\overline{\mathbf{F}}_{1,i-\frac{1}{2},j} = \mathscr{F}(\langle \mathbf{W}^{L_1}\rangle _{i-\frac{1}{2},j},\langle \mathbf{W}^{R_1}\rangle_{i-\frac{1}{2},j} ) \, . \label{eq:flux-bar-face-ave}$$
4. Compute the Laplacian of the $\mathcal{O}(\Delta x^2) $ estimate of the face-averaged fluxes in directions orthogonal to the interface normal. Then, transform the face-centered fluxes to fourth-order accurate face-averaged fluxes $$\langle \mathbf{F}_1\rangle _{i-\frac{1}{2}, j} = \mathbf{F}_{1,i-\frac{1}{2},j} -\frac{h^2}{24} \Delta^{\perp,1} \overline{\mathbf{F}}_{1,i-\frac{1}{2},j} \, . \label{eq:flux-face-ave}$$ Note, the decisions to apply the Laplacian operator to $ \overline{\mathbf{F}} $ in Equation and to $ \overline{\mathbf{W}} $ in Equation are made to reduce the stencil size at the cost of an additional Riemann solve and variable inversion, respectively [@Guzik2015].
After the flux averages are known for all faces, the flux divergence is computed and the conserved variables are updated using Equation .
Validation of fourth-order finite volume implementation for hydrodynamics subsystem {#sec:hydro-tests}
===================================================================================
For the purpose of the results in this section, the high-order algorithm described in Section \[sec:fv-hydro\] will be designated as RK4+PPM. We begin by validating that RK4+PPM exhibits the expected behavior in the hydrodynamics limit when $\mathbf{B}=0$. We emphasize comparisons to the second-order counterpart to the high-order algorithm. This scheme, below referred to using the shorthand VL2+PLM, uses:
- $ \mathcal{O}(\Delta t^2) $ predictor-corrector time integrator based on the method in [@Falle1991] which uses a first-order scheme for the predictor step.
- $ \mathcal{O}(\Delta x^2) $ piecewise linear method (PLM) reconstruction of primitive variables using the limiter defined in 3.1 of [@Mignone2014].
- The midpoint approximation is assumed everywhere in the algorithm; hence the above Laplacian conversions from face-/cell-centered to face-/cell-averaged variables are skipped.
VL2+PLM is currently used in Athena++, a new version of the Athena astrophysics code [@Stone2008]. We refer the reader to the corresponding Athena method paper [@StoneGardiner2009] for a summary of the VL2+PLM scheme for MHD. The HLLC Riemann solver was used in both VL2+PLM and RK4+PPM to produce the hydrodynamics results.
We use $(x_1, x_2)$ to refer to the lab-frame coordinates in the below results. All results are from uniform, Cartesian grids with resolution $(N_{x_1}, N_{x_2})$. For some problem descriptions and analysis, a rotated vector frame is employed, and $(x,y)$ will refer to the rotated frame coordinates. In such cases, the coordinate transformation will be explicitly given from the lab-frame coordinates.
2D slotted cylinder circular advection {#subsec:slotted-cylinder}
--------------------------------------
Before considering the fully nonlinear hydrodynamics regime, the first problem we consider is the slotted cylinder scalar advection test presented in [@ColellaDorrHittingerMartin2011] Section 4.4.5. We model scalar advection in our solver by considering a 2D domain governed by an isothermal equation of state, uniform density $\rho=1$, and rotational flow about $ \mathbf{x}^c = (0.5,0.5) $ defined by $$(v_1(\mathbf{x}),v_2(\mathbf{x})) = 2\pi \omega (-(x_2 - x_2^c), (x_1 - x_1^c)) \, , \label{eq:cylinder-velocity}$$ with $\omega=1$. Under these assumptions, the out-of-plane velocity component, $v_3$, is passively advected counterclockwise by the fluid.
Figure \[fig:slotted-cylinder-ic-quad\] plots the initial condition of a slotted cylinder with radius $R=0.15$, slot width $W=0.05$, and slot height $H=0.25$ centered on $\mathbf{x}^* = (0.5, 0.75)$ and $$v_3(\mathbf{x}) = \left\{
\begin{array}{ll}
0 & 0 \leq R < r \\
0 & |2z_1| \leq W \text{ and } 0 < z_2 + R < H \\
1 & \text{ otherwise } \\
\end{array}
\right. \, , \label{eq:cylinder-profile}$$ where $\mathbf{z} = \mathbf{x} - \mathbf{x}^*, r =|\mathbf{z}| $. At a $100 \times 100$ resolution, the slot width is exactly five cells wide. We note that the initial condition in [@ColellaDorrHittingerMartin2011] is asymmetric; the slot has two cells to the right of and three cells to the left of the center of the domain and cylinder.
Figure \[fig:slotted-cylinder-comparisons-2x\] compares the VL2+PLM and RK4+PPM advected solutions after one rotation. Both the PLM and PPM limiters exhibit excellent preservation of the monotonicity of $v_3$. When an unlimited second- or fourth-order reconstruction is used, unphysical oscillations of the same order of magnitude as the cylinder height appear in the solution. The fourth-order limited solution preserves the discontinuous slot of the cylinder while the second-order limited solution does not. The numerical diffusivity of the two methods can also be compared at the cylinder edge. The smooth transition (shown in white in the diverging color map of Figure \[fig:slotted-cylinder-comparisons-2x\]) to the $v_3=0$ background state is much narrower in the RK4+PPM solution.
Figure \[fig:slotted-cylinder-hst\] compares the time-series data of the domain-averaged $x_3$ component of the kinetic energy. While the total domain $v_3$ is constant within machine precision throughout the simulation (as is guaranteed by the global conservation property of the finite volume method), the decay of the volume-averaged $v_3^2$ from the initial reference value provides a measure of the algorithm’s numerical dissipation per time step. RK4+PPM causes a much slower initial dissipation of the solution than VL2+PLM, which indicates that high-order reconstruction can be effective even for highly discontinuous data.
![The initial condition and reference solution for the slotted cylinder advection test, where $v_3$ is treated as a passive scalar. The Cartesian grid is composed of $100 \times 100$ cells, and the boundary conditions are periodic.[]{data-label="fig:slotted-cylinder-ic-quad"}](slotted-cylinder-ic-quad.pdf){width="\textwidth"}
![Second-order accurate VL2+PLM and the fourth-order accurate RK4+PPM computed solutions at $t=1.0 $ for the slotted cylinder advection test. While both limiters prevent unphysical oscillations, the high-order solution preserves most of the 5-cell slot while VL2+PLM completely fills it in. []{data-label="fig:slotted-cylinder-comparisons-2x"}](slotted-cylinder-comparisons-2x.pdf){width="\textwidth"}
![Plot of the decay of the volume-averaged concentration of the passive scalar $v_3^2$ over time. The values are normalized by the initial value, $ E_0$[]{data-label="fig:slotted-cylinder-hst"}](slotted-cylinder-hst.pdf){width="\textwidth"}
2D oblique hydrodynamic linear wave convergence {#subsec:hydro-linwave}
-----------------------------------------------
The next test problem increases the difficulty of the underlying dynamics by introducing the adiabatic hydrodynamics equation of state and smooth waves. However, the full nonlinearity of the Euler system is avoided by restricting the problem to the evolution of planar waves of small, linear amplitude $\varepsilon$. The conserved variable profiles are initialized using the exact eigenvectors $\mathbf{R}_k $ (for each mode $k$) of the Euler system linearized about the background state $ \overline{\mathbf{U}}_{k} $. They are best described in the coordinate system that is rotated to be aligned with the wave propagation direction with
$$\begin{aligned}
x =& x_1\cos\theta + x_2\sin\theta \, , \label{eq:coord-rotation1} \\
y =& -x_1\sin\theta + x_2\cos\theta \, . \label{eq:coord-rotation2}\end{aligned}$$
The conserved quantities vary sinusoidally with $x$, the coordinate along the parallel rays of the wavefront that are obliquely oriented relative to the $N_{x_1}\times \frac{N_{x_1}}{2}$ grid. The periodic domain extends from $ 0 \leq x_1 \leq \sqrt{5} $ and $ 0 \leq x_2 \leq \frac{\sqrt{5}}{2} $ to ensure square cells. The wavevector direction is set to $\theta = \tan^{-1}(2) \approx 63.43^\circ$ inclined with respect to the $x_1$-axis, and the wavelength is $ \lambda=1$. With these parameters, exactly one wavelength propagates along each domain boundary in one period. Furthermore, the problem is truly multidimensional, as there is no symmetry between the $ x_1 $ and $x_2$ fluxes.
The eigenfunctions for the sound and entropy wave modes must be initialized at fourth-order or greater accuracy on the mesh. While the exact cell-averaged initial condition could be calculated analytically in this case, the existing second-order accurate initialization of the linear wave problem in Athena++ is extended to fourth-order accuracy using a two step process. A similar procedure can be applied to correct other smooth initial conditions approximated by cell-centered approximations. First, the eigenfunctions are evaluated at the cell center position in the rotated coordinate frame, resulting in a midpoint approximation to the cell-averaged conserved variables, $$\mathbf{U}^0 = \overline{\mathbf{U}}_{k} + \varepsilon \mathbf{R}_k \cos(2\pi x) \, . \label{eq:linwave-ic}$$ Then, the Laplacian operator in Equation is applied to get a fourth-order accurate approximation to cell-averaged initial conserved data $$\langle \mathbf{U}^0 \rangle_{i,j} = \mathbf{U}^0_{i,j}+ \frac{h^2}{24} \Delta \mathbf{U}^0_{i,j} \, . \label{eq:laplacian-ic4}$$ We let $\varepsilon=10^{-6}$ for all the results shown here. The uniform background is $ \rho = 1, P = 3/5, \gamma = 5/3 $, thus the sound speed is $c_s=1$. For the sound wave, the background flow velocity is $v_1 = 0 $; for the entropy wave $v_1 = 1 $.
Due to the global smoothness of the wave, this problem does not test the behavior of the limiter at discontinuities. Nevertheless, the linear wave test is a discriminating challenge of the algorithm’s formal order of accuracy. The wave propagates for one wavelength, and the evolved solution is compared to the initial condition. The vector of $L_1$ errors of each $s$ of the $ N_{hydro} $ conserved variables at timestep $n$ is $$\delta \mathbf{U}^n = \frac{1}{N_{x_1}N_{x_2}} \sum_{i,j} | \langle \mathbf{U}^n \rangle_{i,j} - \langle \mathbf{U}^0 \rangle_{i,j} | \, . \label{eq:l1-vector}$$ Figure \[fig:hydro-linwave\] displays in logarithmic scale the convergence of the root mean square norm of the $L_1$ error vector $${\left\lVert\delta \mathbf{U}^n\right\rVert} = \sqrt{\sum_s^{N_{hydro}} (\delta U^n_s)^2} \, , \label{eq:rms-l1}$$ for the sound and entropy wave modes for resolutions spanning $8\times 4$ to $128\times 64 $ cells. Example second, third, and fourth-order convergence rates are juxtaposed as dashed lines. In Figure \[fig:hydro-linwave\], the plots show that the errors for both the RK3+PPM and RK4+PPM methods converge much faster than for VL2+PLM. The RK4+PPM solution on a $16\times8$ grid has similar accuracy as the VL2+PLM solution at the $128\times 64 $ resolution.
Despite initially converging at fourth-order, the RK3+PPM sound wave error converges at only third-order with the fixed CFL number of 0.4 for most resolutions, which indicates that the temporal error dominates for this test and solver configuration. The third-order convergence turn-over point decreases to before the smallest $N_{x_1}=8$ when the CFL number is increased to 0.8. The RK3+PPM error lines nearly exactly match the RK4+PPM errors when the CFL is decreased to 0.2. At the largest resolution considered, the convergence stops as nonlinear steepening effects invalidate the linear approximation. This occurs at errors near $ L_1 \approx \varepsilon^2 = 10^{-12} $. Floating-point round-off error may affect the error convergence at smaller wave amplitudes and larger resolutions.
![Hydrodynamic linear wave convergence plots of the adiabatic sound and entropy modes. The high-order schemes reduce the error nearly by three orders of magnitude relative to the second-order scheme at the largest resolution.[]{data-label="fig:hydro-linwave"}](hydro-linwave.pdf){width="\textwidth"}
1D Shu-Osher shock tube {#subsec:shu-osher}
-----------------------
The final hydrodynamics validation test is a fully nonlinear shock tube problem. The Shu-Osher problem involves the interaction of a discontinuous shock front propagating from $x_1=-0.8$ with a sinusoidal smooth flow [@ShuOsher1989]. The adiabatic index is $ \gamma=\frac{7}{5} $ and the domain spans $ x_1 \in [-1, 1] $, with initial condition to the left and right of $x_1=-0.8$ given by $$\begin{pmatrix}
\rho^L \\
v_1^L \\
v_2^L \\
v_3^L \\
P^L
\end{pmatrix}
=
\begin{pmatrix}
3.857143 \\
2.629369 \\
0 \\
0 \\
10.3333
\end{pmatrix} \, ,
\begin{pmatrix}
\rho^R \\
v_1^R \\
v_2^R \\
v_3^R \\
P^R
\end{pmatrix}
=
\begin{pmatrix}
1+ 0.2\sin(5\pi x_1)\\
0 \\
0 \\
0 \\
1.0
\end{pmatrix}
\, . \label{eq:shu-osher-states}$$ The interaction produces a density profile containing both discontinuities and smooth structure composed of a large range of wavelengths.
Figure \[fig:shu-osher\] compares the $N_{x_1}=200$ low-resolution results from the second-order algorithm VL2+PLM with the results from the fourth-order algorithm RK4+PPM. The solid line is a high-resolution $N_{x_1}=8000$ reference solution computed with the RK4+PPM method. The solutions are shown at $t_f=0.47$. The smooth extremum preserving PPM limiters are essential to prevent the clipping of the many short wavelength peaks at the low-resolution. The second-order VL2+PLM solution experiences large dissipation near the short wavelength structures due to the frequent and severe extremum clipping of the sinusoidal profile at this resolution.
![Second-order and fourth-order accurate solutions of the density in the Shu-Osher shock tube at $t_f=0.47$ are shown for $N_{x_1}=200 $ cells are shown in comparison to the $N_{x_1}=8000$ reference solution. Both algorithms capture the long wavelength smooth features, but the fourth-order method yields the largest improvements in solution accuracy in regions where the profile changes rapidly over a few grid cells. []{data-label="fig:shu-osher"}](shu-osher.pdf){width="\textwidth"}
Upwind constrained transport implementation {#sec:uct}
===========================================
Having validated the fourth-order finite volume method for the hydrodynamics subsystem, Equation , we reintroduce the full MHD system of equations and focus on the treatment of the magnetic field. The induction equation enters the conservative system of equations of MHD, but it is not evolved using the techniques from Section \[sec:fv-hydro\]. In differential form, the equation is $$\frac{\partial \mathbf{B}}{\partial t} + \nabla \times \mathscr{E} = 0 \, , \label{eq:induction}$$ where $\mathscr{E} = -\mathbf{v} \times \mathbf{B} $ is the electric field under the assumption of ideal MHD. Constrained transport ensures a strictly solenoidal magnetic field by evolving field quantities averaged on cell faces. By applying Stoke’s theorem to Equation in 2D, the following difference formulas in semi-discrete form are derived
$$\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t} \langle B_{1} \rangle_{i-\frac{1}{2},j}& =- \frac{1}{h} (\langle \mathscr{E}_{3} \rangle_{i-\frac{1}{2},j+\frac{1}{2}} - \langle \mathscr{E}_{3} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} )\label{eq:induction-difference-1}\\
\frac{\mathrm{d}}{\mathrm{d}t} \langle B_{2} \rangle_{i,j-\frac{1}{2}}& = \frac{1}{h} (\langle \mathscr{E}_{3} \rangle_{i+\frac{1}{2},j-\frac{1}{2}} - \langle \mathscr{E}_{3} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} ) \, . \label{eq:induction-difference-2}\end{aligned}$$
In this formulation, the magnetic flux is a conserved quantity while the fluxes are line-averaged corner (edges in 3D) electric fields $\langle \mathscr{E}_{3} \rangle $, or emf; therefore, exact maintenance of the divergence-free condition in Equation is a result of the local conservation property of the numerical method. While this advantage of the CT technique is well known, the staggered discretization of the field on cell faces introduces a dual, independent representation of the magnetic field quantities. Approaches to coupling CT with upwinded, cell-averaged quantities of the Godunov scheme are highly varied and may not prevent the onset of numerical monopoles in Equation [@Londrillo2004].
Gardiner and Stone [@GardinerStone2005] (hereafter GS05) present a 2D CT scheme that couples to the underlying corner transport upwind (CTU) method by transversely upwinding the electric fields of the flux vectors returned by the Godunov-type method. The algorithm was extended to 3D for CTU in [@GardinerStone2008] (hereafter GS08) and to a simplified Godunov-type method, VL2 in [@StoneGardiner2009]. The multidimensional construction of CT schemes in 3.2 of GS05 proceeds by considering schemes that reduce to the analytic solution for plane-parallel, grid-aligned flow. They begin by pointing out that the viscous flux contribution of the CT algorithm based on arithmetic averaging of the emf [@BalsaraSpicer1999] must be doubled for stability. The modified scheme is referred to as $\mathscr{E}_z^\circ $. The authors then construct two other novel CT schemes $\mathscr{E}_z^\alpha $ and $\mathscr{E}_z^c $, motivated by the Local Lax-Friedrichs and upwinding methods, respectively, applied to the emf derivatives in the differentiated induction equations. We refer the reader to Section 3.2.2 of GS05 for the derivations [@GardinerStone2005]. After a rigorous comparison of the three proposed CT schemes, $\mathscr{E}_z^\circ, \mathscr{E}_z^\alpha, \mathscr{E}_z^c, $ the authors concluded that $\mathscr{E}_z^c $, which upwinds the emf by the contact mode direction, produces a stable, non-oscillatory CT scheme with optimal numerical viscosity.
In this method, the upwinding of the face-averaged emf contained in the Godunov fluxes is completed from the nearest four cell faces and averaged for each corner. Equation 41 of GS05 expresses the spatial average of the four $\mathcal{O}(\Delta x^2)$ estimates to the corner emf as $$\begin{gathered}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} = \frac{1}{4} ( \mathscr{E}_{3,i,j-\frac{1}{2}} + \mathscr{E}_{3,i-1,j-\frac{1}{2}} +\mathscr{E}_{3,i-\frac{1}{2},j} +\mathscr{E}_{3,i-\frac{1}{2},j-1} ) \\
+\frac{h}{8} \left( \left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-\frac{1}{2},j-\frac{3}{4}} - \left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-\frac{1}{2},j-\frac{1}{4}} \right)
+\frac{h}{8} \left( \left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{3}{4},j-\frac{1}{2}} - \left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j-\frac{1}{2}} \right) \, . \label{eq:gs05-emf-average}\end{gathered}$$ The approximations to the derivatives are upwinded in the transverse direction, and the upwind directions are based on the fluid contact mode for both $x_1$ and $x_2$. For example, the expression for upwinding the $\partial_2$ spatial derivatives to $x_1$ faces is $$\left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-\frac{1}{2},j-\frac{1}{4}} =
\begin{cases}
\left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-1,j-\frac{1}{4}} & v_{1,i-\frac{1}{2}} > 0 \\
\left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i,j-\frac{1}{4}} & v_{1,i-\frac{1}{2}} < 0 \\
\frac{1}{2} \left[ \left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-1,j-\frac{1}{4}} + \left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-1,j-\frac{1}{4}} \right] & \text{ otherwise}
\end{cases} \, . \label{eq:gs05-ct-upwind}$$
The $\mathscr{E}_z^c $ GS05 CT scheme is the exclusive CT discretization used in subsequent publications by the authors [@GardinerStone2008; @Stone2008; @StoneGardiner2009] and in the Athena astrophysics code [@Stone2008]. We remark that $\mathscr{E}_z^\circ $ has been occasionally misidentified as the final GS05 CT scheme in subsequent literature such as [@Balsara2014a].[^1] We also note that there are typos in subsequent formulations of GS05 Equation 41 due to a change in indexing from the upper corner $\mathscr{E}_{z,i+\frac{1}{2},j+\frac{1}{2}}$ to the lower corner $\mathscr{E}_{z,i-\frac{1}{2},j-\frac{1}{2}}$. The signs of the derivative terms in [@Stone2008] Equation 79 and [@StoneGardiner2009] Equation 22 are all incorrect.
While this CT scheme is consistent with the underlying Godunov-type algorithm for 1D plane parallel solutions, the consistency and accuracy are at most second-order in spatial resolution. The algorithm contains several steps and assumptions that limit the overall spatial accuracy of the scheme to $\mathcal{O}(\Delta x^2)$, even when combined with a higher-order base finite volume scheme:
1. By truncating higher-order derivative terms, Equation 40 of GS05 provides a single spatial estimate for the emf at a grid cell corner using a second-order Taylor-series expansion across a face $$\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} = \mathscr{E}_{3,i-\frac{1}{2},j} - \frac{h}{2}\left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-\frac{1}{2},j-\frac{1}{4}} + \mathcal{O}(\Delta x^2) \, . \label{eq:gs05-taylor-series}$$
2. The face-centered emf quantities in Equations and are midpoint approximations.
3. The cell-averaged magnetic field components, necessary for the transverse reconstruction steps in the Godunov-type finite volume scheme, are derived at second-order accuracy using the average of the longitudinal face-averaged values. Equations 19 and 20 of GS05 define this final step in the CT scheme after the update of the face-averaged magnetic field components in induction Equations and :
$$\begin{aligned}
\langle B_{1} \rangle_{i,j} =& \frac{1}{2}(\langle B_{1} \rangle_{i+\frac{1}{2},j} + \langle B_1 \rangle_{i-\frac{1}{2},j}) \, , \label{eq:average-b1-face} \\
\langle B_{2} \rangle_{i,j} =& \frac{1}{2}(\langle B_{2} \rangle_{i,j+\frac{1}{2}} + \langle B_2 \rangle_{i,j-\frac{1}{2}}) \, . \label{eq:average-b2-face}\end{aligned}$$
4. Most subtly, the $ \mathscr{E}_z^c $ scheme only captures dimensionally split approximations to the multidimensional Riemann fan at the cell corner. The upwinded emf slopes in Equation are approximations given by GS05 Equation 45 as $$\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{3}{4},j} = \frac{2}{h} (\mathscr{E}_{3,i-\frac{1}{2},j} - \mathscr{E}^r_{3,i-1,j}) \, , \label{eq:emf-slope}$$ where $ \mathscr{E}_{3,i-\frac{1}{2},j} $ is the magnetic flux returned by an approximate 1D Riemann solver and $\mathscr{E}^r_{3,i,j} = v_{2,i,j}B_{1,i,j} - v_{1,i,j}B_{2,i,j} $ is the cell-centered reference electric field. Regardless of the order of accuracy of the reconstruction method, this difference between the cell-centered reference electric field and the face-centered magnetic flux limits the approximation to second-order accuracy.
While the first three second-order accurate assumptions of the $ \mathscr{E}_z^c $ scheme can all be addressed by replacing them with higher-order approximations, the final limitation cannot be generalized to fourth-order accuracy in a straightforward fashion. See \[sec:appendix-gs05-uct\] for detailed analysis, including a proof demonstrating that the upwinding of fluxes from approximate Riemann solutions on $x_1,x_2$ faces (instead of smooth approximations to $\mathscr{E}_3$) in each dimension causes the scheme to fail to reduce to the 2D planar wave modes along the $x-y, x+y$ cell diagonals. The upwind constrained transport framework of Londrillo and Del Zanna, developed in [@Londrillo2000] (hereafter LD2000) and generalized in [@Londrillo2004] (hereafter LD2004), is an approach to implementing a CT discretization that is not subject to the $\mathcal{O}(\Delta x^2)$ limitations above. In \[sec:appendix-gs05-uct\], we show that the $ \mathscr{E}_z^c $ CT scheme is consistent to within second-order approximations to UCT, but the derivation of the scheme in GS05 is not extensible to higher-order for the above reasons.
UCT defines two phases for the treatment of the MHD subsystem of Equation : the reconstruction phase and the upwind phase [@Londrillo2000]. This dichotomy is analogous to the steps for computing fluxes in a high-order generalization of Godunov’s scheme for hydrodynamics [@Londrillo2000]. Whereas the reconstruction and upwind phases are typically formulated to approximate the fluxes at cell faces in hydrodynamics, the UCT analogy is made to compute the fluxes (electric fields in induction Equation ) at cell corners in 2D. Furthermore, the UCT steps require special handling to ensure that the divergence-free and field line continuity constraints of MHD are maintained.
Therefore, the implementation of a fourth-order accurate UCT method consistent with the finite volume hydrodynamics subsystem of Section \[sec:fv-hydro\] requires specification of two main algorithmic components:
1. Reconstruction of quantities along cell faces to cell corners
2. Corner upwinding procedure that approximates the solution to a multidimensional Riemann problem for the magnetic fluxes
As in the steps of the hydrodynamics subsystem of Section \[sec:fv-hydro\], each component of the UCT implementation must treat all approximations at fourth-order accuracy. In the following subsections, we specify these techniques used in our overall scheme at the end of each integrator substage, after the calculation of the Godunov fluxes in Section \[subsubsec:fv-flux\] but before the flux-divergence is applied to update the cell-averaged conserved variables in Equation . For the 2D algorithm, $\langle B_3\rangle$ is evolved using the finite volume techniques of Section \[sec:fv-hydro\].
Reconstruction phase {#subsec:uct-reconstruction}
--------------------
Just as limiting may introduce dual $L/R $ states collocated at cell faces, limited piecewise polynomial reconstruction may introduce four discontinuous states at cell corners. LD2004 establishes notation for the four-state functions by referring to the orientation of each state relative to the center of its reconstructed cell: $Q^{NW}, Q^{NE}, Q^{SE}, Q^{SW} $ [@Londrillo2000; @Londrillo2004]. Note, these are counter-intuitive if you consider the cardinal directions relative to the corner, and the $N/S$ states corresponding to discontinuities in the $x_2$ direction come before the $E/W$ states used for jumps across $x_1$ in the superscript. Therefore, we adopt the notation in Equation 39 of [@Amano2015], which uses superscripted states relative to $x_1, x_2 $ interfaces. Figure \[fig:corner-states-diagram\] summarizes the locations of these states. The states equivalent to the above LD2004 states are $Q^{R_1L_2}, Q^{L_1L_2}, Q^{L_1R_2}, Q^{R_1R_2} $, respectively.
![2D slice of $x_1-x_2$ plane that shows the locations of cell-centered conserved variables, face-centered 1D Riemann states and upwind flux components, and the four-state function of a quantity $Q$ reconstructed in 2D for each of the nearest cells at the $i-\frac{1}{2},j-\frac{1}{2}$ corner. The cell-averaged conserved variables are shown in the $i,j$ cell as a shaded gray square to emphasize that all of the other displayed cell-/face-centered quantities are distinct from the corresponding cell-/face-averaged quantities (not shown).[]{data-label="fig:corner-states-diagram"}](corner-states-diagram.pdf){width="50.00000%"}
Unlike normal reconstructed quantities in hydrodynamics, the magnetic field quantities cannot be freely represented by the basis of piecewise polynomials. The components of the field must have single-valued states at longitudinal faces in order to ensure that the field lines are continuous [@Londrillo2000]. For example, $$\langle B_1^{L_1} \rangle = \langle B_1^{R_1} \rangle = \langle B_1 \rangle \, \label{eq:B1-face-continuity}$$ must be satisfied at an $x_1$ face. At an $ x_1 $-$x_2$ corner, discontinuous states of $B_1(x_1,x_2)$ can only occur across the $x_2$ jump as
$$\begin{aligned}
\langle B_1^{L_1R_2} \rangle =& \langle B_1^{R_1R_2} \rangle = \langle B_1^{R_2} \rangle \, , \label{eq:B1R2-corner-continuity} \\
\langle B_1^{L_1L_2} \rangle =& \langle B_1^{R_1L_2} \rangle = \langle B_1^{L_2} \rangle \, . \label{eq:B1L2-corner-continuity}\end{aligned}$$
In UCT the longitudinal face-averages for each magnetic field component, such as $\langle B\rangle_{i\pm\frac{1}{2},j} $ for the $ x_1$-component, become the primary representation of the field; these quantities are never reconstructed. From stencils of these quantities, both the cell-averaged $\langle {B_1}\rangle_{i,j}$ required by the finite volume hydrodynamics subsystem in Section \[sec:fv-hydro\] and the corner states $ \langle B_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}}, \langle B_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} $ required by the CT scheme below are reconstructed.
### Reconstruction of corner electric fields {#subsubsec:uct-reconstruction-emf}
After completing the reconstruction step of the hydrodynamics subsystem in Section \[subsubsec:fv-reconstruct\] for all face-averages, we compute fourth-order accurate reconstructions of the emf at each cell corner in 2D from the necessary velocity and magnetic field components using $$\langle \mathscr{E}_{3} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} = \langle v_{2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \langle B_{1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle v_{1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \langle B_{2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \, . \label{eq:corner-emf-unlimited}$$ We continue to use the angled bracket notation in Equation to emphasize that the emfs are line-averaged along cell edges in 3D and to distinguish the quantities from their second-order counterparts from the GS05 CT scheme.
The above calculation at each corner of each cell is accomplished by performing transverse reconstructions of previously reconstructed $L/R$ Riemann states of $\mathbf{v},\mathbf{B}$ along cell faces. We apply PPM4 as in Section \[subsubsec:fv-reconstruct\] to suppress spurious oscillations that may arise from physically admissible discontinuities. However, there is nothing inherent to this UCT formulation that requires the use of PPM for these reconstructions. Future work will compare the computational efficiency and accuracy of results from using alternative methods for non-oscillatory reconstructions such as WENO in this step. For example, the $\langle \mathscr{E}^{R_1R_2}_3\rangle_{i-\frac{1}{2}, j-\frac{1}{2}} = \langle \mathscr{E}^{SW}_3\rangle_{i-\frac{1}{2}, j-\frac{1}{2}} $ state at the $i-\frac{1}{2},j-\frac{1}{2} $ corner in the $i,j$ cell is approximated by:
1. $ \langle B_1^{L_2}\rangle_{i-\frac{1}{2}, j-\frac{1}{2}}, \langle B_1^{R_2}\rangle_{i-\frac{1}{2}, j-\frac{1}{2}} $ are reconstructed along the $x_2 $ coordinate from $x_1$-face-averaged field data, $ \langle B_1 \rangle_{i-\frac{1}{2}, j} $.
2. $ \langle B_2^{L_1}\rangle_{i-\frac{1}{2}, j-\frac{1}{2}}, \langle B_2^{R_1}\rangle_{i-\frac{1}{2}, j-\frac{1}{2}} $ are reconstructed along the $x_1 $ coordinate from $x_2$-face-averaged field data, $ \langle B_2 \rangle_{i, j-\frac{1}{2}} $.
3. Both components of $\mathbf{v} $ are independently reconstructed along both $x_1$ and $ x_2 $ face-averaged data. Because limited 1D reconstruction operations may not commute, the independent velocity estimates are averaged to approximate $\langle v_1^{R_1R_2} \rangle_{i-\frac{1}{2}, j-\frac{1}{2}}, \langle v_2^{R_1R_2}\rangle_{i-\frac{1}{2}, j-\frac{1}{2}}$.
4. Finally, the approximation to the emf is computed using $$\langle \mathscr{E}^{R_1R_2}_3\rangle_{i-\frac{1}{2}, j-\frac{1}{2}} = \langle v_2^{R_1R_2} \rangle_{i-\frac{1}{2}, j-\frac{1}{2}} \langle B_1^{R_2}\rangle_{i-\frac{1}{2}, j-\frac{1}{2}} - \langle v_1^{R_1R_2}\rangle_{i-\frac{1}{2}, j-\frac{1}{2}}\langle B_2^{R_1} \rangle_{i-\frac{1}{2}, j-\frac{1}{2}} \, ,\label{eq:corner-emf-limited}$$ the limited version of Equation .
The quantities used to compute these corner reconstructions are illustrated in Figure \[fig:corner-ppm-diagram\]. After the four-state emf function is approximated at each cell corner, the UCT upwinding step discussed in Section \[subsec:uct-upwind\] is used to select a single valued $ \langle \mathscr{E}^{U}_3\rangle_{i-\frac{1}{2}, j-\frac{1}{2}} $.
![The stencils of face-averaged quantities used to compute the four-state emf at a corner to fourth-order accuracy using PPM4. The magnetic field components are single-valued on longitudinal faces because these face-averages are never reconstructed, but the limiting in the transverse reconstruction may introduce two discontinuous Riemann states at the corner. Unlike $\mathbf{B}$, the 1D reconstructions of face-averaged velocity components are dual-valued at all faces, so the subsequent application of PPM4 results in a four-state function of $\mathbf{v} $ at the corner for both $x_1$ and $x_2$ face-averages. The independent corner reconstructions of velocity are averaged in our method.[]{data-label="fig:corner-ppm-diagram"}](corner-ppm-diagram.pdf){width="50.00000%"}
### Reconstruction of cell-averaged $\langle\mathbf{B} \rangle_{i,j} $ {#uct-reconstruction-B}
Before discussing the UCT corner upwinding procedure, we discuss the final reconstruction step necessary to relate the primary field representations of longitudinal face averages $ \langle B_1 \rangle_{i\pm\frac{1}{2}, j}, \langle B_2 \rangle_{i, j\pm\frac{1}{2}} $ to the derived cell-averaged field at $\mathcal{O}(\Delta x^4)$ accuracy. While this is the final UCT step in a single integration substage and occurs after the content of Section \[subsec:uct-upwind\], it is a reconstruction procedure, so we provide the details in this section.
After evolving the face-averaged magnetic field quantities in the induction difference Equations and , the cell-averaged $\langle\mathbf{B} \rangle_{i,j} $ must be updated to be consistent with the evolved field. At second-order accuracy, this consistency relationship is typically maintained by taking the average of the longitudinal face-averaged components as seen in Equations and . Using the techniques from Section \[subsubsec:eos\] and the finite difference Laplacian operator in Equation , we can perform an “inverse-reconstruction” of the magnetic field at fourth-order accuracy. For example, the procedure for the $x_1$ field component follows:
1. Using the Laplacian operator consisting of transverse derivatives, convert the face-averaged field to an approximation of the face-centered field on longitudinal faces $$B_{1,i-\frac{1}{2},j} = \langle B_1 \rangle_{i-\frac{1}{2},j} - \frac{h^2}{24} \Delta^{\perp,1} \langle B_1 \rangle_{i-\frac{1}{2},j} \, . \label{eq:B1-face-center}$$
2. Interpolate along longitudinal face-centers to cell-centered field components $$B_{1,i,j} = -\frac{1}{16} (B_{1,i-\frac{3}{2},j} + B_{1,i+\frac{3}{2},j} ) + \frac{9}{16}(B_{1,i-\frac{1}{2},j} + B_{1,i+\frac{1}{2},j} ) \, . \label{eq:B1-cell-center}$$
3. Apply Laplacian operator to convert face-centered fields to cell-averaged fields $$\langle B_1 \rangle_{i,j} = B_{1,i,j} + \frac{h^2}{24} \Delta B_{1,i,j} \, . \label{eq:B1-cell-average}$$
As in the unlimited equation of state conversions in Section \[subsubsec:eos\], no limiting is used in the above conversions. Future work may consider using non-oscillatory second-derivative approximations as in LD2000.
Figure \[fig:cell-averaged-B1-reconstruction\] summarizes the $x_1$ face-averaged magnetic field input data and intermediate approximations computed in Equations , and necessary for the inverse reconstruction of a single cell-averaged $\langle B_1 \rangle_{i,j} $ at fourth-order accuracy. While this wide stencil contains many quadrature points for the approximation of a single cell-averaged value, the intermediate quantities are reused in calculations of the surrounding cell-averages, as is expected for such approaches to high-order accuracy. However, optimizing data reuse and on-node performance of these stencils with finite cache sizes generally requires large box sizes and careful loop scheduling techniques [@Olschanowsky2014]. The stencil is wider in $x_1$ than in $x_2$ because the upwind constrained transport framework treats the longitudinal faces as primary representations of the magnetic field. The transverse stencil quantities are used for the Laplacian corrections in Equations and , which is only approximated at second-order accuracy. In contrast, the pointwise interpolation in the longitudinal direction $x_1$ must be performed at $\mathcal{O}(\Delta x^4)$.
![The red shaded $x_1$ interfaces indicate the requisite longitudinal face-averaged $\langle B_1\rangle $ input data (indices suppressed in diagram) for the $\mathcal{O}(\Delta x^4) $ inverse reconstruction of cell-averaged $\langle B_1\rangle_{i,j} $ using the procedure described in Section \[uct-reconstruction-B\]. The empty circles are the cell-centered $B_{1,i,j} $ used in Equation when approximating the 2D Laplacian. The filled circles are the face-centered $ B_{1,i-\frac{1}{2},j} $ used in the longitudinal interpolation in Equation . The black dashed line connects the face-centered points necessary to interpolate the cell-centered $B_{1,i,j}$ at fourth-order accuracy. The blue dash-dotted lines denote the additional stencil points necessary to interpolate to the four nearest cell centers.[]{data-label="fig:cell-averaged-B1-reconstruction"}](cell-averaged-B1-reconstruction.pdf){width="100.00000%"}
Upwind phase: collocated corner flux functions for Roe-type and HLL-type Riemann solvers {#subsec:uct-upwind}
----------------------------------------------------------------------------------------
The upwinding phase for Godunov schemes for nonlinear systems of conservation laws is a generalization of the trivial upwinding procedure of the scalar advection equation. In Godunov-type schemes for hydrodynamics, the solution of the Riemann problem typically encapsulates the selection of the upwind state and the calculation of the single valued flux from the reconstructed L/R Riemann states collocated at the interface between two cells. Since solving Riemann problems typically encapsulates the majority of the computational demand of a Godunov-type scheme, approximate Riemman solvers are used to simplify the problem while maintaining accuracy. Approximate Riemann solvers provide a *direct* approximation to the numerical flux (as opposed to approximating a state and then evaluating the flux function). Two popular classes of approximate Riemann solvers include:
1. Roe-type solvers are based on the linearization of the flux Jacobian of the system of equations. The Roe solver requires the characteristic decomposition of the variables at the interface, which may be expensive [@Roe1981]. Also, Roe-type solvers may return unphysical states in certain pathological cases [@Einfeldt1991].
2. Introduced by Harten, van Leer, and Lax [@HartenLaxLeer1983], HLL-type Riemann solvers compute the fluxes by averaging over an approximate Riemann fan. These Riemann solvers are computationally efficient and guarantee positivity but may produce less accurate solutions than the Roe Riemann solver. However, the greater numerical dissipation of simple approximate Riemann solvers can be ameliorated by high-order reconstructions; for such numerical methods, the one-state HLL or even the Local Lax-Friedrichs (LLF) Riemann solvers may be sufficient to produce the desired level of accuracy. An illustration of this phenomenon is shown in Section \[subsec:mhd-rj2a\].
While they were originally developed for hydrodynamics, both types of Riemann solvers have been extended to the MHD system. The Roe Riemann solver was extended to MHD by Cargo and Gallice [@CargoGalice1997]. Miyoshi and Kusano [@MiyoshiKusano2005] developed HLLD, which restores the Alfvén wave and contact modes in the MHD Riemann fan approximation.
As discussed above, the Godunov fluxes for the magnetic field variables should not be evaluated at the face-centers in the constrained transport context. Rather, the cell corners are the proper locations for evaluating the Riemann problem of the MHD subsystem. UCT generalizes this upwinding procedure from cell faces to cell corners in 2D (edges in 3D). Upwinding at cell-corners requires extending the underlying Riemann solver from a two-state to a four-state selection rule [@Londrillo2000].
The UCT formulation for these corner magnetic fluxes depends on the type of approximate Riemann solver used in the underlying finite volume subsystem. LD2000 first presents a UCT formulation based on Roe-linearized fluxes [@Londrillo2000]. The derivation of the formula requires averaging only the dissipative fluxes in each direction using a flux-vector splitting (FVS) formalism. LD2004 3.2 provides two central-upwind implementations of the UCT framework based on the one-state HLL Riemann solver: a second-order accurate scheme, MC-HLL-UCT, and a third-order accurate scheme, CENO-HLL-UCT [@Londrillo2004].
For our applications, we are primarily interested in the one-state HLL-type UCT formulation. Averaging the two overlapping $x_1$ and $x_2$ approximate Riemann fans at the corner results in the flux formula in Equation 56 of LD2004 $$\begin{gathered}
\langle \mathscr{E}^U_{3} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} = \frac{\alpha^+_1\alpha^+_2 \langle \mathscr{E}^{L_1L_2}_{3}\rangle_{i-\frac{1}{2},j-\frac{1}{2}} + \alpha^-_1 \alpha^+_2 \langle \mathscr{E}_{3}^{R_1L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} + \alpha^+_1\alpha^-_2 \langle \mathscr{E}_{3}^{L_1R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} + \alpha^-_1 \alpha^-_2 \langle \mathscr{E}_{3}^{R_1R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}}}{(\alpha^+_1 + \alpha^-_1)(\alpha^+_2 + \alpha^-_2)} \\
- \frac{\alpha^+_2 \alpha^-_2}{\alpha^+_2 + \alpha^-_2}( \langle {B}_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} )+ \frac{\alpha^+_1 \alpha^-_1}{\alpha^+_1 + \alpha^-_1}(\langle {B}_2^{R_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_2^{L_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} ) \, , \label{eq:uct-hll}\end{gathered}$$ where $\alpha^\pm_1$ are the nonnegative dissipative terms computed from $S^{L_1}, S^{R_1} $, the minimum and maximum wavespeed estimates in the $x_1$ direction, for example. See \[sec:appendix-gs05-uct\] for more details. The wavespeed estimates are properly evaluated at the $i-\frac{1}{2},j-\frac{1}{2}$ corner along with the reconstructed emf states. However, in practice they are computed by taking the extrema of the existing wavespeed estimates from the four 1D Riemann solutions computed at the nearest faces for the Godunov fluxes in Section \[sec:fv-hydro\]. The implementations of Equation in LD2004 use simple wavespeed estimates from Davis, Equation 4.5 of [@Davis1988]. We use the Einfeldt wavespeed estimates which reference the Roe-averaged wavespeeds [@Einfeldt1988]. This is a generalization of the GS05 four-state upwinding and averaging procedure encapsulated in Equation .
The HLL-UCT solver is a simple multidimensional Riemann solver applied to the induction equation. Recent work has involved the development of multidimensional Riemann solvers for MHD [@Balsara2010; @Balsara2012]. It automatically reduces to the 1D HLL fluxes for the magnetic fluxes for plane parallel solutions.
MHD numerical results {#sec:mhd-tests}
=====================
In this section, we present the results from a series of numerical experiments designed to test the accuracy, stability, shock-capturing ability, and numerical monopole suppression of the overall upwind constrained transport finite volume scheme, RK4+PPM. As in Section \[sec:hydro-tests\], we emphasize comparisons to the second-order counterpart to the high-order algorithm. This scheme, again referred to using the shorthand VL2+PLM, uses the second-order hydrodynamics scheme augmented with the second-order UCT scheme from GS05 [@GardinerStone2005; @StoneGardiner2009]. All MHD results are generated using the HLLD Riemann solver for the hydrodynamic subsystem fluxes unless otherwise noted. The CFL number for the MHD tests is defined by $$C_0 = \frac{\Delta t}{ \min \left( \frac{h}{|\lambda_1^{max}|}, \frac{h}{|\lambda_2^{max}|} \right)} \, , \label{eq:cfl-mhd}$$ where $\lambda_{1,2}^{max} $ is the fastest wave mode speed in the $x_1$- or $x_2$- direction over all of the cells. For a single cell, the wavespeed estimate is defined using the cell-averaged states as $ \lambda_1 = |\langle v_1\rangle| + c_1^f $, where $ c_1^f \left(\langle \mathbf{W} \rangle, \langle \mathbf{B} \rangle \right) $ is the fast magnetosonic wavespeed in the $x_1$-direction, for example. The same linear stability analysis as in [@ColellaDorrHittingerMartin2011] was used as reference to estimate the CFL restriction for the MHD problems, but a conservative value of $ C_0 = 0.4 $ was used for all of the following tests.
For the convergence plots in this section, the root mean square error metric of Equation of the $L_1$ error vector is extended to $N_{hydro} + N_{field} $ cell-averaged variables: $${\left\lVert\delta \mathbf{U}^n\right\rVert} = \sqrt{\sum_s^{N_{hydro}} (\delta U^n_s)^2 + \sum_s^{N_{field}} (\delta B^n_s)^2} \, . \label{eq:rms-l1-mhd}$$
2D oblique MHD linear wave convergence {#subsec:mhd-linwave}
--------------------------------------
As in the hydrodynamics linear wave test of Section \[subsec:hydro-linwave\], the conserved variable profiles are initialized along an oblique direction on a 2D Cartesian grid using the exact eigenvectors of the linearized MHD system. The domain size, wavelength, and wave propagation direction remain unmodified from the earlier section. The same uniform background fluid variables as the hydrodynamics test are used with $ \rho = 1, P = 3/5, \gamma = 5/3 $, and a background magnetic field is introduced with $\mathbf{B} = (1, \sqrt{2}, 1/2)$ specified in the wavevector rotated frame. The background velocity $v_1=1$ only for the entropy wave mode test, and it is $\mathbf{v}=0$ for all others.
With these parameters, the wavespeeds are: $ c_f=2, c_{a,x}=1, c_s=1/2, c_{v_1}=1 $ for the fast magnetosonic, Alfvén, slow magnetosonic, and entropy wave modes, respectively. See Appendix A of GS05 for the exact eigenvectors used for each wave family [@GardinerStone2005]. We note that the 3D MHD linear wave convergence results presented in [@Stone2008] Section 8.6 and Section 6.1 of [@StoneGardiner2009] both reference GS08 Appendix A [@GardinerStone2008] for the eigenvectors used in the test, but the correct eigenvector values are those in GS05 Appendix A [@GardinerStone2005]. The earlier references are incorrect since GS08 eigenvectors are derived from the linearization around a different wavevector-frame background field, $\mathbf{B} = (1, \frac{3}{2}, 0)$ [@GardinerStone2008].
The $\mathcal{O}(\Delta x^4) $ accurate initialization of the fluid variables follows Equations and . The average magnetic field is initialized for each component on longitudinal cell faces using the differences of the analytic magnetic vector potential computed at cell corners, which ensures that the initial condition satisfies $ \nabla \cdot \mathbf{B}=0$ to machine precision. However, the initialization of the face-averaged fields $ \langle B_1 \rangle_{i\pm \frac{1}{2}, j, k} , \langle B_2 \rangle_{i, j\pm \frac{1}{2}, k} , \langle B_3 \rangle_{i, j,k\pm \frac{1}{2}} $ required some care to ensure fourth-order accurate solutions at errors near machine precision. We originally observed that the errors prematurely ceased converging when naively applying Stoke’s theorem to $\mathbf{B} = \nabla \times \mathbf{A} $ on cell faces after computing analytic $\mathbf{A}$ at cell corners. The convergence issues were caused by a numerical loss of significance at double precision in the differencing operations. For example, when initializing the $x_2$ face-averaged longitudinal field component with $$\langle B_2 \rangle_{i, j\pm \frac{1}{2}, k} = \frac{1}{\Delta x_3} (\langle A_1 \rangle_{i,j\pm \frac{1}{2},k+\frac{1}{2}} - \langle A_1 \rangle_{i,j\pm \frac{1}{2},k-\frac{1}{2}} ) - \frac{1}{\Delta x_1} (\langle A_3 \rangle_{i+\frac{1}{2},j\pm \frac{1}{2},k} - \langle A_3 \rangle_{i-\frac{1}{2},j\pm \frac{1}{2},k} ) \, , \label{eq:stokes-linwave}$$ two differences are evaluated. The particularly large relative error of this operation in finite-precision arithmetic for the linear wave can be attributed to the relative sizes $$\label{eq:linwave-terms}
\begin{split}
A_1(\mathbf{x}) &= \bar{A}_1(\mathbf{x}) + \tilde{A}_1(\mathbf{x}) \\
&\approx O(1) + O(\varepsilon)
\end{split}$$ of the linear and perturbative terms in the vector potential profiles. The loss of significance due to the rounding of intermediate floating-point values was ameliorated by changing the order of operations. For example, the first term in Equation was reordered as $$\label{eq:reordered-linwave-diff}
\frac{1}{\Delta x_3} \left((\langle \bar{A}_1 \rangle_{i,j\pm \frac{1}{2},k+\frac{1}{2}} - \langle \bar{A}_1 \rangle_{i,j\pm \frac{1}{2},k-\frac{1}{2}} ) + (\langle \tilde{A}_1 \rangle_{i,j\pm \frac{1}{2},k+\frac{1}{2}} - \langle \tilde{A}_1 \rangle_{i,j\pm \frac{1}{2},k-\frac{1}{2}} ) \right) \,$$ in order to first difference similarly sized quantities.
Figure \[fig:mhd-linwave\] demonstrates formal fourth-order convergence of the UCT implementation in conjunction with the hydrodynamics finite volume subsystem. The root mean square of the $L_1$ error vector is shown for resolutions spanning $8 \times 4 $ to $128 \times 64$ for all MHD wave modes. When combined with the fourth-order spatially accurate single stage algorithm, the third-order temporally accurate integrator RK3 produces errors that are nearly identical to the fourth-order RK4 results. The only significant difference occurs for the fast magnetosonic mode, for which the RK3+PPM solution converges at only third-order for the majority of resolutions. Approximately four times fewer timesteps are required to evolve the wave for one period in the fast magnetosonic wave test when compared to the slow magnetosonic wave test. Therefore, the spatial truncation error associated with application of the operators in single stage contributes (in total) approximately four times more error for the evolution of the slow magnetosonic wave. For this fixed CFL of $0.4$, the spatial error then dominates the global solution error at all resolutions in the slow magnetosonic wave test.
The difference in behavior of the integrators illustrates the flexibility of the semi-discrete formulation. The same complicated single stage algorithm can easily be combined with many potential temporal integrators depending on the demands of the particular application. RK3 can be used at (potentially significantly) reduced computational expense relative to RK4 if the dynamics indicate that the error will likely be dominated by sources other than the finite temporal resolution.
![MHD linear wave convergence plots for the fast magnetosonic, Alfvén, slow magnetosonic, and entropy wave modes. With nearly four orders of magnitude smaller error than the second-order VL2+PLM results at $N_{x_1}=128$, the high-order results demonstrate even greater improvement than in the hydrodynamics test of Figure \[fig:hydro-linwave\]. The RK3+PPM error lines are obscured by the RK4+PPM lines in the slow and entropy wave plots.[]{data-label="fig:mhd-linwave"}](mhd-linwave.pdf){width="\textwidth"}
3D and 2D oblique circularly polarized Alfvén waves {#subsec:mhd-cpaw}
---------------------------------------------------
The circularly polarized Alfvén wave test was first described by Tóth in Section 6.3.1 of [@Toth2000]. The same domain setup, wavelength, and propagation direction as in the 2D MHD and hydrodynamics linear wave tests of Sections \[subsec:mhd-linwave\], \[subsec:hydro-linwave\] are used. However, no restriction to small amplitude perturbations is made for the initial condition. Unlike the linear wave test, these wave profiles are exact nonlinear solutions to the ideal MHD equations. We use the parameters from the subsequent formulation in GS05 Section 3.3.2 [@GardinerStone2005], using background $\rho=1, P=0.1$ with velocity and magnetic field components
$$\begin{aligned}
\mathbf{B} =& (1, 0.1\sin(2\pi x), 0.1\cos(2\pi x)) \, , \label{eq:cpaw-B} \\
\mathbf{v} =& (v_x, 0.1\sin(2\pi x), 0.1\cos(2\pi x)) \, , \label{eq:cpaw-v}\end{aligned}$$
specified in the wavevector rotated coordinate system. These parameters produce a circularly polarized wave that is not subject to a parametric instability that may cause other numerically evolved Alfvén waves to decay into magnetosonic waves [@GardinerStone2005].
As in the MHD linear wave test, the average magnetic field components are initialized on longitudinal cell faces using the differences of the analytic magnetic vector potential at cell corners. The background flow velocity is set to $v_x=0$ for the traveling wave test. For a standing wave profile, the background flow is $v_x=1$ to exactly oppose the wave propagation to the left. The evolution of the standing wave is a challenging variant of the test because the multidimensional operators for updating the face-averaged magnetic field must exactly cancel to preserve the wave field profile.
Figure \[fig:cpaw-By\] plots the transverse, in-plane magnetic field component $B_y$ in the wavevector-frame of all cells for a $32 \times 16$ grid at $t=5$. RK4+PPM nearly exactly reproduces the initial condition, whereas VL2+PLM results in diffusion of more than half of the peak height. The smooth extrema preserving PPM limiter was instrumental in preventing such dissipation in the fourth-order solver. At lower resolutions such as $N_{x_1}= 16$, dispersion error dominates the second-order solutions. In contrast, the high-order solutions have negligible dispersion error for the all tested resolutions.
Unlike line plots showing a subset of cells produced by a 1D slice of the domain, a scatter plot of all cells based on the cell-centered positions along the wave, $x$, can reveal the presence of multidimensional grid noise in the solution. VL2+PLM and RK4+PPM both produce solutions with negligible spread in the distribution of $B_y$ samples from nearby phases. Therefore, both the second- and fourth-order schemes preserve uniformity along the planar wavefronts. The fourth-order results in the right plot of Figure \[fig:cpaw-By\] can be compared to Figure A.2 of [@Mignone2010a], which also showed nearly perfect recovery of $B_y$ with the fifth-order WENO-Z and MP5 schemes for the traveling circularly polarized Alfvén wave test at $32\times16 \times 16$ resolution.
Because the wave is globally smooth, the errors should converge at fourth-order as the mesh is resolved with fixed CFL number, as in the linear wave tests. Figure \[fig:cpaw-convergence-t1\] plots the convergence of the RMS-L1 error for the standing and traveling circularly polarized waves at $t=1$ from $16\times 8 $ to $256 \times 128$ cells. When compared to the linear Alfén wave convergence results in Figure \[fig:mhd-linwave\], the curves in Figure \[fig:cpaw-convergence-t1\] are nearly identical when scaled by $10^{-5}$, the ratio of wave amplitudes. As Stone and Gardiner identified in Section 6.2 of [@StoneGardiner2009] for the VL2+PLM scheme, the relative dissipation of the waves does not depend on the wave amplitude or the presence of nonlinear effects. Figure \[fig:cpaw-convergence-t1\] confirms that the same invariance holds for the fourth-order RK4+PPM scheme; the resolution of the grid is the only factor that controls the numerical diffusivity of the overall scheme in these tests.
Figure \[fig:cpaw-convergence-t1\] also juxtaposes the errors of the RK4+PPM method applied to a 3D variant of the problem. We refer the reader to Section 5.3 of GS08 [@GardinerStone2008] for the details on the problem setup. The same fixed CFL number of 0.4 was used, and a range of resolutions from $ 16 \times 8 \times 8 $ to $ 128\times 64 \times 64 $ cells was tested. The results again demonstrate fourth-order convergence of the method in both the standing and traveling wave tests. For the standing wave case, the errors are all between 48-50% greater than their 2D counterparts. For the traveling wave, the errors grew by 13-15% for this problem and solver configuration. There are no new algorithmic components in the 3D method, but the computational expense relative to VL2+PLM is significantly greater than for 2D problems. The growth in the performance cost from 2D to 3D for the fourth-order MHD algorithm is largely dominated by the additional transverse PPM reconstructions and upwinding of $\mathscr{E}_1, \mathscr{E}_2 $ necessary to compute the induction equation.
Low resolution, large $t_f=5$ comparisons are displayed in the Figure \[fig:cpaw-By\] scatter plots to highlight the differences in numerical diffusivity between the second-order and fourth-order algorithms. However, shorter $t_f=1$ test results are used in the error convergence plots of Figure \[fig:cpaw-convergence-t1\], since we follow the conventions of earlier publications [@GardinerStone2005; @GardinerStone2008; @Stone2008; @StoneGardiner2009; @Londrillo2004]. We have observed (plot not shown) that the second-order VL2+PLM solution fails to converge at second-order for longer $t_f=5$ propagation tests at the initial resolutions due to the large dispersion error. RK4+PPM does not suffer from such a slow transition to the asymptotic fourth-order convergence regime. The same difference in convergence behavior was observed when comparing the second-order and fourth-order schemes in Figure 1 of Susanto (2013) [@Susanto2013].
![Scatter plots of one component of the transverse magnetic field of all points in the $32 \times 16$ resolution 2D circularly polarized Alfvén wave test. All cells are shown according to their cell center positions in the wavevector-aligned coordinate frame.[]{data-label="fig:cpaw-By"}](cpaw-By.pdf){width="\textwidth"}
![Convergence of the root mean square $L_1$ error of the circularly polarized Alfvén plane wave profiles after the traveling wave has propagated for one wavelength. The fourth-order convergence of the RK4+PPM errors of a 3D formulation of the problem are juxtaposed in green; the errors are uniformly larger than the 2D problem due to the increased spatial error in also resolving the wave’s oblique orientation relative to $x_3$. As observed in Figure 4 of GS08, the traveling wave mode errors are larger than the standing wave mode errors, and the increase is fairly uniform over the components of the error vector (not shown). [@GardinerStone2008]. []{data-label="fig:cpaw-convergence-t1"}](cpaw-convergence-t1.pdf){width="\textwidth"}
3D diagonal advection of a field loop {#mhd-field-loop}
-------------------------------------
The dynamic advection of a field loop, described for a 2D domain in GS05 and for a 3D domain in GS08, is a rigorous test of the CT discretization [@GardinerStone2005; @GardinerStone2008]. In this test, a cylinder of constant, weak magnetic pressure $ P_B $ is passively transported by the fluid for two periods along the domain diagonal. We let $B_z=0$ everywhere, and initialize a uniform poloidal magnetic field in a cylinder with radius $R=0.3$ centered on the origin via the out-of-plane component of the vector potential $$A_3(x_1, x_2) \equiv \left\{
\begin{array}{ll}
A_0(R-r) & r\leq R \\
0 & r > R \\
\end{array}
\right. \, , \label{eq:field-loop-a3}$$ where $A_0= 10^{-3}$. This potential corresponds to a plasma $\beta = \frac{2P}{B^2} = 2 \times 10^{-6} $ inside the cylinder. Again, the average magnetic field is initialized on longitudinal cell faces by differencing this analytic magnetic vector potential across cell corners.
Following GS05, we initialize a $2N \times N $ uniform 2D grid with periodic boundary conditions, where $N=64$ [@GardinerStone2005]. The uniform fluid has $\rho=1, P=1$, with an adiabatic MHD equation of state and $\gamma=5/3$. Unlike GS05 or GS08, the grid spans $ -1 \leq x_1 \leq 1, -0.5 \leq x_2 \leq 0.5, -1 \leq x_3 \leq 1 $, and the velocity vector points along the space diagonal of the 3D rectangular domain with $ \mathbf{v} = (2, 1, 2) $. Hence, $ |v| = 3 $ and the cylinder returns to its original position every $t=1$ periods.
Several comparisons of the advection test results at $t_f=2$ produced by the second-order and fourth-order schemes are shown in Figure \[fig:field-loop-comparison\]. The first row displays 2D plots of the magnetic pressure represented with the same color scale. Overall, the fourth-order scheme produces a more uniform advected $P_B$ cylinder than the second-order counterpart. This comparison also highlights the sharper resolution of the cylinder edge in the RK4+PPM result. Additionally, the hole that has formed due to magnetic reconnection in the center of the field loop is much larger in the VL2+PLM result.
The second row considers a subset of the $P_B$ data that is produced by taking 1D slices through the center of the domain along four directions of particular interest: the two coordinate axes and the two diagonals of the uniform Cartesian mesh. The corresponding slicing of the analytic reference solution (equivalent to the initial condition) is shown as the dashed black line. The results show reflective symmetry about the domain diagonal that is parallel to the fluid velocity vector. The leading edge of the cylinder (upper right) overshoots the initial maximum by nearly 20% in the VL2+PLM result. In contrast, the RK4+PPM solution has a much smaller maximum overshoot on the trailing edge (bottom left), which is approximately 8% greater than the initial condition. Both solutions exhibit small leading-edge versus trailing-edge asymmetries that may occur in solutions to advection problems; this can be observed by comparing the $ R > 0.5 $ and $ R <0.5$ profiles along the $ \theta = \frac{\pi}{2} $ slice in Figure \[fig:field-loop-comparison\].
The improved resolution of field discontinuities, namely the magnetically-reconnected hole and the outer cylinder boundary, can be quantified using these profiles. Along a diagonal ray, for example, the edges are resolved by 6-8 cells in the VL2+PLM solution, whereas RK4+PPM resolves the edges with only 4 cells. Although minor oscillations are present in the fourth-order solution, they are relatively minor, and the fourth-order UCT scheme better approximates the flat-top $P_B$ profile of the cylinder despite the oscillations.
Finally, the third row of Figure \[fig:field-loop-comparison\] provides an MHD counterpart to Figure \[fig:slotted-cylinder-hst\] of the slotted cylinder advection test. The time-series of the domain-averaged magnetic energy $B_p^2$ provides a measure of the numerical diffusivity of the UCT implementation. The rate of decay is significantly slower with the fourth-order scheme than in VL2+PLM with the second-order UCT implementation. The PPM reconstruction of field quantities at cell-corners can be credited for significantly decreasing the diffusion of the $P_B=5\times 10^{-7}$ profile in the UCT upwinding step.
Most importantly, $B_z$ remains zero to within round-off error for the lifetime of the simulation even as $v_z \neq 0$ and nonzero terms enter the induction equation updates. Figure \[fig:field-loop-Bcc3\] illustrates the final spatial distribution of the deviations from $B_3=0.0$. The nonzero values remain largely concentrated near the path of the field loop where $B_1,B_2 \neq 0 $, but they remain far smaller in magnitude than double-precision machine epsilon $\approx 2.22 \times 10^{-16} $. Figure \[fig:field-loop-me3-t\] compares the time-series of the domain-integrated $B^2_3$ of the RK4+PPM and VL2+PLM solutions. While initially the RK4+PPM value grows much faster owing to the greater number of stages and induction equation evaluations per timestep, the number of cells with nonzero $B_3$ grows at a similar rate to those in the VL2+PLM solution.
Advection in the $x_3$ direction of a poloidal field is nontrivial for general-purpose constrained transport algorithms [@Stone2008]. This test confirms the built-in divergence-free property of the fourth-order UCT implementation. While not shown here, the fourth-order UCT implementation also preserves the geometry of the concentric field lines.
![Comparisons of magnetic field quantities advected by the second-order and fourth-order schemes at $t_f=2$. The high-order algorithm demonstrates improved preservation of the original cylinder symmetry and overall domain magnetic energy density.[]{data-label="fig:field-loop-comparison"}](field-loop-comparison.pdf){width="\textwidth"}
![Plot of the out-of plane $B_3$ at $t_f=2$. The differences in the induction equation updates of $B_3$ are guaranteed to be 0.0 within floating-point round-off error by the $\nabla \cdot \mathbf{B}=0$ preservation of the constrained transport method.[]{data-label="fig:field-loop-Bcc3"}](field-loop-2D-xorder4-ssprk5_4-Bcc3.pdf){width="\textwidth"}
![Time-series growth of the out-of-plane component of the magnetic energy averaged over the domain.[]{data-label="fig:field-loop-me3-t"}](field-loop-me3-vs-t-v2.pdf){width="\textwidth"}
1D Brio-Wu shock tube {#subsec:mhd-bw}
---------------------
![Density, pressure, non-constant velocity and magnetic field components, and specific internal energy density scaled by $(\gamma -1)$ profiles of the Brio-Wu shock tube problem at $t=0.1$. The dashed lines show for $N_{x_1}=256 $ solutions of the VL2+PLM second-order scheme (blue) and the RK4+PPM fourth-order scheme (red). The reference solution is shown with a solid black line and was computed with RK4+PPM at a resolution of 8192 cells.[]{data-label="fig:bw-shock-1a"}](bw-shock-1a.pdf){width="\textwidth"}
![Comparison of the Brio-Wu shock tube results from the second-order algorithm using piecewise linear reconstruction of primitive MHD variables and results from the fourth-order algorithm using piecewise parabolic reconstruction of characteristic MHD variables. The largest spurious oscillations occur in the velocity component profiles when using primitive reconstruction.[]{data-label="fig:bw-shock-1b"}](bw-shock-1b.pdf){width="\textwidth"}
In following the analogous increasing complexity of the hydrodynamics tests of Section \[sec:hydro-tests\], we now introduce an MHD Riemann problem to test the solver’s ability to capture shocks and complex nonlinear waves. The Brio-Wu shock tube is an MHD analog to the classical Sod shock tube of hydrodynamics [@BrioWu1988; @Sod1978]. For this shock tube problem, the background longitudinal magnetic field is $B_1 = 0.75 $ and $\gamma=2$. The left and right states are given by $$\begin{pmatrix}
\rho^L \\
v_1^L \\
v_2^L \\
v_3^L \\
P^L \\
B_1^L \\
B_2^L \\
B_3^L \\
\end{pmatrix}
=
\begin{pmatrix}
1 \\
0 \\
0 \\
0 \\
1 \\
\frac{3}{4} \\
1 \\
0 \\
\end{pmatrix} \, ,
\begin{pmatrix}
\rho^R \\
v_1^R \\
v_2^R \\
v_3^R \\
P^R \\
B_1^R \\
B_2^R \\
B_3^R \\
\end{pmatrix}
=
\begin{pmatrix}
0.125 \\
0 \\
0 \\
0 \\
0.1 \\
\frac{3}{4} \\
-1 \\
0 \\
\end{pmatrix}
\, . \label{eq:bw-states}$$
Figure \[fig:bw-shock-1a\] compares the global solutions of the second-order and fourth-order schemes to a high-resolution reference solution at $t=0.1$. Characteristic reconstruction as described in Section \[subsubsec:fv-reconstruct\] was used to produce the RK4+PPM results in Figures \[fig:bw-shock-1a\] and \[fig:bw-shock-1b\]. In this test, the characteristic projection procedure was necessary for the fourth-order scheme to avoid spurious oscillations that exceeded 10% of the solution range. Even at second order, primitive PLM reconstruction causes nonphysical oscillations to appear.
Figure \[fig:bw-shock-1b\] provides a closer view of all solutions. The velocity profile in between the slow shock front at $x_1\approx 0.14$ and the fast rarefaction at $x_1\approx 0.36$ exhibited the worst oscillations. This phenomenon is well-known for high-order reconstruction; Figure 5 of [@Matsumoto2016] provides an analogous comparison with fifth-order MP5 reconstruction. The RK4+PPM results shown here avoid the anomalous staircasing present in the compound slow wave near $x_1\approx -0.02$ produced by the MP5 method. Figure \[fig:bw-shock-1b\] also shows slight improvement in the resolution of the slow shock front with the RK4+PPM solver.
1D RJ2a shock tube {#subsec:mhd-rj2a}
------------------
The next MHD test we consider is the shock tube problem introduced by Ryu & Jones in Figure 2a (RJ2a) [@RyuJones1995]. In this test, all 7 MHD wave modes propagate from the discontinuous initial data given by $$\begin{pmatrix}
\rho^L \\
v_1^L \\
v_2^L \\
v_3^L \\
P^L \\
B_1^L \\
B_2^L \\
B_3^L \\
\end{pmatrix}
=
\begin{pmatrix}
1.08 \\
1.2 \\
0.01 \\
0.5 \\
0.95 \\
0.5641895835477562\\
1.0155412503859613 \\
0.5641895835477562 \\
\end{pmatrix} \, ,
\begin{pmatrix}
\rho^R \\
v_1^R \\
v_2^R \\
v_3^R \\
P^R \\
B_1^R \\
B_2^R \\
B_3^R \\
\end{pmatrix}
=
\begin{pmatrix}
1 \\
0 \\
0 \\
0 \\
1 \\
0.5641895835477562\\
1.1283791670955125 \\
0.5641895835477562 \\
\end{pmatrix}
\, . \label{eq:rj2a-states}$$
Figure \[fig:rj2a-1\] shows that the fourth-order scheme captures all of the features and resolves the discontinuities with at most 5 cells for the resolution $N_{x_1}=512$. Figure \[fig:rj2a-2\] considers a single profile, $B_2$, given a lower resolution mesh and a more diffusive Riemann solver, HLLE, instead of the default HLLD solver. The RK4+PPM result compares favorably to the second-order VL2+PLM result, especially when comparing the rotational discontinuities. If stability or computational constraints necessitate the use of HLLE in a low-resolution mesh, the low numerical diffusivity of high-order schemes may significantly improve the results, even in a problem dominated by discontinuities.
Reconstruction of characteristic variables was again used for the fourth-order RJ2a results in both Figure \[fig:rj2a-1\] and \[fig:rj2a-2\]. Unlike the Brio-Wu shock tube problem in Section \[subsec:mhd-bw\], RK4+PPM with primitive reconstruction produced tolerable oscillations on the same order of magnitude as the VL2+PLM results in Figure \[fig:rj2a-2\].
![RJ2a profiles of density, pressure, total energy, velocity components, transverse magnetic field components, and rotation angle $\Phi=\tan^{-1}(\frac{\langle B_3 \rangle}{\langle B_2 \rangle}) $ of the magnetic field produced by the fourth-order scheme with $N_{x_1}=512 $. The low-resolution solution is superimposed on a reference solution produced by the same scheme at high-resolution $N_{x_1}=8192$ . All MHD discontinuities are resolved with 2-5 cells.[]{data-label="fig:rj2a-1"}](rj2a-1.pdf){width="\textwidth"}
![The transverse magnetic field component $B_2$ profile of $ N_{x_1}=128$ cells illustrates the advantages of using the high-order scheme in highly diffusive settings. The HLLE Riemann solver is used to generate both the second-order VL2+PLM and fourth-order RK4+PPM results. The high-order solution shows much better resolution of the rotational discontinuities.[]{data-label="fig:rj2a-2"}](rj2a-2.pdf){width="\textwidth"}
2D Orszag-Tang vortex {#subsec:orszag-tang}
---------------------
The vortex problem of Orszag & Tang [@OrszagTang1979] is a common test of the robustness of MHD schemes. The turbulence that results in this problem tests the ability of the numerical method to resolve the MHD shock-shock interactions while maintaining strict suppression of magnetic monopoles. The initial condition is described in Section 8.4 of [@Stone2008] and elsewhere; they are simply rescaled here for a domain $ [-0.5, 0.5]^2$ .
Figure \[fig:ot-half\] shows the evolution of the vortex at $ t =0.5$ evolved by the fourth-order RK4+PPM scheme with UCT on a uniform grid of $500^2$ cells. Figure \[fig:ot-1\] shows only the pressure and density at a later $t=1.0$. When compared with the second-order solution (not shown), the fourth-order solution shows improved resolution of the vortex at the origin.
![Clockwise from top-left subplot: density, pressure, specific kinetic energy, and magnetic pressure of the Orszag-Tang vortex at $t=\frac{1}{2}$. Thirty contours, linearly spaced between the minimum and maximum values, are overlaid on each plot.[]{data-label="fig:ot-half"}](ot-tlim-half.pdf){width="\textwidth"}
![Density and pressure of the Orszag-Tang vortex solution at $t=1 $. Since the late-time evolution of this problem develops into highly turbulent features, no contours are shown as they would obscure much of the plot.[]{data-label="fig:ot-1"}](ot-tlim-1.pdf){width="\textwidth"}
2D MHD blast wave {#subsec:mhd-blast}
-----------------
In this test problem, a strongly magnetized medium with uniform $B_0=1$ aligned with the main diagonal, $$\mathbf{B} =
\begin{pmatrix}
\frac{B_0}{\sqrt{2}} \\
\frac{B_0}{\sqrt{2}} \\
0
\end{pmatrix}
\, , \label{eq:b-blast}$$ of a square periodic domain spanning $ [-0.5, 0.5]^2 $ is initialized with $\rho=1$ and $ \gamma=5/3 $. The ambient $P=0.1$ while an overpressure $P=10$ region is set for cells within a radius $r=0.1$ of the origin.
A resolution of $500^2$ cells is used for this test. Figure \[fig:mhd-blast\] displays the blast wave at $t=0.2$, right before the shock wave has crossed the periodic boundary. The results can be compared to the second-order accurate results in Figure 28 of the Athena method paper [@Stone2008] and Figure 8 of the VL2+PLM method paper [@StoneGardiner2009]. The expanding shell is correctly collimated into an ellipse of low density gas oriented with the background magnetic field. The density and pressure contours are well-resolved relative to the second-order results.
Perpendicular to the in-plane $\mathbf{B}$, the outermost blast wave is a fast-mode that is dominated by the magnetic pressure, and it quickly establishes a large separation from the contact discontinuity of the initial overpressure cylinder. Along the domain diagonal parallel to the magnetic field, the slow-mode shock front is closer to the elongated contact discontinuity, and the fast-mode wave front disappears from the density and gas pressure plots. For the background plasma $\beta = 0.2 $ in this blast wave setup, the fast-mode wave becomes a non-compressional Alfvén mode along the magnetic field lines and appears with only a slight separation from the slow-mode shock front in the $P_B$ plot of Figure \[fig:mhd-blast\]. The correct resolution of these features is a important test for a numerical MHD scheme.
![Clockwise from top-left subplot: density, pressure, specific kinetic energy, and magnetic pressure of the MHD blast wave at $t=0.2 $. Thirty linearly spaced contours between the minimum and maximum values of each quantity are shown.[]{data-label="fig:mhd-blast"}](mhd-blast-diag.pdf){width="\textwidth"}
2D MHD rotor {#subsec:mhd-rotor}
------------
The 2D MHD rotor test was introduced by Balsara and Spicer [@BalsaraSpicer1999]; it considers the creation of strong rotational discontinuities in the magnetic field resulting from the shearing of a rapidly rotating disk of dense fluid. We use the same initial conditions used to generate the Athena method paper’s Figure 25 results, also described as “Rotor Test 1” in [@Toth2000]. Uniform background density $\rho=1$, pressure $P=1$, and $B_1=\frac{5}{2\sqrt{\pi}}$ are initialized with $ \gamma=7/5$. Within a radius of $r=0.1$ of the origin, a dense gas of $\rho=10$ is set to rotate with initial angular velocity $\omega=20$. No smoothing is used for the density nor velocity of the initial condition.
Figure \[fig:mhd-rotor\] shows the result at $t=0.15$. Again, a resolution of $500^2$ cells is used; symmetry is well-maintained in the solution, especially for the Mach number plot’s concentric circles of the rarefaction from the origin.
![Clockwise from top-left subplot: density, pressure, Mach number, and magnetic pressure of the rotor problem at $t=0.15$. Thirty contours, linearly spaced between the minimum and maximum values, are overlaid on each plot. Note, density is shown using a logarithmic color scale and linear contours to clearly show the density maxima.[]{data-label="fig:mhd-rotor"}](mhd-rotor.pdf){width="\textwidth"}
Conclusion {#sec:conclusion}
==========
We have presented a fourth-order accurate method of lines method for the numerical solution of the ideal MHD equations. Using the upwind constrained transport framework, we were able to implement a divergence-free staggered-mesh CT scheme that is consistent with a fourth-order accurate finite volume scheme for the hydrodynamics subsystem. The underlying finite volume scheme is based the quadrature rules and $\mathcal{O}(\Delta x^4) $ intermediate calculations of McCorquodale and Colella [@McCorquodaleColella2011] for nonlinear systems of hyperbolic conservation laws. In comparison to the second-order constrained transport scheme of GS05, the fourth-order method yields orders of magnitude of improvement in error in globally smooth linear and nonlinear MHD problems. The overall scheme also exhibits excellent robustness for discontinuous features in multidimensional tests.
Future work will involve the extension of the four-state flux upwinding rule of HLL-UCT in Section \[subsec:uct-upwind\] to be consistent with the multidimensionally-averaged approximate Riemann fan of the HLLD solver. Comprehensive analysis will be made in comparing these methods to the broader multidimensional Riemann solver literature. Additionally, the extension of our fourth-order accurate UCT implementation to adaptive mesh refinement is a high priority for the practical use in demanding astrophysics simulations. To that end, we will formulate the necessary $\mathcal{O}(\Delta x^4) $ prolongation and restriction operators that are compatible with AMR and the mapped grid formalism discussed in Section \[sec:fv-hydro\].
High-order methods offer the greatest advantages relative to conventional second-order accurate methods when applied to demanding problems with long simulation times and complex smooth features. Therefore, the fourth-order method will soon be applied to the simulation of shearing box approximations of accretion disk dynamics. The high arithmetic intensity of the fourth-order method can ameliorate the increasing performance and power costs of memory references relative to floating-point operations on modern computing architectures. These performance and accuracy tradeoffs and the scaling trends will be quantitatively evaluated on emerging manycore architectures.
Acknowledgments {#sec:acknowledgements .unnumbered}
===============
The authors thank Thomas Gardiner for discussion and comments on an early draft of this manuscript. K.G.F. was supported by the Department of Energy Computational Science Graduate Fellowship (CSGF), grant number DE-FG02-97ER25308. J.M.S. was supported by National Science Foundation, grant number AST-1715277.
$\mathscr{E}_z^c $ scheme of GS05 scheme is equivalent to a second-order accurate UCT method {#sec:appendix-gs05-uct}
============================================================================================
This appendix directly compares the $\mathscr{E}_z^c $ constrained transport algorithm of GS05 [@GardinerStone2005] and the UCT framework of LD2004 [@Londrillo2004]. In particular, we consider several limiting cases to show that the UCT methodology, when implemented with an $\mathcal{O}(\Delta x^2) $ reconstruction method, returns upwind corner emf values $\mathscr{E}_z$ that are consistent with the GS05 algorithm when coupled to an underlying Godunov-type method based on HLL fluxes. The analysis below highlights the differences between the GS05 and LD2004 approaches to constructing a constrained transport discretization, and it provides further context for the $\mathcal{O}(\Delta x^2) $ limitations that are inherent to the GS05 derivation, as discussed in Section \[sec:uct\].
We begin by assuming that the two-speed flux approximation is made by using the one-state HLL Riemann solver to compute the Godunov fluxes at all faces in the GS05 implementation. Then, the numerical fluxes are approximated by a single intermediate (subsonic) and two supersonic states $$\mathbf{F}^{HLL}_1 = \begin{cases}
\mathbf{F}^{L_1}_1 & S^{L_1} \geq 0 \\
\mathbf{F}^*_1 & S^L \leq 0 \leq S^R \\ \mathbf{F}^{R_1}_1 & S^{R_1} \leq 0
\end{cases} \, , \label{eq:hll-states}$$ for fluxes in the $x_1$ direction. For further detail, we refer the reader to Equation 11 of Miyoshi and Kusano [@MiyoshiKusano2005], which uses the same notation but with subscripted $L/R$. In particular, the nontrivial intermediate state of the $ B_2 $ flux component at an $x_1$-face (an [**upwinded**]{} approximation to the emf $\mathscr{E}_{3} $) can be written as $$\mathbf{e}_{B_2} \cdot \bm{\mathrm{F}}_1^* = \mathscr{E}_{3,i-\frac{1}{2},j} = \frac{S^{R_1} \mathscr{E}_{3,i-\frac{1}{2},j}^{L_1} - S^{L_1} \mathscr{E}_{3,i-\frac{1}{2},j}^{R_1}}{S^{R_1} - S^{L_1}} - \frac{S^{R_1} S^{L_1}}{S^{R_1} - S^{L_1}} (\langle B_2^{R_1} \rangle_{i-\frac{1}{2},j} - \langle B_2^{L_1} \rangle_{i-\frac{1}{2},j}) \,, \label{eq:hll-star-emf-x1}$$ where we have separated the expression into two terms: the first term consists of smooth flux approximations and the second term encapsulates the explicit numerical dissipation of the HLL solver. A well-known trick to unify Equations , is to replace the wavespeed estimates with nonnegative quantities that are both nonzero only for the subsonic case [@MiyoshiKusano2005]. We use the notation for the dissipative terms in LD2004 Equation 55 [@Londrillo2004]. By letting $ \alpha^+_1 \equiv \max(0, S^{R_1}), \alpha^-_1 \equiv -\min(0, S^{L_1}) $, we can express the HLL flux $ F^{HLL} $ for all possible cases of wavespeed estimates with a single expression $$\mathscr{E}_{3,i-\frac{1}{2},j} = \frac{\alpha^+_1 \mathscr{E}_{3,i-\frac{1}{2},j}^{L_1} + \alpha^-_1 \mathscr{E}_{3,i-\frac{1}{2},j}^{R_1}}{\alpha^+_1 + \alpha^-_1} + \frac{\alpha^+_1 \alpha^-_1}{\alpha^+_1 + \alpha^-_1} (\langle B_2^{R_1} \rangle_{i-\frac{1}{2},j} - \langle B_2^{L_1} \rangle_{i-\frac{1}{2},j}) \, . \label{eq:hll-emf-x1}$$ Due to the antisymmetry of the curl operator, the counterpart of the previous equation for the emf component of the HLL flux on $x_2$-faces is $$\mathscr{E}_{3,i,j-\frac{1}{2}} = \frac{\alpha^+_2 \mathscr{E}_{3,i,j-\frac{1}{2}}^{L_2} + \alpha^-_2 \mathscr{E}_{3,i,j-\frac{1}{2}}^{R_2}}{\alpha^+_2 + \alpha^-_2} - \frac{\alpha^+_2 \alpha^-_2}{\alpha^+_2 + \alpha^-_2} (\langle B_1^{R_2} \rangle_{i,j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i,j-\frac{1}{2}}) \, . \label{eq:hll-emf-x2}$$ We now separately consider two possibilities for the upwind directions at the four 1D interfaces used in the $\mathscr{E}_z^c $ scheme.
Stationary domain $\mathbf{v}=0$ {#app:gs05-uct-stationary}
--------------------------------
The comparison of GS05 and LD2004 in the case of a stationary domain is nontrivial due to the upwinding of the 1D conserved variable fluxes that occurs in GS05 in Equation but does not occur in UCT. Nevertheless, the analysis is greatly simplified because all smooth reconstructions of the emf in both the 2D upwinding in UCT Equation and the 1D upwinding in Equations and are zero since $\mathscr{E}_{3} = v_{2}B_{1} - v_{1}B_{2} =0 $. By symmetry, the estimates of the minimum and maximum wavespeed bounds must be equal in magnitude separately in each direction, with $\alpha_1 \equiv \alpha_1^+ = \alpha_1^-$ and $ \alpha_2 \equiv \alpha_2^+ = \alpha_2^-$. Hence, only the HLL explicit dissipation terms remain in the GS05 and UCT Riemann solver fluxes: $$\begin{aligned}
\mathscr{E}_{3,i-\frac{1}{2},j} =& \frac{\alpha_1}{2} (\langle B_2^{R_1} \rangle_{i-\frac{1}{2},j} - \langle B_2^{L_1} \rangle_{i-\frac{1}{2},j}) \, , \label{eq:hll-emf-x1-stationary} \\
\mathscr{E}_{3,i,j-\frac{1}{2}} =& - \frac{\alpha_2}{2} (\langle B_1^{R_2} \rangle_{i,j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i,j-\frac{1}{2}}) \, , \label{eq:hll-emf-x2-stationary} \\
\langle \mathscr{E}^U_{3} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} =& - \frac{\alpha_2}{2}( \langle {B}_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} ) \nonumber \\
&+ \frac{\alpha_1 }{2}(\langle {B}_2^{R_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_2^{L_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} ) \, . \label{eq:uct-hll-stationary}\end{aligned}$$ The GS05 method’s upwinding of the emf derivatives in Equation reduces to averaging in both the $x_1,x_2$ directions $$\begin{aligned}
\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j-\frac{1}{2}} =& \frac{1}{2}\left[\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j-1} + \left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j}\right] \, , \label{eq:gs05-ave-x1}\\
\left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-\frac{1}{2},j-\frac{1}{4}} =& \frac{1}{2}\left[\left(\frac{\partial \mathscr{E}_3}{\partial x_2}\right)_{i-1, j-\frac{1}{4}} + \left(\frac{\partial \mathscr{E}_3}{\partial x_2}\right)_{i, j-\frac{1}{4}} \right] \, , \label{eq:gs05-ave-x2}\end{aligned}$$ and the four-way average in Equation becomes $$\label{eq:gs05-emf-average-stationary}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} =& \frac{1}{4} ( \mathscr{E}_{3,i,j-\frac{1}{2}} + \mathscr{E}_{3,i-1,j-\frac{1}{2}} +\mathscr{E}_{3,i-\frac{1}{2},j} +\mathscr{E}_{3,i-\frac{1}{2},j-1} ) \\
+&\frac{h}{16} \left(
\left(\frac{\partial \mathscr{E}_3}{\partial x_2}\right)_{i-1, j-\frac{3}{4}} + \left(\frac{\partial \mathscr{E}_3}{\partial x_2}\right)_{i, j-\frac{3}{4}}
-\left(\frac{\partial \mathscr{E}_3}{\partial x_2}\right)_{i-1, j-\frac{1}{4}} - \left(\frac{\partial \mathscr{E}_3}{\partial x_2}\right)_{i, j-\frac{1}{4}}
\right) \\
+&\frac{h}{16} \left(
\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{3}{4},j-1} + \left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{3}{4},j}
-\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j-1} - \left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j}
\right) \, .
\end{split}$$ These averaged slopes are approximated at $\mathcal{O}(\Delta x^2) $ accuracy in GS05 by Equation . Due to the sharing of the same quadrature point of the face-centered flux, the difference of two linear slope stencils at the same index in the transverse direction can be slightly condensed as $$\label{eq:gs05-slope-diff}
\begin{split}
\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{3}{4},j} - \left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j} =& \frac{2}{h} (\mathscr{E}_{3,i-\frac{1}{2},j} - \mathscr{E}^r_{3,i-1,j} ) - \frac{2}{h} (\mathscr{E}^r_{3,i,j} - \mathscr{E}_{3,i-\frac{1}{2},j} ) \\
= & \frac{2}{h} (2\mathscr{E}_{3,i-\frac{1}{2},j} - \mathscr{E}^r_{3,i-1,j} - \mathscr{E}^r_{3,i,j} ) \, .
\end{split}$$ In this specific case of a stationary domain, the cell-centered reference electric fields are 0, but the HLL upwinded quantities may not be zero, so Equation is reduced to $$\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{3}{4},j} - \left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j} = \frac{4}{h}\mathscr{E}_{3,i-\frac{1}{2},j} \, . \label{eq:gs05-slope-diff-stationary}$$ Thus, the GS05 expression for the corner emf in Equation can be written solely in terms of the 1D face-centered fluxes as $$\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} = \frac{1}{2} ( \mathscr{E}_{3,i,j-\frac{1}{2}} + \mathscr{E}_{3,i-1,j-\frac{1}{2}} +\mathscr{E}_{3,i-\frac{1}{2},j} +\mathscr{E}_{3,i-\frac{1}{2},j-1} ) \, , \label{eq:gs05-emf-average-stationary2}$$ which is the directionally unbiased formula with correct numerical viscosity, corresponding to GS05 Equation 39 under the assumption of a stationary domain. Substituting the expressions for the face-centered fluxes in Equations and , we get $$\label{eq:gs05-emf-average-stationary-dissipation}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} =& \frac{\alpha_1}{4} (\langle B_2^{R_1} \rangle_{i-\frac{1}{2},j} - \langle B_2^{L_1} \rangle_{i-\frac{1}{2},j}
+ \langle B_2^{R_1} \rangle_{i-\frac{1}{2},j-1} - \langle B_2^{L_1} \rangle_{i-\frac{1}{2},j-1}) \\
-& \frac{\alpha_2}{4} (\langle B_1^{R_2} \rangle_{i,j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i,j-\frac{1}{2}}
+ \langle B_1^{R_2} \rangle_{i-1,j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i-1,j-\frac{1}{2}}) \, .
\end{split}$$ Finally, to show equivalence to the UCT expression in Equation , we use the second-order reconstruction assumption to relate the cell-corner state of $\langle {B}_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}}$ to the transverse face-centered reconstructed states $\langle B_1^{L_2} \rangle_{i,j-\frac{1}{2}}, \langle B_1^{L_2} \rangle_{i-1,j-\frac{1}{2}}$. While the magnetic field is never explicitly reconstructed at cell-corners in GS05, the continuity of $B_1$ along $x_1$ demands that $$\langle {B}_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} = \frac{1}{2} \left(\langle B_1^{L_2} \rangle_{i,j-\frac{1}{2}} + \langle B_1^{L_2} \rangle_{i-1,j-\frac{1}{2}}\right) + \mathcal{O}(\Delta x^2) \, . \label{eq:gs05-corner-b}$$ Hence, Equation is equivalent to $$\label{eq:gs05-equiv-uct2}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} =& \frac{\alpha_1}{2} \left(\frac{\langle B_2^{R_1} \rangle_{i-\frac{1}{2},j} + \langle B_2^{R_1} \rangle_{i-\frac{1}{2},j-1} }{2} - \frac{\langle B_2^{L_1} \rangle_{i-\frac{1}{2},j} - \langle B_2^{L_1} \rangle_{i-\frac{1}{2},j-1}}{2}\right) \\
-& \frac{\alpha_2}{2} \left(\frac{\langle B_1^{R_2} \rangle_{i,j-\frac{1}{2}} + \langle B_1^{R_2} \rangle_{i-1,j-\frac{1}{2}}}{2} - \frac{\langle B_1^{L_2} \rangle_{i,j-\frac{1}{2}} + \langle B_1^{L_2} \rangle_{i-1,j-\frac{1}{2}}}{2} \right) \\
=& \frac{\alpha_1 }{2}(\langle {B}_2^{R_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_2^{L_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} ) - \frac{\alpha_2}{2}( \langle {B}_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} ) + \mathcal{O}(\Delta x^2) \\
=& \langle\mathscr{E}_{3}^U \rangle_{i-\frac{1}{2},j-\frac{1}{2}} + \mathcal{O}(\Delta x^2) \, .
\end{split}$$
Grid-aligned plane-parallel flow: $v_1\neq 0, v_2=0$ {#app:gs05-uct-x1-flow}
----------------------------------------------------
Cases involving non-stationary background flows follow similar lines of reasoning as \[app:gs05-uct-stationary\], but the GS05 formulas are significantly more complicated owing to the introduction of $L/R$ reconstructed emf terms in the HLL expressions. Here, we allow for grid-aligned flow in the positive $x_1$ direction while assuming no motion in the $x_2$ direction. Hence, smooth approximations to the emf satisfy $\mathscr{E}_{3} = - v_{1}B_{2} $. Initially, we consider a magnetic field $\mathbf{B}(x_1,x_2)$ that may have any physically admissible discontinuities.
By symmetry, the estimates of the minimum and maximum wavespeed bounds must be equal in magnitude in the $x_2$ direction, with $ \alpha_2 \equiv \alpha_2^+ = \alpha_2^-$. The fluxes upwinded by the Riemann solver in the two methods can be written as $$\begin{aligned}
\mathscr{E}_{3,i-\frac{1}{2},j} =& \frac{\alpha^+_1 \mathscr{E}_{3,i-\frac{1}{2},j}^{L_1} + \alpha^-_1 \mathscr{E}_{3,i-\frac{1}{2},j}^{R_1}}{\alpha^+_1 + \alpha^-_1} + \frac{\alpha^+_1 \alpha^-_1}{\alpha^+_1 + \alpha^-_1} (\langle B_2^{R_1} \rangle_{i-\frac{1}{2},j} - \langle B_2^{L_1} \rangle_{i-\frac{1}{2},j}) \, , \label{eq:hll-emf-x1-x1-flow} \\
\mathscr{E}_{3,i,j-\frac{1}{2}} =& \frac{1}{2} \left(\mathscr{E}_{3,i,j-\frac{1}{2}}^{L_2} +\mathscr{E}_{3,i,j-\frac{1}{2}}^{R_2} \right)
- \frac{\alpha_2}{2} (\langle B_1^{R_2} \rangle_{i,j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i,j-\frac{1}{2}}) \, , \label{eq:hll-emf-x2-x1-flow} \\
\langle \mathscr{E}^U_{3} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} =& \frac{
\alpha^+_1\left( \langle \mathscr{E}^{L_1L_2}_{3}\rangle_{i-\frac{1}{2},j-\frac{1}{2}} + \langle \mathscr{E}_{3}^{L_1R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right) + \alpha^-_1\left(\langle \mathscr{E}_{3}^{R_1L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} + \langle \mathscr{E}_{3}^{R_1R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right) }{2(\alpha^+_1 + \alpha^-_1)} \nonumber \\
& - \frac{\alpha_2}{2}( \langle {B}_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} ) + \frac{\alpha^+_1 \alpha^-_1}{\alpha^+_1 + \alpha^-_1}(\langle {B}_2^{R_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_2^{L_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} ) \, .\label{eq:uct-hll-x1-flow}\end{aligned}$$ The GS05 upwinding of the emf derivatives in Equation reduces to the selection of the lower index of the approximations of $\partial_{2}$ in the $x_1$ direction $$\left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-\frac{1}{2},j-\frac{1}{4}} = \left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-1, j-\frac{1}{4}} \, , \label{eq:gs05-lower-x1select-x1-flow}$$ and central averaging of the $\partial_1$ approximations in the $x_2$ direction $$\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j-\frac{1}{2}} = \frac{1}{2}\left[\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j-1} + \left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j}\right] \, .\label{eq:gs05-ave-x2-x1-flow}$$ The four-way average in Equation becomes $$\label{eq:gs05-emf-average-x1-flow}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} =& \frac{1}{4} ( \mathscr{E}_{3,i,j-\frac{1}{2}} + \mathscr{E}_{3,i-1,j-\frac{1}{2}} +\mathscr{E}_{3,i-\frac{1}{2},j} +\mathscr{E}_{3,i-\frac{1}{2},j-1} ) \\
+&\frac{h}{8} \left( \left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-1,j-\frac{3}{4}} - \left(\frac{\partial \mathscr{E}_3}{\partial x_2} \right)_{i-1,j-\frac{1}{4}} \right) \\
+&\frac{h}{16} \left(
\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{3}{4},j-1} + \left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{3}{4},j}
-\left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j-1} - \left(\frac{\partial \mathscr{E}_3}{\partial x_1} \right)_{i-\frac{1}{4},j}
\right) \, .
\end{split}$$ Using Equation , the GS05 expression reduces to $$\label{eq:gs05-emf-average-x1-flow-2}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} =& \frac{1}{4} ( \mathscr{E}_{3,i,j-\frac{1}{2}} + \mathscr{E}_{3,i-1,j-\frac{1}{2}} +\mathscr{E}_{3,i-\frac{1}{2},j} +\mathscr{E}_{3,i-\frac{1}{2},j-1} ) + \frac{h}{8}\frac{2}{h} \left( 2\mathscr{E}_{3,i,j-\frac{1}{2}} - \mathscr{E}^r_{3,i-1,j} - \mathscr{E}^r_{3,i-1,j-1} \right) \\
+&\frac{h}{16}\frac{2}{h} \left(
(2\mathscr{E}_{3,i-\frac{1}{2},j-1} - \mathscr{E}^r_{3,i-1,j-1} - \mathscr{E}^r_{3,i,j-1} )
+ (2\mathscr{E}_{3,i-\frac{1}{2},j} - \mathscr{E}^r_{3,i-1,j} - \mathscr{E}^r_{3,i,j} )
\right) \, .
\end{split}$$ After collecting the terms in the expression, we observe that the GS05 upwinding scheme has essentially acted as a switch between the two intercell $x_2$ face HLL fluxes, with $$\label{eq:gs05-emf-average-x1-flow-3}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} =& \frac{3}{4}\mathscr{E}_{3,i-1,j-\frac{1}{2}} + \frac{1}{4}\mathscr{E}_{3,i,j-\frac{1}{2}} + \frac{1}{2}(\mathscr{E}_{3,i-\frac{1}{2},j} +\mathscr{E}_{3,i-\frac{1}{2},j-1} ) \\
-& \frac{3}{8}\left(\mathscr{E}^r_{3,i-1,j} +\mathscr{E}^r_{3,i-1,j-1} \right) -\frac{1}{8}\left(\mathscr{E}^r_{3,i,j-1} +\mathscr{E}^r_{3,i,j} \right) \, ,
\end{split}$$ where the shared upwind $x_1$ direction at both $i-\frac{1}{2},j$ and $i-\frac{1}{2},j-1$ interfaces biases the terms above with larger $\frac{3}{4},-\frac{3}{8}$ coefficients relative to the downstream quantities with $\frac{1}{4},-\frac{1}{8}$ coefficients. The interface quantities in the previous equation have been upwinded in 1D by a Riemann solver. Therefore, in general they cannot be described exclusively in terms of the reconstructed quantities of a single cell. We now insert the expressions for the 1D HLL fluxes in Equations and in order to separate out the HLL explicit dissipation terms as $$\label{eq:gs05-emf-average-x1-flow-4}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} =& \frac{3}{8}\left(\left(\mathscr{E}_{3,i-1,j-\frac{1}{2}}^{L_2} + \mathscr{E}_{3,i-1,j-\frac{1}{2}}^{R_2} \right)
- \alpha_2(\langle B_1^{R_2} \rangle_{i-1,j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i-1,j-\frac{1}{2}}) \right) \\
+& \frac{1}{8} \left(\left(\mathscr{E}_{3,i,j-\frac{1}{2}}^{L_2} + \mathscr{E}_{3,i,j-\frac{1}{2}}^{R_2} \right)
- \alpha_2(\langle B_1^{R_2} \rangle_{i,j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i,j-\frac{1}{2}}) \right) \\
+& \frac{\alpha^+_1 \mathscr{E}_{3,i-\frac{1}{2},j}^{L_1} + \alpha^-_1 \mathscr{E}_{3,i-\frac{1}{2},j}^{R_1} + \alpha^+_1 \mathscr{E}_{3,i-\frac{1}{2},j-1}^{L_1} + \alpha^-_1 \mathscr{E}_{3,i-\frac{1}{2},j-1}^{R_1}}{2(\alpha^+_1 + \alpha^-_1)} \\
+& \frac{\alpha^+_1 \alpha^-_1}{2(\alpha^+_1 + \alpha^-_1)} (\langle B_2^{R_1} \rangle_{i-\frac{1}{2},j} - \langle B_2^{L_1} \rangle_{i-\frac{1}{2},j} + \langle B_2^{R_1} \rangle_{i-\frac{1}{2},j-1} - \langle B_2^{L_1} \rangle_{i-\frac{1}{2},j-1}) \\
-& \frac{3}{8}\left(\mathscr{E}^r_{3,i-1,j} +\mathscr{E}^r_{3,i-1,j-1} \right) -\frac{1}{8}\left(\mathscr{E}^r_{3,i,j-1} +\mathscr{E}^r_{3,i,j} \right) \, .
\end{split}$$ These terms include Riemann states of $\mathbf{B}$ that are reconstructed in adjacent cells. We then use Equation to combine the $B_2$ terms from the $x_1$ interface Riemann solutions, resulting in $$\label{eq:gs05-emf-average-x1-flow-5}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} =& -\frac{3\alpha_2}{8}\left(\langle B_1^{R_2} \rangle_{i-1,j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i-1,j-\frac{1}{2}} \right) - \frac{\alpha_2}{8} \left(\langle B_1^{R_2} \rangle_{i,j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i,j-\frac{1}{2}} \right) \\
+& \frac{\alpha^+_1 \left( \mathscr{E}_{3,i-\frac{1}{2},j}^{L_1} + \mathscr{E}_{3,i-\frac{1}{2},j-1}^{L_1} \right) + \alpha^-_1 \left( \mathscr{E}_{3,i-\frac{1}{2},j}^{R_1} + \mathscr{E}_{3,i-\frac{1}{2},j-1}^{R_1}\right)}{2(\alpha^+_1 + \alpha^-_1)} \\
+& \frac{\alpha^+_1 \alpha^-_1}{(\alpha^+_1 + \alpha^-_1)} \left(\langle {B}_2^{R_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_2^{L_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right) \\
+& \frac{3}{8}\left(\left(\mathscr{E}_{3,i-1,j-\frac{1}{2}}^{L_2} + \mathscr{E}_{3,i-1,j-\frac{1}{2}}^{R_2} \right) - \left(\mathscr{E}^r_{3,i-1,j} +\mathscr{E}^r_{3,i-1,j-1} \right) \right) \\
+& \frac{1}{8}\left( \left(\mathscr{E}_{3,i,j-\frac{1}{2}}^{L_2} + \mathscr{E}_{3,i,j-\frac{1}{2}}^{R_2} \right) - \left(\mathscr{E}^r_{3,i,j-1} +\mathscr{E}^r_{3,i,j} \right) \right) \, .
\end{split}$$ Next, we consider only the $B_1$ terms $$\label{eq:b1-continuous-bias}
\begin{split}
&-\frac{\alpha_2}{2}\left(
\left(\frac{3}{4}\langle B_1^{R_2} \rangle_{i-1,j-\frac{1}{2}} + \frac{1}{4}\langle B_1^{R_2} \rangle_{i,j-\frac{1}{2}} \right)
- \left(\frac{3}{4}\langle B_1^{L_2} \rangle_{i-1,j-\frac{1}{2}} + \frac{1}{4}\langle B_1^{L_2} \rangle_{i,j-\frac{1}{2}} \right)
\right) \\
& \approx -\frac{\alpha_2}{2}\left(\langle B_1^{R_2} \rangle_{i-\frac{3}{4},j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i-\frac{3}{4},j-\frac{1}{2}} \right) + \mathcal{O}(\Delta x^2) \\
& \approx -\frac{\alpha_2}{2}\left(\langle B_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right) + \mathcal{O}(\Delta x^2) \, ,
\end{split}$$ which as in the previous proof, are guaranteed to provide a continuous approximation to an intermediate linear reconstructed value since $B_1$ is continuous in $x_1$. Unlike Equation , this linear interpolation is biased to the left of the $i-\frac{1}{2},j-\frac{1}{2}$ corner position, which corresponds to the upstream direction. Nevertheless, it is a consistent approximation at second-order accuracy, so Equation can be written as $$\label{eq:gs05-approx-uct-x1-flow}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} \approx & -\frac{\alpha_2}{2}\left(\langle B_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right) \\
+& \frac{\alpha^+_1 \left( \mathscr{E}_{3,i-\frac{1}{2},j}^{L_1} + \mathscr{E}_{3,i-\frac{1}{2},j-1}^{L_1} \right) + \alpha^-_1 \left( \mathscr{E}_{3,i-\frac{1}{2},j}^{R_1} + \mathscr{E}_{3,i-\frac{1}{2},j-1}^{R_1}\right)}{2(\alpha^+_1 + \alpha^-_1)} \\
+& \frac{\alpha^+_1 \alpha^-_1}{(\alpha^+_1 + \alpha^-_1)} \left(\langle {B}_2^{R_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_2^{L_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right) \\
+& \frac{3}{8}\left(\left(\mathscr{E}_{3,i-1,j-\frac{1}{2}}^{L_2} + \mathscr{E}_{3,i-1,j-\frac{1}{2}}^{R_2} \right) - \left(\mathscr{E}^r_{3,i-1,j} +\mathscr{E}^r_{3,i-1,j-1} \right) \right) \\
+& \frac{1}{8}\left( \left(\mathscr{E}_{3,i,j-\frac{1}{2}}^{L_2} + \mathscr{E}_{3,i,j-\frac{1}{2}}^{R_2} \right) - \left(\mathscr{E}^r_{3,i,j-1} +\mathscr{E}^r_{3,i,j} \right) \right) \, .
\end{split}$$ It is impossible for the general expression in Equation to exactly match the UCT solution in Equation due to the variable factors of $\alpha_1^\pm$ in the $x_1$ face-centered emf flux states that are absent for the other terms. However, we can show second-order accurate agreement under further simplifying assumptions.
We first assume usage of the simple HLL wavespeed estimates of Davis [@Davis1988], which are defined as
\[eq:davis-hll-mhd\] $$\begin{aligned}
S^{L_1} =& \min \left(\lambda^-_1(\mathbf{W}^{L_1}, B_1), \lambda^-_1(\mathbf{W}^{R_1}, B_1)\right) \, , \\
S^{R_1} =& \max \left(\lambda^+_1(\mathbf{W}^{L_1}, B_1), \lambda^+_1(\mathbf{W}^{R_1}, B_1) \right) \, ,\end{aligned}$$
where $ \lambda^\pm_1 $ are the largest and smallest eigenvalues of the system in the $x_1$ direction. For the MHD system, these are related to the fast magnetosonic wavespeeds $\lambda_1^\pm = v_1 \pm c_1^f$. We refer the reader to Equation 55 of LD2004 for comparison.
Next, we consider the limiting case of $v_1 \gg 0$ such that $\alpha_1^- \equiv -\min(0,S^{L_1}) = 0$. For the above two-speed flux approximation applied to the MHD system, the supersonic limiting case occurs when $v_1 \geq c_1^f$. In such a case, the $x_1$ explicit dissipation term goes to zero in Equation , and the expression becomes $$\label{eq:gs05-supersonic-x1-flow}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} \approx & -\frac{\alpha_2}{2}\left(\langle B_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right)
+ \frac{\mathscr{E}_{3,i-\frac{1}{2},j}^{L_1} + \mathscr{E}_{3,i-\frac{1}{2},j-1}^{L_1} }{2} \\
+& \frac{3}{8}\left(\left(\mathscr{E}_{3,i-1,j-\frac{1}{2}}^{L_2} + \mathscr{E}_{3,i-1,j-\frac{1}{2}}^{R_2} \right) - \left(\mathscr{E}^r_{3,i-1,j} +\mathscr{E}^r_{3,i-1,j-1} \right) \right) \\
+& \frac{1}{8}\left( \left(\mathscr{E}_{3,i,j-\frac{1}{2}}^{L_2} + \mathscr{E}_{3,i,j-\frac{1}{2}}^{R_2} \right) - \left(\mathscr{E}^r_{3,i,j-1} +\mathscr{E}^r_{3,i,j} \right) \right) \, .
\end{split}$$ Now, recall that the reconstructed emf is simplified in this 1D flow case. The continuity of $B_2$ across the $x_2$ interface further simplifies the reconstructed emf states
\[eq:emf-x2-interface-x1-flow\] $$\begin{aligned}
\mathscr{E}_{3,i,j-\frac{1}{2}}^{L_2} =& -v_{1,i,j-\frac{1}{2}}^{L_2} B_{2,i,j-\frac{1}{2}} \, , \\
\mathscr{E}_{3,i,j-\frac{1}{2}}^{R_2} =&-v_{1,i,j-\frac{1}{2}}^{R_2} B_{2,i,j-\frac{1}{2}} \, ,\end{aligned}$$
since they only depend on the $v_1$ reconstruction. Rearranging the emf terms in Equation and using the continuity of $B_2$ in $x_2$ results in $$\label{eq:gs05-supersonic-x1-flow-2}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} \approx & -\frac{\alpha_2}{2}\left(\langle B_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right)
+ \frac{\mathscr{E}_{3,i-\frac{1}{2},j}^{L_1} + \mathscr{E}_{3,i-\frac{1}{2},j-1}^{L_1} }{2} \\
+& \frac{1}{2}\left(\frac{3}{4}\left(-v_{1,i-1,j-\frac{1}{2}}^{L_2} B_{2,i-1,j-\frac{1}{2}} - \mathscr{E}^r_{3,i-1,j} \right) + \frac{1}{4} \left(-v_{1,i,j-\frac{1}{2}}^{L_2} B_{2,i,j-\frac{1}{2}}- \mathscr{E}^r_{3,i,j}\right) \right) \\
+& \frac{1}{2}\left( \frac{3}{4}\left(-v_{1,i-1,j-\frac{1}{2}}^{R_2} B_{2,i-1,j-\frac{1}{2}}- \mathscr{E}^r_{3,i-1,j-1} \right)
+ \frac{1}{4}\left(-v_{1,i,j-\frac{1}{2}}^{R_2} B_{2,i,j-\frac{1}{2}} - \mathscr{E}^r_{3,i,j-1} \right) \right) \, .
\end{split}$$ In contrast, no such simplification can be made for the emf across the $x_1$ interfaces, which satisfy
\[eq:emf-x1-interface-x1-flow\] $$\begin{aligned}
\mathscr{E}_{3,i-\frac{1}{2},j}^{L_1} =& -v_{1,i-\frac{1}{2},j}^{L_1} B_{2,i-\frac{1}{2},j}^{L_1} \, , \\
\mathscr{E}_{3,i-\frac{1}{2},j}^{R_1} =&-v_{1,i-\frac{1}{2},j}^{R_1} B_{2,i-\frac{1}{2},j}^{R_1} \, .\end{aligned}$$
Because $B_2$ may be discontinuous in $x_1$, combining these terms in a similar fashion as the $B_1$ terms in Equation is impossible without making further simplifying assumptions about the field. The general expression in terms of the velocity and magnetic field is $$\label{eq:gs05-supersonic-x1-flow-3}
\begin{split}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} \approx & -\frac{\alpha_2}{2}\left(\langle B_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right)
- \frac{1}{2} \left(v_{1,i-\frac{1}{2},j}^{L_1} B_{2,i-\frac{1}{2},j}^{L_1} + v_{1,i-\frac{1}{2},j-1}^{L_1} B_{2,i-\frac{1}{2},j-1}^{L_1} \right) \\
-& \frac{1}{2}\left(\frac{3}{4}\left(v_{1,i-1,j-\frac{1}{2}}^{L_2} B_{2,i-1,j-\frac{1}{2}} - v_{1,i-1,j}B_{2,i-1,j} \right) + \frac{1}{4} \left(v_{1,i,j-\frac{1}{2}}^{L_2} B_{2,i,j-\frac{1}{2}}- v_{1,i,j}B_{2,i,j}\right) \right) \\
-& \frac{1}{2}\left( \frac{3}{4}\left( v_{1,i-1,j-\frac{1}{2}}^{R_2} B_{2,i-1,j-\frac{1}{2}}- v_{1,i-1,j-1}B_{2,i-1,j-1} \right)
+ \frac{1}{4}\left(v_{1,i,j-\frac{1}{2}}^{R_2} B_{2,i,j-\frac{1}{2}} - v_{1,i,j-1}B_{2,i,j-1} \right) \right) \, .
\end{split}$$ If the $x_1$-reconstruction step produces identical Riemann states $B_{2,i-\frac{1}{2},j}^{L_1}, B_{2,i-\frac{1}{2},j-1}^{L_1} $, then the GS05 approximation to the emf at the cell corner finally reduces to $$\begin{gathered}
\mathscr{E}_{3,i-\frac{1}{2},j-\frac{1}{2}} \approx -\frac{\alpha_2}{2}\left(\langle B_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle B_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right) \\
- \frac{\left( \langle v^{L_1L_2}_{1}\rangle_{i-\frac{1}{2},j-\frac{1}{2}} + \langle v_{1}^{L_1R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right) \langle {B}_2^{L_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}}}{2} \, . \label{eq:gs05-supersonic-x1-flow-4}\end{gathered}$$
Having considered the GS05 algorithm’s behavior in this limiting case, we turn our attention to the upwind constrained transport approach. Under the assumption of supersonic wavespeed estimates, the UCT formula in Equation becomes $$\label{eq:uct-hll-supersonic-x1-flow}
\langle \mathscr{E}^U_{3} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} = \frac{
\langle \mathscr{E}^{L_1L_2}_{3}\rangle_{i-\frac{1}{2},j-\frac{1}{2}} + \langle \mathscr{E}_{3}^{L_1R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} }{2}
- \frac{\alpha_2}{2}( \langle {B}_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} ) \, .$$ Since $v_2=0$ in this case, $$\begin{gathered}
\langle \mathscr{E}^U_{3} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} = \frac{
- \left( \langle v^{L_1L_2}_{1}\rangle_{i-\frac{1}{2},j-\frac{1}{2}} + \langle v_{1}^{L_1R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} \right) \langle {B}_2^{L_1} \rangle_{i-\frac{1}{2},j-\frac{1}{2}}}{2} \\
- \frac{\alpha_2}{2}( \langle {B}_1^{R_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} - \langle {B}_1^{L_2} \rangle_{i-\frac{1}{2},j-\frac{1}{2}} ) \, , \label{eq:uct-hll-supersonic-x1-flow-2}\end{gathered}$$ which matches Equation .
[^1]: See Balsara (2014) 9.5 for results from a field loop advection test generated by the GS05 $\mathscr{E}_z^\circ $ method.
|
---
abstract: |
Suppose $\la$ and $\mu$ are integer partitions with $\la\supseteq\mu$. Kenyon and Wilson have introduced the notion of a *[cover-inclusive Dyck tiling]{}* of the skew Young diagram ${\ensuremath{\la\backslash\mu}}$, which has applications in the study of double-dimer models. We examine these tilings in more detail, giving various equivalent conditions and then proving a recurrence which we use to show that the entries of the transition matrix between two bases for a certain permutation module for the symmetric group are given by counting [cover-inclusive Dyck tiling]{}s. We go on to consider the inverse of this matrix, showing that its entries are determined by what we call *[cover-expansive Dyck tiling]{}s*. The fact that these two matrices are mutual inverses allows us to recover the main result of Kenyon and Wilson.
We then discuss the connections with recent results of Kim et al, who give a simple expression for the sum, over all $\mu$, of the number of [cover-inclusive Dyck tiling]{}s of ${\ensuremath{\la\backslash\mu}}$. Our results provide a new proof of this result. Finally, we show how to use our results to obtain simpler expressions for the *homogeneous Garnir relations* for the universal Specht modules introduced by Kleshchev, Mathas and Ram for the cyclotomic quiver Hecke algebras.
title: |
[Dyck tilings and the homogeneous Garnir\
relations for graded Specht modules]{}
---
=3mu plus 1mu minus 1mu
1
Introduction
============
The motivation for this paper is the study of the modular representation theory of the symmetric group, and more generally the representation theory of the cyclotomic Hecke algebra of type $A$. This area has recently been revolutionised by the discovery by Brundan and Kleshchev of new presentations for these algebras, which show in particular that the algebras are non-trivially $\bbz$-graded. The contribution in the present paper concerns the definition of the *Specht modules*, which play a central role in the representation theory of cyclotomic Hecke algebras. These modules have been studied within the graded setting by Brundan, Kleshchev and Wang, and developed further by Kleshchev, Mathas and Ram, who have given a presentation for each Specht module with a single generator and a set of homogeneous relations. These relations include homogeneous analogues of the classical Garnir relations for the symmetric group, which allow the Specht module to be expressed as a quotient of a ‘row permutation module’. Although the homogeneous Garnir relations are in some sense simpler than their classical counterparts, their statement in [@kmr] is awkward in that the ‘Garnir elements’ involved are given as linear combinations of expressions in the standard generators $\psi_1,\dots,\psi_{n-1}$ for the row permutation module which are not always reduced. Our main result concerning Specht modules is an expression for each Garnir relation as a linear combination of reduced expressions; this simplifies calculations with Specht modules, both theoretically and computationally.
But this result is a by-product of the main work in this paper, which is to consider tilings of skew Young diagrams by Dyck tiles. These tilings were introduced by Kenyon and Wilson, who defined in particular the notion of a *cover-inclusive Dyck tiling*. They used these tilings to give a formula for the inverse of a certain matrix $M$ arising in the study of double-dimer models. We re-interpret the entries of $M$ in terms of what we call *cover-expansive Dyck tilings*, and then, by proving recurrence relations for the numbers of cover-inclusive and cover-expansive Dyck tilings, we show that (sign-modified versions of) $M$ and $M^{-1}$ are in fact change-of-basis matrices for two natural bases for a certain permutation representation of the symmetric group. The fact that the two change-of-basis matrices are obviously mutually inverse provides a new proof of Kenyon and Wilson’s result. Along the way we give a result showing several different equivalent conditions to the cover-inclusive condition.
In order to derive our result on Garnir relations, we then express the sum of the elements of one of our two bases in terms of the other; this involves defining a certain function ${F}$ on partitions, and proving a similar recurrence to the recurrence for [cover-inclusive Dyck tiling]{}s. Combining this with our results on change-of-basis matrices means that ${F}(\la)$ equals the sum over all partitions $\mu\subseteq\la$ of the number of [cover-inclusive Dyck tiling]{}s of ${\ensuremath{\la\backslash\mu}}$. In fact, this had already been shown by Kim, and then by Kim, Mészáros, Panova and Wilson, verifying a conjecture of Kenyon and Wilson. As well as providing a new proof of this result, our results working directly with the function ${F}$ allow us to derive our application to Garnir relations without using Dyck tilings.
We now describe the structure of this paper. \[defnsec\] is devoted to definitions. In \[cidtsec\] we study [cover-inclusive Dyck tiling]{}s, giving equivalent conditions for cover-inclusiveness and then proving several bijective results which allow us to deduce recurrences for the number ${\operatorname{i}_{\la\mu}}$ of [cover-inclusive Dyck tiling]{}s of ${\ensuremath{\la\backslash\mu}}$. In \[cedtsec\] we prove similar, though considerably simpler, recurrences for [cover-expansive Dyck tiling]{}s. In \[ypmsec\] we recall the Young permutation module ${\mathscr{M}^{(f,g)}}$ for the symmetric group; we define our two bases for this module, and use Dyck tilings to describe the change-of-basis matrices. We then introduce the function ${F}$ and use it to express the sum of the elements of the first basis in terms of the second, before summarising the relationship between our work and that of Kenyon, Kim, Mészáros, Panova and Wilson. Finally in \[klrsec\] we give the motivating application of this work, introducing the Specht modules in the modern setting, and using our earlier results to give a new expression for the homogeneous Garnir relations.
The author is greatly indebted to David Speyer and Philippe Nadeau for comments on the online forum MathOverflow which inspired the author to introduce Dyck tilings into this paper.
Definitions {#defnsec}
===========
Partitions and Young diagrams
-----------------------------
As usual, a *partition* is a weakly decreasing sequence $\la=(\la_1,\la_2,\dots)$ of non-negative integers with finite sum. We write this sum as $|\la|$, and say that $\la$ is a partition of $|\la|$. When writing partitions, we may group equal parts together with a superscript, and omit trailing zeroes, and we write the partition $(0,0,\dots)$ as $\varnothing$.
The *Young diagram* of a partition $\la$ is the set $$\lset{(a,b)\in\bbn^2}{b\ls\la_a}.$$ We may abuse notation by identifying $\la$ with its Young diagram; for example, we may write $\la\supseteq\mu$ to mean that $\la_i\gs\mu_i$ for all $i$. If $\la\supseteq\mu$, then the *skew Young diagram* ${\ensuremath{\la\backslash\mu}}$ is simply the set difference between the Young diagrams for $\la$ and $\mu$.
We draw (skew) Young diagrams as arrays of boxes in the plane, and except in the final section of this paper we use the *Russian convention*, where $a$ increases from south-east to north-west, and $b$ increases from south-west to north-east. For example, the Young diagram of ${\ensuremath{(7^2,4,3,2^2)\backslash(2,1^2)}}$ is as follows. $$\begin{tikzpicture}[scale=1,rotate=45]
\tgyoung(0cm,0cm,::;;;;;,:;;;;;;,:;;;,;;;,;;,;;)
\end{tikzpicture}$$ The *conjugate partition* to $\la$ is the partition $\la'$ obtained by reflecting the Young diagram for $\la$ left to right; thus $\la'_i=\left|\lset{j\gs1}{\la_j\gs i}\right|$ for all $i$.
We define a *node* to be an element of $\bbn^2$, and a node of $\la$ to be an element of the Young diagram of $\la$. The *height* of the node $(a,b)$ is $a+b$. The $j$th *column* of $\bbn^2$ is the set of all nodes $(a,b)$ for which $b-a=j$.
We use compass directions to label the neighbours of a node; for example, if $\fkn$ is a node, then we write ${\ensuremath{\mathtt{SW}(\fkn)}}=\fkn-(0,1)$ and refer to this as the *SW neighbour* of $\fkn$; we also write ${\ensuremath{\mathtt{N}(\fkn)}}=\fkn+(1,1)$, and similarly for the other compass directions.
A node $\fkn$ of $\la$ is *removable* if it can be removed from $\la$ to leave the Young diagram of a partition (i.e. if neither nor is a node of $\la$), while a node $\fkn$ not in $\la$ is an *addable node of $\la$* if it can be added to $\la$ to leave the Young diagram of a partition.
Tiles and tilings
-----------------
We define a *tile* to be a finite non-empty set $t$ of nodes that can be ordered $\fkn_1,\dots,\fkn_r$ such that $\fkn_{i+1}\in\{{\ensuremath{\mathtt{NE}(\fkn_i)}},{\ensuremath{\mathtt{SE}(\fkn_i)}}\}$ for each $i=1,\dots,r-1$. We say that $t$ *starts* at its leftmost node, which we denote t, and *ends* at its rightmost, which we denote t. The *height* t of $t$ is defined to be $\max\lset{{\ensuremath{\operatorname{ht}(\fkn)}}}{\fkn\in t}$, and we say that $t$ is a *Dyck tile* if this maximum is achieved at the start and end nodes of $t$, i.e. ${\ensuremath{\operatorname{ht}(t)}}={\ensuremath{\operatorname{ht}({\ensuremath{\operatorname{st}(t)}})}}={\ensuremath{\operatorname{ht}({\ensuremath{\operatorname{en}(t)}})}}$. $t$ is *big* if it contains more than one node, and is a *singleton* otherwise. The *depth* of a node $\fkn\in t$ is $${\ensuremath{\operatorname{dp}(\fkn)}}:={\ensuremath{\operatorname{ht}(t)}}-{\ensuremath{\operatorname{ht}(\fkn)}}.$$
Now suppose $\la$ and $\mu$ are partitions with $\la\supseteq\mu$. A *Dyck tiling* of ${\ensuremath{\la\backslash\mu}}$ is a partition of ${\ensuremath{\la\backslash\mu}}$ into Dyck tiles. Given a Dyck tiling $T$ of ${\ensuremath{\la\backslash\mu}}$ and a node $\fkn\in{\ensuremath{\la\backslash\mu}}$, we write for the tile containing $\fkn$. We say that $\fkn$ is *attached to* if $\fkn$ and lie in the same tile in $T$, and similarly for , and . If $t$ is a tile in $T$, the *NE neighbour of $t$* is the tile starting at [$\mathtt{NE}({\ensuremath{\operatorname{en}(t)}})$]{}, if there is one, while the SW neighbour of $t$ is the tile ending at [$\mathtt{SW}({\ensuremath{\operatorname{st}(t)}})$]{}, if there is one. SE and NW neighbours of tiles are defined similarly.
Now we recast the main definition from [@kw]. Say that a Dyck tiling $T$ of ${\ensuremath{\la\backslash\mu}}$ is *left-cover-inclusive* if whenever $\fka$ and ${\ensuremath{\mathtt{N}(\fka)}}$ are nodes of ${\ensuremath{\la\backslash\mu}}$, [$\operatorname{st}({\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})}})$]{} lies weakly to the left of [$\operatorname{st}({\ensuremath{\operatorname{tile}(\fka)}})$]{}. Similarly, $T$ is *right-cover-inclusive* if whenever $\fka$ and ${\ensuremath{\mathtt{N}(\fka)}}$ are nodes of ${\ensuremath{\la\backslash\mu}}$, [$\operatorname{en}({\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})}})$]{} lies weakly to the right of [$\operatorname{en}({\ensuremath{\operatorname{tile}(\fka)}})$]{}. Say that $T$ is *cover-inclusive* if it is both left- and right-cover-inclusive.
We let ${\scri(\la,\mu)}$ denote the set of cover-inclusive Dyck tilings of ${\ensuremath{\la\backslash\mu}}$ if $\la\supseteq\mu$, and set ${\scri(\la,\mu)}=\emptyset$ otherwise. Let ${\operatorname{i}_{\la\mu}}=\left|{\scri(\la,\mu)}\right|$.
Next we make a definition which is in some sense dual to the notion of cover-inclusiveness and which appears heavily disguised in [@kw]. Say that a Dyck tiling is *left-cover-expansive* if whenever $\fka$ and are nodes of ${\ensuremath{\la\backslash\mu}}$, [$\operatorname{st}({\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{SE}(\fka)}})}})$]{} lies weakly to the left of [$\operatorname{st}({\ensuremath{\operatorname{tile}(\fka)}})$]{}, and *right-cover-expansive* if whenever $\fka$ and are nodes of ${\ensuremath{\la\backslash\mu}}$, [$\operatorname{en}({\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{SW}(\fka)}})}})$]{} lies weakly to the right of [$\operatorname{en}({\ensuremath{\operatorname{tile}(\fka)}})$]{}. A Dyck tiling is *cover-expansive* if it is both left- and right-cover-expansive.
We write ${\operatorname{e}_{\la\mu}}$ for the number of [cover-expansive Dyck tiling]{}s of ${\ensuremath{\la\backslash\mu}}$, setting ${\operatorname{e}_{\la\mu}}=0$ if $\la\nsupseteq\mu$. We shall see later that ${\operatorname{e}_{\la\mu}}\ls1$ for all $\la,\mu$.
We illustrate four Dyck tilings of ${\ensuremath{(6^2,4,3,1^2)\backslash(4,1^2)}}$. Only the first one is cover-inclusive, and only the second is cover-expansive. $$\begin{tikzpicture}[scale=.25]
\draw(3,5)--++(1,-1)--++(1,1);
\draw(4,6)--++(1,-1)--++(1,1);
\draw(3,7)--++(1,-1)--++(1,1);
\draw(-6,6)--++(1,-1)--++(1,1);
\draw(-2,4)--++(1,-1)--++(1,-1)--++(1,1)--++(1,1);
\draw(-2,6)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)--++(1,1);
\draw(-5,5)--++(1,-1)--++(1,-1)--++(1,1)--++(1,1);
\draw(6,6)--++(-1,1)--++(0,0)--++(-1,1)--++(-2,-2)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-2,-2)--++(-1,1)--++(0,0)--++(-1,1)
--++(-1,-1);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=.25]
\draw(3,5)--++(1,-1)--++(1,1);
\draw(4,6)++(1,-1)--++(1,1);
\draw(3,7)--++(1,-1)--++(1,1);
\draw(-6,6)--++(1,-1);
\draw(-2,4)--++(1,-1)--++(1,-1)--++(3,3);
\draw(-2,6)--++(1,-1)--++(1,-1)--++(2,2);
\draw(-5,5)--++(1,-1)--++(1,-1)--++(1,1);
\draw(6,6)--++(-1,1)--++(0,0)--++(-1,1)--++(-2,-2)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-2,-2)--++(-1,1)--++(0,0)--++(-1,1)
--++(-1,-1);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=.25]
\draw(3,5)--++(1,-1)--++(1,1);
\draw(4,6)--++(1,-1)--++(1,1);
\draw(-1,3)--++(1,-1)--++(1,1);
\draw(0,4)++(1,-1)--++(1,1);
\draw(1,5)--++(1,-1)--++(1,1);
\draw(2,6)++(1,-1)--++(1,1);
\draw(3,7)--++(1,-1)--++(1,1);
\draw(-2,4)--++(1,-1);
\draw(-1,5)--++(1,-1)--++(1,1);
\draw(0,6)--++(1,-1);
\draw(-4,4)--++(1,-1)--++(1,1);
\draw(-3,5)++(1,-1)--++(1,1);
\draw(-2,6)++(1,-1)--++(1,1);
\draw(-5,5)--++(1,-1);
\draw(-6,6)--++(1,-1);
\draw(6,6)--++(-1,1)--++(0,0)--++(-1,1)--++(-2,-2)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-2,-2)--++(-1,1)--++(0,0)--++(-1,1)
--++(-1,-1);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=.25]
\draw(3,5)--++(1,-1)--++(1,1);
\draw(4,6)++(1,-1)--++(1,1);
\draw(-1,3)--++(1,-1)--++(1,1);
\draw(0,4)++(1,-1)--++(1,1);
\draw(1,5)--++(1,-1)--++(1,1);
\draw(2,6)--++(1,-1);
\draw(3,7)--++(1,-1)--++(1,1);
\draw(-2,4)--++(1,-1);
\draw(-1,5)--++(1,-1)--++(1,1);
\draw(0,6)++(1,-1)--++(1,1);
\draw(-4,4)--++(1,-1)--++(1,1);
\draw(-3,5)++(1,-1)--++(1,1);
\draw(-2,6)--++(1,-1);
\draw(-5,5)--++(1,-1);
\draw(-6,6)--++(1,-1)--++(1,1);
\draw(6,6)--++(-1,1)--++(0,0)--++(-1,1)--++(-2,-2)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-2,-2)--++(-1,1)--++(0,0)--++(-1,1)
--++(-1,-1);
\end{tikzpicture}$$
We end this section with some notation which we shall use repeatedly later. Suppose $j\in\bbz$ is fixed, and $\la$ is a partition with an addable node $\fkl$ in column $j$. We define ${X_{\la}}$ to be the set of all integers $x$ such that $\la$ has a removable node $\fkn$ in column $j+x$, with ${\ensuremath{\operatorname{ht}(\fkn)}}={\ensuremath{\operatorname{ht}(\fkl)}}-1$, and ${\ensuremath{\operatorname{ht}(\fkp)}}<{\ensuremath{\operatorname{ht}(\fkl)}}$ for all nodes $\fkp\in\la$ in all columns between $j$ and $j+x$. We set ${{X_{\la}}^+}$ to be the set of positive elements of ${X_{\la}}$. Note that $x\in{{X_{\la}}^+}$ precisely when there is a Dyck tile $t\subset\la$, starting in column $j+1$ and ending in column $j+x$, which can be removed from $\la$ to leave a smaller partition; we denote this smaller partition ${\la^{[x]}}$. We define ${\la^{[x]}}$ for $x\in{{X_{\la}}^-}={X_{\la}}\setminus{{X_{\la}}^+}$ similarly.
Take $\la=(6,4^2,3,2^2)$. Then $\la$ has an addable node in column $0$, and for this node we have ${X_{\la}}=\{-1,1,5\}$. The partitions ${\la^{[x]}}$ are as follows. $$\begin{array}{ccccc}
\tikz[rotate=45,scale=.8]{\tyng(0cm,0cm,6,4^2,3,2^2)}&
\tikz[rotate=45,scale=.8]{\tyng(0cm,0cm,6,4^2,2^3)}&
\tikz[rotate=45,scale=.8]{\tyng(0cm,0cm,6,4,3^2,2^2)}&
\tikz[rotate=45,scale=.8]{\tyng(0cm,0cm,4^4,2^2)}\\
\la&
{\la^{[-1]}}=(6,4^2,2^3)&
{\la^{[1]}}=(6,4,3^2,2^2)&
{\la^{[5]}}=(4^4,2^2)
\end{array}$$
Cover-inclusive Dyck tilings {#cidtsec}
============================
In this section we examine [cover-inclusive Dyck tiling]{}s in more detail. We give some equivalent conditions to the cover-inclusive condition, and then we prove a recurrence for the number ${\operatorname{i}_{\la\mu}}$ of [cover-inclusive Dyck tiling]{}s.
Equivalent conditions
---------------------
\[citfae\] Suppose $\la$ and $\mu$ are partitions with $\la\supseteq\mu$, and $T$ is a Dyck tiling of ${\ensuremath{\la\backslash\mu}}$. The following are equivalent.
1. \[:ci\] $T$ is cover-inclusive.
2. \[:de\] If $\fka$ and ${\ensuremath{\mathtt{N}(\fka)}}$ are nodes of ${\ensuremath{\la\backslash\mu}}$, then ${\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{N}(\fka)}})}}\gs{\ensuremath{\operatorname{dp}(\fka)}}$.
3. \[:11\] If $\fka$ and ${\ensuremath{\mathtt{N}(\fka)}}$ are nodes of ${\ensuremath{\la\backslash\mu}}$, then ${\ensuremath{\operatorname{tile}(\fka)}}+(1,1)\subseteq{\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})}}$.
4. \[:nw\] If $\fka$ and ${\ensuremath{\mathtt{N}(\fka)}}$ are nodes of ${\ensuremath{\la\backslash\mu}}$ and $\fka$ is attached to ${\ensuremath{\mathtt{NW}(\fka)}}$, then [$\mathtt{NW}({\ensuremath{\mathtt{N}(\fka)}})$]{} is a node of ${\ensuremath{\la\backslash\mu}}$ and ${\ensuremath{\mathtt{N}(\fka)}}$ is attached to [$\mathtt{NW}({\ensuremath{\mathtt{N}(\fka)}})$]{}.
5. \[:en\] If $\fka$ and ${\ensuremath{\mathtt{N}(\fka)}}$ are nodes of ${\ensuremath{\la\backslash\mu}}$ and is the end node of its tile, then $\fka$ is the end node of its tile.
6. \[:rci\] $T$ is right-cover-inclusive.
7. \[:ne\] If $\fka$ and ${\ensuremath{\mathtt{N}(\fka)}}$ are nodes of ${\ensuremath{\la\backslash\mu}}$ and $\fka$ is attached to ${\ensuremath{\mathtt{NE}(\fka)}}$, then [$\mathtt{NE}({\ensuremath{\mathtt{N}(\fka)}})$]{} is a node of ${\ensuremath{\la\backslash\mu}}$ and ${\ensuremath{\mathtt{N}(\fka)}}$ is attached to [$\mathtt{NE}({\ensuremath{\mathtt{N}(\fka)}})$]{}.
8. \[:st\] If $\fka$ and ${\ensuremath{\mathtt{N}(\fka)}}$ are nodes of ${\ensuremath{\la\backslash\mu}}$ and is the start node of its tile, then $\fka$ is the start node of its tile.
9. \[:lci\] $T$ is left-cover-inclusive.
[(\[:ci\]$\Rightarrow$\[:de\])]{}
: Take $\fka\in{\ensuremath{\la\backslash\mu}}$ such that ${\ensuremath{\mathtt{N}(\fka)}}\in{\ensuremath{\la\backslash\mu}}$, and let $\fkc={\ensuremath{\operatorname{st}({\ensuremath{\operatorname{tile}(\fka)}})}}$. Since $T$ is cover-inclusive, there is a node $\fkb$ in [$\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})$]{} in the same column as $\fkc$, and $${\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{N}(\fka)}})}}\gs{\ensuremath{\operatorname{ht}(\fkb)}}-{\ensuremath{\operatorname{ht}({\ensuremath{\mathtt{N}(\fka)}})}}\gs{\ensuremath{\operatorname{ht}(\fkc)}}+2-({\ensuremath{\operatorname{ht}(\fka)}}+2)={\ensuremath{\operatorname{dp}(\fka)}}.$$
[(\[:de\]$\Rightarrow$\[:11\])]{}
: Suppose (\[:11\]) is false, and take $\fka\in{\ensuremath{\la\backslash\mu}}$ such that ${\ensuremath{\mathtt{N}(\fka)}}\in{\ensuremath{\la\backslash\mu}}$ and ${\ensuremath{\operatorname{tile}(\fka)}}+(1,1)\nsubseteq{\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})}}$. There is a node $\fkb\in{\ensuremath{\operatorname{tile}(\fka)}}$ such that $\fkb+(1,1)\notin{\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})}}$, and we may assume that $\fkb$ is attached to $\fka$; in fact, by symmetry, we may assume $\fkb$ is either or . If $\fkb={\ensuremath{\mathtt{NE}(\fka)}}$, then is attached to neither its NE nor its SE neighbour, so is the end node of its tile, and in particular ${\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{N}(\fka)}})}}=0$. On the other hand ${\ensuremath{\operatorname{dp}(\fka)}}>{\ensuremath{\operatorname{dp}(\fkb)}}\gs0$, contradicting (\[:de\]). If instead $\fkb={\ensuremath{\mathtt{SE}(\fka)}}$, then is attached to neither its NW nor its SW neighbour, so is the start node of its tile, and has depth $0$; but ${\ensuremath{\operatorname{dp}(\fkb)}}>{\ensuremath{\operatorname{dp}(\fka)}}\gs0$, and again (\[:de\]) is contradicted.
[(\[:11\]$\Rightarrow$\[:nw\])]{}
: This is trivial.
[(\[:nw\]$\Rightarrow$\[:en\])]{}
: Suppose (\[:en\]) is false, and take $\fka\in{\ensuremath{\la\backslash\mu}}$ as far to the left as possible such that ${\ensuremath{\mathtt{N}(\fka)}}\in{\ensuremath{\la\backslash\mu}}$ and is the end node of its tile, while $\fka$ is not the end node of its tile.
Since $\fka$ is not the end node of its tile, it is attached to either or . If $\fka$ is attached to , then is attached to its NW neighbour, but ${\ensuremath{\mathtt{N}({\ensuremath{\mathtt{SE}(\fka)}})}}={\ensuremath{\mathtt{NE}(\fka)}}{}$ is not attached to its NW neighbour, contradicting (\[:nw\]).
So assume that $\fka$ is attached to . This implies in particular that $\fka$ has positive depth.
1 If $\fkc\in{\ensuremath{\operatorname{tile}(\fka)}}$ and ${\ensuremath{\mathtt{N}(\fkc)}}\in{\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})}}$, then $\fkc$ is not the start node of . Since [$\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})$]{} is a Dyck tile and is its end node, we have ${\ensuremath{\operatorname{ht}({\ensuremath{\mathtt{N}(\fkc)}})}}\ls{\ensuremath{\operatorname{ht}({\ensuremath{\mathtt{N}(\fka)}})}}$. Hence ${\ensuremath{\operatorname{ht}(\fkc)}}\ls{\ensuremath{\operatorname{ht}(\fka)}}$, so ${\ensuremath{\operatorname{dp}(\fkc)}}\gs{\ensuremath{\operatorname{dp}(\fka)}}>0$.
2 For every node $\fkb\in{\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})}}$ we have ${\ensuremath{\mathtt{S}(\fkb)}}\in{\ensuremath{\operatorname{tile}(\fka)}}$. If the claim is false, let $\fkb$ be the rightmost counterexample. Obviously $\fkb\neq{\ensuremath{\mathtt{N}(\fka)}}$, and in particular $\fkb$ is not the end node of its tile, so $\fkb$ is attached to either or . In the first case, the choice of $\fkb$ means that we have ${\ensuremath{\mathtt{SE}(\fkb)}}\in{\ensuremath{\operatorname{tile}(\fka)}}$; neither $\fkb$ nor is in , so is the start node of , contradicting Claim 1.
So we can assume $\fkb$ is attached to [$\mathtt{SE}(\fkb)$]{}. The choice of $\fkb$ means that ${\ensuremath{\mathtt{S}({\ensuremath{\mathtt{SE}(\fkb)}})}}\in{\ensuremath{\operatorname{tile}(\fka)}}$ and is not attached to . By Claim 1 [$\mathtt{S}({\ensuremath{\mathtt{SE}(\fkb)}})$]{} is not the start node of , so is attached to [$\mathtt{S}({\ensuremath{\mathtt{S}(\fkb)}})$]{}. But now is not attached to either its NE or SE neighbour, so is the end node of its tile. [$\mathtt{S}({\ensuremath{\mathtt{S}(\fkb)}})$]{} is not the end node of its tile, and this contradicts the choice of $\fka$.
Now let $\fkb={\ensuremath{\operatorname{st}({\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})}})}}$. Then by Claim 2, $\fkc:={\ensuremath{\mathtt{S}(\fkb)}}\in{\ensuremath{\operatorname{tile}(\fka)}}$, and by Claim 1 $\fkc$ is not the start node of . So $\fkc$ is attached to either or . The first possibility contradicts (\[:nw\]), since $\fkb$ is not attached to , so assume that $\fkc$ is attached to . But then [$\mathtt{NW}(\fkc)$]{} is attached to neither $\fkb$ nor $\fkc$, so is the end node of its tile, while is not the end node of its tile, and this contradicts the choice of $\fka$.
[(\[:en\]$\Rightarrow$\[:rci\])]{}
: Suppose $T$ is not right-cover-inclusive, and take $\fka\in{\ensuremath{\la\backslash\mu}}$ such that ${\ensuremath{\mathtt{N}(\fka)}}\in{\ensuremath{\la\backslash\mu}}$ and ends strictly to the right of [$\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})$]{}. Let $\fkb={\ensuremath{\operatorname{en}({\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})}})}}$; then there is a node in in the same column as $\fkb$, which we can write as $\fkb-(h,h)$ for some $h>0$. If we let $i\in\{1,\dots,h\}$ be minimal such that $\fkb-(i,i)$ is not the end node of its tile, then ${\ensuremath{\mathtt{N}(\fkb-(i,i))}}$ is the end node of its tile, contradicting (\[:en\]).
[(\[:rci\]$\Rightarrow$\[:ne\])]{}
: Suppose $\fka,{\ensuremath{\mathtt{N}(\fka)}}\in{\ensuremath{\la\backslash\mu}}$ and $\fka$ is attached to . Then [$\operatorname{en}({\ensuremath{\operatorname{tile}(\fka)}})$]{} lies to the right of $\fka$, so by (\[:rci\]) [$\operatorname{en}({\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})}})$]{} does too. So is attached to either [$\mathtt{NE}({\ensuremath{\mathtt{N}(\fka)}})$]{} or [$\mathtt{SE}({\ensuremath{\mathtt{N}(\fka)}})$]{}. But ${\ensuremath{\mathtt{SE}({\ensuremath{\mathtt{N}(\fka)}})}}={\ensuremath{\mathtt{NE}(\fka)}}$ is attached to $\fka$, so is attached to [$\mathtt{NE}({\ensuremath{\mathtt{N}(\fka)}})$]{}.
[(\[:ne\]$\Rightarrow$\[:st\])]{}
: This is symmetrical to the argument that \[:nw\]$\Rightarrow$\[:en\].
[(\[:st\]$\Rightarrow$\[:lci\])]{}
: This is symmetrical to the argument that \[:en\]$\Rightarrow$\[:rci\].
[(\[:lci\]$\Rightarrow$\[:ci\])]{}
: Since \[:rci\]$\Rightarrow$\[:ne\]$\Rightarrow$\[:st\]$\Rightarrow$\[:lci\], right-cover-inclusive implies left-cover-inclusive. Symmetrically, left-cover-inclusive implies right-cover-inclusive, and hence cover-inclusive.
We shall use these equivalent definitions of cover-inclusiveness, often without comment, in what follows. We also observe the following property of [cover-inclusive Dyck tiling]{}s, which we shall use without comment.
Suppose $T$ is a [cover-inclusive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$. If $\fkn\in{\ensuremath{\la\backslash\mu}}$ is the highest node in its column in ${\ensuremath{\la\backslash\mu}}$, then every node in is the highest node in its column in ${\ensuremath{\la\backslash\mu}}$.
Suppose not, and take $\fkm\in{\ensuremath{\operatorname{tile}(\fkn)}}$ which is not the highest node in its column. Without loss of generality we may assume $\fkm$ lies in the column immediately to the right of $\fkn$, i.e. $\fkm$ is either or . But if $\fkm={\ensuremath{\mathtt{NE}(\fkn)}}$, then the assumption ${\ensuremath{\mathtt{N}(\fkm)}}\in{\ensuremath{\la\backslash\mu}}$ means that ${\ensuremath{\mathtt{N}(\fkn)}}\in{\ensuremath{\la\backslash\mu}}$ (otherwise ${\ensuremath{\la\backslash\mu}}$ would not be a skew Young diagram), a contradiction. So assume $\fkm={\ensuremath{\mathtt{SE}(\fkn)}}$. But now $\fkm$ is attached to ${\ensuremath{\mathtt{NW}(\fkm)}}$ and ${\ensuremath{\mathtt{N}(\fkm)}}\in{\ensuremath{\la\backslash\mu}}$, so by \[citfae\](\[:nw\]) ${\ensuremath{\mathtt{N}(\fkn)}}\in{\ensuremath{\la\backslash\mu}}$, contradiction.
Recurrences
-----------
Now we consider recurrences. We start with a simple result.
\[noadd\] Suppose $\la$ and $\mu$ are partitions and $j\in\bbz$, and that $\mu$ has an addable node $\fkm$ in column $j$, but $\la$ does not have an addable node in column $j$. Let $\mu^+=\mu\cup\{\fkm\}$. Then ${\operatorname{i}_{\la\mu}}={\operatorname{i}_{\la\mu^+}}$.
The fact that $\la$ does not have an addable node in column $j$ implies that $\la\supseteq\mu$ if and only if $\la\supseteq\mu^+$, so we may as well assume that both of these conditions hold. Given a [cover-inclusive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu^+}}$, we can obtain a tiling of ${\ensuremath{\la\backslash\mu}}$ simply by adding the singleton tile $\{\fkm\}$, and it is clear that this tiling is a [cover-inclusive Dyck tiling]{}.
In the other direction, suppose $T$ is a [cover-inclusive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$; then we claim that $\fkm$ forms a singleton tile. If we let $\fkn$ denote the highest node in column $j$ of ${\ensuremath{\la\backslash\mu}}$, then (since $\la$ does not have an addable node in column $j$) and are not both nodes of ${\ensuremath{\la\backslash\mu}}$; suppose without loss that ${\ensuremath{\mathtt{NE}(\fkn)}}\notin{\ensuremath{\la\backslash\mu}}$. Then in particular $\fkn$ is not attached to in $T$, and so (using \[citfae\](\[:ne\])) $\fkm$ is not attached to . $\fkm$ cannot be attached to or (since these are not nodes of ${\ensuremath{\la\backslash\mu}}$), and a node in a Dyck tile cannot be attached only to its NW neighbour. So $\fkm$ is not attached to any of its neighbours, i.e. $\{\fkm\}$ is a singleton tile as claimed. We can remove this tile, and we clearly obtain a [cover-inclusive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu^+}}$. So we have a bijection between ${\scri(\la,\mu)}$ and ${\scri(\la,\mu^+)}$.
Now we prove a more complicated result, for which it will help us to fix some notation.
**Notation in force for the remainder of Section \[cidtsec\]:**
$j$ is a fixed integer, and $\la,\mu$ are partitions such that $\la$ has an addable node $\fkl$ in column $j$, while $\mu$ has an addable node $\fkm$ in column $j$. We define ${\la^+}=\la\cup\{\fkl\}$ and $\mu^+=\mu\cup\{\fkm\}$.
If $T\in{\scri(\la,\mu^+)}$; then ${{\scriptstyle\overrightarrow{\displaystyle T}}}$ denotes the size of the highest tile starting in column $j+1$, and ${{\scriptstyle\overleftarrow{\displaystyle T}}}$ denotes the size of the highest tile ending in column $j-1$.
Note that if $T\in{\scri(\la,\mu^+)}$, then there must be at least one tile starting in column $j+1$, since there are more nodes in column $j+1$ of ${\ensuremath{\la\backslash\mu^+}}$ than in column $j$; so ${{\scriptstyle\overrightarrow{\displaystyle T}}}$ is well-defined, and similarly ${{\scriptstyle\overleftarrow{\displaystyle T}}}$ is well-defined.
To prove our main recurrence result for the numbers ${\operatorname{i}_{\la\mu}}$, we give three results in which we construct bijections between sets of [cover-inclusive Dyck tiling]{}s. The first of these is as follows.
\[bij1\] $$\left|\rset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overrightarrow{\displaystyle T}}}\notin{X_{\la}}}\right|=\left|\rset{T\in{\scri(\la^+,\mu^+)}}{\textup{there is a big tile in $T$ starting in column }j}\right|.$$
Note first that if $\la^+\nsupseteq\mu^+$ then $\la\nsupseteq\mu^+$, so both sides equal zero. Conversely, if $\la\nsupseteq\mu^+$, then either $\la^+\nsupseteq\mu^+$ or there are no nodes in column $j$ of ${\ensuremath{\la^+\backslash\mu^+}}$, so again both sides are zero. So we may assume that $\la\supseteq\mu^+$. We’ll construct a bijection $$\phi_1:\rset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overrightarrow{\displaystyle T}}}\notin{X_{\la}}}\longrightarrow\rset{T\in{\scri(\la^+,\mu^+)}}{\textup{there is a big tile in $T$ starting in column }j}.$$ Given $T$ in the domain, let $t$ be the highest tile in $T$ starting in column $j+1$, and let $A$ be the set of nodes in column $j$ which are higher than t. No node in column $j+1$ can be attached to its NW neighbour, because the highest node in this column (namely ) is not, so (by the choice of $t$) every node $\fkb$ higher than t in column $j+1$ is attached to ; hence each $\fka\in A$ is attached to . In a Dyck tiling a node cannot be attached only to its NE neighbour, so each $\fka\in A$ is attached to either its NW or SW neighbour. But a node in column $j-1$ cannot be attached to its NE neighbour (because the highest node in column $j-1$ is not), so every $\fka\in A$ is attached to ${\ensuremath{\mathtt{NW}(\fka)}}$.
Now we consider columns $x$ and $x+1$, where $x={{\scriptstyle\overrightarrow{\displaystyle T}}}+j=|t|+j$. Let $\fkd={\ensuremath{\mathtt{SE}(\fkl)}}$. Then $\fkd={\ensuremath{\operatorname{st}(t)}}+(h,h)$ for some $h\gs0$, so by \[citfae\](\[:11\]) contains $t+(h,h)$. In particular, includes the node $\fke={\ensuremath{\operatorname{en}(t)}}+(h,h)$ in column $x$, and (if ${\ensuremath{\operatorname{st}(t)}}\neq{\ensuremath{\operatorname{en}(t)}}$) includes . Since $\fkd$ is the highest node in its column, the same is true for every node in , and in particular neither nor is a node of $\la$. However, $\fke$ cannot be a removable node of $\la$, since then ${{\scriptstyle\overrightarrow{\displaystyle T}}}=|t|$ would lie in ${X_{\la}}$. So is a node of ${\ensuremath{\la\backslash\mu}}$. Now is not attached to its NW neighbour (since this is ${\ensuremath{\mathtt{N}(\fke)}}\notin\la$), so no node in column $x+1$ is attached to its NW neighbour, and so no node in column $x$ is attached to its SE neighbour. This implies that any node in column $x$ of positive depth must be attached to its NE neighbour, while any node in column $x$ of depth $0$ is the end node of its tile, and this tile has a NE neighbour. In particular, $t$ has a NE neighbour.
Now suppose $\fka\in A$ has depth $1$, and write $\fka={\ensuremath{\operatorname{st}(t)}}+(i,i-1)$ for $i>0$. Then contains ${\ensuremath{\operatorname{st}(t)}}+(i,i)$, and hence contains ${\ensuremath{\operatorname{en}(t)}}+(i,i)$. Since $\fka$ has depth $1$, ${\ensuremath{\operatorname{en}(t)}}+(i,i)$ has depth $0$, and so from the last paragraph ${\ensuremath{\operatorname{en}(t)}}+(i,i)$ is the end node of , and has a NE neighbour.
Now we can construct $\phi_1(T)$, as follows.
- Add the node $\fkl$ to the tiling (as a singleton tile).
- For each $\fka\in A$:
- if $\fka$ has depth $1$ in $T$, attach to its NE neighbour;
- detach $\fka$ from and ;
- attach $\fka$ to ;
- if $\fka$ has depth at least $2$ in $T$, attach to ;
- Attach $t$ to is NE neighbour.
- Finally, attach $\fkl$ to .
Now let’s check that $\phi_1(T)$ is a [cover-inclusive Dyck tiling]{}. First we check that every tile is a Dyck tile. Passing from $T$ to $\phi_1(T)$, the tiles change in the following ways:
- given a tile $u\in T$ containing a node $\fka\in A$ of depth at least $2$, $\phi_1(T)$ contains the tile obtained by replacing $\fka$ with in $u$;
- given a tile $u$ containing a node $\fka\in A$ of depth $1$, $u$ has a NE neighbour $v$; $\phi_1(T)$ contains the tile consisting of the portion of $u$ ending at , and another tile obtained by combining , the portion of $u$ starting at , and $v$;
- let $v$ denote the NE neighbour of $t$; then $\phi_1(T)$ contains the tile obtained by joining [$\mathtt{NW}({\ensuremath{\operatorname{st}(t)}})$]{} to $t$ and $v$.
Clearly, all these new tiles are Dyck tiles. It’s also clear that the last tile mentioned is a big tile starting in column $j$. So to see that $\phi_1(T)$ lies in the codomain, all that remains is to check that $\phi_1(T)$ is cover-inclusive. But this is easy using (for example) \[citfae\](\[:nw\]).
As an example, we give a [cover-inclusive Dyck tiling]{} $T$ for $\la=(12^2,10,9,6^2,5,4,3^3)$, $\mu=(1)$ and $j=-1$, and its image under $\phi_1$; the nodes $\fkl$ and $\fkm$ are marked, and the nodes in $A$ are shaded and labelled with their depths.
$$\begin{array}{c@{\qquad}c}
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\draw(8,10)--++(1,-1)--++(1,1);
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\draw(-3,7)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,-1)
--++(1,1)--++(1,1)--++(1,1);
\draw(4,10)++(1,-1)--++(1,-1)--++(1,1)--++(1,1);
\draw(-9,13)--++(1,-1)--++(1,-1)--++(1,-1)--++(1,-1)--++(1,1)
--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)--++(1,-1)
--++(1,-1)--++(1,1)--++(1,1)--++(1,1)--++(1,-1)
--++(1,1)--++(1,-1)--++(1,1)--++(1,1)--++(1,1);
\draw(-7,9)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)
--++(1,-1)--++(1,1)--++(1,-1)--++(1,-1)--++(1,1)
--++(1,1)--++(1,1);
\draw(-7,7)--++(1,-1)--++(1,-1)--++(1,1)--++(1,1);
\draw(12,12)--++(-1,1)--++(-1,1)--++(-2,-2)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-3,-3)--++(-1,1)--++(-1,1)
--++(-1,-1)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-1,1)--++(-1,1)
--++(-3,-3);
\draw(-1,2)node{$\fkm$};
\draw(-1,12)node{$\fkl$};
\draw(-1,6)node{1};
\draw(-1,8)node{2};
\draw(-1,10)node{3};
\end{tikzpicture}
&
\begin{tikzpicture}[scale=0.31]
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\draw(-9,9)--++(1,-1)--++(1,1);
\draw(-8,10)--++(1,-1)++(1,1);
\draw(-10,10)--++(1,-1)--++(1,1);
\draw(-9,11)--++(1,-1)--++(1,1);
\draw(-11,11)--++(1,-1)--++(1,1);
\draw(-10,12)--++(1,-1)--++(1,1);
\draw(-1,5)--++(1,-1)--++(1,-1)--++(1,-1)--++(1,1)--++(1,1)--++(1,1);
\draw(-3,7)--++(1,-1)--++(1,-1)++(1,1)--++(1,-1)--++(1,-1)
--++(1,1)--++(1,1)--++(1,1);
\draw(4,10)++(1,-1)--++(1,-1)--++(1,1)--++(1,1);
\draw(-9,13)--++(1,-1)--++(1,-1)--++(1,-1)--++(1,-1)--++(1,1)
--++(1,-1)--++(1,1)++(1,-1)++(1,1)--++(1,-1)
--++(1,-1)--++(1,1)--++(1,1)--++(1,1)--++(1,-1)
--++(1,1)--++(1,-1)--++(1,1)--++(1,1)--++(1,1);
\draw(-7,9)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)
--++(1,-1)++(1,1)--++(1,-1)--++(1,-1)--++(1,1)
--++(1,1)--++(1,1);
\draw(-7,7)--++(1,-1)--++(1,-1)--++(1,1)--++(1,1);
\draw(12,12)--++(-1,1)--++(-1,1)--++(-2,-2)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-3,-3)--++(-1,1)--++(-1,1)
--++(-1,1)--++(-1,-1)--++(-1,-1)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-1,1)--++(-1,1)
--++(-3,-3);
\draw(-2,6)--++(1,1)--++(1,-1);
\draw(-2,8)--++(1,1)--++(1,-1);
\draw(-2,10)--++(1,1)--++(1,-1);
\draw(-1,2)node{$\fkm$};
\draw(-1,12)node{$\fkl$};
\end{tikzpicture}\\
T&\phi_1(T)
\end{array}$$
To show that $\phi_1$ is a bijection, we construct the inverse map $$\psi_1:\rset{T\in{\scri(\la^+,\mu^+)}}{\textup{there is a big tile in $T$ starting in column }j}\longrightarrow\rset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overrightarrow{\displaystyle T}}}\notin{X_{\la}}}.$$ Suppose $T$ is a [cover-inclusive Dyck tiling]{} of ${\ensuremath{\la^+\backslash\mu^+}}$ in which there is at least one big tile starting in column $j$. Since the highest node in column $j$ (namely $\fkl$) is not attached to its NE or NW neighbour, no node in column $j$ is attached to its NE or NW neighbour. So from top to bottom, column $j$ of $T$ consists of:
- (possibly) some nodes attached to both their SE and SW neighbours;
- at least one node attached only to its SE neighbour;
- (possibly) some singleton tiles.
Construct $\psi_1(T)$ as follows.
- For each node $\fka$ in column $j$ which is attached to :
- if $\fka$ is not attached to , let $\fkb$ be the first node in to the right of column $j$ which has the same height as $\fka$, and detach $\fkb$ from ;
- detach $\fka$ from ;
- detach $\fka$ from (if it is attached);
- (if $\fka\neq\fkl$) attach $\fka$ to and .
- Remove $\fkl$.
Let’s check that $\psi_1(T)$ is a Dyck tiling. The tiles have changed in the following ways:
- if $u\in T$ contains a node $\fka$ in column $j$ and also contains and , then $\psi_1(T)$ contains the tile obtained by replacing $\fka$ with in $u$;
- if $u\in T$ is a big tile starting at a node $\fka$ in column $j$, let $\fkb$ be the first node of $u$ to the right of column $j$ which has the same height as $\fka$. Then:
- $\psi_1(T)$ contains the tile consisting of the portion of $u$ starting at $\fkb$;
- if $u$ is the lowest big tile in $T$ starting in column $j$, then $\psi_1(T)$ also contains the tile consisting of the portion of $u$ running from ${\ensuremath{\mathtt{SE}(\fka)}}$ to ${\ensuremath{\mathtt{SW}(\fkb)}}$;
- if $u$ is not the lowest big tile starting in column $j$, then $\psi_1(T)$ contains the tile constructed by joining together the SE neighbour of $u$, the node ${\ensuremath{\mathtt{S}(\fka)}}$ and the portion of $u$ running from ${\ensuremath{\mathtt{SE}(\fka)}}$ to ${\ensuremath{\mathtt{SW}(\fkb)}}$.
Again, we can see that these are all Dyck tiles, and checking that $\psi_1(T)$ is cover-inclusive is straightforward using \[citfae\](\[:nw\]).
So $\psi_1(T)$ is a [cover-inclusive Dyck tiling]{}. It remains to check that ${{\scriptstyle\overrightarrow{\displaystyle \psi_1(T)}}}\notin{X_{\la}}$. Let $t$ be the highest tile in $\psi_1(T)$ starting in column $j+1$; then $t$ starts at , where $\fka$ is the lowest node in column $j$ which is the start of a big tile in $T$, and ends at , where $\fkb$ is the first node in to the right of column $j$ with the same height as $\fka$. In $T$, is attached to $\fkb$, and so every node above in the same column is attached to its NE neighbour in $T$. In particular, the highest node in this column has a NE neighbour in $\la$, and so is not removable; so ${{\scriptstyle\overrightarrow{\displaystyle \psi_1(T)}}}\notin{X_{\la}}$.
So our two maps $\phi_1,\psi_1$ really do map between the specified sets. It is easy to see from the construction that they are mutual inverses.
Symmetrically, we have the following result.
\[bij1a\] $$\left|\rset{T\in{\scri(\la,\mu^+)}}{-{{\scriptstyle\overleftarrow{\displaystyle T}}}\notin{X_{\la}}}\right|=\left|\rset{T\in{\scri(\la^+,\mu^+)}}{\textup{there is a big tile in $T$ ending in column }j}\right|.$$
Our next bijective result is the following.
\[bij2\] For each $x\in{{X_{\la}}^+}$, $$\left|\lset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overrightarrow{\displaystyle T}}}=x}\right|={\operatorname{i}_{{\la^{[x]}}\mu^+}}.$$
Fix $x\in{{X_{\la}}^+}$. We define a bijection $$\phi_2:\lset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overrightarrow{\displaystyle T}}}=x}\longrightarrow{\scri({\la^{[x]}},\mu^+)}.$$ Since $x\in{{X_{\la}}^+}$, there is a Dyck tile $\rho\subset\la$ consisting of the highest nodes in columns $j+1,\dots,j+x$ of $\la$. Now take $T\in{\scri(\la,\mu^+)}$ with ${{\scriptstyle\overrightarrow{\displaystyle T}}}=x$, and let $t$ be the highest tile in $T$ starting in column $j+1$. Then every northward translate of $t$ is an interval in a tile; in particular, [$\operatorname{tile}({\ensuremath{\mathtt{SE}(\fkl)}})$]{} contains a translate $t+(h,h)$ for some $h\gs0$; since is the highest node in its column, the same is true for every node in [$\operatorname{tile}({\ensuremath{\mathtt{SE}(\fkl)}})$]{}, so the translate $t+(h,h)$ coincides with $\rho$.
Consider the nodes in column $j$ of ${\ensuremath{\la\backslash\mu^+}}$. Arguing as in the proof of \[bij1\], every node $\fka$ in column $j$ which is higher than $t$ is attached to and .
Next we consider nodes in column $j+x$. The highest node in column $j+x$ of ${\ensuremath{\la\backslash\mu^+}}$ is the end node of $\rho$, and so is not attached to its NE neighbour (since this node is not in $\la$); hence no node in column $j+x$ is attached to its NE neighbour. So no node in column $j+x+1$ is attached to its SW neighbour. So from top to bottom , the nodes in column $j+x+1$ higher than t comprise:
- (possibly) some nodes attached to their NW neighbours;
- (possibly) some nodes attached to neither their NW nor SW neighbours.
Now we can construct $\phi_2(T)$, as follows.
- For each pair $(\fka,\fkd)$ of nodes with $\fka$ in column $j$, $\fkd$ in column $j+x+1$ and ${\ensuremath{\operatorname{ht}(\fka)}}={\ensuremath{\operatorname{ht}(\fkd)}}>{\ensuremath{\operatorname{ht}(t)}}$:
- detach $\fka$ from ;
- attach $\fka$ to ;
- if $\fkd$ is not attached to , then detach $\fka$ from ;
- detach $\fkd$ from (if attached);
- attach $\fkd$ to .
- Remove $\rho$.
Let’s check that $\phi_2(T)$ is a Dyck tiling. The new tiles are as follows. For every pair $(\fka,\fkd)$ as above,
- if $\fkd$ is attached to , then $\fka$ and $\fkd$ lie in the same tile in $T$, and $\phi_2(T)$ contains this tile but with the portion lying in columns $j+1$ to $j+x$ translated downwards;
- if $\fkd$ is not attached to , then ${\ensuremath{\operatorname{en}({\ensuremath{\operatorname{tile}(\fka)}})}}={\ensuremath{\mathtt{NW}(\fkd)}}$, and $\phi_2(T)$ contains the portion of ending at , and the tile constructed by joining the portion of [$\operatorname{tile}({\ensuremath{\mathtt{SE}(\fka)}})$]{} between columns $j+1$ and $j+x$ to $\fka$ and .
It is easy to see that these tiles are Dyck tiles, and the cover-inclusive property follows very easily from the cover-inclusive property for $T$ and the construction.
We give an example with $\la=(13^2,11,9^2,8,7,6,5^2,2^2)$, $\mu^+=(2,1)$, $s=-1$ and $x=5$. The nodes $\fkl$ and $\fkm$ are marked, and the nodes lying above $t$ in columns $s$ and $s+x+1$ are shaded.
$$\begin{array}{c@\qquad c}
\begin{tikzpicture}[scale=0.3]
\foreach\x in{-2,4}\foreach\y in{4,6,8,10}\path[fill=black!10!white](\x,\y)--++(1,1)--++(-1,1)--++(-1,-1)--cycle;
\draw(-1,3)--++(-2,-2)--++(-10,10)--++(1,1);
\foreach\x in{4,5,6,7,8,9,12}\draw(-\x,\x-2)--++(1,1);
\draw(-2,2)--++(-1,1);
\draw(0,2)--++(1,-1)--++(1,1);
\draw(-2,2)--++(1,-1)--++(1,1);
\draw(3,5)--++(1,-1)--++(1,1);
\draw(6,8)--++(1,-1)--++(1,1);
\draw(7,9)--++(1,-1)--++(1,1);
\draw(8,10)--++(1,-1)--++(1,1);
\draw(9,11)--++(1,-1)--++(1,1);
\draw(10,12)--++(1,-1)--++(1,1);
\draw(-4,4)--++(1,-1)--++(1,1);
\draw(-5,5)--++(1,-1)--++(1,1);
\draw(-6,6)--++(1,-1)--++(1,1);
\draw(-7,7)--++(1,-1)--++(1,1);
\draw(-8,8)--++(1,-1)--++(1,1);
\draw(-7,9)--++(1,-1)--++(1,1);
\draw(-9,9)++(1,-1)--++(1,1);
\draw(-8,10)++(1,-1)--++(1,1);
\draw(-7,11)--++(1,-1)--++(1,1);
\draw(-2,4)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)--++(1,1);
\draw(3,7)--++(1,-1)--++(1,-1)--++(1,1)--++(1,1);
\draw(-4,6)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)
--++(1,-1)--++(1,1)--++(1,1);
\draw(-6,10)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)
--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)--++(1,-1)
--++(1,-1)--++(1,1)--++(1,1)--++(1,1);
\draw(-7,13)--++(1,-1)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)
--++(1,1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)
--++(1,-1)--++(1,-1)--++(1,1)--++(1,1)--++(1,-1)
--++(1,1)--++(1,1)--++(1,1);
\draw(-6,8)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)
--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)--++(1,1);
\draw(-12,12)--++(1,-1)--++(1,-1)--++(1,-1)--++(1,1)--++(1,1)--++(1,1);
\draw(12,12)--++(-1,1)--++(-1,1)--++(-2,-2)--++(-1,1)--++(-2,-2)--++(-1,1)--++(-1,1)--++(-1,-1)--++(-1,1)
--++(-1,-1)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-1,-1)--++(-1,1)--++(-1,1)--++(-3,-3)--++(-1,1)
--++(-1,1)--++(-1,-1);
\draw(-2,1)node{$\fkm$};
\draw(-2,13)node{$\fkl$};
\end{tikzpicture}
&
\begin{tikzpicture}[scale=0.3]
\draw(-2,1)node{$\fkm$};
\draw(-2,13)node{$\fkl$};
\foreach\x in{-2,4}\foreach\y in{4,6,8,10}\path[fill=black!10!white](\x,\y)--++(1,1)--++(-1,1)--++(-1,-1)--cycle;
\draw(0,2)--++(1,-1)--++(1,1);
\draw(-1,3)--++(-2,-2)--++(-10,10)--++(1,1);
\foreach\x in{4,5,6,7,8,9,12}\draw(-\x,\x-2)--++(1,1);
\draw(-2,2)--++(-1,1);
\draw(-2,2)--++(1,-1)--++(1,1);
\draw(3,5)++(1,-1)--++(1,1);
\draw(6,8)--++(1,-1)--++(1,1);
\draw(7,9)--++(1,-1)--++(1,1);
\draw(8,10)--++(1,-1)--++(1,1);
\draw(9,11)--++(1,-1)--++(1,1);
\draw(10,12)--++(1,-1)--++(1,1);
\draw(-4,4)--++(1,-1)--++(1,1);
\draw(-5,5)--++(1,-1)--++(1,1);
\draw(-6,6)--++(1,-1)--++(1,1);
\draw(-7,7)--++(1,-1)--++(1,1);
\draw(-8,8)--++(1,-1)--++(1,1);
\draw(-7,9)--++(1,-1)--++(1,1);
\draw(-9,9)++(1,-1)--++(1,1);
\draw(-8,10)++(1,-1)--++(1,1);
\draw(-7,11)--++(1,-1)--++(1,1);
\foreach\y in{5,7}\draw(-3,\y)--++(1,1)--++(1,-1);
\foreach\y in{10,12}\draw(-2,\y)--++(1,-1);
\draw(3,9)--++(1,1);
\draw(3,11)--++(1,1);
\draw(-2,4)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)--++(1,1);
\draw(3,7)++(1,-1)--++(1,-1)--++(1,1)--++(1,1);
\draw(-4,6)--++(1,-1)--++(1,-1)++(1,1)--++(1,-1)--++(1,1)
--++(1,-1)--++(1,1)--++(1,1);
\draw(-6,10)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)++(1,1)
--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)++(1,-1)
--++(1,-1)--++(1,1)--++(1,1)--++(1,1);
\draw(-7,13)--++(1,-1)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)
++(1,1)--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)
++(1,-1)--++(1,-1)--++(1,1)--++(1,1)--++(1,-1)
--++(1,1)--++(1,1)--++(1,1);
\draw(-6,8)--++(1,-1)--++(1,-1)--++(1,1)--++(1,-1)++(1,1)
--++(1,-1)--++(1,1)--++(1,-1)--++(1,1)--++(1,1);
\draw(-12,12)--++(1,-1)--++(1,-1)--++(1,-1)--++(1,1)--++(1,1)--++(1,1);
\draw(12,12)--++(-1,1)--++(-1,1)--++(-2,-2)--++(-1,1)--++(-2,-2)--++(-1,1)++(-6,0)
--++(-1,1)--++(-1,-1)--++(-1,1)--++(-1,1)--++(-3,-3)--++(-1,1)
--++(-1,1)--++(-1,-1);
\end{tikzpicture}\\
T&\phi_2(T)
\end{array}$$ Now we construct the inverse map $$\psi_2:{\scri({\la^{[x]}},\mu^+)}\longrightarrow\lset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overrightarrow{\displaystyle T}}}=x}.$$ Suppose $T$ is a [cover-inclusive Dyck tiling]{} of ${\ensuremath{{\la^{[x]}}\backslash\mu^+}}$. No node in column $j-1$ or column $j$ of ${\la^{[x]}}$ can be attached to its NE neighbour (since the NE neighbours of the highest nodes in these columns are not nodes of ${\la^{[x]}}$). Hence from top to bottom the nodes in column $j$ comprise:
- (possibly) some nodes attached to their NW and SE neighbours;
- (possibly) some nodes attached only to their SE neighbours;
- (possibly) some singleton nodes.
Note that if $\fka$ is a node in column $j$ which is not attached to , then is the end node of its tile; in particular, has a NW neighbour.
Suppose $r\gs1$, and the node $\fka:=\fkl-(r,r)$ is attached to . Then reaches column $j+x+1$ and includes all the nodes in $\rho-(r,r)$. We proceed by induction on $r$. First suppose $r=1$. Since $x\in{X_{\la}}$, every node in columns $j+1,\dots,j+x$ of $\la$ has height less than . Hence every node in columns $j+1,\dots,j+x$ of ${\la^{[x]}}$ has height less than ${\ensuremath{\operatorname{ht}(\fkl)}}-2={\ensuremath{\operatorname{ht}(\fka)}}$; since is a Dyck tile, it must reach at some point to the right of , and so must reach column $x+j+1$. Furthermore, $\fka$ is the highest node in its column, so every node in is the highest in its column, and in particular the portion of between columns $j+1$ and $j+x$ must consist of the highest nodes in these columns, i.e. the nodes in $\rho-(1,1)$.
Now suppose $r>1$. By induction cannot include any of the nodes in $\rho-(v,v)$ for any $v<r$; all the remaining nodes in columns $j+1,\dots,j+r$ have height less than , and so (since must reach at some point to the right of ) must reach column $j+x+1$. The translate ${\ensuremath{\operatorname{tile}(\fka)}}+(1,1)$ is contained in [$\operatorname{tile}({\ensuremath{\mathtt{N}(\fka)}})$]{}, and by induction this includes the nodes in $\rho-(r-1,r-1)$; so includes the nodes in $\rho-(r,r)$.
Now we consider the nodes in column $j+x+1$. If $\fkd$ is a node in column $j+x+1$, then by similar arguments to those used above, $\fkd$ cannot be attached to , and if $\fkd$ is attached to , then includes a node in column $j$ and also includes all the nodes in $\rho-(r,r)$, where $r=\frac12({\ensuremath{\operatorname{ht}(\fkl)}}-{\ensuremath{\operatorname{ht}(\fkd)}})$.
So we find that, if $\fka$ and $\fkd$ are nodes in columns $j$ and $j+x+1$ respectively with the same height, then $\fka$ is attached to if and only if $\fkd$ is attached to , and that in this case ${\ensuremath{\operatorname{tile}(\fka)}}={\ensuremath{\operatorname{tile}(\fkd)}}$. Say that such a pair $(\fka,\fkd)$ is a *connected* pair. Now we can construct $\psi_2(T)$ as follows.
- Add $\rho$ as a tile.
- For each connected pair $(\fka,\fkd)$:
- detach $\fkd$ from ;
- if $\fka$ is attached to , then attach $\fkd$ to ;
- detach $\fka$ from ;
- attach $\fka$ to ;
- attach $\fka$ to (if it is not already attached).
To check that the resulting tiling is a Dyck tiling, we examine the new tiles:
- for each connected pair $(\fka,\fkd)$ with $\fka=\fkl-(r,r)$ and $\fka$ attached to , $\psi_2(T)$ contains the tile obtained from by replacing the segment $\rho-(r,r)$ with $\rho-(r-1,r-1)$;
- for each connected pair $(\fka,\fkd)$ with $\fka=\fkl-(r,r)$ and $\fka$ not attached to , $\psi_2(T)$ contains the tile comprising $\fka$, the NW neighbour of and $\rho-(r-1,r-1)$, and another tile consisting of the portion of starting at $\fkd$;
- $\psi_2(T)$ contains the tile $\rho-(r,r)$, where $r$ is the number of connected pairs.
Once more, we can see that these are all Dyck tiles, and the cover-inclusive property is easy to check from the construction. Furthermore, the last tile mentioned above is the highest tile starting in column $j+1$, and has size $x$.
So $\psi_2$ really does map ${\scri({\la^{[x]}},\mu^+)}$ to $\lset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overrightarrow{\displaystyle T}}}=x}$. And it is easy to see that $\phi_2$ and $\psi_2$ are mutual inverses.
Symmetrically, we have the following.
\[bij2a\] For each $x\in{{X_{\la}}^-}$, $$\left|\lset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overleftarrow{\displaystyle T}}}=-x}\right|={\operatorname{i}_{{\la^{[x]}}\mu^+}}.$$
Our third bijection involves the partition ${\la^+}$.
\[bij3\] $$\left|\rset{T\in{\scri(\la,\mu)}}{\parbox{85pt}{\rm$\fkm$ does not lie in a singleton tile of $T$}}\right|=\left|\rset{T\in{\scri(\la^+,\mu^+)}}{\parbox{129pt}{\rm there is no big tile in $T$ starting or ending in column $j$}}\right|.$$
First observe that $\la\supseteq\mu$ if and only if $\la^+\supseteq\mu^+$, so we may as well assume that both these conditions hold. Next note that if $\fkl=\fkm$, then ${\ensuremath{\la\backslash\mu}}={\ensuremath{\la^+\backslash\mu^+}}$, and this skew Young diagram contains no nodes in column $j$; in particular, it does not include $\fkm$. So in this case, \[bij3\] amounts to the trivial statement ${\operatorname{i}_{\la\mu}}={\operatorname{i}_{\la^+\mu^+}}$.
So we assume that $\la\supset\mu$ and $\fkl\neq\fkm$; so $\fkm$ is a node of ${\ensuremath{\la\backslash\mu}}$. Now we want to construct a bijection $$\phi_3:\lset{T\in{\scri(\la,\mu)}}{\fkm\textup{ lies in a big tile in $T$ }}\longrightarrow\rset{T\in{\scri(\la^+,\mu^+)}}{\parbox{129pt}{there is no big tile in $T$ starting or ending in column $j$}}.$$
Suppose we have $T\in{\scri(\la,\mu)}$ with $\fkm$ lying in a big tile. Then $\fkm$ is attached to both and , and hence every node $\fka$ in column $j$ is attached to both and . In particular, every node in column $j$ has depth at least $1$. Now we construct $\phi_3(T)$ from $T$ as follows.
- Add $\fkl$ as a singleton tile.
- For each node $\fka$ in column $j$:
- if $\fka$ has depth at least $2$, attach to both and ;
- (if $\fka\neq\fkl$) detach $\fka$ from and .
- Remove $\fkm$.
The new tiles in $\phi_3(T)$ are as follows:
- for $u\in T$ containing a node $\fka$ in column $j$ of depth at least $2$, $\phi_3(T)$ contains the tile obtained from $u$ by replacing $\fka$ with ;
- for $u\in T$ containing a node $\fka$ in column $j$ of depth $1$, $\phi_3(T)$ contains the portion of $u$ ending in column $j-1$, the portion of $u$ starting in column $j+1$, and the singleton tile $\{{\ensuremath{\mathtt{N}(\fka)}}\}$.
Clearly all these tiles are Dyck tiles. The cover-inclusive property follows easily from the corresponding property of $T$, and in particular the fact (\[citfae\](\[:de\])) that depth weakly decreases down columns in a cover-inclusive tiling. Moreover, $\phi_3(T)$ has no big tile starting or ending in column $j$, so lies in the codomain.
To show that $\phi_3$ is a bijection, we construct its inverse $$\psi_3:\rset{T\in{\scri(\la^+,\mu^+)}}{\parbox{129pt}{there is no big tile in $T$ starting or ending in column $j$}}\longrightarrow\lset{T\in{\scri(\la,\mu)}}{\fkm\textup{ lies in a big tile in $T$}}.$$ Suppose $T\in{\scri(\la^+,\mu^+)}$, and that there is no big tile in $T$ starting or ending in column $j$. The highest node in column $j$ of ${\ensuremath{\la^+\backslash\mu^+}}$, namely $\fkl$, is not attached to or (since these are not nodes of ${\la^+}$) and so no node in column $j$ is attached to its NE or NW neighbour. So every node in column $j$ is either a singleton or attached to both its SE and SW neighbours. Now we can construct $\psi_3(T)$ from $T$ as follows.
- Add $\fkm$ as singleton tile.
- For each node $\fka$ in column $j$:
- detach $\fka$ from and (if it is attached);
- (if $\fka\neq\fkl$) attach $\fka$ to and .
- Remove $\fkl$.
Again, it is easy to check that $\psi_3(T)$ is a [cover-inclusive Dyck tiling]{}, and clearly $\fkm$ lies in a big tile in $\psi_3(T)$. Furthermore, it is easy to see that $\phi_3$ and $\psi_3$ are mutually inverse.
Now we combine \[bij1,bij1a,bij2,bij2a,bij3\] to prove our main recurrence. We retain the assumptions and notation from above.
Our main result is as follows. Recall that ${\operatorname{i}_{\la\mu}}$ denotes the total number of [cover-inclusive Dyck tiling]{}s of ${\ensuremath{\la\backslash\mu}}$.
\[recur\] With notation as above, $${\operatorname{i}_{\la\mu}}+{\operatorname{i}_{\la\mu^+}}={\operatorname{i}_{\la^+\mu^+}}+\sum_{x\in{X_{\la}}}{\operatorname{i}_{{\la^{[x]}}\mu^+}}.$$
By \[bij2\], we have $$\left|\lset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overrightarrow{\displaystyle T}}}\in{X_{\la}}}\right|=\sum_{x\in{{X_{\la}}^+}}{\operatorname{i}_{{\la^{[x]}}\mu^+}}.$$ So $$\begin{aligned}
{\operatorname{i}_{\la\mu^+}}&=\left|\lset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overrightarrow{\displaystyle T}}}\in{X_{\la}}}\right|\ +\ \left|\lset{T\in{\scri(\la,\mu^+)}}{{{\scriptstyle\overrightarrow{\displaystyle T}}}\notin{X_{\la}}}\right|\\
&=\sum_{x\in {{X_{\la}}^+}}{\operatorname{i}_{{\la^{[x]}}\mu^+}}\ +\ \left|\rset{T\in{\scri(\la^+,\mu^+)}}{\textup{there is a big tile in $T$ starting in column }j}\right|\end{aligned}$$ by \[bij1\].
Symmetrically, we have $${\operatorname{i}_{\la\mu^+}}=\sum_{x\in{{X_{\la}}^-}}{\operatorname{i}_{{\la^{[x]}}\mu^+}}\ +\ \left|\rset{T\in{\scri(\la^+,\mu^+)}}{\textup{there is a big tile in $T$ ending in column }j}\right|.$$
Now consider ${\operatorname{i}_{\la\mu}}$. Obviously we have $$\left|\lset{T\in{\scri(\la,\mu)}}{\fkm\textup{ lies in a singleton tile in }T}\right|={\operatorname{i}_{\la\mu^+}},$$ and by \[bij3\] $$\left|\rset{T\in{\scri(\la,\mu)}}{\parbox{85pt}{\rm$\fkm$ does not lie in a singleton tile of $T$}}\right|=\left|\rset{T\in{\scri(\la^+,\mu^+)}}{\parbox{129pt}{\rm there is no big tile in $T$ starting or ending in column $j$}}\right|.$$ Hence $$\begin{aligned}
{\operatorname{i}_{\la\mu}}+{\operatorname{i}_{\la\mu^+}}&=\left|\lset{T\in{\scri(\la,\mu)}}{\fkm\textup{ lies in a singleton tile in }T}\right|+\left|\rset{T\in{\scri(\la,\mu)}}{\parbox{85pt}{\rm$\fkm$ does not lie in a singleton tile of $T$}}\right|+{\operatorname{i}_{\la\mu^+}}\\
&=\left|\rset{T\in{\scri(\la^+,\mu^+)}}{\parbox{129pt}{there is no big tile in $T$ starting or ending in column $j$}}\right|+2{\operatorname{i}_{\la\mu^+}}\\
&=\sum_{x\in{X_{\la}}}{\operatorname{i}_{{\la^{[x]}}\mu^+}}+\left|\rset{T\in{\scri(\la^+,\mu^+)}}{\textup{there is a big tile in $T$ starting in column }j}\right|\\
&\phantom{=\sum_{x\in{X_{\la}}}{\operatorname{i}_{{\la^{[x]}}\mu^+}}\ }+\left|\rset{T\in{\scri(\la^+,\mu^+)}}{\textup{there is a big tile in $T$ ending in column }j}\right|\\
&\phantom{=\sum_{x\in{X_{\la}}}{\operatorname{i}_{{\la^{[x]}}\mu^+}}\ }+\left|\rset{T\in{\scri(\la^+,\mu^+)}}{\parbox{129pt}{there is no big tile in $T$ starting or ending in column $j$}}\right|.\end{aligned}$$ Since a [cover-inclusive Dyck tiling]{} cannot contain big tiles starting and ending in the same column, the sum of the last three terms is ${\operatorname{i}_{\la^+\mu^+}}$, and we are done.
Cover-expansive Dyck tilings {#cedtsec}
============================
In this section we consider [cover-expansive Dyck tiling]{}s, proving similar (though considerably simpler) recurrences to those in \[cidtsec\].
Basic properties
----------------
We begin by studying left- and right-[cover-expansive Dyck tiling]{}s. In contrast to [cover-inclusive Dyck tiling]{}s, it is not the case that the left-cover-expansive and right-cover-expansive conditions are equivalent. For example, the unique Dyck tiling of ${\ensuremath{(2)\backslash\varnothing}}$ is left- but not right-cover-expansive.
We begin with equivalent conditions to the left- and right-cover-expansive conditions.
\[cetfae\] Suppose $\la$ and $\mu$ are partitions with $\la\supseteq\mu$, and $T$ is a Dyck tiling of ${\ensuremath{\la\backslash\mu}}$.
1. $T$ is left-cover-expansive if and only if for every tile $t$ in $T$, we have ${\ensuremath{\mathtt{NW}({\ensuremath{\operatorname{st}(t)}})}}\notin{\ensuremath{\la\backslash\mu}}$.
2. $T$ is right-cover-expansive if and only if for every tile $t$ in $T$, we have ${\ensuremath{\mathtt{NE}({\ensuremath{\operatorname{en}(t)}})}}\notin{\ensuremath{\la\backslash\mu}}$.
We prove (1); the proof of (2) is similar.
Suppose $T$ is left-cover-expansive. Given a tile $t$, let $\fka={\ensuremath{\mathtt{NW}({\ensuremath{\operatorname{st}(t)}})}}$. If $\fka\in{\ensuremath{\la\backslash\mu}}$, then by the left-cover-expansive property ${\ensuremath{\operatorname{st}({\ensuremath{\operatorname{tile}(\fka)}})}}$ lies weakly to the right of t; but $\fka$ lies strictly to the left of t, a contradiction.
Conversely, suppose the given property holds, and $\fka,{\ensuremath{\mathtt{SE}(\fka)}}$ are nodes of ${\ensuremath{\la\backslash\mu}}$. Let $\fkb={\ensuremath{\operatorname{st}({\ensuremath{\operatorname{tile}({\ensuremath{\mathtt{SE}(\fka)}})}})}}$, and suppose $\fkb$ lies in column $i$. Then ${\ensuremath{\mathtt{NW}(\fkb)}}\notin{\ensuremath{\la\backslash\mu}}$, and hence there are no nodes in column $i-1$ of ${\ensuremath{\la\backslash\mu}}$ higher than $\fkb$ (otherwise ${\ensuremath{\la\backslash\mu}}$ would not be a skew Young diagram). In particular, there are no nodes of in column $i-1$, and hence [$\operatorname{st}({\ensuremath{\operatorname{tile}(\fka)}})$]{} lies weakly to the right of $\fkb$.
\[leftlowest\] Suppose $\la,\mu$ are partitions with $\la\supseteq\mu$, and $T$ is a left-[cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$. If $\fka\in{\ensuremath{\la\backslash\mu}}$ is the lowest node in its column, then every node in to the right of $\fka$ is the lowest node in its column.
Suppose $\fka$ lies in column $i$, and proceed by induction on the number of nodes to the right of $\fka$ in . Assuming $\fka$ is not the end node of its tile, there is a node $\fkb$ in in column $i+1$, which must be either or . The only way $\fkb$ can fail to be the lowest node in column $i+1$ is if $\fkb={\ensuremath{\mathtt{NE}(\fka)}}$ and ${\ensuremath{\mathtt{SE}(\fka)}}\in{\ensuremath{\la\backslash\mu}}$. But in this case, is not attached to $\fka$ or to ${\ensuremath{\mathtt{S}(\fka)}}$ (which is not a node of ${\ensuremath{\la\backslash\mu}}$), and so is the start of its tile; but this contradicts \[cetfae\].
So $\fkb$ is the lowest node in its column. By induction every node to the right of $\fkb$ in the same tile is the lowest node in its column, and we are done.
\[ls1l\] Suppose $\la,\mu$ are partitions with $\la\supseteq\mu$. Then ${\ensuremath{\la\backslash\mu}}$ admits at most one left-[cover-expansive Dyck tiling]{}, and at most one right-[cover-expansive Dyck tiling]{}. If ${\ensuremath{\la\backslash\mu}}$ admits both a left- and a right-[cover-expansive Dyck tiling]{}, then these tilings coincide.
We use induction on $|{\ensuremath{\la\backslash\mu}}|$. If $\la=\mu$ then the result is trivial, so assume $\la\supset\mu$. Let $\fka$ be the unique leftmost node of ${\ensuremath{\la\backslash\mu}}$, and suppose $\fka$ lies in column $i$. Let $m\gs0$ be maximal such that columns $i,\dots,i+m$ each contain a node of height at most . Suppose $T$ is a left-[cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$. 1 consists of the lowest nodes in columns $i,\dots,i+m$. Since $\fka$ is the start node of its tile, every node in has height at most , and in particular cannot contain a node in column $i+m+1$ or further to the right.
Now we prove by induction on $l$ that contains the lowest node in column $l$, for $i\ls l\ls i+m$. Let $\fkb$ be the lowest node in column $l$; then (assuming $l>i$) the lowest node in column $l-1$ is either or . In the first case, $\fkb$ cannot be the start of its tile, by \[cetfae\]; $\fkb$ cannot be attached to , since this is not a node of ${\ensuremath{\la\backslash\mu}}$, and so $\fkb$ is attached to , and hence lies in . In the second case, the fact that ${\ensuremath{\operatorname{ht}({\ensuremath{\mathtt{SW}(\fkb)}})}}<{\ensuremath{\operatorname{ht}(\fkb)}}\ls{\ensuremath{\operatorname{ht}(\fka)}}$ means that cannot be the end node of , so is attached to either $\fkb$ or ${\ensuremath{\mathtt{S}(\fkb)}}$; but is not a node of ${\ensuremath{\la\backslash\mu}}$, and hence is attached to $\fkb$, i.e. $\fkb\in{\ensuremath{\operatorname{tile}(\fka)}}$. The definition of $m$ means that the nodes in can be removed to leave a smaller skew Young diagram ${\ensuremath{\la\backslash\nu}}$, and the fact that $T$ is left-cover-expansive means that $T\setminus\{{\ensuremath{\operatorname{tile}(\fka)}}\}$ is a left-[cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\nu}}$. By induction on $|{\ensuremath{\la\backslash\mu}}|$ there is at most one such tiling, and so $T$ is uniquely determined.
So there is at most one left-[cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$, and similarly at most one right-[cover-expansive Dyck tiling]{}. To prove the final statement, we continue to assume that ${\ensuremath{\la\backslash\mu}}$ is non-empty; we choose a connected component $\calc$ of ${\ensuremath{\la\backslash\mu}}$, and let $\fka$ denote the unique leftmost node of $\calc$, and $\fkc$ the unique rightmost node of $\calc$. (So there is no node in the column to the left of the column containing $\fka$ or in the column to the right of the column containing $\fkc$, but there are nodes in all columns in between.) 2 Suppose there exists a left-[cover-expansive Dyck tiling]{} $T$ of ${\ensuremath{\la\backslash\mu}}$. Then ${\ensuremath{\operatorname{ht}(\fka)}}\ls{\ensuremath{\operatorname{ht}(\fkc)}}$, and if equality occurs then $\fka$ and $\fkc$ lie in the same tile in $T$. Let $\fkb_1,\dots,\fkb_r$ be the nodes in $\calc$ which are both the end nodes of their tiles and the lowest nodes in their columns, numbering them so that they appear in order from left to right. Then $\fkb_1={\ensuremath{\operatorname{en}({\ensuremath{\operatorname{tile}(\fka)}})}}$ by Claim 1, and $\fkb_r=\fkc$. We claim that ${\ensuremath{\operatorname{ht}(\fkb_1)}}<\dots<{\ensuremath{\operatorname{ht}(\fkb_r)}}$. Given $1\ls l<r$, let $\fkd$ be the lowest node in the column to the right of $\fkb_l$. Then $\fkd$ is either [$\mathtt{NE}(\fkb_l)$]{} or [$\mathtt{SE}(\fkb_l)$]{}; but in the latter case, $\fkd$ must be the start node of its tile, and this contradicts \[citfae\]. So $\fkd={\ensuremath{\mathtt{NE}(\fkb_l)}}$, and in particular ${\ensuremath{\operatorname{ht}(\fkd)}}>{\ensuremath{\operatorname{ht}(\fkb_l)}}$. By \[leftlowest\] ${\ensuremath{\operatorname{en}({\ensuremath{\operatorname{tile}(\fkd)}})}}=\fkb_{l+1}$, and hence ${\ensuremath{\operatorname{ht}(\fkb_{l+1})}}={\ensuremath{\operatorname{ht}(\fkd)}}>{\ensuremath{\operatorname{ht}(\fkb_l)}}$.
So we have ${\ensuremath{\operatorname{ht}(\fka)}}={\ensuremath{\operatorname{ht}(\fkb_1)}}<\dots<{\ensuremath{\operatorname{ht}(\fkb_r)}}={\ensuremath{\operatorname{ht}(\fkc)}}$, so ${\ensuremath{\operatorname{ht}(\fka)}}\ls{\ensuremath{\operatorname{ht}(\fkc)}}$, with equality if and only if $r=1$, in which case $\fkc=\fkb_1={\ensuremath{\operatorname{en}({\ensuremath{\operatorname{tile}(\fka)}})}}$. Now we can complete the proof. Assume there is a left-[cover-expansive Dyck tiling]{} $T$ of ${\ensuremath{\la\backslash\mu}}$ and a right-[cover-expansive Dyck tiling]{} $U$. By Claim 2, we have ${\ensuremath{\operatorname{ht}(\fka)}}\ls{\ensuremath{\operatorname{ht}(\fkc)}}$, and symmetrically (since $U$ exists) we have ${\ensuremath{\operatorname{ht}(\fka)}}\gs{\ensuremath{\operatorname{ht}(\fkc)}}$. Hence ${\ensuremath{\operatorname{ht}(\fka)}}={\ensuremath{\operatorname{ht}(\fkc)}}$, and so $\fka,\fkc$ lie in the same tile $t\in T$, which consists of the lowest node in every column of $\calc$. Similarly, $t$ is a tile in $U$. Removing $t$ from ${\ensuremath{\la\backslash\mu}}$ yields a smaller skew Young diagram, and $T\setminus\{t\}$ is a left-[cover-expansive Dyck tiling]{} of this diagram, while $U\setminus\{t\}$ is a right-[cover-expansive Dyck tiling]{}. By induction $T\setminus\{t\}=U\setminus\{t\}$, and hence $T=U$.
Now we restrict attention to [cover-expansive Dyck tiling]{}s. We shall need the following lemma, which examines the effect of the cover-expansive property on depths of nodes.
\[cicedepth\] Suppose $\la\supseteq\mu$, $T$ is a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$, and $\fka,{\ensuremath{\mathtt{N}(\fka)}}\in{\ensuremath{\la\backslash\mu}}$. Then ${\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{N}(\fka)}})}}<{\ensuremath{\operatorname{dp}(\fka)}}$.
Suppose the lemma is false, and take $\fka\in{\ensuremath{\la\backslash\mu}}$ as far to the left as possible such that ${\ensuremath{\mathtt{N}(\fka)}}\in{\ensuremath{\la\backslash\mu}}$ and ${\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{N}(\fka)}})}}\gs{\ensuremath{\operatorname{dp}(\fka)}}$. We have ${\ensuremath{\mathtt{NW}(\fka)}}\in{\ensuremath{\la\backslash\mu}}$, and cannot be the end node of its tile by \[cetfae\], so is attached to either $\fka$ or .
If is attached to , then $\fka$ must be attached to since it cannot be the start node of its tile (again by \[cetfae\]). But now ${\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{NW}(\fka)}})}}-{\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{SW}(\fka)}})}}=({\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{N}(\fka)}})}}+1)-({\ensuremath{\operatorname{dp}(\fka)}}+1)\gs0$, contradicting the choice of $\fka$.
So suppose instead that is attached to $\fka$. Then ${\ensuremath{\mathtt{N}(\fka)}}$ cannot be the start node of its tile, since ${\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{N}(\fka)}})}}\gs{\ensuremath{\operatorname{dp}(\fka)}}={\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{NW}(\fka)}})}}+1>0$. So is attached to [$\mathtt{NW}({\ensuremath{\mathtt{N}(\fka)}})$]{}. But now ${\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{NW}({\ensuremath{\mathtt{N}(\fka)}})}})}}-{\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{NW}(\fka)}})}}=({\ensuremath{\operatorname{dp}({\ensuremath{\mathtt{N}(\fka)}})}}-1)-({\ensuremath{\operatorname{dp}(\fka)}}-1)\gs0$, and again the choice of $\fka$ is contradicted.
We remark that, in contrast to the similar condition in \[citfae\](\[:de\]) for [cover-inclusive Dyck tiling]{}s, the condition in \[cicedepth\] does not imply the cover-expansive condition. For example, the unique Dyck tiling of ${\ensuremath{(2)\backslash\varnothing}}$ satisfies this condition (trivially) but is not right-cover-expansive.
The following lemma will be useful in the next section.
\[addrem\] Suppose $\la\supseteq\mu$, and that $\mu$ has an addable node in column $j$. If there exists a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$, then $\la$ has either an addable or a removable node in column $j$.
Suppose not, and let $T$ be the [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$. There must be at least one node in column $j$ of ${\ensuremath{\la\backslash\mu}}$ (otherwise the addable node of $\mu$ would also be an addable node of $\la$). Let $\fka$ be the highest node in column $j$, and consider the nodes attached to $\fka$ in $T$.
The lowest node in column $j$ of ${\ensuremath{\la\backslash\mu}}$, namely $\fkm$, is not attached to its SW neighbour (since this is not a node of ${\ensuremath{\la\backslash\mu}}$). Now let $\fkb$ be the highest node in column $j$ which is not attached to its SW neighbour, and suppose $\fkb\neq\fka$. Then the choice of $\fkb$ means that is attached to , so $\fkb$ is the start node of its tile; but ${\ensuremath{\mathtt{NW}(\fkb)}}\in{\ensuremath{\la\backslash\mu}}$, and this contradicts \[cetfae\]. So $\fkb=\fka$.
So $\fka$ is not attached to , and symmetrically is not attached to . If $\fka$ is attached to either or , then (since $T$ is a Dyck tiling) it is attached to both and , and hence both of these nodes belong to $\la$, so $\la$ has an addable node in column $j$. The remaining possibility is that $\fka$ is a singleton tile. But now \[cetfae\] implies that neither nor is a node of ${\ensuremath{\la\backslash\mu}}$, so $\la$ has a removable node in column $j$.
Recurrences
-----------
We now reassume the notation in \[cidtsec\]: $\la,\mu$ are partitions with addable nodes $\fkl,\fkm$ respectively in column $j$, and ${\la^+},\mu^+$ are the partitions obtained by adding these nodes.
\[exp1\] Suppose $\la\supseteq\mu^+$, and let $\fka={\ensuremath{\mathtt{S}(\fkl)}}$. Then:
1. there exists a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$ in which $\fka$ has depth $1$ if and only if there exists a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu^+}}$;
2. there exists a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$ in which $\fka$ has depth greater than $1$ if and only if there exists a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la^+\backslash\mu^+}}$.
Suppose there is a [cover-expansive Dyck tiling]{} $T$ of ${\ensuremath{\la\backslash\mu}}$. Arguing as in the proof of \[addrem\], $\fka$ cannot be attached to either or (since $\mu$ has an addable node in column $j$), and $\fka$ cannot be a singleton tile (since $\la$ has an addable node in column $j$). So $\fka$ is attached to both and . So by the cover-expansive conditions, every node in column $j$ is attached to its NE and NW neighbours.
1. \[bj1\] Suppose $\fka$ has depth $1$ in $T$. Construct a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu^+}}$ $U$ from $T$ as follows:
- detach each node in column $j$ from its NW and NE neighbours;
- remove $\fkm$;
- attach each node in column $j$ to its SE and SW neighbours.
It is easy to check that $U$ really does give a [cover-expansive Dyck tiling]{}; the tiles in $U$ but not in $T$ are obtained as follows:
- if $u$ is the tile containing $\fka$ in $T$, then the portions of $u$ ending at and starting at are both tiles in $U$; since $\fka$ has depth $1$ in $T$, these are both Dyck tiles;
- if $u$ is a tile in $T$ containing a node $\fkc\neq\fka$ in column $j$, then $U$ contains the tile obtained by replacing $\fkc$ with in $u$; this is a Dyck tile, since $\fkc$ has depth greater than $1$ in $T$ by \[cicedepth\].
So $U$ is a Dyck tiling, and the cover-expansive property follows easily from the corresponding property of $T$.
For the other direction, suppose $T$ is a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu^+}}$. We claim that every node in column $j$ must be attached to its SW and SE neighbours. If there are no nodes in column $j$ (i.e. if $\fkm={\ensuremath{\mathtt{S}(\fkl)}}$) then this statement is trivial, so suppose otherwise; then $\fka\in{\ensuremath{\la\backslash\mu^+}}$. $\fka$ cannot be attached to in $T$, since then every node in column $j-1$ would be attached to its SE neighbour; but the SE neighbour of the bottom node in column $j-1$ is not a node of ${\ensuremath{\la\backslash\mu^+}}$. Since $\fka$ is not attached to , it must be attached to . Similarly $\fka$ is attached to , and the cover-expansive property implies that every node in column $j$ is attached to its SE and SW neighbours as claimed. Now we construct a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$ as follows:
- detach each node in column $j$ from its SW and SE neighbours;
- add $\fkm$;
- attach each node in column $j$ to its NE and NW neighbours.
Again, it is easy to see that we have a [cover-expansive Dyck tiling]{}. Moreover, $\fka$ has depth $1$ in this tiling, since is the end node of its tile in $T$, and so has depth $0$ (in both tilings).
2. \[bj2\] Suppose $\fka$ has depth greater than $1$ in $T$. Construct a [cover-expansive Dyck tiling]{} $U$ of ${\ensuremath{\la^+\backslash\mu^+}}$ from $T$ as follows:
- detach each node in column $j$ from its NW and NE neighbours;
- add $\fkl$ and remove $\fkm$;
- attach each node in column $j$ to its SE and SW neighbours.
Again, $U$ is a Dyck tiling, since every node in column $j$ of ${\ensuremath{\la\backslash\mu}}$ has depth greater than $1$ in $T$. And the cover-expansive property for $U$ follows from that for $T$.
For the other direction, suppose $T$ is a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la^+\backslash\mu^+}}$. Arguing as in (\[bj1\]), every node in column $j$ is attached to its SW and SE neighbours. Now we construct a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$ $U$ as follows:
- detach each node in column $j$ from its SW and SE neighbours;
- add $\fkm$ and remove $\fkl$;
- attach each node in column $j$ to its NE and NW neighbours.
Again, it is easy to see that $U$ is a [cover-expansive Dyck tiling]{}. Moreover, since is attached to $\fkl$ in $T$, it has positive depth (in both tilings), so $\fka$ has depth at least $2$ in $U$.
\[etrec1\] ${\operatorname{e}_{\la\mu}}={\operatorname{e}_{\la\mu^+}}+{\operatorname{e}_{\la^+\mu^+}}$.
First suppose $\la\supseteq\mu^+$. The first paragraph of the proof of \[exp1\] shows that the node $\fka={\ensuremath{\mathtt{S}(\fkl)}}$ must have positive depth in a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$. Hence the result follows from \[exp1\].
Alternatively, suppose $\la\nsupseteq\mu^+$. Then we have $\la\supseteq\mu$ if and only if $\la^+\supseteq\mu^+$, and if these conditions hold then ${\ensuremath{\la\backslash\mu}}={\ensuremath{\la^+\backslash\mu^+}}$; so ${\operatorname{e}_{\la\mu}}={\operatorname{e}_{\la^+\mu^+}}$.
\[exp2\] ${\operatorname{e}_{\la^+\mu}}={\operatorname{e}_{\la\mu}}$.
Suppose $T$ is a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la^+\backslash\mu}}$. Then $\fkl$ cannot be attached to , since then every node in column $j$ would be attached to its SE neighbour; but is not a node of ${\ensuremath{\la^+\backslash\mu}}$. Similarly $\fkl$ is not attached to , so forms a singleton tile in $T$. Removing this tile yields a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$.
In the other direction, suppose $T$ is a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$. From the proof of \[exp1\], ${\ensuremath{\mathtt{S}(\fkl)}}$ is attached to both and . Hence if we add $\fkl$ as a singleton tile, the resulting tiling is a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la^+\backslash\mu}}$.
Young permutation modules for two-part compositions {#ypmsec}
===================================================
In this section we apply our results on Dyck tilings to compute change-of-basis matrices for a certain module for the symmetric group.
The Young permutation module ${\mathscr{M}^{(f,g)}}$ {#ypmsubsec}
----------------------------------------------------
Suppose $k\gs0$, and let $\sss k$ denote the symmetric group of degree $k$, with $\{t_1,\dots,t_{k-1}\}$ the set of Coxeter generators (so $t_i$ is the transposition $\perc(i{i+1})$). Let $\bbf$ be any field, and consider the group algebra $\bbf\sss k$. Define $s_i=t_i-1\in\bbf\sss k$ for $i=1,\dots,n-1$.
Now write $k=f+g$ with $f,g$ non-negative integers. Consider the *Young permutation module* ${\mathscr{M}^{(f,g)}}$ for $\bbf\sss k$ indexed by the composition $(f,g)$. This is just the permutation module on the set of cosets of the maximal Young subgroup $\sss f\times\sss g$, and has the following presentation: $${\mathscr{M}^{(f,g)}}=\left\lan{m}\ \left|\ t_i{m}={m}\text{ for }i\neq f\right.\right\ran.$$
\[ann\] For $1\ls i\ls k-2$, the element $s_is_{i+1}s_i-s_i=t_is_{i+1}s_i-s_{i+1}s_i-s_i$ annihilates ${\mathscr{M}^{(f,g)}}$.
In terms of permutations, the given element is $$\perc(i{i+2})-\perc(i{i+1}{i+2})-\perc(i{i+2}{i+1})+\perc(i{i+1})+\perc({i+1}{i+2})-1.$$ ${\mathscr{M}^{(f,g)}}$ may be viewed as the permutation module on the set of $f$-subsets of the set $\{1,\dots,k\}$; it is easily seen that the given element kills any such subset.
${\mathscr{M}^{(f,g)}}$ has a basis indexed by the set of minimal left coset representatives of $\sss f\times\sss g$ in $\sss k$; this set is in one-to-one correspondence with the set ${\scrp_{f,g}}$ of partitions $\la$ for which $\la_1\ls f$ and $\la_1'\ls g$; given $\la\in{\scrp_{f,g}}$, we write the corresponding basis element as $t_\la{m}$, where $$\begin{aligned}
t_\la&=\left(t_{\la_1+g}t_{\la_1+g+1}\dots t_{f+g-1}\right)\left(t_{\la_2+g-1}t_{\la_2+g}\dots t_{f+g-2}\right)\dots\left(t_{\la_g+1}t_{\la_g+2}\dots t_f\right).
\\
\intertext{Our objective here is to study the elements}
s_\la&=\left(s_{\la_1+g}s_{\la_1+g+1}\dots s_{f+g-1}\right)\left(s_{\la_2+g-1}s_{\la_2+g}\dots s_{f+g-2}\right)\dots\left(s_{\la_g+1}s_{\la_g+2}\dots s_f\right)\end{aligned}$$ for $\la\in{\scrp_{f,g}}$. It is easy to see that $\rset{s_\la{m}}{\la\in{\scrp_{f,g}}}$ is also a basis for ${\mathscr{M}^{(f,g)}}$, since when each each $s_\la{m}$ is expressed as a linear combination of the elements $t_\mu{m}$, the matrix of coefficients is unitriangular with respect to a suitable ordering.
We shall use cover-expansive Dyck tilings to describe this change-of-basis matrix explicitly, and then describe its inverse using cover-inclusive Dyck tilings. We also give a simple expression for the sum $\sum_{\la\in{\scrp_{f,g}}}t_\la{m}$ as a linear combination of the elements $s_\la{m}$, which will be useful in \[klrsec\].
Change of basis
---------------
Our first result on change-of-basis matrices is the following.
\[mainsn1\] Suppose $f,g$ are non-negative integers, and $\mu\in{\scrp_{f,g}}$. Then $$s_\mu{m}=\displaystyle{\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}(-1)^{|\la|+|\mu|}{\operatorname{e}_{\la\mu}} t_\la{m}.$$
We begin with some simple observations concerning the actions of the generators $t_1,\dots,t_{k-1}$ on the basis elements $t_\la$. We continue to use the Russian convention for Young diagrams.
\[actiontt\] Suppose $\la\in{\scrp_{f,g}}$ and $1\ls i<k$. Then:
- if $\la$ has an addable node $\fkl$ in column $i-g$, then $t_it_\la{m}=t_{\la^+}{m}$, where $\la^+$ is the partition obtained by adding $\fkl$ to $\la$;
- if $\la$ has a removable node $\fkl$ in column $i-g$, then $t_it_\la{m}=t_{\la^-}{m}$, where $\la^-$ is the partition obtained by removing $\fkl$ from $\la$;
- if $\la$ has neither an addable nor a removable node in column $i-g$, then $t_it_\la{m}=t_\la{m}$.
This is an easy consequence of the definitions and the Coxeter relations.
We proceed by downwards induction on $|\mu|$. If $\mu=(f^g)$, then $s_\mu=t_\mu=1$ and the result follows. Assuming $\mu\neq(f^g)$, $\mu$ has an addable node in column $j$, for some $-g<j<f$. We let ${\mu^+}$ denote the partition obtained by adding this addable node; then we have $s_\mu=s_{j+g}s_{\mu^+}$, and by induction $$s_{\mu^+}{m}={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}(-1)^{|\la|+|{\mu^+}|}{\operatorname{e}_{\la\mu^+}}t_\la{m}.$$ Let ${{\scrp_{f,g}}^+}$ denote the set of $\la\in{\scrp_{f,g}}$ having an addable node in column $j$, and for $\la\in{{\scrp_{f,g}}^+}$ let $\la^+$ denote the partition obtained by adding this addable node; similarly, let ${{\scrp_{f,g}}^-}$ denote the set of partitions in ${\scrp_{f,g}}$ having a removable node in column $j$, and for each $\la\in{{\scrp_{f,g}}^-}$ let $\la^-$ denote the partition obtained by removing this removable node. Note that the functions $\la\mapsto\la^+$ and $\la\mapsto\la^-$ define mutually inverse bijections between ${{\scrp_{f,g}}^+}$ and ${{\scrp_{f,g}}^-}$. Now $$\begin{aligned}
s_\mu{m}&=(t_{j+g}-1){\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}(-1)^{|\la|+|{\mu^+}|}{\operatorname{e}_{\la\mu^+}} t_\la{m}\\
&={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^+}}$\hfil}}}(-1)^{|\la|+|{\mu^+}|}{\operatorname{e}_{\la\mu^+}}(t_{\la^+}-t_\la){m}+{\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^-}}$\hfil}}}(-1)^{|\la|+|{\mu^+}|}{\operatorname{e}_{\la\mu^+}}(t_{\la^-}-t_\la){m}\tag*{by \cref{actiontt}}\\
&={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^+}}$\hfil}}}(-1)^{|\la|+|\mu|}\big({\operatorname{e}_{\la\mu^+}}+{\operatorname{e}_{\la^+\mu^+}}\big)t_\la{m}+{\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^-}}$\hfil}}}(-1)^{|\la|+|\mu|}\big({\operatorname{e}_{\la\mu^+}}+{\operatorname{e}_{\la^-\mu^+}}\big)t_\la{m}\\
&={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^+}}$\hfil}}}(-1)^{|\la|+|\mu|}{\operatorname{e}_{\la\mu}} t_\la{m}+{\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^-}}$\hfil}}}(-1)^{|\la|+|\mu|}{\operatorname{e}_{\la^-\mu}} t_\la{m}\tag*{by \cref{exp1}}\\
&={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^+}\cup{{\scrp_{f,g}}^-}}$\hfil}}}(-1)^{|\la|+|\mu|}{\operatorname{e}_{\la\mu}} t_\la{m}\tag*{by \cref{exp2}}\\
&={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}(-1)^{|\la|+|\mu|}{\operatorname{e}_{\la\mu}} t_\la{m}\tag*{by \cref{addrem}.}\end{aligned}$$ So \[mainsn1\] follows by induction.
Now we give our second main result on change-of-basis matrices.
\[mainsn2\] Suppose $f,g$ are non-negative integers, and $\mu\in{\scrp_{f,g}}$. Then $$t_\mu{m}=\displaystyle{\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}{\operatorname{i}_{\la\mu}} s_\la{m}.$$
The proof of this result is rather more difficult. To begin with, we compute the actions of $s_1,\dots,s_{k-1}$ on the basis elements $s_\la{m}$.
\[actionts\] Suppose $\mu\in{\scrp_{f,g}}$ and $1\ls i<k$. Then exactly one of the following occurs.
1. $\mu$ has an addable node in column $i-g$. In this case $s_is_\mu{m}=-2s_\mu{m}$.
2. $\mu$ has a removable node in column $i-g$. In this case $s_is_\mu{m}=s_{\mu^-}{m}$, where $\mu^-$ denotes the partition obtained by removing this node.
3. For some $0\ls a\ls g$ we have $\mu_a>i-g+a>\mu_{a+1}$ (where the left-hand inequality is regarded as automatically true in the case $a=0$).
1. If $\mu_w<i-g+2a-w$ for all $a<w\ls g$, then $s_is_\mu{m}=0$.
2. Otherwise, let $w>a$ be minimal such that $\mu_w=i-g+2a-w$, and set $$\mu^{a,w}=(\mu_1,\dots,\mu_a,i-g+a,\mu_{a+1}+1,\dots,\mu_{w-1}+1,\mu_{w+1},\dots,\mu_g).$$ Then $s_is_\mu{m}=s_{\mu^{a,w}}{m}$.
4. For some $1\ls a<g$ we have $\mu_a=\mu_{a+1}=i-g+a$.
1. If $i+2a>k$ and $\mu_w<i-g+2a-w$ for $w=1,\dots,a-1$, then $s_is_\mu{m}=0$.
2. Otherwise, let $w<a$ be maximal such that $\mu_w\gs i-g+2a-w$ (taking $w=0$ if there is no such $w$), and define $$\mu^{w,a}=(\mu_1,\dots,\mu_w,i-g+2a-w,\mu_{w+1}+1,\dots,\mu_{a-1}+1,\mu_{a+1},\dots,\mu_g).$$ Then $s_is_\mu{m}=s_{\mu^{w,a}}{m}$.
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1. Suppose $f=4$, $g=5$ and $\mu=(4,2^4)$. Then $$s_\mu=s_6s_7s_5s_6s_4s_5s_3s_4.$$ Taking $i=3$, we find that $\mu$ satisfies condition 4(a) of \[actionts\], with $a=4$. And indeed $$\begin{aligned}
s_3s_\mu m&=s_3s_6s_7s_5s_6s_4s_5s_3s_4m\\
&=s_6s_7s_5s_6(s_3s_4s_3)s_5s_4m\\
&=s_6s_7s_5s_6s_3s_5s_4m\tag*{by \cref{ann}}\\
&=s_6s_7(s_5s_6s_5)s_3s_4m\\
&=s_6s_7s_5s_3s_4m\tag*{by \cref{ann}}\\
&=s_6s_5s_3s_4s_7m\\
&=0\tag*{since $s_7m=0$.}\end{aligned}$$
2. Now suppose $f=4$, $g=5$ and $\mu=(4,2,1^3)$. Then $$s_\mu=s_6s_7s_4s_5s_6s_3s_4s_5s_2s_3s_4.$$ Taking $i=2$, we find that $\mu$ satisfies condition 4(b) of \[actionts\], with $a=4$ and $w=1$, giving $\mu^{w,a}=(4^2,3,2,1)$. Now we have $$\begin{aligned}
s_2s_\mu m&=s_2s_6s_7s_4s_5s_6s_3s_4s_5s_2s_3s_4m\\
&=s_6s_7s_4s_5s_6(s_2s_3s_2)s_4s_5s_3s_4m\\
&=s_6s_7s_4s_5s_6s_2s_4s_5s_3s_4m\\
&=s_6s_7(s_4s_5s_4)s_6s_2s_5s_3s_4m\\
&=s_6s_7s_4s_6s_2s_5s_3s_4m\\
&=(s_6s_7s_6)s_4s_2s_5s_3s_4m\\
&=s_6s_4s_2s_5s_3s_4m\\
&=m_{(4^2,3,2,1)}m.\end{aligned}$$
To see that $\mu$ satisfies exactly one of the four conditions, let $\fkl$ denote the lowest node in column $j$ which is not a node of $\mu$, and consider whether and lie in $\mu$; for the purposes of this argument we regard all nodes of the form $(a,0)$ or $(0,b)$ as lying in $\mu$. If both and lie in $\mu$, then $\fkl$ is an addable node and we are in case (1). If neither lies in $\mu$, then is a removable node, and we are in case (2). If ${\ensuremath{\mathtt{SE}(\fkl)}}\in\mu\notni{\ensuremath{\mathtt{SW}(\fkl)}}$, then we are in case (3), while if ${\ensuremath{\mathtt{SE}(\fkl)}}\notin\mu\ni{\ensuremath{\mathtt{SW}(\fkl)}}$ then we are in case (4).
Now we analyse the four cases.
1. Since $\mu$ has an addable node in column $i-g$, $s_\mu$ can be written in the form $s_is_{\mu^+}$. We have $s_i^2=-2s_i$, and the result follows.
2. By definition (and the fact that $s_i$ and $s_j$ commute when $j\neq i\pm1$) $s_{\mu^-}=s_is_\mu$.
3. \[case3\]
1. We proceed by induction on $i$. Consider the case $a=g$. In this case $s_\mu$ only involves terms $s_j$ for $j\gs i+2$, so (since $s_i{m}=0$) we get $s_is_\mu{m}=0$.
Now suppose $a<g$, and set $$\bar\mu=(\mu_1,\dots,\mu_a,i-g+a,\mu_{a+2},\dots,\mu_g).$$ Then $\bar\mu$ satisfies the inductive hypothesis, with $a$ replaced by $a+1$ and $i$ replaced by $i-2$. Furthermore, we can write $$s_\mu=s_{\mu_{a+1}+g-a}\dots s_{i-1}s_{\bar\mu}.$$ $\bar\mu$ has an addable node $(a+1,i-g+a+1)$ in column $i-g$, so $s_{\bar\mu}{m}$ can be written in the form $s_ix$ for some $x\in{\mathscr{M}^{(f,g)}}$. So we have $$\begin{aligned}
s_is_\mu{m}&=s_is_{\mu_{a+1}+g-a}\dots s_{i-1}s_ix\\
&=s_{\mu_{a+1}+g-a}\dots s_{i-2}s_is_{i-1}s_ix\\
&=s_{\mu_{a+1}+g-a}\dots s_{i-2}s_ix\tag*{by \cref{ann}}\\
&=s_{\mu_{a+1}+g-a}\dots s_{i-2}s_{\bar\mu}{m}\\
&=0\tag*{by induction.}\end{aligned}$$
2. We use induction on $w-a$. If $w=a+1$, then we have $s_\mu=s_{i-1}s_{\mu^{a,w}}$, and $\mu^{a,w}$ has an addable node $(a+1,i-g+a+1)$ in column $i-g$, so $s_{\mu^{a,w}}{m}$ can be written as $s_ix$ for some $x\in{\mathscr{M}^{(f,g)}}$. So by \[ann\] $$s_is_\mu{m}=s_is_{i+1}s_ix=s_ix=s_{\mu^{a,w}}{m}.$$
Now assume that $w>a+1$, and as above set $$\bar\mu=(\mu_1,\dots,\mu_a,i-g+a,\mu_{a+2},\dots,\mu_g).$$ Then $\bar\mu$ satisfies the inductive hypothesis, with $a$ replaced by $a+1$ and $i$ replaced by $i-2$, and with the same value of $w$, yielding $$\bar\mu^{a+1,w}=(\mu_1,\dots,\mu_a,i-g+a,i-g+a-1,\mu_{a+2}+1,\dots,\mu_{w-1}+1,\mu_{w+1},\dots,\mu_g).$$ As above we write $s_{\bar\mu}=s_ix$, and obtain $$\begin{aligned}
s_is_\mu{m}&=s_{\mu_{a+1}+g-a}\dots s_{i-2}s_{\bar\mu}{m}\\
&=s_{\mu_{a+1}+g-a}\dots s_{i-3}s_{\bar\mu^{a+1,w}}{m}\tag*{by induction}\\
&=s_{\mu^{a,w}}{m}.\end{aligned}$$
4. This is symmetrical to case (\[case3\]), replacing partitions with their conjugates and swapping $f$ and $g$.
We now seek to re-phrase \[actionts\] so that for a given $\la,\mu$ we can write down the coefficient of $s_\la{m}$ in $s_is_\mu{m}$. Fix $i$, and suppose that as in part (3) of \[actionts\] we have $\mu_a>i-g+a>\mu_{a+1}$ for some $1\ls a\ls g$. Let $w>a$ be minimal such that $\mu_w=i-g+2a-w$, assuming there is such a $w$, and write $\mu{\stackrel{i}\rightharpoondown}\mu^{a,w}$, where $\mu^{a,w}$ is as defined in \[actionts\].
Now recall some notation from \[defnsec\]: if $j$ is fixed and $\la$ is a partition with an addable node $\fkl$ in column $j$, then ${X_{\la}}$ denotes the set of all integers $x$ such that $\la$ has a removable node $\fkn$ in column $j+x$, with ${\ensuremath{\operatorname{ht}(\fkn)}}={\ensuremath{\operatorname{ht}(\fkl)}}-1$, and ${\ensuremath{\operatorname{ht}(\fkp)}}<\fkl$ for all nodes $\fkp$ in all columns between $j$ and $j+x$. ${{X_{\la}}^+}$ denotes the set of positive elements of ${X_{\la}}$, and ${{X_{\la}}^-}$ the set of negative elements. Given $x\in{{X_{\la}}^+}$, ${\la^{[x]}}$ is the partition obtained from $\la$ by removing the highest node in each column from $j+1$ to $j+x$; ${\la^{[x]}}$ is defined similarly for $x\in{{X_{\la}}^-}$.
\[harp\] Suppose $\la,\mu\in{\scrp_{f,g}}$ and $1\ls i<k$, and set $j=i-g$. Then $\mu{\stackrel{i}\rightharpoondown}\la$ if and only if $\la$ has an addable node in column $j$ and $\mu={\la^{[x]}}$ for some $x\in{{X_{\la}}^-}$.
$(\Rightarrow)$
: If $\mu{\stackrel{i}\rightharpoondown}\la$ then we have $$\la=\mu^{a,w}=(\mu_1,\dots,\mu_a,j+a,\mu_{a+1}+1,\dots,\mu_{w-1}+1,\mu_{w+1},\dots,\mu_g),$$ where $\mu_a>j+a>\mu_{a+1}$ and $w>a$ is minimal such that $\mu_w\gs j+2a-w$. Observe that $\la$ has an addable node $\fkl=(a+1,j+a+1)$ in column $j$. The choice of $w$ implies that $\mu_{w-1}=\mu_w=j+2a-w$, and so $\la$ has a removable node $\fkn=(w,j+2a-w+1)$ in column $j+x$, where $x=2(a-w)+1$. Furthermore, ${\ensuremath{\operatorname{ht}(\fkn)}}=j+2a={\ensuremath{\operatorname{ht}(\fkl)}}-1$, and we claim that all the nodes in columns $j+x+1,\dots,j-1$ of $\la$ have height at most $j+2a$: if this condition fails, then it must fail for a node of the form $(x,\la_x)$ with $a+2\ls x\ls w-1$, but for $x$ in this range we have ${\ensuremath{\operatorname{ht}((x,\la_x))}}=x+\la_x=x+\mu_{x-1}+1\ls x+j+2a-x$. So $x\in{{X_{\la}}^-}$.
To construct $\mu$ from $\la$ we remove the nodes $$\begin{aligned}
&(w,\mu_w+1),(w,\mu_w+2),\dots,(w,\mu_{w-1}+1),\\
&(w-1,\mu_{w-1}+1),(w-1,\mu_{w-1}+2),\dots,(w-1,\mu_{w-2}+1),\\
&\dots\\
&(a+2,\mu_{a+2}+1),(a+2,\mu_{a+2}+2),\dots,(a+2,\mu_{a+1}+1),\\
&(a+1,\mu_{a+1}+1),(a+1,\mu_{a+1}+2),\dots,(a+1,j+a),\end{aligned}$$ which lie in columns $j+x,\dots,j-1$ respectively. So $\mu={\la^{[x]}}$.
$(\Leftarrow)$
: Suppose $\la$ has an addable node in column $j$. Writing this node as $(a+1,\la_{a+1}+1)$, we have $a=0$ or $\la_a>\la_{a+1}$, and $\la_{a+1}-a=j$.
Now suppose $x\in{{X_{\la}}^-}$. Then $\la$ has a removable node $\fkn=(w,\la_w)$ in column $j+x$, and $w>a$ since $x<0$. Then $\la_w>\la_{w+1}$ and $\la_w-w=j+x$, and (since ${\ensuremath{\operatorname{ht}(\fkn)}}={\ensuremath{\operatorname{ht}(\fkl)}}-1$) $w+\la_w=a+1+\la_{a+1}$. The fact that ${\ensuremath{\operatorname{ht}(\fkp)}}<{\ensuremath{\operatorname{ht}(\fkl)}}$ for all nodes $\fkp$ in columns $j+x+1,\dots,j-1$ implies in particular that ${\ensuremath{\operatorname{ht}((x,\la_x))}}<{\ensuremath{\operatorname{ht}(\fkl)}}$ for $a+2\ls x\ls w-1$, i.e. $x+\la_x\ls a+1+\la_{a+1}$.
Constructing ${\la^{[x]}}$ involves removing the nodes $$\begin{aligned}
(w,\la_w),{}&(w-1,\la_w),(w-1,\la_w+1),\dots,(w-1,\la_{w-1}),\\
&(w-2,\la_{w-1}),(w-2,\la_{w-1}+1),\dots,(w-2,\la_{w-2}),\\
&\dots\\
&(a+1,\la_{a+2}),(a+1,\la_{a+2}+1),\dots,(a+1,\la_{a+1}),\end{aligned}$$ so $${\la^{[x]}}=(\la_1,\dots,\la_a,\la_{a+2}-1,\dots,\la_w-1,\la_w-1,\la_{w+1},\dots,\la_g).$$ Setting $\mu={\la^{[x]}}$, the (in)equalities observed above for $\la$ give $\mu_a>j+a>\mu_{a+1}$, $\mu_x<j+2a-x$ for $a<x<w$ and $\mu_w=j+2a-w$. Furthermore, $\la$ equals the partition $\mu^{a,w}$, so $\mu{\stackrel{i}\rightharpoondown}\la$.
We now note a counterpart to this for part (4) of \[actionts\], which follows by conjugating partitions. Suppose that $\mu\in{\scrp_{f,g}}$ satisfies the conditions of (4), and that $\mu^{w,a}$ is defined, and write $\mu{\stackrel{i}\rightharpoonup}\mu^{w,a}$.
\[prah\] Suppose $\la,\mu\in{\scrp_{f,g}}$ and $1\ls i<k$, and set $j=i-g$. Then $\mu{\stackrel{i}\rightharpoonup}\la$ if and only if $\la$ has a removable node in column $j$ and $\mu={\la^{[x]}}$ for some $x\in{{X_{\la}}^+}$.
The following result is now immediate from \[actionts,harp,prah\].
\[appear\] Suppose $\mu\in{\scrp_{f,g}}$, and write $s_is_\mu{m}$ as a linear combination ${\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}a_\la s_\la{m}$. Then $a_\la$ equals:
- $-2$, if $\la$ has an addable node in column $i-g$ and $\mu=\la$;
- $1$, if $\la$ has an addable node in column $i-g$, and
- $\mu$ is the partition obtained by adding this node, or
- $\mu={\la^{[x]}}$ for some $x\in{X_{\la}}$;
- $0$, otherwise.
As with \[mainsn1\], we proceed by downwards induction on $|\mu|$. If $\mu=(f^g)$, then $s_\mu=t_\mu$ and the result follows, since clearly ${\operatorname{i}_{\la\la}}=1$.
Assuming $\mu\neq(f^g)$, $\mu$ has an addable node in column $j$ for some $-g<j<f$. We let $\mu^+$ denote the partition obtained by adding this addable node. Then $t_\mu=t_{j+g}t_{\mu^+}$, and by induction $$t_{\mu^+}{m}={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}{\operatorname{i}_{\la\mu^+}} s_\la{m}.$$ As before, we write ${{\scrp_{f,g}}^+}$ for the set of $\la\in{\scrp_{f,g}}$ having an addable node in column $j$, and for $\la\in{{\scrp_{f,g}}^+}$ we let $\la^+$ denote the partition obtained by adding this addable node. We have $$\begin{aligned}
t_\mu{m}&={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}{\operatorname{i}_{\la\mu^+}}(s_{j+g}+1)s_\la{m}\\
&={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}{\operatorname{i}_{\la\mu^+}} s_\la{m}+{\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^+}}$\hfil}}}\left({\operatorname{i}_{\la^+\mu^+}}-2{\operatorname{i}_{\la\mu^+}}+\sum_{x\in{X_{\la}}}{\operatorname{i}_{{\la^{[x]}}\mu^+}}\right)s_\la{m}\tag*{by \cref{appear}}\\
&={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}\setminus{{\scrp_{f,g}}^+}}$\hfil}}}{\operatorname{i}_{\la\mu^+}} s_\la{m}+{\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^+}}$\hfil}}}\left({\operatorname{i}_{\la^+\mu^+}}-{\operatorname{i}_{\la\mu^+}}+\sum_{x\in{X_{\la}}}{\operatorname{i}_{{\la^{[x]}}\mu^+}}\right)s_\la{m}\\
&={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}\setminus{{\scrp_{f,g}}^+}}$\hfil}}}{\operatorname{i}_{\la\mu}} s_\la{m}+{\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^+}}$\hfil}}}{\operatorname{i}_{\la\mu}} s_\la{m}\tag*{by \cref{noadd,recur}}\\
&={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}{\operatorname{i}_{\la\mu}} s_\la{m}.\end{aligned}$$ \[mainsn2\] follows by induction.
The sum of the elements $t_\mu{m}$
----------------------------------
It will be critical in the next section to be able to express ${\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}t_\la{m}$ in terms of the basis elements $s_\la{m}$. To enable us to do this, we define an integer-valued function ${F}$ on partitions.
If $\la$ is a partition, we let ${\partial^{a}\la}$ denote the partition obtained by removing the first $a$ parts of $\la$, i.e. the partition $(\la_{a+1},\la_{a+2},\dots)$, and let ${\partial_{b}\la}$ denote the partition obtained by reducing all the parts by $b$ (and deleting all negative parts), i.e. the partition $(\max\{\la_1-b,0\},\max\{\la_2-b,0\},\dots)$. We write ${\partial^{a}_{b}\la}={\partial^{a}{\partial_{b}\la}}$.
Now we can define ${F}$ recursively. Set ${F}(\varnothing)=1$. Given a partition $\la\neq\varnothing$, take a node $(a,b)$ of $\la$ for which $a+b$ is maximal (call this a *highest* node of $\la$) and define $$\sigma={\partial^{a}\la},\qquad\tau={\partial_{b}\la}.$$ Now set $${F}(\la)=\binom{a+b}a{F}(\sigma){F}(\tau).$$
\[fwelld\] ${F}(\la)$ is well-defined, i.e. does not depend on the choice of $(a,b)$.
Suppose $(a,b)$ and $({\hat a},{\hat b})$ are both highest nodes of $\la$; in particular, $a+b={\hat a}+{\hat b}$. Assume without loss that $a<{\hat a}$. Let $\sigma={\partial^{a}\la}$ and $\tau={\partial_{b}\la}$ as above, and define $\hat\sigma,\hat\tau$ correspondingly. By induction ${F}(\sigma),{F}(\tau),{F}(\hat\sigma),{F}(\hat\tau)$ are well-defined, and we must show that $${F}(\sigma){F}(\tau)\binom{a+b}a={F}(\hat\sigma){F}(\hat\tau)\binom{{\hat a}+{\hat b}}{\hat a}.$$ Now $({\hat a}-a,{\hat b})$ is a highest node of $\sigma$, so we have $${F}(\sigma)={F}(\hat\sigma){F}(\xi)\binom{{\hat a}-a+{\hat b}}{{\hat a}-a},$$ where $\xi={\partial^{a}_{{\hat b}}\la}$. Also, $(a,b-{\hat b})$ is a highest node of $\hat\tau$, so that $${F}(\hat\tau)={F}(\xi){F}(\tau)\binom{a+b-{\hat b}}a.$$ So (since $a+b={\hat a}+{\hat b}$) we obtain $${F}(\sigma){F}(\tau){F}(\xi)\binom{\hat a}a={F}(\hat\sigma){F}(\hat\tau){F}(\xi)\binom b{\hat b}.$$ Dividing both sides by ${F}(\xi){\hat a}!b!$ (which is obviously positive) and multiplying by $(a+b)!({\hat a}-a)!=({\hat a}+{\hat b})!({\hat b}-b)!$ gives the result.
In the notation of the above lemma, we have $${F}(\la)={F}(\hat\sigma){F}(\xi){F}(\tau){(a,{\hat a}-a,{\hat b})!},$$ where we use ${(x,y,z)!}$ to denote the trinomial coefficient $\mfrac{(x+y+z)!}{x!y!z!}$. We will refer to this as ‘factorising using the nodes $(a,b)$ and $({\hat a},{\hat b})$’. This readily generalises to factorising using any number of highest nodes, with a corresponding multinomial coefficient.
Our main objective is to prove the following statement.
\[snsum\] Suppose $f,g$ are non-negative integers. Then $$\sum_{\mu\in{\scrp_{f,g}}}t_\mu{m}={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}{F}(\la)s_\la{m}.$$
We prove this using the following recurrence, in which we again use the notation ${X_{\la}}$ and ${\la^{[x]}}$ from \[defnsec\].
\[fprecur\] Assume that $\la$ is a partition with an addable node $\fkl$ in column $j$, and let $\la^+$ denote the partition obtained by adding this node. Then $${F}(\la^+)+\sum_{x\in{X_{\la}}}{F}({\la^{[x]}})=2{F}(\la).$$
Write $\fkl$ as $(a,b)$. Suppose first of all that $\fkl$ is not the unique highest node of $\la^+$. Take a highest node $\fkm=(\hat a,\hat b)\neq\fkl$, and suppose without loss that $\hat a<a$. Since the height of $\fkm$ is at least the height of $\fkl$, $\hat b-\hat a-j$ is strictly larger than any element of ${X_{\la}}$. So $\fkm$ is a highest node of ${\la^{[x]}}$ for every $x\in{X_{\la}}$, as well as of $\la$ and $\la^+$. By induction the result holds for the partition $(\la_{\hat a+1},\la_{\hat a+2},\dots)$, and so it holds for $\la$, factorising ${F}(\la)$ (and also ${F}(\la^+)$ and ${F}({\la^{[x]}})$ for each $x$) using the node $(\hat a,\hat b)$.
So we can assume that $\fkl$ is the unique highest node of $\la^+$. Let $(a_1,b_1),\dots,(a_r,b_r)$ be the nodes of $\la$ to the right of $\fkl$ for which $a_x+b_x=a+b-1$; order these nodes so that $a_1>\dots>a_r$. Note that if $r\gs1$, then automatically $(a_1,b_1)=(a-1,b)$. Similarly, let $(c_1,d_1),\dots,(c_s,d_s)$ be the nodes to the left of $\fkl$ for which $c_x+d_x=a+b-1$.
We compute ${F}(\la^+)$ by first factorising using the node $\fkl$. We may also factorise ${F}(\la)$ using the nodes $(a-1,b)$ and $(a,b-1)$ (omitting either or both of these, if $a$ or $b$ equals $1$), and we readily obtain $${F}(\la^+)=\frac{a+b}{ab}{F}(\la).\tag*{{(\textdagger)}}$$ Now we consider the partitions ${\la^{[x]}}$, for $x\in{{X_{\la}}^+}$. From our assumptions so far, ${{X_{\la}}^+}$ is the set of integers $x_i=b_i-a_i-b+a$, for $i=1,\dots,r$. Choose such an $i$, and set $\kappa={\la^{[x_i]}}$. We wish to compare ${F}(\kappa)$ with ${F}(\la)$. To do this, we define $$\gamma={\partial^{a_{i+1}}_{b-1}\la},\qquad
\delta={\partial^{a_{i+1}}_{b-1}\kappa}$$ (where we put $a_{i+1}=0$ if $i=r$), and we claim that ${F}(\kappa)/{F}(\la)={F}(\delta)/{F}(\gamma)$. If $s=0$ and $i=r$, then this is trivial since $\gamma=\la$ and $\delta=\kappa$; assuming instead that $s\gs1$ or $i<r$, factorising both ${F}(\kappa)$ and ${F}(\la)$ at the nodes $(c_1,d_1),\dots,(c_s,d_s),(a_{i+1},b_{i+1}),\dots,(a_r,b_r)$ (i.e. at all the highest nodes of $\kappa$) gives the result.
Now $\gamma$ has highest nodes $(a-1-a_{i+1},1)$ and $(a_i-a_{i+1},b_i-b+1)$ (which coincide if $i=1$), and we factorise ${F}(\gamma)$ at these nodes, obtaining $${F}(\gamma)={F}(\mu){F}(\nu)\frac{(a-a_{i+1})!}{(a-1-a_i)!(a_i-a_{i+1})!},$$ where $\mu={\partial^{a_i-a_{i+1}}_{1}\gamma}$ and $\nu={\partial_{b_i-b+1}\gamma}$. $\dfrac{{F}(\delta)}{{F}(\gamma)}=\dfrac{a_i-a_{i+1}}{(a-a_i)(a-a_{i+1})}$. In the case where $a_i=a_{i+1}+1$, we have $\mu=\delta$ and $\nu=\varnothing$, so the factorisation of ${F}(\gamma)$ above gives the result. If $a_i>a_{i+1}+1$, then $\delta$ has a highest node $(a_i-a_{i+1}-1,b_i-b+1)$, and factorising at this node we get $${F}(\delta)={F}(\mu){F}(\nu)\binom{a_i-a_{i+1}+b_i-b}{a_i-a_{i+1}-1}.$$ The claim follows from this factorisation and the factorisation of ${F}(\gamma)$ above, using the fact that $a_i+b_i=a+b-1$. This claim yields $$\frac{{F}(\kappa)}{{F}(\la)}=\frac{{F}(\delta)}{{F}(\gamma)}=\frac1{a-a_i}-\frac1{a-a_{i+1}},$$ and summing over $i$ we obtain $$\frac{\sum_{x\in{{X_{\la}}^+}}{F}({\la^{[x]}})}{{F}(\la)}=\frac1{a-a_1}-\frac1{a-a_{r+1}}=1-\frac1a.$$ A symmetrical calculation applies to partitions ${\la^{[x]}}$ for $x\in{{X_{\la}}^-}$, yielding $$\frac{\sum_{x\in{{X_{\la}}^-}}{F}({\la^{[x]}})}{{F}(\la)}=1-\frac1b.$$ Adding these two results together and combining with [()]{} gives the result.
\[sikills\] For $1\ls i<k$ we have $$s_i{\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}{F}(\la)s_\la{m}=0.$$
As above, write ${{\scrp_{f,g}}^+}$ for the set of $\la\in{\scrp_{f,g}}$ having an addable node in column $i-g$, and for each such $\la$ let $\la^+$ denote the partition obtained by adding this addable node. Then by \[appear\] we have $$s_i{\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}{F}(\la)s_\la{m}={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{{\scrp_{f,g}}^+}}$\hfil}}}\left(-2{F}(\la)+{F}(\la^+)-\sum_{x\in{X_{\la}}}{F}({\la^{[x]}})\right)s_\la{m},$$ and each summand on the right-hand side is zero by \[fprecur\].
shows that $\sum_\la{F}(\la)s_\la{m}$ is killed by all of $s_1,\dots,s_{k-1}$, and hence lies in a trivial submodule of ${\mathscr{M}^{(f,g)}}$. But ${\mathscr{M}^{(f,g)}}$ is a permutation module for a transitive action of $\sss k$, and so has a unique trivial submodule, spanned by the sum of the elements of the permutation basis, i.e. $\sum_\la t_\la{m}$. So the two sides agree up to multiplication by a scalar. On the other hand, it is clear that when the right-hand side is expressed as a linear combination of the $t_\la{m}$, the coefficient of $t_\varnothing{m}$ is ${F}(\varnothing)=1$, so in fact the two sides are equal.
Results of Kenyon, Kim, Mészáros, Panova and Wilson
---------------------------------------------------
As we have mentioned, cover-inclusive Dyck tilings were introduced by Kenyon and Wilson in [@kw], and in fact the main result in their paper has a bearing on the results in the present paper.
First we explain how [cover-expansive Dyck tiling]{}s appear in disguise in [@kw]. Given a partition $\la$, we define a sequence of parentheses [`(`]{} and [`)`]{}, by ‘reading along the boundary’ of $\la$, as follows. Working from left to right, for each node $\fka$ such that is not a node of $\la$, we write a [`)`]{}, and for each node $\fka$ such that is not a node of $\la$ we write a [`(`]{}. (If neither nor is a node of $\la$, then we write [`)`]{}[`(`]{}.) We then append an infinite string of [`(`]{}s at the start of the sequence, and an infinite sequence of [`)`]{}s at the end. The resulting doubly infinite sequence is called the *parenthesis sequence* of $\la$.
Let $\la=(5,3^2,1)$. Then the parenthesis sequence of $\la$ is $\cdots{\texttt(}{\texttt(}{\texttt(}{\texttt)}{\texttt(}{\texttt)}{\texttt)}{\texttt(}{\texttt(}{\texttt)}{\texttt)}{\texttt(}{\texttt)}{\texttt)}\cdots$, as we see from the Young diagram, in which we mark the boundary in bold, extending it infinitely far to the north-west and north-east; segments $\diagdown$ contribute a [`(`]{} to the parenthesis sequence, while segments $\diagup$ contribute a [`)`]{}. $$\begin{tikzpicture}[scale=.45]
\foreach\x in{0,1}\draw(-\x,\x)--++(5,5);
\foreach\x in{2,3}\draw(-\x,\x)--++(3,3);
\draw(0,0)--(-4,4)--++(1,1)--++(4,-4);
\foreach\x in{2,3}\draw(\x,\x)--++(-3,3);
\foreach\x in{4,5}\draw(\x,\x)--++(-1,1);
\draw[very thick](-7,7)--++(3,-3)--++(1,1)--++(1,-1)--++(2,2)--++(2,-2)--++(2,2)--++(1,-1)--++(2,2);
\draw[very thick,dashed](-8,8)--(-7,7);
\draw[very thick,dashed](7,7)--(8,8);
\end{tikzpicture}$$
Now partition the parenthesis expression for $\la$ into pairs, in the usual way for pairing up parentheses: each [`(`]{} is paired with the first subsequent [`)`]{} for which there are an equal number of [`)`]{}s and [`(`]{}s in between.
Continuing the last example, we illustrate the pairs by numbering the ${\texttt)}$s in increasing order, and numbering each ${\texttt(}$ with the same number as its corresponding ${\texttt)}$: $$\begin{array}{cccccccccccccccccc}
\cdots&{\texttt(}&{\texttt(}&{\texttt(}&{\texttt(}&{\texttt)}&{\texttt(}&{\texttt)}&{\texttt)}&{\texttt(}&{\texttt(}&{\texttt)}&{\texttt)}&{\texttt(}&{\texttt)}&{\texttt)}&{\texttt)}&\cdots\\
\cdots&8&7&3&1&1&2&2&3&5&4&4&5&6&6&7&8&\cdots
\end{array}$$
We remark that in [@kw] and elsewhere, finite parenthesis expressions are used: instead of appending infinite strings of [`(`]{} and [`)`]{} at the start and end of the expression obtained from the partition, one appends sufficiently long finite strings that the resulting expression is *balanced*, meaning that it contains equally many [`(`]{}s and [`)`]{}s, and that any initial segment contains at least as many [`(`]{}s as [`)`]{}s. We find our convention more straightforward in the present context, and translation between the two conventions is very easy.
Now we recall a definition from [@kw]; it is phrased there in terms of finite Dyck paths, but it is more convenient for us to phrase it in terms of partitions. Given two partitions $\la,\mu$, write $\la{\stackrel{\textup{\texttt{()}}}\leftarrow}\mu$ if the parenthesis expression for $\la$ can be obtained from that for $\mu$ by taking some of the pairs ${\texttt(}\cdots{\texttt)}$ and reversing them to get ${\texttt)}\cdots{\texttt(}$.
With this definition, we can describe the connection to [cover-expansive Dyck tiling]{}s. The following is not hard to prove, and we leave it as an exercise for the reader.
\[cedtkw\] Suppose $\la,\mu$ are partitions. Then $\la{\stackrel{\textup{\texttt{()}}}\leftarrow}\mu$ if and only if $\la\supseteq\mu$ and there is a [cover-expansive Dyck tiling]{} of ${\ensuremath{\la\backslash\mu}}$.
Now we can describe the relationship between our results and those of Kenyon and Wilson. Suppose $f,g$ are positive integers as in \[ypmsubsec\], and define matrices $N,P$ with rows and columns indexed by ${\scrp_{f,g}}$, and with $$N_{\la\mu}=(-1)^{|\la|+|\mu|}{\operatorname{e}_{\la\mu}},\qquad P_{\la\mu}={\operatorname{i}_{\la\mu}}.$$ Then \[mainsn1,mainsn2\] show that $N$ and $P$ are mutual inverses. In fact, Kenyon and Wilson prove this result, but in a slightly different form. They define a matrix $M$ with $M_{\la\mu}=1$ if $\la{\stackrel{\textup{\texttt{()}}}\leftarrow}\mu$ and $0$ otherwise, and show (using a different recursion for ${\operatorname{i}_{\la\mu}}$ from ours) that $(M^{-1})_{\la\mu}=(-1)^{|\la|+|\mu|}{\operatorname{i}_{\la\mu}}$ [@kw Theorem 1.5]. In view of \[cedtkw\], we have $M_{\la\mu}={\operatorname{e}_{\la\mu}}$, and so the main result of [@kw] shows that \[mainsn1,mainsn2\] are equivalent. (There is another minor difference with our results, in that instead of the set ${\scrp_{f,g}}$ of partitions lying inside a rectangle, they consider the set of partitions lying inside a partition of the form $(c,c-1,\dots,1)$; however, one can easily show the equivalence of the two results by taking very large bounding partitions and appropriate submatrices.)
Of course, using this result we could remove a lot of the work in this paper, but we prefer to keep our proofs, thereby keeping our paper self-contained and providing a new (albeit longer) proof of the main result of [@kw].
We remark that Kim [@kim §7] observes a $q$-analogue of the above result (for which he also credits Konvalinka); in this, the integer ${\operatorname{i}_{\la\mu}}$ is replaced with the polynomial $${\operatorname{i}_{\la\mu}}(q)=\sum_{T\in{\scri(\la,\mu)}}q^{|T|},$$ where $|T|$ denotes the number of tiles in the tiling $T$, and similarly for ${\operatorname{e}_{\la\mu}}$. It should be possible to give a new proof of this result using our techniques: in our bijective results (\[noadd,bij1,bij1a,bij2,bij2a,bij3,exp1\]) it is easy to keep track of how the number of tiles in a tiling changes, so $q$-analogues of our recursive results can easily be proved. Replacing the group algebra of the symmetric group with the Hecke algebra of type $A$ should enable $q$-analogues of \[mainsn1,mainsn2\] to be proved, yielding a new proof of Kim’s and Konvalinka’s result. We leave the details to the proverbial interested reader.
Another result from the literature relates to \[snsum\]. To describe this relationship, we give an alternative characterisation of the function ${F}$, for which we need another definition. Given a partition $\la$, construct the parenthesis sequence of $\la$ and divide it into pairs as above, and define a partial order on the set of pairs by saying that the pair $\underset a{{\texttt(}}\cdots\underset a{{\texttt)}}$ is larger than the pair $\underset b{{\texttt(}}\cdots\underset b{{\texttt)}}$ if the former is nested inside the latter: $\underset b{\texttt(}\cdots\underset{\vphantom ba}{\texttt(}\cdots\underset{\vphantom ba}{\texttt)}\cdots\underset b{\texttt)}$. Call this partially ordered set $\calp(\la)$.
Continuing the last example and retaining the labelling for the pairs in the parenthesis sequence for $\la$, we have the following Hasse diagram for $\calp(\la)$. $$\begin{tikzpicture}[scale=1.2]
\draw(0,0)--++(2,-2)--++(0,2);
\draw(1,0)--++(0,-1);
\draw(2,-2)--++(1,1);
\draw(2,-2)--++(0,-1);
\draw[dashed](2,-3)--++(0,-1);
\draw(0,0)node[fill=white]{1};
\draw(1,0)node[fill=white]{2};
\draw(2,0)node[fill=white]{4};
\draw(1,-1)node[fill=white]{3};
\draw(2,-1)node[fill=white]{5};
\draw(3,-1)node[fill=white]{6};
\draw(2,-2)node[fill=white]{7};
\draw(2,-3)node[fill=white]{8};
\end{tikzpicture}$$
Now we have the following.
\[fpchord\] Suppose $\la$ is a partition. Then ${F}(\la)$ is the number of linear extensions of the poset $\calp(\la)$.
A proof of this was sketched by David Speyer in a response to the author’s MathOverflow question [@mo]. In a comment to the same answer, Philippe Nadeau pointed out that this provides a non-recursive expression for ${F}(\la)$. To state this, we let $\calp_N(\la)$ denote the poset obtained by taking the $N$ largest elements of $\calp(\la)$. Then for large enough $N$ the number of linear extensions of $\calp(\la)$ equals the number of linear extensions of $\calp_N(\la)$. Given a pair $p\in\calp(\la)$, define its *length* $l(p)$ to be $1$ plus the number of intervening pairs. Now an easy exercise [@knuth p.70, Exercise 20] gives the following.
\[noext\] Suppose $\la$ is a partition, and $N\gg0$. Then the number of linear extensions of the poset $\calp_N(\la)$ equals $$\frac{N!}{\prod_{p\in\calp_N(\la)}l(p)}.$$
Taking $N=8$ in the last example and labelling each pair by its length, we have $$\begin{tikzpicture}[scale=1.2]
\draw(0,0)--++(2,-2)--++(0,2);
\draw(1,0)--++(0,-1);
\draw(2,-2)--++(1,1);
\draw(2,-2)--++(0,-1);
\draw[dashed](2,-3)--++(0,-1);
\draw(0,0)node[fill=white]{1};
\draw(1,0)node[fill=white]{1};
\draw(2,0)node[fill=white]{1};
\draw(1,-1)node[fill=white]{3};
\draw(2,-1)node[fill=white]{2};
\draw(3,-1)node[fill=white]{1};
\draw(2,-2)node[fill=white]{7};
\draw(2,-3)node[fill=white]{8};
\end{tikzpicture}$$ So ${F}(\la)$ is the number of linear extensions of $\calp_8(\la)$, which is $$\frac{8!}{2\times3\times7\times8}=120.$$
The author is very grateful to David Speyer and Philippe Nadeau for these comments (and also to Gjergji Zaimi for similar comments in the same thread), which inspired the introduction of Dyck tilings in the present work.
Given \[mainsn2\] and \[fpchord\], \[snsum\] is equivalent to the following theorem.
\[kimmain\] Suppose $\la$ is a partition and $N\gg0$. Then $$\sum_{\mu\subseteq\la}{\operatorname{i}_{\la\mu}}=\frac{N!}{\prod_{p\in\calp_N(\la)}l(c)}.$$
This statement (in fact, a $q$-analogue) was conjectured by Kenyon and Wilson [@kw Conjecture 1]. It was proved inductively by Kim [@kim Theorem 1.1], and then a bijective proof was given by Kim, Mészáros, Panova and Wilson [@kmpw Theorem 1.1]. We retain our proof of \[snsum\] to keep the paper self-contained (and so that \[snsum\] can be proved without reference to tilings), and we note in passing that this yields a new proof of \[kimmain\].
The homogeneous Garnir relations {#klrsec}
================================
In this section we apply our earlier results to the study of representations of the (cyclotomic) quiver Hecke algebras (also known as *KLR algebras*) of type $A$, and in particular the homogeneous Garnir relations for the *universal graded Specht modules* of Kleshchev, Mathas and Ram [@kmr].
The quiver Hecke algebra
------------------------
Suppose $\bbf$ is a field of characteristic $p$, $e\in\{0,2,3,4,\dots\}$ and $n\in\bbn$. The *quiver Hecke algebra* $\calr_n$ of type $A$ is a unital associative $\bbf$-algebra with a generating set $$\{y_1,\dots,y_n\}\cup\{\psi_1,\dots,\psi_{n-1}\}\cup\lset{e(\bfi)}{\bfi\in({\mathbb{Z}/e\mathbb{Z}})^n}$$ and a somewhat complicated set of defining relations, which may be found in [@bk] or [@kmr], for example. These relations allow one to write down a basis for $\calr_n$: to do this, choose and fix a reduced expression $w=t_{i_1}\dots t_{i_r}$ for each $w\in\sss n$, and set $\psi_w=\psi_{i_1}\dots\psi_{i_r}$. Then the set $$\lset{\psi_wy_1^{c_1}\dots y_n^{c_n}e(\bfi)}{w\in\sss n,\ c_1,\dots,c_n\gs0,\ \bfi\in({\mathbb{Z}/e\mathbb{Z}})^n}$$ is a basis for $\calr_n$.
The quiver Hecke algebra has attracted considerable attention in recent years, thanks to the astonishing result of Brundan and Kleshchev [@bk Main Theorem] that when $p$ does not divide $e$, a certain finite-dimensional ‘cyclotomic’ quotient of $\calr_n$ is isomorphic to the cyclotomic Hecke algebra of type $A$ (as introduced by Ariki–Koike [@arko] and Broué–Malle [@brma]) defined at a primitive $e$th root of unity in $\bbf$; when $e=p$, there is a corresponding isomorphism to the degenerate cyclotomic Hecke algebra (which includes the group algebra $\bbf\sss n$ as a special case). This in particular shows that these Hecke algebras (which include the group algebra of the symmetric group) are non-trivially $\bbz$-graded, and has initiated the study of the graded representation theory of these algebras.
A crucial role in the representation theory of Hecke algebras is played by the *Specht modules*. Brundan, Kleshchev and Wang [@bkw] showed how to work with the Specht modules in the quiver Hecke algebra setting, demonstrating in particular that these modules are graded. Kleshchev, Mathas and Ram [@kmr] gave a presentation for each Specht module with a single generator and a set of homogeneous relations. These relations include ‘homogeneous Garnir relations’; although these are in general simpler than the classical Garnir relations for ungraded Specht modules (which go back to [@garni] in the symmetric group case), the expressions given for these relations in [@kmr] are quite complicated. The purpose of this section is to use the results of the previous section to give a simpler expression for each homogeneous Garnir relation. In computations with graded Specht modules using the author’s GAP programs, implementing these simpler expressions has been observed to have some benefits in terms of computational efficiency.
We now define the cyclotomic quotients of the quiver Hecke algebras and their Specht modules. Choose a positive integer $l$, and an $l$-tuple $(\kappa_1,\dots,\kappa_l)\in({\mathbb{Z}/e\mathbb{Z}})^l$. For each $a\in{\mathbb{Z}/e\mathbb{Z}}$ define $\kappa(a)$ to be the number of values of $j$ for which $\kappa_j=a$, and define $\calr^\kappa_n$ to be the quotient of $\calr_n$ by the ideal generated by the elements $y_1^{\kappa(i_1)}e(\bfi)$ for all $i\in({\mathbb{Z}/e\mathbb{Z}})^n$. We use the same notation for the standard generators of $\calr_n$ and their images in $\calr^\kappa_n$.
Row permutation modules and Specht modules
------------------------------------------
From now on, we stick to the case $l=1$, which corresponds to the Iwahori–Hecke algebra of type $A$; there is no essential difference in the homogeneous Garnir relations for arbitrary $l$, and we save on notation and terminology by restricting to this special case. We also assume that $e>0$, since the difficulties we describe below do not arise in the case $e=0$. We work only with ungraded modules, since the grading plays no part in our results.
In the case $l=1$, the Specht modules (by which we mean the *row Specht modules* of [@kmr §5.4]) are labelled by partitions. For the sake of alignment with the rest of the literature, we now change to using the *English* convention for Young diagrams in this section, where the first coordinate increases down the page and the second increases from left to right.
Suppose $\pi$ is a partition of $n$. If $(a,b)$ is a node, its *residue* is defined to be ${\operatorname{res}}((a,b))=b-a+\kappa_1\ppmod e$. A *$\pi$-tableau* is a bijection from the Young diagram of $\pi$ to the set $\{1,\dots,n\}$. We specify a particular $\pi$-tableau $\ttt^\pi$ by assigning the numbers $1,\dots,n$ to the nodes $$(1,1),(1,2),\dots,(1,\pi_1),\ (2,1),(2,2),\dots,(2,\pi_2),\ (3,1),(3,2),\dots$$ in order. for example, when $\pi=(6,4,1^2)$ we have $$\ttt^\pi=\young(123456,789{10},{11},{12}).$$ We define an element $\bfi^\pi$ of $({\mathbb{Z}/e\mathbb{Z}})^n$ by setting $\bfi^\pi_j={\operatorname{res}}((\ttt^\pi)^{-1}(j))$. For example, with $\pi=(6,4,1^2)$, $e=4$ and $\kappa_1=0$, $\bfi^\pi$ is the sequence $(0,1,2,3,0,1,3,0,1,2,2,1)$.
Now we can define the *row permutation module* ${\operatorname{M}^{\pi}}$ from [@kmr §5.3]. This has a single generator $m^\pi$, and relations as follows:
- $e(\bfi^\pi)m^\pi=m^\pi$;
- $y_jm^\pi=0$ for all $j$;
- $\psi_jm^\pi=0$ whenever $j$ and $j+1$ lie in the same row of $\ttt^\pi$.
It is easy to write down a basis for ${\operatorname{M}^{\pi}}$. Say that a $\pi$-tableau $\ttt$ is *row-strict* if the entries increase from left to right along the rows. If $\ttt$ is row-strict, let $w^\ttt$ be the permutation taking $\ttt^\mu$ to $\ttt$, and define $\psi^\ttt:=\psi_{w^\ttt}$, and $m^\ttt=\psi^\ttt m^\pi$. Then by [@kmr Theorem 5.6] the set $\lset{m^\ttt}{\ttt\text{ a row-strict $\pi$-tableau}}$ is a basis for ${\operatorname{M}^{\pi}}$.
The Specht module ${\operatorname{S}^{\pi}}$ is the quotient of ${\operatorname{M}^{\pi}}$ by the homogeneous Garnir relations, which we now define. Say that a node $\fkn=(a,b)$ of $\pi$ is a *Garnir node* if $(a+1,b)$ is also a node of $\pi$. If $\fkn$ is a Garnir node, define the *Garnir belt* ${\mathbf B}^{\fkn}$ to be the set of nodes $$\{(a,b),(a,b+1),\dots,(a,\pi_a),\ (a+1,1),(a+1,2),\dots,(a+1,b)\}.$$ The *Garnir tableau* $\ttg^{\fkn}$ is the tableau defined by taking $\ttt^\pi$ and rearranging the entries within ${\mathbf B}^{\fkn}$ so that they increase from bottom left to top right. For example, with $\pi=(6,4,1^2)$ and $\fkn=(1,3)$, the tableau $\ttg^{\fkn}$ (with the Garnir belt ${\mathbf B}^\fkn$ shaded) is as follows. $$\begin{tikzpicture}[scale=1]
\Yfillcolour{white!80!black}
\tgyoung(0cm,0cm,::;;;;,;;;)
\Yfillcolour{white}
\Yfillopacity0
\tyoung(0cm,0cm,126789,345{10},{11},{12})
\end{tikzpicture}$$ Now we define *bricks*: an $\fkn$-brick is defined to be a set of $e$ consecutive nodes in the same row of ${\mathbf B}^{\fkn}$ of which the leftmost has the same residue as $\fkn$. For example, take $\pi=(6,4,1^2)$ and $\fkn=(1,3)$ as above. If $e=2$, then the bricks are $$\{(1,3),(1,4)\},\{(1,5),(1,6)\},\{(2,2),(2,3)\},$$ while if $e=3$ the bricks are $$\{(1,3),(1,4),(1,5)\},\{(2,1),(2,2),(2,3)\}.$$ Let $f$ denote the number of bricks in row $a$ and $g$ the number of bricks in row $a+1$, and set $k=f+g$. If $k>0$, let $d=d^{\fkn}$ be the smallest number which is contained in a brick in $\ttg^\fkn$, and for $1\ls i<k$ define $w_i$ to be the permutation $$(d+ie-e,\ d+ie)(d+ie-e+1,\ d+ie+1),\dots,(d+ie-1,\ d+ie+e-1),$$ Recalling the notation $\psi_w$ for $w\in\sss n$ from above, set $\sigma^\fkn_i=\psi_{w_i}$; unlike the situation for arbitrary $w$, $\psi_{w_i}$ does not depend on the choice of reduced expression for $w_i$. Also set $\tau^\fkn_i=\sigma^\fkn_i+1$.
In the definition [@kmr (5.7)], the elements $\tau^\fkn_i,\sigma^\fkn_i$ include an idempotent $e(\bfi^\fkn)$ as a factor; but this factor is unnecessary for our purposes, so we prefer the simpler version.
Now let $\ttt^{\fkn}$ be the tableau obtained from $\ttg^{\fkn}$ by re-ordering the bricks so that their entries increase along row $a$ and then row $a+1$.
Continuing the example above, we illustrate the tableau $\ttt^\fkn$ for $e=2,3$, shading the Garnir belt and outlining the bricks: $$\ttt^{\fkn}=
\begin{tikzpicture}[scale=1,baseline=-17pt]
\Yfillcolour{white!80!black}
\Yfillopacity1
\tgyoung(0cm,0cm,::;;;;,;;;)
\Yfillopacity0
\tyoung(0cm,0cm,124567,389{10},{11},{12})
\Ylinethick{1.3pt}
\tgyoung(0cm,0cm,::_2_2,:_2\ )
\Ylinethick{.3pt}
\end{tikzpicture}
\text{ if }e=2,\qquad
\ttt^{\fkn}=
\begin{tikzpicture}[scale=1,baseline=-17pt]
\Yfillcolour{white!80!black}
\Yfillopacity1
\tgyoung(0cm,0cm,::;;;;,;;;)
\Yfillopacity0
\tyoung(0cm,0cm,123459,678{10},{11},{12})
\Ylinethick{1.3pt}
\tgyoung(0cm,0cm,::_3,_3\ )
\Ylinethick{.3pt}
\end{tikzpicture}
\text{ if }e=3.$$ Define $\psi^{\ttt^\fkn}$ as above; as noted in [@kmr §5.4], $\psi^{\ttt^\fkn}$ is independent of the choice of reduced expression for $w_{\ttt^\fkn}$.
Now we define elements of $\calr^\kappa_n$ corresponding to the elements $t_\la$ and $s_\la$ of $\bbf\sss k$ defined in \[ypmsec\]. For $\la\in{\scrp_{f,g}}$, define $$\tau^\fkn_\la=\left(\tau^\fkn_{\la_1+g}\tau^\fkn_{\la_1+g+1}\dots \tau^\fkn_{f+g-1}\right)\left(\tau^\fkn_{\la_2+g-1}\tau^\fkn_{\la_2+g}\dots \tau^\fkn_{f+g-2}\right)\dots\left(\tau^\fkn_{\la_g+1}\tau^\fkn_{\la_g+2}\dots \tau^\fkn_f\right),$$ and define $\sigma^\fkn_\la$ similarly. Then the elements $\sigma^\fkn_\la\psi^{\ttt^\fkn}$ are precisely the elements $\psi^\ttt$ for row-strict tableaux $\ttt$ obtained from $\ttg^\fkn$ by permuting the bricks.
Now we can define the homogeneous Garnir relations and the Specht module. Given a Garnir node $\fkn$, the corresponding *Garnir element* is $$g^\fkn={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}\tau^\fkn_\la\psi^{\ttt^\fkn}.$$ The Specht module ${\operatorname{S}^{\pi}}$ is defined to be the quotient of ${\operatorname{M}^{\pi}}$ by the relations $g^\fkn m^\pi=0$ for all Garnir nodes $\fkn$ of $\pi$.
Re-writing the Garnir relations
-------------------------------
The difficulty with the definition of the Specht module given above, especially from a computational point of view, is that the individual terms $\tau^\fkn_\la\psi^{\ttt^\fkn}m^\pi$ appearing in the Garnir relation are not readily expressed in terms of the standard basis $\lset{m^\ttt}{\ttt\text{ row-strict}}$ for ${\operatorname{M}^{\pi}}$; to express it in these terms, some quite involved reduction is required using the defining relations in $\calr_n$.
Take $\pi=(11,5,3,1)$, $\fkn=(1,5)$ and $e=3$. Then we have $f=2$, $g=1$, and $$\sigma^\fkn_1=\psi_9\psi_8\psi_7\psi_{10}\psi_9\psi_8\psi_{11}\psi_{10}\psi_9,\qquad\sigma^\fkn_2=\psi_{12}\psi_{11}\psi_{10}\psi_{13}\psi_{12}\psi_{11}\psi_{14}\psi_{13}\psi_{12},$$ while $\psi^{\ttt^\fkn}=\psi_{6}\psi_{5}\psi_{7}\psi_{6}\psi_{8}\psi_{7}\psi_{9}\psi_{8}\psi_{10}\psi_{9}\psi_{11}\psi_{10}\psi_{15}\psi_{14}\psi_{13}\psi_{12}\psi_{11}$. Hence the Garnir element $g^\fkn$ is $$\begin{aligned}
\left(\tau^\fkn_\varnothing+\tau^\fkn_{(1)}+\tau^\fkn_{(2)}\right)\psi^{\ttt^\fkn}&=\left(\left(\sigma^\fkn_1+1\right)\left(\sigma^\fkn_2+1\right)+\left(\sigma^\fkn_2+1\right)+1\right)\psi^{\ttt^\fkn}\\
&=\left(\sigma^\fkn_1\sigma^\fkn_2+\sigma^\fkn_1+2\sigma^\fkn_2+3\right)\psi^{\ttt^\fkn}.\end{aligned}$$ A non-trivial calculation using the defining relations for $\calr_n$ and the fact that $m^\pi$ is killed by $\psi_5$, $\psi_6$ and $\psi_7$ shows that $\sigma^\fkn_1\psi^{\ttt^\fkn}m^\pi=0$. Hence $$g^\fkn m^\pi=\left(\sigma^\fkn_\varnothing+2\sigma^\fkn_{(1)}+3\sigma^\fkn_{(2)}\right)\psi^{\ttt^\fkn}m^\pi=m^{\ttg^\fkn}+2m^\ttu+3m^{\ttt^\fkn},$$ where $$\begin{aligned}
\ttg^\fkn&=\young(1234{10}{11}{12}{13}{14}{15}{16},56789,{17}{18}{19},{20}),\\
\ttu&=\young(1234789{13}{14}{15}{16},56{10}{11}{12},{17}{18}{19},{20}),\\
\ttt^\fkn&=\young(1234789{10}{11}{12}{16},56{13}{14}{15},{17}{18}{19},{20}).\end{aligned}$$
We wish to generalise the above example, to re-write $g^\fkn m^\pi$ for an arbitrary Garnir node $\fkn$ as a linear combination of elements $\sigma^\fkn_\la\psi^{\ttt^\fkn}m^\pi$. Fortunately, all the computations that we need with the defining relations for $\calr_n$ have already been done in [@kmr].
Given a Garnir node $\fkn$ of $\pi$, define the *brick permutation space* $T^{\pi,\fkn}$ to be the $\bbf$-subspace of ${\operatorname{M}^{\pi}}$ spanned by all elements of the form $\sigma^\fkn_{i_1}\dots\sigma^\fkn_{i_s}\psi^{\ttt^\fkn}m^\pi$. Then the $\sigma^\fkn_i$, and hence the $\tau^\fkn_i$, act on $T^{\pi,\fkn}$, and we have the following.
[kmr]{}[Theorem 5.11]{}\[brickspace\] As operators on $T^{\pi,\fkn}$, the elements $\tau^\fkn_1,\dots,\tau^\fkn_{k-1}$ satisfy the Coxeter relations for the symmetric group $\sss k$, and hence $T^{\pi,\fkn}$ can be considered as an $\bbf\sss k$-module. In fact, as an $\bbf\sss n$-module $T^{\pi,\fkn}$ is isomorphic to the Young permutation module ${\mathscr{M}^{(f,g)}}$, with an isomorphism given by mapping $\psi^{\ttt^\fkn}m^\pi$ to the standard generator ${m}$.
By \[brickspace\] and the discussion at the start of \[ypmsec\], the sets $$\lset{\tau^\fkn_\la\psi^{\ttt^\fkn}m^\pi}{\la\in{\scrp_{f,g}}}\qquad\text{and}\qquad\lset{\sigma^\fkn_\la\psi^{\ttt^\fkn}m^\pi}{\la\in{\scrp_{f,g}}}$$ are $\bbf$-bases of $T^{\pi,\fkn}$. \[mainsn1,mainsn2\] give the change-of-basis matrices between these bases, while \[snsum\] gives the sum of the elements of the first basis in terms of the second basis. So we can immediately deduce the following, which is the main result of this section.
\[maingarn\] Suppose $\pi$ is a partition and $\fkn=(a,b)$ is a Garnir node of $\pi$, and let $f,g$ denote the numbers of bricks in rows $a$ and $a+1$ of ${\mathbf B}^\fkn$ respectively. Define $\sigma^\fkn_\la$ and $\tau^\fkn_\la$ for $\la\in{\scrp_{f,g}}$ as above, and define the *modified Garnir element* $$\hat g^\fkn={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}{F}(\la)\sigma^\fkn_\la\psi^{\ttt^\fkn}.$$ Then:
1. $\sigma^\fkn_\mu\psi^{\ttt^\fkn}m^\pi={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}(-1)^{|\la|+|\mu|}{\operatorname{e}_{\la\mu}}\tau^\fkn_\la\psi^{\ttt^\fkn}m^\pi$;
2. $\tau^\fkn_\mu\psi^{\ttt^\fkn}m^\pi={\sum_{\hbox to \sumwid{\hfil$\mathclap{\scriptstyle\la\in{\scrp_{f,g}}}$\hfil}}}{\operatorname{i}_{\la\mu}}\sigma^\fkn_\la\psi^{\ttt^\fkn}m^\pi$;
3. $\hat g^\fkn m^\pi=g^\fkn m^\pi$.
Hence the Specht module ${\operatorname{S}^{\pi}}$ is the quotient of ${\operatorname{M}^{\pi}}$ by the relations $\hat g^\fkn m^\pi=0$ for all Garnir nodes $\fkn$ of $\pi$.
1. Statements (1) and (2) in the theorem are not necessary for the Garnir relations, but may be of interest for computation in the row permutation modules; indeed, the coefficients in (1) are essentially the coefficients $c_w$ appearing in [@kmr Corollary 5.12].
2. We emphasise that the identities in \[maingarn\] are not true if the terms $m^\pi$ are omitted; in general, $\tau^\fkn_\mu$ is not a linear combination of $\sigma^\fkn_\la$s (even if the idempotent $e(\bfi^\fkn)$ from [@kmr (5.7)] is reinstated in the definition of $\sigma^\fkn_i$ and $\tau^\fkn_i$).
We end with an example which illustrates how the Garnir relations are used in Specht module computations. Say that a $\pi$-tableau is *standard* if the entries are increasing both along the rows and down the columns. If we let $v^\ttt$ denote the image of $m^\ttt$ in the Specht module, then by [@kmr Corollary 6.24] the set $\lset{v^\ttt}{\ttt\text{ a standard $\pi$-tableau}}$ is a basis for ${\operatorname{S}^{\pi}}$. The Garnir relations allow one to write $v^\ttt$ as a linear combination of this basis when $\ttt$ is a row-strict, but not standard, tableau. The foundation for this is [@kmr Lemma 5.13] which does this in the case where $\ttt=\ttg^\fkn$ for a Garnir node $\fkn$.
For our example, take $\pi=(8,4)$ and $e=2$. Let $\fkn=(1,4)$. Then $f=g=2$, and $$\ttg^\fkn=\young(12389{10}{11}{12},4567).$$ As with any Garnir node, $\ttg^\fkn$ is row-strict but not standard. The modified Garnir element is $$\hat g^\fkn=\left(\sigma^\fkn_\varnothing+2\sigma^\fkn_{(1)}+3\sigma^\fkn_{(2)}+3\sigma^\fkn_{(1^2)}+6\sigma^\fkn_{(2,1)}+6\sigma^\fkn_{(2^2)}\right)\psi^{\ttt^\fkn},$$ and the corresponding Garnir relation yields $$v^{\ttg^\fkn}=-2v^\ttu-3v^\ttv-3v^\ttw-6v^\ttx-6v^\tty,$$ where $\ttu,\ttv,\ttw,\ttx,\tty$ are standard tableaux that we describe as follows. We represent the elements $\psi^\ttu,\psi^\ttv,\psi^\ttw,\psi^\ttx,\psi^\tty$ using the braid diagrams of Khovanov and Lauda [@kl]; by using shaded squares, we illustrate (reverting to the Russian convention!) how the labelling partitions $\la\in\scrp_{2,2}$ arise. [$$\begin{aligned}
3
\ttg^\fkn&=\young(12389{10}{11}{12},4567),&\qquad\psi^{\ttg^\fkn}&=\sigma^\fkn_\varnothing\psi^{\ttt^\fkn}&&=\
\begin{tikzpicture}[scale=.3,baseline=1.7cm,rounded corners=2pt]
\foreach\x in{1,2,3}\draw(\x,0)--++(0,9);
\foreach\x in{0,1,2,3}\draw(4+\x,0)--++(0,3.5-\x)--++(5,5)--++(0,.5+\x);
\foreach\x in{0,1,2,3,4}\draw(8+\x,0)--++(0,.5+\x)--++(-4,4)--++(0,4.5-\x);
\end{tikzpicture}
\\
\ttu&=\young(12367{10}{11}{12},4589),&\psi^{\ttu}&=\sigma^\fkn_{(1)}\psi^{\ttt^\fkn}&&=\
\begin{tikzpicture}[scale=.3,baseline=1.7cm]
\draw[fill=white!85!black](7.5,0)--++(2,2)--++(-2,2)--++(-2,-2);
\foreach\x in{0,2}{\draw[line width=2pt,white](7.5+\x,\x)--++(-2,2);\draw[line width=2pt,white](7.5-\x,\x)--++(2,2);}
\foreach\x in{1,2,3}\draw[rounded corners](\x,0)--++(0,9);
\foreach\x in{0,1}\draw[rounded corners](4+\x,0)--++(0,3.5-\x)--++(5,5)--++(0,.5+\x);
\foreach\x in{2,3,4}\draw[rounded corners](8+\x,0)--++(0,.5+\x)--++(-4,4)--++(0,4.5-\x);
\foreach\x in{0,1}\draw[rounded corners](6+\x,0)--++(0,2.5+\x)--++(-2,2)--++(0,4.5-\x);
\foreach\x in{0,1}\draw[rounded corners](8+\x,0)--++(0,3.5-\x)--++(3,3)--++(0,2.5+\x);
\end{tikzpicture}
\\
\ttv&=\young(1236789{12},45{10}{11}),&\psi^{\ttv}&=\sigma^\fkn_{(2)}\psi^{\ttt^\fkn}&&=\
\begin{tikzpicture}[scale=.3,baseline=1.7cm]
\draw[fill=white!85!black](7.5,0)--++(4,4)--++(-2,2)--++(-4,-4);
\foreach\x in{0,2,4}\draw[line width=2pt,white](7.5+\x,\x)--++(-2,2);
\foreach\x in{0,2}\draw[line width=2pt,white](7.5-\x,\x)--++(4,4);
\foreach\x in{1,2,3}\draw[rounded corners](\x,0)--++(0,9);
\foreach\x in{0,1}\draw[rounded corners](4+\x,0)--++(0,3.5-\x)--++(5,5)--++(0,.5+\x);
\foreach\x in{4}\draw[rounded corners](8+\x,0)--++(0,.5+\x)--++(-4,4)--++(0,4.5-\x);
\foreach\x in{0,1,2,3}\draw[rounded corners](6+\x,0)--++(0,2.5+\x)--++(-2,2)--++(0,4.5-\x);
\foreach\x in{0,1}\draw[rounded corners](10+\x,0)--++(0,5.5-\x)--++(1,1)--++(0,2.5+\x);
\end{tikzpicture}
\\
\ttw&=\young(12345{10}{11}{12},6789),&\psi^{\ttw}&=\sigma^\fkn_{(1^2)}\psi^{\ttt^\fkn}&&=\
\begin{tikzpicture}[scale=.3,baseline=1.7cm]
\draw[fill=white!85!black](7.5,0)--++(2,2)--++(-4,4)--++(-2,-2);
\foreach\x in{0,2}\draw[line width=2pt,white](7.5+\x,\x)--++(-4,4);
\foreach\x in{0,2,4}\draw[line width=2pt,white](7.5-\x,\x)--++(2,2);
\foreach\x in{1,2,3,4,5}\draw[rounded corners](\x,0)--++(0,9);
\foreach\x in{0,1}\draw[rounded corners](6+\x,0)--++(0,5.5-\x)--++(3,3)--++(0,.5+\x);
\foreach\x in{2,3,4}\draw[rounded corners](8+\x,0)--++(0,.5+\x)--++(-4,4)--++(0,4.5-\x);
\foreach\x in{0,1}\draw[rounded corners](8+\x,0)--++(0,3.5-\x)--++(3,3)--++(0,2.5+\x);
\end{tikzpicture}
\\
\ttx&=\young(1234589{12},67{10}{11}),&\psi^{\ttx}&=\sigma^\fkn_{(2,1)}\psi^{\ttt^\fkn}&&=\
\begin{tikzpicture}[scale=.3,baseline=1.7cm]
\draw[fill=white!85!black](7.5,0)--++(4,4)--++(-2,2)--++(-2,-2)--++(-2,2)--++(-2,-2);
\foreach\x in{0,2}{\draw[line width=2pt,white](7.5+\x,\x)--++(-4,4);\draw[line width=2pt,white](7.5-\x,\x)--++(4,4);}
\foreach\x in{4}{\draw[line width=2pt,white](7.5+\x,\x)--++(-2,2);\draw[line width=2pt,white](7.5-\x,\x)--++(2,2);}
\foreach\x in{1,2,3,4,5}\draw[rounded corners](\x,0)--++(0,9);
\foreach\x in{0,1}\draw[rounded corners](6+\x,0)--++(0,5.5-\x)--++(3,3)--++(0,.5+\x);
\foreach\x in{4}\draw[rounded corners](8+\x,0)--++(0,.5+\x)--++(-4,4)--++(0,4.5-\x);
\foreach\x in{0,1}\draw[rounded corners](10+\x,0)--++(0,5.5-\x)--++(1,1)--++(0,2.5+\x);
\foreach\x in{0,1}\draw[rounded corners](8+\x,0)--++(0,4.5+\x)--++(-2,2)--++(0,2.5-\x);
\end{tikzpicture}
\\
\tty&=\young(1234567{12},89{10}{11}),&\psi^{\tty}&=\sigma^\fkn_{(2^2)}\psi^{\ttt^\fkn}&&=\
\begin{tikzpicture}[scale=.3,baseline=1.7cm]
\draw[fill=white!85!black](7.5,0)--++(4,4)--++(-4,4)--++(-4,-4);
\foreach\x in{0,2,4}{\draw[line width=2pt,white](7.5+\x,\x)--++(-4,4);\draw[line width=2pt,white](7.5-\x,\x)--++(4,4);}
\foreach\x in{1,2,3,4,5,6,7}\draw[rounded corners](\x,0)--++(0,9);
\foreach\x in{4}\draw[rounded corners](8+\x,0)--++(0,.5+\x)--++(-4,4)--++(0,4.5-\x);
\foreach\x in{-2,-1,0,1}\draw[rounded corners](10+\x,0)--++(0,5.5-\x)--++(1,1)--++(0,2.5+\x);
\end{tikzpicture}\end{aligned}$$]{}
[99]{}
H. Garnir, ‘Théorie de la representation lineaire des groupes symétriques’, *Mém. Soc. Roy. Sci. Liège* **10** (1950), 100pp.
D. Knuth, *The art of computer programming. Volume 3: sorting and searching.* Addison–Wesley Series in Computer Science and Information Processing. Addison–Wesley Publishing Co., Reading, Mass.–London–Don Mills, Ont., 1973.
D. Speyer, *A function from partitions to natural numbers – is it familiar?*, mathoverflow.net/q/132548.
|
---
abstract: 'A variational method is proposed for calculating the percolation threshold, the real-space structure, and the ground-state energy of a disordered two-dimensional electron liquid. Its high accuracy is verified against exact asymptotics and prior numerical results. The inverse thermodynamical density of states is shown to have a strongly asymmetric minimum at a density that is approximately the triple of the percolation threshold. This implies that the experimentally observed metal-insulator transition takes place well before the percolation point is reached.'
author:
- 'Michael M. Fogler'
title: 'Nonlinear screening and percolative transition in a two-dimensional electron liquid'
---
The discovery that a two-dimensional (2D) electron liquid can be a metal at moderate electron density $n_e$ and an insulator at small $n_e$ was a major surprise that questioned our fundamental understanding of the role of disorder in such systems. Today, almost a decade later, it remains a subject of an intense debate. [@Abrahams_01] One important reason why the conventional theory fails could be its flawed basic premise of the “good” metal, i.e., a uniform electron liquid slightly perturbed by impurities and defects. Indeed, modern nanoscale imaging techniques [@Eytan_98; @Finkelstein_00; @Ilani_00; @Morgenstern_00] unambiguously showed that low-density 2D electron systems are strongly inhomogeneous, “bad” metals, where effects of disorder are nonperturbatively strong. In particular, depletion regions (DR), i.e., regions where $n(\textbf{r})$ is effectively zero, exist. They appear when $n_e$ is too small to adequately compensate fluctuating charge density of randomly positioned impurities. As $n_e$ is reduced in the experiment, e.g., in order to approach the vicinity of the metal-insulator transition (MIT), the DRs are expected to grow in size and concentration and eventually merge below some percolation threshold $n_e = n_p$. An important and controversial issue is whether or not this percolation transition plays any role in the observed MIT. [@Abrahams_01] To resolve it one needs to have a theory that is able to calculate $n_p$ and that can describe the inhomogeneous structure of the 2D metal at $n_e \sim n_p$. A great progress in this direction has been achieved by Efros, Pikus, and Burnett, [@Efros_93] whose paper is intellectually tied to earlier work on nonlinear screening by Efros and Shklovskii. [@Efros_Shklovskii_book] Still, analytical results remained scanty and numerical simulations [@Efros_93; @Shi_02] were the only known way to quantitatively study the $n_e \sim n_p$ regime. These simulations are very time consuming and redoing them in order to get any information beyond what is published [@Efros_93; @Shi_02] or to study novel experimental setups seems impracticable. Below I will show that a variational approach to the problem can be a viable alternative. Comparing it with the available numerical results for a typical model of the experimental geometry (Fig. \[Fig\_bilayer\]), I establish that it correctly predicts the value of $n_p$ and accurately reproduces the energetics of the ground state, in particular, the density dependence of the electrochemical potential $\mu$ and of the inverse thermodynamical density of states (ITDOS), $\chi^{-1} = d \mu /
d n_e$, over a broad range of $n_e$. The most striking feature of the resultant function $\chi^{-1}(n_e)$ is a strongly asymmetric minimum, which is observed in real experiments. [@Eisenstein_94; @Dultz_00; @Dultz_xxx; @Allison_02] I will elaborate on the origin of this feature and show that it occurs at $n_e
\approx 3 n_p$ largely independently of the parameters of the system. Recently, this minimum attracted much interest when Dultz *et al.* [@Dultz_00] reported that in some samples it virtually coincides with the apparent MIT. The proposed theory indicates that, at least for these samples, any connection between the MIT and the percolation of the DRs can be ruled out. Thus, the explanation of the MIT lies elsewhere.
![ The geometry of the theoretical model. The 2D layer of interest is sandwiched between the top and the bottom metallic gates. The dopants reside in the plane with the dashed border. The depletion regions (shown as holes in the probed layer) enhance the penetrating electric field $E_p$. \[Fig\_bilayer\] ](comperc_1.ps){width="2.1in"}
The Hamiltonian of the model is adopted from Efros, Pikus, and Burnett (EPB) [@Efros_93] (see also Fig. \[Fig\_bilayer\]), $$H = \int d^2 r \left\{\frac12 [n(\textbf{r}) - n_e] \Phi(\textbf{r})
+ H_0(n)\right\}
\label{H}$$ where $\Phi(\textbf{r})$ is the electrostatic potential, $$\Phi = \int d^2 r^\prime \frac{e^2}{\kappa} \left[
\frac{n(\textbf{r}^\prime) - n_e}{|\textbf{r}^\prime - \textbf{r}|}
- \frac{n_d(\textbf{r}^\prime) - n_i}
{\sqrt{(\textbf{r}^\prime - \textbf{r})^2 + s^2}}\right]
\label{Phi}$$ $\kappa$ is the dielectric constant, and $H_0(n)$ is the energy density of the uniform liquid of density $n$, $$H_0(n) = - (e^2 / \kappa) n^{3/2} h_0(n).
\label{H_0}$$ At low densities, $h_0(n) \sim 1$ is a slow function [@Tanatar_89] of $n$. The negative sign in Eq. (\[H\_0\]) reflects the prevalence of the exchange-correlation energy over the kinetic one in this regime. The potential created by the random concentration of dopants $n_d({\bf
r})$ can be expressed in terms of the effective *in-plane* background charge $\tilde{\sigma}(\textbf{q}) = \tilde{n}_d(\textbf{q})
\exp(-q s)$ (the tildes denote the Fourier transforms). With these definition, $\Phi(\textbf{r})$ coincides with the potential created by the charge density $\sigma(\textbf{r}) = n(\textbf{r}) - n_e - \sigma(\textbf{r})$. I will assume that the dopants have the average concentration $\langle n_d
\rangle \equiv n_i \gg s^{-2}$ and are uncorrelated in space. In this case the one- and two-point distribution functions of $\sigma(\textbf{r})$ have the Gaussian form, $$\begin{aligned}
&P_1(\sigma) = (2 \pi K_0)^{-1/2} \exp (-\sigma^2 / 2 K_0),\quad&
\label{P_1}\\
&\displaystyle P_2 = \frac{1}{2 \pi \sqrt{K_0^2 - K_{r^\prime}^2}}
\exp\left[\frac{2 K_{r^\prime} \sigma \sigma^\prime - K_0 (\sigma^2 + \sigma^{\prime\,2})}
{2 (K_0^2 - K_{r^\prime}^2)}
\right]\quad&
\label{P_2}\end{aligned}$$ where $\sigma^\prime = \sigma(\textbf{r} + \textbf{r}^\prime)$ and $K_r$ is given by $$K_r \equiv \langle \sigma(r) \sigma(0) \rangle = {n_i s} /
\pi {(r^2 + 4 s^2)^{3/2}}.
\label{K}$$ The characteristic scales in the problem are as follows. [@Efros_93] The typical amplitude of fluctuations in $\sigma$ is $\sqrt{n_i} / s$, see Eqs. (\[P\_1\]) and (\[K\]). Their characteristic spatial scale is the spacer width $s$ \[cf. Eqs. (\[P\_1\]–\[K\]) and Fig. \[Fig\_bilayer\]\]. In the cases studied below, $n_e \gtrsim n_p$ and $n_e \gg n_p$, $s$ exceeds the average interelectron separation $a_0
= n_e^{-1/2}$. As usual in Coulomb problems, the energy is dominated by the longest scales, in this case $s$. The fluctuations $H_0(n) -
H_0(n_e)$ of the the local energy density come from the interactions on the much shorter scale of $a_0 \ll s$ and can be treated as a perturbation. [@Efros_93] For the purpose of calculating the ground-state density profile $n(\textbf{r})$ I neglect $H_0$. Once such a ground state is known, I correct the total energy $H$ by adding to it $H_0$ averaged over $n(\textbf{r})$.
To find $n(\textbf{r})$ one needs to solve the electrostatic problem with the following dual boundary conditions: if $n(\textbf{r}) > 0$, then $\Phi(\textbf{r}) = \mu = {\rm const}$; otherwise, if $n = 0$ (DR), then [@Efros_Shklovskii_book; @Efros_93; @Fogler_comp_cb] $\Phi > \mu$. I start with the analysis of the large-$n_e$ case, which clarifies why $\chi^{-1}(n_e)$ dependence is nonmonotonic and which provides a formula for the the density $n_m$ where $\chi^{-1}$ has the minimum.
In the limit $n_e \gg \sqrt{n_i} / s$, the asymptotically exact treatment is possible because the DRs appear only in rare places where $\sigma(\textbf{r})$ dips below $-n_e$. (In practice, this limit is realized when $n_e > K_0^{1/2} \approx 0.2 \sqrt{n_i} / s$). The corresponding electrostatic problem is analogous to that of the metallic sheet perforated by small holes, see Fig. \[Fig\_bilayer\]. The most elegant way to derive $\chi^{-1}$ in this regime is to calculate the fraction $E_p / E_0$ of the electric field that reaches the bottom layer in the geometry of Fig. \[Fig\_bilayer\]. Indeed, $E_p / E_0$ is nonzero only if the probe layer is not a perfect metal, $\chi^{-1} \neq
0$. In the simplest case, where distances $s_1$ and $s_2$ are large, the following formula holds: $\chi^{-1} = 4 \pi (e^2 s_2 / \kappa)(d E_p / d
E_0)$ (cf. Ref. ). This is essentially the formula used to deduce $\chi^{-1}$ in the experiment [@Eisenstein_94; @Dultz_00; @Dultz_xxx; @Allison_02]). It is immediately obvious that holes in the metallic sheet enhance the penetrated field $E_p$. For example, the field leaking through a round hole of radius $a$ is the field of a dipole [@Jackson_book] $\textbf{p} = (a^3 / 3 \pi) \textbf{E}_0$. If there is a finite but small concentration $N_h$ of such holes, their fields are additive, leading to $\chi^{-1} = (8 \pi e^2 / 3 \kappa) N_h a^3$. From here the exact large-$n_e$ asymptotics of $\chi^{-1}$ and $\mu$ can be obtained by substituting the proper $N_h$ and averaging over the distribution of $a$. This can be done by noting that these holes appear around the minima of $\sigma(\textbf{r})$ whose statistics is fixed by Eqs. (\[P\_1\]–\[K\]). It is easy to see that the most probable holes are nearly perfect circles with radii [@Efros_93] $a \sim s \sqrt{K_0} / n_e$. The charge distribution around a single hole at distances $r > a$ is given by the formula $$n(r) = \frac{\sigma_{x x} a^2}{\pi}
\left[ \sqrt{\frac{r^2}{a^2} - 1} - \left(\frac{r^2}{a^2} +
\frac{2}{3}\right) \arccos \frac{a}{r}\right],\:
\label{sigma_hole_out}$$ where the hole is assumed to be centered at $r = 0$ and $a^2 = -3 [n_e +
\sigma(0)] / \sigma_{x x}$. Equation (\[sigma\_hole\_out\]) can be obtained, e.g., by generalizing the textbook solution [@Jackson_book] for the hole in the metallic sheet [@Comment_on_hole] and is also the limiting form of Eq. (11) in Ref. . Previously, Eq. (\[sigma\_hole\_out\]) was used for study of quantum dots in Ref. .
In the current problem the main factor that determines the net contribution $\chi^{-1}_{\rm DR}$ of the depletion holes to $\chi^{-1}$ is their exponentially small concentration, proportional to $P_1(-n_e)
\propto \exp(-n_e^2 / 2 K_0)$. The final result, $$\chi^{-1}_{\rm DR} \simeq ({3 \sqrt{2}}/{8 \pi})
({e^2 n_i}/{\kappa s n_e^2})
\exp(-{4 \pi s^2 n_e^2}/{n_i}),
\label{chi_asymp}$$ agrees with that of EPB but has no numerical coefficients left undetermined. To finalize the calculation, one needs to augment $\chi^{-1}_{\rm DR}$ by the local term $\chi_0^{-1} = d^2 \langle H_0(n)
\rangle / d n_e^2$. In the present case, fluctuations around the average density are small. Hence, $\langle H_0(n) \rangle \simeq H_0(n_e)$ and $$\chi^{-1}_0(n_e) \simeq -(e^2 / \kappa) h_1(n_e) / \sqrt{n_e},\quad
n_e \gg \sqrt{n_i} / s,
\label{chi_0_large_n}$$ where $h_1(n_e) = (3 / 4) h_0(n_e) + 3 h_0^\prime n_e + h_0^{\prime\prime}
n_e^2 \sim 1$.
Formula (\[chi\_asymp\]) implies a sharp upturn of the ITDOS as $n_e
\to 0$ caused by the exponential growth of the DRs. At the boundary of its validity, $n_e \sim \sqrt{n_i} / s$, Eq. (\[chi\_asymp\]) gives $\chi^{-1}_{\rm DR} \sim e^2 s / \kappa$, which is large and *positive*. On the other hand, the local term $\chi^{-1}_0$ \[Eq. (\[chi\_0\_large\_n\])\] shows a weak dependence on $n_e$, remaining small and *negative*. Combined, they produce a strongly asymmetric minimum in $\chi^{-1}(n_e) = \chi^{-1}_{\rm DR} +
\chi^{-1}_0$ at the density $$n_m = \frac{1}{4 \sqrt{\pi}} \frac{\sqrt{n_i}}{s}
\ln^{1/2} \left(\frac{4096}{\pi h_1^4} n_i s^2\right).
\label{n_m}$$ The log-factor in Eq. (\[n\_m\]) is rather insensitive to $n_i s^2$ and $h_1$. For $n_i s^2 = h_1 = 1$, one gets $n_m \approx
0.38 {\sqrt{n_i}}/\,{s}$. In comparison, (see Ref. and below) the percolation threshold is $$n_p \approx 0.12 {\sqrt{n_i}}/\,{s},
\label{n_p}$$ so that $n_m \approx 3 n_p$. In accord with EPB’s heuristic argument, at such density a very small area fraction is depleted (see inset in Fig. \[Fig\_comparison\]a), and so the use of the asymptotic formula (\[chi\_asymp\]) is justified.
![ (a) Electrochemical potential *vs.* electron density for the electrostatic problem, $H_0 \equiv 0$, according to the present theory (solid line) and EPB’s numerical simulations (dots). Inset: Fraction of the depleted area *vs.* density. (b) $\chi^{-1}$ [*vs.*]{} density according to the numerical simulations of Shi and Xie [@Shi_02] (squares), the present theory (solid line), and for the uniform electron liquid (dashed line). \[Fig\_comparison\] ](comperc_2.ps){width="2.5in"}
Let us now proceed to the case $n_e \sim n_p$. Again, I start with the electrostatic problem ($H_0 \equiv 0$). One expects DRs to be abundant and irregularly shaped. For a given $n_e$, the ground-state $n(\textbf{r})$ is some nonlocal functional of $\sigma$ and there is no hope to find it exactly. What I wish to report here is that a variational solution sought within the class of purely local functionals, $n(\textbf{r}) = n[\sigma(\textbf{r})]$ remarkably accurately reproduces the $\mu(n_e)$ and $\chi^{-1}(n_e)$ dependencies found in numerical work. [@Efros_93; @Shi_02] More interestingly, it predicts the correct value of $n_p$ \[Eq. (\[n\_p\])\].
The system of equation that defines such a variational state $n(\sigma)$ follows from the fact that for a Gaussian random function the averages over the total area $L^2$ of the system and over the distribution function are the same. This yields \[cf. Eqs. (\[H\]–\[P\_2\])\] $$\begin{aligned}
&\displaystyle \frac{H_v}{L^2} = \frac12 \int \int d \sigma d \sigma^\prime
\rho(\sigma) G(\sigma, \sigma^\prime) \rho(\sigma^\prime),\quad\quad&
\label{H_v}\\
&\displaystyle G(\sigma, \sigma^\prime) = \int\limits d^2 r^\prime V(r^\prime)
[P_2(\sigma, \sigma^\prime) - P_1(\sigma) P_1(\sigma^\prime)],\quad\quad&
\label{G}\end{aligned}$$ where $\rho(\sigma) = n(\sigma) - n_e - \sigma$ and $V(r^\prime) = e^2 / \kappa
r^\prime$. The energy $H_v$ needs to be minimized with respect to all functions $n(\sigma)$ that obey the constraints $n(\sigma) \geq 0$ and $$\int d \sigma P_1(\sigma) n(\sigma) = n_e.
\label{f_condition}$$ The latter ensures that the average density is equal to $n_e$. Introducing the Lagrange multiplier $\mu_v$ (variational estimate of the electrochemical potential), one obtains that $H_v$ is minimized if, for all $\sigma > f$, $\rho(\sigma^\prime)$ satisfies $$\int\limits d \sigma^\prime
G(\sigma, \sigma^\prime) \rho(\sigma^\prime) = \mu_v(n_e) P_1(\sigma)
\label{var_equation}$$ Here $f$ is such that $n(f) = 0$. \[I found that $n(\sigma)$ is always a monotonically increasing function, so that $n > 0$ corresponds to $\sigma > f$\]. The kernel $G(\sigma, \sigma^\prime)$ \[Eq. (\[G\])\] is log-divergent, $G \propto -\ln|\sigma - \sigma^\prime|$ at $\sigma \to
\sigma^\prime$ and decays exponentially at large $\sigma$ and $\sigma^\prime$. There is a certain analogy between Eq. (\[var\_equation\]) and the integral equations of 1D electrostatics, which also have log-divergent kernels. [@Jackson_book; @Fogler_94] This analogy entails that $n(\sigma) \sim \sqrt{\sigma - f}$ at $\sigma$ close to $f$. Note that $\sigma(\textbf{r}) - f$ is proportional to the distance in the real space between the given point $\textbf{r}$ and the boundary of the nearby DR. Thus, the variational principle renders correctly the square-root singularity in $n(\textbf{r})$ at the edge of the DRs \[cf., e.g., Eq (\[sigma\_hole\_out\])\]. I was not able to establish the analytical form of the solution beyond this property and resorted to finding $n(\sigma)$ numerically. To do so the integral in Eq. (\[var\_equation\]) was converted into a discrete sum over 101 points on the interval $|\sigma| < 1.5 \sqrt{n_i} / s$ the resultant system of 101 linear equations was solved on the computer. The solution can be approximated by a simple analytical *ansatz* $$n_a(\sigma) = [(n_e + \sigma)^2 - (n_e + f)^2]^{1/2} \theta(\sigma - f),
\label{n_var}$$ where $\theta(z)$ is the step-function. For example, the energy $H_v$ is nearly the same whether it is calculated using $n_a$ or using the actual solution of Eq. (\[var\_equation\]). So, in principle, Eq. (\[n\_var\]) obviates the need to solve Eq. (\[var\_equation\]). The only equation that needs to be solved is Eq. (\[f\_condition\]) for $f$.
At this point one can compare the predictions of the variational method for $\mu_v(n_e)$ with EPB’s numerical results that were also obtained for the $H_0 = 0$ problem. As Fig. \[Fig\_comparison\]a illustrates, they are in a good agreement. [@Comment_on_ansatz]
To test the theory further I compare it next with the numerical data of Shi and Xie. [@Shi_02] To this end, one needs to calculate $\chi^{-1}$ including the effect of a finite $H_0$. The first step is to take the derivative $\chi^{-1}_{\rm DR} = d \mu_v / d n_e$, which is easily done numerically. An accurate fit to the result is provided by the interpolation formula $$\chi^{-1}_{\rm DR} \approx \frac{e^2 s}{\kappa}
\frac{3 \sqrt{2}}{8 \pi \eta}
\frac{0.30 + \eta}{0.036 + 0.12 \eta + \eta^2}
\exp(-4 \pi \eta^2),
\label{chi_DR_interp}$$ where $\eta \equiv n_e s / \sqrt{n_i}$. This particular form is devised to match Eq. (\[chi\_asymp\]) at large $n_e$ and is consistent up to log-corrections with the behavior expected [@Efros_Shklovskii_book; @Gergel_78] at very small $n_e$. The next step is to evaluate the quantity $$\chi^{-1}_0 = \frac{d^2}{d n_e^2} \langle H_0 \rangle
= -\frac{d^2}{d n_e^2} \int_f^\infty d \sigma P_1(\sigma) H_0[n(\sigma)],
\label{chi_0}$$ which is also easily done on the computer. The total ITDOS, $\chi^{-1}(n_e) = \chi^{-1}_{\rm DR} + \chi^{-1}_0$, calculated from Eqs. (\[chi\_DR\_interp\]–\[chi\_0\]) and the theoretical value of $n_m \sim 0.7 \times 10^{-3} a_B^{-2}$ \[per Eq. (\[n\_m\])\] compare very well with the simulations, see Fig. \[Fig\_comparison\]b. The parameters I used are $n_i = 6.25 \times 10^{-4} a^{-2}_B$ and $s = 10
a_B$, where $a_B$ is the effective Bohr radius. [@Comment_on_Shi] In agreement with the exact results presented above, $\chi_0^{-1} \ll
\chi_{\rm DR}^{-1}$ at $n_e < n_m$, and so the upturn of $\chi^{-1}$ at low $n_e$ is driven the growth of DRs.
Within the variational method the boundaries of the DRs coincide with the level lines $\sigma(\textbf{r}) = f$ of the zero-mean random function $\sigma$. Consequently, the DRs percolate at $f \geq 0$. Solving Eq. (\[var\_equation\]) with $f = 0$, one arrives at Eq. (\[n\_p\]), which is in excellent agreement with EPB’s result $n_p
\approx 0.11 \sqrt{n_i} / s$. One important quantity not reported in the published numerical works [@Efros_93; @Shi_02] is the area fraction of the DR. Within the variational method, it is equal to ${\rm erfc}(-f /
\sqrt{2 K_0}) / 2$, where ${\rm erfc}(z)$ is the complementary error function. It is exactly $1 / 2$ at $n_e = n_p$ and increases as $n_e \to
0$ as shown in Fig. \[Fig\_comparison\]a (inset).
Let us now compare our results with the experimental data. Taking a rough number $n_i = 3 \times 10^{11} {\rm cm}^{-2}$ and a typical spacer width $s = 40\,{\rm nm}$ in Eq. (\[n\_m\]), one gets $n_m \approx 5.2
\times 10^{10} {\rm cm}^{-2}$, in agreement with observed values. [@Dultz_00] The estimate for the percolation point is $n_p
\approx n_m / 3 \approx 1.7 \times 10^{10} {\rm cm}^{-2}$. Despite some small uncertainty in the last number (due to the uncertainties in $n_i$ and $h_1$), $n_p$ and $n_m$ differ substantially, and so they can be easily distinguished experimentaly. Therefore, the observed apparent MIT [@Dultz_00] at $n = n_m$ has nothing to do with the percolation of the DRs and moreover with breaking of the electron liquid into droplets (the latter occurs at $n_e \ll n_p$). From Fig. \[Fig\_comparison\]a, one can estimate that the DR occupy mere 6% of the total area at $n = n_m$.
One qualitative prediction that follows from Eqs. (\[chi\_0\_large\_n\]) and (\[chi\_DR\_interp\]) is that the upturn of $\chi^{-1}(n_e)$ should be sharper in samples with larger $s$, which seems to be the case if the data of Ref. ($s = 14\,{\rm nm}$) are compared with those of Refs. ($s \leq 4\,{\rm
nm}$). Detailed fits are left for future.
I conclude with mentioning some other theoretical work on the subject. It was suggested [@Chakravarty_99] that near the MIT, function $\chi^{-1}(n_e)$ may contain both regular and singular parts. My theory can be considered the calculation of the former. Its detailed comparison with experiment may furnish an estimate of the putative singular term. The effect of disorder on $\chi^{-1}$ was also studied in Refs. and but the electron density inhomogeneity was not accounted for. Finally, a nonmonotonic $\chi^{-1}(n_e)$ dependence was found in a model [@Orozco_03] without disorder.
This work is supported by C. & W. Hellman Fund and A. P. Sloan Foundation. I am grateful to B. I. Shklovskii for indispensable comments and insights, and also to H. W. Jiang, A. K. Savchenko, and J. Shi for discussions.
[99]{}
For review, see E. Abrahams, S. V. Kravchenko, and M. P. Sarachik, **73**, 251 (2001).
G. Eytan, Y. Yayon, M. Rappaport, H. Shtrikman, and I. Bar-Joseph, **81**, 1666 (1998).
G. Finkelstein, P. I. Glicofridis, R. C. Ashoori, and M. Shayegan, Science **289**, 90 (2000).
S. Ilani, A. Yacoby, D. Mahalu, and H. Shtrikman, **84**, 3133 (2000); Science **292**, 1354 (2001).
M. Morgenstern, Chr. Wittneven, R. Dombrowski, and R. Wiesendanger, **84**, 5588 (2000).
A. L. Efros, F. G. Pikus, and V. G. Burnett, **47**, 2233 (1993) and references therein.
A. L. Efros and B. I. Shklovskii, *Electronic Properties of Doped Semiconductors* (Springer, New York, 1984); B. I. Shklovskii and A. L. Efros, JETP Lett. **44**, 669 (1986).
J. Shi and X. C. Xie, **88**, 86401 (2002).
J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, **50**, 1760 (1994).
S. C. Dultz and H. W. Jiang, **84**, 4689 (2000).
S. C. Dultz, B. Alavi, and H. W. Jiang, cond-mat/0210584.
G. D. Allison, A. K. Savchenko, and E. H. Linfield, Proceedings the 26th Intl. Conf. on Physics of Semiconductors (Edinburgh, 2002).
B. Tanatar and D. M. Ceperley, **39**, 5005 (1989).
M. M. Fogler, .
L. D. Landau and E. M. Lifshitz, *Electrodynamics of Continuous Media* (Pergamon, New York, 1960). Note that $E_0 = 4 \pi \sigma(0) / \kappa +
\text{const}$ and that $p$ can be read off the asymptotic law $2 \pi \sigma(r) \sim -p / r^3$ valid at $r \gg a$. This leads to the same formula for $\chi^{-1}$ as in the case of a hole in a metallic sheet, see more details in Ref. .
V. G. Burnett, A. L. Efros, and F. G. Pikus, **48**, 14365 (1993).
M. M. Fogler, E. I. Levin, and B. I. Shklovskii, **49**, 13767 (1994).
This is also true in serveral other geometries, see Ref. .
V. A. Gergel’ and R. A. Suris, Sov. Phys. JETP **48**, 95 (1978).
The data in Ref. must be divided by two to correct a notational error (J. Shi, private communication); $n_i$ is four times smaller than in Ref. to account for a peculiar choice $q = e / 2$ of the impurity charge made there.
S. Chakravarty, S. Kivelson, C. Nayak, and K. Voelker, Phil. Mag. B **79**, 859 (1999).
Q. Si and C. M. Varma, **81**, 4951 (1998).
R. Asgari and B. Tanatar, **65**, 085311 (2002).
S. Orozco, R. M. Méndez-Moreno, and M. Moreno, **67**, 195109 (2003).
|
=1200 =5.5truein =8.15truein 0.25truein 1.5em
**GRADED CONTRACTIONS OF BILINEAR INVARIANT**
**FORMS OF LIE ALGEBRAS**
Marc de Montigny
Department of Physics
McGill University
Montreal (Quebec)
H3A 2T8, CANADA
**February 1994**
ABSTRACT. We introduce a new construction of bilinear invariant forms on Lie algebras, based on the method of graded contractions. The general method is described and the $\Bbb Z_2$-, $\Bbb Z_3$-, and $\Bbb Z_2\otimes\Bbb Z_2$-contractions are found. The results can be applied to all Lie algebras and superalgebras (finite or infinite dimensional) which admit the chosen gradings. We consider some examples: contractions of the Killing form, toroidal contractions of $su(3)$, and we briefly discuss the limit to new WZW actions.
Contractions of Lie algebras were introduced in physics forty years ago by Wigner and Inönü in order to get a formal way of passing from the Poincaré group to the Galilei group [@1]. In general, contractions consist of the introduction of parameters in the basis of a Lie algebra such that, for some singular value of these parameters, we get a different ([*i.e.*]{} nonisomorphic) algebra. The interest of this method in physics stems from the fact that it relates different symmetry groups, from which the [*contracted*]{} group can be understood as an “approximation” of the “exact”, or noncontracted group. (A discussion of the concept of contraction and some generalizations of the Wigner-Inönü method are given in Ref. [@2]).
A different approach has been developed recently, which is based on the preservation of some [*grading*]{} of the Lie algebra through the contraction procedure. The general method of so-called [*graded contractions*]{} is presented in Ref. [@3]. The analogous theory for the representations (which contains the contractions of algebras as a particular case) and their Casimir operators is found in Refs [@4] and [@5], respectively.
In this paper we use the concept of graded contractions of Lie algebras to construct new symmetric bilinear invariant forms of a Lie algebra (in general, non-semisimple), starting from the known bilinear form of a noncontracted algebra. Even within a single Lie algebra, one can obtain different bilinear forms starting from one bilinear invariant form. As for the method of graded contractions of algebras, the present procedure is very general and can be applied to any Lie algebra or superalgebra, of finite or infinite dimension. Also the results are universal in the sense that given a grading group (which is abelian for our purpose), the problem is solved simultaneously for all Lie algebras of any type and dimension which admit the chosen grading. The construction is presented in the next section. In Section 3 we find the general bilinear invariant form obtained from preserving $\Bbb Z_2$-, $\Bbb Z_3$-, and $\Bbb Z_2\otimes\Bbb Z_2$-gradings. In Section 4 we present some examples: contractions of the Killing form, toroidal contractions of $su(3)$ (or, more precisely, of its complexification $A_2$), and the “deformation” of Wess-Zumino-Witten (WZW) actions (defined over some group manifold) into modified WZW actions, defined over a contracted group.
Consider a Lie algebra $g$ (corresponding to a Lie group $G$) defined over the field $\Bbb K \ (=\Bbb R$ or $\Bbb C\ )$. A symmetric bilinear invariant form $\Omega$ on $g$ is a mapping, $$\Omega : \quad g\times g\rightarrow \Bbb K\ ,\tag2.1$$ which satisfies the symmetry property, $$\Omega (X, Y) = \Omega (Y, X),\tag2.2$$ and is $G$-invariant, $$\Omega (\vartheta X\vartheta^{-1},\vartheta Y\vartheta^{-1}) =
\Omega (X, Y),\tag2.3$$ with $X, Y\in g$, and $\vartheta\in G$.
In terms of the generator $Z$ of $\vartheta$ in the Lie algebra, (2.3) is equivalent to $$\Omega (X, [Z, Y]) + \Omega (Y, [Z, X]) = 0.\tag2.4$$
Given a basis of generators $\{X_1,\dots ,X_{dim[g]}\}$ of $g$ such that $[X_A, X_B] = f^C_{AB} X_C$, the bilinear form is often written as a matrix $\Omega_{AB}\equiv
\Omega (X_A, X_B)$ which is symmetric (from (2.2)) and satisfies $f^D_{AB}\Omega_{CD} + f^D_{AC}\Omega_{BD} = 0$ (from (2.4)).
Now let us recall some definitions and properties that we need from the theory of graded contractions of Lie algebras (more details are in Refs [@3; @4]). A $\Gamma$-grading of a Lie algebra $g$ consists of the decomposition $$g=\bigoplus_{\mu\in\Gamma} g_\mu,\tag2.5$$ into eigenspaces under the action of a set of automorphisms of finite order on $g$ (for our purposes we can take $\Gamma$ to be an abelian finite group). Each of these automorphisms provides $g$ with a grading by a cyclic group having the same order. Thus the grading group $\Gamma$ is the tensor product of all the cyclic groups associated with every automorphism: $\Gamma = \bigotimes \Bbb Z_N$.
The commutators in $g$ inherit a grading structure from the automorphisms of finite order, that is, if $X\in g_\mu$ and $Y\in g_\nu$, then $$[X, Y] = Z,\tag2.6$$ with $Z\in g_{\mu +\nu}$ as long as the commutator is not zero. (The addition $\mu +\nu$ denotes the product of elements $\mu$ and $\nu$ in the grading group $\Gamma$.) This grading structure can be written symbolically, $$0\neq [g_\mu , g_\nu ]\subseteq g_{\mu + \nu },
\qquad \mu ,\ \nu ,\ \mu +\nu\in\Gamma.\tag2.7$$
A graded contraction $g^\varepsilon$ of $g$ is defined by modifying the commutators, $$[g_\mu ,g_\nu ]_\varepsilon\equiv\varepsilon_{\mu ,\nu}
[g_\mu ,g_\nu ]\subseteq \varepsilon_{\mu ,\nu}g_{\mu +\nu},
\tag2.8$$ where $\varepsilon\in\Bbb K$ . We have kept the $\varepsilon$-parameter in the right-hand side of (2.8) in order to show that if $\varepsilon =0$, then the commutator of an element of $g_\mu$ with an element of $g_\nu$ vanishes. From the definition (2.8) we see that the $\varepsilon$ parameters are symmetric ([*i.e.*]{} $\varepsilon_{\mu ,\nu} = \varepsilon_{\nu ,\mu}$), and by enforcing the Jacobi identities on the new commutators, one finds that the $\varepsilon$-parameters must satisfy the [*contraction relations*]{}, $$\varepsilon_{\mu ,\nu}\varepsilon_{\mu +\nu , \lambda} =
\varepsilon_{\nu ,\lambda}\varepsilon_{\nu +\lambda ,\mu},\tag2.9$$ for all the values of the indices. A solution $\varepsilon$ of (2.9) defines a contraction, that is, a new Lie algebra.
Now we define the contractions of the bilinear invariant form $\Omega$. Using the same notation as in (2.7), we define the [*contracted symmetric bilinear invariant form*]{} $\Omega^\gamma$ as $$\Omega^\gamma (g_\mu , g_\nu ) \equiv
\gamma_{\mu , \nu}\Omega (g_\mu , g_\nu ),\tag2.10$$ where $g_\mu \ (g_\nu )$ represents any element $X\in g_\mu
\ (\in g_\nu)$.
From the properties of symmetry and invariance, and given a contraction matrix $\varepsilon$ ([*i.e.*]{} a contracted algebra), we get the restrictions upon $\gamma$: $$\gamma_{\mu ,\nu} = \gamma_{\nu ,\mu}\tag2.11$$ and $$\varepsilon_{\lambda ,\mu}\gamma_{\lambda +\mu ,\nu} =
\varepsilon_{\lambda ,\nu}\gamma_{\lambda +\nu ,\mu}.\tag2.12$$
The relation (2.11) follows from equations (2.2) and (2.10).
The condition (2.12) is obtained from substituting (2.10) into (2.4):
$$\aligned
&\Omega^\gamma (g_\mu ,[g_\lambda ,g_\nu ]_\varepsilon )
+ \Omega^\gamma (g_\nu ,[g_\lambda ,g_\mu ]_\varepsilon ) = 0,\\
&\Omega^\gamma (g_\mu ,\varepsilon_{\lambda ,\nu}[g_\lambda ,
g_\nu]) +
\Omega^\gamma (g_\nu ,\varepsilon_{\lambda ,\mu}[g_\lambda ,
g_\mu]) = 0,\\
&\varepsilon_{\lambda ,\nu}\gamma_{\mu ,\lambda +\nu}
\Omega (g_\mu ,[g_\lambda ,g_\nu ]) +
\varepsilon_{\lambda ,\mu}\gamma_{\nu ,\lambda +\mu}
\Omega (g_\nu ,[g_\lambda ,g_\mu ]) = 0.
\endaligned$$ By comparing the last line with the invariance condition relation, $$\Omega (g_\mu ,[g_\lambda ,g_\nu ]) + \Omega
(g_\nu ,[g_\lambda ,g_\mu ])=0,$$ before contraction, we see that one must have $\varepsilon_{\lambda ,\nu}\gamma_{\mu ,\lambda +\nu}
= \varepsilon_{\lambda ,\mu}\gamma_{\nu ,\lambda +\mu}$. The form (2.12) is obtained by using the symmetry properties of $\varepsilon$ and $\gamma.\ \ \bullet$
The solutions of (2.11, 12) provide new invariant bilinear forms, obtained by substituting the $\gamma$-parameters back into (2.10). As for the contractions of algebras and representations, there is a trivial contracted bilinear form, for which all $\gamma$’s are equal to zero. However, the solution with all the $\gamma$’s equal to one is not trivial. As explained in Ref. [@3], the equations in (2.9) which contains an undefined $\varepsilon$ (when the corresponding commutator vanishes already in the noncontracted algebra) does not appear in the system of equations to be solved. The same occurs for the $\gamma$’s. If $\mu$ and $\nu$ are such that $\Omega (g_\mu , g_\nu) =0$, then the equations containing the corresponding $\gamma_{\mu ,\nu}$ must be removed from the system (2.12). If it does not happen, we say that this grading is [*generic*]{} regarding the invariant bilinear form. Note that definitions (2.8) and (2.10) allow one to consider also [*deformations*]{}, that is, processes after which some commutators or bilinear forms –initially zero– become non-trivial, but satisfy the grading property nevertheless. We shall not consider this type of process below.
Note also that the “composition” $(\gamma_1\bullet\gamma_2)_{\mu ,\nu}
\equiv (\gamma_1)_{\mu ,\nu} (\gamma_2)_{\mu ,\nu}$ (without summation over repeated indices) of solution matrices, which we have defined for the contractions of algebras in equations (2.8, 9) of Ref. [@3], here does not yield a new solution in general. Thus, given two solutions of (2.12), $\gamma_1$ and $\gamma_2$, their composition will not be a solution unless extremely particular conditions on the $\varepsilon$’s are satisfied. The same applies for the “normalization” of the $\gamma$’s, because it is tied up to the normalization of the $\varepsilon$’s. If we perform a change of basis $g_\mu\rightarrow g'_\mu = a_\mu g_\mu$, then the invariant bilinear form becomes $\Omega^\gamma (g'_\mu ,g'_\nu )
= {\gamma_{\mu ,\nu}\over {a_\mu a_\nu}}\Omega (g_\mu ,g_\nu )$. Unless we consider very particular values of $\gamma$’s and $\varepsilon$’s it will not be possible to satisfy ${\gamma_{\mu ,\nu}
\over {a_\mu a_\nu}} = 1$ for all $\gamma$’s. Unlike the contractions of representations, in which case we can normalize the parameters $\psi$ (see Ref. [@4]) by changing the basis of the representation vector space independently of the basis of the algebra, here the normalizations of $\varepsilon$ and $\gamma$ are interdependent.
In this section we find the contractions of the invariant bilinear forms, using the results of Ref. [@3], for Lie algebras over the field of [*complex*]{} numbers. As in Ref. [@3] we consider the contractions with gradings of [*generic*]{} type. We have not normalized the $\gamma$’s in order to display the possible zero parameters. (Note that the $\varepsilon$’s of the present paper are the $\gamma$’s in Ref. [@3].)
The grading group consists of two elements, $\Gamma = \{0,1\}$, with the product (denoted additively): $$0+0=0,\quad 0+1=1+0=1,\quad 1+1=0.$$ There are three independent $\gamma$-parameters: $\gamma_{0,0}, \gamma_{0,1}\ (=\gamma_{1,0})$ and $\gamma_{1,1}$. We shall cast them into a matrix form: $\gamma =\pmatrix
\gamma_{0,0} & \gamma_{0,1}\\
\gamma_{0,1} & \gamma_{1,1}\endpmatrix$, although it should not be taken formally as a matrix, since none of the properties of matrices –multiplication, inverse, determinant, etc.– are shared by the present objects. (Hereafter, we write only the upper diagonal of $\gamma$ and $\varepsilon$, remembering that they are symmetric).
The condition (2.12) reads, for $\Bbb Z_2$, $$\varepsilon_{0,0}\gamma_{0,1} = \varepsilon_{0,1}\gamma_{0,1},\quad
\varepsilon_{0,1}\gamma_{1,1} = \varepsilon_{1,1}\gamma_{0,0}.
\tag3.1$$
For the trivial contraction $\varepsilon =\pmatrix
\varepsilon_{0,0} & \varepsilon_{0,1} \\
& \varepsilon_{1,1}\endpmatrix = \pmatrix
1 & 1 \\
& 1 \endpmatrix = (1)$, the relations (3.1) become $\gamma_{0,1} =
\gamma_{0,1}$ and $\gamma_{0,0}=\gamma_{1,1}$ (which imposes no further restriction upon $\gamma_{0,1}$). For the other trivial contraction $\varepsilon = \pmatrix
0 & 0 \\
& 0\endpmatrix =(0)$, there are no restrictions on $\gamma$, which are thus completely free. If $\varepsilon_{0,0}\neq\varepsilon_{0,1}$, then $\gamma_{0,1}$ must vanish. This happens for the Wigner-Inönü-like contraction matrix $\varepsilon = \pmatrix
1 & 0 \\
& 0\endpmatrix$.
In summary, we find the following contractions, $$\aligned
\varepsilon &= \pmatrix
1 & 1 \\
& 1 \endpmatrix ,\
\gamma = \pmatrix
a & b \\
& a \endpmatrix ,\\
\varepsilon &= \pmatrix
0 & 0 \\
& 0 \endpmatrix ,\
\gamma = \pmatrix
a & b \\
& c \endpmatrix ,\\
\varepsilon &= \pmatrix
1 & 0 \\
& 0 \endpmatrix ,\
\gamma = \pmatrix
a & 0 \\
& b \endpmatrix ,\\
\varepsilon &= \pmatrix
0 & 0 \\
& 1 \endpmatrix ,\
\gamma = \pmatrix
0 & a \\
& b \endpmatrix ,\\
\varepsilon &= \pmatrix
1 & 1 \\
& 0 \endpmatrix ,\
\gamma = \pmatrix
a & b \\
& 0 \endpmatrix .\endaligned\tag3.2$$ where $a, b$ and $c$ are arbitrary (possibly zero) complex numbers.
The $\Bbb Z_3$-contractions ($\Bbb Z_3$ consists of three elements $0, 1, 2$ on which the product is the usual addition modulo 3) are determined by six parameters (cast into matrix form), $$\gamma =\pmatrix
\gamma_{0,0} & \gamma_{0,1} & \gamma_{0,2} \\
& \gamma_{1,1} & \gamma_{1,2} \\
& & \gamma_{2,2} \endpmatrix .$$
In terms of these parameters, the restrictions (2.12) are $$\aligned
&\varepsilon_{0,0}\gamma_{0,1} = \varepsilon_{0,1}\gamma_{0,1},\quad
\varepsilon_{1,1}\gamma_{2,2} = \varepsilon_{1,2}\gamma_{0,1},\\
&\varepsilon_{0,0}\gamma_{0,2} = \varepsilon_{0,2}\gamma_{0,2},\quad
\varepsilon_{0,2}\gamma_{1,2} = \varepsilon_{1,2}\gamma_{0,0},\\
&\varepsilon_{0,1}\gamma_{1,2} = \varepsilon_{0,2}\gamma_{1,2},\quad
\varepsilon_{0,2}\gamma_{2,2} = \varepsilon_{2,2}\gamma_{0,1},\\
&\varepsilon_{0,1}\gamma_{1,1} = \varepsilon_{1,1}\gamma_{0,2},\quad
\varepsilon_{1,2}\gamma_{0,2} = \varepsilon_{2,2}\gamma_{1,1},\\
&\varepsilon_{0,1}\gamma_{1,2} = \varepsilon_{1,2}\gamma_{0,0}.
\endaligned\tag3.3$$
And the solutions are (with the $\varepsilon$’s given in the Section 4 of Ref. [@3]), $$\aligned
&\gamma (\varepsilon^I) = \pmatrix
a & b & c\\
& c & 0\\
& & 0\endpmatrix ,\
\gamma (\varepsilon^{II}) = \pmatrix
a & b & c\\
& 0 & 0\\
& & b\endpmatrix ,\\
&\gamma (\varepsilon^{III}) = \pmatrix
a & b & c\\
& 0 & 0\\
& & 0\endpmatrix ,\
\gamma (\varepsilon^{IV}) = \pmatrix
a & 0 & 0\\
& b & c\\
& & 0\endpmatrix ,\\
&\gamma (\varepsilon^{V}) = \pmatrix
a & 0 & 0\\
& 0 & b\\
& & c\endpmatrix ,\
\gamma (\varepsilon^{VI}) = \pmatrix
a & 0 & 0\\
& b & c\\
& & d\endpmatrix ,\\
&\gamma (\varepsilon^{VII}) = \pmatrix
a & b & 0\\
& 0 & 0\\
& & c\endpmatrix ,\
\gamma (\varepsilon^{VIII}) = \pmatrix
a & 0 & b\\
& c & 0\\
& & 0\endpmatrix ,\\
&\gamma (\varepsilon^{IX}) = \pmatrix
0 & a & 0\\
& b & c\\
& & a\endpmatrix ,\
\gamma (\varepsilon^{X}) = \pmatrix
0 & 0 & a\\
& a & b\\
& & c\endpmatrix ,\\
&\gamma (\varepsilon^{XI}) = \pmatrix
0 & 0 & 0\\
& a & b\\
& & c\endpmatrix ,\
\gamma (\varepsilon^{XII}) = \pmatrix
a & b & 0\\
& c & d\\
& & 0\endpmatrix ,\\
&\gamma (\varepsilon^{XIII}) = \pmatrix
a & 0 & b\\
& 0 & c\\
& & d\endpmatrix,\
\gamma (\varepsilon =(1)) = \pmatrix
a & b & c\\
& c & a\\
& & b\endpmatrix ,\endaligned
\tag3.4$$ where $a, b, c$ and $d$ are arbitrary. The matrix $\varepsilon =(1)$ has all its entries equal to 1 (it is not the identity matrix). Obviously, the $\gamma$’s corresponding to $\varepsilon =(0)$ are free.
The grading group consists of four elements, that we name $a=00,\ b=01,\ c=10$ and $d=11$, so that their product is $a+k=k,\ 2k=a\ (k=a,\dots ,d)$ and $b+c=d, c+d=b, b+d=c$. In the generic case ([*i.e.*]{} when all the $\varepsilon$’s are defined), the equations (2.12) are: $$\aligned
&\varepsilon_{a,a}\gamma_{a,k} = \varepsilon_{a,k} \gamma_{a,k},\quad
\varepsilon_{a,k}\gamma_{k,k} = \varepsilon_{k,k} \gamma_{a,a},\\
&\varepsilon_{a,b}\gamma_{b,c} = \varepsilon_{a,c}\gamma_{b,c}
= \varepsilon_{b,c}\gamma_{a,d},\\
&\varepsilon_{a,b}\gamma_{b,d} = \varepsilon_{a,d}\gamma_{b,d}
= \varepsilon_{b,d}\gamma_{a,c},\\
&\varepsilon_{a,c}\gamma_{c,d} = \varepsilon_{a,d}\gamma_{c,d}
= \varepsilon_{c,d}\gamma_{a,b},\\
&\varepsilon_{b,c}\gamma_{d,d} = \varepsilon_{b,d}\gamma_{c,c}
= \varepsilon_{c,d}\gamma_{b,b},\\
&\varepsilon_{b,b}\gamma_{a,c} = \varepsilon_{b,c}\gamma_{b,d},\quad
\varepsilon_{b,b}\gamma_{a,d} = \varepsilon_{b,d}\gamma_{b,c},\\
&\varepsilon_{b,c}\gamma_{c,d} = \varepsilon_{c,c}\gamma_{a,b},\quad
\varepsilon_{c,c}\gamma_{a,d} = \varepsilon_{c,d}\gamma_{b,c},\\
&\varepsilon_{c,d}\gamma_{b,d} = \varepsilon_{d,d}\gamma_{a,c},\quad
\varepsilon_{b,d}\gamma_{c,d} = \varepsilon_{d,d}\gamma_{a,b},
\endaligned\tag3.5$$ where $k=b, c, d$. The $\gamma$-solutions (corresponding to the $\varepsilon$-solutions given in Ref. [@3]) are in Tables I and II.
First, consider the algebra $A_1$ (the complexification of $sl(2)$), with elements $\{X, Y, Z\}$ and commutation relations $$[X,Y]=Z,\ [Y,Z]=X,\ [Z,X]=Y.$$ From these we find that the Killing form $\Omega^{Killing}(\cdot ,\cdot )
\equiv Tr (ad(\cdot ) ad(\cdot ))$ is equal to $$\Omega^{Killing} = diag (-2, -2, -2).\tag4.1$$
Now consider the $\Bbb Z_2\otimes\Bbb Z_2$-grading $$L_b = \{X\},\ L_c = \{Y\},\ L_d = \{Z\},\tag4.2$$ where we use the same notation for the grading group elements as in subsection 3.3. Following (2.8), the commutators are modified to $$[X,Y]_\varepsilon =\varepsilon_{b,c}Z,\
[Y,Z]_\varepsilon =\varepsilon_{c,d}X,\
[Z,X]_\varepsilon =\varepsilon_{b,d}Y,\tag4.3$$ so that the Killing form that we get from the adjoint representation is $$\Omega^{Killing,\varepsilon} =
-2\ diag (\varepsilon_{b,c}\varepsilon_{b,d},\
\varepsilon_{b,c}\varepsilon_{c,d},\
\varepsilon_{b,d}\varepsilon_{c,d},).\tag4.4$$
Now if we apply the definition (2.10) for the grading (4.2), the form (4.1) becomes $$\Omega^{Killing, \gamma} = -2\ diag (\gamma_{b,b}, \gamma_{c,c},
\gamma_{d,d}),\tag4.5$$ which is to be compared to (4.4). This is done by noting that the equations (2.12) which are relevant here ([*i.e.*]{} in which $\varepsilon$ and $\gamma$ do not contain $a$ as an index) are $$\varepsilon_{b,c}\gamma_{d,d} = \varepsilon_{c,d}\gamma_{b,b} =
\varepsilon_{b,d}\gamma_{c,c},\tag4.6$$ for which $$\gamma_{b,b}=\varepsilon_{b,c}\varepsilon_{b,d},\
\gamma_{c,c}=\varepsilon_{b,c}\varepsilon_{c,d},\
\gamma_{d,d}=\varepsilon_{b,d}\varepsilon_{c,d}\tag4.7$$ is a possible solution. By comparing (4.5) to (4.4), we see that the form obtained by graded contractions of $A_1$ ([*i.e.*]{} by the introduction of $\varepsilon$-parameters into the commutators) can be compatible with the $\gamma$-contracted form obtained by the present procedure, [*i.e.*]{} using (2.10).
A similar compatibility exists in any general Lie algebra $g$ with basis $\{X_1,\dots ,X_{dim [g]}\}$ such that $$[X_A, X_B] = f^C_{AB} X_C = ad(X_A)_{CB} X_C.\tag4.8$$ Suppose that $X_A, X_B$ belong to the $\Gamma$-grading subspaces $g_\mu ,g_\nu$, respectively, so that $X_C\in g_{\mu +\nu}$. Then the Killing form is $$\aligned
\Omega^{Killing} (X_{\mu A}, X_{\nu B}) &=
Tr \left [ad(X_{\mu A}) ad(X_{\nu B})\right ], \\
&= \sum_{C,D} ad(X_{\mu A})_{(\mu +\sigma )C;\sigma D}\
ad(X_{\nu B})_{(\mu +\nu +\sigma )D;(\mu +\sigma )C},
\endaligned\tag4.9$$ from which we see that $\mu$ must be the $\Gamma$-inverse of $\nu$. (We have written the grading indices along with the algebra indices).
By introducing $\varepsilon$-parameters as in (2.8), this becomes $$\Omega^{Killing, \varepsilon} (X_{\mu A}, X_{\nu B}) =
\sum_{C,D} \varepsilon_{\mu ,\sigma}\varepsilon_{\nu ,\mu +\sigma}
ad(X_{\mu A})_{(\mu +\sigma )C;\sigma D}
ad(X_{\nu B})_{\sigma D;(\mu +\sigma )C},\tag4.10$$ where the $\varepsilon$’s are relevant if the corresponding commutators are defined. One can compare (4.10) to the form obtained via our definition (2.10), [*i.e.*]{} $$\Omega^{Killing,\gamma} (X_{\mu A}, X_{\nu B}) =
\gamma_{\mu ,\nu} \Omega^{Killing} (X_{\mu A}, X_{\nu B}),\tag4.11$$ where $\mu + \nu = 0$ (the identity element in $\Gamma$), if the algebra is be such that $\varepsilon^2$ in (4.10) can be factored out. Then one can identify $$\gamma_{\mu ,\nu} = \sum_\sigma \varepsilon_{\mu ,\sigma}
\varepsilon_{\nu ,\mu +\sigma},\tag4.12$$ where the sum is over the $\sigma$ indices such that the $\varepsilon$’s are defined, and with $\mu +\nu =0$. In the particular case of $A_1$, (4.12) is just (4.7).
There are four different fine gradings of $A_2$ (the complexification of $su(3)$) [@6--7], one of which is the [*toroidal*]{} (or [*Cartan*]{}) grading: $$A_2 = \oplus_{\mu =-3}^3\ g_\mu,$$ with $$\aligned
g_1 &= \{e_\alpha\},\ g_{-1} = \{e_{-\alpha}\},\\
g_2 &= \{e_\beta\},\ g_{-2} = \{e_{-\beta}\},\\
g_3 &= \{e_{\alpha +\beta}\},\ g_{-3} =
\{e_{-(\alpha +\beta )}\},\\
g_0 &= \{h_\alpha ,h_\beta\}.\endaligned\tag4.13$$ for which the grading group is $\Bbb Z_7$.
Consider the following invariant bilinear form on $A_2$; $$\aligned
\Omega (a_\delta ,a_{-\delta}) &= \Omega_\delta, \\
\Omega (h_{\alpha_i}, h_{\alpha_j}) &= <\alpha_i |\alpha_j >,
\endaligned\tag4.14$$ where $i, j = 1, 2$; $\delta$ is any root of $A_2$, and all other elements of $\Omega$ are zero. The only relevant $\gamma$-parameters are $\gamma_{0,0}$ and $\gamma_{\mu ,-\mu}$.
The grading (4.13) is such that the relevant $\varepsilon$-parameters are: $$\aligned
&\varepsilon_{1,2},\ \varepsilon_{1,-3},\ \varepsilon_{1,-1},\
\varepsilon_{-2,-1},\\
&\varepsilon_{2,-3},\ \varepsilon_{2,-2},\ \varepsilon_{3,-3},\
\varepsilon_{3,-2},\\
&\varepsilon_{3,-1},\ \varepsilon_{0, \mu}\ \ (\mu = -3, \dots ,3).
\endaligned$$
From (2.12), we get the following relations: $$\varepsilon_{\mu ,-\mu} \gamma_{0,0}
= \varepsilon_{0,\mu} \gamma_{\mu ,-\mu}
= \varepsilon_{0,-\mu} \gamma_{\mu ,-\mu},\tag4.15$$ for $\mu = 1, 2, 3$. The possible $\varepsilon$-parameters have been found and the associated algebras identified in Ref. [@8]. The $\gamma$ solutions are to be substituted into $$\aligned
&\Omega^\gamma (e_\alpha , e_{-\alpha}) = \gamma_{1,-1}\
\Omega_\alpha,\\
&\Omega^\gamma (e_\beta ,e_{-\beta}) = \gamma_{2,-2}\
\Omega_\beta,\\
&\Omega^\gamma (e_{\alpha +\beta}, e_{-(\alpha +\beta )}) =
\gamma_{3,-3}\ \Omega_{\alpha +\beta},\\
&\Omega^\gamma (h_{\alpha_i}, h_{\alpha_j}) =
\gamma_{0,0}\ <\alpha_i |\alpha_j>.\endaligned\tag4.16$$
The method can be applied readily to Kac-Moody algebras. Some examples of graded contractions of Kac-Moody algebras are given in Ref. [@3]. (The particular case of Wigner-Inönü contractions of Kac-Moody and Virasoro algebras has been studied explicitly in Refs [@9] and [@10], respectively.)
In this subsection, we exploit the fact that, starting from a Lie algebra $g$ with invariant bilinear form $\Omega$, one may define a WZW action on the corresponding group manifold [@11]. Suppose that the group $G$, which corresponds to the Lie algebra $g$, is simply connected. Consider a three-manifold $B$ with boundary $\Sigma = \partial B$ and $\vartheta$, a map from $\Sigma$ to $G$, extended to a map from $B$ to $G$. The WZW action on a Riemann surface $\Sigma$ is [@11] $$S_{WZW} (\vartheta ) = {1\over{4\pi}}\int_\Sigma d^2 y\
\Omega_{AB} A^A_aA^{Ba}\ +\ {i\over{12\pi}} \int_B d^3 y\
\epsilon_{abc}A^{Aa}A^{Bb}A^{Cc}\Omega_{CD}f^D_{AB},
\tag4.17$$ where $a, b, c$ represent indices over $\Sigma$ and $B$, and $\vartheta^{-1}
\partial_a\vartheta = A^A_aX_A$ (the $X$’s being the generators of the Lie algebra $g$).
Now if we consider a $\Gamma$-grading of $g$, such that $X_A\in
g_{\Cal A}, X_B\in g_{\Cal B}$, etc. ($\Cal A, \Cal B, \dots$ denote the grading labels of the subspace to which $X_A, X_B, \dots$ belong, respectively) then we get a new WZW action: $$\aligned
S^{\varepsilon ,\gamma}_{WZW} (\vartheta ) &=
{1\over {4\pi}}\int_\Sigma d^2y\ \gamma_{\Cal A,\Cal B} \Omega_{AB}
A^A_aA^{Ba}\ + \\
& \ \ + \ {i\over{12\pi}}\int_B d^3 y\ \epsilon_{abc}
A^{Aa}A^{Bb}A^{Cc}\gamma_{\Cal C,\Cal D}\Omega_{CD}
\varepsilon_{\Cal A,\Cal B} f^D_{AB}.\endaligned\tag4.18$$ The invariant bilinear form $\Omega^\gamma$ must be non-degenerate, [*i.e.*]{} the inverse $\Omega^{\gamma ,AB}$ of $\Omega^\gamma_{AB}$ must be defined such that $\Omega^{\gamma ,AB}\Omega^\gamma_{BC} = \delta^A_C$. The construction of a WZW model based on a non-semisimple group has been discussed recently in Refs [@12] and [@13], for ungauged and gauged models, respectively.
From that point of view, we see that the contraction procedure may provide a new geometry (in particular, a new spacetime), by starting from the one associated to the noncontracted WZW model, and by “deforming” it compatibly with the deformation of the algebra. The concept of contraction has not been mentioned explicitly in Refs [@12; @13], although the invariant bilinear form used therein was obtained from a contraction procedure [@14]. To our knowledge, contraction methods have been used explicitly for the first time in Refs [@15; @16]. The contraction used in [@15] leaves the contracted algebras with the particular $\Bbb Z_3$ structure, $$\varepsilon = \pmatrix
1 & 1 & 1\\
& 1 & 0\\
& & 0\endpmatrix ,$$ which suggests a generalization along our point of view. A detailed discussion about this topic is postponed to a future publication.
The deformation of a geometry into another one, using the concept of contraction, has been studied in Ref. [@17] (although, using a completely different approach). Group contractions, interpreted as quasi-catastrophical connections between different geometries, or topological fluctuations in spacetime, have been considered in Ref. [@18], establishing a relation between different models of the universe.
The author is indebted to Drs R. C. Myers and R. T. Sharp for reading the manuscript, and to the Natural Sciences and Engineering Research Council of Canada for financial support.
The $\gamma$-solutions corresponding to the contractions of algebras given in Table 1 of Ref. [@3], on which they must be superposed.
$$\aligned
&\pmatrix
a & b & c & d\\
& a & d & c\\
& & a & b\\
& & & a\endpmatrix
\pmatrix
a & b & c & d\\
& 0 & d & 0\\
& & a & b\\
& & & 0\endpmatrix
\pmatrix
0 & a & 0 & b\\
& c & b & d\\
& & 0 & a\\
& & & e\endpmatrix
\pmatrix
a & 0 & b & 0\\
& c & 0 & d\\
& & a & 0\\
& & & e\endpmatrix\\
&\pmatrix
a & b & c & d\\
& a & d & c\\
& & 0 & 0\\
& & & 0\endpmatrix
\pmatrix
a & b & c & d\\
& 0 & d & 0\\
& & 0 & 0\\
& & & 0\endpmatrix
\pmatrix
0 & a & 0 & b\\
& c & b & d\\
& & 0 & 0\\
& & & e\endpmatrix
\pmatrix
a & 0 & b & 0\\
& c & 0 & d\\
& & 0 & 0\\
& & & e\endpmatrix \\
&\pmatrix
0 & 0 & a & b\\
& 0 & b & a\\
& & c & d\\
& & & e\endpmatrix
\pmatrix
0 & 0 & a & b\\
& 0 & b & 0\\
& & c & d\\
& & & e\endpmatrix
\pmatrix
0 & 0 & 0 & a\\
& b & c & d\\
& & e & f\\
& & & g\endpmatrix
\pmatrix
0 & 0 & a & 0\\
& b & c & d\\
& & e & f\\
& & & g\endpmatrix \\
&\pmatrix
a & b & 0 & 0\\
& a & 0 & 0\\
& & c & d\\
& & & e\endpmatrix
\pmatrix
a & b & 0 & 0\\
& 0 & 0 & 0\\
& & c & d\\
& & & e\endpmatrix
\pmatrix
0 & a & 0 & 0\\
& b & c & d\\
& & e & f\\
& & & g\endpmatrix
\pmatrix
a & 0 & 0 & 0\\
& b & c & d\\
& & e & f\\
& & & g\endpmatrix\endaligned$$
The $\gamma$-solutions corresponding to the contractions of algebras given in Table 2 of Ref. [@3], on which they must be superposed.
$$\aligned
&\pmatrix
a & b & c & 0\\
& 0 & 0 & 0\\
& & 0 & 0\\
& & & d\endpmatrix
\pmatrix
a & 0 & 0 & b\\
& c & d & 0\\
& & e & 0\\
& & & 0\endpmatrix
\pmatrix
a & b & 0 & c\\
& 0 & 0 & 0\\
& & d & 0\\
& & & 0\endpmatrix\\
&\pmatrix
a & 0 & 0 & 0\\
& b & c & 0\\
& & d & 0\\
& & & 0\endpmatrix
\pmatrix
0 & a & b & 0\\
& c & d & b\\
& & e & 0\\
& & & 0\endpmatrix
\pmatrix
a & b & 0 & c\\
& d & 0 & e\\
& & 0 & 0\\
& & & f\endpmatrix\\
&\pmatrix
0 & a & b & 0\\
& c & d & 0\\
& & e & a\\
& & & 0\endpmatrix
\pmatrix
0 & a & b & 0\\
& c & d & b\\
& & e & a\\
& & & 0\endpmatrix
\pmatrix
a & 0 & b & c\\
& 0 & 0 & 0\\
& & d & e\\
& & & f\endpmatrix\\
&\pmatrix
0 & 0 & 0 & 0\\
& a & b & c\\
& & d & e\\
& & & f\endpmatrix
\pmatrix
0 & a & 0 & b\\
& c & 0 & d\\
& & 0 & a\\
& & & e\endpmatrix
\pmatrix
0 & 0 & 0 & 0\\
& a & b & c\\
& & d & e\\
& & & f\endpmatrix\endaligned$$
last three columns:
$$\aligned
&\pmatrix
a & 0 & b & c\\
& d & 0 & 0\\
& & 0 & 0\\
& & & 0\endpmatrix
\pmatrix
a & b & c & d\\
& 0 & 0 & 0\\
& & 0 & 0\\
& & & 0\endpmatrix
\pmatrix
a & b & c & 0\\
& d & e & 0\\
& & f & 0\\
& & & 0\endpmatrix\\
&\pmatrix
a & 0 & 0 & 0\\
& b & 0 & c\\
& & 0 & 0\\
& & & d\endpmatrix
\pmatrix
a & b & c & d\\
& 0 & 0 & c\\
& & 0 & 0\\
& & & 0\endpmatrix
\pmatrix
0 & 0 & 0 & 0\\
& a & b & c\\
& & d & e\\
& & & f\endpmatrix\\
&\pmatrix
a & 0 & 0 & 0\\
& 0 & 0 & 0\\
& & b & c\\
& & & d\endpmatrix
\pmatrix
a & b & c & d\\
& 0 & 0 & 0\\
& & 0 & b\\
& & & 0\endpmatrix
\pmatrix
a & 0 & 0 & b\\
& c & d & 0\\
& & e & 0\\
& & & 0\endpmatrix\\
&\pmatrix
0 & 0 & 0 & 0\\
& a & b & c\\
& & d & e\\
& & & f\endpmatrix
\pmatrix
0 & 0 & a & b\\
& 0 & 0 & a\\
& & c & d\\
& & & e\endpmatrix
\pmatrix
a & b & c & d\\
& 0 & 0 & c\\
& & 0 & b\\
& & & a\endpmatrix\endaligned$$
\[\]1E. Inönü and E. P. Wigner Proc. Nat. Acad. Sci. US391953 510–524
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|
---
abstract: 'Influence maximization has found applications in a wide range of real-world problems, for instance, viral marketing of products in an online social network, and information propagation of valuable information such as job vacancy advertisements and health-related information. While existing algorithmic techniques usually aim at maximizing the total number of people influenced, the population often comprises several socially salient groups, e.g., based on gender or race. As a result, these techniques could lead to disparity across different groups in receiving important information. Furthermore, in many of these applications, the spread of influence is time-critical, i.e., it is only beneficial to be influenced before a time deadline. As we show in this paper, the time-criticality of the information could further exacerbate the disparity of influence across groups. This disparity, introduced by algorithms aimed at maximizing total influence, could have far-reaching consequences, impacting people’s prosperity and putting minority groups at a big disadvantage. In this work, we propose a notion of *group fairness* in *time-critical influence maximization*. We introduce surrogate objective functions to solve the influence maximization problem under fairness considerations. By exploiting the submodularity structure of our objectives, we provide computationally efficient algorithms with guarantees that are effective in enforcing fairness during the propagation process.'
author:
- |
Junaid Ali$^\dagger$ $\qquad$ Mahmoudreza Babaei$^\dagger$ $\qquad$ Abhijnan Chakraborty$^\dagger$ Baharan Mirzasoleiman$^\ddagger$ $\qquad$ Krishna P. Gummadi$^\dagger$ $\qquad$ Adish Singla$^\dagger$\
\
$^\dagger$Max Planck Institute for Software Systems (MPI-SWS),\
{junaid, babaei, achkrab, gummadi, adishs}@mpi-sws.org\
$^\ddagger$Stanford University, baharanm@stanford.edu
bibliography:
- 'fair\_inf\_max.bib'
title: 'On the Fairness of Time-Critical Influence Maximization in Social Networks'
---
Introduction
============
The problem of [*Influence Maximization*]{} has been widely studied due to its application in multiple domains such as viral marketing [@richardson2002mining], social recommendations [@ye2012exploring], propagation of information related to jobs, financial opportunities or public health programs [@banerjee2013diffusion; @yadav2016using]. Over the years, extensive research efforts have focused on the cascading behavior, diffusion and spreading of ideas, or containment of diseases, [@leskovec2007cost]; [@kempe2003maximizing]; [@richardson2002mining]; [@wallinga2004different]; and [@leskovec2010inferring]. The idea is to identify a set of initial sources (i.e., [*seed nodes*]{}) in a social network who can influence other people (e.g., by propagating key information), and traditionally the goal has been to maximize the total number of people influenced in the process (e.g., who received the information being propagated) [@li2018influence]; [@kourtellis2013identifying]; [@kempe2003maximizing]; [@goyal2013minimizing]; [@babaei2013revenue]; [@bharathi2007competitive]; [@budak2011limiting]; [@carnes2007maximizing].
Real-world social networks, however, are often not homogeneous and comprise different groups of people. Due to the disparity in group sizes, the potentially high propensity towards creating within-group links [@mcpherson2001birds], and differences in dynamics of influences among different groups [@DBLP:conf/wsdm/SinglaW09], the structure of the social network can cause a disparity in the influence maximization process. For example, selecting most of the seed nodes from the majority group might maximize the total number of influenced nodes, but very few members of the minority group may get influenced. In many application scenarios such as propagation of job or health-related information, such disparity can end up impacting people’s livelihood and some groups may become impoverished in the process.
Moreover, some applications are also [*time-critical*]{} in nature [@chen2012time]. For example, many job applications typically have a deadline by which one needs to apply; if information related to the application reaches someone after the deadline, it is not useful. Similarly, in viral marketing, many companies offer discount deals only for few days (hours); getting this information late doesn’t serve the recipient(s). More worryingly, if one group of people gets influenced (i.e., they get the information) faster than other groups, it could end up exacerbating the inequality in information access. This is possible if the majority group is better connected than the minority group, and they are more central in the network. Thus, in time-critical application scenarios, focusing on the traditional criteria of maximizing the number of influenced nodes can have a disparate impact on different groups present in the network. This disparity in time-critical applications, in turn, can bring minority and under-represented groups at a big disadvantage with far-reaching consequences. In this paper, we attempt to mitigate such unfairness in time-critical influence maximization ([<span style="font-variant:small-caps;">TCIM</span>]{}), and we focus on two settings: (i) where the budget (i.e., the number of seeds) is fixed and the goal is to find a seed set which maximizes the time-critical influence, we call this as [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem, and (ii) where a certain quota or fraction of the population should be influenced under the prescribed time deadline, and the goal is to find such a seed set of minimal size, we call this as [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem.
[[[**Our Contributions.**]{}]{}]{} Our first contribution is to formally introduce the notion of fairness in time-critical influence maximization, which requires that [*within a prescribed time deadline, the fraction of influenced nodes should be equal across different groups*]{}. We highlight, via experiments and illustrative example, that the standard algorithmic techniques for solving [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}and [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problems lead to unfair solutions, and the disparity across groups could get worse with tighter time deadline.
We introduce two formulations of [<span style="font-variant:small-caps;">TCIM</span>]{}problems under fairness considerations, namely [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}and [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}. However, directly optimizing [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem and [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem are computationally challenging, and comes without any structural properties. As our second contribution, we propose [*monotone submodular*]{} surrogates for solving both of these problems that capture the tradeoff between maximizing total influence and minimizing unfairness (i.e., disparity across groups). Although the surrogate problems are still NP-Hard, we propose a greedy approximation with provable guarantees.
We evaluate our proposed solutions over a real-world social network and show that they are successful in enforcing the aforementioned fairness notion. As expected, enforcing this fairness comes at the cost of a reduction in performance, i.e., lowering the total influence, in case of [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem, and solutions with larger seed set sizes, in case of [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem. However, as guaranteed by our theoretical results, our experiments indeed demonstrate that this cost of fairness, i.e., reduction in performance, is bounded for our approach.
[[[**Contemporary Works.**]{}]{}]{} Fairness has mostly been studied in algorithmic decision-making system, e.g, [@speicher2018unified]; [@chakraborty2019equality]; [@dwork2012fairness]; [@hajian2016algorithmic]; [@feldman2015certifying]; and [@hardt2016equality], however, only a few contemporary works have investigated fairness in influence maximization, as described next.
Very recently, [@fish_dmt] proposed a notion of individual fairness in information access, but did not consider the group fairness aspects. In addition, some prior works have proposed constrained optimization problems to encourage diversity in selecting the most influential nodes [@bredereck2018multiwinner; @faliszewski2017multiwinner; @benabbou2018diversity; @aghaei2019learning]. [@yu2017fair] proposed a greedy heuristic to enforce group fairness where multiple sources of influence are in competition.
In a concurrent work, [@Tsang_dmt] propose a method to achieve group fairness in influence maximization. However, their work is different from our approach in three ways: i) they propose a different problem formulation with objective that does not have submodular structural properties; ii) they only study the problem under budget constraint; and iii) they do not consider the time-critical aspect of influence in their definition of fairness for influence maximization. This could result in majority groups being influenced before the minority, and can lead to disparity in applications where the timing of being influenced/informed is critical. In our work, we introduce a submodular objective that directly addresses the time-criticality in influence maximization problem under budget constraint as well as coverage constraint.
{width="\linewidth"} \[fig:example\_graph\]
Disparity in [<span style="font-variant:small-caps;">TCIM</span>]{}and Introducing Notions of Fairness {#sec:notions}
======================================================================================================
In this section, we first introduce the utility funciton of time-critical influence function, then, we highlight the disparity in utility across population resulting from the solution to the standard [<span style="font-variant:small-caps;">TCIM</span>]{}problem formulations, and introduce a notion of fairness in [<span style="font-variant:small-caps;">TCIM</span>]{}.
Utility of Time-Critical Influence
----------------------------------
Given the influence propagation model and the notion of time-critical aspect via a deadline $\tau$, we quantify the utility of time-critical influence for a given seed set $S$ on a set of target nodes $Y \subseteq \Vcal$ via the following: $$\label{eq:time_critical_utility}
f_{\tau} (S; Y, \Gcal) = \EE \Big[ \sum_{v \in Y, t_v \geq 0} \II(t_v \le \tau) \Big],$$ where the expectation is w.r.t. the activation probabilities associated to each edge under independant cascade model. The function is parametrized by deadline $\tau$, set $Y \subseteq \Vcal$ representing the set of nodes over which the utility is measured (by default, one can consider $Y = \Vcal$), and the underlying graph $\Gcal$ along with edge activation probabilities $p_{\Ecal}$. Given a fixed value of these parameters, the utility function $f_{\tau}: 2^\Vcal \rightarrow \mathbb{R}_{\geq 0}$ is a set function defined over the seed set $S \subseteq \Vcal$. Note that the constraint $t_v \geq 0$ represents the node was activated and the constraint $t_v \le \tau$ represents that the activation happened before the deadline $\tau$.
Socially Salient Groups and Their Utilities {#sec:notions:groups}
-------------------------------------------
The current approaches to [<span style="font-variant:small-caps;">TCIM</span>]{}consider all the nodes in $\Vcal$ to be homogeneous. We capture the presence of different socially salient groups in the population by dividing individuals into $k$ disjoint groups. Here, socially salient groups could be based on some sensitive attribute such as gender or race. We denote the set of nodes in each group $i \in \{1, 2, \ldots, k\}$ as $\Vcal_i \subseteq \Vcal$, and we have $\Vcal = \cup_{i} \Vcal_i$. For any given seed set $S$, we define the utilities for a group $i$ as $f_{\tau} (S; \Vcal_{i}, \Gcal)$ by setting target nodes $Y = \Vcal_{i}$ in Eq. \[eq:time\_critical\_utility\].
Disparity in Utility Across Groups {#sec:notions:example}
----------------------------------
In the standard formulations for [<span style="font-variant:small-caps;">TCIM</span>]{}problem, i.e., [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem and [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem, the utility $f_{\tau} (S; \Vcal, \Gcal)$ is optimized for the whole population $ \Vcal$ without considering their groups. Clearly, a solution to [<span style="font-variant:small-caps;">TCIM</span>]{}problem can, in general, lead to high disparity in utilities of different groups.
In particular, this disparity in utility across groups arises from several factors in which two groups differ from each other. One of the factors is that the groups are of different sizes, i.e., one group is a minority. The different group sizes could, in turn, lead to selecting seed nodes from the majority group when optimizing for utility $f_{\tau} (S; \Vcal, \Gcal)$ in standard [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem and [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem. Another factor is related to the connectivity and centrality of nodes from different groups. The solution to these standard optimization problems tend to favor nodes which are more central and have high-connectivity. Finally, given the above two factors, we note that the disparity in influence across groups can be further exacerbated for lower values of deadline $\tau$ in the time-critical influence maximization.
In Figure \[fig:example\_graph\], we provide an example to illustrate the disparity across groups in the standard approaches to [<span style="font-variant:small-caps;">TCIM</span>]{}. In particular, to show this disparity, we consider the [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem, and it is easy to extend this example to show disparity in [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem. The graph that we consider in this example (see Figure \[fig:example\_graph\] caption for details) has the two characteristic properties that we discussed above: (i) group $V_2$ is in minority with less than half of the size of group $V_1$, (ii) group $V_1$ has more central nodes compared to group $V_2$, and (iii) nodes in group $V_1$ have higher connectivity than nodes in group $V_2$. We consider the probability of influence in the graph to be $p_e = 0.7$ for all edges, and study the [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem for budget $B = 2$.
For different time critical deadlines $\tau$, we report the following normalized utilities: $\frac{f(S; \Vcal, \Gcal)}{|\Vcal|}$ for the whole population $\Vcal$, $\frac{f(S; \Vcal_1, \Gcal)}{|\Vcal_1|}$ for the group $\Vcal_1$, and $\frac{f(S; \Vcal_2, \Gcal)}{|\Vcal_2|}$ for the group $\Vcal_2$. Here, normalization captures the notion of “average" utility per node in a group, and automatically allows us to account for the differences in the group sizes. As can be seen in Figure \[fig:example\_graph\], the optimal solution to the problem consistently picks set $S = \{a, b\}$ comprising of the most central and high-connectivity nodes. While these nodes maximize the total utility, they lead to a high disparity in the normalized utilities across groups. As the influence becomes more time-critical, i.e., $\tau$ is reduced, we see an increasing disparity as discussed above. For $\tau=2$, the utility of group $\Vcal_2$ reduces to 0.
Notion of Fairness {#sec:notions:fairnessdef}
------------------
Next, in order to guide the design of fair solutions to [<span style="font-variant:small-caps;">TCIM</span>]{}problems, we introduce a formal notion of group fairness in [<span style="font-variant:small-caps;">TCIM</span>]{}. In particular, we measure the (un-)fairness or disparity of an algorithm by the maximum *disparity in normalized utilities* across all pairs of socially salient groups, given by: $$\label{eq:measure_fair}
\max_{i,j \in \{1, 2, \ldots, k\}} \bigg| \frac{f_\tau(S;\Vcal_{i},\Gcal)}{|\Vcal_{i}|} - \frac{f_\tau(S;\Vcal_{j},\Gcal)}{|\Vcal_{j}|} \bigg|.$$ As discussed above (see Section \[sec:notions:example\]), normalization w.r.t. group sizes captures the notion of average utility per node in a group and hence makes the measure agnostic to the group size. In the next section, we seek to design fair algorithms for [<span style="font-variant:small-caps;">TCIM</span>]{}problems that have low disparity (or more fairness) as measured by Eq. \[eq:measure\_fair\].
Solutions and Results {#solutions}
=====================
Due to lack of space, we only provide a sketch of our proposed solution. We solve the [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem and the [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem with additionally considering the fairness notion introduced in section \[sec:notions:fairnessdef\], which we call [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem and [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem, respectively. However, adding the fairness measure yields objectives which remain NP-Hard as the original problem, but, additionally, they are no longer submodular—a property used to approximately solve the traditional problem with guarantees using the greedy heuristic. We propose submodular proxies for both [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}and [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problems. In the case of [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}, we propose to maximize sum of concave functions whose arguments are group influences. The resultant function encourages selecting seeds which influence underrepresented groups. The penalty of disparity in the group influences would be dependent on the curvature of the concave function. For [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem, we propose that *every group should reach the required quota of influence* instead of just the entire population. This way every group is ensured a minimum reach. Since our proposed proxies are submodular and increasing functions of the seed set, we can approximate the solutions of our proposed proxy objectives using a greedy heuristic. We, additionally, provide bounds: (i) on the minimum expected influence achieved by our solution, in the case of [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem, and (ii) on the maximum length of the solution set size, in the case of [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem. We evaluate our proposed solutions using a real and a synthetic dataset and show that: (i) our methods are effective in enforcing the proposed fairness notion, and (ii) the cost of achieving this fairness is bounded as guaranteed by our theoretical results. We also study the effect of disparity of influence between groups: (i) by varying graph properties, such as connectivity and relative group sizes etc., and (ii) by varying [<span style="font-variant:small-caps;">TCIM</span>]{}algorithmic properties, such as seed budget, reach quota and time deadline etc. The proposed solutions, the theoretical guarantees and the corresponding empirical evaluations can be found in detail in the supplementary material.
Background on Time-Critical Influence Maximization ([<span style="font-variant:small-caps;">TCIM</span>]{}) {#sec:background}
===========================================================================================================
In this section, we provide the necessary background on the problem of time-critical influence maximization (henceforth, referred to as [<span style="font-variant:small-caps;">TCIM</span>]{}for brevity). First, we formally introduce a well-studied influence propagation model and specify the notion of time-critical influence that we consider in this paper. Then, we discuss two discrete optimization formulations to tackle the [<span style="font-variant:small-caps;">TCIM</span>]{}problem.
Influence Propagation in Social Network
---------------------------------------
Consider a directed graph $\Gcal = (\Vcal, \Ecal)$, where $\Vcal$ is the set of nodes and $\Ecal$ is the set of directed edges connecting these nodes. For instance, in a social network the nodes could represent people and edges could represent friendship links between people. An undirected link between two nodes can be represented by simply considering two directed edges between these nodes.
There are two classical influence propogation models that are studied in the literature [@kempe2003maximizing]: (i) Independent Cascade model ([<span style="font-variant:small-caps;">IC</span>]{}) and (ii) Linear Threshold ([<span style="font-variant:small-caps;">LT</span>]{}) model. In this paper, we will consider [<span style="font-variant:small-caps;">IC</span>]{}model and our results can easily be extended to the [<span style="font-variant:small-caps;">LT</span>]{}model.
In the [<span style="font-variant:small-caps;">IC</span>]{}model, there is a probability of influence associated with each edge denoted as $p_{\Ecal} := \{p_e \in [0, 1]: e \in \Ecal\}$. Given an initial seed set $S \subseteq \Vcal$, the influence propagation proceeds in discrete time steps $t= \{0, 1, 2, \ldots, \}$ as follows. At $t=0$, the initial seed set $S$ is “activated" (i.e., influenced). Then, at any time step $t > 0$, a node $v \in \Vcal$ which was activated at time $t-1$ gets a chance to influence its neighbors (i.e., set of nodes $\{w : (v, w) \in \Ecal\}$). The influence propagation process stops at time $t > 0$ if no new nodes get influenced at this time. Under the [<span style="font-variant:small-caps;">IC</span>]{}model, once a node is activated it stays active throughout the process and each node has only one chance to influence its neighbors.
-1 Note that the influence propagation under [<span style="font-variant:small-caps;">IC</span>]{}model is a stochastic process: the stochasticity here arises because of the random outcomes of a node $v$ influencing its neighbor $w$ based on the Bernoulli distribution $p_{(v, w)}$. An outcome of the influence propagation process can be denoted via a set of timestamps $\{t_v \geq 0 : v \in \Vcal\}$ where $t_v$ represents the time at which a node $v \in \Vcal$ was activated. We have $t_v = 0 \textnormal{ iff } v \in S$ and for convenience of notation, we define $t_v = -1$ to indicate that the node $v$ was not activated in the process.
Utility of Time-Critical Influence
----------------------------------
As motivated in the introduction, we focus on the application settings where the spread of influence is time-critical, i.e., it is more beneficial to be influenced earlier in the process. In particular, we adopt the well-studied notion of time-critical influence as proposed by [@chen2012time]. Their time-critical model is captured via a deadline $\tau$: If a node is activated before the deadline, it receives a utility of $1$, otherwise it receives no utility. This simple model captures the notion of timing in many important real-world applications such as viral marketing of an online product with limited availability, information propagation of job vacancy information, etc.
Given the influence propagation model and the notion of time-critical aspect via a deadline $\tau$, we quantify the utility of time-critical influence for a given seed set $S$ on a set of target nodes $Y \subseteq \Vcal$ via the following: $$\label{eq:time_critical_utility}
f_{\tau} (S; Y, \Gcal) = \EE \Big[ \sum_{v \in Y, t_v \geq 0} \II(t_v \le \tau) \Big],$$ where the expectation is w.r.t. the randomness of the outcomes of the [<span style="font-variant:small-caps;">IC</span>]{}model. The function is parametrized by deadline $\tau$, set $Y \subseteq \Vcal$ representing the set of nodes over which the utility is measured (by default, one can consider $Y = \Vcal$), and the underlying graph $\Gcal$ along with edge activation probabilities $p_{\Ecal}$. Given a fixed value of these parameters, the utility function $f_{\tau}: 2^\Vcal \rightarrow \mathbb{R}_{\geq 0}$ is a set function defined over the seed set $S \subseteq \Vcal$. Note that the constraint $t_v \geq 0$ represents the node was activated and the constraint $t_v \le \tau$ represents that the activation happened before the deadline $\tau$.
[<span style="font-variant:small-caps;">TCIM</span>]{}as Discrete Optimization Problem
--------------------------------------------------------------------------------------
Next, we present two settings under which we study [<span style="font-variant:small-caps;">TCIM</span>]{}by casting it as a discrete optimization problem.
### Maximization under Budget Constraint ([<span style="font-variant:small-caps;">TCIM-Budget</span>]{})
In the maximization problem under budget constraint, we are given a fixed budget $B > 0$ and the goal is to find an optimal set of seed nodes that maximize the expected utility. Formally, we state the problem as $$\begin{aligned}
\label{eq:budget_problem}
\max_{S \subseteq \Vcal}{f_{\tau}(S; \Vcal, \Gcal)}
\quad \text{ subject to } |S| \leq B. \tag{P1} \end{aligned}$$
### Minimization under Coverage Constraint ([<span style="font-variant:small-caps;">TCIM-Cover</span>]{})
In the minimization problem under coverage constraint, we are given a quota $Q \in [0, 1]$ representing the minimal fraction of nodes that must be activated or “covered" by the influence propagation in expectation. The goal is then to find an optimal set of seeds of minimal size that achieves the desired coverage constraint. We formally state the problem as $$\begin{aligned}
\label{eq:cover_problem}
\min_{S \subseteq \Vcal} |S|
\quad \text{ subject to } \frac{f_{\tau}(S; \Vcal, \Gcal)}{|\Vcal|} \geq Q. \tag{P2} \end{aligned}$$
Submodularity and Approximate Solutions {#sec:background:submodular}
---------------------------------------
Next, we present some key properties of the utility function $f_\tau(.)$ to get a better understanding of the above-mentioned optmization problems. In their seminal work, [@kempe2003maximizing] showed that the utility function without time-critical deadline, i.e., $f_\infty(.): S \rightarrow \mathbb{R}_{+}$, is a non-negative, monotone, submodular set function w.r.t. the optimization variable $S \subseteq \Vcal$. Submodularity is an intuitive notion of diminishing returns and optimization of submodular set functions finds numerous applications in machine learning and social networks, such as influence maximization [@kempe2003maximizing], sensing [@DBLP:conf/aaai/KrauseG07], information gathering [@DBLP:conf/ijcai/SinglaHKWK15], and active learning [@DBLP:conf/uai/GuilloryB11] (see [@krause2014submodular] for a survey on submodular function optimization and its applications).
@chen2012time showed that the utility function for the general time-critical setting for any $\tau$ also satisfies these properties. Submodularity is an intuitive notion of diminishing returns, stating that, for any sets $A \subseteq A' \subseteq \Vcal$, and any node $a \in \Vcal \setminus A'$, it holds that (omitting the parameters $\Vcal$ and $\Gcal$ for brevity): $$\begin{aligned}
f_\tau(A \cup \{a\}) - f_\tau(A) \geq f_\tau(A' \cup \{a\}) - f_\tau(A').\end{aligned}$$
The fact that the utility function is submodular in turn implies that these two problems \[eq:budget\_problem\] and \[eq:cover\_problem\] are NP-Hard and hence finding the optimization solution is intractable [@1978-_nemhauser_submodular-max; @1998-_feige_threshold-of-ln-n; @krause2014submodular]. However, on a positive note, one can exploit the submodularity property of the function to design efficient approximation algorithms with provable guarantees [@1978-_nemhauser_submodular-max; @krause2014submodular]. In particular, we can run the following greedy heuristic: start from an empty set, iteratively add a new node to the set that provides the maximal marginal gain in terms of utility, and stop the algorithm when the desired constraint on budget or coverage is met. This greedy algorithm provides the following guarantees for these two problems:.
- for the [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\], the greedy algorithm returns a set $\hat{S}$ that guarantees the following lower bound on the utility: $f_\tau(\hat{S}; \Vcal, \Gcal) \geq (1 - \frac{1}{e}) \cdot f_\tau(S^*; \Vcal, \Gcal)$ where $S^*$ is an optimal solution to problem \[eq:budget\_problem\].
- for the [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem \[eq:cover\_problem\], the greedy algorithm returns a set $\hat{S}$ that guarantees the following upper bound on the seed set size: $|\hat{S}| \leq \ln(1+ |\Vcal|) \cdot |S^*|$ where $S^*$ is an optimal solution to problem \[eq:cover\_problem\].
Achieving Fairness in [<span style="font-variant:small-caps;">TCIM</span>]{} {#sec:algorithms}
============================================================================
In this section, we seek to develop efficient algorithms for [<span style="font-variant:small-caps;">TCIM</span>]{}problems under fairness considerations that have low disparity measured by Eq. \[eq:measure\_fair\] while maintaining high performance.
Fair Maximization under Budget Constraint ([<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}) {#sec:algorithms:budget}
-------------------------------------------------------------------------------------------------------------
### Introducing fairness considerations in [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}
A fair [<span style="font-variant:small-caps;">TCIM</span>]{}algorithm under budget constraint should seek to achieve the following two objectives: (i) maximizing total influence for the whole population $\Vcal$ as was done in the standard [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\], and (ii) enforcing fairness by ensuring that disparity across different groups as per Eq. \[eq:measure\_fair\] is low. Clearly, enforcing fairness would lead to a reduction in total influence, and we seek to design algorithms that can achieve a good trade-off between these two objectives. We formulate the following fair variant of [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\] that captures this trade-off: $$\begin{aligned}
\label{eq:fair_budget_problem}
&\max_{S \subseteq \Vcal} \bigg({f_{\tau}(S; \Vcal, \Gcal)} - \gamma \cdot \max_{i,j} \Big| \frac{f_\tau(S;\Vcal_{i},\Gcal)}{|\Vcal_{i}|} - \frac{f_\tau(S;\Vcal_{j},\Gcal)}{|\Vcal_{j}|} \Big| \bigg) \tag{P3} \\
&\text{ subject to } |S| \leq B,\end{aligned}$$ where $\gamma \geq 0$ is a fixed regularization factor to trade-off the total influence and disparity of the solution.
### Surrogate problem for [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}with guarantees
We note that problem \[eq:fair\_budget\_problem\] is a challenging discrete optimization problem and the objective function does not have structural properties of submodularity as was the case for the standard [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\]. Instead of directly solving problem \[eq:fair\_budget\_problem\], we introduce a novel surrogate problem that would allow us to indirectly trade-off the two objectives of maximizing total influence and minimizing disparity across groups, as follows: $$\begin{aligned}
\label{eq:surrogate_fair_budget_problem}
\max_{S \subseteq \Vcal} \sum_{i=1}^k \lambda_i \Hcal(f_\tau(S;\Vcal_{i},\Gcal))
\quad \text{ subject to } |S| \leq B, \tag{P4}\end{aligned}$$ where $\Hcal$ is a non-negative, monotone concave function and $\lambda_i \ge 0$ are fixed scalars.
The key idea of using the surrogate objective function in problem \[eq:surrogate\_fair\_budget\_problem\] is the following: Passing the group influence functions through a monotone concave function $\Hcal$ rewards selecting seeds that would lead to higher influence on under-represented groups early in the selection process; this in turn helps in reducing disparity across groups. It is important to note that controlling the curvature of the concave function $\Hcal$ provides an indirect way to trade-off the total influence and the disparity of the solution. For instance, using $\Hcal(z) := \log(z)$ has higher curvature than using $\Hcal(z) := \sqrt z$ and hence leads to lower disparity (this is demonstrated in the experimental results in Figure \[fig:synth\_budget\_fairness\]). For our illustrative example from Section \[sec:notions\], we report the results for an optimal solution to [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem \[eq:surrogate\_fair\_budget\_problem\] with $\Hcal(z) := \log(z)$. As can be seen in Figure \[fig:example\_graph\], the solution leads to a drastic reduction in disparity across groups for different values of deadline $\tau$ compared to an optimal solution of the standard [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\].
While it is intuitively clear that using the concave function $\Hcal(z)$ in problem \[eq:surrogate\_fair\_budget\_problem\] reduces disparity, we also need to ensure that the solution to this problem has high influence for the whole population $\Vcal$ and that the solution can be computed efficiently. As proven in the theorem below, we can find an approximate solution to problem \[eq:surrogate\_fair\_budget\_problem\], with guarantees on the total influence, by running the greedy heuristic (as was introduced in Section \[sec:background:submodular\]).
\[theorem:fair\_budget\] Let $\hat{S}$ denote the output of the greedy algorithm for problem \[eq:surrogate\_fair\_budget\_problem\] with $\lambda_i = 1 \ \forall i \in [k]$. Let $S^*$ be an optimal solution to problem \[eq:budget\_problem\]. Then, the total influence of the greedy algorithm is guaranteed to have the following lower bound: $f_\tau(\hat{S}; \Vcal, \Gcal) \geq (1 - \frac{1}{e}) \cdot \Hcal\big(f_\tau(S^*; \Vcal, \Gcal)\big)$.
This is equivalent to the fact that the multiplicative approximation factor of the utility of [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}using greedy algorithm w.r.t. the utility of an optimal solution to [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}scales as $\big((1 - \frac{1}{e}) \cdot \frac{\Hcal(f_\tau(S^*; \Vcal, \Gcal))}{f_\tau(S^*; \Vcal, \Gcal)}\big)$. Note that as the curvature of the concave function $\Hcal$ increases, the approximation factor gets worse—this further highlights how the curvature of the function $\Hcal$ provides a way to trade-off the total influence and disparity of the solution. We provide sketches of proofs below.
There are two key steps in proving the above theorem. The first step is showing that the objective function in problem \[eq:surrogate\_fair\_budget\_problem\] is monotone submodular function, where we use a result from [@lin2011class] that composition of a non-decreasing concave and a non-decreasing submodular function is submodular. The second step is to bound the utility of the optimal solution of problem \[eq:surrogate\_fair\_budget\_problem\] w.r.t. the optimal solution of problem \[eq:budget\_problem\]. We prove that this can be bounded by a multiplicative factor of $\frac{\Hcal(f_\tau(S^*; \Vcal, \Gcal))}{f_\tau(S^*; \Vcal, \Gcal)}$. Combining bounds from individual steps and by applying inequalities resulting from the concavity of $\Hcal$ gives us the final result.
Fair Minimization under Coverage Constraint ([<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}) {#sec:algorithms:cover}
--------------------------------------------------------------------------------------------------------------
### Introducing fairness considerations in [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}
A fair [<span style="font-variant:small-caps;">TCIM</span>]{}algorithm under coverage constraint should seek to achieve the following two objectives: (i) minimizing the size of the seed set that achieves the desired coverage constraint as was done in the standard [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem \[eq:cover\_problem\], and (ii) enforcing fairness by ensuring that disparity across different groups as per Eq. \[eq:measure\_fair\] is low. As was the case for [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem above, enforcing fairness would lead to increasing the size of the required seed set, and we seek to design algorithms that can achieve a good trade-off between these two objectives. We formulate a fair variant of [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem \[eq:cover\_problem\] that captures this trade-off as follows:
$$\begin{aligned}
\label{eq:fair_cover_problem}
&\min_{S \subseteq \Vcal} \bigg(|S| + \gamma \cdot \max_{i,j} \Big| \frac{f_\tau(S;\Vcal_{i},\Gcal)}{|\Vcal_{i}|} - \frac{f_\tau(S;\Vcal_{j},\Gcal)}{|\Vcal_{j}|} \Big|\bigg) \tag{P5} \\
&\text{ subject to } \frac{f_{\tau}(S; \Vcal, \Gcal)}{|\Vcal|} \geq Q,\end{aligned}$$
where $\gamma \geq 0$ is a fixed regularization factor to trade-off the size of seed set and disparity of the solution.
\[fig:rice\_budget\]
### Surrogate problem for [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}with guarantees
As in Section \[sec:algorithms:budget\], we note that problem \[eq:fair\_cover\_problem\] is a challenging discrete optimization problem and does not have structural properties as was the case for the standard [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem \[eq:cover\_problem\]. Instead of directly solving problem \[eq:fair\_cover\_problem\], we introduce a novel surrogate problem that indirectly trade-offs the two objectives of minimizing the size of selected seed set and minimizing disparity, as follows: $$\begin{aligned}
\label{eq:surrogate_fair_cover_problem}
&\min_{S \subseteq \Vcal} |S|
\quad \text{ subject to } &\sum_{i=1}^k \min\bigg\{\frac{f_{\tau}(S; \Vcal_i, \Gcal)}{|\Vcal_i|}, Q\bigg\} \tag{P6} \geq k \cdot Q.\end{aligned}$$
The key idea of using the surrogate objective function in problem \[eq:surrogate\_fair\_cover\_problem\] is the following: the problem has a constraint that enforces that at least $Q$ fraction of nodes in each group are influenced by the selected seed set $S$; this in turn directly provides a bound on the disparity of any feasible solution to the problem as $(1 - Q)$.
While it is intuitively clear that the solution to problem \[eq:surrogate\_fair\_cover\_problem\] reduces disparity, we also would like to bound the size of the final seed set and that the solution can be computed efficiently. As proven in the theorem below, we can find an approximate solution to problem \[eq:surrogate\_fair\_cover\_problem\], with guarantees on the final seed set size, by running the greedy heuristic (as was introduced in Section \[sec:background:submodular\]).
\[theorem:fair\_cover\] Let us denote the output of the greedy algorithm for problem \[eq:surrogate\_fair\_cover\_problem\] by set $\hat{S}$. For group $i \in \{1, \ldots, k\}$, let $S^*_i$ denote an optimal solution to the coverage problem \[eq:cover\_problem\] for the target nodes set to $\Vcal_i$, i.e., solving problem \[eq:cover\_problem\] with constraint given by $\frac{f_{\tau}(S; \Vcal_i, \Gcal)}{|\Vcal_i|} \geq Q$. Then, the size of the seed set $\hat{S}$ returned by the greedy algorithm is guaranteed to have the following upper bound: $|\hat{S}| \leq \ln(1+ |\Vcal|) \Big(\sum_{i=1}^{k} |S^*_i|\Big)$.
There are two key steps in proving the above theorem. The first step is showing that the objective function in the constraint of problem \[eq:surrogate\_fair\_cover\_problem\] is monotone submodular function by using several composition properties of submodular functions [@krause2014submodular]. The second step is to provide an upper bound on the optimal solution of problem \[eq:surrogate\_fair\_cover\_problem\] as $\Big(\sum_{i=1}^{k} |S^*_i|\Big)$.
\[fig:synth\_budget\_1\]
\[fig:synth\_budget\_2\]
Evaluation on Synthetic Dataset {#sec:eval_synth}
===============================
In this section, we compare the solutions of different problems on a synthetic dataset. We show the effects of different properties of the synthetic graph and parameters of the algorithms.
Dataset and Experimental Setup {#sec:eval_synth_setup}
------------------------------
First, we discuss the synthetic dataset generation process and the setup used in our experiments.
[[[**Synthetic dataset.**]{}]{}]{} We consider an undirected graph with $500$ nodes, where each node belongs to either group $\Vcal_1$ or group $\Vcal_2$. The fraction of nodes belonging to each group is determined by a parameter $g$ (e.g., setting $g= 0.7$ results in $70\%$ of the nodes to be randomly assigned to group $\Vcal_1$). Nodes are connected based on two probabilities: (i) within-group edge probability ([*Homophily*]{}) $p_{hom}$ and (ii) across-group edge probability ([*Heterophily*]{}) $p_{het}$. Placing an edge between two nodes goes as follows: given a pair of nodes $(v,w)$, if they belong to the same group, we perform a Bernoulli trial with parameter $p_{hom}$; otherwise we use the parameter $p_{het}$. If the outcome of the trial is $1$, we place an undirected edge $e$ between these two nodes. Each edge has a probability of activation, $p_{e} \in [0,1]$, with which the nodes can activate each other.
[[[**Experimental Setup.**]{}]{}]{} In our experiments, we used $g = 0.7$ yielding $350$ nodes in $\Vcal_1$ and $150$ nodes in $\Vcal_2$. We set $p_{hom} = 0.025$ and $p_{het} = 0.001$, which yielded $3606$ total edges, out of which $2965$ edges were within group $\Vcal_1$, $514$ within $\Vcal_2$, and $127$ edges connecting nodes across two groups. We used a constant activation probability on all edges given by $p_e = 0.05$. Finally, we consider the time deadline $\tau = 20$, unless explicitly stated otherwise.
Evaluating utilities, as described in Eq. \[eq:time\_critical\_utility\], in closed form is intractable, so we used Monte Carlo sampling to estimate these utilities. We used $200$ samples for this estimation, which yielded a stable estimation of the utility function. In all the experiments, we pick a seed set by solving the corresponding problem. Then, we use this seed set to estimate the expected number of nodes influenced in the graph using [<span style="font-variant:small-caps;">TCIM</span>]{}. We report the following normalized utilities: $\frac{f(S; \Vcal, \Gcal)}{|\Vcal|}$ for the whole population $\Vcal$, $\frac{f(S; \Vcal_1, \Gcal)}{|\Vcal_1|}$ for the group $\Vcal_1$, and $\frac{f(S; \Vcal_2, \Gcal)}{|\Vcal_2|}$ for the group $\Vcal_2$.
\[fig:synth\_cover\]
TCIM under Budget Constraints {#sec:eval_synth_budget}
-----------------------------
Next, we compare the solutions of [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\] with our solution to [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem \[eq:surrogate\_fair\_budget\_problem\], obtained through the greedy algorithm, , by greedily picking $B$ seeds which maximize the objective functions. In all the figures discussed in this section, red color represents the results of [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\], and blue color represents the results of our solution to the [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem \[eq:surrogate\_fair\_budget\_problem\]. For the experiments in this section, we used a budget of $B = 30$ seeds.
[[[**Effect of different $\Hcal(z)$.**]{}]{}]{} Figure \[fig:synth\_budget\_fairness\] presents the comparison of three algorithms: one solving [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\], using the greedy heuristic; the other two solving [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem \[eq:surrogate\_fair\_budget\_problem\], where we use two realizations of the concave monotone function, $\Hcal(z)$, given by: (i) $\Hcal(z):= \log(z)$ and (ii) $\Hcal(z):= \sqrt{z}$. Figure \[fig:synth\_budget\_fairness\] shows the fraction of population influenced, both overall and for every group. We can observe that solving the traditional [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem leads to large disparity between the fraction of nodes influenced from each group: while $30\%$ of nodes in group $\Vcal_1$ are influenced, this fraction is only $2\%$ for group $\Vcal_2$.
On the other hand, our proposed solution to [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem results in lower disparity between the groups, ensuring similar fraction of influenced nodes. We can further see that $\sqrt{z}$ performs worse than $\log(z)$ in removing the disparity, however incurring lower loss in total influence, which is expected as it has lower curvature than $\log(z)$. One could consider higher powers of the root to increase the curvature or increase the weights $\lambda$ in problem \[eq:surrogate\_fair\_budget\_problem\] for the under-represented group. The *key point* to notice is that the reduction in the total influence is only marginal as guaranteed by Theorem \[theorem:fair\_budget\]. In the subsequent figures, we only show the results of $\Hcal(z):= \log(z)$ for the solution to problem \[eq:surrogate\_fair\_budget\_problem\]. [[[**Effect of seed budget.**]{}]{}]{} Figure \[fig:synth\_budget\_cost\] shows the effect of different seed budgets on the number of influenced nodes (from different groups). Dotted and dash-dotted lines correspond to groups $\Vcal_2$ and $\Vcal_1$ respectively, while solid lines represent the total influence. We can see that problem \[eq:surrogate\_fair\_budget\_problem\], with a small reduction in total influence, reduces a lot of disparity between the groups. We also observe that higher budget could lead to more disparity in an imbalanced graph.
[[[**Effect of deadline.**]{}]{}]{} Figure \[fig:synth\_budget\_deadline\] compares disparity as we vary the value of the deadline $\tau$, in problems \[eq:budget\_problem\] and \[eq:surrogate\_fair\_budget\_problem\]. Disparity is computed as the absolute difference between the fraction of individuals influenced in each group, given by Eq. \[eq:measure\_fair\]. The figure shows that the disparity depends on the value of $\tau$ and, in this case, our method is more effective for lower values of $\tau$.
[[[**Effect of activation probabilities.**]{}]{}]{} Figure \[fig:synth\_budget\_activation\] shows the disparity in influence for different activation probabilities $p_e\in\{0.01,0.05,0.1,$ $0.2, 0.3,0.5,0.7,1.0\}$. The results show that lower values of $\tau$ and lower activation probabilities could result in larger disparity and our method consistently performs better.
[[[**Effect of group sizes.**]{}]{}]{} Figure \[fig:synth\_budget\_groups\] shows the effect of group sizes $g \in \{0.55, 0.6, 0.7,0.8\}$. x-axis represents ratio of the nodes belonging to different groups and y-axis represents disparity. The figure confirms our hypothesis that *imbalance in a graph could lead to disparate influence*, as motivated in the illustrative example given in Figure \[fig:example\_graph\].
[[[**Effect of graph structure.**]{}]{}]{} Figure \[fig:synth\_budget\_clique\] demonstrates the importance of the graph structure $(p_{het},p_{hom})\in\{(0.025, 0.025),(0.015, 0.025),$ $(0.01,0.025), (0.001,0.025)\}$. x-axis shows the ratio of across and within group edge probabilities. The figure validates our hypothesis that the majority group containing more influential nodes fares better in [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem, as proposed in Figure \[fig:example\_graph\].
[[[**Takeaways.**]{}]{}]{} In this section we demonstrated that: (i) solving [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem can lead to disparity of influence in different groups; (ii) the amount of disparity depends on the time limit, activation probability, relative group sizes, budget, and connectivity of the graph; and (iii) instead, solving [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}results in lower disparity of influence, with marginal reduction in overall influence, as guaranteed by Theorem \[theorem:fair\_budget\].
TCIM under Coverage Constraints {#sec:eval_synth_cover}
-------------------------------
Next, we compare solutions of [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem \[eq:cover\_problem\], and our solution to [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem \[eq:surrogate\_fair\_cover\_problem\]. The goal is to reach the prescribed quota $Q$, while trying to minimize the number of seeds. In all the figures discussed in this section, red color represents the results of [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem \[eq:cover\_problem\], and blue color represents the results of our solution to [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem \[eq:surrogate\_fair\_cover\_problem\].
[[[**Effect of iterations.**]{}]{}]{} Figure \[fig:synth\_cover\_classic\_v\_fair\] shows how the fraction of population influenced changes with seed selection at each iteration. Solid lines represent total influence while dash-dotted lines and dotted lines represent groups $\Vcal_1$ and $\Vcal_2$, respectively. In this experiment, $Q$ was set to $0.2$ which is represented by the horizontal green line. The figure demonstrates that in [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}, the fraction of influenced individuals are more than $Q$ in both groups. On the other hand, in the [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}, the fraction of influenced individuals in group $\Vcal_1$ is larger than $Q$; however, the fraction of influenced individuals in group $\Vcal_2$ is much smaller than $Q$. The figure also shows that this fairness is achieved with only marginal increase in the solution set sizes, as guaranteed in Theorem \[theorem:fair\_cover\].
[[[**Effect of quota Q.**]{}]{}]{} Figure \[fig:synth\_cover\_fairness\] shows fractions of individuals that are influenced for different quota $Q$: (i) for the [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem, the disparity in the fraction of influenced individuals in different groups persists with increased quota; (ii) however, our proposed method results in both groups reaching the specified coverage $Q$. Figure \[fig:synth\_cover\_cost\] shows the coverage $Q$ on the x-axis and number of seeds used on the y-axis. The figure shows that our method results in only slightly larger seed sets compared to the [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem, as guaranteed by Theorem \[theorem:fair\_cover\].
[[[**Takeaways.**]{}]{}]{} We compared the results of [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem, \[eq:cover\_problem\], and our solution to [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem, \[eq:surrogate\_fair\_cover\_problem\]. The results demonstrate that: (i) both methods influence $Q$ fraction of the total population; (ii) however, the [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}approach can result in disparate coverage, i.e., the coverage in some groups could be much lower than the prescribed fraction $Q$; (iii) [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}, on the other hand, ensures coverage of at least $Q$ fraction in both groups; and (iv) additionally, [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}only yields slightly larger solution set sizes, as guaranteed by Theorem \[theorem:fair\_cover\].
\[fig:rice\_cover\]
Evaluation on Real Dataset {#sec:eval_real}
==========================
In this section, we evaluate our proposed solutions using a real-world dataset. We first describe the dataset and the details of the experiments, and then we present our findings.
Dataset and Experimental Setup {#sec:eval_real_setup}
------------------------------
Next, we describe the dataset we used to evaluate our proposed methods, followed by the experimental setup.
[[[**Rice-Facebook dataset.**]{}]{}]{} To evaluate our proposed methods, we used *Rice-Facebook* dataset collected by @mislove2010you, where they capture the connections between students at the Rice University. The resulting network consists of $1205$ nodes and $42443$ undirected edges. Each node has $3$ attributes: (i) the residential college id (a number between $[1-9]$), (ii) age (a number between $[18-22]$), and (iii) a major ID (which is in the range $[1-60]$).
We grouped the nodes (students) into two groups based on their age attributes. We experimented with all four groups while running our algorithms but present the results using only 2 groups which showed the *highest disparity*. We considered nodes with ages $18$ and $19$ as group $\Vcal_1$ and age $20$ as group $\Vcal_2$. Group $\Vcal_1$ has $97$ nodes and $513$ within-group edges. Whereas, group $\Vcal_2$ has $344$ nodes and $7441$ within-group edges. Overall, there are $3350$ across-group edges going between nodes in $\Vcal_1$ and $\Vcal_2$.
[[[**Experimental Setup.**]{}]{}]{} In all the experiments in this section, we used activation probability $p_e = 0.01$. All the other parameter were the same as described in Section \[sec:eval\_synth\_setup\].
TCIM under Budget Constraint {#sec:eval_real_budget}
----------------------------
In this section, we compare the results of [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\] and our solution to [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem \[eq:surrogate\_fair\_budget\_problem\]. Red color in all the figures discussed in this section corresponds to the solution of [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\] and the blue color corresponds to our solution of [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem \[eq:surrogate\_fair\_budget\_problem\]. In all the experiments in this section we used a seed budget $B = 30$.
[[[**Effect of different $\Hcal(z)$.**]{}]{}]{} In Figure \[fig:rice\_budget\_fairness\], we compare the results of [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem \[eq:budget\_problem\] and [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem \[eq:surrogate\_fair\_budget\_problem\] using two realizations of $\Hcal(z)$, given by: (i) $\Hcal(z):= \log(z)$ and (ii) $\Hcal(z):= \sqrt{z}$. The figure demonstrates that: (i) At a marginal reduction of total influence, as guaranteed by Theorem \[theorem:fair\_budget\], our proposed method reduces disparity in influence; (ii) as hypothesized in Section \[sec:algorithms:budget\], a higher curvature function, $\Hcal(z):= \log(z)$, leads to a bigger reduction in disparity compared to $\Hcal(z):= \sqrt{z}$. [[[**Effect of seed budget.**]{}]{}]{} Figure \[fig:rice\_budget\_cost\] demonstrates the effect of allowed seed budget on the group and total influences. Groups $\Vcal_1$ and $\Vcal_2$ are represented by dash-dotted lines and dotted lines respectively and solid lines correspond to total influence. The figure demonstrates that: (i) Disparity in the utility between both the groups increases with increase in allowed seed budget. A reason for these differences could be the imbalances in groups sizes and average degrees, between both the groups. If a very big seed budget is allowed the disparity in influence might also reduce, however in many applications due to limited resources it is not practical to allow a big budget; (ii) [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem results in a lower disparate utility between the two groups compared to [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem; (iii) this reduction in disparity is achieved at a very low cost to the total influence.
[[[**Effect of time deadline.**]{}]{}]{} Figure \[fig:rice\_budget\_deadline\] shows the effect of different time deadlines on the disparity between group influences, as calculated by Eq. \[eq:measure\_fair\]. It demonstrates that: (i) our proposed method, given by problem \[eq:surrogate\_fair\_budget\_problem\], yields solutions which result in lower disparity; (ii) in this case disparity in group utilities increases as the time deadline is increased. One explanation could be that the seed nodes are propagating influence in both the groups, but as we increase the time deadline Group $V_2$, with more nodes and edges, is more efficient at propagating influence compared to Group $V_1$, so it results in a larger disparity. One could imagine a case, as shown in the motivating example in Figure \[fig:example\_graph\], where seed nodes are surrounded by nodes of only one group, in this case a longer time deadline could yield a lower disparity.
[[[**Takeaways.**]{}]{}]{} We demonstrated that: (i) [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}, our proposed method, yields more fair solutions; (ii) this fairness is achieved at a very small reduction of the total influence compared to [<span style="font-variant:small-caps;">TCIM-Budget</span>]{}problem, as guaranteed by Theorem \[theorem:fair\_budget\].
TCIM under Coverage Constraint {#sec:eval_real_cover}
------------------------------
Next, we compare [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem \[eq:cover\_problem\] and our solution to [<span style="font-variant:small-caps;">FairTCIM-Budget</span>]{}problem \[eq:surrogate\_fair\_cover\_problem\]. Red color in all the figures discussed in this section corresponds to the solution of [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem \[eq:cover\_problem\] and the blue color corresponds to our solution of [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem \[eq:surrogate\_fair\_cover\_problem\].
[[[**Effect of iterations.**]{}]{}]{} In Figure \[fig:rice\_cover\_classic\_v\_fair\] we compare iterations of problem \[eq:cover\_problem\] and problem \[eq:surrogate\_fair\_cover\_problem\], realized with the log function. In each iteration, one seed is selected. Green line represents the required quota of coverage. Dashed-dotted lines, dotted lines and solid lines represent group $\Vcal_1$, group $\Vcal_2$ and total population, respectively. The figure demonstrates that: (i) both methods reach the required quota of the population; (ii) however, only the solution set of [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem \[eq:surrogate\_fair\_cover\_problem\] reaches the required quota in both the groups; (iii) while maintaining roughly similar utility for both the groups throughout the iterations; (iv) and it does so *at a small expense of additional seeds*.
[[[**Effect of quota.**]{}]{}]{} Figure \[fig:rice\_cover\_fairness\] demonstrates that: (i) for different values of the required quota, problem \[eq:cover\_problem\] results in disparate utility between both the groups; (ii) while seeds selected by solving problem \[eq:surrogate\_fair\_cover\_problem\] result in a more equal utility; (iii) [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem \[eq:surrogate\_fair\_cover\_problem\] uses only a small number of additional seeds, as shown by Figure \[fig:rice\_cover\_cost\].
[[[**Takeaways.**]{}]{}]{} We compared the result of [<span style="font-variant:small-caps;">TCIM-Cover</span>]{}problem \[eq:cover\_problem\] and our solution to [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem \[eq:surrogate\_fair\_cover\_problem\]. The results show that: (i) both methods reach the same fraction of the population; (ii) however, only [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}problem results in seed sets influencing the required quota in all the groups; and (iii) lastly, [<span style="font-variant:small-caps;">FairTCIM-Cover</span>]{}yields only slightly larger solution sets as guaranteed by Theorem \[theorem:fair\_cover\].
|
---
abstract: 'It is very challenging for speech enhancement methods to achieves robust performance under both high signal-to-noise ratio (SNR) and low SNR simultaneously. In this paper, we propose a method that integrates an SNR-based teachers-student technique and time-domain U-Net to deal with this problem. Specifically, this method consists of multiple teacher models and a student model. We first train the teacher models under multiple small-range SNRs that do not coincide with each other so that they can perform speech enhancement well within the specific SNR range. Then, we choose different teacher models to supervise the training of the student model according to the SNR of the training data. Eventually, the student model can perform speech enhancement under both high SNR and low SNR. To evaluate the proposed method, we constructed a dataset with an SNR ranging from -20dB to 20dB based on the public dataset. We experimentally analyzed the effectiveness of the SNR-based teachers-student technique and compared the proposed method with several state-of-the-art methods.'
address: |
Inner Mongolia Key Laboratory of Mongolian Information Processing Technology,\
College of Computer Science, Inner Mongolia Univeristy, Hohhot, China\
haoxiangsnr@gmail.com, cssxd@imu.edu.cn
bibliography:
- 'icme2020template.bib'
title: 'SNR-based teachers-student technique for speech enhancement'
---
Ł[[L]{}]{}
speech enhancement, knowledge distillation, teacher-student technique, low SNR
Introduction {#sec:intro}
============
Speech enhancement is to separate clean speech from noisy speech [@se_book]. It is an essential branch of speech signal processing and has been widely studied in the past few decades. It can be used in hearing aids, voice recorders, and smart speakers, as well as the front end of tasks such as speech recognition [@speech_se] and speaker recognition [@speaker_se]. In recent years, a large number of speech enhancement methods based on deep learning have been proposed [@se_overview] [@regression_approch] [@crn] [@wavenet] [@iconip_2019], showing stronger robustness than traditional signal-based methods. These methods can generally be divided into time-domain methods and frequency-domain methods. The time-domain methods [@unetgan] [@tcnn] use the neural network to directly map noisy speech waveform to clean speech waveform and usually do not require any preprocessing. The frequency-domain methods generally use short-term Fourier transform (STFT) to convert the noisy speech from the time domain to the frequency domain. They then use the neural network to map the magnitude spectrum of the noisy speech to some masking [@ibm_as_goal_wang_2005] or the magnitude spectrum of the clean speech [@regression_approch]. Compared with the chaotic time-domain sampling points, the magnitude spectrum contains more geometric information, which makes it easier to calculate losses and analyze frequency components. As the SNR of noisy speech decreases, the correct phase becomes more and more important for speech intelligibility and quality [@important_phase]. However, since the mapping of the phase spectrum is complicated (no obvious geometric structure), the speech enhancement methods in the time domain are also widely used.
With the increasing demand for speech-related services in recent years, the scenarios that need to be addressed for speech enhancement are becoming more and more challenging. A noticeable trend is that the range of SNR of the noisy speech is significantly expanded. Typical scenarios include stations, factories, subways, and shopping malls. The increased range of SNR means that speech enhancement needs to have the ability to enhance a wide range of background noises of different intensities. When the background noise is low, the main goal of the speech enhancement is to improve the hearing sense of the speech. When the background noise is high (in extreme cases (below -10dB), noisy speech is even hard to be heard), the speech enhancement needs to enhance the speech that is difficult to be heard to a clearer speech.
To better perform speech enhancement in the above scenarios, this paper proposes a speech enhancement method based on knowledge distillation [@hinton2015distilling] and time-domain U-Net. Our motivation is as follows. Speech enhancement can be viewed as a particular task of speech separation. In speech separation, Wave-U-Net [@wave_u_net], a model in the time domain, has achieved state-of-the-art performance. It can perform feature mapping directly in the time domain to avoid processing the phase. Inspired by it, we built a powerful time-domain model suitable for speech enhancement.
To enable the speech enhancement model to handle both high SNR and low SNR, there are usually two solutions. The first solution is providing a large amount of training data at each SNR and training a large-scale neural network. The second solution is to integrate multiple different models for a specific SNR range to process the noisy speech in parallel or serially. However, the disadvantage of the former is that large-scale neural networks will consume a lot of computing resources, memories, and times. The problem of the latter is that it needs to train multiple models, which is very troublesome to deploy and severely limits the application. To deal with the above problems, we introduce the knowledge distillation, which has been widely used in image recognition [@object_detection_knowledge_distillation] and speech recognition [@danpovey_distilling]. Knowledge distillation can extract knowledge from a large teacher model and improve the performance of a small student model. It is also called the teacher-student technique. In this paper, we extend the traditional teacher-student technique and propose an SNR-based teachers-student technique. We first build multiple teacher speech enhancement models and train them independently with the datasets of small SNR range. Then we build a student model. To make the student model have the ability to handle both high SNR and low SNR, we will use different teacher models to guide its training according to the SNR of the training data.
To evaluate the proposed method, we construct a challenging speech enhancement dataset that covers a wide range of SNR (-20dB to 20dB). We experimentally analyzed the effectiveness of the SNR-based teachers-student technique and compared the proposed method with several state-of-the-art (SOTA) methods.
Method
------
SNR-based teachers-student technique
------------------------------------
![An illustration of the standard knowledge distillation.[]{data-label="fig:teacher_and_student"}](figures/teacher_and_student.png){width="\linewidth"}
![An illustration of the SNR-based teachers-student technique.[]{data-label="fig:teachers_and_student"}](figures/teachers.png){width="\linewidth"}
We show the standard knowledge distillation in Figure \[fig:teacher\_and\_student\]. The idea behind knowledge distillation is that the soft probability output $p'$ of a pre-trained teacher model contains more information than the correct class label $p''$. If the teacher model gives higher probabilities for specific categories, then this indicates that the categories of the input image should be near these categories. Knowledge distillation forces the student model to minimize the difference between its probability output $p$ and the teacher’s probability output $p'$ to extract the additional knowledge that the teacher model obtained when calculating the correct probability. We usually use knowledge distillation to distill the information learned by large networks and pass it to networks with small parameters and weak learning capabilities. In this paper, we extend the standard knowledge distillation. The workflow are shown in Figure \[fig:teachers\_and\_student\]. First, we train multiple teacher models who are proficient in speech enhancement at a specific small SNR range. Then we use the teacher models to guide the training of the student model so that the student model can perform speech enhancement under both high SNR and low SNR. Below we describe the entire process.
**Train the teacher models.** We use $ f_T = \{ f_{t_1}, f_{t_2}, ... \}$ to represent the set of teacher models. We use $f_{t_i}$ represents the $i$-th teacher model. The dataset of each teacher model covers only a small range of SNRs, and the range of SNRs covered by each teacher’s dataset does not overlap each other. After sufficient training, we get multiple teacher models that are good at dealing with the SNR covered by their dataset. We fixed the weights of the teacher models.
**Forward calculation.** We use $x$ to represent a noisy speech, and $y$ to represent the clean speech corresponding to the $x$. We input the $x$ to the SNR-based teacher models. According to the SNR of $x$, the training algorithm will select a teacher model $f_{t_i}$ to obtain the enhanced speech $f_{t_i} (x)$. We also input $x$ to the student model $f_s$ to get the enhanced speech $f_s (x)$.
**Distill knowledge from the teacher model**. We use a $L_2$ loss function to minimize the difference between the output of the student model and the teacher model. This loss can extract additional knowledge from the teacher model during the student training process, and strengthen the student model’s ability to enhance speech at a specific SNR. We still need to preserve the loss between the enhanced speech of the student model $f_s (x)$ and the clean speech $y$. The overall loss $L$ is as follows: $$\begin{aligned}
L = \alpha \frac{1}{2} {|| f_s (x) - f_{t_i} (x) ||}^2 + (1 - \alpha) \frac{1}{2} {|| f_s (x) - y||}^2
\label{eq:loss}\end{aligned}$$ where $\alpha$ is the weight of the knowledge obtained from the teachers, which is determined based on the validation dataset.
{width="\textwidth"}
Model structure
---------------
As the SNR of noisy speech decreases, the correct phase becomes more and more critical for speech enhancement. However, since the phase spectrum mapping is complicated, we conduct speech enhancement in the time domain. Inspired by Wave-U-Net [@wave_u_net], we designed a time-domain speech enhancement model based on one-dimensional (1D) convolutional neural networks. The teacher models and student model both take such the network structure. The input of the model is a fixed-length noisy speech signal, and the output is an enhanced speech signal. The model contains three parts: Encoder, Bottleneck, and Decoder, and its structure is shown in Figure \[fig:basic\_network\]. Below we describe it in detail.
The Encoder part consists of 12 consecutive encoder blocks. The first seven encoder blocks use the same structure: “1D convolutional layer + batch normalization + leaky ReLU + downsampling layer". 1D convolutional layers can be used for feature extraction and integration. The parameters of the convolutional layers are listed above each encoder block. These seven encoder blocks all contain a downsampling layer, which can select one sampling point from two adjacent sampling points as output. After their processing, the size of the input signal in the time dimension will gradually become smaller. The structure of the remaining five encoder blocks in the Encoder is similar to that of the previous seven encoder blocks, but they do not contain a downsampling layer. For the time-domain model, since the first convolution layer of the model plays a critical role [@sincnet], we set the number of convolution kernels in the first convolution layer to a larger value (48), so that the model can better extract the features of the speech waveform. The number of convolution kernels in the remaining convolutional layers increases with the step size of 24 as the number of layers of the network increases. The structure of the Bottleneck section is the same as that of the Encoder.
The Decoder part contains 12 consecutive decoder blocks. The beginning of each decoder block contains the concatenation operation, which is used to concatenate the underlying features passed through the skip-connection. In order to maintain symmetry, the last seven decoder blocks in the decoder part all include the upsampling layer. The upsampling layer doubles the size of the feature map in the time dimension by linear interpolation, and finally restores the size corresponding to the input speech waveform. The remaining structure of the decoder blocks are similar to the encoder blocks.
\[tab:model\_performance\]
Experiment
==========
Dataset
-------
We use public datasets to evaluate the proposed method.
**For SNR-based teacher models:** In this paper, we set up four teacher models. In order to train the teacher models for different SNR ranges, we randomly select 950 clean speeches from the TIMIT [@timit] training dataset, and mixed speech shaped noise, babble, destroy engine, destroyer ops and factory floor 1 (from NoiseX-92 dataset [@noisex92]) under different SNRs to generate the noisy speech dataset. The SNRs of the four teacher models are {-20, -17, -13, -11}dB, {-10, -7, -3, 1}dB, {0, 3, 7, 9}dB, and {10, 13, 17, 20}dB. In the end, we generated 19,000 noisy speeches for each teacher model, of which 18,000 speeches were used as the training dataset and the rest were used as the validation dataset.
**For student model:** We randomly selected 950 speeches from the TIMIT training dataset and mixed them with the five noises that appeared in the training dataset of the teacher models at {-20, -10, 0, 10, 20}dB. This produces 23,750 noisy speeches. Among them, 22,000 were used as the training dataset, and 1,750 were used the validation dataset. To evaluate the student model, we randomly selected 100 speeches from the TIMIT test dataset, mixing with five noises that appeared in the above two datasets and four noises from the CHiME-4 [@chime4] dataset (pedestrian area noise, on the bus noise, cafe noise, and street noise) at {-20, -15, -10, -5, 0, 5, 10, 15, 20}dB. The resulting 8,100 noisy speeches act as the test dataset. It is worth mentioning that the test dataset of the student model contains noises, SNRs, and speakers that have not appeared in the training dataset, which makes speech enhancement very challenging.
We use PESQ [@pesq] and STOI [@stoi] to measure speech quality and intelligibility, respectively.
Implementation and training details
-----------------------------------
The sampling rate of all speeches is 16,000Hz. The input length of all models is fixed. Before the beginning of each training epoch, we will select consecutive 16,384 sampling points (1.024 seconds) from the random position of noisy speech as the input of the model. Except for the learning rate, other experimental parameters are the same for all models. We use Adam optimizer [@adam] (decay rates $\beta_{1} = 0.9$, $\beta_{2} = 0.999$) and set the batch size to 16. we set $\alpha$ of leaky ReLU to 0.1. We set the learning rate of the teacher models to a small constant of 0.0002. We set the initial learning rate of the student model to 0.002, which is reduced twice by every 300 epochs until the validation loss does not decrease. According to the validation dataset, we set $\alpha$ in Equation \[eq:loss\] to $0.5$.
Baseline methods
----------------
We compared with the three methods, which are Wavenet-denoising [@wavenet], PSM-BiLSTM [@phase_sensitive_bilstm], and CRN [@crn]. The first one is to mapping features on the time domain, and the last two are mapping features on the frequency domain.
- Wavenet-denoising: It is an end-to-end method for speech denoising based on wavenet [@wavenet_sn]. It retains wavenet’s powerful acoustic modeling capabilities, while significantly reducing time complexity by eliminating its autoregressive nature. We used its official implementation.
- PSM-BiLSTM: It is a bidirectional LSTM network for speech enhancement, using a phase-sensitive spectrum approximation (PSA) cost function. We implemented it and used the same hyperparameters as that in the paper.
- CRN: It contains a convolutional encoder-decoder and a LSTM bottleneck. We implemented it and used the same hyperparameters as that in the original paper.
Result
======
Performance of the proposed method
----------------------------------
Table \[tab:model\_performance\] presents the noisy speech and the enhanced speech from -20dB to 20dB in terms of PESQ and STOI. The “Noisy” lines in the table represent the noisy speech, and the “Enhanced” lines represent the enhanced speech using the proposed method. The “Seen” columns mean the SNRs exist in the training dataset of the student model while the “Unseen” columns indicate the SNRs that do not exist in the training dataset of the student model.
When comparing the noisy and enhanced speech, we can found that the PESQ and STOI are improved after speech enhancement with the proposed method even some noises do not appear in the training dataset (the last four noises). It is also clear that our method improves the PESQ and STOI for both “Seen” SNR and “Unseen” SNRs. The SNRs lower than -5dB are generally considered as extremely low SNRs, and the speech enhancement under such conditions is very challenging. However, our method works well, even if the range of the SNR of the dataset is so wide (-20dB to 20dB). The average improvement of the PESQ and STOI is 38.71% and 12.73%, respectively. This improvement indicates that the proposed method is effective.
Effectiveness of SNR-based teachers
-----------------------------------
To demonstrate the effectiveness of the SNR-based teacher models, Table \[tab:compare\_student\] lists the performance of the four SNR-based teacher models (T1, T2, T3, T4), the student model trained without teacher supervision (S1), and the proposed method (S2). As shown in Table \[tab:compare\_student\], we only test the teacher models on the trained small range of SNRs. For clarity, the PESQ and STOI of the noisy speech are also listed in the table.
Comparing with the noisy speech, there are notable improvements on the PESQ and STOI for the teacher models on their corresponding SNRs. This suggests that the teacher models can perform well on a small range of SNRs. The S1 and the S2 use the same training dataset and network structure. Compared to all the above models, the S1 performs the worst no matter which SNR. We guess that this is because the SNR range of the dataset is too large. With the learning ability of the S1, it is very difficult to consider all the SNRs from low to high.
The PESQ and STOI of the proposed method S2 approximate those of the teacher models at the corresponding SNR, and are far better than the S1 at all SNRs. These results indicate that the proposed SNR-based teachers-student technique can help the student model improve its ability to handle high SNR and low SNR at the same time. We calculated the PESQ and STOI on the validation dataset for every ten epochs during the training process of the S1 and S2. The growth curves are shown in Figure \[fig:validation\_metrics\]. The horizontal axis indicates the epoch of training, and the vertical axis is the average metric. We can clearly notice that from about the 200th epoch, the PESQ and STOI of S2 exceed the PESQ and STOI of S1. In the subsequent training process, the performance gap between the two models is still increasing.
\[tab:compare\_student\]
![The effectiveness of the SNR-based teachers-student technique.[]{data-label="fig:validation_metrics"}](./figures/metric_PESQ.png){width="1\linewidth"}
![The effectiveness of the SNR-based teachers-student technique.[]{data-label="fig:validation_metrics"}](./figures/metric_STOI.png){width="1\linewidth"}
Comparison of our method with the baselines
-------------------------------------------
Table \[tab:compare\_with\] reports the performances of the proposed method and the baseline methods on the test dataset (-20dB to 20dB). We can easily notice that the proposed method (S2) achieves the best performance, whether it is PESQ or STOI. Compared to the most robust method CRN in the baseline methods, we still improved 0.143 on PESQ and 0.021 on STOI. In the scenario that contains both high SNR and low SNR noisy speeches, our proposed method based on SNR-based teachers-student technique and time-domain U-Net is undoubtedly more advantageous.
Conclusion
==========
Speech enhancement for both low SNR and high SNR is a very challenging task. This paper proposes a method that integrates an SNR-based teachers-student technique and time-domain U-Net to deal with this problem. The student model is trained with the supervision of multiple teacher models. Each teacher model is well trained in an SNR-based way, which means they are only responsible for speech enhancement of a small SNR range. The experimental result shows that our method achieves state-of-the-art performance and suggests the effectiveness of SNR-based knowledge distillation in speech enhancement. The result also proves that our method is robust on “Seen" and “Unseen" noise and SNRs. To our best knowledge, this is the first time that knowledge distillation is investigated in speech enhancement.
ACKNOWLEDGMENTS
===============
This work was funded by National Natural Science Foundation of China (Grant No.61762069, 61773224), Natural Science Foundation of Inner Mongolia Autonomous Region (Grant No. 2017BS0601), and Inner Mongolia University Research and Innovation Project (Grant No. 10000-15010109).
|
---
abstract: 'A linear ordering is called context-free if it is the lexicographic ordering of some context-free language and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal, called its rank. It is known that the isomorphism problem of scattered context-free orderings is undecidable, if one of them has a rank at least two. In this paper we show that it is decidable whether a context-free ordering has rank at most one, and if so, its order type is effectively computable.'
author:
- 'Kitti Gelle, Szabolcs Iván'
bibliography:
- 'biblio.bib'
title: 'The order type of scattered context-free orderings of rank one is computable'
---
Introduction
============
If an alphabet $\Sigma$ is equipped by a linear order $<$, this order can be extended to the lexicographic ordering $<_\ell$ on $\Sigma^*$ as $u<_\ell v$ if and only if either $u$ is a proper prefix of $v$ or $u=xay$ and $v=xbz$ for some $x,y,z\in\Sigma^*$ and letters $a<b$. So any language $L\subseteq \Sigma^*$ can be viewed as a linear ordering $(L,<_\ell)$. Since $\{a,b\}^*$ contains the dense ordering $(aa+bb)^*ab$ and every countable linear ordering can be embedded into any countably infinite dense ordering, every countable linear ordering is isomorphic to one of the form $(L,<_\ell)$ for some language $L\subseteq\{a,b\}^*$. A linear ordering (or an order type) is called *regular* or *context-free* if it is isomorphic to the linear ordering (or, is the order type) of some language of the appropriate type. It is known [@DBLP:journals/fuin/BloomE10] that an ordinal is regular if and only if it is less than $\omega^\omega$ and is context-free if and only if it is less than $\omega^{\omega^\omega}$. Also, the Hausdorff rank [@rosenstein] of any scattered regular (context-free, resp.) ordering is less than $\omega$ ($\omega^\omega$, resp) [@ITA_1980__14_2_131_0; @10.1007/978-3-642-29344-3_25].
It is known [@GelleIvanTCS] that the order type of a well-ordered language generated by a prefix grammar (i.e. in which each nonterminal generates a prefix-free language) is computable, thus the isomorphism problem of context-free ordinals is decidable if the ordinals in question are given as the lexicograpic ordering of *prefix* grammars. Also, the isomorphism problem of regular orderings is decidable as well [@DBLP:journals/ita/Thomas86; @BLOOM200555]. At the other hand, it is undecidable for a context-free grammar whether it generates a dense language, hence the isomorphism problem of context-free orderings in general is undecidable [@ESIK2011107].
Algorithms that work for the well-ordered case can in many cases be “tweaked” somehow to make them work for the scattered case as well: e.g. it is decidable whether $(L,<_\ell)$ is well-ordered or scattered [@10.1007/978-3-642-22321-1_19] and the two algorithms are quite similar. In an earlier paper [@GelleIvanGandalf] we showed that it is undecidable for a scattered context-free ordering of rank $2$ whether its order type is $\omega+(\omega+\zeta)\times\omega$, even if it is given by a prefix grammar – so the complexity of the isomorphism problem is quite different when one makes the step from well-ordered languages to scattered ones.
In the current paper we complement this result by showing that if the rank of a scattered context-free ordering is at most one (we also show that this property is decidable as well), then its order type is effectively computable (as a finite sum of the order types $1$, $\omega$ and $-\omega$).
Notation
========
A *linear ordering* is a pair $(Q,<)$, where $Q$ is some set and the $<$ is a transitive, antisymmetric and connex (that is, for each $x,y\in Q$ exactly one of $x<y$, $y<x$ or $x=y$ holds) binary relation on $Q$. The pair $(Q,<)$ is also written simply $Q$ if the ordering is clear from the context. A (necessarily injective) function $h: Q_1\to Q_2$, where $(Q_1,<_1)$ and $(Q_2,<_2)$ are some linear orderings, is called an *(order) embedding* if for each $x,y\in Q_1$, $x<_1 y$ implies $h(x) <_2 h(y)$. If $Q_1$ can be embedded into $Q_2$, then this is denoted by $Q_1\preceq Q_2$. If $h$ is also surjective, $h$ is an *isomorphism*, in which case the two orderings are *isomorphic*. An isomorphism class is called an *order type*. The order type of the linear ordering $Q$ is denoted by $o(Q)$.
For example, the class of all linear orderings contain all the finite linear orderings and the orderings of the integers ($\mathbb{Z}$), the positive integers ($\mathbb{N}$) and the negative integers ($\mathbb{N}_{-}$) whose order type is denoted $\zeta$, $\omega$ and $-\omega$ respectively. Order types of the finite sets are denoted by their cardinality, and $[n]$ denotes $\{1,\ldots,n\}$ for each $n\geq 0$, ordered in the standard way.
The ordered sum $\sum_{x\in Q} Q_x$, where $Q$ is some linear ordering and for each $x \in Q$, $Q_x$ is a linear ordering, is defined as the ordering with domain $\{(x,q):x\in Q,q\in Q_x\}$ and ordering relation $(x,q)<(y,p)$ if and only if either $x<y$, or $x=y$ and $q<p$ in the respective $Q_x$. If each $Q_x$ has the same order type $o_1$ and $Q$ has order type $o_2$, then the above sum has order type $o_1\times o_2$. If $Q=[2]$, then the sum is usally written as $Q_1+Q_2$.
If $(Q,<)$ is a linear ordering and $Q'\subseteq Q$, we also write $(Q',<)$ for the subordering of $(Q,<)$, that is, to ease notation we also use $<$ for the restriction of $<$ to $Q'$.
A linear ordering $(Q,<)$ is called *dense* if it has at least two elements and for each $x,y\in Q$ where $x<y$ there exists a $z\in Q$ such that $x<z<y$. A linear ordering is *scattered* if no dense ordering can be embedded into it. It is well-known that every scattered sum of scattered linear orderings is scattered, and any finite union of scattered linear orderings is scattered. A linear ordering is called a *well-ordering* if it has no subordering of type $-\omega$. Clearly, any well-ordering is scattered. Since isomorphism preserves well-orderedness or scatteredness, we can call an order type well-ordered or scattered as well, or say that an order type embeds into another. We also write $o_1\preceq o_2$ to denote $o_1$ embeds into $o_2$. The well-ordered order types are called *ordinals*. For any set $\Omega$ of ordinals, $(\Omega,<)$ is well-ordered by the relation $o_1\prec o_2~\Leftrightarrow~\hbox{``}o_1\hbox{ can be embedded injectively into }o_2$ but not vice versa”. Amongst ordinals, it is common to use the notation $o_1<o_2$ instead of $o_1\prec o_2$. The principle of well-founded induction can be formulated as follows. Assume $P$ is a property of ordinals such that for any ordinal $o$, if $P$ holds for all ordinals smaller than $o$, then $P$ holds for $o$. Then $P$ holds for all the ordinals.
For standard notions and useful facts about linear orderings see e.g. [@rosenstein] or [@jalex].
Hausdorff classified the countable scattered linear orderings with respect to their rank. We will use the definition of the Hausdorff rank from [@10.1007/978-3-642-29344-3_25], which slightly differs from the original one (in which $H_0$ contains only the empty ordering and the singletons, and the classes $H_\alpha$ are not required to be closed under finite sum, see e.g. [@rosenstein]). For each countable ordinal $\alpha$, we define the class $H_\alpha$ of countable linear orderings as follows. $H_0$ consists of all finite linear orderings, and when $\alpha> 0$ is a countable ordinal, then $H_\alpha$ is the least class of linear orderings closed under finite ordered sum and isomorphism which contains all linear orderings of the form $\sum_{i\in\mathbb{Z}} Q_i$, where each $Q_i$ is in $H_{\beta_i}$ for some $\beta_i < \alpha$.
By Hausdorff’s theorem, a countable linear order $Q$ is scattered if and only if it belongs to $H_\alpha$ for some countable ordinal $\alpha$. The *rank* $r(Q)$ of a countable scattered linear ordering is the least ordinal $\alpha$ with $Q \in H_\alpha$.
As an example, $\omega$, $\zeta$, $-\omega$ and $\omega+\zeta$ or any finite sum of the form $\mathop\sum\limits_{i\in[n]}o_i$ with $o_i\in\{\omega, -\omega,1\}$ for each $i\in[n]$ each have rank $1$ while $(\omega+\zeta)\times\omega$ has rank $2$.
Let $\Sigma$ be an alphabet (a finite nonempty set) and let $\Sigma^*$ ($\Sigma^+$, resp) stand for the set of all (all nonempty, resp) finite words over $\Sigma$, $\varepsilon$ for the empty word, $|u|$ for the length of the word $u$, $u\cdot v$ or simply $uv$ for the concatenation of $u$ and $v$. A *language* is an arbitrary subset $L$ of $\Sigma^*$. We assume that each alphabet is equipped by some (total) linear order. Two (strict) partial orderings, the strict ordering $<_s$ and the prefix ordering $<_p$ are defined over $\Sigma^*$ as follows:
- $u <_s v$ if and only if $u = u_1au_2$ and $v = u_1bv_2$ for some words $u_1, u_2, v_2\in\Sigma^*$ and letters $a < b$,
- $u <_p v$ if and only if $v = uw$ for some nonempty word $w\in\Sigma^*$.
The union of these partial orderings is the lexicographical ordering $<_\ell = <_s \cup <_p$. We call the language $L$ well-ordered or scattered, if $(L,<_\ell)$ has the appropriate property and we define the rank $r(L)$ of a scattered language $L$ as $r(L,<_\ell)$. The order type $o(L)$ of a language $L$ is the order type of $(L,<_\ell)$. For example, if $a<b$, then $o\Bigl(\{a^kb:k\geq 0\}\Bigr)=-\omega$ and $o\Bigl(\{(bb)^ka:k\geq 0\}\Bigr)=\omega$.
When $\varrho$ is a relation over words (like $<_\ell$ or $<_s$), we write $K\varrho L$ if $u\varrho v$ for each word $u\in K$ and $v\in L$.
An *$\omega$-word* over $\Sigma$ is an $\omega$-sequence $a_1a_2\ldots$ of letters $a_i\in\Sigma$. The set of all $\omega$-words over $\Sigma$ is denoted $\Sigma^\omega$. The orderings $<_\ell$ and $<_p$ are extended to $\omega$-words. An $\omega$-word $w$ is called *regular* if $w=uv^\omega=uvvvv\ldots$ for some finite words $u\in\Sigma^*$ and $v\in\Sigma^+$. When $w$ is a (finite or $\omega$-) word over $\Sigma$ and $L\subseteq\Sigma^*$ is a language, then $L_{<w}$ stands for the language $\{u\in L:u<w\}$. Notions like $L_{\geq w}$, $L_{<_sw}$ are also used as well, with the analogous semantics.
A *context-free grammar* is a tuple $G=(N,\Sigma,P,S)$, where $N$ is the alphabet of the *nonterminal symbols*, $\Sigma$ is the alphabet of *terminal symbols* (or *letters*) which is disjoint from $N$, $S\in N$ is the *start symbol* and $P$ is a finite set of *productions* of the form $A\to\alpha$, where $A\in N$ and $\alpha$ is a *sentential form*, that is, $\alpha = X_1X_2\ldots X_k$ for some $k\geq 0$ and $X_1,\ldots,X_k\in N\cup \Sigma$. The derivation relations $\Rightarrow$, $\Rightarrow_\ell$, $\Rightarrow^*$ and $\Rightarrow_\ell^*$ are defined as usual (where the subscript $\ell$ stands for “leftmost”). The *language generated by* a grammar $G$ is defined as $L(G) = \{u\in\Sigma^* ~|~ S\Rightarrow^*u\}$. Languages generated by some context-free grammar are called *context-free languages*. For any set $\Delta$ of sentential forms, the language generated by $\Delta$ is $L(\Delta) = \{u \in \Sigma^* ~|~ \alpha\Rightarrow^* u\hbox{ for some } \alpha\in\Delta\}$. As a shorthand, we define $o(\Delta)$ as $o(L(\Delta))$. When $X,Y\in N\cup\Sigma$ are symbols of a grammar $G$, we write $Y\preceq X$ if $X\Rightarrow^*uYv$ for some words $u$ and $v$; $X\approx Y$ if $X\preceq Y$ and $Y\preceq X$ both hold; and $Y\prec X$ if $Y\preceq X$ but not $X\preceq Y$. A production of the form $X\to X_1\ldots X_n$ with $X_i\prec X$ for each $i\in[n]$ is called an *escaping production*.
A *regular language* over $\Sigma$ is one which can be built up from the singleton languages $\{a\}$, $a\in\Sigma$ and the empty language $\emptyset$ with finitely many applications of taking (finite) union, concatenation $KL=\{uv:u\in K,v\in L\}$ and iteration $K^*=\{u_1\ldots u_n:n\geq 0,u_i\in K\}$. For standard notions on regular and context-free languages the reader is referred to any standard textbook, such as [@Hopcroft+Ullman/79/Introduction].
Linear orderings which are isomorphic to the lexicographic ordering of some context-free (regular, resp.) language are called *context-free (regular, resp.) orderings*.
Limits of languages
===================
In this section we introduce the notion of a limit of a language and establish a connection: the main contribution of this concept is that one can decide whether a context-free language has a finite number of limits and if so, one can effectively compute the limits themselves (Lemma \[lem-limx-finite\]), and that a language has a finite number of limits if and only if its order type is scattered of rank at most one (Theorem \[thm-finite-limits-imply-computability\]).
Firstly, we recall (and prove for the sake of completeness) that $\Sigma^\infty$ forms a complete lattice with the partial ordering $\leq_\ell$ (which can be turned into a metric space as well).
\[lem-lattice\] $(\Sigma^\infty, \leq_\ell)$ is a complete lattice.
Let $L$ be an infinite language. If $L$ has a maximal element, then it is the supremum, otherwise we generate the word $a_1a_2a_3\ldots\in\Sigma^\omega$ in the following way: let $u_0 = \varepsilon$ and $u_i = a_1\ldots a_i$. We choose the largest possible letter $a_{i+1}$ with $(u_ia_{i+1})^{-1}L$ being nonempty. The word generated by this way is the supremum of $L$.
Limits in general
-----------------
Though limits of a language could be defined as limits of Cauchy sequences in the aforementioned metric space, the following (equivalent) definition is more convenient for our purposes.
The word $w\in\Sigma^\omega$ is a *limit* of an infinite language $L$, if for each $w_0<_p w$ there exists a word $u\in L$ such that $w_0<_p u$.
If $L \subseteq \Sigma^*$ is a language, then we denote the limits of $L$ with $\blim{L}$.
If $w$ is a limit of an infinite language $L$, then for each $w_0<_p w$ there exist infinitely many words $u\in L$ with $w_0<_p u$.
We construct two sequences $w_0 <_p w_1 <_p \ldots <_p w$ and $u_0,\ u_1, \ldots \in L$ such that $w_i <_p u_i$ for each $i$ with mutual induction. Now $w_0$ is given. By definition for each $i$ there exists an $u_i\in L$ such that $w_i<_p u_i$ and $w_i\in \mathbf{Pref}(w)$ is constructed such that $|u_{i-1}|+1<|w_i|$.
It is clear the words $u_i$ are pairwise different, since each $w_i$ has different length. So we get that $w_0$ is a prefix of each $u_i$, so $w_0$ is a prefix of infinitely many words in $L$.
\[lem-infinite\] For each infinite language $L$, the set $\blim{L}$ is nonempty.
We construct a limit word $w=a_1a_2\ldots\in\Sigma^\omega$. Let $u_0 = \varepsilon$ and $u_i = a_1\ldots a_i$ and we choose $a_{i+1}\in\Sigma$ such that $u_ia_{i+1}$ is a prefix of infinitely many words in $L$. Since $L$ is infinite there exists such a letter. Thus we can construct an infinite word which is a limit of $L$.
Now we justify using the name “limit”: any supremum or infimum of a chain (which is a Cauchy sequence) of words of a language is a limit of the language.
\[lem-sup-is-limit\] If $w_0, w_1, \ldots$ is a $<_\ell$ (or $>_\ell$ respectively) chain in $L$, then its supremum (infimum, resp.) is a limit of $L$.
Let $w$ be the supremum of the $<_\ell$ chain in $L$. We only have to show that for each $w'<_p w$ there exists a member $u$ of the chain with $w'<_p u$. First, as the chain is infinite, there are infinitely many words $w_i$ with $|w_i|>|w'|$ and the supremum of these words $w_i$ is $w$ as well. For these words we cannot have $w'<_pw_i$. As $w>w'$ is the supremum of the words $w_i$, we have $w_i<_\ell w$ for each of them, moreover, there exists some word $w_i$ with $w'<_\ell w_i<_\ell w$. Since we know that $w'<_pw$, it has to be the case that $w'<_pw_i<_\ell w$ and the claim is proved.
For the other case, let $w$ be the infimum of the $>_\ell$ chain in $L$ and let $w'<_pw$ be a prefix of $w$. We again have to show that $w'<_pw_i$ for some $i$. Again, we can take the subchain consisting of those words $w_i$ with $|w_i|>|w'|$, this does not change their infimum. Now we have $w'<_pw<_sw_i$ for each of these words $w_i$. We claim that $w'<_pw_i$ for at least one index $i$. Assume to the contrary that $w'<_sw_i$ for each $i$ and let us write $w'=u_0az^t$ where $z$ is the largest letter of $\Sigma$, and $a<z$. (Since there exist words with $w'<_sw_i$, $w'$ cannot have the form $z^t$). Let $b$ be the successor letter of $a$ and consider the word $w''=u_0b$. Obviously, $w''$ is the least word with $w'<_sw''$, thus $w''\leq_\ell w_i$ for each $i$. Since $w$ is the infimum of these words $w_i$, we should have $w''\leq_\ell w$ but this contradicts to $w'<_pw$ and $w'<_sw''$ as these two imply $w<_sw''$.
Now we recall from [@GelleIvanGandalf] that for any context-free language, we can compute a supremum or infimum of the language.
\[lem-sequence\] For each sentential form $\alpha$ with $L(\alpha)$ being infinite, we can generate a sequence $w_0,w_1,\ldots \in L(\alpha)$ and a regular word $w\in\Sigma^\omega$ satisfying one of the following cases:
- $w_1<_sw_2<_s\ldots$ and $w=\mathop\bigvee\limits_{i\geq 0}w_i$
- $w_1>_sw_2>_s\ldots$ and $w=\mathop\bigwedge\limits_{i\geq 0}w_i$
- $w_1<_pw_2<_p\ldots$ and $w=\mathop\bigvee\limits_{i\geq 0}w_i$
Hence, Lemma \[lem-sup-is-limit\] in conjunction with Lemma \[lem-sequence\] ensure that whenever $L$ is an infinite context-free language, then one of its limits can be effectively computed, and this particular limit will be a regular word.
Next, we show how to compute limits of unions and products:
\[lem-union\] For any languages $K$ and $L$ $\blim{K\cup L} = \blim{K}\cup\blim{L}$.
Assume $w$ is a limit of $L$. Then for each $w_0<_pw$, there exists some $u\in L$ with $w_0<_p u$. Since then $u\in K\cup L$ as well, $w$ is a limit of $K\cup L$ as well.
For the other direction, assume $w$ is a limit of $K\cup L$. Then for each $w_0<_pw$, there exists some $u\in K\cup L$ with $w_0<_pu$. Thus, either there exists infinitely many prefixes $w_0$ of $w$ for which there exists some $u\in K$ with $w_0<_pu$ or there exists infinitely many prefixes $w_0$ of $w$ for which there exists some $u\in L$ with $w_0<_pu$. In the former case, $w$ is a limit of $K$, in the latter, $w$ is a limit of $L$.
\[lem-limits-of-product\] $\blim {KL} = \blim{K} \cup K\blim{L}$ if $K, L\neq \emptyset$.
Let $u\in K$ be a word and $w$ be a limit of $L$. To prove that $uw$ is a limit of $KL$ we only have to show that for each prefix $w'$ of $uw$ there exist a word $w^*\in KL$ with $w' <_p w^*$. Let $w'\in \mathbf{Pref}(uw)$, and since it is enough to see the prefixes which are longer than $u$, the word $w'$ can be written as $uw_0$. Since $w$ is a limit of $L$ there exists a word $v\in L$ such that $w_0<_p v$. Thus there is a word $uv\in KL$ such that $w'<_puv$.
Now let $w$ be a limit of $K$. To prove that $w$ is a limit of $KL$, let $u$ be a word in $L$ and $w_0$ be a prefix of $w$. Since $w$ is a limit of $K$ there exists a word $v\in K$ with $w_0 <_p v$ by definition. So $vu \in KL$ and $w_0$ is a prefix of $vu$ as well.
For the other direction, we have to prove that there are no more limits of $KL$. Let $w$ be a limit of $KL$ and $w_i$ be the prefix of $w$ with length $i$. Since $w$ is a limit of $KL$ there exist a word $u_iv_i\in KL$ such that $w_i <_p u_iv_i$ where $u_i\in K$ and $v_i\in L$. Consider for each $i>0$ the lengths of these words $u_i$. There are two cases: either there is a finite upper bound on $|u_i|$ or there is not.
If $|u_i|$ is not upperbounded, then $w$ is a limit of $K$, since for each prefix $w_i$ there exists some long enough $u_j$ with $w_i<_pu_j$.
In the case where the lengths of these $u_i$ words is bounded, let $\ell = \max|u_i|$ be the maximal length. Since there are only finitely many words of length at most $\ell$, there has to be some $u=u_j$ such that $u=u_i$ for infinitely many indices $i$. Hence in particular, $w=uw'$ for some $w'\in\Sigma^\omega$. We show that $w'\in\blim L$, yielding $w\in K\blim L$. Indeed, if $w^*<_pw'$ is a prefix of $w'$, then $uw^*<_pw$ and thus there exists some $v_i$ with $uw^*<_puv_i$, that is, $w^*<_pv_i$ and so $w^*$ is a limit of $L$.
\[cor-la-al\] For any language $L$ and letter $a\in\Sigma$, $\blim{L}=\blim{La}$ and $a\cdot\blim{L}=\blim{aL}$.
\[cor-uLv\] For any language $L\subseteq\Sigma^*$ and words $u,v\in\Sigma^*$, $\blim{uLv}=u\cdot \blim{L}$.
Unique limits
-------------
In this part we establish the decidability of the problem whether a context-free language has a unique limit. (In this case, the limit itself is computable as well, thanks to Lemma \[lem-sequence\].)
In the rest of the paper when grammars are involved, we assume the grammar $G=(N,\Sigma,P,S)$ contains no left recursive nonterminals, and for each $X\in N$, $X$ is usable and $L(X)$ is an infinite language of nonempty words. Moreover, each nonterminal but possibly $S$ is assumed to be recursive. Any context-free grammar can effectively be transformed into such a form, see e.g. [@GelleIvanTCS].
It is also known [@10.1007/978-3-642-22321-1_19] that if the context-free grammar $G$ generates a scattered language, then for each recursive nonterminal $X$ there exists a unique (and computable) primitive word $u_X$ such that whenever $X\Rightarrow^+uX\alpha$ for some $u\in\Sigma^*$ and sentential form $\alpha$, then $u\in u_X^+$. Moreover, for each pair $X\approx Y$ of recursive nonterminals there exists a (computable) word $u_{X,Y}\in\Sigma^*$ such that whenever $X\Rightarrow^+ uYv$, then $u\in u_{X,Y}u_Y^*$.
\[lem-finite-strict-unique-limit\] The word $w\in\Sigma^\omega$ is the unique limit of an infinite language $L$ if and only if for each $w_0<_p w$ there exists only finitely many words $u\in L$ such that $u<_s w_0$ or $w_0<_s u$.
We will see just the case where $w_0 <_s u$, the other one can be done analogously.
Suppose for the sake of contradiction there exist infinitely many words $u\in L$ with $w_0 <_s u$. Let $w_0$ be the shortest such word, it can be written as $w_0 = w_0'a$. Since there are just finitely many words $u\in L$ with $w_0'<_s u$, it has to be the case that $w_0'<_p u$ and $w_0'a<_s u$ for infinitely many $u\in L$. Then there exists a letter $b\in\Sigma$ such that $b>a$ and infinitely many words $u\in L$ such that $w_0'b <_p u$. But any limit of these words is in $w_0'b\Sigma^\omega$ (and by Lemma \[lem-infinite\] at least one limit exists), which cannot be equal to $w$, so the language $L$ has two different limits which is a contradiction.
\[lem-unique-limit-embeds-into-omega-plus-minusomega\] If $L$ has a unique limit $w$, then $o(L_{<w}) \preceq \omega$ and $o(L_{>w}) \preceq -\omega$. Moreover, in this case both $o(L_{<w})$ and $o(L_{>w})$ are effectively computable (and hence so is $o(L)=o(L_{<w})+o(L_{>w})$).
In the $o(L_{<w})$ case, if $L_{<w}$ is finite (and thus its size is computable), we are done. Otherwise $L_{<w}$ is infinite, which means it has a limit, and this limit has to be the $w$ (since it is unique for $L$). By Lemma \[lem-sup-is-limit\] the supremum $\bigvee L_{<w}$ of this language is $w$ as well, moreover, each infinite $K\subseteq L_{<w}$ has $\bigvee K = w$, so the order type of $L_{<w}$ has to be $\omega$.
In the $o(L_{>w})$ case, if $L_{>w}$ is finite, it can be embedded into $-\omega$ so we are done. Otherwise $L_{>w}$ is infinite, so it has to have a limit which is $w$. Since this limit should be an infimum of a descending chain, each infinite $K\subseteq L_{>w}$ has $\bigwedge K = w$, so the order type of $L_{>w}$ has to be $-\omega$.
Before proceeding to the case of concatenation, we recall the notion of prefix chains from [@GelleIvanGandalf]. For a word $w\in\Sigma^\omega$, let $\mathbf{Pref}(w)$ stand for the set $\{u\in\Sigma^*:u<_pw\}$ of proper prefixes of $w$. A language $L\subseteq\Sigma^*$ is called a *prefix chain* if $L\subseteq\mathbf{Pref}(w)$ for some $\omega$-word $\omega$. Lemma 2 from [@GelleIvanGandalf] states that it is decidable for any context-free language $L$ whether $L$ is a prefix chain and if so, a suitable $w\in\Sigma^\omega$ can be effectively computed.
\[lem-x1x2\] Assume we know that for the nonterminals $X_1$ and $X_2$ whether the languages $L_1=L(X_1)$ and $L_2=L(X_2)$ have a unique limit. Then it is computable whether $L=L(X_1X_2)$ has a unique limit.
By assumption, the nonterminals $X_1$ and $X_2$ each generate an infinite language so they have at least one limit by Lemma \[lem-infinite\]. By Lemma \[lem-limits-of-product\], if either $L_1$ or $L_2$ has at least two limits, then so have $L$ and we can stop.
Assume both $L_1$ and $L_2$ have a unique limit. As they are both context-free languages, their limits are computable regular words by Lemma \[lem-sequence\]. Let $u_1v_1^\omega$ and $u_2v_2^\omega$ respectively be the limits of $L_1$ and $L_2$. If $L_1$ is not a prefix chain, that is, $x<_sy$ for some $x,y\in L_1$, then both $xu_2v_2^\omega$ and $yu_2v_2^\omega$ are limits of $L$ by Lemma \[lem-limits-of-product\] and these two words are different by $x<_sy$. So in this case we can stop.
From this point we can assume that $L_1$ is a prefix chain, that is, $L_1\subseteq\mathbf{Pref}(u_1v_1^\omega)$. By Lemma \[lem-limits-of-product\], $u_1v_1^\omega$ is also a limit of $L$, and for each $u\in L_1$, the word $uu_2v_2^\omega$ is also a limit of $L$. Thus, we have to decide whether $u_1v_1^\omega=uu_2v_2^\omega$ holds for each $u\in L_1$.
Now consider the direct product automaton $M=M_{u_1,v_1}\times M_{u_2,v_2}$, where in the automaton corresponding to $L_2$ we use primed states $q'$ in place of each state $q$.
Obviously, $u_1v_1^\omega\neq uu_2v_2^\omega$ if and only if from the state $(q_0\cdot u,q'_0)$ some state of the form $(\bot,q')$ or $(q,\bot')$ is reachable for some $q\neq\bot$. Thus, it suffices to determine the set of states $q$ in $M_1$ for which $q_0\cdot u=q$ for some $u\in L_1$, that is, for which $M_1(q)\cap L_1$ is nonempty, which is decidable since $L_1$ is context-free and $M_1(q)$ is regular. Then, for each such state $q$ we test whether a state of the form $(\bot,p')$ or $(p,\bot')$ is reachable from $(q,q'_0)$ and if so, then $uu_2v_2^\omega\neq u_1v_1^\omega$ for some $u\in L_1$ and thus $L$ has at least two different limits.
Now assume $uu_2v_2^\omega=u_1v_1^\omega$ for each $u\in L_1$. We claim that in this case $L$ has the unique limit $u_1v_1^\omega$. To see this, we first prove that for each prefix $w_0<_pu_1v_1^\omega$, there are only a finite number of words $v\in L$ with either $w_0<_sv$ or $v<_sw_0$. Suppose for the sake of contradiction that for some prefix $w_0$ of $u_1v_1^\omega$ there are infinitely many such words and let $w_0$ be the shortest such prefix. By assumption, there are infinitely many words $xy\in L$, $x\in L_1$, $y\in L_2$, with either $xy<_sw_0$ or $w_0<_sxy$. Since $L_1\subseteq\mathbf{Pref}(u_1v_1^\omega)$ and $w_0<_pu_1v_1^\omega$, this can happen only if $x<_pw_0$. Since there are only finitely many prefixes of $w_0$, for some $u<_pw_0$ in $L_1$ there has to be an infinite number of words $y\in L_2$ with either $uy<_sw_0$ or $w_0<_suy$. Let us write $w_0=uv$. The condition $uy<_sw_0$ or $w_0<_suy$ is then equivalent $y<_sv$ or $v<_sy$ for infinitely many $y\in L_2$. But since we know that $uu_2v_2^\omega=u_1v_1^\omega$, $v$ is a prefix of $u_2v_2^\omega$, which is the unique limit of $L_2$ and thus only finitely many words $y\in L_2$ exist with $y<_sv$ or $v<_sy$ by Lemma \[lem-finite-strict-unique-limit\], yielding a contradiction.
Hence in this case, $u_1v_1^\omega$ is the unique limit of $L$.
\[cor-unique-alpha-limit\] Assume $n\geq 0$ and $X_1,\ldots,X_n\in N\cup\Sigma$ are symbols so that for each $X_i$ we know whether $L(X_i)$ has a unique limit. Then it is computable whether $L(X_1\ldots X_n)$ has a unique limit.
Let us introduce the fresh nonterminals $Y_1,\ldots,Y_{n-1}$ and productions $Y_1\to X_1Y_2$, $Y_2\to X_2Y_3$,…, $Y_{n-1}\to X_{n-1}X_n$. Applying Lemma \[lem-x1x2\] or Corollary \[cor-la-al\] (depending on whether $X_i$ is a nonterminal or a letter) for the nonterminals $Y_{n-1}$, $Y_{n-2}$, …, $Y_1$ in this order we can successively decide whether each $L(Y_i)$ has a unique limit, proving the statement since $L(X_1\ldots X_n)=L(Y_1)$.
\[cor-alpha\] Let $X$ be a nonterminal $X\to\alpha_1~|~\ldots~|~\alpha_k$ be the collection of all the escaping productions with left-hand side $X$. Assume we already know for each $Y\prec X$ whether $L(Y)$ has a unique limit. Then it is computable whether $L(\{\alpha_1,\ldots,\alpha_k\})$ has a unique limit.
By Corollary \[cor-unique-alpha-limit\], it is computable for each $\alpha_i$ whether each $L(\alpha_i)$ is finite or has a unique limit. If not, then neither has their union (by Lemma \[lem-union\]). If each of the languages $L(\alpha_i)$ is either finite or has a unique limit, then their union has a unique limit if and only if all the limits are the same. But this is decidable since these languages are context-free, hence their unique limit is a computable regular word by Lemma \[lem-sequence\], and the equivalence of these words is decidable.
\[lem-recursive-nonprefix-alpha-x-beta\] Assume $X$ is a recursive nonterminal, $L(X)$ is not a prefix chain and for some nonterminal $X'\approx X$ there exists a production $X'\to \alpha X''\beta$ with $\beta$ containing at least one nonterminal.
Then $L(X)$ has at least two limits.
Let $u<_sv$ be members of $L(X)$. Since $\beta$ contains a nonterminal, $L(\beta)$ is infinite and has a limit $w$ by Lemma \[lem-infinite\]. By the conditions on the recursive nonterminal $X$, we get $X\Rightarrow^*u_1Xu_2\beta u_3$ for some words $u_1,u_2,u_3\in\Sigma^*$. By Lemma \[cor-uLv\], both $u_1uu_2w$ and $u_1vu_2w$ are limits of $L(X)$ and they are distinct by $u<_sv$.
\[lem-recursive-x\] Assume $L(G)$ is scattered. Then it is decidable for each nonterminal $X$ whether $L(X)$ has a unique limit.
We prove the statement by induction on $\prec$. So let $X$ be a nonterminal and assume we already know for each $Y\prec X$ whether $L(Y)$ has a unique limit.
If $X$ is nonrecursive, and $X\to\alpha_1~|~\ldots~|~\alpha_k$ are all the alternatives of $X$, then the question is decidable by Corollary \[cor-alpha\].
So let $X$ be a recursive nonterminal. If $L(Y)$ has at least two limits for some $Y\prec X$, then by Corollary \[cor-uLv\] so does $L(X)$ and we are done. So we can assume from now on that each $L(Y)$ with $Y\prec X$ has exactly one limit. This limit is a computable regular word.
If $L(X)$ is a prefix chain, then its supremum is its unique limit and we can stop. So we can assume that $L(X)$ is not a prefix chain. Now if there exist some production of the form $X'\to \alpha X''\beta$ with $X'\approx X''\approx X$ and $\beta$ containing at least one nonterminal, then by Lemma \[lem-recursive-nonprefix-alpha-x-beta\], $L(X)$ has at least two limits and we can stop.
Otherwise, we can assume that each component production in the component of $X$ has the form $X'\to \alpha X''u$ for some $u\in\Sigma^*$, $X'\approx X''\approx X$. Since $L(X)$ is scattered, for each such $\alpha$ it has to be the case that $L(\alpha)\subseteq u_{X',X''}^{}u_{X''}^*$.
Now let $X'\approx X$ be a nonterminal and $\alpha_1,\ldots,\alpha_k$ all the escaping alternatives of $X'$. If $L(\{\alpha_1,\ldots,\alpha_k\})$ has at least two limits, then so does $L(X')$ and $L(X)$ and we are done. Otherwise, if $L(\{\alpha_1,\ldots,\alpha_k\})$ is infinite, then its unique limit is a computable word. On the other hand, for each recursive nonterminal $X'$ the word $u_{X'}^\omega$ is a limit of $L(X')$, and by $L(\{\alpha_1,\ldots,\alpha_k\})\subseteq L(X')$, the two limits has to coincide (which is decidable). If they are not the same, then again, $L(X)$ has at least two limits and we can stop.
Hence we can assume that for each $X'\approx X$, the language $L(X')$ has the limit $u_{X'}^\omega$ which is the same as the unique limit of $L(\{\alpha_1,\ldots,\alpha_k\})$ if this latter language is infinite.
We claim that in this case, $L(X)$ has the unique limit $u_X^\omega$. To see this, we apply Lemma \[lem-finite-strict-unique-limit\] and show that for each prefix $w_0$ of $u_X^\omega$, there are only finitely many words $u\in L(X)$ with either $w_0<_su$ or $u<_sw_0$.
Assume to the contrary that $w_0<_pu_X^\omega$ and there are infinitely many words $u\in L(X)$ with either $w_0<_su$ or $u<_sw_0$. Each word $u\in L(X)$ can be derived from $X$ using a leftmost derivation sequence resulting in a sentential form $u_X^tu_{X,X'}\alpha v$ for some $t\geq 0$ so that $X'\to\alpha$ is an escaping production from the component of $X$ and $u\in u_X^tu_{X,X'}L(\alpha)v$. Since $u$ and $w_0<_pu_X^\omega$ are not related by $<_p$, we have an upper bound for $t$, which, as $G$ does not contain left-recursive nonterminals, places an upper bound for $|v|$. Hence, there are only finitely many possibilities for picking $t\geq 0$, $X'$, $\alpha$ and $v$, thus for some combination of them, there are infinitely many such words $u$ belonging to the same language $u_X^tu_{X,X'}L(\alpha)v$. So we can write each such $u$ as $u=u_X^tu_{X,X'}u'v$ with $u'\in L(\alpha)$, and we can write $w_0$ as $w_0=u_X^tu_{X,X'}w'_0$, that is, $w'_0<_p u_{X'}^\omega$. This yields that $u'v<_sw'_0$ or $w'_0<_su'v$ for infinitely many words $u'\in L(\alpha)$. Thus, there are infinitely words $u'\in L(\alpha)$ of length at least $|w'_0|$ with either $u'v<_sw'_0$ or $w'_0<_su'v$, hence with either $u'<_sw'_0$ or $w'_0<_su'$, which is a contradiction, since by Lemma \[lem-finite-strict-unique-limit\] this would yield that $L(\alpha)$ has at least two limits, which we already handled in a former case.
Finitely many limits
--------------------
In this part we extend the results of the previous subsection for the context-free languages having a finite number of limits and get the main result of the paper: it is decidable whether a context-free language has a scattered order type of rank at most one, and if so, then its order type is effectively computable.
\[lem-uvomega-islimit-decidable\] It is decidable for any context-free language $L$ and regular word $w=uv^\omega$ whether $w$ is a limit of $L$.
Let $L\subseteq\Sigma^*$ be a context-free language and consider the generalized sequential mapping $f:\Sigma^*\to a^*$ defined as $$f(x)=\begin{cases}a\cdot f(y)&\hbox{if }x=vy\hbox{ for some }y\in\Sigma^*\\\varepsilon&\hbox{otherwise}.\end{cases}$$ Now for any word $x$, $f(x)=a^n$ for the unique $n$ such that $x=v^ny$ for some $y$ not having $v$ as prefix. Thus, $v^\omega$ is a limit of a language $L'$ if and only if $f(L')$ is infinite; hence, by Corollary \[cor-uLv\], $w=uv^\omega$ is a limit of $L$ if and only if $f(u^{-1}L)$ is infinite. Since the class of context-free languages is effectively closed under left quotients and generalized sequential mappings, and their finiteness problem is decidable, the claim is proved.
\[lem-recursive-one-or-infinite\] If $X$ is a recursive nonterminal, then $u_X^\omega \in \blim{X}$ and if some $w\neq u_X^\omega$ is also a member of $\blim{X}$, then $\blim{X}$ is infinite.
Since $X\Rightarrow^* u_X^nX\alpha$ holds for each recursive nonterminals, there exists a word $v\in L(X)$ for each $u\in\mathbf{Pref}(u_X^\omega)$ such that $u<_p v$.
If $w\neq u_X^\omega$ is also a limit of $L(X)$, then it can be written as $w=ubw'$, where $u<_pu_X^\omega$, $ub\not<_pu_X^\omega$ and $w'\in\Sigma^\omega$. So if we consider a derivation of the form $X\Rightarrow^* u_X^nX\alpha$, we get that each $(u_x^n)^kubw'$, $k>0$ is a limit, thus $\blim{X}$ is infinite as these words are pairwise different.
\[lem-kvomega-is-finite\] Assume $K$ is a context-free language and $v$ is a nonempty word. Then it is decidable whether $Kv^\omega$ is finite and if so, its members (which are regular words) can be effectively enumerated.
Consider the generalized sequential mapping $f: \Sigma^*\to \Sigma^*$ defined as $$f(x) = \begin{cases}
f(y)&\hbox{if } x=v^Ry \hbox{ for some } y\in\Sigma^*\\
x&\hbox{otherwise.}
\end{cases}$$
Now for any word $x$, $f(x)=y$ for some $y$ not having $v^R$ as prefix such that $x=(v^R)^ky$ for some $k\geq 0$, that is, $f$ strips away the leading $v^R$s of its input. So, we have that $\bigl(f(K^R)\bigr)^R$ consists of those words we get from members of $K$, stripping away their trailing $v$s. Now $u\in \bigl(f(K^R)\bigr)^R$ if and only if $u$ does not end with $v$ and $uv^\omega\in Kv^\omega$. Moreover, $Kv^\omega$ is finite if and only if there exist some $u_1, u_2, \ldots, u_n\in \Sigma^*$ such that for each $i\in[n]$ the word $u_i$ does not end with $v$ and $Kv^\omega = \{u_iv^\omega ~|~ i\in [n]\}$. So we get that $Kv^\omega$ is finite if and only if so is $\bigl(f(K^R)\bigr)^R$ which is decidable since the class of context-free languages is effectively closed under reversal and generalized sequential mappings, and their finiteness problem is also decidable. In this case, members of $\bigl(f(K^R)\bigr)^R=\{u_1,\ldots,u_n\}$ can also be effectively enumerated and $Kv^\omega=\{u_jv^\omega:j\in[n]\}$.
\[lem-kl-finite\] Assume $K$ and $L$ are context-free languages such that $\blim{K} = \{u_iv_i^\omega ~|~ 1\leq i\leq k\}$ and $\blim{L} = \{u'_jw_j^\omega ~|~ 1\leq j\leq \ell \}$ are finite sets of regular words. Then it is decidable whether $\blim{KL}$ is finite and if so, then it is a computable (finite) set of regular words.
By Lemma \[lem-limits-of-product\] $\blim{KL}=\blim{K}\cup K\blim{L}$. Since $\blim{K}$ is a finite set (and is of course computable since it is given as input), we only have to deal with $K\cdot\blim{L}$. Since $$K\cdot\blim{L}=K\cdot\bigl\{u_j'w_j^\omega:j\in[\ell]\bigr\}=\mathop\bigcup\limits_{j\in[\ell]}(Ku_j')w_j^\omega$$ and this union is finite if and only if so is each language $(Ku'_j)w_j^\omega$, which is decidable by Lemma \[lem-kvomega-is-finite\] (since the languages $Ku'_j$ are each context-free), we get decidability and even computability if each of them is finite.
\[cor-finite-alpha-limit\] Assume $n\geq 0$ and $X_1,\ldots,X_n\in N\cup\Sigma$ are symbols so that for each $X_i$, $\blim{X_i}$ is a known finite set. Then it is decidable whether $L(X_1\ldots X_n)$ has a finite number of limits and if so, $\blim{X_1\ldots X_n}$ is effectively computable.
Let us introduce the fresh nonterminals $Y_1,\ldots,Y_{n-1}$ and productions $Y_1\to X_1Y_2$, $Y_2\to X_2Y_3$,…, $Y_{n-1}\to X_{n-1}X_n$. Applying Lemma \[lem-kl-finite\] or Corollary \[cor-la-al\] (depending on whether $X_i$ is a nonterminal or a letter) for the nonterminals $Y_{n-1}$, $Y_{n-2}$, …, $Y_1$ in this order we can decide whether each $L(Y_i)$ has a finite number of limits, and if so, we compute $\blim{Y_i}$ as well, proving the statement since $L(X_1\ldots X_n)=L(Y_1)$.
\[lem-limx-finite\] It is decidable for any nonterminal $X$ whether $L(X)$ has a finite number of limits.
There are two cases: either $X$ is recursive or not. If $X$ is recursive, then by Lemma \[lem-recursive-one-or-infinite\] $L(X)$ has a finite number of limits if and only it has a unique limit which is decidable due to Lemma \[lem-recursive-x\]. Hence we can decide for each nonterminal $X\neq S$ whether $\blim{ L(X) }$ is finite.
Now suppose $X$ is not recursive (thus $X=S$ as $G$ is in normal form) and let $\balpha_1,\ldots,\balpha_t$ be all the alternatives of $S$. By Corollary \[cor-finite-alpha-limit\] for each $\alpha_i$ $(1\leq i\leq t)$ it is decidable whether $L(\alpha_i)$ has a finite number of limits. If one of them has infinite limits than so has $X$. So we can assume that each $L(\alpha_i)$ has a finite number of limits and we can even compute each $\blim{\alpha_i}$. By Lemma \[lem-union\], $\blim{X} = \bigcup_{i\in [t]}\blim{\alpha_i}$.
\[thm-finite-limits-imply-computability\] Suppose $L$ is a context-free language having a finite number of limits. Then $o(L)$ is effectively computable and is scattered of rank at most one.
We prove the statement by induction on the number of limits.
If $L$ has no limits, then it is finite by Lemma \[lem-infinite\], and so $o(L)=|L|$ is computable.
If $L$ has a unique limit, then $o(L)$ can be embedded into $\omega+-\omega$ and is computable by Lemma \[lem-unique-limit-embeds-into-omega-plus-minusomega\]. Moreover, it is decidable whether $L$ has a unique limit by Lemma \[lem-recursive-x\].
Now assume $L$ has at least two limits. Since $L$ is infinite, we can compute a regular limit of the form $w=uv^\omega$ for $L$ by Lemma \[lem-sequence\]. By Lemma \[lem-uvomega-islimit-decidable\], it is decidable whether $w$ is a limit of either $L_{<w}$ or $L_{>w}$ or both of them. (By Lemma \[lem-union\], $w$ is a limit of at least one of them.) If $w$ is not a limit of $L_{<w}$ ($L_{>w}$, resp.), then this language has a smaller number of limits than $L$ and we can proceed by induction. Suppose now $w$ is a limit of $L_{<w}$ – it has to be $w=\bigvee L_{<w}$ then. If $L$ has a limit which is larger than $w$ (that is, $L_{>w}$ is infinite and either $w$ is not a limit of $L_{>w}$ or $L_{>w}$ has at least two limits – this is decidable as well), then $L_{<w}$ has a smaller number of limits than $L$ (since no limit of $L_{<w}$ can be strictly larger than its supremum) and we can proceed again by induction and get that $L_{<w}$ is computable. It is also decidable whether $L_{<w}$ has only one limit and if so, its order type is also computable and we are done.
The last case is when $w=\bigvee L_{<w}$ is the largest limit of $L$ and $L_{<w}$ has at least two limits. Thus, there exists some limit $w'$ of $L_{<w}$ and an integer $n\geq 0$ such that $w'<_suv^n$, or equivalently, $L_{<uv^n}$ is infinite for some $n\geq 0$. We can compute (say, the least) such $n$ by starting from $n=0$ and iterating, eventually we will find an integer $n$ with this property. Then, $L_{<uv^n}$ has a smaller number of limits than $L_{<w}$ so we can use induction and compute $o(L_{<uv^n})$; also, ${L_{<w}}_{\geq uv^n}$ also has a smaller number of limits than $L_{<w}$ (since $w'$ is missing) and we can apply induction to this half as well and compute its order type. Then, $o(L_{<w})$ is the sum of the two already computed order types.
Repeating the same argument (by appropriate modifications: taking infimum instead of supremum, splitting the case when $w$ is the least limit of $L$) we get that $o(L_{>w})$ is also computable, and $o(L)$, being the sum $o(L_{<w})+o(L_{>w})$, is hence computable as well.
We also got that the order type of such a language has to be a finite sum of the order types $\omega$, $-\omega$ and $1$, that is, has to have rank at most $1$.
\[cor-hausdorff-one-computable\] Suppose $L$ is a scattered context-free language of rank at most one. Then $o(L)$ is effectively computable.
If $o(L)\in\{\omega,-\omega\}$, then $L$ has one limit, while if $o(L)$ is finite, then it has no limits. Since scattered order types of rank at most one are finite sums of the order types $\omega$, $-\omega$ and $1$, thus scattered languages of rank at most one are finite unions of languages of order type $\omega$, $-\omega$ or $1$, by Lemma \[lem-union\] we get that such languages have a finite number of limits, and thus their order type is effectively computable by Theorem \[thm-finite-limits-imply-computability\].
\[cor-main\] For any context-free language $L$, it is decidable whether $L$ is a scattered language of rank at most one, and if so, $o(L)$ can be effectively computed (as a finite sum of the order types $1$, $\omega$ and $-\omega$).
Conclusion
==========
We showed that it is decidable whether a context-free ordering is scattered of rank at most one, and if so, then its order type is effectively computable as a finite sum of the order types $1$, $\omega$ and $-\omega$. This complements our earlier result [@GelleIvanGandalf] that for scattered context-free orderings of rank (at least) two, it is undecidable whether their order type is $\omega+(\omega+\zeta)\times\omega$, thus the order type is not computable, even if the grammar in question is a so-called prefix grammar.
An interesting question for further study is whether the rank of a scattered context-free ordering is computable. Another, maybe easier one is to determine which rank-two scattered orderings are context-free (as there are uncountably many such orderings, the vast majority of them cannot be context-free).
A relatied notion is that of tree automatic orderings: these are the order types of regular tree languages equipped with the lexicographic ordering (on trees). Through derivation trees, there is a tight connection between context-free string languages and regular tree languages but as the two orderings differ (lexicographic ordering of trees vs their frontiers), it is unclear whether there is a nontrivial inclusion between these two classes of orderings (or at least for the scattered case).
|
---
abstract: 'Representation learning is a central challenge across a range of machine learning areas. In reinforcement learning, effective and functional representations have the potential to tremendously accelerate learning progress and solve more challenging problems. Most prior work on representation learning has focused on generative approaches, learning representations that capture all the underlying factors of variation in the observation space in a more disentangled or well-ordered manner. In this paper, we instead aim to learn functionally salient representations: representations that are not necessarily complete in terms of capturing all factors of variation in the observation space, but rather aim to capture those factors of variation that are important for decision making – that are “actionable.” These representations are aware of the dynamics of the environment, and capture only the elements of the observation that are necessary for decision making rather than all factors of variation, eliminating the need for explicit reconstruction. We show how these learned representations can be useful to improve exploration for sparse reward problems, to enable long horizon hierarchical reinforcement learning, and as a state representation for learning policies for downstream tasks. We evaluate our method on a number of simulated environments, and compare it to prior methods for representation learning, exploration, and hierarchical reinforcement learning.'
author:
- |
Dibya Ghosh, Abhishek Gupta, & Sergey Levine [^1]\
Department of Electrical Engineering and Computer Science\
University of California, Berkeley\
Berkeley, CA 94703, USA\
bibliography:
- 'iclr2019\_conference.bib'
title: 'Learning Actionable Representations with Goal-Conditioned Policies'
---
Introduction
============
Representation learning refers to a transformation of an observation, such as a camera image or state observation, into a form that is easier to manipulate to deduce a desired output or perform a downstream task, such as prediction or control. In reinforcement learning (RL) in particular, effective representations are ones that enable generalizable controllers to be learned quickly for challenging and temporally extended tasks. While end-to-end representation learning with full supervision has proven effective in many scenarios, from supervised image recognition [@alexnet] to vision-based robotic control [@levinefinn16JMLR], devising representation learning methods that can use unlabeled data or experience effectively remains an open problem.
Much of the prior work on representation learning in RL has focused on generative approaches. Learning these models is often challenging because of the need to model the interactions of *all* elements of the state. We instead aim to learn functionally salient representations: representations that are not necessarily complete in capturing all factors of variation in the observation space, but rather aim to capture factors of variation that are relevant for decision making – that are actionable.
How can we learn a representation that is aware of the dynamical structure of the environment? We propose that a basic understanding of the world can be obtained from a *goal-conditioned policy*, a policy that can knows how to reach arbitrary goal states from a given state. Learning how to execute shortest paths between all pairs of states suggests a deep understanding of the environment dynamics, and we hypothesize that a representation incorporating the knowledge of a goal-conditioned policy can be readily used to accomplish more complex tasks. However, such a policy does not provide a readily usable state representation, and it remains to choose *how* an effective state representation should be extracted. We want to extract those factors of the state observation that are critical for deciding which action to take. We can do this by comparing which actions a goal-conditioned policy takes for two different goal states. Intuitively, if two goal states require different actions, then they are functionally different and vice-versa. This principle is illustrated in the diagram in Figure \[fig:act\_rep\]. Based on this principle, we propose actionable representations for control (ARC), representations in which Euclidean distances between states correspond to expected differences between actions taken to reach them. Such representations emphasize factors in the state that induce significant differences in the corresponding actions, and de-emphasize those features that are irrelevant for control.
[R]{}[0.35]{} {width="\linewidth"}
While learning a goal-conditioned policy to extract such a representation might itself represent a daunting task, it is worth noting that such a policy can be learned without any knowledge of downstream tasks, simply through unsupervised exploration of the environment. It is reasonable to postulate that, without active exploration, *no* representation learning method can possibly acquire a dynamics-aware representation, since understanding the dynamics requires experiencing transitions and interactions, rather than just observations of valid states. As we demonstrate in our experiments, representations extracted from goal-conditioned policies can be used to better learn more challenging tasks than simple goal reaching, which cannot be easily contextualized by goal states. The process of learning goal-conditioned policies can also be made recursive, so that the actionable representations learned from one goal-conditioned policy can be used to quickly learn a better one.
Actionable representations for control are useful for a number of downstream tasks: as representations for task-specific policies, as representations for hierarchical RL, and to construct well-shaped reward functions. We show that ARCs enable these applications better than representations that are learned using unsupervised generative models, predictive models, and other prior representation learning methods. We analyze structure of the learned representation, and compare the performance of ARC with a number of prior methods on downstream tasks in simulated robotic domains such as wheeled locomotion, legged locomotion, and robotic manipulation.
Preliminaries {#sec:prelims}
=============
#### Goal-conditioned reinforcement learning.
In RL, the goal is to learn a policy $\pi_{\theta}(a_t|s_t)$ that maximizes the expected return $R_t = \mathbb{E}_{\pi_{\theta}}[ \sum_t r_t]$. Typically, RL learns a single task that optimizes for a particular reward function. If we instead would like to train a policy that can accomplish a variety of tasks, we might instead train a policy that is conditioned on another input – a goal. When the different tasks directly correspond to different states, this amounts to conditioning the policy $\pi$ on both the current and goal state. The policy $\pi_{\theta}(a_t|s_t, g)$ is trained to reach goals from the state space $g \sim \states$, by optimizing $\mathbb{E}_{g \sim S} [\mathbb{E}_{\pi_{\theta}(a|s,g)}(R_g))]$, where $R_g$ is a reward for reaching the goal $g$.
#### Maximum entropy RL.
Maximum entropy RL algorithms modify the RL objective, and instead learns a policy to maximize the reward as well as the entropy of the policy [@sql; @lmdp], according to . In contrast to standard RL, where optimal policies in fully observed environments are deterministic, the solution in maximum entropy RL is a stochastic policy, where the entropy reflects the sensitivity of the rewards to the action: when the choice of action has minimal effect on future rewards, actions are more random, and when the choice of action is critical, the actions are more deterministic. In this way, the action distributions for a maximum entropy policy carry more information about the dynamics of the task.
Learning Actionable Representations {#sec:method}
===================================
![An illustration of actionable representations. For a pair of states $s_1, s_2$, the divergence between the goal-conditioned action distributions they induce defines the actionable distance $D_{\text{Act}}$, which in turn is used to learn representation $\phi$.[]{data-label="fig:my_label"}](figures/actrepmethods.png){width="0.8\linewidth"}
In this work, we extract a representation that can distinguish states based on actions required to reach them, which we term an actionable representation for control (ARC). In order to learn state representations $\phi$ that can capture the elements of the state which are important for decision making, we first consider defining actionable distances $D_{\text{Act}}(s_1, s_2)$ between states. Actionable distances are distances between states that capture the differences between the actions required to reach the different states, thereby implicitly capturing dynamics. If actions required for reaching state $s_1$ are very different from the actions needed for reaching state $s_2$, then these states are *functionally* different, and should have large actionable distances. This subsequently allows us to extract a feature representation($\phi(s)$) of state, which captures elements that are important for decision making.
To formally define actionable distances, we build on the framework of goal-conditioned RL. We assume that we have already trained a maximum entropy goal-conditioned policy $\pi_{\theta}(a|s, g)$ that can start at an arbitrary state $s_0 \in S$ in the environment, and reach a goal state $s_g \in \mathcal{S}$. Although this is a significant assumption, we will discuss later how this is in fact reasonable in many settings. We can extract actionable distances by examining how varying the goal state affects action distributions for goal-conditioned policies. Formally, consider two different goal states $s_1$ and $s_2$. At an intermediate state $s$, the goal-conditioned policy induces different action distributions $\pi_{\theta}(a|s, s_1)$ and $\pi_{\theta}(a|s, s_2)$ to reach $s_1$ and $s_2$ respectively. If these distributions are similar over many intermediate states $s$, this suggests that these states are functionally similar, while if these distributions are different, then the states must be functionally different. This motivates a definition for actionable distances $D_{\text{Act}}$ as $$\label{eqn:act_distance}
D_{\text{Act}}(s_1, s_2) = \mathbb{E}_{s}\Bigg[ D_{KL}(\pi(a|s, s_1)|| \pi(a|s, s_2)) + D_{KL}(\pi(a|s, s_2)|| \pi(a|s, s_1))\Bigg].$$ The distance consists of the expected divergence over *all* initial states $s$ (refer to Section \[sec:appendix\_representation\_details\] for how we do this practically). If we focus on a subset of states, the distance may not capture action differences induced elsewhere, and can miss functional differences between states. Since maximum entropy policies learn *unique* optimal stochastic policies, the actionable distance is well-defined and unambiguous. Furthermore, because max-ent policies capture sensitivity of the value function, goals are similar under ARC if they require the same action *and* they are equally “easy" to reach.
We can use $D_{\text{Act}}$ to extract an actionable representation of state. To learn this representation $\phi(s)$, we optimize $\phi$ such that Euclidean distance between states in representation space corresponds to actionable distances $D_{\text{Act}}$ between them. This optimization yields good representations of state because it emphasizes the functionally relevant elements of state, which significantly affect the actionable distance, while suppressing less functionally relevant elements of state. The problem is: $$\label{eqn:arc_objective}
\min_{\phi} \mathbb{E}_{s_1, s_2}\Bigg[\|\phi(s_1) - \phi(s_2)\|_2 - D_{\text{Act}}(s_1, s_2)\Bigg]^2$$ This objective yields representations where Euclidean distances are meaningful. This is not necessarily true in the state space or in generative representations (Section \[sec:exps\_analysis\]). These representations are meaningful for several reasons. First, since we are leveraging a goal-conditioned policy, they are aware of dynamics and are able to capture local connectivity of the environment. Secondly, the representation is optimized so that it captures only the *functionally* relevant elements of state.
#### Requirement for Goal Conditioned Policy:
A natural question to ask is whether needing a goal-conditioned policy is too strong of a prerequisite. However, it is worth noting that the GCP can be trained with existing RL methods (TRPO) using a sparse task-agnostic reward (Section \[sec:exps\_details\], Appendix \[appendix:gcp\_details\]) – obtaining such a policy is not especially difficult, and existing methods are quite capable of doing so ( [@RIG]). Furthermore, it is likely not possible to acquire a functionality-aware state representation *without* some sort of active environment interaction, since dynamics can only be understood by observing outcomes of actions, rather than individual states. Importantly, we discuss in the following section how ARCs help us solve tasks beyond what a simple goal-conditioned policy can achieve.
Using Actionable Representations for Downstream Tasks {#sec:downstream}
=====================================================
A natural question that emerges when learning representations from a goal-conditioned policy pertains to what such a representation enables over the goal-conditioned policy itself. Although goal-conditioned policies enable reaching between arbitrary states, they suffer from fundamental limitations: they do not generalize very well to new states, and they are limited to solving only goal-reaching tasks. We show in our empirical evaluation that the ARC representation expands meaningfully over these limitations of a goal-conditioned policy - to new tasks and to new regions of the environment. In this section, we detail how ARCs can be used to generalize beyond a goal-conditioned policy to help solve tasks that cannot be expressed as goal reaching (Section \[sec:downstream\_vf\]), tasks involving larger regions of state space (Section \[sec:reward\_shaping\]), and temporally extended tasks which involve sequences of goals (Section \[sec:downstream\_hierarchy\]).
Features for Learning Policies {#sec:downstream_vf}
------------------------------
Goal-conditioned policies are trained with only a goal-reaching reward, and so are unaware of reward structures used for other tasks in the environment which do not involve simple goal-reaching. Tasks which cannot be expressed as simply reaching a goal are abundant in real life scenarios such as navigation under non-uniform preferences or manipulation with costs on quality of motion, and for such tasks, using the ARC representation as input for a policy or value function can make the learning problem easier. We can learn a policy for a downstream task, of the form $\pi_{\theta}(a | \phi(s))$, using the representation $\phi(s)$ instead of state $s$. The implicit understanding of the environment dynamics in the learned representation prioritizes the parts of the state that are most important for learning, and enables quicker learning for these tasks as we see in Section \[sec:exps\_vf\].
Reward Shaping {#sec:reward_shaping}
--------------
We can use ARC to construct better-shaped reward functions. It is common in continuous control to define rewards in terms of some distance to a desired state, oftentimes using Euclidean distance [@trpo; @ddpg]. However, Euclidean distance in state space is not necessarily a meaningful metric of *functional* proximity. ARC provides a better metric, since it directly accounts for reachability. We can use the actionable representation to define better-shaped reward functions for downstream tasks. We define a shaping of this form to be the negative Euclidean distance between two states in ARC space: $-|| \phi(s_1) - \phi(s_2) ||_2$: for example, on a goal-reaching task $r(s) = r_{\text{sparse}}(s, s_g) -||\phi(s) - \phi(s_g)||_2$. This allows us to explore and learn policies even in the presence of sparse reward functions.
One may wonder whether, instead of using ARCs for reward shaping, we might directly use the goal-conditioned policy to reach a particular goal. As we will illustrate in Section \[sec:exps\_rewardshaping\], the representation typically generalizes better than the goal-conditioned policy. Goal-conditioned policies typically can be trained on small regions of the state space, but don’t extrapolate well to new parts of the state space. We observe that ARC exhibits better generalization, and can provide effective reward shaping for goals that are very difficult to reach with the goal-conditioned policy.
Hierarchical Reinforcement Learning {#sec:downstream_hierarchy}
-----------------------------------
[R]{}[0.4]{} {width="0.9\linewidth"}
We can use a goal-conditioned policy as a low-level controller for hierarchical tasks that may not be expressible as a single goal-reaching objective. These tasks are not just reaching one particular subgoal, but a particular sequence of behaviors. Such tasks attempt to learn a high-level controller $\pi_{\text{meta}}(g|s)$ via RL that produces desired goal states for a goal-conditioned policy to reach sequentially as described in [@hiro]. The high-level controller suggests a goal directly in state space, the goal conditioned policy executes for several time-steps till it achieves the goal, following which the high level controller picks a goal again. For many tasks, naively training such a high-level controller which outputs goals directly in state space is unlikely to perform well, since such a controller must disentangle the relevant attributes in the *goal* for long horizon reasoning. In this work, we consider two schemes to use ARC representations for hierarchical RL - learning a high level policy which commands directly in ARC space or commands in a clustered latent space.
#### HRL directly in ARC space:
We can use ARC representations as a better goal space for high-level controllers, since it de-emphasizes components of the goal space irrelevant for determining the optimal action. In this scheme, the high-level controller $\pi_{\text{meta}}(z|s)$ observes the current state and generates a distribution over points in the latent space. At every meta-step, a sample $z_h$ is taken from $\pi_{\text{meta}}(z|s)$ which represents the high level command. $z_h$ is then translated into a goal $g_h$ via a decoder which is trained to reconstruct states from their corresponding goals. This goal $g_h$ can then be used to command the goal conditioned policy for several time steps, before we take another sample from $\pi_{\text{meta}}$. A high-level controller producing outputs in ARC space does not need to also find salient features from the goal space, making hierarchical RL easier since the search problem is less noisy. We show that for tasks such as waypoint navigation, using ARC as a hierarchical goal space enables significant improvement.
#### Clustering in ARC space:
We can also more directly use the learned structure in the representation space. Since ARC captures the topology of the environment, clusters in ARC space often correspond to semantically meaningful state abstractions. We utilize these clusters, with the intuition that a meta-controller searching in “cluster space” should learn faster. In this scheme, we first build a discrete number of clusters by clustering the points that the goal conditioned policy is trained on. This is done using the k-means algorithm, with a hand-chosen number of clusters (as a hyperparameter), and using the Euclidean distance in corresponding ARC representation space as a distance metric between points. We then train a high-level controller $\pi_{\text{meta}}(c|s)$ which observes a state $s$ and generates a distribution over discrete clusters $c$. At every meta-step, a cluster sample $c_h$ is taken from $\pi_{\text{meta}}(c|s)$. A goal in state space $g_h$ is then chosen uniformly at random from points within the cluster $c_h$ and used to command the GCP for several time steps before the next cluster is sampled from $\pi_{\text{meta}}$. We train a meta-policy to output clusters, instead of states: $\pi_{meta}(\text{cluster}|s)$. We see that for hierarchical tasks with less granular reward functions such as room navigation, performing RL in “cluster space” induced by ARC outperform cluster spaces induced by other representations, since the distance metric is much more meaningful in ARC space.
Related Work {#sec:related}
============
The capability to learn effective representations is a major advantage of deep neural network models. These representations can be acquired implicitly, through end-to-end training [@goodfellow2016deep], or explicitly, by formulating and optimizing a representation learning objective. A classic approach to representation learning is generative modeling, where a latent variable model is trained to model the data distribution, and the latent variables are then used as a representation [@laddernet; @ale; @vae; @dsae; @dppt; @curran; @goroshin; @darla]. In the context of control and sequence models, generative models have also been proposed to model transitions [@e2c; @assael; @solar; @causalinfogan]. While generative models are general and principled, they must not only explain the entirety of the input observation, but must also generate it. Several methods perform representation learning without generation, often based on contrastive losses [@tcn; @cpc; @mine; @contrastive; @killiancontrastive]. While these methods avoid generation, they either still require modeling of the entire input, or utilize heuristics that encode user-defined information. In contrast, ARCs are directly trained to focus on decision-relevant features of input, providing a broadly applicable objective that is still selective about which aspects of input to represent.
In the context of RL and control, representation learning methods have been used for many downstream applications [@SRLoverview], including representing value functions [@successorfeatures] and building models [@e2c; @assael; @solar]. Our approach is complementary: it can also be applied to these applications. Several works have sought to learn representations that are specifically suited for physical dynamical systems [@jonschbrock] and that use interaction to build up dynamics-aware features [@bengio; @laversanne-finot]. In contrast to @jonschbrock, our method does not attempt to encode all physically-relevant features of state, only those relevant for choosing actions. In contrast to @bengio [@laversanne-finot], our approach does not try to determine which features of the state can be independently controlled, but rather which features are relevant for *choosing* controls. Related methods learn features that are predictive of actions based on pairs of sequential states (so-called *inverse models*) [@poking; @icm; @zhanginverse]. More recent work such as [@burda] perform a large scale study of these types of methods in the context of exploration. Unlike ARC, which is learned from a policy performing long-horizon control, inverse models are not obliged to represent all relevant features for multi-step control, and suffer from greedy reasoning.
Experiments {#sec:exps}
===========
The aim of our experimental evaluation is to study the following research questions:
1. Can we learn ARCs for multiple continuous control environments? What are the properties of these learned representations?
2. Can ARCs be used as feature representations for learning policies quickly on new tasks?
3. Can reward shaping with ARCs enable faster learning?
4. Do ARCs provide an effective mechanism for hierarchical RL?
Full experimental and hyperparemeter tuning details are presented in the appendix.
Domains {#sec:domains}
-------
We study six simulated environments as illustrated in Figure \[fig:tasks\]: 2D navigation tasks in two settings, wheeled locomotion tasks in two settings, legged locomotion, and object pushing with a robotic gripper. The 2D navigation domains consist of either a room with a central divider wall or four rooms. Wheeled locomotion involves a two-wheeled differential drive robot, either in free space or with four rooms. For legged locomotion, we use a quadrupedal ant robot, where the state space consists of all joint angles, along with the Cartesian position of the center of mass (CoM). The manipulation task uses a simulated Sawyer arm to push an object, and the state consists of end-effector and object positions. Further details are presented in Appendix \[sec:appendix\_task\_descriptions\].
These environments present interesting representation learning challenges. In 2D navigation, the walls impose structure similar to those in Figure 1: geometrically proximate locations on either side of a wall are far apart in terms of reachability, and we expect a good representation to pick up on this. The locomotion environments present an additional challenge: an effective representation must account for the fact that the internal joints of each robot (legs or wheel orientation) are less salient for long-horizon tasks than CoM. The original state representation does not reflect this structure: joint angles expressed in radians carry as much weight as CoM positions in meters. In the object manipulation task, a key representational challenge is to distinguish between pushing the block and simply moving the arm in free space.
[0.15]{} ![The tasks in our evaluation. The 2D navigation tasks allow for easy visualization and analysis, while the more complex tasks allow us to investigate how well ARC and prior methods can discern the most functionally-relevant features of the state.[]{data-label="fig:tasks"}](figures/point_maze.png "fig:"){width="\linewidth"}
[0.15]{} ![The tasks in our evaluation. The 2D navigation tasks allow for easy visualization and analysis, while the more complex tasks allow us to investigate how well ARC and prior methods can discern the most functionally-relevant features of the state.[]{data-label="fig:tasks"}](figures/point_rooms.png "fig:"){width="\linewidth"}
[0.15]{} ![The tasks in our evaluation. The 2D navigation tasks allow for easy visualization and analysis, while the more complex tasks allow us to investigate how well ARC and prior methods can discern the most functionally-relevant features of the state.[]{data-label="fig:tasks"}](figures/wheeled_task.png "fig:"){width="\linewidth"}
[0.2]{} ![The tasks in our evaluation. The 2D navigation tasks allow for easy visualization and analysis, while the more complex tasks allow us to investigate how well ARC and prior methods can discern the most functionally-relevant features of the state.[]{data-label="fig:tasks"}](figures/wheeled_rooms.png "fig:"){width="\linewidth"}
[0.15]{} ![The tasks in our evaluation. The 2D navigation tasks allow for easy visualization and analysis, while the more complex tasks allow us to investigate how well ARC and prior methods can discern the most functionally-relevant features of the state.[]{data-label="fig:tasks"}](figures/ant_task.png "fig:"){width="\linewidth"}
[0.15]{} ![The tasks in our evaluation. The 2D navigation tasks allow for easy visualization and analysis, while the more complex tasks allow us to investigate how well ARC and prior methods can discern the most functionally-relevant features of the state.[]{data-label="fig:tasks"}](figures/sawyer.jpg "fig:"){width="\linewidth"}
Learning the Goal-Conditioned Policy and ARC Representation {#sec:exps_details}
-----------------------------------------------------------
We first learn a goal-conditioned policy, a stochastic policy parametrized by a neural network which output actions given the current state and the desired goal. This goal-conditioned policy is trained using a sparse reward using entropy-regularized Trust Region Policy Optimization (TRPO) [@trpo]. Exact details about the training procedure, the reward function, and the hyperparameters used are presnted in Appendix \[appendix:gcp\_details\]. For a discussion of the assumption about the existence of a goal-conditioned policy, please refer to Section \[sec:method\]. We collect a dataset of 500 trajectories with horizon 100 from the trained goal-conditioned policy, where each trajectory has an arbitrary start state and intended goal state. We train the ARC representation using this dataset, defining all expectations over states to be uniformly sampled from states in the dataset, by optimizing Eqn \[eqn:arc\_objective\] as a supervised learning problem. A detailed outline of the training procedure, along with hyperparameter and architecture choices, is presented in Appendix \[appendix:trainingrep\_details\].
Comparisons with Prior Work {#sec:exps_comparisons}
---------------------------
We compare ARC to other representation learning methods used in previous works for control: variational autoencoders [@vae] (VAE), variational autoencoders trained for feature slowness [@jonschbrock] (slowness), features extracted from a predictive model [@acvp] (predictive model), features extracted from inverse models [@poking; @burda18], and a naïve baseline that uses the full state space as the representation (state). Details of the exact objectives used to train these methods is provided in Appendix \[sec:appendix\_representation\_details\].
For each downstream task in Section \[sec:downstream\], we also compare with specific approaches for solving the task that do not involve representation learning. For reward shaping, we compare with VIME [@vime], an exploration method based on novelty bonuses. For hierarchical RL, we compare with option critic [@ppooc] and an on-policy adaptation of HIRO [@hiro]. We also compare to model-based reinforcement learning with MPC [@nagabandi], a method which explicitly learns and uses environment dynamics, as compared to how ARC implicitly learns the dynamics. Because sample complexity of model-based and model-free methods differ, all results with model-based reinforcement learning indicate final performance.
To ensure a fair comparison between the methods, we provide the *same* information and trajectory data that ARC receives to all of the representations in this evaluation. Precisely, each representation is trained on the same dataset of trajectories collected from the goal-conditioned policy, ensuring that each representation receives data from the full state distribution and meaningful transitions. More details are in Section \[appendix:trainingrep\_details\].
Analysis of Learned Actionable Representations {#sec:exps_analysis}
----------------------------------------------
[0.16]{} ![Visualization of ARC for 2D navigation. The states in the environment are colored to help visualize their position in representation space. For the wall task, points on opposite sides of the wall are clearly separated in ARC space (c). For four rooms, we see that ARCs provide a clear decomposition into room clusters (f), while VAEs do not (e).[]{data-label="fig:analysis_2d"}](analysis/wall_orig.png "fig:"){width="\linewidth"}
[0.16]{} ![Visualization of ARC for 2D navigation. The states in the environment are colored to help visualize their position in representation space. For the wall task, points on opposite sides of the wall are clearly separated in ARC space (c). For four rooms, we see that ARCs provide a clear decomposition into room clusters (f), while VAEs do not (e).[]{data-label="fig:analysis_2d"}](analysis/wall_vae.png "fig:"){width="\linewidth"}
[0.16]{} ![Visualization of ARC for 2D navigation. The states in the environment are colored to help visualize their position in representation space. For the wall task, points on opposite sides of the wall are clearly separated in ARC space (c). For four rooms, we see that ARCs provide a clear decomposition into room clusters (f), while VAEs do not (e).[]{data-label="fig:analysis_2d"}](analysis/wall_actionable.png "fig:"){width="\linewidth"}
[0.16]{} ![Visualization of ARC for 2D navigation. The states in the environment are colored to help visualize their position in representation space. For the wall task, points on opposite sides of the wall are clearly separated in ARC space (c). For four rooms, we see that ARCs provide a clear decomposition into room clusters (f), while VAEs do not (e).[]{data-label="fig:analysis_2d"}](analysis/fourroom_orig.png "fig:"){width="\linewidth"}
[0.16]{} ![Visualization of ARC for 2D navigation. The states in the environment are colored to help visualize their position in representation space. For the wall task, points on opposite sides of the wall are clearly separated in ARC space (c). For four rooms, we see that ARCs provide a clear decomposition into room clusters (f), while VAEs do not (e).[]{data-label="fig:analysis_2d"}](analysis/fourroom_vae.png "fig:"){width="\linewidth"}
[0.16]{} ![Visualization of ARC for 2D navigation. The states in the environment are colored to help visualize their position in representation space. For the wall task, points on opposite sides of the wall are clearly separated in ARC space (c). For four rooms, we see that ARCs provide a clear decomposition into room clusters (f), while VAEs do not (e).[]{data-label="fig:analysis_2d"}](analysis/fourroom_actionable.png "fig:"){width="\linewidth"}
We analyze the structure of ARC space for the tasks described in Section \[sec:domains\], to identify which factors of state ARC chooses to emphasize, and how system dynamics affect the representation.
To examine the 2D navigation tasks, we visualize the original state and learned representations in Figure \[fig:analysis\_2d\]. In both tasks, ARC reflects the dynamics: points close by in Euclidean distance in the original state space are distant in representation space *when they are functionally distinct*. For instance, there is a clear separation in the latent space where the wall should be, and points on opposite sides of the wall are much further apart in ARC space (Figure \[fig:analysis\_2d\]) than in the original environment and in the VAE representation. In the room navigation task, the passages between rooms are clear bottlenecks, and the ARC representation separates the rooms in representation space according to these bottlenecks.
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The representations learned in more complex domains, such as wheeled or legged locomotion and block manipulation, also show meaningful patterns. We aim to understand which elements of state are being emphasized by the representation, and which are being suppressed. To do so, we analyze how distances in the latent space change as we perturb different elements of state. (Fig \[fig:analysis\_all\]). We determine two factors in a state: one which we consider important for decision making (in orange), and one which is secondary (in purple). We expect a good representation to have a larger variation in distance as we perturb the important factor than when we perturb the secondary factor. For instance, in the legged locomotion environment, the CoM is the important factor and the joint angles are secondary. As we vary the CoM, the representation should vary significantly, while the effect should be muted as we vary the joints. For the wheeled environment, position of the car should cause large variations while the orientation should be secondary. For the object pushing, we’d expect block position to be the important factor, while end-effector position is secondary. Since distances in high-dimensional representation space are hard to visualize, we project \[ARC, VAE, State\] representations of perturbed states into 2 dimensions (Fig \[fig:analysis\_all\]) using multi-dimensional scaling (MDS) [@mds]. MDS is a technique that projects points from a high dimensional space into 2-D space such that Euclidean distances are preserved. From Fig \[fig:analysis\_all\], we see that ARC captures the factors of interest; as the important factor is perturbed the representation changes significantly (spread out orange points), while when the secondary factor is perturbed the representation changes minimally (close together purple points). This implies that for Ant, ARC captures CoM while suppressing joint angles; for wheeled, ARC captures position while suppressing orientation; for block pushing, ARC captures block position, suppressing arm movement. Both VAE representations and original state space are unable to capture this.
Leveraging Actionable Representations for Reward Shaping {#sec:exps_rewardshaping}
--------------------------------------------------------
ARCs can be used for reward shaping to solve tasks that present a large exploration challenge with sparse reward functions. We evaluate this on two challenging exploration tasks for wheeled locomotion and legged locomotion (seen in Fig \[fig:reward\_shaping\]). We acquire an ARC from a goal-conditioned policy in the region $\states$ where the CoM is within a 2m square. The learned representation is then used to guide learning via reward shaping for learning a goal-conditioned policy on a larger region $\mathcal{S}'$, where the CoM is within a square of 8m. The task is to reach arbitrary goals in $\mathcal{S}'$, but with only a *sparse* goal completion reward, so exploration is challenging. As described in Section \[sec:downstream\], we consider shaping the reward with a term corresponding to distance between the representation of the current and desired state: $r(s, g) = r_{\text{sparse}} - \| \phi(s) - \phi(g)\|_2$. To ensure fairness, all comparisons initialize from the goal-conditioned policy on small region $\mathcal{S}$ and train on the same data. Further details on the experimental setup for this domain can be found in Appendix \[appendix:rewardshaping\_details\].
![Learning new tasks with reward shaping in representation space. ARC representations are more effective than other methods, and match the performance of a hand-specified shaping. []{data-label="fig:reward_shaping"}](figures/wheeled_generalization.png "fig:"){width="0.19\linewidth"} ![Learning new tasks with reward shaping in representation space. ARC representations are more effective than other methods, and match the performance of a hand-specified shaping. []{data-label="fig:reward_shaping"}](graphs/generalization/wheeled.pdf "fig:"){width="0.3\linewidth"} ![Learning new tasks with reward shaping in representation space. ARC representations are more effective than other methods, and match the performance of a hand-specified shaping. []{data-label="fig:reward_shaping"}](figures/ant_generalization.png "fig:"){width="0.19\linewidth"} ![Learning new tasks with reward shaping in representation space. ARC representations are more effective than other methods, and match the performance of a hand-specified shaping. []{data-label="fig:reward_shaping"}](graphs/generalization/ant.pdf "fig:"){width="0.3\linewidth"}
The plots in Fig \[fig:reward\_shaping\] show learning curves for the goal-conditioned policy using shaping from a variety of learned representations and alternative methods. ARC demonstrates a faster learning speed and better asymptotic performance over all compared methods, when all are initialized from the goal conditioned policy trained on the small region. This is likely because the ARC representation explicitly optimizes for a meaningful distance in latent space, whereas other representations do not directly optimize for this structure. The performance of ARC is similar to a hand-designed reward shaping corresponding to distance in COM space, further corroborating Figure \[fig:analysis\_all\] that ARC considers CoM to be important. ARC and the predictive model outperform VIME, which uses a novelty-based exploration strategy without considering environment dynamics, indicating that dynamics-aware representations are performant over those that are not.
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Leveraging Actionable Representations as Features for Learning Policies {#sec:exps_vf}
-----------------------------------------------------------------------
We consider using the ARC representation as a feature space for learning policies for a quadruped ant robot task which requires the agent to reach a target (shown in green in Fig \[fig:vf\_result\]) *while avoiding* a dangerous region (shown in red in Fig \[fig:vf\_result\]). Specifically, instead of learning a policy on state features $\pi(a|s)$, we learn a policy using the representation as features $\pi(a|\phi(s))$ for a representation $\phi$. The reward function for this task and other experimental details are noted in Appendix \[appendix:valuefn\_details\]. It is important to note that this task cannot be solved directly by a goal-conditioned policy (GCP), and a GCP attempting to reach the specified goal will go straight through the dangerous region and receive a reward of -760, far below the reward for ARC.
Figure \[fig:vf\_result\] shows that although all the methods ultimately learn how to solve the task, policies using ARC features learn much faster. The ARC policy solves the task with high success by Iteration 100, when the next best performing method has learned to reach the goal only 5% of the time. We attribute this to ARC emphasizes those elements of the state that are important for multi-timestep control, and not greedy features used for reconstruction or one-step prediction. Features which emphasize elements important for control make learning easier because they reduce redundancy and noise in the input, and allows the RL algorithm to effectively assign credit. We further note that other representation learning methods learn only as fast as the original state representation. We also provide a comparison to a model-based MPC controller [@nagabandi], but it does not perform well. Planning with learned models is difficult and often fails to match asymptotic performance of model-free methods. It is important to note that the same representation can be used to quickly train many different tasks, amortizing the cost of training a GCP.
Building Hierarchy from Actionable Representations {#sec:exps_hierarchy}
--------------------------------------------------
We consider using ARC representations to control high-level controllers for learning temporally extended navigation tasks in room and waypoint navigation settings, as described in Section \[sec:downstream\]. In the multi-room environments, the agent must navigate through a sequence of 50 rooms in order, receiving a sparse reward when it enters the correct room. In waypoint navigation, the ant must reach a sequence of waypoints in order with a similar sparse reward. These tasks are illustrated in Fig \[fig:task\_hierarchy\], and are described in detail in Appendix \[appendix:state\_hrl\_details\].
[R]{}[0.22]{} {width="0.5\linewidth"} {width=".6\linewidth"}
We evaluate two different strategies for hierarchical reasoning with ARCs (as discussed in Section \[sec:downstream\_hierarchy\]): commanding by specifying a target point or a target cluster picked from a $k$-means clustering in representation space. We train a high-level controller $\pi_h$ with TRPO which outputs as actions either a direct point in the latent space $z_h$ which are decoded into goals $g_h$, or a cluster index from which a goal $g_h$ is uniformly sampled from and decoded. For each step of the high-level policy, a low-level controller executes the goal-conditioned policy with goal $g_h$ for 50 timesteps. Exact specifications and details are in Appendix \[appendix:state\_hrl\_details\] and \[appendix:cluster\_hrl\_details\].
![Comparison on hierarchical tasks. ARCs perform significantly better than other representation methods, option-critic, and commanding goals in state space[]{data-label="fig:hierarchy_results"}](graphs/hierarchy/biased_rooms_cluster.pdf "fig:"){width="0.25\linewidth"} ![Comparison on hierarchical tasks. ARCs perform significantly better than other representation methods, option-critic, and commanding goals in state space[]{data-label="fig:hierarchy_results"}](graphs/hierarchy/wheeled_rooms_cluster.pdf "fig:"){width="0.25\linewidth"} ![Comparison on hierarchical tasks. ARCs perform significantly better than other representation methods, option-critic, and commanding goals in state space[]{data-label="fig:hierarchy_results"}](graphs/hierarchy/ant_waypoint_latent.pdf "fig:"){width="0.25\linewidth"}
As shown in Fig \[fig:hierarchy\_results\], using a hierarchical meta-policy with ARCs performs significantly better than those using alternative representations. These representations are unable to capture abstraction and environment dynamics. For multi-rooms, ARC clusters very clearly capture different rooms (Fig \[fig:analysis\_2d\]), so commanding in cluster space reduces redundancy in action space, allowing for effective exploration. ARC likely works better than commanding goals in spaces learned by other representation learning algorithms, because the learned ARC space is more structured for high-level control, which makes search and clustering much easier. Semantically similar states like different rooms end up in the same cluster, which reduces burden for the high level planning for ARC. We compared to learning from scratch via TRPO, and standard HRL methods such as option critic [@ppooc] and an on-policy adaptation of HIRO [@hiro] which sets full state goals for a low-level GCP. TRPO and option-critic fail to make progress on any of the tasks, because they do not use a goal-conditioned policy. This failure emphasizes the task difficulty and indicates that a goal-conditioned policy trained on simple reaching tasks can be re-used to solve long-horizon problems. Commanding in ARC space is better than in state space using HIRO because state space has redundancies which makes search challenging. Importantly, the methods which are able to learn are ones which use the learned goal conditioned policy with a meta-controller. These methods are all trained using the same information, and perform much better than alternatives like TRPO and Option-Critic, with ARC outperforming all other methods.
Discussion
==========
In this work, we introduce ARC, which capture representations of state important for decision making. We build on the framework of goal-conditioned RL to extract state representations that emphasize features of state that are functionally relevant. The learned state representations are implicitly aware of the dynamics, and capture meaningful distances in representation space. ARCs are useful for tasks such as learning policies, HRL and exploration. While ARC are learned by first training a goal-conditioned policy, learning this policy using off-policy data is a promising direction for future work. Interleaving the process of representation learning and learning of the goal-conditioned policy promises to scale ARC to more general tasks.
#### Acknowledgements
This research was supported by Berkeley DeepDrive, Honda, an ONR Young Investigator Program Award, Google, and computational resources from Amazon. Abhishek Gupta was supported by an NSF Graduate Research Fellowship. We thank Pim de Haan, Aviv Tamar, Vitchyr Pong, and Ignasi Clavera for helpful insights and discussions.
Experimental Details
====================
Training the Goal-Conditioned Policy {#appendix:gcp_details}
------------------------------------
We train a stochastic goal-conditioned policy $\pi(\cdot | s,g)$ using TRPO with an entropy regularization term, where the goal space $\mathcal{G}$ coincides with the state space $\mathcal{S}$. In every episode, a starting state and a goal state $s,g \in \mathcal{S}$ are sampled from a uniform distribution on states, with a sparse reward given of the form below, where $\epsilon$ is task-specific, and listed in the table below.
$$r(s,g) = \begin{cases}
0 & \|s-g\|_{\infty} > \epsilon\\
\eps - \|s-g\|_{\infty} & \|s-g\|_{\infty} < \eps
\end{cases}$$
For the Sawyer environment, although this sparse reward formulation can learn a goal-conditioned policy, it is highly sample inefficient, so in practice we use a shaped reward as detailed in Appendix \[sec:appendix\_task\_descriptions\]. For all of the other environments, in the free space and rooms environments, the goal-conditioned policy is trained using a sparse reward.
The goal-conditioned policy is parameterized as $\pi_{\theta}(a \vert s,g) \sim \mathcal{N}(\mu_{\theta}(s,g), \Sigma_{\theta})$. The mean, $\mu_{\theta}(\cdot,~\cdot)$ is a fully-connected neural network which takes in the state and the desired goal state as a concatenated vector, and has three hidden layers containing 150, 100, and 50 units respectively. $\Sigma$ is a learned diagonal covariance matrix, and is initially set to $\Sigma = I$.
[Navigation]{} [Wheeled Navigation]{} [Ant Navigation]{} [Sawyer Pushing]{}
---------------------------------- ---------------- ------------------------ -------------------- -------------------- -- --
State/Goal Space Dimension 2 6 15 6
Action Space Dimension 2 2 7 3
Sparse Reward Threshold ($\eps$) 0.1 0.5 1 0.3
\# Trajectories per Iteration 100 200 250 500
\# Steps in Trajecotry 50 100 200 100
\# Iterations 200 1000 2000 2000
Entropy Penalty 1 1 0.1 0.1
Learning Rate .01 .02 .02 .01
Training the Representation {#appendix:trainingrep_details}
---------------------------
After training a goal-conditioned policy $\pi$ on the specified region of interest, we collect 500 trajectories each of length 100 timesteps, where each trajectory starts at an arbitrary start state, going towards an arbitrary goal state, selected exactly as the goal-conditioned policy was trained in Appendix \[appendix:gcp\_details\]. This dataset was chosen to be large enough so that the collected dataset has full coverage of the entire state space. Each of the representation learning methods evaluated is trained on this dataset, which means that each learning algorithm receives data from the full state space, and witnesses meaningful transitions between states $(s_t,s_{t+1})$.
We evaluate ARCs against representations minimizing reconstruction error (VAE, slowness) and representations performing one-step prediction (predictive model, inverse dynamics). For each representation, each component is parametrized by a neural network with ReLU activations and linear outputs, and the objective function is optimized using Adam with a learning rate of $10^{-3}$, holding out 20% of the trajectories as a validation set. We perform coarse hyperparameter sweeps over various hyperparameters for all of the methods, including the dimensionality of the latent state, the size of the neural networks, and the parameters which weigh the various terms in the objectives. The exact objective functions for each representation are detailed further in Appendix \[sec:appendix\_representation\_details\].
Reward Shaping {#appendix:rewardshaping_details}
--------------
We test the reward shaping capabilities of the learned representations with a set of navigation tasks on the `Wheeled` and `Ant` tasks. A goal-conditioned policy $\pi$ is trained on a $n \times n$ meter square of free space, and representations are learned (as specified above) on trajectories collected in this small region. We then attempt to generalize to an $m \times m$ meter square (where $m >> n$), and consider the set of tasks of reaching an arbitrary goal in the larger region: a start state and goal state are chosen uniformly at random every episode. The environment setup is identical to that in Appendix \[appendix:gcp\_details\], although with a larger region, and policy training is done the same with two distinctions. Instead of training with the sparse reward $r_{sparse}(s,g)$, we train on a “shaped” surrogate reward $$r_{shaped,\phi}(s,g) = r_{sparse}(s,g) - \alpha \|\phi(s) - \phi(g)\|_2$$
where $\alpha$ weights between the euclidean distance and the sparse reward terms. Second, the policy is initialized to the parameters of the original goal-conditioned policy $\pi$ which was previously trained on the small region to help exploration. As a heuristic for the best possible shaping term, we compare with a “hand-specified” In addition to reward shaping with the various representations, we also compare to a dedicated exploration algorithm, VIME [@vime], which also uses TRPO as a base algorithm. Understanding that different representation learning methods may learn representations with varying scales, we performed a hyperparameter sweep on $\alpha$ for all the representation methods. For VIME, we performed a hyperparameter sweep on $\eta$. The parameters used for TRPO are exactly those in Appendix \[appendix:gcp\_details\], albeit for 3000 iterations.
Features for Policies {#appendix:valuefn_details}
---------------------
We test the ability of the representation to be used as features for a policy learning some downstream task within the `Ant` environment. The downstream task is a “reach-while-avoid” task, in which the Ant requires the quadruped robot to start at the point $(-1.5,-1.5)$ and reach the point $(1.5,1.5)$ while avoiding a circular region centered at the origin with radius $1$ (all units in meters). Letting $d_{goal}(s)$ be the distance of the agent to $(1.5,1.5)$ and $d_{origin}(s)$ to be the distance of the agent to the origin, the reward function for the task is
$$r(s) = -d_{goal}(s) - 4*\mathbf{1}\{d_{origin}(s) < 1\}$$
For any given representation $\phi(s)$, we train a policy which uses the feature representation as input as follows. We use TRPO to train a stochastic policy $\pi(a|\phi(s))$, which is of the form $\mathcal{N}(\mu_{\theta}(\phi(s)), \Sigma_{\theta})$. The mean is a fully connected neural network which takes in the representation, and has two layers of size 50 each, and $\Sigma$ is a learned diagonal covariance matrix initially set to $\Sigma=I$. Note that gradients do not flow through the representation, so only the policy is adapted and the representation is fixed for the entirety of the experiment.
Hierarchical Reinforcement Learning in Latent Space {#appendix:state_hrl_details}
---------------------------------------------------
We provide comparisons on using the learned representation to direct a goal-conditioned policy for long-horizon sequential tasks. In particular, we consider a waypoint reaching task for the Ant, in which the agent must navigate to a sequence of $50$ target locations in order: $\{(x_1,y_1), (x_2,y_2), \dots (x_{50}, y_{50})\}$. The agent receives as input the state of the ant and the checkpoint number that it is currently trying to reach (encoded as a one-hot vector). When the agent gets within $0.5$m of the checkpoint, it receives $+1$ reward, and the checkpoint is moved to the next point, making this a highly sparse reward. Target locations are sampled uniformly at random from a $8 \times 8$ meter region, but are fixed for the entirety of the experiment.
We consider learning a high-level policy $\pi_h(z_h \vert s)$ which outputs goals in latent space, which are then executed by a goal-conditioned policy as described in Appendix \[appendix:gcp\_details\]. Specifically, when the high-level policy outputs a goal in latent space $z_h$, we use a reconstruction network $\psi$, which is described below, to receive a goal state $g_h = \psi(z_h)$. The goal-conditioned policy executes for $50$ timesteps according to $\pi(a \vert s, g_h)$. The high-level policy is trained with TRPO with the reward being equal to the sum of the rewards obtained by running the goal-conditioned policy for every meta-step. We parametrize the high-level policy $\pi_h(z_h \vert s)$ as having a Gaussian distribution in the latent space, with the mean being specified as a MLP with two layers of 50 units and Tanh activations, and the covariance as a learned diagonal matrix independent of state.
To allow the latent representation $z$ to provide commands for the goal-conditioned policy, we separately train a reconstruction network $\psi$ which minimizes the loss function $\E_{s}[\|\psi(\phi(s)) - s\|_{2}]$. For any latent $z$, we can now use $\psi(z)$ as an input into the goal-conditioned policy. Note that an alternative method of providing commands in latent space is to train a new goal-conditioned policy $\pi_{\phi}$, which is trained to minimize the loss $\E_{s,g}[D_{KL}(\pi_{\phi}(\cdot | s, \phi(g)) \| \pi(\cdot \vert s,g))]$, however to maintain abstraction between the representation and the goal-conditioned policy, we choose the former approach.
Hierarchical Reinforcement Learning in Cluster Space {#appendix:cluster_hrl_details}
----------------------------------------------------
We provide comparisons on using the learned representation to direct a goal-conditioned policy in **cluster space**, as described in Section \[sec:downstream\]. We consider navigation through *a sequence of rooms in order* in the rooms and wheeled rooms environment, as visualized in Figure \[fig:tasks\]. A sequence of 50 checkpoints are sampled uniformly from the four rooms with the extra constraint that the same room is never repeated two checkpoints in a row (that is, each checkpoint is chosen to be any of the four rooms), and held fixed for the entirety of the experiment. The agent is tasked with going through these rooms in order, receiving a $+1$ reward every time it enters the appropriate room. The policy receives as input the state of the agent, and which number checkpoint the agent is currently trying to reach (encoded as a 50-dimensional one-hot vector).
After having learned a representation $\phi$ using some set of trajectory data, as described in Appendix \[appendix:trainingrep\_details\], we run $k$-means clustering on states in the trajectory data to cluster latent states in the representation into $k$ components. We then consider learning a high-level policy $\pi_h(c_h|s)$ which outputs a cluster between $\{1 \dots k\}$. Given a cluster number $c_h$ from the high-level policy, the low-level policy samples a latent state $z_h$ uniformly from the cluster, and then proceeds to command a learnt goal-conditioned policy exactly as described in Appendix \[appendix:state\_hrl\_details\].
Specifically, we learn a high-level policy of the form $\pi_h(c_h \vert s) \sim \text{Categorical}(p_{\theta}(s))$ using TRPO where the probabilities for each cluster are specified by a neural network $\pi_{\theta}$ which has two layers of 50 units each, with Tanh activations, and a final Softmax activation to normalize outputs into the probability simplex.
We performed hyperparameter sweeps over $k$ - the number of clusters - for each representation method.
Benchmark Representations {#sec:appendix_representation_details}
=========================
We provide the loss functions that are used to train each of the representations evaluated in our work. All representations are trained on a dataset of trajectories $\mathcal{D} = \{\tau_{i}\}_{i=1}^n$. We use the notation $s \sim \mathcal{D}$ to denote sampling a state uniformly at random from a trajectory uniformly at random from the dataset. We use the notation $s_t,s_{t+1} \sim \mathcal{D}$ to denote sampling a state and the state right after it according to the same uniform sampling scheme.
- **ARC** - After precomputing $D_{act}$:a matrix of actionable distances, we train a neural network $\phi$ to minimize $$D_{act}(s_i, s_j) = \E_{s \sim \mathcal{D}}\left[D_{KL}(\pi(a|s,s_i) \| \pi(a|s,s_j))\right]$$ $$\mathcal{L}(\phi) = \E_{s \sim \mathcal{D}}\left[\E_{s' \sim \mathcal{D}}\left[\|\|\phi(s) - \phi(s')\| - D_{act}(s,s')\|^2\right]\right]$$
- **VAE** [@vae] - Given $q_{\phi}(z|x) = \mathcal{N}(\mu_{\phi}(x), \sigma_{\phi}(x))$, $p_{\theta}(x|z) = \mathcal{N}(\psi_{\theta}(z), I)$, and $p(z) = \mathcal{N}(0,I)$ $$\mathcal{L}(\phi,\theta) = \E_{s \sim \mathcal{D}}\left[\E_{z \sim q_{\phi}(x)} \left[\log p_{\theta}(x|z) - \beta D_{KL}(q_{\theta}(z|x) \| p(z)\right]\right]$$
Here $\mu_{\phi}, \sigma_{\phi}, \psi_{\theta}$ are all neural networks, and $\beta$ is a tunable hyperparameter. The log-likelihood term is equivalent to minimizing mean squared error.
- **Slowness** [@jonschbrock] - Given $q_{\phi}(z|x) = \mathcal{N}(\mu_{\phi}(x), \sigma_{\phi}(x))$, $p_{\theta}(x|z) = \mathcal{N}(\psi_{\theta}(z), I)$, and $p(z) = \mathcal{N}(0,I)$ $$\mathcal{L}(\phi,\theta) = \E_{(s_t,s_{t+1}) \sim \mathcal{D}}\left[\E_{z \sim q_{\phi}(s_t)} \left[\log p_{\theta}(s_t|z) - \beta D_{KL}(q_{\theta}(z|x) \| p(z)) - \alpha\|\mu_{\theta}(s_{t+1} - \mu_{\theta}(s_t)\| \right]\right]$$
Here $\mu_{\phi}, \sigma_{\phi}, \psi_{\theta}$ are all neural networks, and $\alpha, \beta$ are tunable hyperparameters. The log-likelihood terms are equivalent to minimizing mean squared error.
- **Predictive Model** [@acvp] - Given $z = \phi(s_{t})$, $\hat{z}_{t+1} = f(z,a)$ and $\psi(z) = \hat{s}$, we $$\mathcal{L}(\phi,f,\psi) = \E_{(s_t,a_t,s_{t+1}) \sim \mathcal{D}}\left[\|s_{t+1} - \psi(f(\phi(s_t), a_t))\|_2^2\right]$$ where $\phi$ is the learnt representation,$f$ a model in representation space, and $\psi$ a reconstruction network retrieving are all neural networks.
- **Inverse Model** [@burda18] - Given $z_t = \phi(s_t), \hat{z}_{t+1} = f(z_t,a), \hat{a}_{t+1} = g(z_{t},z_{t+1})$
$$\mathcal{L}(\phi, f, g) = \E_{(s_t,a_t,a_{t+1}) \sim \mathcal{D}}\left[\|a_t - g(\phi(s_t),\phi(s_{t+1}))\|_2^2 + \beta \|\phi(s_{t+1}) - f(\phi(s_{t}),a_t)\|_2^2\right]$$
Here, $\phi$ is the learnt representation, $f$ is a learnt model in the representation space, and $g$ is a learnt inverse dynamics model in the representation space. $\beta$ is a hyperparameter which controls how forward prediction error is balanced with inverse prediction error.
Task Descriptions {#sec:appendix_task_descriptions}
=================
- **2D Navigation** This environment consists of an agent navigating to points in an environment, either with a wall as in Figure \[fig:tasks\]a or with four rooms, as in Figure \[fig:tasks\]b. The state space is 2-dimensional, consisting of the Cartesian coordinates of the agent. The agent has acceleration control, so the action space is 2-dimensional. Downstream tasks for this environment include reaching target locations in the environment and navigating through a sequence of 50 rooms.
- **Wheeled Navigation** This environment consists of a car navigating to locations in an empty region, or with four rooms, as illustrated in Figure \[fig:tasks\]. The state space is 6-dimensional, consisting of the Cartesian coordinates, heading, forward velocity, and angular velocity of the car. The agent controls the velocity of both of its wheels, resulting in a 2-dimensional action space. Goal-conditioned policies are trained within a $3 \times 3$ meter square.
Downstream tasks for wheeled navigation include reaching target locations in the environment, navigating through sequences of rooms, and navigating through sequences of waypoints.
- **Ant** This task requires a quadrupedal ant robot navigating in free space.The state space is 15-dimensional, consisting of the Cartesian coordinates of the ant, body orientation as a quaternion, and all the joint angles of the ant. The agent must use torque control to control it’s joints, resulting in an 8-dimensional action space. Goal conditioned policies are trained within a $2 \times 2$ meter square.
Downstream tasks for the ant include reaching target locations in the environment, navigating through sequences of waypoints, and reaching target locations while avoiding other locations.
- **Sawyer** This environment involves a Sawyer manipulator and a freely moving block on a table-top. The state space is 6-dimensional, consisting of the Cartesian coordinates of the end-effector of the Sawyer, and the Cartesian coordinates of the block. The Sawyer is controlled via end-effector position control with a 3-dimensional action space.
Because training a goal-conditioned policy takes an inordinate number of samples for the Sawyer environment, we instead use the following shaped reward to train the GCVF where $h(s)$ is the position of the hand and $o(s)$ is the position of the object
$$r_{shaped}(s,g) = r_{sparse}(s,g) - \|h(s) - o(s)\| - 2\|o(s) - o(g)\|$$
Hyperparameter Tuning
=====================
We perform hyperparameter tuning on three ends: one to discover appropriate parameters for each representation for each environment which are then held constant for the experimental analysis, then on the downstream applications, to choose a scaling factor for reward shaping (see Appendix \[appendix:rewardshaping\_details\]), and to choose the number of clusters for the hierarchical RL experiments in cluster space (see Appendix \[appendix:cluster\_hrl\_details\]).
To discover appropriate parameters for each representation for the legged and wheeled locomotion environments, we evaluate representations on the downstream reward-shaping task, performing a hyperparameter sweep on latent dimension and the parameters which weigh the various terms in the representation learning objectives. We keep the network architecture fixed for each representation and each task. We emphasize carefully here that the ARC representation requires *no parameters* to tune beyond the size of the latent dimension, and we perform a hyperparameter sweep on the penalty terms to ensure that other methods aren’t improperly penalized. On the size of the latent dimension, we sweep over $\{2,3,4\}$ for wheeled locomotion and $\{3,5,7,9,11\}$ for the ant. For the relative weighting for the penalty terms for the comparison representations (defined by $\beta$ in Appendix \[sec:appendix\_representation\_details\]), we evaluate possible values $\beta \in \{4^{-2},4^{-1},1,4^1,4^2\}$. These representations are then fixed and used for all the downstream applications.
For reward shaping, we tune the relative scales between the sparse reward and the shaping term, (denoted by $\alpha$ in Appendix \[appendix:rewardshaping\_details\]) over possible values $\alpha \in \{1,4^1,4^2,4^3,4^4\}$ for each representation on both the legged and wheeled locomotion environments. Tuning for $\alpha$ is required because the representations may have different latent dimensions and different scales, and chose to perform this hyperparameter sweep instead of adding a term to the representation learning objectives to ensure uniformity in scale. For performing $k$-means clustering on the HRL cluster experiments, we sweep over possible values $k \in \{4,5,6,7,8\}$ for each representation on the room navigation tasks for 2D and wheeled navigation, but however found that most representations were robust to choice of the number of clusters.
[^1]: dibya.ghosh@berkeley.edu
|
---
abstract: 'In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the idea of parameterizing homotopies by $\boxx$; the projective line with respect to its compactifying logarithmic structure at infinity. Hodge cohomology of log schemes is an example of an $\boxx$-invariant theory that is representable in the category of logarithmic motives. This bears a resemblance to Voevodsky’s category of motives and $\A^{1}$-invariant theories. Assuming resolution of singularities, we identify the latter with the full subcategory comprised of $\A^{1}$-local objects in the category of logarithmic motives. Palpable properties such as $\boxx$-homotopy invariance, Mayer-Vietoris for coverings, and a symmetric monoidal structure witness the robustness of the setup. Moreover, we show that logarithmic motives satisfy fundamental properties such as a projective bundle theorem, a blow-up distinguished triangle, and a Gysin distinguished triangle.'
address:
- |
Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano\
Via Cesare Saldini 50, 20133 Milano, Italy
- |
Institut für Mathematik, Universität Zürich\
Winterthurerstr. 190, 8057 Zürich, Switzerland
- 'Department of Mathematics, University of Oslo, Niels Henrik Abels hus, Moltke Moes vei 35, 0851 Oslo, Norway'
author:
- Federico Binda
- Doosung Park
- 'Paul Arne [Ø]{}stv[æ]{}r'
bibliography:
- 'biblogmotives.bib'
title: Triangulated categories of logarithmic motives over a field
---
Introduction
============
Overview
--------
Logarithmic structures and log geometry developed by Deligne, Faltings, Fontaine, Illusie, Kato, Tsuji, and many others [@Ogu] can be thought of as an enlargement of algebraic geometry. Having its origin in the theory of logarithmic differentials, log schemes make precise the notion of schemes with a boundary and have implications, inter alia, for moduli spaces, deformation theory, and $p$-adic Hodge theory. The main purpose of this work is to build a theory of motives in the setting of logarithmic algebraic geometry.
Depending on the point of view there are several desiderata for a theory of motives of log schemes. For starters one should find a way to linearize log schemes such that the values of standard cohomology theories, e.g., crystalline, de Rham-Witt, and Hodge cohomology, occur as hom-groups in the same linear tensor category. One can hope that log motives give rise to new invariants for logarithmic schemes, as well as a better understanding of “old” invariants for usual schemes. For sanity checks one should have available various realization functors for log motives and their regulator maps. By equipping schemes with trivial log structures one should be able to recover motives in the sense of algebraic geometry. Finally, an advanced understanding of the basic properties should involve the formalism of six operations. In this work we begin this exploration by constructing the triangulated category of effective log motives $\ldmeff$, extending Voevodsky’s category $\dmeff$, where $\Lambda$ is a commutative unital ring.
There have been a number of constructions of triangulated categories of motives in algebraic geometry, including Levine [@LevineMixedMotives] and Voevodsky [@MR1764202], which together with motivic homotopy theory have led to Voevodsky’s spectacular proofs of the Milnor and Bloch-Kato conjectures [@voevodsky-milnor], [@voevodsky-BK]. The fact that homotopies are parametrized by the affine line restricts the theory to $\A^{1}$-homotopy invariant phenomena. Since there are many nontrivial étale $\Z/p$-coverings of the affine line, where $p$ is the characteristic of the base field $k$, this restriction limits the reach of the theory. Indeed, the derived category $\mathbf{DM}_{\text{\'et}}(k,\Z/p)$ of étale motives with mod-$p$ coefficients is trivial [@AyoICM].
In Voevodsky’s construction of motivic categories, one of the first problems is to find an algebraic avatar for the unit interval $[0,1]\subset \mathbb{R}$ in topology. In our approach to log motives we encounter the same problem. One option is to proceed with the affine line $\A^{1}$ equipped with the trivial log structure, see [@Howellthesis], [@ParThesis]. There is, however, another less obvious choice: in the log geometric setting it is natural to compactify the affine line, i.e., replace it by the projective line $\P^{1}$ pointed at infinity, denoted by $\boxx= (\P^1,\{\infty\})$. A priori this is not a unit interval in the traditional sense since the multiplication map on $\A^{1}$ extends only to a rational map $$\P^{1}\times\P^{1}\dashrightarrow\P^{1}.$$ This pronounced difference is also noticeable in the theory of algebraic cycles with moduli conditions [@BS].
Let us write $lSm/k$ for the category of fine and saturated log schemes that are log smooth and separated over a fixed field $k$ equipped with the trivial log structure. To carry out our constructions we turn $lSm/k$ into a Grothendieck site by means of the [*dividing Nisnevich topology*]{}. The coverings in this topology arise from distinguished squares of different origins: [*dividing squares*]{}, typically involving a log blow-up or the star subdivision of a monoid, and [*strict Nisnevich squares*]{}, an adaption of the Nisnevich topology to log geometry. Conceptually one can view the strict Nisnevich coverings as a witness for the scheme-theoretic data and the dividing coverings as a witness for the log structures. The properties of these topologies are used throughout in the proofs of our results.
Another fundamental ingredient in the theory of log motives is the notion of [*finite log correspondences*]{} in $lSm/k$. With this in hand we define the additive category $lCor/k$ of finite log correspondences over $k$, having the same objects as $lSm/k$ and with morphisms given by finite log correspondences. There is a faithful embedding $lSm/k\hookrightarrow lCor/k$ induced by taking the graph of morphisms in $lSm/k$. Passing to the underlying schemes yields a functor to the category $Cor/k$ of finite correspondences defined by Suslin-Voevodsky [@VSF]. Moreover, the compositions in $lCor/k$ and $Cor/k$ are compatible in the obvious sense.
Let $\Lambda$ be a commutative unital ring. A [*presheaf of $\Lambda$-modules with log transfers*]{} on $lSm/k$ is a contravariant additive functor from $lCor/k$ to $\Lambda$-modules. By the linear setup every presheaf equipped with log transfers determines “wrong way” maps between $\Lambda$-modules parametrized by finite log correspndences. As in the case of usual schemes, the structure sheaf $\mathcal{O}$ and the sheaf of global units $\mathcal{O}^{\times}$ are basic examples of presheaves with log transfers. For any log scheme $(X,\mathcal{M}_{X})$ in $lSm/k$ the group completion $\mathcal{M}^{gp}$ is also a presheaf with log transfers.
\[A.0.2\] Suppose $Z=n_1Z_1+\cdots+n_rZ_r$ is an effective Cartier divisor on a scheme $X$ over $k$. Let $I_i$ be the invertible sheaf of ideals defining $n_iZ_i$ for each irreducible component $Z_i$. Then $(X,Z)$ is an fs log scheme over $k$ whose log structure is given by the Deligne-Faltings structure [@Ogu §III.1.7] associated to the inclusions $I_i\rightarrow \mathcal{O}_X$, $i=1,\ldots,r$. In particular, for the projective line, we set $$\label{equation:box}
\boxx:=(\P^1,\{\infty\})\in lSm/k.$$
For most part of this paper we use the multiplicity one case $n_1=\cdots=n_r=1$. In this case the log structure of $(X,Z)$ is the compactifying log structure [@Ogu §III.1.6] associated to the open immersion $X-Z\rightarrow X$.
\[exm:box\] Following Definition \[A.0.2\] in the case of a normal strict crossing divisor, we set $$\boxx^{n}
=
((\P^1)^{n},\underbrace{\infty\times\P^{1}\times\cdots\times\P^{1}}_{n}
+
\underbrace{\P^{1}\times\infty\times\P^{1}\times\cdots\times\P^{1}}_{n}
+
\underbrace{\P^{1}\times\cdots\times\P^{1}\times\infty}_{n}).$$ This is the $n$th self product of $\boxx$ for the standard monoidal structure on $lSm/k$. The notation means that on $(\P^1)^{n}$ we consider the strict normal crossing divisor given by setting the coordinate $z_{i}=\infty$ for $i=1,\ldots n$.
We note there are canonically defined morphisms $$(\A^1,\emptyset)
\to
\boxx
\to
(\P^1,\emptyset)$$ between distinctly different objects in $lSm/k$. We emphasize that $\boxx$ has a nontrivial log structure.
The triangulated category of effective log motives of $k$ with $\Lambda$-coefficients $$\ldmeff$$ is obtained from the category of chain complexes of presheaves with log transfers on $lSm/k$ by imposing descent with respect to all dividing Nisnevich coverings, and homotopy invariance with respect to $\boxx$. To every fs log scheme $X\in lSm/k$ we can associate the motive $$M(X)\in \ldmeff.$$ This construction is our main object of study in this work.
Notwithstanding the technicalities, the next four statements are straightforward consequences of the construction of $\ldmeff$.
- [(Monoidal structure)]{} For every $X,Y\in lSm/k$ there is a naturally induced isomorphism of log motives $$M(X\times Y)\cong M(X)\otimes M(Y).$$
- [(Homotopy invariance)]{} For every $X\in lSm/k$ there is a naturally induced isomorphism of log motives $$M(X\times \boxx)\cong M(X).$$
- [(Mayer-Vietoris)]{} For every strict Nisnevich distinguished square in $lSm/k$ $$\begin{tikzcd}
Y'\arrow[d]\arrow[r]&Y\arrow[d]\\X'\arrow[r]&X
\end{tikzcd}$$ there is a naturally induced homotopy cartesian square of log motives $$\begin{tikzcd}
M(Y')\arrow[d]\arrow[r]&M(Y)\arrow[d]\\
M(X')\arrow[r]&M(X).
\end{tikzcd}$$
- [(Log modification)]{} Every log modification $f:Y\rightarrow X$ of fs log schemes log smooth over $k$ in the sense of Kato [@FKato] induces an isomorphism of log motives $$M(f)\colon M(Y)\cong M(X).$$
To an fs log scheme $X\in lSm/k$ and a vector bundle $\xi\colon \cE\to X$ we associate the Thom motive $$MTh_{X}(\cE)
\in
\ldmeff$$ via the blow up of $\cE$ along its $0$-section. We refer to Definitions \[A.5.7\], \[df:logvectorbundle\], and \[A.3.22\] for precise statements. The construction of Thom motives turns out to be an important one for several reasons. One should note that the Betti realization of the log motivic Thom space of $\xi\colon\cE\to X$ is homotopy equivalent to the quotient of the unit disk bundle by the unit sphere bundle for the Betti realization of $\xi\colon\cE\to X$. In the presence of a Euclidean metric the latter is one formulation of Thom spaces in topology.
We employ motivic Thom spaces of log vector bundles to show a Gysin isomorphism in Theorem \[A.3.36\]. A closely related result is the Gysin triangle for strict normal crossing divisors.
- [(Gysin triangle)]{} Suppose $Z,Z_1,\ldots,Z_r$ are smooth divisors forming a strict normal crossing divisor on $X\in lSm/k$, and set $Y:=(X,Z_1+\cdots+Z_r)$. Let $E$ be the exceptional divisor of the blow-up $B_Z Y$ of $Y$ along $Z$, and let $N_ZY$ denote the normal bundle of $Z$ in $Y$. Then there is a distinguished triangle $$M(B_Z Y,E)\rightarrow M(Y)\rightarrow MTh(N_Z Y)\rightarrow M(B_Z Y,E)[1]$$ of log motives in $\ldmeff$.
In Theorem \[A.3.13\] we show the following blow-up exact triangle.
- [(Blow-up triangle)]{} Let $X$ be a smooth scheme over $k$, and let $X'$ be the blow-up of $X$ along a smooth center $Z$. Then there is a distinguished triangle $$M(Z\times_X X')\rightarrow M(X')\oplus M(Z)\rightarrow M(X)\rightarrow M(Z\times_X X')[1]$$ of log motives in $\ldmeff$.
In $\A^1$-homotopy theory, the affine space $\A^{n}$ can be viewed as a disk. Similarly, $\boxx$ in Definition \[A.0.2\] and more generally $\boxx^{n}$ in Example \[exm:box\] can be viewed as a disk in our setting. For our purposes, however, it is also natural to consider the pair of projective spaces $(\P^{n},\P^{n-1})$, where $\P^{n-1}$ is considered as a hyperplane in $\P^{n}$.
In Proposition \[A.6.1\] we prove the following fundamental invariance for log motives.
- [($(\P^{n},\P^{n-1})$-invariance)]{} For every $X\in lSm/k$ there is a naturally induced isomorphism of log motives $$M(X\times (\P^{n},\P^{n-1}))\cong M(X).$$
Following the convention in [@MR1764202] we set $$\Lambda(0):=M(\Spec k)=\Lambda,$$ and define the Tate object $\Lambda(1)$ to be the shifted cone $$\Lambda(1)
:=
M(\Spec k\stackrel{i_0}\rightarrow \P^1)[-2]$$ in $\ldmeff$. Here $i_0:\Spec k \rightarrow \P^1$ is the $0$-section, and both schemes are equipped with a trivial log structure. Moreover, for $n\in\N$, we form the derived tensor product $$\Lambda(n)
:=
\underbrace{\Lambda(1)\otimes \cdots \otimes \Lambda(1)}_{n}.$$ More generally, the $n$th Tate twist of an object $M\in\ldmeff$ is defined as $$M(n)
:=
M\otimes \Lambda(n).$$
In Theorem \[A.6.2\] we conclude a projective bundle theorem for the projectivization of log vector bundles. We refer the reader to Definition \[A.3.8\] for the precise meaning of resolution of singularities over fields as in [@MR2289519].
- [(Projective bundle theorem)]{} Assume that $k$ admits resolution of singularities. Let $\cE$ be a log vector bundle of rank $n+1$ over $X\in lSm/k$. In $\ldmeff$ there is a canonical isomorphism $$M(\P(\cE))
\cong
\bigoplus_{i=0}^{n} M(X)(i)[2i].$$
As a consequence of the projective bundle theorem we can relate Thom motives to Tate twists via the Thom isomorphism, see Theorem \[A.6.7\].
- [(Thom isomorphism)]{} Assume that $k$ admits resolution of singularities. Let $\cE$ be a log vector bundle of rank $n$ over $X\in lSm/k$. In $\ldmeff$ there is a canonical isomorphism $$MTh(\cE)
\cong
M(X)(n)[2n].$$
Associated to any log structure on a scheme there is a corresponding notion of a boundary enabling us to think about log schemes as being analogous to manifolds with boundaries. In our setting, the boundary corresponds to an effective Cartier divisor on the underlying scheme.
\[A.0.1\] For an fs log scheme $X$ over $k$, let $\partial X$ denote the set of points of $X$ with a nontrivial log structure. That is, $x\in\partial X$ if the stalk $\overline{\cM}_{X,x}$ of the characteristic monoid $$\overline{\cM}_{X}:=\cM_{X}/\cM_{X}^{\ast}$$ at $x\in X$ is nontrivial.
We note that $\partial X$ is a closed subset of $X$ according to [@Ogu Proposition III.1.2.8] so that $X-\partial X$ is an open subscheme of $X$. If $X\in lSm/k$, then $X-\partial X$ is smooth over $k$ and $\partial X$ is an effective Cartier divisor on $X$. The following fundamental admissible blow-up property telling us how the motive of $X$ depends on the boundary $\partial X$ is shown in Theorem \[A.3.12\].
- [(Admissible blow-ups)]{} Assume that $k$ admits resolution of singularities. Let $f:Y\rightarrow X$ be a proper morphism of fs log schemes that are log smooth over $k$. If the naturally induced morphism $$Y-\partial Y\rightarrow X-\partial X$$ is an isomorphism of $k$-schemes, then there is a naturally induced isomorphism $$M(Y)\rightarrow M(X)$$ of log motives in $\ldmeff$.
In Section \[section:cvvmotives\] we compare $\ldmeff$ with Voevodsky’s triangulated category of effective motives $\dmeff$ introduced in [@MR1764202] and reviewed in [@MR2181839]. For proper smooth schemes we are able to identify certain hom objects in $\ldmeff$ and $\dmeff$. More precisely, Theorem \[A.4.10\] shows the following useful comparison result.
- [(Comparison with Voevodsky’s derived category of effective motives)]{} Assume that $k$ admits resolution of singularities. Let $X$ and $Y$ be fs log schemes that are log smooth over $k$. If $X$ is proper over $k$, then for every $i\in \Z$ there is a naturally induced isomorphism of abelian groups $$\hom_{\ldmeff}(M(Y)[i],M(X))
\cong
\hom_{\dmeff}(M(Y-\partial Y)[i],M(X-\partial X)).$$
As an immediate application we can identify the endomorphism ring of the unit object in $\ldmeff$ with the coefficients $$\label{equation:endomorphismringofunit}
\hom_{\ldmeff}(\Lambda(0),\Lambda(0))
\cong
\Lambda.$$
Assuming $k$ admits resolution of singularities we show there is an adjoint functor pair $$\omega_\sharp:
\ldmeff
\rightleftarrows
\dmeff:
R\omega^*.$$
Theorem \[A.4.14\] shows that the right adjoint functor $R\omega^*$ is fully faithful. However, as witnessed by the sheaf of logarithmic differentials $\Omega^{i}_{/k}$, the functor $R\omega^*$ is not essentially surjective.
While $\dmeff$ admits a fully faithful embedding into $\ldmeff$, the latter also contains interesting invariants that are not $\A^1$-invariant. This is one way of justifying our choice of $\boxx$ as the preferred unit interval in $\ldmeff$.
To identify the essential image of $R\omega^*$ we follow [@MR1764202] and declare that $$\cF\in\ldmeff$$ is [*$\A^1$-local*]{} if for every $X\in lSm/k$ and $i\in\Z$ the projection $X\times \A^1\rightarrow X$ induces an isomorphism of abelian groups $$\hom_{\ldmeff}(M(X)[i],\cF)
\rightarrow
\hom_{\ldmeff}(M(X\times \A^1)[i],\cF).$$ Theorem \[A.4.22\] shows that every $\A^1$-local effective log motive is in the essential image of $R\omega^*$. That is, Voevodsky’s category of derived motives $\dmeff$ is equivalent to the [*$\A^1$-localization*]{} of $\ldmeff$.
Let $\ldmeffprop$ be the smallest triangulated subcategory of $\ldmeff$ that is closed under small sums and contains every motive $M(X)$, where $X\in lSm/k$ is proper as a scheme over $\Spec{k}$. We identity $\ldmeffprop$ with the essential image of $R\omega^*$ under the assumption that $k$ admits resolution of singularities. This shows there is an equivalence of triangulated categories $$\label{eq:equivpropintro}
\ldmeffprop\simeq \dmeff,$$ which gives another log geometric description of Voevodsky’s category $\dmeff$. Note that the equivalence *does not hold* with integral coefficients if we replace the dividing Nisnevich topology with its étale counterpart. See Remark \[Etalemot.5\] and \[ssec:specul\] below for a discussion about this.
Outline of the paper
--------------------
In what follows we give a more detailed overview of our work.
### Correspondences on logarithmic schemes
In Section \[sec:logtransfers\] we introduce the notion of *finite log correspondences*. Our approach is fairly classical: informally, an elementary finite log correspondence $Z$ from $X$ to $Y$ (smooth log schemes over $k$) is the datum of an elementary finite correspondence in the sense of Suslin-Voevodsky between the underlying subschemes $\ul{X}$ and $\ul{Y}$ that is equipped with the minimal log structure necessary to ensure the projection $Z\to X$ is a strict morphism. However, to get a morphism to the product $X\times Y$ in the category of log schemes we require the existence of a morphism from the normalization $Z^N$ to $Y$ as part of the data. The said condition on the normalization of an elementary finite log correspondence is reminiscent of the modulus condition on correspondences discussed in [@KSY2]. We work with the normalizations in order to establish a well-defined associative composition of finite log correspondences and turn $lCor/k$ into a category. In turn, this requires a certain amount of technicalities involving solid log schemes; a notion with strong permanence inherited from $X$ and $Y$ by the log structure on the normalization $Z^N$ of the elementary finite log correspondence. We refer the reader to Section \[sec:logtransfers\] (more specifically Definition \[A.5.2\]) for details.
### cd structures on log schemes and sheaves with log transfers
Suppose $X$ is a log scheme with log structure given by an open immersion $j\colon U\to X$ corresponding to the complement of an effective Cartier divisor $\partial X$. Then the motive of $X$ is expected to help us understand the cohomology of $U$. To achieve this we need to impose a suitable invariance of the motives under the choice of the “compactification” $X$. The solution we offer involves the notion of *a dividing cover*, i.e., a surjective proper log étale monomorphisms $f\colon Y\to X$ (see Definition \[A.5.14\] and [@Par]).
Intuitively, we can think of such morphisms as blow-ups with centre in the boundary $\partial X$. We also consider strict Nisnevich covers, i.e., Nisnevich covers coming from the underlying schemes, These two types of coverings conspire into a cd-structure in the sense of [@Vcdtop], and we call the associated topology the *dividing Nisnevich topology*. As discussed in Section \[sec:topologies\], the dividing Nisnevich cd-structure is not bounded. For this reason we need to generalise Voevodsky’s results to *quasi-bounded density structures*, which is the main technical notion of the section (see Definition \[A.9.65\]). Theorem \[A.10.4\] proves that for any quasi-bounded, regular and complete cd structures on a category $\mathcal{C}$ the associated cohomology groups vanish above the *density dimension* of any object. Corollary \[A.10.10\] allows us to define a suitable descent structure in the sense of [@CD09].
In Section \[sec.sheavestransfer\] we study how finite log correspondences relate to topologies. Our main findings can be summarized in the following combination of Proposition \[A.8.7\], Proposition \[A.5.22\], Proposition \[A.8.8\] and Theorem \[A.5.23\] (see below for a reminder on Kummer étale maps).
\[thm-intro\] Let $t$ be one of the following topologies on $lSm/k$: strict or dividing Nisnevich, strict or dividing étale, Kummer étale, or log étale.
- The topology $t$ is compatible with log transfers on $lSm/k$.
- The category $\mathbf{Shv}_{t}^{\rm ltr}(k,\Lambda)$ is a Grothendieck abelian category.
- For every complex of $t$-sheaves $\mathcal{F}$ with log transfers and for every $X\in lSm/k$, there is a canonical isomorphism $$\hom_{\Deri(\Shvltrtkl)}(a_t^*\Zltr(X),\cF[i])
\cong
\bH_t^i(X,\cF).$$
The dividing topology on $lSm/k$ is not subcanonical (like the $h$-topology on $Sm/k$). However, sections of the sheafification of the representable presheaf $a_{t}^*\Zltr(X)$ of any $X$ (here $t$ is one of the dividing topologies on $lSm/k$) can be described in terms of algebraic cycles. To that end, we introduce the notion of *dividing log correspondences* between fs log schemes in Definition \[A.5.47\], which up to log modifications can be viewed as finite log correspondences in the sense of Definition \[A.5.2\]. In fact, it turns out that the resulting categories of sheaves are equivalent, see Proposition \[A.5.56\].
Dividing log correspondences allow us to further restrict the class of log schemes that is needed to build $\ldmeff$. For $t=dNis$, $d\acute{e}t$, $l\acute{e}t$, the results in Section \[Subsec::Sheaves.SmlSm\] give an equivalence of categories $$\label{eq-smlsmintro}
\iota^*
\colon
\Shv_{t}^{\rm ltr}(k,\Lambda)
\xrightarrow{\simeq}
\Shv_{t}^{\rm ltr}(SmlSm/k,\Lambda).$$ Here $\iota\colon SmlSm/k\hookrightarrow lSm/k$ is the full subcategory of $lSm/k$ consisting of fs log schemes of the form $(X,\partial X)$, where $X$ is smooth over $k$ and $\partial X$ is a strict normal crossing divisor on $X$ (see Lemma \[lem::SmlSm\]). This gives a way of computing dividing cohomology groups, see Theorem \[Div.4\].
Let $\cF$ be a bounded below complex of strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaves on $SmlSm/k$. Then for every $X\in SmlSm/k$ and integer $i\in \Z$ there is an isomorphism $$\begin{aligned}
\bH_{dNis}^i(X,a_{dNis}^*\cF)&\cong \colimit_{Y\in X_{div}^{Sm}}\bH_{sNis}^i(Y,\cF)
\\
\text{(resp.\ }\bH_{d\acute{e}t}^i(X,a_{d\acute{e}t}^*\cF)&\cong \colimit_{Y\in X_{div}^{Sm}}\bH_{s\acute{e}t}^i(Y,\cF),
\\
\text{resp.\ }\bH_{l\acute{e}t}^i(X,a_{l\acute{e}t}^*\cF)&\cong \colimit_{Y\in X_{div}^{Sm}}\bH_{k\acute{e}t}^i(Y,\cF)\text{)}.\end{aligned}$$
We remark that the equivalence provides a canonical extension to $lSm/k$ of any sheaf with log transfers a priori defined on $SmlSm/k$.
### Motivic categories and properties of motives
Defining $\ldmeff$ is straightforward at this point. The local objects for the $\boxx$-descent model structure are precisely the complexes of strictly $\boxx$-invariant $t$-sheaves with log transfer (especially for $t=dNis$), and the properties established in Sections \[sec:topologies\]-\[sec.constructldmeff\] automatically give the first set of axioms ($\boxx$-invariance, Mayer-Vietoris, log modification invariance). We present next a leitfaden for the less transparent properties.
First we extend the log modification invariance of motives to blow-ups along smooth centres contained in $\partial X$. This is the content of Section \[ssec:invariancesmoothblowups\], and it is required for the following reason. Like any other smooth curve, the affine line $\A^1$ affords a unique compactification to an object in $lSm/k$, namely $\boxx$. In higher dimensions this construction is no longer canonical. For example, in dimension $2$, the product $\A^1\times \A^1 = \A^2$ can be compactified to $\boxx\times \boxx = \boxx^2$, or to $(\P^2, \P^1)$, where the $\P^1$ is viewed as a line at infinity.
The construction of $\ldmeff$ makes the first choice automatically contractible, but from the construction it is unclear what happens to $(\P^2, \P^1)$. Inspired by the properties of log differentials observed in [@BS §6.2], see also Section \[PnPn-1invariancediff\], it is reasonable to expect that $(\P^2, \P^1)$ is contractible too. In order to compare $\boxx^2$ and $(\P^2, \P^1)$, we consider the blow-up $X$ of $\P^2$ at a point contained in its boundary, $\P^1$. This latter object can more easily be compared with the blow-up of $\P^1\times \P^1$ at the point $(\infty, \infty)$. The map $X\to \P^2$, however, is not a log modification. Our crucial observation is that a combination of Mayer-Vietoris, $\boxx$-invariance, and dividing-invariance properties are in fact enough to establish that this map is in fact an equivalence in $\ldmeff$. See Theorem \[A.3.7\] and Theorem \[A.3.13\] for precise statements. A key point of the proof is to deal with the case of surfaces, see Proposition \[A.3.39\]. This finally allow us to prove the $(\P^n, \P^{n-1})$-invariance property of motives and also the blow-up formula in Proposition \[A.6.1\] and Theorem \[A.3.13\], respectively.
Section \[ssec:ThomMotives\] discusses Thom motives, i.e., the motives of Thom spaces of vector bundles $\mathcal{E}$ in the logarithmic setting. To get the correct homotopy type, the definition we use involves the blow-up of $\mathcal{E}$ along the zero section (and it is therefore quite different from the corresponding notion in Voevodsky’s category). The key technical result here is Proposition \[A.3.42\], in which we show the existence of a canonical isomorphism $$MTh_{X_1}(\cE_1)\otimes MTh_{X_2}(\cE_2)\rightarrow MTh_{X_1\times_S X_2}(\cE_1\times_S \cE_2).$$ for vector bundles $\cE_i$ on $X_i$. This part is quite involved in the log setting compared to the $\A^1$-invariant setting.
Having Thom motives at our disposal, we finally prove the Gysin isomorphism and the existence of Gysin triangles in Theorem \[A.3.36\]. Assuming resolution of singularities, we show an even finer invariance under admissible blow-ups in Section \[ssec:invadmblowrefined\]. The latter is used in our comparison with Voevodsky’s category of motives, from which we deduce the projective bundle formula in Theorem \[A.6.2\] as well as the Thom isomorphism Theorem \[A.6.7\].
In Section \[section:calculus\] we review the construction of a $Sing$ functor, inspired by the Suslin complex, for a symmetric monoidal category $\mathcal{A}$ equipped with a weak interval object $I$. This seemingly detour is necessary since $\boxx$ is not an interval in the sense of [@MV] for $lCor/k$: the graph of the multiplication map $\mu\colon \A^1\times \A^1\to \A^1$ does not extend to an admissible correspondence in our sense.
Our solution, see also [@AdditiveHomotopy §4], is to enlarge the category using calculus of fractions $\mathcal{A}[T^{-1}]$ for a suitable class of maps $T$ in order to include the multiplication for $I$, build the singular functor there, and then restrict back to the original category. Applied to the category of chain complexes $\mathbf{C}(\mathbf{Psh}^{\rm ltr}(k))$, the said construction produces a $\boxx$-local object, albeit not very computationally-friendly. Nevertheless, it plays a key role in the comparison between $\ldmeff$ and $\dmeff$, see Propositions \[A.4.12\] and \[A.4.29\].
By ignoring transfer structures altogether, we present a simpler construction $\mathbf{logDA}^{\rm eff}(k,\Lambda)$ that is a direct analogue of Ayoub’s $\mathbf{DA}^{\rm eff}(k, \Lambda)$ (see [@Ayo07] or [@AyoICM] for a short overview). However, making use of correspondences allow us to prove much stronger results including the comparison with Voevodsky’s $\dmeff$ and the existence of Gysin triangles.
### Applications: Hodge sheaves and cyclic homology
Prototype examples of objects that are not available in Voevodsky’s derived category of motives are the additive group scheme $\mathbb{G}_{a}$ and the differentials $\Omega^i_{-/k}$ (or their absolute counterpart). We show in Section \[sec:Hodge\] that this problem disappears in the log setting (see Corollary \[Pn-invariance of Log differentials\], Theorem \[thm:Omega\_log\_transfer\]).
Suppose $k$ is a perfect field and $X\in lSm/k$. Let $t$ be one of the topologies on log schemes in Theorem \[thm-intro\] or the Zariski topology. Then the assignment $$X\mapsto \Omega^j_{X/k}$$ extends to a $t$-sheaf with log transfers. Moreover, for every $i$, $j$, $n\geq 0$, there is an isomorphism $$H_{t}^i(X\times (\P^n,\P^{n-1}),\Omega_{X\times (\P^n,\P^{n-1})}^j)\cong H_{t}^i(X,\Omega_X^j).$$ In particular, the sheaves $\Omega^j_{-/k}$ are strictly $\boxx$-invariant $t$-sheaves with log transfers.
The transfer structure is obtained by extending to the logarithmic setting the work of Rülling-Chatzistamatiou [@ChatzistamatiouRullingANT]. By extending from $SmlSm/k$ to $lSm/k$ via Construction \[SmlSm.3\], we conclude there exists a natural isomorphism $$\label{eq.introhodge}
\hom_{\ldmeff}(M(X),\Omega^j_{-/k}[i])
\cong H^i_{Zar}(\ul{X},\Omega^j_{X/k}).$$ The fact that Hodge cohomology is representable in $\ldmeff$ can be seen as a refinement of the representability of de Rham cohomology in Voevodsky’s $\mathbf{DM}^{\rm eff}(k)$, see [@lecomtewach].
When the characteristic of the base field $k$ is zero, combining with the Hochschild-Kostant-Rosenberg theorem yields an isomorphism for every $X\in Sm/k$ $$HC_n(X)
\cong
\hom_{{\mathbf{logDM}^{\rm eff}(k, \mathbb{Q})}}(M(X), \bigoplus_{p\in \Z} \Omega^{\leq 2p}_{-/k}[2p-n]).$$ This shows representability of cyclic homology in ${\mathbf{logDM}^{\rm eff}(k, \mathbb{Q})}$, see Section \[ssec-Hodgecyclic\], an example of a non-$\A^{1}$-invariant theory on $Sm/k$ and thus out of reach of $\mathbf{DM}^{\rm eff}(k)$. We plan to address the question of representing the cohomology of Hodge-Witt sheaves in characteristic $p>0$ in a later work.
### Appendices
Besides reminding the reader of some necessary background material in log geometry, the main aim of Appendix A is to minimize confusion in other chapters. Appendix A should be consulted if the reader is in doubt about notation or the precise form of basic log geometric notions. Appendix B gives a concise introduction to model structures on chain complexes, and thereby introduces notation employed elsewhere in the text.
Outlook and future developments
-------------------------------
We conclude the introduction with some remarks and conjectures that put our theory in a broader framework.
### Relationship with reciprocity sheaves
The idea of extending Voevodsky’s framework in order to encompass non-$\A^{1}$-invariant phenomena has been investigated by many authors in the past decade, starting with the work of Bloch-Esnault [@BEAdditiveChow], [@BEAdditive] on additive Chow groups and Chow groups with modulus.
The theory of *reciprocity sheaves* by Kahn-Saito-Yamazaki [@KSY], and the closely related theory of *modulus sheaves* by Kahn-Miyazaki-Saito-Yamazaki [@KSY2], [@KSY3], has provided a unified framework for both $\mathbb{A}^1$-invariant objects as well as interesting non-$\mathbb{A}^1$-invariant ones. The category $\mathbf{RSC}_{\rm Nis}$ of (Nisnevich) reciprocity sheaves is a full subcategory of $\Shv^{\rm tr}_{Nis}(k, \mathbb{Z})$ that contains $\mathbf{HI}_{\rm Nis}$, i.e., $\mathbb{A}^1$-invariant Nisnevich sheaves with transfers but also the additive group scheme $\mathbb{G}_a$, the sheaf of absolute Kähler differentials $\Omega^i$, and the de Rham-Witt sheaves $W_m\Omega^i$. The connection between $\mathbf{RSC}_{\rm Nis}$ and our theory has recently been provided by Saito [@SaitoRSClog].
\[saitothmintro\] There exists a fully faithful exact functor $$\mathcal{L}og
\colon
\mathbf{RSC}_{\rm Nis} \to \Shv_{dNis}^{\rm ltr}(k, \mathbb{Z})$$ such that $\mathcal{L}og(F)$ is strictly $\boxx$-invariant for every $F\in \mathbf{RSC}_{\rm Nis}$. Moreover, for each $X\in Sm/k$, there is a natural isomorphism $$H^i_{Nis}(X, F_X) \cong \hom_{\ldmeff}(M(X),\mathcal{L}og(F)[i]).$$
This shows that Nisnevich cohomology of reciprocity sheaves is representable in $\ldmeff$, giving log motivic interpretations (at least for reduced modulus) of properties such as the Gysin exact sequence, the projective bundle formula, the blow-up formula, and the $(\P^n, \P^{n-1})$-invariance of the cohomology of reciprocity sheaves provided in [@BRS]. This applies, for example, to some of the results on Hodge cohomology discussed in Section \[sec:Hodge\], like $(\P^n, \P^{n-1})$-invariance or the existence of a pushforward, that can be seen, at least for $X\in SmlSm/k$, as a consequence of Theorem \[saitothmintro\] and of the corresponding formulas in [@BRS]. This result is an important step in connecting the theory of motives with modulus to logarithmic motives.
### Conjectures and speculations {#ssec:specul}
Suppose that $\Lambda$ is an $N$-torsion ring, where $N>0$ is coprime to the exponential characteristic of the ground field $k$. Voevodsky proved the following version of Suslin’s rigidity theorem for the small étale site of $k$, see [@MVW Theorem 9.35].
The morphism of sites $\pi\colon k_{\acute{e}t} \to (Sm/k)_{\acute{e}t}$ induces an equivalence $$\pi_\sharp \colon \mathbf{D}(k_{\acute{e}t},\Lambda)\rightarrow \dmeffet.$$
There are many other instances of rigidity theorems comparing small and big étale sites in the motivic context. See [@AyoubEtale] for $\mathbf{DA}_{\acute{e}t}(S, \Lambda)$, [@CDEtale] for motives with transfer, and [@BambVezz] for the rigid analytic setting.
Recall that a morphism of fs log schemes $f\colon X\to Y$ is called Kummer étale if it is exact and log étale, see Proposition \[kummer-is-logetandstrict\]. A typical example is given by $\A^1_k\to \A^1_k$, $t\mapsto t^n$, where $n\geq 1$ and $(n, {\rm char}(k))=1$. Here $\A^1_k$ is endowed with the log structure given by the origin, written $\A_\N = \Spec{k[\N]}$ in the following. Proposition \[ketcomp.4\] shows that every locally constant log étale sheaf on $lSm/k$, see Definition \[ketcomp.3\], admits a unique transfer structure. This allow us to define a functor $$\label{eq:conjrig}
\eta_\sharp
\colon
\mathbf{D}(\Shv(k_{\acute{e}t},\Lambda))
\rightarrow
\ldmefflet,$$ which is easily seen to be fully faithful. On the right-hand side we consider the log étale topology, i.e., the smallest Grothendieck topology finer than the Kummer étale topology and the dividing topology (on the left-hand side there are no non-trivial dividing covers on the small site, so it makes no difference). Conjecture \[conjrigidity\] is an analogue of the Suslin-Voevodsky rigidity theorem.
The functor is an equivalence.
More generally, if $\Lambda$ is a torsion ring of exponent invertible in $k$, we expect that the functors $$\ldmeffet\leftarrow \ldmeffetprop\rightarrow \dmeffet,$$ are equivalences (and similarly in the log étale topology, see Conjecture \[ketcomp.7\]). This expectation is justified by the fact that examples of strictly $\boxx$-invariant complexes of sheaves that are not strictly $\A^1$-invariant, such as $\Omega^i_{-}$ and $W_m\Omega^i_{-}$, are nullified under our assumption on $\Lambda$. This is similar in spirit to [@tor-div-rec Theorem 3.5] and [@MiyazakiCI].
Let us discuss the case when the ground field $k$ has positive characteristic $p>0$. Then the category $\mathbf{DM}_{\acute{e}t}(k, \mathbb{Z})$ of étale motives has trivial $p$-torsion and is in fact a $\mathbb{Z}[1/p]$-linear category. On the other hand, the étale (or Lichtenbaum) motivic cohomology groups with $\Z/p^r$-coefficients are not $\mathbb{A}^1$-invariant, so they cannot be hom-groups in one of Voevodsky’s categories.
Currently there is no interpretation of the category of *integral* étale motivic complexes in terms of algebraic cycles. Milne-Ramachandran [@MRpreprint] proposed a construction of $\mathbf{DM}^{\rm gm}_{\acute{e}t}(k, \mathbb{Z})$ as a dg-pullback of $\mathbf{DM}_{\acute{e}t}(k, \mathbb{Z}[1/p])$ and $\mathbf{D}^b_c(R)$, the derived category of coherent complexes of graded modules over the Raynaud ring $R$, along the rigid realisation functor. The category of étale *log* motives constructed in this work can be a candidate for a cycle-theoretic incarnation of Milne-Ramachandran’s category $\mathbf{DM}^{\rm gm}_{\acute{e}t}(k, \mathbb{Z})$. To support this belief, note that $\mathbf{logDM}_{d\acute{e}t, prop}^{\rm eff}(k, \Z/p^r)$ is nontrivial because $\Z/p\not\cong 0$, while its $\mathbb{A}^1$-invariant counterpart is trivial because of the Artin-Schreier sequence. It would be interesting to compare $\mathbf{logDM}_{d\acute{e}t, prop}^{\rm eff}(k, \Z/p^r)$ with $\mathbf{D}^b_c(R, \Z/p^r)$ via a log-crystalline realisation functor.
Notations and terminology
-------------------------
Every log scheme in this paper is equipped with a Zariski log structure unless otherwise stated. Recall that for every fs log scheme $X$ with an étale log structure, there exists a log blow-up $Y\rightarrow X$ such that $Y$ has a Zariski log structure thanks to work by Niziol [@Niz Theorem 5.10]. Since motives are invariant under log blow-ups, this shows that the restriction to fs log schemes with Zariski log structures is no loss of generality.
Throughout the paper we employ the following notation (see the Index for a more comprehensive list).
------------------------------- --------------------------------------------------------------------------------
$k$ (perfect) field
$Sm/S$ category of smooth and separated schemes of finite type over
a scheme $S$
$lSm/S$ category of log smooth and separated fs log schemes of finite
type over an fs log scheme $S$
$lSch/S$ category of separated fs log schemes of finite type over an fs
log scheme $S$
$\LCor$ category of finite log correspondences over $k$
$SmlSm/S$ the full subcategory of $lSm/S$ consisting of $X$ such that $\underline{X}$ is
smooth over $\underline{S}$
$\Lambda$ commutative unital ring
$\Lambda\text{-}\mathbf{Mod}$ category of $\Lambda$-modules
$\cA$ abelian category
$\Co(\cA)$ category of [unbounded]{} chain complexes of $\cA$
${\bf K}(\cA)$ $\Co(\cA)$ modulo chain homotopy equivalences
$\Deri(\cA)$ derived category of $\cA$
------------------------------- --------------------------------------------------------------------------------
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Joseph Ayoub for reading a preliminary version of this paper and providing helpful suggestions. We also thank Shuji Saito for his interest in this work, a detailed reading of the manuscript, and for sharing the first version of [@SaitoRSClog]. It is a pleasure to acknowledge the generous hospitality and support from Institut Mittag-Leffler in Djursholm (2017), University of Oslo (2017-2019), and Forschungsinstitut f[ü]{}r Mathematik at ETH Zurich (2019) during the various stages of this project. Binda was partially supported by the SBF 1085 “Higher Invariants” at the University of Regensburg and the PRIN “Geometric, Algebraic and Analytic Methods in Arithmetic” at MIUR. [Ø]{}stv[æ]{}r was partially supported by the Friedrich Wilhelm Bessel Research Award from the Humboldt Foundation, a Nelder Visiting Fellowship from Imperial College London, the Professor Ingerid Dal and sister Ulrikke Greve Dals prize for excellent research in the humanities, and a Guest Professorship under the auspices of The Radbound Excellence Initiative. The authors gratefully acknowledge support by the RCN Frontier Research Group Project no. 250399 “Motivic Hopf Equations.”
Finite logarithmic correspondences {#sec:logtransfers}
==================================
Definition of finite log correspondences
----------------------------------------
In this section we introduce the notion of finite log correspondences relative to any base field $k$ equipped with its trivial log structure. [^1] This is an adaptation to the log setting of the original definition of finite correspondences due to Suslin and Voevodsky [@VSF], [@MVW]. As for usual schemes, one can intuitively regard finite log correspondences as multivalued functions, but there are some subtleties arising involving the log structure. Building on this concept we turn $lSm/k$ into an additive category $\LCor$ of finite log correspondences over $k$. The objects of $\LCor$ are fine and saturated log schemes that are log smooth over $k$, i.e., the same objects as in $lSm/k$. As morphisms we take the free abelian group generated by elementary log correspondences (see [@MVW Lecture 1]).
If $X\in lSm/k$ we write $\underline{X}$ for the corresponding underlying scheme.
\[A.5.2\] For $X,Y\in lSm/k$, an [*elementary log correspondence*]{} $Z$ from $X$ to $Y$ consists of
1. an integral closed subscheme $\underline{Z}$ of $\underline{X}\times \underline{Y}$ that is finite and surjective over a connected component of $\underline{X}$, and
2. a morphism $Z^N\rightarrow Y$ of fs log schemes, where $Z^N$ denotes the fs log scheme whose underlying scheme is the normalization of $\underline{Z}$ and whose log structure is induced by the one on $X$. More precisely, if $p:\underline{Z^N}\rightarrow \underline{X}$ denotes the induced scheme morphism, then the log structure $\cM_{Z^N}$ is given as $p_{log}^*\cM_X$.
A [*finite log correspondence*]{} from $X$ to $Y$ is a formal sum $\Sigma n_{i}Z_{i}$ of elementary log correspondences $Z_{i}$ from $X$ to $Y$. Let $\lCor_k(X,Y)$ denote the free abelian group of finite log correspondences from $X$ to $Y$.
The above definitions coincide with the ones in [@VSF] and [@MVW] when $X$ and $Y$ have trivial log structures. We shall omit the subscript $k$ and write $\lCor(X,Y)$ when no confusion seems likely to arise. Note that if $X=\amalg_{i\in I} X_i$ is a decomposition into connected components, then there is a direct sum decomposition $$\lCor(X,Y)
=
\bigoplus_{i\in I} \lCor(X_i,Y).$$
\[A.5.29\]
1. With reference to Definition \[A.5.2\], let $p:\underline{Z^N}\rightarrow \underline{X}$ and $q:\underline{Z^N}\rightarrow \underline{Y}$ denote the induced scheme morphisms. Then condition (ii) is equivalent to the existence of a morphism of log structures $$q_{log}^*\cM_Y\rightarrow p_{log}^*\cM_X$$ on the underlying scheme $\underline{Z^N}$ (see Definition \[A.9.13\]). Indeed, to give a morphism $Z^N\rightarrow Y$, we only need to specify a morphism $q_{log}^*\cM_Y\rightarrow \cM_{Z^N}$. We have that $\cM_{Z^N}=p_{log}^*\cM_X$.
2. Our notion of an elementary log correspondence involves considering a log structure on the normalization of the underling elementary correspondence between the underling schemes. The reason for this extra complication will be explained below, in the construction of compositions of finite log correspondences.
3. Let $V\in \lCor(X,Y)$ be an elementary log correspondence. Then condition (ii) in Definition \[A.5.2\] implies the inclusion $V\cap ((X-\partial X)\times Y) \subset V\cap (X\times (Y\setminus \partial Y))$.
4. Suppose $F/k$ is a finite separable field extension. Then for every $U\in lSm/F$, we have $U\times_k Y\cong U\times_F Y_F$, where $Y_F:=\Spec F\times_k X$. Hence there is a base change isomorphism $$\lCor_k(U,X)
\cong
\lCor_F(U,X_F).$$
\[A.5.31\]
1. Suppose $X$ has a trivial log structure. Then $Z^N$ has a trivial log structure, and $Z^N\rightarrow Y$ factors through $Y-\partial Y$. Hence there is an isomorphism of abelian groups $$\lCor(X,Y)
\cong
\Cor(X,Y-\partial Y).$$ In particular, $\lCor(\Spec k,Y)$ can be identified with the group of $0$-cycles on the open complement $Y-\partial Y$.
2. If $Y$ has trivial log structure, then the condition (ii) is automatically satisfied. It follows that $\lCor(X,Y)\cong\Cor(\underline{X},Y)$. In particular, $\lCor(X,\Spec k)$ is the free group generated by the irreducible components of $X$.
3. By combining (1) and (2) we see that if $X$ and $Y$ have trivial log structures, then $\lCor(X,Y)\cong\Cor(X,Y)$.
4. Let $i:Z\rightarrow X\times Y$ be a closed immersion of fs log schemes, i.e., $\underline{Z}$ is a closed subscheme of $\underline{X}\times \underline{Y}$ and the induced homomorphism $i_{log}^*(\cM_{X\times Y})\rightarrow \cM_Z$ is surjective. In addition, assume that $Z$ is strict (i.e., $p_{log}^*(\cM_X)\cong \cM_Z$, where $p$ is the projection from $X\times Y$ onto $X$), finite, and surjective over a component of $X$. Then the induced morphism $Z\rightarrow Y$ gives rise to a morphism $Z^N\rightarrow Y$. Hence we can consider $Z$ as a finite log correspondence from $X$ to $Y$.
The reader might object that our definition of log correspondence is too restrictive since we require the underlying subscheme $\ul{Z}$ of any elementary log correspondence to be finite over $\ul{X}$ (and not, for example, only over $X-\partial X$, cf. [@KSY2]). This will be “corrected” thanks to the introduction of a suitable topology on log schemes, the dividing topology. By sheafifying the representable presheaf $\lCor(-,X)$ with respect to the dividing topology, we are able to consider correspondences that are finite over $X- \partial X$ and become finite globally after a suitable log modification. This will be discussed in Section \[subsec:divdinglogcor\].
Solid log schemes {#ss:sls}
-----------------
For the notion of finite log correspondences to have any meaning, we need to define a sensible composition of finite log correspondences $V\in \lCor(X,Y)$ and $W\in \lCor(Y,Z)$, where $X,Y,Z\in lSm/k$. To that end it turns out that we need to construct log structures on normalizations of cycles on $\underline{X}\times \underline{Z}$. This can be achieved using cycles on $\underline{X}\times\underline{Y}\times \underline{Z}$ provided that $X$, $Y$, and $Z$ are solid log schemes, as defined below.
\[A.9.1\] Let $X$ be a coherent log scheme over $k$. We say that $X$ is [*solid*]{} if for any point $x\in X$, the induced map $$\Spec {\cO_{X,x}}
\rightarrow
\Spec {\cM_{X,x}}$$ is surjective.
According to [@Ogu Proposition III.1.10.8] this definition of solidness is equivalent to the one given in [@Ogu Definition III.1.10.1(2)].
\[A.9.2\] Let $f:Y\rightarrow X$ be a strict open morphism of coherent log schemes. If $X$ is solid, then $Y$ is solid.
Consider the induced commutative diagram $$\begin{tikzcd}
\Spec{\cO_{Y,y}}\arrow[r]\arrow[d]&\Spec{\cM_{Y,y}}\arrow[d]\\
\Spec{\cO_{X,x}}\arrow[r]&\Spec{\cM_{X,x}}
\end{tikzcd}$$ of topological spaces where $y\in Y$ is a point and $x:=f(y)$. The lower horizontal arrow is surjective since $X$ is solid. The left vertical arrow is surjective by [@EGA IV.1.10.3] since $f$ is open. Moreover, the right vertical arrow is an isomorphism since $f$ is strict. Thus the upper horizontal arrow is surjective.
\[A.9.6\] Let $X$ and $Y$ be log schemes, and let $g:\underline{X}\rightarrow \underline{Y}$ be a morphism of schemes. If the log structure homomorphisms $\alpha_X:\cM_X\rightarrow \cO_X$ and $\alpha_Y:\cM_Y \rightarrow \cO_Y$ are injective, then there is at most one morphism $f:X\rightarrow Y$ of log schemes such that $\underline{f}=g$.
Such a morphism $f$ consists of $\underline{f}:\underline{X}\rightarrow \underline{Y}$ and a commutative diagram $$\begin{tikzcd}
\cM_Y\arrow[d,"\alpha_Y"]\arrow[r,"f^\flat"]&{\underline{f}}_*\cM_X\arrow[d,"{{\underline{f}}_*\alpha_X}"]\\
\cO_Y\arrow[r,"{\underline{f}}^\sharp"]&{\underline{f}}_*\cO_X
\end{tikzcd}$$ of sheaves of monoids on $X$. The vertical arrows are injective by assumption. Thus there are at most one possible $f^\flat$ if $\underline{f}=g$.
\[A.9.9\] Let $\theta:P\rightarrow Q$ be an exact homomorphism of integral monoids. If $\theta^{\rm gp}$ is an isomorphism, then $\theta$ is an isomorphism.
Since $\theta$ is exact, the induced monoid homomorphism $P\rightarrow P^{\rm gp}\times_{Q^{\rm gp}}Q$ is an isomorphism. Thus $\theta$ is an isomorphism since $\theta^{\rm gp}$ is an isomorphism.
\[A.9.4\] Suppose that $X$ is a solid fs log scheme. Then the log structure homomorphism $\alpha_X:\cM_X\rightarrow \cO_X$ is injective.
Assume for contradiction that $\alpha_X$ is not injective. Then there exists a point $x$ of $X$ such that $\alpha_{X,x}:\cM_{X,x}\rightarrow \cO_{X,x}$ is not injective. We choose different elements $p$ and $p'$ of $\cM_{X,x}$ such that $\alpha_{X,x}(p)=\alpha_{X,x}(p')$. Let $P$ be the submonoid of $\cM_{X,x}^{\rm gp}$ generated by $\cM_{X,x}$, $p-p'$, and $p'-p$. Note that $\alpha_{X,x}$ factors through $P$.
Since $X$ is solid, the induced map $\Spec {\cO_{X,x}}\rightarrow \Spec {\cM_{X,x}}$ is surjective. Thus the induced map $\Spec P\rightarrow \Spec {\cM_{X,x}}$ is surjective. Then by [@Ogu Proposition I.4.2.2], the inclusion $\cM_{X,x}\rightarrow P$ is exact, which is an isomorphism by Lemma \[A.9.9\]. This contradicts the assumption that $p$ and $p'$ are different elements.
\[A.9.12\] Let $K$ be a field of characteristic $p>0$. If an element $x$ of the multiplicative group of units $K^*$ has order $r$, then $p$ does not divide $r$.
If $r=pr'$, then $(x^{r'}-1)^p=0$ and hence $x^{r'}=1$. This contradicts the assumption that $x$ has order $r$.
\[A.9.3\] Let $X$ be a fs log scheme log smooth over $k$. Then $X$ is solid.
The question is Zariski local on $X$, so we may assume that $X$ has a neat fs chart $f:X\rightarrow \A_P$ at a point $x\in X$ such that $f$ is smooth by [@Ogu Theorem IV.3.3.1] and Lemma \[A.9.12\]. Then $f$ is an open dominant morphism. Let $F$ be a face of $P$, and let $Y$ be the corresponding strict closed subscheme of $\A_P$. Then $f(X)\cap Y$ is a dense open subset of $Y$, so the image of $f$ contains the generic point $y$ of $Y$.
The prime ideal $P-F$ of $P$ is equal to $g(y)$ for the induced map $g:\A_P\rightarrow \Spec P$. Since $y$ is also in the image of $f$, $P-F$ is in the image of the composite $gf$. This shows that $gf$ is surjective.
\[A.9.18\] Let $X$ be a fs log scheme log smooth over $k$. Then $X$ is normal.
The question is Zariski local on $X$, so we may assume that $X$ has a fs chart $f:X\rightarrow \A_P$ such that $f$ is étale by [@Ogu Theorem IV.3.3.1]. Since $\A_P$ is normal by [@Ogu Proposition I.3.4.1(2)], $X$ is normal due to [@EGA IV.11.3.13].
\[A.9.17\] Let $f:X\rightarrow Y$ be a finite and surjective morphism of fs log schemes. If $Y$ is log smooth over $k$, then $X$ is solid.
According to Lemmas \[A.9.2\] and \[A.9.3\], it suffices to show that $f$ is an open morphism. Note that $f$ is equidimensional by assumption. Since $Y$ is normal by Lemma \[A.9.18\], $f$ is universally open by Chevalley’s criterion ([@EGA IV.14.4.4]).
\[A.9.14\] Let $X$ be a fs log scheme with a neat fs chart $P\rightarrow \cO_X$ at a point $x\in X$. Then $\cM_X\cong \cO_X^*\oplus P$ as sheaves of monoids.
Consider the exact sequence of sheaves of monoids $$0\rightarrow \cO_X^*\rightarrow \cM_X\rightarrow \overline{\cM}_X\rightarrow 0.$$ By assumption, $\overline{\cM}_X\cong P$, and the homomorphism $\cM_X\rightarrow \overline{\cM}_X$ admits a section $P\rightarrow \cM_X$.
\[A.9.11\] Let $f:X\rightarrow Y$ be finite dominant morphism of integral schemes, and let $\alpha_Y:\cM_Y\rightarrow \cO_Y$ be a fs log structure on $Y$. Assume that $\alpha_Y$ is injective. If $Y$ is normal, then the unit of the adjunction $$\cM_Y\rightarrow f_*^{log}f_{log}^*\cM_Y$$ is an isomorphism.
The question is Zariski local on $Y$, so we may assume that $Y$ is an affine scheme $\Spec B$ and that $\cM_Y\cong B^*\oplus P$ by Lemma \[A.9.14\]. Since $f$ is a dominant affine morphism, $\underline{f}=\Spec g$ for some injective homomorphism $g:B\rightarrow A$ of rings. We only need to show that the induced commutative diagram $$\label{A.9.11.1}\begin{tikzcd}
P\oplus B^*\arrow[r]\arrow[d]&B\arrow[d,"g"]\\
P\oplus A^*\arrow[r]&A
\end{tikzcd}$$ of monoids is cartesian.
To that end, let $(p,a)$ be an element of $P\oplus A^*$ such that $pa\in B$. Since $\alpha_Y$ is injective we see that $p\neq 0$ in $B$. Thus $a,a^{-1}\in {\rm Frac}(B)$, and hence $a,a^{-1}\in B$ since $B$ is normal and $f$ is finite. We conclude that $a\in B^*$ and the assertion follows.
\[A.9.16\] Let us give an example showing that the conclusion in Lemma \[A.9.11\] is false when $Y$ is non-normal. We set $X=\Spec A$, $Y=\Spec B$, where $$B:=k[x,y]/(y^2-x^3)_{(x-1)} \text{ and } A:=k[t]_{(t^2-1)}.$$ Set $f=\Spec g$, where $g:B\rightarrow A$ is the inclusion mapping $x$ to $t^2$ and $y$ to $t^3$.
Let $\alpha:\N\oplus B^*\rightarrow B$ be the log structure mapping $(n,f)$ to $x^nf$. Then $x(1+t)$ is in the image of the induced homomorphism $\N\oplus A^*\rightarrow A$. But since $x(1+t)=x+y\in B$ and $1+t\notin B^*$, the diagram of monoids is not cartesian.
Compositions of finite log correspondences
------------------------------------------
Using the observations in Section \[ss:sls\] we are now ready to define the composition of finite log correspondences. We begin with the following Lemma.
\[A.5.9\] Let $X$ and $Y$ be fs log schemes log smooth over $k$, and suppose that $\underline{Z}$ is a closed subscheme of $\underline{X}\times \underline{Y}$. Then there exists at most one elementary log correspondence from $X$ to $Y$ whose underlying scheme is $\underline{Z}$.
Let $Z^N$ be the fs log scheme whose underlying scheme is the normalization $\underline{Z}^N$ of $\underline{Z}$. We equip it with the log structure induced from $X$ via the canonical map $\underline{Z}^N\to \underline{Z}\to \underline{X}$, as in Definition \[A.5.2\] (ii). Let $q:\underline{Z^N}\rightarrow \underline{Y}$ be the induced morphism of schemes. We need to show that there is at most one morphism $r\colon Z^N\rightarrow Y$ of fs log schemes such that $\underline{r}=q$. Lemma \[A.9.3\] shows that $Y$ is solid. Since $Z^N$ is finite over $X$, we see that $Z^N$ is solid on account of Lemma \[A.9.17\]. To conclude we apply Lemmas \[A.9.6\] and \[A.9.4\].
\[A.5.10\] Let $X$ and $Y$ be fs log schemes log smooth over $k$. Then the homomorphism $$(-)^\circ
\colon
\lCor(X,Y)
\rightarrow \
Cor(X-\partial X,Y-\partial Y)$$ of abelian groups mapping $V\in \lCor(X,Y)$ to $V^\circ :=V-\partial V$ is injective.
We may assume $X$ and $Y$ are integral. Let $$V
=
n_1V_1+\cdots+n_rV_r\in \lCor(X,Y)$$ be a finite log correspondence with $r>0$ where $n_i\neq 0$ for every $i$ and the $V_i$’s are distinct. If $V-\partial V=0$, then $V_i-\partial V_i=V_j-\partial V_j$ for some $i\neq j$. Since the closures of $V_i-\partial V_i$ and $V_j-\partial V_j$ in $\underline{X}\times \underline{Y}$ are equal to $\underline{V_i}$ and $\underline{V_j}$ respectively, we have that $\underline{V_i}=\underline{V_j}$. By uniqueness, Lemma \[A.5.9\] implies $V_i=V_j$. Since this contradicts our assumption, we conclude that $V-\partial V\neq 0$.
\[lem:composition\] For every $X,Y,Z\in lSm/k$ and finite log correspondences $\alpha\in \lCor(X,Y)$, $\beta\in \lCor(Y,Z)$, there exists a well-defined composition $\beta\circ \alpha\in\lCor(X,Z)$. The assignment $$(\alpha,\beta)\mapsto \beta\circ \alpha$$ is associative. Under the morphism $(-)^\circ$ of Lemma \[A.5.10\], the composite $\beta\circ \alpha$ is sent to the corresponding composite of finite correspondences.
We may assume $\alpha=[V]$ and $\beta=[W]$ are elementary log correspondences. In that case we can form the finite correspondences $$V-\partial V
\in
\Cor(X-\partial X,Y-\partial Y) \text{ and } W-\partial W\in \Cor(Y-\partial Y,Z-\partial Z).$$ The intersection product $$(V-\partial V\times Z)\cdot (X\times W-\partial W)$$ can be expressed as a formal sum $$n_1U_1+\cdots+n_r U_r$$ of reduced cycles in $X-\partial X\times Y-\partial Y\times Z-\partial Z$. Let $\underline{T_i}$ be the closure of $U_i$ in $\underline{X}\times \underline{Y}\times \underline{Z}$. By Remark \[A.5.29\] (iv), the irreducible components of $$(\underline{V}\times \underline{Z})\cap (\underline{X}\times \underline{W})$$ are precisely the closures of the irreducible components of $$(V-\partial V \times Z- \partial Z) \cap (X-\partial X \times W-\partial W).$$
We claim that $\underline{T_i}$ is finite and surjective over a component of $\underline{X}$. To that end we may assume $X$, $Y$, and $Z$ are irreducible. Since $Y$ is an fs log scheme, log smooth over $k$, $\underline{Y}$ is normal by Lemma \[A.9.18\]. Thus by [@MVW Lemma 1.6], $\underline{T_i}$ is finite and surjective over $\underline{V}$, and so $\underline{T_i}$ is finite and surjective over $\underline{X}$. In particular, we can consider $$[\underline{W}]\circ [\underline{V}] = \sum_{i=1}^r n_i p_*[\underline{T_i}],$$ where $p\colon X\times Y\times Z\to X\times Z$ is the projection, and $$p_*[\underline{T_i}] = [k(\underline{T_i}): k(p(\underline{T_i}))] \cdot p(\underline{T_i}) =: d_i \cdot p(\underline{T_i})$$ is the corresponding pushforward. Note that $p(\underline{T_i})$ is finite and surjective over a component of $\underline{X}$ by [@MVW Lemma 1.4]. It remains to verify that $p(\underline{T_i})$ is admissible in the sense of Definition \[A.5.2\], which is a condition on the log structure.
To simplify our notation, we write $\underline{R_i}$ for the image $p(\underline{T_i})$ in $\underline{X}\times\underline{Z}$. Let $R_i^N$ denote the fs log scheme whose underlying scheme is the normalization of $\underline{R_i}$ and whose log structure is induced by $X$. There is an induced commutative diagram of schemes $$\begin{tikzcd}
&\underline{V^N}\times \underline{W^N}\arrow[ld,"r_1"']\arrow[d,"r_2"]\arrow[rd,"r_3"]\\
\underline{V^N}\arrow[d,"p_1"']\arrow[rd,"q_1"',very near start]&\underline{R_i^N}\arrow[ld,"p_2"',very near start,crossing over]\arrow[rd,"q_2", very near start]
&\underline{W^N}\arrow[ld,"p_3",crossing over, very near start]\arrow[d,"q_3"]\\
\underline{X}&\underline{Y}&\underline{Z}.
\end{tikzcd}$$ By Remark \[A.5.29\] (i), part (ii) in Definition \[A.5.2\] of elementary log correspondences is tantamount to the existence of structure morphisms $$\alpha_{12}
:
(q_1)_{log}^*\cM_Y
\rightarrow
(p_1)_{log}^*\cM_X
\text{ and }
\alpha_{23}:(q_3)_{log}^*\cM_Z
\rightarrow
(p_3)_{log}^*\cM_Y.$$ The composite morphism $\alpha_{123}$ given by $$(q_3r_3)_{log}^*\cM_Z
\stackrel{\alpha_{23}}\longrightarrow
(p_3r_3)_{log}^*\cM_Y
\stackrel{\sim}\longrightarrow
(q_1r_1)_{log}^*\cM_Y
\stackrel{\alpha_{12}}\longrightarrow
(p_1r_1)_{log}^*\cM_X$$ can be rewritten as $$\alpha_{123}
\colon
(q_2r_2)_{log}^* \cM_Z
\to
(p_2 r_2)_{log}^* \cM_X.$$ Since $\underline{R^N}$ is finite and surjective over a component of $\underline{X}$, and $X$ is log smooth over $k$ by assumption, $R^N$ is solid by Lemma \[A.9.17\]. Thus by Lemma \[A.9.4\], the log structure map $\alpha_{R_i^N}$ is injective. Since $\underline{R_i}^N$ is normal we apply Lemma \[A.9.11\] to deduce there exists a unique morphism $$\alpha_{13}
\colon
(q_2)_{log}^*\cM_Z
\rightarrow
(p_2)_{log}^*\cM_X$$ of log structures on $\underline{R_i^N}$ such that $(r_2)_{log}^*\alpha_{13}=\alpha_{123}$. Thus the pair $(R_i^N,\alpha_{13})$ forms an elementary log correspondence $R_i$ from $X$ to $Z$.
Owing to the above we obtain the finite log correspondence $$[W]\circ [V]
:=
\sum_{i=1}^r n_i d_i[R_i]
\in
\lCor(X,Z).$$ By construction, there is a commutative diagram $$\label{equation1:lem:composition}
\begin{tikzcd}
\lCor(Y,Z)\times \lCor(X,Y)\arrow[d]\arrow[r,"(-)\circ(-)"]&\lCor(X,Z)\arrow[d]\\
\Cor(Y-\partial Y,Z-\partial Z)\times \Cor(X-\partial X,Y-\partial Y)\arrow[r,"(-)\circ(-)"]&\Cor(X-\partial X,Z-\partial Z)
\end{tikzcd}$$ where the vertical morphisms are induced by the homomorphism $(-)^\circ$ in Lemma \[A.5.10\]. Since the vertical morphisms are injective by Lemma \[A.5.10\], the associative law for the composition follows from the case of finite correspondences (for references see [@MVW Lecture 1]).
For every $X\in lSm/k$, the *identity* ${\rm id}_X\in \lCor(X,Y)$ is the finite log correspondence associated with the diagonal embedding $X\rightarrow X\times X$.
For every $X,Y\in lSm/k$ and finite log correspondence $\alpha\in \lCor(X,Y)$ we have that $${\rm id}_Y \circ \alpha
=
\alpha
=
\alpha \circ {\rm id}_X.$$
Owing to Lemma \[A.5.10\] we reduce to the case when $X=X-\partial X$ and $Y=Y-\partial Y$, i.e., $X$ and $Y$ are usual schemes and we are done by [@MVW Lecture 1].
\[A.5.3\] Let $\LCor$ denote the category with objects $lSm/k$ and with morphisms given by finite log correspondences between fs log schemes log smooth over $k$. Composition of morphisms are defined in Lemma \[lem:composition\]. This is an additive category with zero object $\emptyset$ (the empty log scheme) and coproduct $\amalg$ (disjoint union).
\[A.5.35\]
1. For $X,Y,Z\in SmlSm/k$, the composition $$\Cor(Y,X)\times \Cor(Z,Y)\rightarrow \Cor(X,Z)$$ in [@MVW §1] coincides with the above composition.
2. If $f\colon X\to Y$ is a morphism in $lSm/k$, its graph $\Gamma_f$ defines an element of $\lCor(X,Y)$. Note that $\underline{X}$ is normal by Lemma \[A.9.18\], and $\underline{\Gamma_f}$ is isomorphic to $\underline{X}$ via the projection morphism.
3. If $W\in \lCor(Y,Z)$ is an elementary log correspondence, then $W\circ \Gamma_f$ is obtained by the compositions $$\underline{X}\times_{\underline{Y}}\underline{W}\rightarrow \underline{X}\times_{\underline{Y}}(\underline{Y}\times \underline{Z})
\cong
\underline{X}\times \underline{Z},\; (W^N\times_Y X)^N\rightarrow W^N\times_Y X\rightarrow X\times Z.$$
4. If $V\in \lCor(U,X)$ is an elementary log correspondence, then $\Gamma_f\circ V$ is obtained by the compositions $$\underline{V}\rightarrow \underline{U}\times \underline{X}\rightarrow \underline{U}\times \underline{Y},\;
V^N\rightarrow U\times X\rightarrow U\times Y.$$
\[A.5.16\] For $X,Y\in SmlSm/k$, there is a canonical homomorphism $$\underline{(-)}:\lCor(X,Y)\rightarrow \Cor(\underline{X},\underline{Y})$$ mapping any elementary log correspondence $V$ to the elementary correspondence $\underline{V}$. This homomorphism is well defined, since the underlying closed subscheme $\ul{V}$ of an elementary log correspondence $V$ is finite and surjective over (a component of) $\ul{X}$ by Definition \[A.5.2\](i).
\[A.5.17\] For $X,Y,Z\in SmlSm/k$, there is a commutative diagram $$\begin{tikzcd}
\lCor(Y,Z)\times \lCor(X,Y)\arrow[r,"(-)\circ (-)"]\arrow[d,"\underline{(-)}\circ \underline{(-)}"']& \lCor(X,Z)\arrow[d,"\underline{(-)}"]\\
\Cor(\underline{Y},\underline{Z})\times \Cor(\underline{X},\underline{Y})\arrow[r,"(-)\circ (-)"]&\Cor(\underline{X},\underline{Z}).
\end{tikzcd}$$
Let $u:X-\partial X\rightarrow X$ and $v:Y-\partial Y\rightarrow Y$ be the structure open immersion. Since $\Cor(-,\underline{Z})$ is an étale sheaf by [@MVW Lemma 6.2], $$(-)\circ \Gamma_{\underline{u}}:\Cor(\underline{X},\underline{Z})\rightarrow \Cor(X-\partial X,\underline{Z})$$ is injective. Hence it suffices to check the commutativity of $$\begin{tikzcd}
\lCor(Y,Z)\times \lCor(X,Y)\arrow[r,"(-)\circ (-)"]\arrow[d,"\underline{(-)}\times (\underline{(-)}\circ \Gamma_{\underline{u}})"']& \lCor(X,Z)\arrow[d,"\underline{(-)}\circ \Gamma_{\underline{u}}"]\\
\Cor(\underline{Y},\underline{Z})\times \Cor(X-\partial X,\underline{Y})\arrow[r,"(-)\circ (-)"]&\Cor(X-\partial X,\underline{Z}).
\end{tikzcd}$$
Due to the commutativity of $$\begin{tikzcd}[column sep=large]
\lCor(Y,Z)\times \lCor(X,Y)\arrow[d,"\underline{(-)}\times (\underline{(-)}\circ \Gamma_{\underline{u}})"']\arrow[r,"(\underline{(-)}\circ \Gamma_{\underline{v}})\times (-)^\circ"]
&\Cor(Y-\partial Y,\underline{Z})\times \Cor(X-\partial X,Y-\partial Y)\arrow[d,"(-)\circ (-)"]\\
\Cor(\underline{Y},\underline{Z})\times \Cor(X-\partial X,\underline{Y})\arrow[r,"(-)\circ (-)"]&\Cor(X-\partial X,\underline{Z}),
\end{tikzcd}$$ it suffices to check the commutativity of $$\begin{tikzcd}
\lCor(Y,Z)\times \lCor(X,Y)\arrow[r,"(-)\circ (-)"]\arrow[d,"(\underline{(-)}\circ \Gamma_{\underline{v}})\times (-)^\circ"']& \lCor(X,Z)\arrow[d,"\underline{(-)}\circ \Gamma_{\underline{u}}"]\\
\Cor(Y-\partial Y,\underline{Z})\times \Cor(X-\partial X,Y-\partial Y)\arrow[r,"(-)\circ (-)"]&\Cor(X-\partial X,\underline{Z}).
\end{tikzcd}$$ Since the left (resp. right) vertical morphism factors through $$\Cor(Y-\partial Y,Z-\partial Z)\times \Cor(X-\partial X,Y-\partial Y)
\textrm{ (resp.\ }\Cor(X-\partial X,Z-\partial Z)),$$ we are done by the commutativity of .
We conclude this section by defining the monoidal structure on $lCor/k$.
\[A.5.36\] The tensor product of $X,Y\in\lCor/k$ is given by the fiber product of fs log schemes $$X\otimes Y
:=
X\times_k Y.$$ If $V$ is an elementary log correspondence from $X$ to $X'$ and $W$ is an elementary log correspondence from $Y$ to $Y'$, then $V\times_k W$ defines an elementary log correspondence from $X\otimes Y$ to $X'\otimes Y'$. The tensor product defines a symmetric monoidal structure on $lCor/k$.
Topologies on fine saturated logarithmic schemes {#sec:topologies}
================================================
We want our construction of motives for fs log schemes to depend on intrinsic local data for logarithmic structures. To achieve this we need to provide an appropriate Grothendieck topology on our ground category categories $lSm/S$ and $lSch/S$, where throughout this section $S$ is a finite dimensional noetherian base fs log scheme. To establish the basic desiderata for log motives we are basically forced to consider a suitable version of the Nisnevich topology for logarithmic schemes called the *strict Nisnevich topology*. We also introduce a new topology in $lSch/S$ which we call the *dividing topology*. The coverings in the dividing topology include log modifications in the sense of F. Kato [@FKato]. The dividing topology arise from a cd-structure in the sense of Voevodsky [@Vcdtop] called the *dividing cd-structure*.
The general theory of cd-structures was developed by Voevodsky [@Vcdtop]. Suppose that $S$ is a scheme. Basic examples are the Zariski and Nisnevich topologies on $Sm/S$, the cdh-topology on $Sch/S$, and their generalizations to algebraic stacks. To prove basic properties about the induced topology, it suffices to check whether a given cd-structure is bounded, complete, and regular. Unfortunately, the dividing cd-structure does not satisfy the boundedness condition, but instead we consider the weaker condition of being *quasi-bounded*. For this reason we revisit Voevodsky’s work in [@Vcdtop] to ensure that the set-up applies to fs log schemes.
cd-structures on fs log schemes {#sec::cdlog}
-------------------------------
Recall the following definition from [@Vcdtop].
\[A.8.25\] A [*cd-structure*]{} $P$ on a category $\cC$ with an initial object is a collection of commutative squares in $\cC$ $$\label{A.8.13.1}
Q
=
\begin{tikzcd}
Y'\arrow[d,"f'"']\arrow[r,"g'"]&Y\arrow[d,"f"]\\
X'\arrow[r,"g"]&X
\end{tikzcd}$$ that are stable under isomorphisms. The squares in the collection $P$ are called the *distinguished squares of $P$*. The topology $t_P$ associated with $P$ is the smallest Grothendieck topology such that $Y\amalg X'$ is a covering sieve of $X$ for every distinguished square $Q$ in $P$, and the empty family is a covering sieve of the initial object.
Let $P$ be a cd-structure on a category $\cC$ with an initial object. The class $S_P$ of [*simple covers*]{} is the smallest class of families of morphisms $\{U_i\rightarrow X\}_{i\in I}$ satisfying the following two conditions.
1. Any isomorphism is in $S_P$.
2. For any distinguished square of the form , if $\{p_i':X_i'\rightarrow X'\}_{i\in I}$ and $\{q_j:Y_j\rightarrow Y\}_{j\in J}$ are in $S_P$, then $\{g\circ p_i'\}_{i\in I}\cup \{f\circ q_j\}_{j\in J}$ is in $S_P$.
We say that $P$ is [*complete*]{} if any covering sieve admits a refinement by a simple cover.
Let $P$ be a cd-structure on a category $\cC$ with an initial object. Let $L(T)$ denote the $t_P$-sheaf associated with the presheaf represented by $T\in \cC$. We say that $P$ is [*regular*]{} if the following conditions are satisfied.
1. For any distinguished square $Q$ of the form , $Q$ is cartesian, $g$ is a monomorphism.
2. The morphism of $t_P$-sheaves $$\Delta\amalg (L(g')\times_{L(g)}L(g')):
L(Y)\amalg (L(Y')\times_{L(X')}L(Y'))
\rightarrow
L(Y)\times_{L(X)}L(Y)$$ is an epimorphism.
\[A.5.14\] A cartesian square of fs log schemes $$\label{A.5.14.1}
Q
=
\begin{tikzcd}
Y'\arrow[d,"f'"']\arrow[r,"g'"]&Y\arrow[d,"f"]\\
X'\arrow[r,"g"]&X
\end{tikzcd}$$ is called a
1. *Zariski distinguished square* if $f$ and $g$ are open immersions.
2. [*strict Nisnevich distinguished square* ]{} if $f$ is strict étale, $g$ is an open immersion, and $f$ induces an isomorphism $$f^{-1}(\underline{X}-g(\underline{X'}))\stackrel{\sim}\rightarrow \underline{X}-g(\underline{X'})$$ with respect to the reduced scheme structures.
3. [*dividing distinguished square*]{} if $Y'=X'=\emptyset$ and $f$ is a surjective proper log étale monomorphism. See §\[subsec::logsmooth\] for properties of log étale morphisms and §\[subsec::mono\] for properties of log étale monomorphisms. According to Proposition \[A.9.75\], this condition is equivalent to asking that $f$ be a log modification in the sense of F. Kato.
Associated to the collection of Zariski distinguished squares (resp. strict Nisnevich distinguished squares, resp. dividing distinguished squares) we have the corresponding *Zariski cd-structure* (resp. [*strict Nisnevich cd-structure*]{}, resp. the [*dividing cd-structure*]{}). These cd-structures give rise to the *Zariski topology*, [*strict Nisnevich topology*]{}, and [*dividing topology*]{} on $lSch/S$, respectively. We let $Zar$ (resp. $sNis$, resp. $div$) be shorthand for the Zariski topology (resp. strict Nisnevich topology, resp. dividing topology).
The *dividing Zariski cd-structure* (resp. [*dividing Nisnevich cd-structure*]{}) is the union of the Zariski (resp. strict Nisnevich) and the dividing cd-structures. We refer to the associated topology as the *dividing Zariski topology* (resp. [*dividing Nisnevich topology*]{}). We let $dZar$ (resp. $dNis$) be shorthand for the dividing Zariski topology (resp. dividing Nisnevich topology).
\[A.5.65\] The [*strict étale topology*]{} (resp. *Kummer étale topology*, resp. log étale topology) on $lSch/S$ is the smallest Grothendieck topology generated by strict étale covers (resp. Kummer étale covers, resp. log étale covers) in $lSch/S$. The [*dividing étale topology*]{} on $lSch/S$ is the smallest Grothendieck topology finer than the strict étale topology and the dividing topology. We let $s\acute{e}t$ (resp. $k\acute{e}t$, $l\acute{e}t$, $d\acute{e}t$) be shorthand for the strict étale topology (resp. Kummer étale topology, resp. log étale topology, resp. dividing étale topology).
We note that none of these étale topologies are associated with a cd-structure. The log étale topology is the smallest Grothendieck topology that is finer than the Kummer étale topology and the dividing topology, see also [@MR1922832 §9.1] whose proof appears in [@MR3658728 Proposition 3.9].
Every Nisnevich distinguished square of usual schemes is also a strict Nisnevich distinguished square. If the square $Q$ in is a strict Nisnevich distinguished square, then the square $$\underline{Q}
=
\begin{tikzcd}
\underline{Y'}\arrow[d,"\underline{f'}"']\arrow[r,"\underline{g'}"]&\underline{Y}\arrow[d,"\underline{f}"]\\
\underline{X'}\arrow[r,"\underline{g}"]&\underline{X}
\end{tikzcd}$$ is a Nisnevich distinguished square of usual schemes since the morphisms $f$ and $g$ in are strict. We also note that $Q$ is a pullback of $\underline{Q}$, i.e., $$Y\cong \underline{Y}\times_{\underline{X}}X,\;
Y'\cong \underline{Y'}\times_{\underline{X}}X,\;
X'\cong \underline{X'}\times_{\underline{X}}X.$$
\[A.9.68\]
1. Suppose $f:Y\rightarrow X$ is a surjective proper log étale monomorphism. Theorem \[KatoStrThm\] shows that Zariski locally on $X$ and $Y$, the restriction $$f^{-1}(X-\partial X)\rightarrow X-\partial X$$ has a chart $\theta:P\rightarrow Q$, where $P=0$ is trivial and $Q$ is finite. Thus $f^{-1}(X-\partial X)$ has a trivial log structure, since $\overline{Q}=0$, so that $Y-\partial Y=f^{-1}(X-\partial X)$ and the induced morphism $Y-\partial Y\rightarrow X-\partial X$ is surjective. The latter morphism is an isomorphism since surjective proper étale monomorphisms of schemes are isomorphisms according to [@EGA Théorème IV.17.9.1].
2. For cdh-distinguished squares of schemes [@Vunst-nis-cdh Section 2], the schemes $Y'$ and $X'$ in can be nonempty. Thus the dNis-structure is not a direct analogue of the cdh-structure [@Vcdtop].
3. We choose the terminology “dividing” because morphisms of fs log schemes induced by subdivisons of fans in toric geometry are surjective proper log étale monomorphisms. For a proof we refer to Proposition \[A.9.75\] and Example \[A.9.76\].
4. A presheaf $\cF$ is a dividing sheaf if and only if for every log modification $Y\rightarrow X$ the induced homomorphism $\cF(X)\rightarrow \cF(Y)$ is an isomorphism.
Cohomology on big and small sites
---------------------------------
If the base scheme $S$ has trivial log structure and $\cF$ is an étale sheaf of $\Lambda$-modules on $Sm/S$, then there are two equivalent ways of defining the cohomology group $H^i(X,\cF)$ for any integer $i\geq 0$, namely $$\hom_{\mathbf{D}(\Shv_{\acute{e}t}(X_{\acute{e}t},\Lambda))}(\Lambda(X),\cF\vert_{X_{\acute{e}t}}[i])
\text{ and }
\hom_{\mathbf{D}(\Shv_{\acute{e}t}(Sm/S,\Lambda))}(\Lambda(X),\cF[i]).$$ In this section we generalize the above comparison to the topologies we consider on $lSm/k$. As an application we shall discuss compact objects in the corresponding categories of sheaves.
A morphism of fs log schemes $Y\rightarrow X$ is a *dividing Nisnevich* (resp. *dividing étale*) morphism if it is contained in the smallest class of morphisms containing log modifications and strict Nisnevich (resp. strict étale) morphisms and closed under compositions.
\[bigsmall.1\] If $X$ is an fs log scheme, let $X_{sNis}$ (resp. $X_{s\acute{e}t}$, $X_{k\acute{e}t}$, $X_{dNis}$, $X_{d\acute{e}t}$, $X_{l\acute{e}t}$) denote the full subcategory of $lSch/X$ consisting of fs log schemes strict Nisnevich (resp. strict étale, Kummer étale, dividing Nisnevich, dividing étale, log étale) over $X$.
Note that these classes of morphisms are closed under compositions and pullbacks. Due to Lemma \[A.9.71\] for every log étale morphism $Z\rightarrow Y$ of fs log schemes, the diagonal morphism $Z\rightarrow Z\times_Y Z$ is an open immersion. Thus all these categories have finite limits.
If $X$ is an fs log scheme, every morphism in $X_{sNis}$ (resp. $X_{s\acute{e}t}$, $X_{k\acute{e}t}$, $X_{dNis}$, $X_{d\acute{e}t}$, $X_{l\acute{e}t}$) is strict Nisnevich (resp. strict étale, Kummer étale, dividing Nisnevich, dividing étale, log étale).
We focus on the dividing Nisnevich case since the proofs are similar. Let $\mathscr{P}$ denote the class of dividing Nisnevich morphisms. Suppose that $g:Z\rightarrow Y$ and $f:Y\rightarrow X$ be morphisms of fs log schemes such that $f,fg\in \mathscr{P}$. We claim that $g\in \mathscr{P}$. There exists a commutative diagram $$\begin{tikzcd}
Z\arrow[r,"\Gamma"]\arrow[d,"g"']&
Z\times_X Y\arrow[r,"q"]\arrow[d,"g\times {\rm id}"]&
Y\arrow[d,"{\rm id}"]
\\
Y\arrow[r,"\Delta"]&
Y\times_X Y\arrow[r,"p"]&
Y,
\end{tikzcd}$$ where $p$ and $q$ are the second projections, $\Delta$ is the diagonal morphism, and the left square is cartesian. Since $f$ is log étale, as noted above $\Delta\in \mathscr{P}$. It follows that $\Gamma,q\in \mathscr{P}$ because $\mathscr{P}$ is closed under pullbacks. This shows that $g=q\circ \Gamma\in \mathscr{P}$.
Let $t$ be one of the following topologies on $lSm/S$: $$t=sNis,\; s\acute{e}t,\; dNis,\;d\acute{e}t,\;k\acute{e}t,\textrm{ and }\;l\acute{e}t.$$ We have an inclusion functor $$\eta\colon X_t\rightarrow lSm/S$$ sending $Y\in X_t$ to $Y$. We regard $lSm/S$ as a site for the topology $t$. Since $X_t$ has fiber products and the functor $\eta$ commutes with fiber products, [@SGA4 Proposition III.1.6] shows that $\eta$ is a continuous functor of sites. If $Y\in X_t$, then every $t$-covering $\{U_i\rightarrow Y\}_{i\in I}$ in $lSm/S$ has a refinement in $X_t$, so $\eta$ is a cocontinuous functor of sites. Then [@SGA4 Sections IV.4.7, IV.4.9] gives adjoint functors $$\label{bigsmall.1.1}
\begin{tikzcd}
\Shv_t(X_t,\Lambda)
\arrow[rr,shift left=1.5ex,"\eta_\sharp"]
\arrow[rr,leftarrow,"\eta^*" description]
\arrow[rr,shift right=1.5ex,"\eta_*"']
&&
\Shv_t(lSm/S,\Lambda),
\end{tikzcd}$$ where $\eta^*$ is the restriction functor, $\eta_\sharp$ is left adjoint to $\eta^*$, and $\eta_*$ is right adjoint to $\eta^*$. Note that $\eta^*$ is exact.
There are naturally induced adjoint functors $$\label{bigsmall.1.2}
\eta_\sharp:\mathbf{C}(\Shv_t(X_t,\Lambda))
\rightleftarrows
\mathbf{C}(\Shv_t(lSm/S,\Lambda)):\eta^*.$$ We impose the descent model structures on both categories, see Proposition \[A.8.15\]. By virtue of Example \[A.8.28\] an object $\cF$ in $\mathbf{C}(\Shv_t(X_t,\Lambda))$ (resp. $\mathbf{C}(\Shv_t(lSm/S,\Lambda))$) is fibrant if for every $t$-hypercover $\mathscr{Y}\rightarrow Y$ in $X_t$ (resp. $lSm/S$) there is a natural quasi-isomorphism $$\label{bigsmall.1.3}
\cF(Y)\xrightarrow{\cong} {\rm Tot}^\pi \cF(\mathscr{Y}).$$ Moreover, a morphism $\cF\rightarrow \cG$ in $\mathbf{C}(\Shv_t(X_t,\Lambda))$ (resp. $\mathbf{C}(\Shv_t(lSm/S,\Lambda))$) is a fibration if it is a degreewise epimorphism and its kernel is fibrant. If $Y\in X_t$, then every $t$-hypercovering of $Y$ in $lSm/S$ is again a $t$-hypercovering in $X_t$. Moreover, since $\eta^*$ is exact, $\eta^*$ commutes with ${\rm Tot}^\pi$. Hence from the description of fibrant objects we deduce that $\eta^*$ preserves fibrant objects. Use again the fact that $\eta^*$ is exact to deduce that $\eta^*$ preserves fibrations and trivial fibraions. Thus is a Quillen adjunction. By passing to homotopy categories we obtain the adjoint functors $$\label{bigsmall.1.4}
L\eta_\sharp:
\mathbf{D}(\Shv_t(X_t,\Lambda))
\rightleftarrows
\mathbf{D}(\Shv_t(lSm/S,\Lambda))
:R\eta^*.$$ Since $\eta^*$ is exact, $R\eta^*\cong \eta^*$.
\[bigsmall.2\] With $t$ is as above, for every complex of $t$-sheaves $\cF$ there is a canonical isomorphism $$\hom_{\mathbf{D}(\Shv_{t}^{\rm log}(k,\Lambda))}(a_t^*\Lambda(X),\eta^*\cF)
\xrightarrow{\cong}
\hom_{\mathbf{D}(\Shv_{t}(X_{t},\Lambda))}(a_t^*\Lambda(X),\cF).$$
Since $a_t^*\Lambda(X)$ is cofibrant there is an isomorphism $$L\eta_\sharp a_t^*\Lambda(X)\cong a_t^*\Lambda(X).$$ To conclude we use the adjunction .
\[bigsmall.3\] With $t$ is as above, suppose that $X$ is a noetherian fs log scheme. Then the topos $\Shv_t(X_t)$ is coherent.
Since $X_t$ has fiber products, owing to [@SGA4 Proposition VI.2.1] it suffices to show that every object of $X_t$ is quasi-compact. The log étale topology is finer than $t$, and $X_t$ is a full subcategory of $X_{l\acute{e}t}$. Hence it remains to show that every object of $X_{l\acute{e}t}$ is quasi-compact. For this we refer to [@MR3658728 Proposition 3.14(1)].
\[bigsmall.4\] Let $\cC$ be a category with a topology $\tau$. We say that $\cC$ has *finite $\tau$-cohomological dimension* for $\Lambda$-linear coefficients if for every fixed $X\in \cC$, there exists an integer $N$ such that for every $\cF\in \Shv_\tau(\cC,\Lambda)$ and integer $i>N$ we have the vanishing $$H_{\tau}^i(X,\cF)=0.$$
\[bigsmall.5\] With $t$ as above, suppose that $S$ is noetherian and $lSm/S$ has finite $t$-cohomological dimension for $\Lambda$-linear coefficients. Then for every $X\in lSm/S$ the functor $$\hom_{\mathbf{D}(\Shv_t(lSm/S,\Lambda))}(a_t^*\Lambda(X),-)
\colon
\mathbf{D}(\Shv_t(lSm/S,\Lambda))\rightarrow \Lambda\text{-}\mathbf{Mod}$$ commutes with small sums. In particular, the category $$\mathbf{D}(\Shv_t(lSm/S,\Lambda)$$ is compactly generated.
By virtue of Proposition \[bigsmall.2\], it suffices to show that the functor $$\hom_{\mathbf{D}(\Shv_t(X_t,\Lambda))}(a_t^*\Lambda(X),-)
\colon
\mathbf{D}(\Shv_t(X_t,\Lambda))\rightarrow \Lambda\text{-}\mathbf{Mod}$$ commutes with small sums. By assumption $\Shv_t(X_t,\Lambda)$ has finite $t$-cohomological dimension for $\Lambda$-linear coefficients, and $X_t$ is coherent due to Proposition \[bigsmall.3\]. Then apply [@CDEtale Proposition 1.9] to conclude.
Dividing density structure
--------------------------
In addition to completeness and regularity, Voevodsky [@Vcdtop] also introduced the notion of a bounded cd-structure. In tandem, these properties ensure the associated topology has finite cohomological dimension. The standard density structure, introduced in [@Vunst-nis-cdh], is used to demonstrate that certain cd-structures are bounded, including the Nisnevich cd-structure.
For fs log schemes, we need to deal with certain non-bounded cd-structures due to the abundance of proper log étale monomorphisms (such morphisms are typically not isomorphisms). In effect, we introduce the notion of a quasi-bounded cd-structure, see Definition \[A.9.65\], and modify the standard density structure so that it fits our situation, see Definition \[A.9.43\]. Moreover, we extend B.G. functors to simplicial objects in order to deduce a finite cohomological dimension result.
\[A.9.42\] A density structure on a category $\cC$ with an initial object is the data of $D_d(X)$ for $d\in \N$ and $X\in \cC$, where $D_d(X)$ is a family of morphisms with codomain $X$ satisfying the following properties.
1. $(\emptyset\rightarrow X)\in D_0(X)$.
2. $D_d(X)$ contains all isomorphisms.
3. $D_{d+1}(X)\subset D_{d}(X)$.
4. If $v:V\rightarrow U$ is in $D_d(U)$ and $u:U\rightarrow X$ is in $D_d(X)$, then $u\circ v$ is in $D_d(X)$.
We often write $U\in D_d(X)$ instead of $(U\rightarrow X)\in D_d(X)$ for short.
\[A.9.62\] A density structure $D_*$ on $\cC$ is [*locally of finite dimension with respect to a topology $t$*]{} on $\cC$ if for any $X\in \cC$, there exists a natural number $n\in \N$ such that for any $f:Y\rightarrow X$ in $D_{n+1}(X)$, there exists a finite $t$-covering sieve $\{f_i:Y_i\rightarrow X\}_{i\in I}$ such that $f=\amalg_{i\in I}f_i$. The smallest such $n$ is called the [*dimension*]{} of $X$ with respect to $D_*$.
\[A.9.63\] A density structure is locally of finite dimension with respect to the trivial topology if and only if it is locally of finite dimension in the sense of [@Vcdtop Definition 2.20]. Indeed, according to [@Vcdtop Definition 2.20], a density structure is locally of finite dimension if for any $X$, there exists $n\in \N$ such that any element of $D_{n+1}(X)$ is an isomorphism.
In the following definition we slightly modify the standard density structure on a finite dimensional scheme introduced by Voevodsky [@Vunst-nis-cdh p. 1401]. Recall that distinct points $x_0,\ldots,x_d$ in $X$ form an [*increasing sequence*]{} if $x_{i+1}$ is a generalization of $x_i$ for every $i$, i.e., the point $x_i$ is in the closure of ${\{s_{i+1}\}}$.
\[A.9.43\] Let $D_d(X)$ denote the family of finite disjoint unions $$U=\amalg_i U_i\rightarrow X$$ of open immersions such that for any point $x\in X$ that is not in the image, there is an increasing sequence $x=x_0,\ldots,x_d$ of length $d$ in $X$. Note that $D_*$ is a density structure.
For a finite dimensional fs log scheme $X$, as in [@Par], we set $$D_d(X):=D_d(\underline{X}).$$
\[A.9.53\] In [@Vunst-nis-cdh Section 2, p. 1401], only open immersions (and not finite disjoint unions of open immersions) are considered for the standard density structure. Our slight weakening of the original conditions allows us to relate the dividing density structure defined in Definition \[A.9.57\] to the standard density structure.
\[A.9.51\] Let $X$ be a finite dimensional scheme, and let $d\in \N$.
1. If $U\in D_d(X)$ and $V$ is an open subscheme of $X$, then $U\times_X V\in D_d(V)$.
2. Let $U\rightarrow X$ be a finite disjoint union of open immersions mapping to $X$, and let $\{V_i\rightarrow X\}_{i\in I}$ be an open cover of $X$. Then $U\times_X V_i\in D_d(V_i)$ for every $i$ if and only if $U\in D_d(X)$.
3. If $U,V\in D_d(X)$, then $U\times_X V\in D_d(X)$.
Parts (1) and (3) are shown in [@Vunst-nis-cdh Lemmas 2.4, 2.5]. Let us prove (2). If $U\in D_d(X)$, then $U\times_X V_i\in D_d(V_i)$ for any $i$ by (1). Conversely, assume that $U\times_X V_i\in D_d(V_i)$ for every $i$. Let $x_0$ be a point of $X$ that is not in the image of $U\rightarrow X$. Choose $i$ such that $x_0\in V_i$. Then there exists an increasing sequence $x_0,\ldots,x_d$ of $V_i$, which is also an increasing sequence of $X$. It follows that $U\in D_d(X)$.
\[A.9.44\] Let $X$ be a finite dimensional noetherian fs log scheme. For $d\in \N$, we denote by $D_d^{div}(X)$ the family $U=\amalg_i U_i\rightarrow X$ of finite disjoint unions of log étale monomorphisms to $X$ with the following property: For any log étale monomorphism $V\rightarrow X$ such that the projection $U\times_X V\rightarrow V$ is a finite disjoint union schemes $\{V_i\}$ with open immersions $V_i\rightarrow V$, then $U\times_X V\in D_d(V)$.
\[A.9.45\] Any finite disjoint union $U\rightarrow X$ of log étale monomorphisms is in $D_0^{div}(X)$.
In the following series of lemmas we study basic properties of $D_d^{div}$. In particular, we will show that $D_d^{div}$ is a density structure.
\[A.9.54\] Let $f:Y\rightarrow X$ be a surjective proper log étale monomorphism of finite dimensional noetherian fs log schemes. Then $Y\in D_d^{div}(X)$ for all $d\in \N$.
Let $V\rightarrow X$ be a log étale monomorphism such that the projection $p:Y\times_X V\rightarrow V$ is a finite disjoint union of open immersions to $V$. By Proposition \[A.9.22\], $p$ is surjective. Thus $Y\times_X V\in D_d(V)$ and $Y\in D_d^{div}(X)$ for all $d\in \N$.
If $X$ has trivial log structure, then every proper log étale monomorphism is an isomorphism, see Remark \[A.9.68\](1). Thus, in this case, the density structure $D_d^{div}(X)$ agrees with the standard one.
Let us review the structure of affine toric varieties that will be used in the proof of Lemma \[A.9.46\]. Suppose $P$ is an fs monoid. Recall from [@Ogu §I.3.3] that for every face $F$ of $P$ there is a naturally induced closed immersion $$\underline{\A_F}=\Spec{\Z[F]}\cong \Spec{\Z[P]/\Z[P-F]}\rightarrow \underline{\A_P}.$$ There is also a naturally induced open immersion $$\underline{\A_F^*}:=\Spec{\Z[F^{\rm gp}]}\rightarrow \underline{\A_F}.$$ The family $\{\underline{\A_F^*}\}$ indexed by all the faces $F$ of $P$ forms a stratification of $\A_P$, see [@Ogu Proposition I.3.3.4]. Moreover, since $\Spec{\Z[P]/\Z[P-F]}$ is naturally identified with $\underline{\A_F}$, the points of $\underline{\A_F^*}$ are exactly identified with the points $x$ of $\underline{\A_P}$ such that $\overline{\cM}_{\A_P,x}$ is naturally given by $P/F$. Suppose that $\theta:P\rightarrow Q$ is a homomorphism of fs monoids. This induces a natural morphism $$\underline{\A_\theta}:\underline{\A_Q}\rightarrow \underline{\A_P}.$$ For every face $G$ of $Q$, we have that $$\label{A.9.46.2}
\underline{\A_\theta}(\underline{\A_G})\cong \underline{\A_{\theta^{-1}(G)}}.$$
\[A.9.46\] Suppose $f:Y\rightarrow X$ be a log étale monomorphism of finite dimensional noetherian fs log schemes. For $x\in X$, let $y$ be a generic point of $f^{-1}(x)$, and let $z\in Y$ be a generalization of $y$. Then $z$ is a generic point of $f^{-1}(f(z))$.
The question is Zariski local on $Y$ and $X$. Thus by Remark \[A.9.10\](2), we may assume that $X$ has a neat chart $P$ at $x$. This means that $P$ is sharp and $\overline{\cM}_{X,x}\cong P$. Moreover, by Lemma \[A.9.19\], we may assume that $f$ has an fs chart $\theta:P\rightarrow Q$ such that $\theta^{\rm gp}$ is an isomorphism and the induced morphism $$Y\rightarrow X\times_{\A_P}\A_Q$$ is an open immersion. By Remark \[A.9.10\](1), we may assume that $Q$ is an exact chart at the point $y$. This means that $\overline{\cM}_{Y,y}\cong Q$, but we do not require that $Q$ is sharp. We may assume $Y\cong X\times_{\A_P}\A_Q$ since the question is Zariski local on $Y$.
Next, we claim that the induced homomorphism $$\overline{\theta}:P\rightarrow \overline{Q}$$ is $\Q$-surjective, i.e., $\overline{\theta}\otimes \Q$ is surjective. Arguing by contradiction, assume that it is not. Let $G_1,G_2,\ldots,G_r$ be all the faces of $Q$ such that $\theta^{-1}(G_i)=0$. Then the corresponding faces $\overline{G_1},\ldots,\overline{G_r}$ in $\overline{Q}$ are all the faces of $\overline{Q}$ such that $\overline{\theta^{-1}}(\overline{G_i})=0$. The implication (6)$\Rightarrow$(1) in [@Ogu Proposition I.4.3.9] means that $r\geq 2$. From we have that $$\label{A.9.46.1}
\{0\}\times_{\underline{\A_P}}\underline{\A_Q}
\cong
\underline{\A_{G_1}}\cup \cdots \cup \underline{\A_{G_r}}.$$ Let $\kappa(-)$ denote the residue field at some point, and set $D_i:=\underline{\A_{G_i}}\times \Spec{\kappa(x)}$ for simplicity. Then from we have that $$f^{-1}(x)
\cong
D_1\cup \cdots \cup D_r.$$ Since $\overline{\cM}_{Y,y}\cong Q$, it follows that $y$ is contained in $D_1$. The isomorphism $$D_2\cong \spec{\kappa(x)[G_2]},$$ shows that $D_2$ is irreducible. We have reached a contradiction since $D_1\subsetneq D_2$ and $y$ is a generic point of $f^{-1}(x)$.
Let $z'$ be a generalization of $z$ in $f^{-1}(f(z))$. Consider the projection $$p:X\times_{\A_P}\A_Q\rightarrow \A_Q,$$ and let $\overline{Q}/G$ (resp. $\overline{Q}/G'$) be the log structure on $z$ (resp. $z'$) where $G$ (resp. $G'$) is a face of $\overline{Q}$. Note that $G'\subset G$. Since $f(z)=f(z')$ we have $$\overline{\theta}^{-1}(G)=\overline{\theta}^{-1}(G').$$ By the implication (1)$\Rightarrow$(2) in [@Ogu Proposition I.4.3.9] it follows that $G=G'$. Set $u:=f(z)=f(z')$, and let $w$ be the generic point of $\underline{\A_G}$ with image $t$ in $\underline{\A_P}$. Then $$z,z'\in\Spec{\kappa(u)\otimes_{\kappa(t)} \kappa(w)}.$$ Thus $z=z'$, so any generalization of $z$ in $f^{-1}(f(z))$ is equal to $z$. This implies that $z$ is a generic point of $f^{-1}(f(z))$.
\[A.9.47\] Let $f:Y\rightarrow X$ be a surjective log étale monomorphism of finite dimensional noetherian fs log schemes, and let $j:U\rightarrow X$ be a finite disjoint union of open immersions to $X$. If $U\times_X Y\in D_d(Y)$, then $U\in D_d(X)$.
Suppose $x\in X$ is not in the image of $j$. Since $f$ is surjective, we can choose a generic point $y_0$ of $f^{-1}(x_0)$ and an increasing sequence $y_0,\ldots,y_d$ of $Y$. Then $f(y_i)$ is a generic point of $f^{-1}(f(y_i))$ according to Lemma \[A.9.46\]. Thus $f(y_0),\ldots,f(y_d)$ is an increasing sequence of $X$. It follows that $U\in D_d(X)$.
\[A.9.56\] Let $X$ be a finite dimensional noetherian fs log scheme, and let $d\in \N$. Then for every $U\in D_d^{div}(X)$ and log étale monomorphism $V\rightarrow X$, we have $U\times_X V\in D_d^{div}(V)$.
Let $W\rightarrow V$ be a log étale monomorphism such that the projection $U\times_X W\rightarrow W$ is a finite disjoint union of open immersions. Since $U\in D_d^{div}(X)$ and the composition $W\rightarrow X$ is a log étale monomorphism, we have $$(U\times_X V)\times_V W=U\times_X W\in D_d(W).$$ This shows that $U\times_X V\in D_d^{div}(V)$.
\[A.9.48\] Let $Y\rightarrow X$ be a surjective proper log étale monomorphism of finite dimensional noetheorian fs log schemes, and let $U\rightarrow X$ be a finite disjoint union of log étale monomorphisms to $X$. Then $U\in D_d^{div}(X)$ if and only if $U\times_X Y\in D_d^{div}(Y)$.
If $U\in D_d^{div}(X)$, then $U\times_X Y\in D_d^{div}(Y)$ by Lemma \[A.9.56\]. Conversely, assume that $U\times_X Y\in D_d^{div}(Y)$. Let $V\rightarrow X$ be a log étale monomorphism such that the projection $U\times_X V\rightarrow V$ is a finite disjoint union of open immersions to $V$. Then $$U\times_X V\times_X Y\in D_d(V\times_X Y),$$ and the projection $V\times_X Y\rightarrow V$ is a surjective proper log étale monomorphism by Proposition \[A.9.22\]. Thus $U\times_X V\in D_d(V)$ by Lemma \[A.9.47\], so $U\in D_d^{div}(X)$.
\[A.9.55\] Let $X$ be a finite dimensional noetherian fs log scheme, and let $d\in \N$.
1. Let $U\rightarrow X$ be a finite disjoint union of log étale monomorphisms to $X$, and let $\{V_i\rightarrow X\}_{i\in I}$ be an open cover of $X$. Then $U\times_X V_i\in D_d^{div}(V_i)$ for any $i$ if and only if $U\in D_d^{div}(X)$.
2. If $U,V\in D_d^{div}(X)$, then $U\times_X V\in D_d^{div}(X)$.
\(1) If $U\in D_d^{div}(X)$, then $U\times_X V_i\in D_d^{div}(V_i)$ for any $i$. Conversely, assume that $U\times_X V_i\in D_d^{div}(V_i)$ for any $i$. Let $W\rightarrow X$ be a log étale monomorphism such that the projection $U\times_X W\rightarrow W$ is a finite disjoint union of open immersions to $W$. Then $$V_i\times_X U\times_X W\in D_d(V_i\times_X W),$$ and hence $U\times_X W\in D_d(W)$ by Lemma \[A.9.51\](3). This shows that $U\in D_d^{div}(X)$.
\(2) The question is Zariski local on $X$ by (1), so we may assume that $X$ has an fs chart $P$. By Proposition \[A.9.21\] there is a subdivision of fans $\Sigma\rightarrow \Spec{P}$ such that the pullbacks $$U\times_{\A_P}\A_\Sigma\rightarrow X\times_{\A_P}\A_\Sigma,
\,\,
V\times_{\A_P}\A_\Sigma\rightarrow X\times_{\A_P}\A_\Sigma$$ are disjoint unions of open immersions to $X\times_{\A_P}\A_\Sigma$. By Lemma \[A.9.48\], $$U\times_{\A_P}\A_\Sigma,V\times_{\A_P}\A_\Sigma\in D_d^{div}(X\times_{\A_P}\A_\Sigma),$$ and we are reduced to showing that $$U\times_S V\times_{\A_P}\A_\Sigma\in D_d^{div}(X\times_{\A_P}\A_\Sigma).$$ We may replace $X$ by $X\times_{\A_P}\A_\Sigma$, and assume that $U$ and $V$ are disjoint union of open subschemes of $X$. Let $W\rightarrow X$ be a log étale monomorphism. Then $$U\times_X W,V\times_X W\in D_d(W)$$ and hence $U\times_X V\times_X W\in D_d(W)$ by Lemma \[A.9.51\](3). This finishes the proof.
\[A.9.52\] If $v:V\rightarrow U$ is in $D_d^{div}(U)$ and $u:U\rightarrow X$ is in $D_d^{div}(X)$, then the composite $v\circ u:V\rightarrow X$ is in $D_d^{div}(X)$.
The question is Zariski local on $X$ by Lemma \[A.9.55\](1), so we may assume that $X$ has an fs chart $P$. By Proposition \[A.9.21\], there is a surjective proper log étale monomorphism $Y\rightarrow X$ such that the projections $U\times_X Y\rightarrow Y$ and $V\times_X Y\rightarrow Y$ are finite disjoint unions of open immersions to $Y$. Then the pullback $V\times_X Y\rightarrow U\times_X Y$ is also a finite disjoint union of open immersions to $U\times_X Y$. By Lemma \[A.9.48\], $$U\times_X Y\in D_d^{div}(Y) \text{ and } V\times_X Y\in D_d^{div}(U\times_X Y).$$ We are reduced to showing that $V\times_X Y\in D_d^{div}(Y)$ again by Lemma \[A.9.48\]. To that end, we may replace $(X,U,V)$ by $(Y,U\times_X Y,V\times_X Y)$ and assume that $u$ and $v$ are disjoint unions of open immersions to $X$.
Let $W\rightarrow X$ be a log étale monomorphism such that the projection $U\times_X W\rightarrow W$ is a finite disjoint union of open immersions. Then $U\times_X W\in D_d(W)$ and $V\times_X W\in D_d(U\times_X W)$, so $V\times_X W\in D_d(W)$ since $D_*$ is a density structure. Thus we have $V\in D_d^{div}(X)$.
Here is a summary of the previous lemmas.
Let $X$ be a finite dimensional noetherian fs log scheme. Then we have the following properties.
1. If $U\rightarrow X$ is a surjective proper log étale monomorphism, then $U\in D_d^{div}(X)$ for all $d\in \N$.
2. Let $Y\rightarrow X$ be a log étale monomorphism. If $U\in D_d^{div}(X)$, then $U\times_X V\in D_d^{div}(V)$.
3. Let $Y\rightarrow X$ be a dividing Zariski cover. Then $U\in D_d^{div}(X)$ if and only if $U\times_X Y\in D_d^{div}(Y)$.
4. If $U,V\in D_d^{div}(X)$, then $U\times_X V\in D_d^{div}(X)$.
5. If $v:V\rightarrow U$ is in $D_d^{div}(U)$ and $u:U\rightarrow X$ is in $D_d^{div}(X)$, then $v\circ u$ is in $D_d^{div}(X)$.
\[A.9.57\] The density structure $D_{\ast}^{div}$ on $lSm/S$ and $lSch/S$ established in Lemma \[A.9.52\] is called the [*dividing density structure*]{}.
\[A.9.58\] Let $X$ be a finite dimensional noetherian fs log scheme with an fs chart $P$, and let $U\rightarrow X$ be a finite disjoint union of log étale monomorphisms to $X$. If $U\times_X V\in D_d(V)$ for any surjective proper log étale monomorphism $V\rightarrow X$ such that the projection $U\times_X V\rightarrow V$ is a finite disjoint union of open immersions to $V$, then $U\in D_d^{div}(X)$.
Let $W\rightarrow X$ be a log étale monomorphism such that the projection $$U\times_X W\rightarrow W$$ is a finite disjoint union of open immersions to $W$. By Proposition \[A.9.21\], there is a surjective proper log étale monomorphism $V\rightarrow X$ such that $$W\times_X V\rightarrow V \text{ and } U\times_X V\rightarrow V$$ is an open immersion and a finite disjoint union of open immersions to $V$, respectively.
Consider the commutative diagram with the evident projection morphisms $$\begin{tikzcd}
U\times_X W\arrow[d]\arrow[r,leftarrow]&U\times_X W\times_X V\arrow[d]\arrow[r]&U\times_X V\arrow[d]\\
W\arrow[r,leftarrow]&W\times_X V\arrow[r]&V.
\end{tikzcd}$$ By assumption we have $U\times_X V\in D_d(V)$. Lemma \[A.9.51\](1) implies that $$U\times_X W\times_X V\in D_d(W\times_X V)$$ since $W\times_X V$ is an open subscheme of $V$. Thus by Lemma \[A.9.47\], $$U\times_X W\in D_d(W)$$ since the projection $W\times_X V\rightarrow W$ is a surjective proper log étale monomorphism by Proposition \[A.9.22\]. This finishes the proof.
\[A.9.49\] Let $f:Y\rightarrow X$ be a strict Nisnevich morphism of finite dimensional noetherian fs log schemes. Assume that $X$ has an fs chart $P$. Then for every $d\in \N$ and $Y_0\in D_d^{div}(Y)$, there exists $X_0\in D_d^{div}(X)$ such that the projection $X_0\times_X Y\rightarrow Y$ factors through $Y_0$.
By definition $Y_0\rightarrow Y$ is a disjoint union of log étale monomorphisms. Proposition \[A.9.21\] says that there exists a subdivision of fans $\Sigma\rightarrow \Spec P$ such that the induced morphism $$Y_0\times_{\A_P}\A_\Sigma\rightarrow Y\times_{\A_P}\A_\Sigma$$ is a finite disjoint union of open immersions to $Y\times_{\A_P}\A_\Sigma$. Now by Lemma \[A.9.48\], $Y_0\times_{\A_P}\A_\Sigma\in D_d^{div}(Y\times_{\A_P}\A_\Sigma)$. Hence replacing $Y_0$ by $Y_0\times_{\A_P}\A_\Sigma$, we may assume $$Y_0\cong Y_0\times_{\A_P}\A_\Sigma.$$ Then the morphism $Y_0\rightarrow Y\times_{\A_P}\A_\Sigma$ is a finite disjoint union of open immersions. It suffices to construct $X_0\in D_d^{div}(X\times_{\A_P}\A_\Sigma)$ since then $X_0\in D_d^{div}(X)$ by Lemma \[A.9.48\]. We may replace $X$ by $X\times_{\A_P}\A_\Sigma$ and assume $Y_0$ is a finite disjoint union of open subschemes of $Y$.
Set $$X_0:=X-cl(f(Y-\im (Y_0\rightarrow Y)),$$ where $cl(-)$ denotes the closure. The construction in the proof of [@Vunst-nis-cdh Lemma 2.9] shows that $X_0\in D_d(X)$. Let $V\rightarrow X$ be a surjective proper log étale monomorphism. Since $V$ is proper over $X$ we have $$X_0\times_X V=V-cl(f_V(Y\times_X V-\im ((Y_0\times_X V)\rightarrow (Y\times_X V)))),$$ where $f_V:Y\times_X V\rightarrow V$ is the projection. Thus again using the proof of [@Vunst-nis-cdh Lemma 2.9], $X_0\times_X V\in D_d(V)$, and $X_0\in D_d^{div}(X)$ by Lemma \[A.9.58\].
\[A.9.64\] Let $P$ be a cd-structure on a category $\cC$ with an initial object, and let $D_*$ be a density structure on $\cC$. A distinguished square $Q$ of the form is called [*reducing*]{} with respect to $D_*$ if for all $d\in \N^+$, $Y_0\in D_d(Y)$, $X_0'\in D_d(X')$, and $Y_0'\in D_{d-1}(Y')$ there is a distinguished square $$Q_1
=
\begin{tikzcd}
Y_1'\arrow[d]\arrow[r]&Y_1\arrow[d]\\
X_1'\arrow[r]&X_1
\end{tikzcd}$$ together a morphism $Q_1\rightarrow Q$ satisfying the following properties.
1. The morphism $X_1\rightarrow X\in D_d(X)$.
2. The morphism $Y_1\rightarrow Y$ (resp. $X_1'\rightarrow X'$, resp. $Y_1'\rightarrow Y'$) factors through $Y_0$ (resp. $X_0'$, resp. $Y_0'$).
We say that a distinguished square $Q'$ is a [*refinement*]{} of $Q$ if there is a morphism $Q'\rightarrow Q$ that is an isomorphism between the lower right corners.
\[A.9.65\] Let $P$ be a cd-structure on a category $\cC$ with an initial object, and let $D_*$ be a density structure on $\cC$. Consider the following conditions.
1. Any distinguished square in $P$ has a refinement that is reducing with respect to $D_*$.
2. $D_*$ is locally of finite dimension with respect to the topology associated with $P$.
If $P$ satisfies (i) (resp. (i) and (ii)), we say that $P$ is [*reducing*]{} ([@Vcdtop Definition 2.22]) (resp. [*quasi-bounded*]{}) with respect to $D_*$.
\[A.9.66\] A cd-structure $P$ is bounded with respect to $D_*$ in the sense of [@Vcdtop Definition 2.22] if and only if $P$ is reducing with respect to $D_*$ and $D_*$ is locally of finite dimension with respect to the trivial topology. The reason why we consider quasi-bounded cd-structures is that $D_*^{div}$ is not bounded. Indeed, if $Y\rightarrow X$ is a surjective proper log étale monomorphism, then $Y\in D_d^{div}(X)$ for any $d\in \N$ by Lemma \[A.9.56\]. Thus for any $d\in \N$, $D_d^{div}(X)$ can contain morphisms other than isomorphisms in general, so $D_*^{div}$ is not bounded.
\[A.9.84\] Let $Y\rightarrow X$ be a log étale morphism in $lSm/k$. Then $\dim Y\leq \dim X$.
Since $X$ and $Y$ are in $lSm/k$, $X-\partial X$ and $Y-\partial Y$ are dense in $X$ and $Y$. To conclude observe that $\dim (Y-\partial X)\leq \dim(X-\partial X)$.
\[A.9.85\] Let $\Sigma$ be the star subdivision of the dual of $\N^2$. If $$X:=\{(0,0)\}\times_{\A^2}\A_{\N^2}
\text{ and }
Y=\{(0,0)\}\times_{\A^2}\A_\Sigma,$$ then the naturally induced morphism $Y\rightarrow X$ is log étale, but $\dim Y=2$ and $\dim X=1$.
\[A.9.61\] The dividing density structures on $lSm/S$ and $lSch/S$ are locally of finite dimension with respect to the dividing Nisnevich topology.
Suppose $X$ is an $n$-dimensional noetherian fs log scheme. Let $f:U\rightarrow X$ be any finite disjoint union of log étale monomorphisms to $X$. We need to show that there exists an integer $m\geq 0$ such that $U\in D_{m}^{div}(X)$ implies that $f$ is a dividing Nisnevich cover. Let $r$ be the maximum rank of $\overline{\cM}_{X,x}^{\rm gp}$, where $x$ ranges over all points of $X$. Since $X$ is an fs log scheme, the rank of $\overline{\cM}_{X,x}^{\rm gp}$ is a locally finite function on $X$. Thus $r$ is a finite number since $X$ is quasi-compact. In order to conclude, it suffices to show that $U\in D_{r+d+1}^{div}(X)$ implies that $f$ is a dividing Zariski cover. This is a Zariski local question on $X$, so that we may assume $X$ has an fs chart $P$ that is exact at a point of $X$, see Remark \[A.9.10\](1). By definition we have $\rank P^{\rm gp}\leq r$.
Proposition \[A.9.21\] shows there exists a subdivision of fans $\Sigma\rightarrow \Spec P$ such that the pullback $$f':U\times_{\A_P}\A_\Sigma\rightarrow X\times_{\A_P}\A_\Sigma$$ is a finite disjoint union of open immersions to $X\times_{\A_P}\A_\Sigma$. For every point $x\in \underline{\A_P}$, the fiber of $x$ under $\underline{\A_\Sigma}\rightarrow \underline{\A_P}$ has dimension $\leq r$. Thus we get that $$\dim (X\times_{\A_P}\A_\Sigma)\leq r+n.$$ It follows that $f'$ is surjective because $$U\times_{\A_P}\A_\Sigma\in D_{r+n+1}(X\times_{\A_P}\A_\Sigma).$$ In [@Vunst-nis-cdh p. 1401], it is shown that the dimension with respect to the standard density structure is equal to the Krull dimension. Thus $f'$ is a Zariski cover. The projection $X\times_{\A_P}\A_\Sigma\rightarrow X$ is a surjective proper log étale monomorphism. In conclusion, $f$ admits a refinement by a dividing Zariski cover, i.e., $f$ is a dividing Zariski cover.
The above proof shows that the dimension of $X$ with respect to $D_*^{div}$ is less than or equal to the maximum possible dimension of $X\times_{\A_P}\A_\Sigma$. If $X$ is log smooth over $k$, we see that the dimension of $X$ with respect to $D_*^{div}$ is $\leq n$ since $\dim X\times_{\A_P}\A_\Sigma\leq n$ due to Lemma \[A.9.84\].
\[A.9.50\] The strict Nisnevich cd-structures on $lSm/S$ and $lSch/S$ are quasi-bounded with respect to the dividing density structure.
By Proposition \[A.9.61\] it remains to show the strict Nisnevich cd-structure is reducing with respect to the dividing density structure. Let $$Q
=
\begin{tikzcd}
Y'\arrow[r,"g'"]\arrow[d,"f'"']&Y\arrow[d,"f"]\\
X'\arrow[r,"g"]&X
\end{tikzcd}$$ be a strict Nisnevich distinguished square of finite dimensional noetherian fs log schemes, where $f$ is strict étale, $g$ is an open immersion, and $f$ induces an isomorphism $$f^{-1}(\underline{X}-g(\underline{X'}))\stackrel{\sim}\rightarrow \underline{X}-g(\underline{X'})$$ with respect to the reduced scheme structures. We will show that $Q$ is reducing.
For elements $X_0'\in D_d^{div}(X')$, $Y_0\in D_d^{div}(Y)$, and $Y_0'\in D_{d-1}^{div}(Y')$ we need to construct $X_1\in D_d^{div}(X)$ and a strict Nisnevich distinguished square $$Q_1
=
\begin{tikzcd}
Y_1'\arrow[r]\arrow[d]&Y_1\arrow[d]\\
X_1'\arrow[r]&X_1
\end{tikzcd}$$ together with a morphism $Q_1\rightarrow Q$ that coincides with $X_1\rightarrow X$ on the lower right corner such that $Y_1\rightarrow Y$ (resp. $X_1'\rightarrow X'$, resp. $Y_1'\rightarrow Y'$) factors through $Y_0$ (resp. $X_0$, resp. $Y_0$). By appealing to Lemma \[A.9.55\](1) it suffices to construct $X_1$ Zariski locally on $X$. Hence we may assume that $X$ has an fs chart $P$.
Owing to Lemma \[A.9.49\] we can choose $V,W\in D_d^{div}(X)$ such that the projections $$V\times_X Y\rightarrow Y \text{ and } W\times_X X'\rightarrow X'$$ factors through $Y_0$ and $X_0'$, respectively. Take $X_0:=V\times_X W$, which is again in $D_d^{div}(X)$ by Lemma \[A.9.55\](2). Then the projections $$X_0\times_X Y\rightarrow Y \text{ and } X_0\times_X X'\rightarrow X'$$ again factors through $Y_0$ and $X_0'$, respectively. It suffices to construct $X_1\in D_d^{div}(X_0)$ with the required property since then $X_1\in D_d^{div}(X)$ by Lemma \[A.9.52\]. Hence we may replace $X$ by $X_0$ and assume that $X'=X_0'$ and $Y=Y_0$.
By Proposition \[A.9.21\], there exists a subdivision of fans $\Sigma\rightarrow \Spec P$ such that the pullback $$Y_0'\times_{\A_P}\A_\Sigma\rightarrow Y'\times_{\A_P}\A_\Sigma$$ is a finite disjoint union of open immersions to $Y'\times_{\A_P}\A_\Sigma$. By Lemma \[A.9.48\] we have $$Y_0'\times_{\A_P}\A_\Sigma\in D_{d-1}^{div}(Y'\times_{\A_P}\A_\Sigma).$$ This reduces the claim to constructing $X_1\in D_d^{div}(X\times_{\A_P}\A_\Sigma)$ with the required property since then $X_1\in D_d^{div}(X)$ again by Lemma \[A.9.48\]. To that end we may replace $X$ by $X\times_{\A_P}\A_\Sigma$, and hence assume that $Y_0'$ is a finite disjoint union of open subschemes of $Y_0$.
The construction in the proof of [@Vunst-nis-cdh Proposition 2.10] shows that $$X_1:=X-((X-X')\cap cl(gf'(Y'-Y_0')))
\in
D_d(X),$$ and the induced square $$\begin{tikzcd}
Y_0'\arrow[r]\arrow[d]&g'(Y_0')\arrow[d]\\
X'\arrow[r]&X_1
\end{tikzcd}$$ is a strict Nisnevich distinguished square.
Now let $V\rightarrow X$ be a surjective proper log étale monomorphism. Then $$Y_0'\times_{X}V\in D_{d-1}(V),$$ and $$X_1\times_X V=V-(V-X'\times_X V)\cap cl(g_Vf_V'(Y'\times_X V-Y_0'\times_X V)),$$ where $f_v':Y'\times_X V\rightarrow X'\times_X V$ and $g_v:X'\times_X V\rightarrow V$ are the pullbacks since $V$ is proper over $X$. Thus $X_1\times_X V\in D_d(X)$ as in the above paragraph, and therefore $X_1\in D_d^{div}(X)$ by Lemma \[A.9.58\].
\[A.9.59\] The dividing cd-structures on $lSm/S$ and $lSch/S$ are quasi-bounded with respect to the dividing density structure.
Let $f:Y\rightarrow X$ be a surjective proper log étale monomorphism, and let $Y_0\in D_d^{div}(Y)$. Then $Y_0\in D_d^{div}(X)$ by Lemma \[A.9.52\], and $$Y_0\times_X Y\cong Y_0\times_Y\times (Y\times_X Y)\cong Y_0$$ since $f$ is a monomorphism. Then $$Q_1=\begin{tikzcd}
\emptyset\arrow[d]\arrow[r]&
Y_0\arrow[d,"{\rm id}"]\arrow[d]
\\
\emptyset \arrow[r]&Y_0
\end{tikzcd}$$ gives the desired distinguished square in Definition \[A.9.64\].
\[A.9.60\] The strict Nisnevich, dividing, and dividing Nisnevich cd-structures on $lSm/S$ and $lSch/S$ are complete, regular, and quasi-bounded with respect to the dividing density structure.
These cd-structures are complete by [@Vcdtop Lemma 2.5]. By Propositions \[A.9.50\] and \[A.9.59\], the strict Nisnevich cd-structure and the dividing cd-structure are quasi-bounded with respect to the dividing density structure. The union of two cd-structures that are quasi-bounded with respect to the same density structure is again a quasi-bounded cd-structure as in [@Vcdtop Lemma 2.24]. Thus the dividing Nisnevich cd-structure is also quasi-bounded with respect to the dividing density structure.
A square is a strict Nisnevich distinguished square if and only if every morphism in the square is strict and the square of underlying schemes is a Nisnevich distinguished square. Therefore the strict Nisnevich cd-structure is regular since the Nisnevich cd-structure is regular by [@Vunst-nis-cdh Lemma 2.14]. The dividing cd-structure is regular since if $f:Y\rightarrow X$ is a log modification, then $f$ is a monomorphism, so the diagonal morphism $Y\rightarrow Y\times_X Y$ is an isomorphism. Thus the dividing Nisnevich cd-structure is regular by [@Vcdtop Lemma 2.12]
Flasque simplicial presheaves
-----------------------------
In this section we generalize Voevodsky’s [@Vcdtop Lemma 3.5] to complete, quasi-bounded, and regular cd-structures. As an application, we discuss a bounded descent structure on sheaves of abelian groups.
\[A.10.1\] A morphism $f:Y\rightarrow X$ in a category $\cC$ is *squareable*[^2] if for every morphism $X'\rightarrow X$ the fiber product $Y\times_X X'$ is representable. If $f$ is squareable, then let $\check{C}(Y/X)$ (or $\check{C}(Y)$ when no confusion seems likely to arise) denote the corresponding Čech nerve $$\check{C}(Y/X):=\left(\cdots \substack{\longrightarrow\\[-1em] \longrightarrow \\[-1em] \longrightarrow} Y\times_X Y\rightrightarrows X\right).$$ Let $\check{C}(f)$ denote the morphism $\check{C}(Y)\rightarrow X$ induced by the structure morphisms $Y\times_X \cdots \times_X Y\rightarrow Y$.
A cd-structure $P$ in a category $\cC$ with an initial object is *squareable* if the following conditions are satisfied.
1. Every morphism from the initial object is an isomorphism.
2. For every distinguished square $Q$ of the form and morphism $X'\rightarrow X$, the square $Q\times_X X'$ is representable and belongs to $P$.
Note that any squareable cd-structure is complete by [@Vcdtop Lemma 2.5]
\[A.10.8\] Let $P$ be a cd-structure on a category $\cC$ with an initial object. A commutative square $$\label{A.10.8.1}
\mathscr{Q}
=
\begin{tikzcd}
\mathscr{Y}'\arrow[d]\arrow[r]&\mathscr{Y}\arrow[d]\\
\mathscr{X}'\arrow[r]&\mathscr{X}
\end{tikzcd}$$ of simplicial objects of $\cC$ is called a [*level-wise distinguished square*]{} if the induced square $$\mathscr{Q}_n
=
\begin{tikzcd}
\mathscr{Y}_n'\arrow[d]\arrow[r]&\mathscr{Y}_n\arrow[d]\\
\mathscr{X}_n'\arrow[r]&\mathscr{X}_n
\end{tikzcd}$$ is a distinguished square for any $n\in \N$.
In the following we consider a variant of the notion of a B.G. functor introduced in [@Vcdtop Definition 3.1] Here B.G. is shorthand for Brown-Gersten.
\[A.10.2\] Let $P$ be a squareable cd-structure on a category $\cC$ with an initial object. An [*extended B.G. functor*]{} on $\cC$ with respect to $P$ is a family of contravariant functors $\{T_q\}_{q\in \Z}$ from the category of simplicial objects in $\cC$ to the category of pointed sets together with pointed maps $$\partial_{\mathscr{Q}}:T_{q+1}(\mathscr{Y}')\rightarrow T_q(\mathscr{X})$$ for any level-wise distinguished square $\mathscr{Q}$ of the form subject to the following conditions.
1. The pointed maps $\partial_{\mathscr{Q}}$ are natural with respect to the morphisms $\mathscr{Q}'\rightarrow \mathscr{Q}$ of level-wise distinguished squares.
2. For any $q\in \Z$ and any level-wise distinguished square $\mathscr{Q}$ of the form , the induced sequence of pointed sets $$T_{q+1}(\mathscr{Y}')\stackrel{\partial_{\mathscr{Q}}}
\rightarrow
T_q(\mathscr{X})\rightarrow T_q(\mathscr{Y})\times T_q(\mathscr{X}')$$ is exact.
3. For any $t_P$-covering sieve $\{U_i\rightarrow X\}_{i\in I}$, the induced morphism $$T_q(X)\rightarrow T_q(\check{C}(U))$$ is an isomorphism, where $U=\amalg_{i\in I}U_i$.
In [@Vcdtop Definition 3.1] only the conditions (i) and (ii) are needed in the definition of B.G. functor. The reason why we introduce the condition (iii) is to take advantage of extending $\{T_q\}_{q\in \Z}$ to simplicial objects.
For any morphism $f:Y\rightarrow X$, let $f^*:T_q(X)\rightarrow T_q(Y)$ denote the induced pullback morphism. Let $D$ be a density structure on $\cC$. For $d\in \N$ and $q\in \Z$, we consider the following condition.
1. For any $X\in \cC$ and $a\in T_q(X)$, there exists $u:U\rightarrow X$ in $D_d(Y)$ such that the restriction of $a$ in $\check{C}(U)$ is trivial.
For every $q\in \Z$, let $a_{t_P}T_q$ be the $t_P$-sheaf on $\cC$ associated with the presheaf $$(X\in \cC)\mapsto T_q(X).$$
\[A.10.3\] Let $P$ be a squareable and regular cd-structure reducing with respect to a density structure $D$ on a category $\cC$ with an initial object, let $\{T_q\}_{q\in \Z}$ be an extended B.G. functor, and let $d\in \N$ and $q\in \Z$. If $a_{t_P}T_{q}$ is trivial, then $({\rm Cov}_{d,q+1})$ implies $({\rm Cov}_{d+1,q})$.
Subject to a few changes we follow the proof of [@Vcdtop Theorem 3.2]. By forming refinements, we may assume that every distinguished square is reducing with respect to a density structure $D$. Let $a\in T_q(X)$, where $X$ is an object of $\cC$.
Consider a distinguished square $$\label{A.10.3.1}
Q
=
\begin{tikzcd}
Y'\arrow[d,"f'"']\arrow[r,"g'"]&Y\arrow[d,"f"]\\
X'\arrow[r,"g"]&X.
\end{tikzcd}$$ First, let us assume there exist morphisms $v:Y_0\rightarrow Y$ in $D_{d+1}(Y)$ and $u':X_0'\rightarrow X'$ in $D_{d+1}(X')$ such that $a$ restructs trivially on $\check{C}(Y_0)$ and $\check{C}(X_0')$. Since $Q$ is reducing, there exists a morphism $Q_1\rightarrow Q$ of distinguished triangles such that $Y_1\rightarrow Y$ (resp. $X_1'\rightarrow X'$) in the upper right (resp. lower left) corner factors through $Y_0$ (resp. $X_0'$), and the morphism $X_1\rightarrow X$ in the lower right corner belongs to $D_{d+1}(X)$. Then $a$ restricts trivially on $\check{C}(Y_1)$ and $\check{C}(X_1')$. According to condition (ii) in Definition \[A.10.2\], there exists an element $b\in T_{q+1}(\check{C}(Y_1'))$ such that $\partial_{\mathscr{Q}}(b)=a$. By $({\rm Cov}_{d,q+1})$, there exists an object $Y_2'\in D_d(Y_1')$ such that $b$ restricts trivially on $\check{C}(Y_1')$. Now $Q_1$ is reducing, so we have a morphism $Q_3\rightarrow Q_1$ of distinguished squares such that $Y_3'\rightarrow Y_1'$ factors through $Y_2'$ and $X_3\in D_{d+1}(X_1)$. Note that $b$ restricts trivially on $\check{C}(Y_3')$, and hence likewise for $a$ and $\check{C}(X_3)$. Since $X_1\in D_{d+1}(X)$ and $X_3\in D_{d+1}(X_1)$, we deduce that $X_3\in D_{d+1}(X)$, i.e., $(\mathrm{Cov}_{d+1,q})$ is satisfied, which concludes the proof in this special case.
In the general case, since $a_{t_P}T_{q}$ is trivial, there exists a $t_P$-covering sieve $$\cF=\{V_j\rightarrow X\}_{j\in J}$$ such that $a$ restricts trivially on $V_j$ for all $j\in J$. Since $P$ is complete, we may assume the covering is a simple covering. By excluding the trivial cover we may assume that $\cF$ is formed by simple coverings of $Y$ and $X'$ for a distinguished square of the form . If there exists morphisms $v:Y_0\rightarrow Y$ in $D_{d+1}(Y)$ and $u':X_0'\rightarrow X'$ in $D_{d+1}(X')$ such that $a$ restricts trivially on $\check{C}(Y_0)$ and $\check{C}(X_0')$, using the above paragraph, there exists a morphism $u:U\rightarrow X$ in $D_d(Y)$ such that $a$ restricts trivially on $\check{C}(U)$. Repeating this argument, we reduce to the case when $\cF$ is the trivial cover. Then clearly $a$ is trivial, and we are done.
\[A.10.4\] Let $P$ be a squareable and regular cd-structure that is quasi-bounded with respect to a density structure $D_{\ast}$ on a category $\cC$ with an initial object, and suppose $\cF$ is a $t_P$-sheaf of abelian groups. Then for any $X\in \cC$ and $n>\dim_D X$, the $n$th $t_P$-cohomology group vanishes $$H_{t_P}^n(X,F)=0.$$
For any simplicial object $\mathscr{X}$ in $\cC$, set $T_q(\mathscr{X}):=\bH_{t_P}^{-q}(\mathscr{X},\cF)$. There is a canonical hypercohomology spectral sequence $$E_1^{pq}:=H_{t_P}^q(\mathscr{X}_p,\cF)\Rightarrow \bH_{t_P}^{p+q}(\mathscr{X},\cF).$$ Together with [@Vcdtop Lemma 2.19] we deduce that $\{T_q\}_{q\in \Z}$ satisfies the conditions (i) and (ii) in Definition \[A.10.2\]. For any finite $t_P$-covering sieve $\{U_i\rightarrow X\}_{i\in I}$, the induced morphism $U\rightarrow X$ forms a $t_P$-covering sieve, where $X:=\amalg_{i\in I}U_i$. Let $\mathscr{U}$ denote the Čech nerve associated with $U\rightarrow X$. By cohomological descent we obtain an isomorphism $$H_{t_P}^{-q}(X,\cF)\rightarrow H_{t_P}^{-q}(\mathscr{U},\cF).$$ That is, $\{T_q\}_{q\in \Z}$ satisfies condition (iii) in Definition \[A.10.2\]. Therefore $\{T_q\}_{q\in \Z}$ is an extended B.G. functor. Moreover, for every $q<0$ we have $$a_{t_P}T_q=a_{t_P}\bH_{t_P}^{-q}=0.$$ Since $\emptyset\in D_0(X)$, we see that ${(\rm Cov}_{0,0})$ holds. Then by Lemma \[A.10.3\], ${(\rm Cov}_{n,-n})$ holds for any $n\in \N$, i.e., there exists $j:U\rightarrow X$ in $D_n(Y)$ such that $a\in H^n(X,F)$ restricts trivially on $\check{C}(U)$. In the range $n>\dim_D X$, the morphism $j$ is a $t_P$-covering sieve. Thus $a$ is trivial owing to condition (iii) in Definition \[A.10.2\].
\[A.10.5\] Let $P$ be a squareable, quasi-bounded, and regular cd-structure on a category $\cC$ with an initial object, and let $\{T_q\}_{q\in \Z}$ be an extended B.G. functor. If $a_{t_P}T_q$ and $T_q(\emptyset)$ are trivial for any $q$, then $T_q$ is trivial for any $q$.
Since $T_q(\emptyset)$ is trivial for any $q$, we see that $({\rm Cov}_{0,q})$ holds for all $q$. Applying Lemma \[A.10.3\] we deduce that $({\rm Cov}_{d,q})$ holds for all $d$ and $q$. Thus for all $d$, $q$, $X\in \cC$, and $a\in T_q(X)$, there exists a morphism $u:U\rightarrow X$ in $D_d(Y)$ such that $a$ restricts trivially on $\check{C}(U)$. In the range $d>\dim_D(X)$, we have $u=\amalg_{i\in I}u_i$ for some finite $t_P$-covering sieve $\{u_i:U_i\rightarrow X\}_{i\in I}$. It follows that $a$ is trivial owing to condition (iii) in Definition \[A.10.2\].
\[A.10.6\] Let $P$ be a cd-structure on a category $\cC$ with an initial object. A simplicial presheaf $\cF$ on $\cC$ is called [*flasque*]{} (with respect to $P$) if $F(\emptyset)$ is contractible and for any distinguished square $Q$, the square $F(Q)$ is homotopy cartesian.
\[A.10.9\] Let $\cC$ be a category equipped with a topology $t$. A morphism of simplicial presheaves $f:\cF\rightarrow \cF'$ is called a [*$t$-local equivalence*]{} if the following conditions are satisfied.
1. The map $a_t\pi_0(\cF)\rightarrow a_t\pi_0(\cF')$ induced by $f$ is an isomorphism.
2. The map $a_t\pi_n(\cF,x)\rightarrow a_t\pi_n(\cF',f(x))$ induced by $f$ is an isomorphism for any $n\in \N^+$, $X\in \cC$, and $x\in F(X)$.
Note that pointwise weak equivalences are exactly local equivalences for the trivial topology.
\[A.10.7\] Let $P$ be a squareable, quasi-bounded, and regular cd-structure on a category $\cC$ with an initial object. Then a morphism $\cF\rightarrow \cG$ of flasque simplicial presheaves is a $t_P$-local equivalence if and only if it is a pointwise weak equivalence.
The if part is trivial, so assume that $\cF\rightarrow \cG$ is a $t_P$-local equivalence. By the first two paragraphs of the proof of [@Vcdtop Lemma 3.5], we reduce to the case when $G={\rm pt}$ and $\cF$ is a Kan simplicial presheaf. The fourth paragraph of the proof of [@Vcdtop Lemma 3.5] shows that $\cF({\rm pt})$ is nonempty. Choose an element $x\in \cF({\rm pt})$ and consider the family of functors $$T_q(\mathscr{X}):=\pi_q(\cF(\mathscr{X}),x|_\mathscr{X}),$$ where $q\in \Z$ and $\mathscr{X}$ is a simplicial object in $\cC$ (by convention, $\pi_0(T,x):=\pi_0(T)$ and $\pi_q(T,x):=0$ for any simplicial set $T$ and $x\in \pi_0(T)$ if $q<0$). Since $\cF$ is flasque, $T_q$ satisfies the conditions (i) and (ii) of Definition \[A.10.2\]. By [@Vcdtop Corollary 5.10], $T_q$ satisfies condition (iii) of Definition \[A.10.2\], i.e., $T_q$ is an extended B.G. functor. Since $\cF\rightarrow {\rm pt}$ is a $t_P$-local equivalence, $a_{t_P}T_q$ is trivial for any $q$. By Theorem \[A.10.5\] we deduce $T_q$ is trivial for all $q$ since $T_q(\emptyset)$ is trivial for all $q$.
Let $P$ be a squareable, quasi-bounded, and regular cd-structure on a category $\cC$ with an initial object, and let $t_P$ be the topology associated with $P$. For an object $T\in \cC$ let $\Lambda(T)$ be the representable presheaf of $\Lambda$-modules given by $$Z\mapsto \Lambda(T)(Z):=\Lambda\hom_{\cC}(Z,T).$$ Let $\cH_{pre}'$ be the family of complexes of the form $$\label{A.10.10.2}
\Lambda(Y')\rightarrow \Lambda(Y)\oplus \Lambda(X')\rightarrow \Lambda(X)$$ induced by the distinguishes squares of $P$, as in .
\[A.10.10\] Let $P$ be a squareable, quasi-bounded, and regular cd-structure on a category $\cC$ with an initial object, and let $t_P$ be the topology associated with $P$. Then a complex of presheaves of $\Lambda$-modules $\cF$ is $\cH_{pre}'$-flasque if and only if for every $X\in \cC$ and $i\in \Z$ there is an isomorphism $$\hom_{\mathbf{K}(\Psh(\cC,\Lambda))}(\Lambda(X)[i],\cF)
\cong
\hom_{\Deri(\Shv_{t}(\cC,\Lambda))}(a_t^*\Lambda(X)[i],a_t^*\cF).$$
This follows by replacing [@Vcdtop Lemma 3.5] with Lemma \[A.10.7\] and adapting [@Vcdtop Lemma 4.3] in the proof of [@MR2415380 Theorem 3.7].
\[A.10.11\] Let $P$ be a squareable, quasi-bounded, and regular cd-structure on a category $\cC$ with an initial object. Then a morphism $f:\cF\rightarrow \cF'$ of $\cH_{pre}'$-flasque complexes of presheaves of $\Lambda$-modules is a quasi-isomorphism if and only if $a_t^*f:a_t^*\cF\rightarrow a_t^*\cF'$ is a quasi-isomorphism.
If $f$ is a quasi-isomorphism, then $a_t^*f$ is a quasi-isomorphism since the sheafification functor $a_t^*$ is exact. Conversely, if $a_t^*f$ is a quasi-isomorphism, then owing to Corollary \[A.10.10\] we have an isomorphism $$\hom_{\mathbf{K}(\Psh(\cC,\Lambda))}(\Lambda(X)[i],\cF)
\cong
\hom_{\mathbf{K}(\Psh(\cC,\Lambda))}(\Lambda(X)[i],\cF')$$ for every $X\in \cC$ and $i\in \Z$. This implies that $f$ is a quasi-isomorphism.
\[A.8.30\] Let $P$ be a squareable, quasi-bounded, and regular cd-structure on a category $\cC$ with an initial object, and let $t_P$ be the topology associated with $P$. Recall the collection $\cG$ in Example \[A.8.28\] and $\cH_{pre}'$ in . Let $\cH'$ be the family of complexes of the form $$a_t^*\Lambda(Y')\rightarrow a_t^*\Lambda(Y)\oplus a_t^*\Lambda(X')\rightarrow a_t^*\Lambda(X)$$ induced by the distinguishes squares of $P$, as in . Suppose that the complex of $t_P$-sheaves $\cF$ is $\cH'$-flasque. By adjunction this implies that $a_{t*}\cF$ is $\cH_{pre}'$-flasque. Owing to Corollary \[A.10.10\] we see that $\cF$ is $\cG$-local. Thus $(\cG,\cH')$ is a bounded descent structure because $\cH'$ consists of bounded complexes of objects that are finite sums of objects of $\cG$. This is genuinely different from the descent structure $(\cG,\cH)$ discussed in Example \[A.8.28\].
Sheaves with logarithmic transfers {#sec.sheavestransfer}
==================================
The following three fundamental results are due to Voevodsky [@MVW §6].
1. The Nisnevich sheafification of a presheaf with transfers has a unique transfer structure.
2. The category of Nisnevich sheaves with transfers $\Shvtrkl$ is a Grothendieck abelian category.
3. For $\cF\in \Shvtrkl$ and $X\in Sm/k$ there is a canonical isomorphism (see below for the definition of $\Ztr$) $${
\rm Ext}_{\Shvtrkl}(\Ztr(X),F)\cong H_{Nis}^i(X,\cF).$$
In this section, we develop the log versions of (1)–(3), and show that the following topologies are compatible with log transfers: $$sNis,\;dNis,\;s\acute{e}t,\;d\acute{e}t,\;k\acute{e}t,\;\text{and }l\acute{e}t.$$
Category of presheaves with log transfers
-----------------------------------------
Recall $\Lambda$ is a commutative unital ring. Having introduced $lCor/k$ the next step is to study presheaves with log transfers in the following sense.
\[A.5.37\] A [*presheaf of $\Lambda$-modules with log transfers*]{} is an additive presheaf of $\Lambda$-modules on the category of finite log correspondences $\LCor$. We let $\Pshltrkl$ denote the category of presheaves of $\Lambda$-modules with log transfers. For $X\in lSm/k$, let $\Zltr(X)$ denote the representable presheaf of $\Lambda$-modules with log transfers given by $$Y\mapsto
\Zltr(X)(Y):=\lCor(Y,X)\otimes \Lambda.$$
\[A.5.38\] Let $F$ be a presheaf of $\Lambda$-modules with log transfers. For every $X,Y\in lSm/k$, by additivity, there exists an induced paring $$\lCor(X,Y)\otimes F(Y)
\rightarrow
F(X).$$ This produces the “wrong way” maps parametrized by finite log correspondences.
Every constant presheaf of $\Lambda$-modules on $lSm/k$ has a canonical log transfer structure.
\[example log transfer Ga\] The presheaves $\cO$ and $\cO^*$ are two examples of presheaves with transfers. Let us extend these to presheaves with log transfers. For $X\in lSm/k$, we simply set $$\cO(X):=\cO(\underline{X}),\;\cO^*(X):=\cO^*(\underline{X}).$$ Owing to Lemma \[A.9.18\], $X$ is normal. For any strict finite surjective morphism $V\rightarrow X$, as observed in [@MVW Example 2.4], there is a canonical trace map and a norm map $${\rm Tr}:\cO(V)\rightarrow \cO(X),\;N:\cO^*(V)\rightarrow \cO^*(X).$$ If $V$ is a finite log correspondence from $X$ to $Y\in lSm/k$, we can define transfer maps as the compositions $$V^*:\cO(Y)\rightarrow \cO(V)\stackrel{\rm Tr}\rightarrow \cO(X),\;V^*:\cO^*(Y)\rightarrow \cO^*(V)\stackrel{N}\rightarrow \cO^*(X)$$ Let us check that these transfer maps are compatible with compositions of finite log correspondences. Since $X$ is reduced, we have the inclusions $$\cO(X)\subset \cO(X-\partial X),\;\cO^*(X)\subset \cO^*(X-\partial X).$$ Thus for $Z\in lSm/k$ and $W\in \lCor(Y,Z)$ to check that $V^*\circ W^*=(W\circ V)^*$, owing to Lemma \[A.5.10\], it suffices to check that $$(V-\partial V)^*\circ (W-\partial W)^*=((W-\partial W)\circ (V-\partial V))^*.$$ This is already observed in [@MVW Example 2.4]. Thus $\cO$ and $\cO^*$ are both examples of presheaves with log transfers.
Let $\cM^{\rm gp}$ denote the presheaf of abelian groups on $lSm/k$ given by $$\cM^{\rm gp}(X):=\cM_X^{\rm gp}(X).$$ By [@Ogu Theorem IV.3.5.1], $X\in lSm/k$ is log regular. By [@Ogu Theorem III.1.11.12], the log structure on $X$ is the compactifying log structure associated with the inclusion $X-\partial X\rightarrow X$. In particular, there is a canonical isomorphism $$\cM^{\rm gp}(X)\cong \cO^*(X-\partial X).$$ For $Y\in lSm/k$ and $V\in \lCor(X,Y)$, we can now define a transfer map $$V^*\colon \cM^{\rm gp}(X)\rightarrow \cM^{\rm gp}(Y)$$ because there is a transfer map $$(V-\partial V)^*\colon \cO^*(X-\partial X)\rightarrow \cO^*(Y-\partial Y).$$ As in the above paragraph, $V^*$ is compatible with compositions of correspondences. Thus $\cM^{\rm gp}$ is an example of a presheaf with log transfers.
Next we give $\Pshltrkl$ the structure of a closed symmetric monoidal category following the work of Day [@MR0272852] in a general setting and [@MR2435654] for presheaves with transfers.
\[A.5.40\] The tensor product of two representable presheaves with log transfers on $lSm/k$ is defined by $$\Zltr(X)\otimes \Zltr(Y)
:=
\Zltr(X\times_{k}Y).$$
More generally, we define the tensor product of $F,G\in \Pshltrkl$ by $$F\otimes G
:=
\varinjlim_{X,Y}\Zltr(X)\otimes \Zltr(X).$$ Here we write $F$ and $G$ as colimits of representable presheaves $$F\cong \varinjlim_X \Zltr(X)
\text{ and }
G\cong \varinjlim_Y \Zltr(Y),$$ This gives a symmetric monoidal structure on $\Pshltrkl$.
The internal hom object $\underline{\hom}_{\Pshltrkl}(F,G)$ is the presheaf with log transfers on $lSm/k$ defined by $$X
\mapsto
\underline{\hom}_{\Pshltrkl}(F,G)(X)
:=
\hom_{\Pshltrkl}(F\otimes \Zltr(X),G).$$ The functor $\underline{\hom}_{\Pshltrkl}(F,-)$ is right adjoint to $(-)\otimes F$.
Let $f\colon X\rightarrow Y$ be a morphism of fs log schemes log smooth over $k$. Then the graph $\gamma_f$ is a finite log correspondence from $X$ to $Y$. To show this we may assume $X$ and $Y$ are irreducible. Note that $\gamma_f$ is a closed subscheme (see Definition \[df::closedimmersion\]) of $X\times Y$. The morphism $\gamma_f\rightarrow X$ is an isomorphism. Thus $\gamma_f$ is an elementary log correspondence from $X$ to $Y$ by Example \[A.5.31\](4). If $g\colon Y\rightarrow Z$ is another morphism of fs log schemes log smooth over $k$, then by construction $\gamma_g\circ \gamma_f=\gamma_{g\circ f}$. Thus there is a faithful functor $$\label{A.5.40.1}
\gamma
\colon
lSm/k\rightarrow \LCor$$ given by $$X\mapsto X,
\,
(f\colon X\rightarrow Y) \mapsto \gamma_f.$$
\[A.5.4\] Let $\Pshlogkl$ be the category of presheaves of $\Lambda$-modules on $lSm/k$. For an fs log scheme $X$ of finite type over $k$, we let $\Lambda(X)$ be the presheaf of $\Lambda$-modules on $lSm/k$ given by $$Y
\mapsto
\Lambda(X)(Y):=\Lambda\hom_{lSm/k}(Y,X).$$ The restriction (forgetting log transfer) functor $$\gamma^*
\colon
\Pshltrkl
\rightarrow
\Pshlogkl$$ is defined by $\gamma^*F(X):=F(\gamma(X))$ for $F\in \Pshltrkl$ and $X\in lSm/k$.
According to [@SGA4 Proposition I.5.1] there exist adjoint functors $$\begin{tikzcd}
\Pshlogkl\arrow[rr,shift left=1.5ex,"\gamma_\sharp "]\arrow[rr,"\gamma^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"\gamma_*"']&&\Pshltrkl.
\end{tikzcd}$$ By convention $\gamma_\sharp$ is left adjoint to $\gamma^*$ and $\gamma^*$ is left adjoint to $\gamma_*$. This sequence of adjoint functors induces a similar sequence of adjoint functors for the categories of chain complexes $$\begin{tikzcd}
\Co(\Pshlogkl)\arrow[rr,shift left=1.5ex,"\gamma_\sharp "]\arrow[rr,"\gamma^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"\gamma_*"']&&\Co(\Pshltrkl).
\end{tikzcd}$$
We shall use similar notations for the category $Cor/k$ of finite correspondences over the field $k$. Note that the functor $\hat{\gamma}_*$ in [@CD12 Definition 10.1.1] corresponds to our $\gamma^*$. Let $\Pshtrkl$ denote the category of presheaves of $\Lambda$-modules with transfers. For any smooth scheme $X$ of finite type over $k$, the representable presheaf of $\Lambda$-modules with transfers $\Ztr(X)$ on $Sm/k$ is given by $$Y\mapsto
\Ztr(X)(Y)
:=
\Cor(Y,X)\otimes \Lambda.$$
We write $\Pshkl$ for the category of presheaves of $\Lambda$-modules on $Sm/k$. For any smooth scheme $X$ of finite type over $k$, we denote by $\Lambda(X)$ the presheaf of $\Lambda$-modules on $Sm/k$ defined by $$Y\mapsto
\Lambda(X)(Y)
:=
\Lambda\hom_{Sm/k}(Y,X)).$$ In this setting there is a faithful functor $$\gamma
\colon
Sm/k
\rightarrow
Cor/k,$$ and corresponding sequences of adjoint functors $$\label{A.5.4.1}
\begin{tikzcd}
\Pshkl\arrow[rr,shift left=1.5ex,"\gamma_\sharp "]\arrow[rr,"\gamma^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"\gamma_*"']&&\Pshtrkl,
\end{tikzcd}$$ $$\begin{tikzcd}
\Co(\Pshkl)\arrow[rr,shift left=1.5ex,"\gamma_\sharp "]\arrow[rr,"\gamma^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"\gamma_*"']&&\Co(\Pshkl).
\end{tikzcd}$$
Next we define functors between the categories of finite log correspondences and finite correspondences.
\[A.4.5\] Let $$\omega
\colon
\LCor
\rightarrow
Cor/k$$ be the functor that sends an object $X$ to $X-\partial X$ and a morphism $V\in \lCor(X,Y)$ to $V-\partial V\in \Cor(X-\partial X,Y-\partial Y)$ for all $X,Y\in lSm/k$.
Conversely, let $$\lambda
\colon
Cor/k
\rightarrow
\LCor$$ be the functor sending an object $X$ to $X\in lSm/k$ equipped with the trivial log structure and a morphism $V\in\Cor(X,Y)$ to $V\in l\Cor(X,Y)$ for all $X,Y\in Sm/k$, see Example \[A.5.31\].
There are induced functors $$\omega^*
\colon
\Pshtrkl\rightarrow \Pshltrkl
\text{ and }
\lambda^*
\colon
\Pshltrkl\rightarrow \Pshtrkl$$ given by $\omega^*F(X):=F(\omega(X))$ for $F\in \Pshtrkl$ and $X\in lSm/k$ and $\lambda^*G(Y):=G(\lambda(Y))$ for $G\in \Pshltrkl$ and $Y\in Sm/k$. Example \[A.5.31\](1) shows that $\lambda^*$ is left adjoint to $\omega^*$.
Again by [@SGA4 Proposition I.5.1] there exist adjoint functors $$\label{A.4.5.1}
\begin{tikzcd}
\Pshtrkl\arrow[rr,shift left=3ex,"\lambda_\sharp "]\arrow[rr,"\lambda^*\cong
\omega_\sharp" description,leftarrow, shift left=1ex]\arrow[rr,shift right=1ex,"\lambda_*\cong
\omega^*" description]\arrow[rr,shift right=3ex,"\omega_*"',leftarrow]&&\Pshltrkl.
\end{tikzcd}$$ Our convention for such diagrams is that $\lambda_\sharp$ is left adjoint to $\lambda^*\cong \omega_\sharp$, which is left adjoint to $\lambda_*\cong \omega^*$, which is left adjoint to $\omega_*$. On chain complexes we use the same notation for the induced adjoint functors $$\begin{tikzcd}
\Co(\Pshtrkl)\arrow[rr,shift left=3ex,"\lambda_\sharp "]\arrow[rr,"\lambda^*\cong
\omega_\sharp" description,leftarrow, shift left=1ex]\arrow[rr,shift right=1ex,"\lambda_*\cong \omega^*" description]\arrow[rr,shift right=3ex,"\omega_*"',leftarrow]&&\Co(\Pshltrkl).
\end{tikzcd}$$
Let $$\omega
\colon
lSm/k\rightarrow Sm/k$$ be the functor that sends an object $X$ to $X-\partial X$ and a morphism $f\colon X\rightarrow Y$ to the naturally induced morphism $X-\partial X\rightarrow Y-\partial Y$ for all $X,Y\in lSm/k$.
Conversely, let $$\lambda
\colon
Sm/k\rightarrow lSm/k$$ be the functor sending $X\in Sm/k$ to $X$ equipped with the trivial log structure and a morphism $f\colon X\rightarrow Y$ to itself for all $X,Y\in Sm/k$.
With these definitions there are associated adjoint functors $$\begin{tikzcd}
\Pshkl\arrow[rr,shift left=3ex,"\lambda_\sharp "]\arrow[rr,"\lambda^*\cong
\omega_\sharp" description,leftarrow, shift left=1ex]\arrow[rr,shift right=1ex,"\lambda_*\cong \omega^*" description]\arrow[rr,shift right=3ex,"\omega_*"',leftarrow]&&\Pshlogkl,
\end{tikzcd}$$ $$\begin{tikzcd}
\Co(\Pshkl)\arrow[rr,shift left=3ex,"\lambda_\sharp "]\arrow[rr,"\lambda^*\cong
\omega_\sharp" description,leftarrow, shift left=1ex]\arrow[rr,shift right=1ex,"\lambda_*\cong \omega^*" description]\arrow[rr,shift right=3ex,"\omega_*"',leftarrow]&&\Co(\Pshlogkl),
\end{tikzcd}$$ and a commutative diagram $$\begin{tikzcd}
lSm/k\arrow[d,"\omega"']\arrow[r,"\gamma"]& lCor/k\arrow[d,"\omega"]\\
Sm/k\arrow[r,"\gamma"]& Cor/k.
\end{tikzcd}$$
\[A.5.24\] Suppose $f\colon Y\rightarrow X$ is a morphism of fs log schemes log smooth over $k$. Let $\Zltr(Y\rightarrow X)$ (resp. $\Lambda(Y\rightarrow X)$) denote the complex $$\Zltr(Y)\stackrel{\Zltr(f)}\longrightarrow \Zltr(X) \;\;(\text{resp.}\ \Lambda(Y)\stackrel{\Lambda(f)}\longrightarrow \Lambda(X)).$$
Suppose $\mathscr{X}$ is a simplicial fs log scheme over $k$. Let $\Zltr(\mathscr{X})$ (resp. $\Lambda(\mathscr{X})$) denote the complex associated with the simplicial object $$i
\mapsto
\Zltr(\mathscr{X}_i)
\;\;(\text{resp.}\
i\mapsto \Lambda(\mathscr{X}_i))$$ under the Dold-Kan correspondence.
Similarly, when $Y\rightarrow X$ is a morphism of schemes smooth over $k$ and $\mathscr{X}$ is a simplicial scheme over $k$, we use the notations $\Lambda(Y\rightarrow X)$, $\Ztr(Y\rightarrow X)$, $\Lambda(\mathscr{X})$, and $\Ztr(\mathscr{X})$.
Compatibility with log transfers
--------------------------------
In this section we shall discuss several notions of compatibility with log transfers for topologies on log schemes. We proceed by adapting to the log setting the analogous notions introduced by Cisinski-Deglise in [@CD12 Definition 10.3.2].
\[A.8.1\] Let $t$ be a topology on $lSm/k$ and $\Lambda$ a commutative unital ring. A presheaf $\cF$ with log transfers is called a [*$t$-sheaf with log transfers*]{} if $\gamma^*\cF\in \Pshlogkl$ is a $t$-sheaf. Here, $\gamma^*\cF(X):=\cF(\gamma(X))$ for $X\in lSm/k$. Let $\Shvltrtkl$ denote the category of $t$-sheaves with log transfers.
Let $\Shvlogtkl$ be the category of $t$-sheaves on $lSm/k$ (without log transfers). The $t$-sheafification functor $a_t^*$ and the forgetful functor $a_{t*}$ form an adjoint functor pair $$a_t^*:\Pshlogkl\rightleftarrows \Shvlogtkl:a_{t*}.$$
By abuse of notation, let $$a_{t*}:\Shvltrtkl\rightarrow \Pshltrkl$$ be the inclusion functor, and let $$\gamma^*\colon \Shvltrtkl\rightarrow \Shvlogtkl$$ be the restriction of the functor $\gamma^*\colon \Pshltrkl\rightarrow \Pshlogkl$. We note that $$\label{A.8.1.1}
\gamma^*a_{t*}\cong a_{t*}\gamma^*.$$
For a topology $t$ on $Sm/k$, let $\Shvtrtkl$ be the category of $t$-sheaves on $Sm/k$ with transfers.
The notion of a topology on $Sm/k$ being compatible with transfers was introduced in [@CD12 Definition 10.3.2]. We adapt this notion to our log setting.
\[A.8.2\] A topology $t$ on $lSm/k$ is *compatible with log transfers* if for any $t$-hypercover $\mathscr{X}\rightarrow X$, the induced morphism $$a_t^*\gamma^*\Zltr(\mathscr{X})\rightarrow a_t^*\gamma^*\Zltr(X)$$ of complexes of presheaves is a quasi-isomorphism (see also Definition \[A.5.24\]).
Following [@CD12 Definition 10.3.5], we say that $t$ is [*mildly compatible with log transfers*]{} if $\gamma_*a_{t*}\cF$ is a $t$-sheaf with log transfers for any $t$-sheaf $\cF$. Recall that $\gamma_*$ is a right adjoint of $\gamma^*$. There is a description of $\gamma_*$: For every presheaf $\cG$ and $X\in lSm/k$, $$\gamma_*\cG(X)=\hom_{\Pshlogkl}(\gamma^*\Zltr(X),\cG).$$
These notions are equivalent to each other if $t$ is a topology associated with a cd-structure, see Proposition \[A.8.13\].
According to [@CD12 10.4.1], the Nisnevich topology on $Sm/k$ is compatible with transfers. One goal in this section is to generalize this to our log setting, i.e., we will show that the strict Nisnevich topology on $lSm/k$ is compatible with log transfers.
\[A.8.3\] Let $t$ be a topology on $lSm/k$ compatible with log transfers. Then $t$ is mildly compatible with log transfers.
Let $\mathscr{X}\rightarrow X$ be a Čech $t$-hypercover, and let $\cF$ be a $t$-sheaf. To show that $\gamma_*a_{t*}\cF$ is a $t$-sheaf, we need to show that the induced sequence of $\Lambda$-modules $$\begin{split}
0\rightarrow \hom_{\Pshltrkl}(\Zltr(X),\gamma_*a_{t*}\cF)\rightarrow &\hom_{\Pshltrkl}(\Zltr(\mathscr{X}_0),\gamma_*a_{t*}\cF)\\
\rightarrow &\hom_{\Pshltrkl}(\Zltr(\mathscr{X}_1),\gamma_*a_{t*}\cF)
\end{split}$$ is exact. This is equivalent to showing exactness of the sequence $$\label{A.8.3.1}
\begin{split}
0\rightarrow \hom_{\Shvltrtkl}(a_t^*\gamma^*\Zltr(X),\cF)\rightarrow &\hom_{\Shvltrtkl}(a_t^*\gamma^*\Zltr(\mathscr{X}_0),\cF)\\
\rightarrow &\hom_{\Shvltrtkl}(a_t^*\gamma^*\Zltr(\mathscr{X}_1),\cF).
\end{split}$$ This follows since by assumption we have a quasi-isomorphism $$a_t^*\gamma^*\Zltr(\mathscr{X})\rightarrow a_t^*\gamma^*\Zltr(X).$$
\[A.8.12\] Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers. Then for any $t$-cover $Y\rightarrow X$, the induced morphism of $t$-sheaves $a_t^*\gamma^*\Zltr(Y)\rightarrow a_t^*\gamma^*\Zltr(X)$ is an epimorphism.
As in the proof of Proposition \[A.8.3\], this follows because the sequence is exact.
\[A.8.4\] Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers. Then the functor $$\gamma^*\colon \Shvltrtkl\rightarrow \Shvlogtkl$$ admits a right adjoint $\gamma_*$ that commutes with $a_{t*}$.
Due to the assumption that $t$ is mildly compatible with log transfers, by abuse of notation we have the functor $$\gamma_*\colon \Shvlogtkl\rightarrow \Shvltrtkl.$$ That is, the restriction of the functor $\gamma_*\colon\Pshlogkl\rightarrow \Pshltrkl$. We note that the functors $a_{t*}$ and $\gamma_*$ commute.
For $\cF\in \Shvltrtkl$ and $\cG\in \Shvlogtkl$ it remains to note that there are natural isomorphisms of $\Lambda$-modules $$\begin{split}
&\hom_{\Shvlogtkl}(\gamma^*\cF,\cG)\\
\cong &\hom_{\Pshlogkl}(a_{t*}\gamma^*\cF,a_{t*}\cG)\quad \text{(since }a_{t*}\text{ is fully faithful)}\\
\cong &\hom_{\Pshlogkl}(\gamma^*a_{t*}\cF,a_{t*}\cG)\quad \text{(since }\gamma^*\text{ commutes with }a_{t*},\text{ see }\eqref{A.8.1.1})\\
\cong &\hom_{\Pshltrkl}(a_{t*}\cF,\gamma_* a_{t*}\cG)\\
\cong &\hom_{\Pshltrkl}(a_{t*}\cF,a_{t*}\gamma_*\cG)\quad \text{(since }\gamma_*\text{ commutes with }a_{t*})\\
\cong &\hom_{\Shvltrtkl}(\cF,\gamma_*\cG)\quad \text{(since }a_{t*}\text{ is fully faithful)}.
\end{split}$$
Let us prove an analogue of [@MVW Lemma 6.16].
\[A.8.23\] Let $t$ be a topology on $lSm/k$ that is mildly compatible with log transfers. Suppose that $p\colon Y'\rightarrow Y$ is a $t$-cover and $f\in \lCor(X,Y)$ is a finite log correspondence, where $X,Y,Y'\in lSm/k$. Then there exists a $t$-cover $p'\colon X'\rightarrow X$ and a commutative diagram in $lCor/k$ $$\label{A.8.23.1}
\begin{tikzcd}
X'\arrow[d,"p'"']\arrow[r,"f'"]&Y'\arrow[d,"p"]\arrow[d]\\
X\arrow[r,"f"]&Y.
\end{tikzcd}$$
By Proposition \[A.8.12\], the induced morphism $$a_t^*\gamma^*\Zltr(Y')\rightarrow a_t^*\gamma^*\Zltr(Y)$$ in $\Shvltrtkl$ is an epimorphism. The implication (ii)$\Rightarrow$(i) in [@SGA4 Proposition II.5.1] shows that $X\times_{\Zltr(Y)}\Zltr(Y')\rightarrow X$ is a covering sieve. From this we immediately deduce the existence of $p'$ along with .
Using this lemma, we prove an analogue of [@MVW Theorem 6.17] as follows.
\[A.8.6\] Let $t$ be a topology on $lSm/k$ that is mildly compatible with log transfers. Then for every $\cF\in \Pshltrkl$ there exists an object $\cG\in \Shvltrtkl$ together with a morphism $u\colon \cF\rightarrow a_{t*}\cG$ in $\Pshltrkl$ such that $\gamma^*u$ is isomorphic to the sheafification $\gamma^*\cF\rightarrow a_{t*}a_t^*\gamma^*\cF$. The pair $(\cG,u)$ is unique up to isomorphism.
Let us first construct $\cG\in \Shvltrtkl$. The unit $\cF\rightarrow \gamma_*a_{t*}a_t^*\gamma^*\cF$ induces a morphism $$\eta\colon \gamma^*\cF\rightarrow \gamma^*\gamma_*a_{t*}a_t^*\gamma^*\cF$$ Here $\gamma^*\gamma_*a_{t*}a_t^*\gamma^*\cF$ is a $t$-sheaf because $t$ is mildly compatible with log transfers. Hence according to the universal property of the sheafification functor the latter morphism admits a factorization $$\gamma^*\cF\rightarrow a_{t*}a_t^* \gamma^* \cF\stackrel{\psi}\rightarrow \gamma^*\gamma_*a_{t*}a_t^*\gamma^*\cF$$ that is unique up to isomorphism. Next we consider the commutative diagram $$\begin{tikzcd}
\gamma^*\cF\arrow[r,"ad"']\arrow[d,"ad"]&\gamma^*\gamma_*\gamma^*\cF\arrow[d,"ad"]\arrow[r,"ad'"]&\gamma^*\cF\arrow[d,"ad"]\\
a_{t*}a_t^*\gamma^*\cF\arrow[r,"\psi"]&\gamma^*\gamma_*a_{t*}a_t^*\gamma^*\cF\arrow[r,"ad'"]&a_{t*}a_t^*\gamma^*\cF.
\end{tikzcd}$$ Here $ad$ (resp. $ad'$) is induced by a unit (resp. counit). Let $\psi'$ be the composition of the lower horizontal morphisms. Since the composition of the upper horizontal morphisms is the identity by the unit-counit identity, there is a commutative diagram $$\begin{tikzcd}
&
\gamma^*\cF\arrow[rd,"ad"]\arrow[ld,"ad"']
\\
a_{t*}a_t^*\gamma^*\cF\arrow[rr,"\psi'"]&&a_{t*}a_t^*\gamma^*\cF.
\end{tikzcd}$$ The diagram still commutes if we replace $\psi'$ by ${\rm id}$. By the universal property of sheafification the morphism $$a_{t*}a_t^*\gamma^*\cF\rightarrow a_{t*}a_t^*\gamma^*\cF$$ making the above diagram commutative is unique, and we deduce $\psi'={\rm id}$. It follows that $\psi$ is a monomorphism.
Let $f\in \lCor(X,Y)$ be a finite log correspondence, where $X,Y\in lSm/k$, and set $\cG=a_t^*\gamma^*\cF$. Then $f$ and $\psi$ induce a diagram of $\Lambda$-modules $$\begin{tikzcd}
\cG(Y)\arrow[r,"{\psi(Y)}"]& \hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(Y),\cG)\arrow[d,"\alpha_f"]\\
\cG(X)\arrow[r,"{\psi(X)}"]& \hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(X),\cG).
\end{tikzcd}$$ Since $\psi$ is a monomorphism, to show the log transfer structure on $\gamma_*\alpha_{t*}\cG$ induces a log transfer structure on $G$ it suffices to show that $$\label{A.8.6.2}
\im (\alpha_f\circ \psi(Y)) \subset \im \psi(X).$$
For every $a\in \cG(Y)$ there exists an element $b'\in \cF(Y')$ for some $t$-cover $q\colon Y'\rightarrow Y$ such that the images of $a$ and $b'$ in $\cG(Y')$ coincide with some $a'\in \cG(Y')$. According to Lemma \[A.8.23\] there exists a $t$-cover $p'\colon X'\rightarrow X$ with a commutative diagram in $lCor/k$ $$\label{A.8.6.1}
\begin{tikzcd}
X'\arrow[d,"p'"']\arrow[r,"f'"]&Y'\arrow[d,"p"]\arrow[d]\\
X\arrow[r,"f"]&Y.
\end{tikzcd}$$ Then there is a diagram of $\Lambda$-modules $$\begin{tikzcd}
\cG(Y')\arrow[r,"{\psi(Y')}"]& \hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(Y'),\cG)\arrow[d,"\alpha_{f'}"]\\
\cG(X')\arrow[r,"{\psi(X')}"]& \hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(X'),\cG).
\end{tikzcd}$$ Suppose that $(\alpha_{f'}\circ \psi(Y'))(a')=\psi(X')(d')$ for some $d'\in \cG(X')$. The two images of $a'$ in $\cG(Y'\times_Y Y')$ are equal since $\cG$ is a sheaf. Thus the two images of $\psi(X')(d')$ in $\hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(X'\times_X X'),\cG)$ are equal. Since $\psi$ is a monomorphism, this implies that the two images of $d'$ in $\cG(X'\times_X X')$ are equal. Thus there exists $d\in \cG(X)$ whose image in $\cG(X')$ is equal to $d'$ since $\cG$ is a sheaf. Using the exactness of we see that $$(\alpha_{f}\circ \psi(Y))(a) = \psi(X)(d).$$ Hence let us show that $(\alpha_{f'}\circ \psi(Y'))(a')=\psi(X')(d')$ for some $d'\in \cG(X')$.
The morphism $\eta\colon \gamma^*\cF\rightarrow \gamma^*\gamma_*a_{t*}a_t^*\gamma^*\cF$ induces a commutative diagram of $\Lambda$-modules $$\label{equation:FGdiagram}
\begin{tikzcd}
\cF(Y')\arrow[r,"\eta(Y')"]\arrow[d,"\cF(f')"']&\hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(Y'),\cG)\arrow[d,"\alpha_{f'}"]\\
\cF(X')\arrow[r,"\eta(X')"]&\hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(X'),\cG).
\end{tikzcd}$$ This implies that $(\alpha_{f'}\circ \psi(Y'))(a')=\psi(X')(d')$ for some $d'\in \cG(X')$. We deduce the claimed log transfer structure on $\cG$.
Since $\psi$ is a monomorphism, combining the diagram for $f\colon X\rightarrow Y$ together with we obtain a canonical commutative diagram of $\Lambda$-modules $$\begin{tikzcd}
\cF(Y)\arrow[d,"\cF(f)"']\arrow[r]&
\cG(Y)\arrow[d,"\cG(f)"]\arrow[d]
\\
\cF(X)\arrow[r]&
\cG(X).
\end{tikzcd}$$ Thus we obtain a morphism $u\colon \cF\rightarrow a_{t*}\cG$ of presheaves with log transfers such that $\gamma^*u$ is isomorphic to the sheafification $\gamma^*\cF\rightarrow a_{t*}a_t^*\gamma^*\cF$.
For uniqueness, suppose $\cG'$ is a $t$-sheaf with log transfers that is equipped with a morphism $u'\colon \cF\rightarrow a_{t*}\cG'$ of presheaves with log transfers such that $\gamma^*u'$ is isomorphic to the sheafification $\gamma^*\cF\rightarrow a_{t*}a_t^*\gamma^*\cF$. Since $\cG$ and $\cG'$ agree on objects, it suffices to check that $\cG(f)=\cG'(f)$ for any $f\in \lCor(X,Y)$ where $X,Y\in lSm/k$. Using the commutative diagram , we see this question is $t$-local on $X$ and $Y$. Hence after shrinking $Y$ we may assume that for any $a\in \cG(Y)$, there is an element $b\in \cF(Y)$ whose image in $\cG(Y)$ is equal to $a$. The morphisms $u\colon \cF\rightarrow a_{t*} \cG$ and $u'\colon \cF\rightarrow a_{t*}\cG'$ induce a commutative diagram of $\Lambda$-modules $$\begin{tikzcd}
\cG'(Y)\arrow[r,leftarrow]\arrow[d,"{\cG'(f)}"']&\cF(Y)\arrow[d,"\cF(f)"']\arrow[r]&\cG(Y)\arrow[d,"\cG(f)"]\\
\cG'(X)\arrow[r,leftarrow,"{u'(X)}"]&\cF(X)\arrow[r,"u(X)"]&\cG(X).
\end{tikzcd}$$ With the identification of $\cG$ and $\cG'$ on objects, the homomorphisms $u(X)$ and $u'(X)$ become equal. This allows us to conclude $$\cG(f)(a)=u(X)(\cF(f)(b))=u'(X)(\cF(f)(b))=\cG'(f)(a).$$
\[A.8.5\] Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers. Then the functor $$a_{t*}\colon \Shvltrtkl\rightarrow \Pshltrkl$$ admits a left adjoint functor $a_t^*$. Moreover, the functor $a_t^*$ is exact.
For $\cF\in \Pshltrkl$ we define $a_t^*\cF\colon =\cH$, where $\cH\in \Shvltrtkl$ is constructed as in Lemma \[A.8.6\]. The uniqueness part in Lemma \[A.8.6\] ensures that we obtain a functor $$a_t^*\colon \Pshltrkl\rightarrow \Shvltrtkl.$$ It remains to show $a_t^*$ is a left adjoint of $a_{t*}$. Again, by uniqueness, the morphism $u\colon \cF\rightarrow a_{t*}\cH$ in $\Pshltrkl$ constructed in Lemma \[A.8.6\] gives rise to a natural transformation $${\rm id}\rightarrow a_{t*}a_t^*.$$ For $\cG\in \Shvltrtkl$, the uniqueness part also implies that there is an isomorphism $$a_t^*a_{t*}\cG\cong \cG.$$ By functoriality of this isomorphism we obtain a natural transformation $$a_t^*a_{t*}\rightarrow {\rm id}.$$ Owing to the uniqueness part in Lemma \[A.8.6\], the unit and counit triangle identities follow from the corresponding identities for $a_t^*\colon \Pshlogkl\rightarrow \Shvlogtkl$ and $a_{t*}\colon \Shvlogtkl\rightarrow \Pshlogkl$.
Since $a_t^*$ is a left adjoint, $a_t^*$ is right exact. It remains to show that $a_t^*$ is left exact. Let $f\colon \cF\rightarrow \cF'$ be a monomorphism of presheaves with log transfers. The functor $\gamma^*$ is exact since $\gamma^*$ has both left and right adjoints, and the functor $$a_t^*\colon \Pshlogkl\rightarrow \Shvlogtkl$$ is known to be exact. It follows that $$a_t^*\gamma^*f\colon a_t^*\gamma^*\cF\rightarrow a_t^*\gamma^*\cF'$$ is a monomorphism of $t$-sheaves. Since $a_t^*$ commutes with $\gamma^*$ by Proposition \[A.8.4\], $$\gamma^*a_t^*f\colon \gamma^*a_t^*\cF\rightarrow \gamma^*a_t^*\cF'$$ is a monomorphism of $t$-sheaves. Let $\cG$ be a kernel of $a_t^*f\colon a_t^*\cF\rightarrow a_t^*\cF'$. It follows that $\gamma^*\cG=0$. This means that $\cG(X)=0$ for all $X\in lSm/k$, so $\cG=0$. Thus $a_t^*f$ is a monomorphism.
\[A.8.9\] Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers. Then the functor $$\gamma^*\colon \Shvltrtkl\rightarrow \Shvlogtkl$$ admits a left adjoint functor $\gamma_\sharp$.
For $\cF\in \Shvlogtkl$ and $\cG\in \Shvltrtkl$, we have isomorphisms $$\begin{split}
&\hom_{\Shvltrtkl}(a_t^*\gamma_\sharp a_{t*}\cF,\cG)\\
\cong &\hom_{\Pshltrkl}(a_{t*}\cF,\gamma^*a_{t*}\cG)\\
\cong &\hom_{\Pshltrkl}(a_{t*}\cF,a_{t*}\gamma^*\cG)\quad \text{(since }\gamma^*\text{ and }a_{t*}\text{ commute)}\\
\cong &\hom_{\Shvltrtkl}(\cF,\gamma^*\cG)\quad \text{(since }a_{t*}\text{ is fully faithful}).
\end{split}$$ By functoriality in $\cF$ and $\cG$, it follows that $a_t^*\gamma_\sharp a_{t*}$ is a left adjoint of $\gamma^*$.
We recall the definition of Grothendieck abelian category (see e.g., [@MR0389953 Chapter V] for details). An abelian category $\cA$ is called a [*Grothendieck abelian category*]{} with the following conditions.
1. $\cA$ admits small sums.
2. Filtered colimits of exact sequences in $\cA$ is exact.
3. There is a generator of $\cA$, i.e., there is an element of $G\in \cA$ such that the functor $\hom_{\cA}(G,-)\colon \cA\rightarrow {\bf Set}$ is faithful.
\[A.8.7\] Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers. Then $\Shvltrtkl$ is a Grothendieck abelian category.
Let us check the above axioms of a Grothendieck abelian category. The categories $\Pshltrkl$ and $\Shvlogtkl$ are Grothendieck abelian categories, and $\gamma^*\colon \Shvltrtkl\rightarrow \Shvlogtkl$ is exact since $\gamma^*$ has both left and right adjoints. Thus $\Shvltrtkl$ is an abelian category, and we have (i) and (ii). Since by the Yoneda lemma $$\hom_{\Shvltrtkl}(a_t^*\Zltr(X),\cF)\cong \hom_{\Pshltrkl}(\Zltr(X),\cF)\cong \cF(X),$$ the object $\bigoplus_{X\in lSm/k} a_t^*\Zltr(X)$ is a generator of $\Shvltrtkl$. Thus we have (iii).
In conclusion, if $t$ is a topology on $lSm/k$ mildly compatible with log transfers, there exist adjunctions between Grothendieck abelian categories $$\label{A.8.7.1}
\begin{tikzcd}
\Pshlogkl\arrow[d,shift left=0.75ex,leftarrow,"a_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']
\arrow[r,shift left=1.5ex,"\gamma_\sharp"]\arrow[r,leftarrow,"\gamma^*" description]
\arrow[r,shift right=1.5ex,"\gamma_*"']&\Pshltrkl
\arrow[d,shift left=0.75ex,leftarrow,"a_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"'] \\
\Shvlogtkl\arrow[r,shift left=1.5ex,"\gamma_\sharp"]
\arrow[r,leftarrow,"\gamma^*" description]\arrow[r,shift right=1.5ex,"\gamma_*"']
&\Shvltrtkl
\end{tikzcd}$$ such that the following properties hold:
1. $a_t^*$ is left adjoint to the corresponding $a_{t*}$
2. $\gamma^*$ is left (resp. right) adjoint to the corresponding $\gamma_*$ (resp. $\gamma_\sharp$)
3. $a_{t*}\gamma_*\cong \gamma_*a_{t*}$
4. $a_{t*}\gamma^*\cong \gamma^*a_{t*}$
\[A.8.11\] Let $t$ be a topology on $lSm/k$. Suppose that for every $t$-cover $f\colon Y\rightarrow X$ the complex $$\label{A.8.11.1}
\cdots\rightarrow a_t^*\gamma^*\Zltr(Y\times_X Y)\rightarrow a_t^*\gamma^*\Zltr(Y)\rightarrow a_t^*\gamma^*\Zltr(X)\rightarrow 0$$ associated with the Čech nerve of $f$ is acyclic. Then $t$ is compatible with log transfers.
Let $p\colon \mathscr{X}\rightarrow X$ be a $t$-hypercover. To check that $$a_t^*\gamma^*\Zltr(\mathscr{X})\rightarrow a_t^*\gamma^*\Zltr(X)$$ is a quasi-isomorphism it suffices to check that $$a_t^*\gamma^*\Zltr(\mathscr{X}_i)\rightarrow \cdots \rightarrow a_t^*\gamma^*\Zltr(\mathscr{X}_0)\rightarrow a_t^*\gamma^*\Zltr(X)\rightarrow 0$$ is exact. Hence we may assume that $\mathscr{X}$ is a bounded $t$-hypercover of height $n$.
To proceed we use the technique in the proof of [@MR2034012 Proposition A4]. If $n=0$, then $\mathscr{X}$ is a Čech nerve, so we are done. Suppose that $n>0$, and consider the bounded $t$-hypercover $$\mathscr{Y}:={\rm cosk}_{n-1} \mathscr{X}$$ of height at most $n-1$. By induction we have that $a_t^*\gamma^*\Zltr(\mathscr{Y})\rightarrow a_t^*\gamma^*\Zltr(X)$ is a quasi-isomorphism. The Čech nerve of the canonical morphism $\mathscr{X}\rightarrow \mathscr{Y}$ is a bisimplicial complex; let $\mathscr{D}$ denote its diagonal. Since is acyclic for every $t$-cover, there is a canonically induced quasi-isomorphism $$a_t^*\gamma^*\Zltr(\mathscr{D})\rightarrow a_t^*\gamma^*\Zltr(\mathscr{Y}).$$ In particular, $a_t^*\gamma^*\Zltr(\mathscr{D})\rightarrow a_t^*\gamma^*\Zltr(X)$ is a quasi-isomorphism. Since $\mathscr{X}$ is a retract of $\mathscr{D}$, see the proof of [@MR2034012 Proposition A4], we deduce that $$a_t^*\gamma^*\Zltr(\mathscr{X})\rightarrow a_t^*\gamma^*\Zltr(X)$$ is a quasi-isomorphism.
Derived categories of sheaves with log transfers {#Subsection:derivedcategories}
------------------------------------------------
For miscellaneous facts about model structures on chain complexes we refer the reader to Appendix \[AppendixB\]. Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers. Passing to chain complexes in we obtain the diagram of adjoint functors $$\label{A.8.0.1}
\begin{tikzcd}
\Co(\Pshlogkl)\arrow[d,shift left=0.75ex,leftarrow,"a_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']
\arrow[r,shift left=0.75ex,"\gamma_\sharp"]\arrow[r,shift right=0.75ex,"\gamma^*"',leftarrow]&\Co(\Pshltrkl)\arrow[d,shift left=0.75ex,leftarrow,"a_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"'] \\
\Co(\Shvlogtkl)\arrow[r,shift left=0.75ex,"\gamma_\sharp"]\arrow[r,leftarrow,shift right=0.75ex,"\gamma^*"']
&\Co(\Shvltrtkl).
\end{tikzcd}$$ We refer to Example \[A.8.28\] for the descent structures on $\Pshlogkl$ and $\Shvlogtkl$. The functors $\gamma^*$ in are exact since they have both left and right adjoints. If the topology $t$ is compatible with log transfers, then for any $t$-hypercover $\mathscr{X}\rightarrow X$, the cone of the morphism $a_t^*\gamma^*\Zltr(\mathscr{X})\rightarrow a_t^*\gamma^*\Zltr(X)$ is quasi-isomorphic to $0$. Thus by Proposition \[A.8.17\], we also obtain descent structures on $\Pshltrkl$ and $\Shvltrtkl$, and the $(\gamma_\sharp,\gamma^*)$’s are Quillen pairs with respect to the said descent model structures. Note also that the $a_t^*$’s satisfy the conditions of Proposition \[A.8.16\] due to exactness. Owing to Propositions \[A.8.16\] and \[A.8.17\], the $(a_t^*,a_{t*})$’s are Quillen pairs with respect to the descent model structures.
Since $\gamma^*$ preserves quasi-isomorphisms and monomorphisms, the adjunctions $$\gamma^*\colon \Co(\Pshltrkl)\rightleftarrows \Co(\Pshlogkl)\colon \gamma_*,$$ $$\gamma^*\colon \Co(\Shvltrtkl)\rightleftarrows \Co(\Shvltrtkl)\colon \gamma_*$$ are Quillen pairs with respect to the injective model structures, see Definition \[A.8.26\] for injective model structures. Thus we have an induced diagram of adjunctions between triangulated categories $$\label{A.8.0.4}
\begin{tikzcd}
\Deri(\Pshlogkl)\arrow[d,shift left=0.75ex,leftarrow,"Ra_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']
\arrow[r,shift left=1.5ex,"L\gamma_\sharp"]\arrow[r,"\gamma^*" description,leftarrow]\arrow[r,shift right=1.5ex,"R\gamma_*"']&\Deri(\Pshltrkl)\arrow[d,shift left=0.75ex,leftarrow,"Ra_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"'] \\
\Deri(\Shvlogtkl)\arrow[r,shift left=1.5ex,"L\gamma_\sharp"]\arrow[r,"\gamma^*" description,leftarrow]\arrow[r,shift right=1.5ex,"R\gamma_*"']
&\Deri(\Shvltrtkl),
\end{tikzcd}$$ where
1. $L\gamma^*\cong \gamma^*\cong R\gamma^*$ since $\gamma^*$ is exact,
2. $a_t^*\cong La_t^*$ since $a_t^*$ is exact, see Proposition \[A.8.5\] for the log transfer case.
Suppose that $\cW$ is an essentially small class of morphisms in $lSm/k$ that is stable by isomorphisms, products, and compositions. We consider $\cW$ as classes of morphisms of representable objects in $\Pshlogkl$, $\Pshltrkl$, $\Shvlogtkl$, and $\Shvltrtkl$. Note that $a_t^*$ preserves $\cW$ because the object of $\Shvlogtkl$ (resp. $\Shvltrtkl$) representable by $X\in lSm/k$ is $a_t^*\Lambda(X)$ (resp. $a_t^*\Zltr(X)$). The four adjunctions in are Quillen adjunctions with respect to the $\cW$-local descent model structures furnished by Propositions \[A.8.20\] and \[A.8.21\]. Thus we have the following induced diagram of adjunctions between triangulated categories $$\label{A.8.0.2}
\begin{tikzcd}
\Deri_{\cW}(\Pshlogkl)\arrow[d,shift left=0.75ex,leftarrow,"Ra_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']
\arrow[r,shift left=0.75ex,"L\gamma_\sharp"]\arrow[r,shift right=0.75ex,"R\gamma^*"',leftarrow]&\Deri_{\cW}(\Pshltrkl)\arrow[d,shift left=0.75ex,leftarrow,"Ra_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"'] \\
\Deri_{\cW}(\Shvlogtkl)\arrow[r,shift left=0.75ex,"L\gamma_\sharp"]\arrow[r,shift right=0.75ex, "R\gamma^*"',leftarrow]
&\Deri_{\cW}(\Shvltrtkl).
\end{tikzcd}$$ Here $a_t^*\cong La_t^*$ since $a_t^*$ is exact by Proposition \[A.8.5\] and $a_t^*$ preserves $\cW$.
\[A.8.10\] Let $t$ be a topology on $lSm/k$, and let $\cF$ be a complex of $t$-sheaves with log transfers. For $X\in lSm/k$ and $i\in \Z$ we define hypercohomology groups by setting $$\bH_t^i(X,\cF)
:=
\bH_t^i(X,\gamma^*\cF).$$
\[A.8.8\] Let $t$ be a topology on $lSm/k$ compatible with log transfers. Then for all $X\in lSm/k$ and every complex $\cF$ of $t$-sheaves with log transfers there is a canonical isomorphism $$\hom_{\Deri(\Shvltrtkl)}(a_t^*\Zltr(X),\cF[i])
\cong
\bH_t^i(X,\cF).$$
The functor $$L\gamma_\sharp\colon \Deri(\Shvlogtkl)\rightarrow \Deri(\Shvltrtkl)$$ in is left adjoint to $$\gamma^*\colon \Deri(\Shvltrtkl)\rightarrow \Deri(\Shvlogtkl).$$ Thus the group $\bH_t^i(X,\cF)$ can be identified with $$\hom_{\Deri(\Shvlogtkl)}(a_t^*\Lambda(X),\gamma^*\cF[i])
\cong
\hom_{\Deri(\Shvltrtkl)}(L\gamma_\sharp a_t^*\Lambda(X),\cF[i]).$$ The object $a_t^*\Lambda(X)$ is cofibrant, see Definition \[A.8.14\]. The functor $a_t^*$ commutes with $\gamma_\sharp$ due to . Thus we deduce $$L\gamma_\sharp a_t^*\Lambda(X)\cong \gamma_\sharp a_t^*\Lambda(X)
\cong
a_t^*\gamma_\sharp\Lambda(X)\cong a_t^*\Zltr(X).$$
We will need the following lemma in our discussion of adjunctions.
\[A.8.33\] Suppose $F\colon C\rightarrow D$ is a left and a right Quillen functor between model categories. Then $F$ preserves cofibrations, fibrations, and weak equivlaences.
For the convenience of the reader we shall outline a proof, see also the discussion in [@CD12 Remark 1.3.23]. By assumption, $F$ preserves cofibrations, fibrations, trivial cofibrations, and trivial fibrations. It remains to show that $F$ preserves weak equivalences. If $f\colon X\rightarrow Y$ be a weak equivalence in $C$ we can choose a commutative diagram $$\begin{tikzcd}
X'\arrow[d,"g'"']\arrow[r,"f'"]&Y'\arrow[d,"g"]\\
X\arrow[r,"f"]&Y,
\end{tikzcd}$$ where $g$ and $g'$ are trivial fibrations and $f'$ is a weak equivalence between cofibrant objects. Then $F(g)$ and $F(g')$ are weak equivalences. Ken Brown’s lemma implies that $F(f')$ is a weak equivalence. It follows that $F(f)$ is also a weak equivalence.
Let $t$ be a topology on $lSm/k$, and let $t'$ be the restriction of $t$ to $Sm/k$. Then the functor $\lambda$ from $Sm/k$ with the topology $t'$ to $lSm/k$ with the topology $t$ is a continuous functor of sites. Thus we have an induced adjunction $$\lambda_\sharp\colon \Shv_{t'}(k,\Lambda)\rightleftarrows \Shvlogtkl\colon \lambda^*.$$ This induces an adjunction $$\lambda_\sharp\colon \Shvtrttkl\rightleftarrows \Shvltrtkl\colon \lambda^*.$$
Assume $t$ is compatible with log transfers and $t'$ is compatible with transfers. By appealing to Proposition \[A.8.16\] we obtain a Quillen pair with respect to the descent model structures $$\lambda_\sharp\colon \Co(\Shvtrttkl)\rightleftarrows \Co(\Shvltrtkl)\colon \lambda^*$$ since $t'$ is the restriction of $t$, for every $t'$-hypercover $\mathscr{X}\rightarrow X$ we obtain a $t$-hypercover $$\lambda(\mathscr{X})\rightarrow \lambda(X).$$ On the level of homotopy categories there is an induced adjunction $$L\lambda_\sharp\colon \Deri(\Shvtrttkl)\rightleftarrows \Deri(\Shvltrtkl)\colon R\lambda^*.$$
Assume that for any $t$-cover $Y\rightarrow X$, the naturally induced morphism $$\omega(Y)
=
Y-\partial Y
\rightarrow
X-\partial X
=
\omega(X)$$ is also a $t$-cover. Then $\omega$ is a continuous functor of sites from $lSm/k$ with the topology $t$ to $Sm/k$ with the topology $t'$. There are induced adjunctions $$\omega_\sharp\colon \Shvlogtkl\rightleftarrows \Shv_t(k,\Lambda)\colon \omega^*$$ and $$\label{eqn::omegaadjunction}
\omega_\sharp\colon \Shvltrtkl\rightleftarrows \Shvtrttkl\colon \omega^*.$$
By assumption $\omega$ maps any $t$-covers to $t'$-covers, and for any $t$-hypercover $\mathscr{X}\rightarrow X$, $\omega(\mathscr{X})\rightarrow \omega(X)$ is a $t'$-hypercover. According to Proposition \[A.8.16\] there is a Quillen pair with respect to the descent model structures $$\omega_\sharp\colon \Co(\Shvltrtkl)\rightleftarrows \Co(\Shvtrttkl)\colon \omega^*.$$ We also note the induced adjunction $$L\omega_\sharp\colon \Deri(\Shvltrtkl)\rightleftarrows \Deri(\Shvtrttkl)\colon R\omega^*.$$ Since $\lambda$ is left adjoint to $\omega$, by Lemma \[A.8.33\] we deduce $$R\lambda^*\cong \lambda^*\cong \omega_\sharp \cong L\omega_\sharp.$$
In conclusion, there is a sequence of adjunctions $$\label{eq::lambdaomegaadj}
\begin{tikzcd}
\Deri(\Shvltrtkl)\arrow[rr,shift left=1.5ex,"L\lambda_\sharp"]\arrow[rr,"\lambda^*\simeq\omega_\sharp" description,leftarrow]\arrow[rr,shift right=1.5ex,"R\omega^*"']&&\Deri(\Shvtrttkl).
\end{tikzcd}$$
As above, suppose $\cW$ is an essentially small class of morphisms in $lSm/k$ that is stable by isomorphisms, products, and compositions. Then, by Proposition \[A.8.20\], we obtain a Quillen pair with respect to the $\cW$-local and $\omega(\cW)$-local descent model structures $$\omega_\sharp\colon \Co(\Shvltrtkl)\rightleftarrows \Co(\Shvtrttkl)\colon \omega^*.$$ Thus we have the induced adjunction $$\label{A.8.0.3}
L\omega_\sharp\colon \Deri_\cW(\Shvltrtkl)\rightleftarrows \Deri_{\omega(\cW)}(\Shvtrttkl)\colon R\omega^*,$$ and similarly for $$L\omega_\sharp\colon \Deri_\cW(\Shvlogtkl)\rightleftarrows \Deri_{\omega(\cW)}(\Shv_t(k,\Lambda))\colon R\omega^*.$$
\[A.8.22\] Let $f\colon X\rightarrow Y$ be a monomorphism of fs log schemes. Then the naturally induced morphism of presheaves $$\Zltr(f)\colon \Zltr(X)\rightarrow \Zltr(Y)$$ is a monomorphism.
For $T\in lSm/k$, there is a naturally induced commutative diagram $$\begin{tikzcd}
\lCor(T,X)\arrow[r]\arrow[d]&
\lCor(T,Y)\arrow[d]&
\\
\Cor(T-\partial T,X-\partial X)\arrow[r]&
\Cor(T-\partial T,Y-\partial Y).
\end{tikzcd}$$ We need to show that the upper horizontal homomorphism is injective. The induced morphism $X-\partial X\rightarrow Y-\partial Y$ is also a monomorphism. Thus the lower horizontal homomorphism is injective, see [@MVW Exercise 12.22]. The vertical homomorphisms are injective due to Lemma \[A.5.10\], which finishes the proof.
Next we specialize to the setting of cd-structures.
\[A.8.13\] Let $P$ be a complete, quasi-bounded, and regular cd-structure on $lSm/k$, and let $t$ be the associated topology. Then the following conditions are equivalent.
1. $t$ is compatible with log transfers.
2. $t$ is mildly compatible with log transfers.
3. For any $P$-distinguished square in $lSm/k$ of the form , the sequence $$\label{A.8.13.2}
0\rightarrow a_t^*\gamma^*\Zltr(Y')\rightarrow a_t^*\gamma^*\Zltr(Y)\oplus a_t^*\gamma^*\Zltr(X')\rightarrow a_t^*\gamma^*\Zltr(X)\rightarrow 0$$ of $t$-sheaves with log transfers is exact.
The implication (i)$\Rightarrow$(ii) is Proposition \[A.8.3\]. Assuming condition (ii) the induced sequence of $t$-sheaves $$\label{A.8.13.5}
0\rightarrow a_t^*\Lambda(Y')\rightarrow a_t^*\Lambda(Y)\oplus a_t^*\Lambda(X')\rightarrow a_t^*\Lambda(X)\rightarrow 0$$ is exact by [@Vcdtop Lemma 2.18]. Let us show that the following naturally induced sequence of $t$-sheaves with log transfers is exact: $$\label{A.8.13.3}
0\rightarrow \gamma_\sharp a_t^*\Lambda(Y')\rightarrow \gamma_\sharp a_t^*\Lambda(Y)\oplus \gamma_\sharp a_t^*\Lambda(X')\rightarrow \gamma_\sharp a_t^*\Lambda(X)\rightarrow 0.$$ Since $\gamma_\sharp$ commutes with $a_t^*$, this is equivalent to showing that the sequence $$\label{A.8.13.4}
0\rightarrow a_t^*\Zltr(Y')\rightarrow a_t^*\Zltr(Y)\oplus a_t^*\Zltr(X')\rightarrow a_t^*\Zltr(X)\rightarrow 0$$ of $t$-sheaves with log transfers is exact. Since $\gamma_\sharp$ is a left adjoint, $\gamma_\sharp$ is right exact. Thus is exact at $\gamma_\sharp a_t^*\Lambda(Y)\oplus \gamma_\sharp a_t^*\Lambda(X')$ and $\gamma_\sharp a_t^*\Lambda(X)$. Since $P$ is regular, the morphism $Y'\rightarrow Y$ is a monomorphism. Thus $\Zltr(Y')\rightarrow \Zltr(Y)$ is a monomorphism owing to Lemma \[A.8.22\]. This gives the exactness of at $a_t^*\Zltr(Y')$, i.e., and are exact. We deduce that is exact since since $\gamma^*$ commutes with $a_t^*$ and $\gamma^*$ is exact. This implies condition (iii).
Let us assume (iii) holds. To conclude (ii), we need to show that is exact when $\mathscr{X}\rightarrow X$ is a Čech nerve associated with a $t$-cover $T\rightarrow X$. Every $t$-cover has a simple cover because $P$ is complete, see [@Vcdtop Definition 2.3]. Thus since $t$ is the topology associated with $P$, it suffices to consider the case when $T\rightarrow X$ is of the form $Y\amalg X'\rightarrow X$. In this case our claim follows from the exactness of .
Now assume (ii) and (iii) hold. It remains to show (i). We work with the descent structure $(\cG,\cH)$ (resp. $(\cG,\cH')$) of $\Shvlogtkl$ given in Example \[A.8.28\] (resp. Example \[A.8.30\]). Let $G\in \Deri(\Shvltrtkl)$ be an injective fibrant complex. Since the sequence is exact, we have $$\hom_{{\bf K}(\Shvlogtkl)}(F',\gamma^*G)=0$$ for all $F'\in \cH'$. Thus $\gamma^*G$ is $\cH'$-flasque, i.e., $\gamma^*G$ is $\cG$-fibrant. By Propositions \[A.8.15\](2) we see that $\gamma^*G$ is $\cH$-flasque. Thus for all $F\in \cH$ we have the vanishing $$\hom_{{\bf K}(\Shvlogtkl)}(F,\gamma^*G)=0.$$ Since $G$ is an injective fibrant complex we also have $$\hom_{{\bf K}(\Shvlogtkl)}(F,\gamma^*G)
\cong
\hom_{{\bf K}(\Shvltrtkl)}(\gamma_\sharp F,G)
\cong
\hom_{\Deri(\Shvltrtkl)}(\gamma_\sharp F,G).$$ Here $G$ is an arbitrary injective fibrant complex. Hence $\gamma_\sharp F$ is quasi-isomorphic to $0$ for all $F\in \cH$. This implies that condition (i) holds.
Structure of dividing Nisnevich covers {#subsec::structure_dNis}
--------------------------------------
In this section we shall study dividing Nisnevich covers and the dividing Nisnevich sheafification functor. We also discuss their étale versions.
Suppose that $X$ is an excellent reduced noetherian scheme. In [@VSelecta Theorem 3.1.9] it is shown that every $h$-cover $f:Y\rightarrow X$ of schemes admits a factorization of the form $$Y\stackrel{g_1}\rightarrow Y_1\stackrel{g_2}\rightarrow Y_2\stackrel{g_3}\rightarrow X,$$ where $g_i$ is a Zariski cover, $g_2$ is a finite surjective morphism, and $g_3$ is a proper surjective birational morphism. This is one of the fundamental tools in the theory of $h$-coverings.
We provide a result with a similar flavor for factorizations of dividing Nisnevich, dividing étale, and log étale covers.
\[A.5.44\] Let $f\colon Y\rightarrow X$ be a dividing Nisnevich (resp. dividing étale, resp. log étale) cover of noetherian fs log schemes. Then there exists a log modification $h\colon X'\rightarrow X$ such that the pullback $$g\colon Y\times_X X'\rightarrow X'$$ is a strict Nisnevich (resp. strict étale, resp. integral Kummer étale) cover.
In particular, $f$ has a refinement of the form $$\label{A.5.44.1}
Y\times_X X'\stackrel{g}\rightarrow X'\stackrel{h}\rightarrow X,$$ where $g$ is a strict Nisnevich (resp. strict étale, resp. integral log étale) cover and $h$ is a log modification.
If $f$ is log étale, see Theorem \[FKatoThm\]. We focus the attention on the dividing Nisnevich topology since the proofs are similar for the dividing Nisnevich and dividing étale cases. First we reduce to the case when $f$ has a refinement in the form $$Y_0\stackrel{f_0}\rightarrow Y_0'\stackrel{f_0'}\rightarrow \cdots \stackrel{f_{n-2}'}\rightarrow Y_{n-1}\stackrel{f_{n-1}}\rightarrow Y_{n-1}'\stackrel{f_{n-1}'}\rightarrow Y_n\stackrel{f_n}\rightarrow X,$$ where each $f_i$ is a strict Nisnevich cover and each $f_i'$ is a log modification. Owing to Proposition \[Fan.12\], there exists a log modification $X'\rightarrow X$ such that the pullback $$Y_{n-1}'\times_X X'\rightarrow Y_n\times_X X'$$ of $f_{n-1}'$ is an isomorphism. In particular, the pullback $$Y_{n-1}\times_X X'\rightarrow Y_n\times_X X'$$ of the composition $Y_{n-1}\rightarrow X$ is a strict Nisnevich cover.
Replacing by $$Y_0'\times_X X'\rightarrow \cdots \rightarrow Y_{n-1}\times_X X'\rightarrow X',$$ we can apply the above process since $X'\in lSm/k$. In this way we obtain a log modification $X''\rightarrow X$ such that the pullback $$Y_0\times_X X''\rightarrow X''$$ of the composition $Y_0\rightarrow X$ is a strict Nisnevich cover.
Recall from Definition \[A.9.79\] that $X_{div}$ denotes the category of log modifications over $X\in lSm/k$, and $X_{div}^{Sm}$ is the full subcategory of $X_{div}$ consisting of log modifications $Y\rightarrow X$ such that $Y\in SmlSm/k$.
\[A.5.46\] Let $\cF$ be a strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaf on $SmlSm/k$. There is an isomorphism $$\label{A.5.46.1}
\begin{split}
a_{dNis}^*\cF(X)\cong &\colimit_{Y\in X_{div}^{Sm}}\cF(Y)
\\
\text{ (resp.\ }
a_{d\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{div}^{Sm}}\cF(Y),
&\text{ resp.\ }
a_{l\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{div}^{Sm}}\cF(Y)
\text{)}.
\end{split}$$
We will only consider the strict Nisnevich case since the proofs are similar. Let $X_{dNis}$ denote the small dividing Nisnevich site of $X$. Recall from [@SGA4 Remarque II.3.3] that there is a functor $L_{dNis}$ such that $a_{dNis}^*=L_{dNis}\circ L_{dNis}$ given by $$L_{dNis}\cF(X)\cong \colimit_{X'\in X_{dNis}}\ker(\cF(X')\stackrel{(+,-)}\longrightarrow \cF(X'\times_X X')).$$ The notation is shorthand for the difference between the two morphisms induced by the projections $X'\times_X X'\longrightarrow X'$.
By Proposition \[A.5.44\], for any dividing Nisnevich covering $X'\rightarrow X$, there is a refinement $$Y'\stackrel{g}\rightarrow Y''\stackrel{h}\rightarrow X$$ such that $g$ is a strict Nisnevich cover and $h$ is a log modification. Further we may assume that $Y''\in SmlSm/k$ by applying Proposition \[A.3.19\]. Since $$Y'\times_X Y'\cong Y'\times_{Y''}Y''\times_X Y''\times_{Y''}Y'\cong Y'\times_{Y''} Y',$$ we have $$\ker(\cF(Y')\stackrel{(+,-)}\longrightarrow \cF(Y'\times_X Y'))
\cong
\ker(\cF(Y')\stackrel{(+,-)}\longrightarrow \cF(Y'\times_{Y''} Y'))
\cong
\cF(Y'')$$ because $\cF$ is assumed to be a strict Nisnevich sheaf. This gives the identification $$L_{dNis}\cF(X)\cong \colimit_{Y\in X_{div}^{Sm}}\cF(Y).$$
Let us show that $L_{dNis}\cF$ is a strict Nisnevich sheaf. For every strict Nisnevich cover $f\colon X'\rightarrow X$ the naturally induced functor $f^*\colon X_{div}^{Sm}\rightarrow X_{div}'^{Sm}$ is cofinal owing to Propositions \[A.9.81\] and \[Fan.12\]. Thus to show that $L_{dNis}\cF$ is a strict Nisnevich sheaf, it suffices to show there is an exact sequence $$0\rightarrow \colimit_{Y\in X_{div}^{Sm}}\cF(Y)\rightarrow \colimit_{Y\in X_{div}^{Sm}}\cF(Y\times_X X')\rightarrow \colimit_{Y\in X_{div}^{Sm}}\cF(Y\times_X X'\times_X X').$$ Owing to Proposition \[A.9.82\] $X_{div}^{Sm}$ is a filtered category. Since filtered colimits preserve exact sequences, it suffices to show there is an exact sequence $$0\rightarrow \cF(Y)\rightarrow \cF(Y\times_X X')\rightarrow \cF(Y\times_X X'\times_X X')$$ for all $Y\in X_{div}$. This follows from the assumption that $\cF$ is a strict Nisnevich sheaf.
Applying the above to $L_{dNis}\cF$ we deduce the isomorphisms $$a_{dNis}^*\cF(X)\cong \colimit_{Z\in Y_{div}^{Sm}}\colimit_{Y\in X_{div}^{Sm}}\cF(X)\cong \colimit_{Z\in X_{div}^{Sm}}\cF(X).$$
\[A.5.45\] Let $S$ be a noetherian fs log scheme, and let $\cF$ be a strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaf on $lSm/S$. There is an isomorphim $$\label{A.5.45.1}
\begin{split}
a_{dNis}^*\cF(X)\cong &\colimit_{Y\in X_{div}}\cF(Y)
\\
\text{ (resp.\ }
a_{d\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{div}}\cF(Y),
&\text{ resp.\ }
a_{l\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{div}}\cF(Y)
\text{)}.
\end{split}$$
The proof is parallel to that of Lemma \[A.5.46\] if we use Proposition \[A.9.80\] instead of Proposition \[A.9.82\].
For the definition of a log modification along a smooth center we refer to Definition \[Fan.39\].
\[A.5.72\] Let $X_{divsc}^{Sm}$ be the full subcategory of $X_{div}^{Sm}$ consisting of $Y\rightarrow X$ that is isomorphic to a composition of log modifications along smooth center.
\[A.5.74\] If $X\in lSm/k$, then the category $X_{divsc}^{Sm}$ is cofinal in $X_{div}^{Sm}$.
Immediate from Theorem \[Fan.16\].
\[A.5.75\] Let $\cF$ be a strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaf on $SmlSm/k$. There is an isomorphism $$\begin{split}
a_{dNis}^*\cF(X)\cong &\colimit_{Y\in X_{divsc}^{Sm}}\cF(Y)
\\
\text{ (resp.\ }
a_{d\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{divsc}^{Sm}}\cF(Y),
&\text{ resp.\ }
a_{l\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{divsc}^{Sm}}\cF(Y)
\text{)}.
\end{split}$$
Immediate from Lemma \[A.5.46\] and Corollary \[A.5.74\].
Examples of topologies compatible with log transfers
----------------------------------------------------
In this section, we show that the strict Nisnevich, strict étale, Kummer étale, dividing Nisnevich, dividing étale, and Kummer étale topologies are compatible with log transfers. We refer to §\[sec::cdlog\] for the definitions of these topologies. For the strict Nisnevich topology, we use arguments similar to [@MVW Lemma 6.2, Proposition 6.12] originating in [@MR1764202].
\[A.4.1\] Let $X$ be an fs log scheme log smooth over $k$. Then $\Zltr(X)$ is a strict étale sheaf.
For fs log schemes $Y_1$ and $Y_2$ log smooth over $k$ we have $$\Zltr(X)(Y_1\amalg Y_2)
=
\Zltr(X)(Y_1)\oplus \Zltr(X)(Y_2).$$ Thus it suffices to check that the sequence of $\Lambda$-modules $$\label{A.4.1.1}
0\rightarrow \Zltr(X)(Y)\rightarrow \Zltr(X)(U)\stackrel{(+,-)}\longrightarrow \Zltr(X)(U\times_Y U)$$ is exact for every strict étale cover $p\colon U\rightarrow Y$.
There is an induced commutative diagram of $\Lambda$-modules $$\begin{tikzcd}
\Zltr(X)(Y)\arrow[d]\arrow[r]&\Zltr(X)(U)\arrow[d]\\
\Zltr(X-\partial X)(Y-\partial Y)\arrow[r]&\Zltr(X-\partial X)(U-\partial U).
\end{tikzcd}$$ By [@MVW Lemma 6.2], $\Zltr(X-\partial X)$ is an étale sheaf. Thus the lower horizontal morphism is injective. By Lemma \[A.5.10\], the vertical morphisms are injective. Thus the upper horizontal morphism is also injective.
It remains to show that the sequence is exact at $\Zltr(X)(U)$. To that end we consider the cartesian square of fs log schemes $$\begin{tikzcd}
(U-\partial U)\times (X-\partial X)\arrow[d]\arrow[r]&U\times X\arrow[d]\\
(Y-\partial Y)\times (X-\partial X)\arrow[r]&Y\times X.
\end{tikzcd}$$
Suppose $W\in \lCor(U,X)$ is a finite log correspondence with trivial image in $\Zltr(X)(U\times_Y U)$, and form the finite correspondence $$W-\partial W\in \Cor(U-\partial U,X-\partial X).$$ Since $\Zltr(X-\partial X)$ is an étale sheaf, there exists a finite correspondence $$V'\in \Cor(Y-\partial Y,X-\partial X)$$ mapping to $W-\partial W$. We let $\underline{V}$ be the closure of $V'$ in $\underline{Y}\times \underline{X}$.
Consider the induced morphism $u\colon \underline{W}\rightarrow \underline{V}\times_{\underline{Y}}\underline{U}$ of closed subschemes of $\underline{U}\times \underline{X}$. By construction its pullback to $(U-\partial U)\times (X-\partial X)$ is the isomorphism $$W-\partial W\rightarrow (V-\partial V)\times_{(Y-\partial Y)}(U-\partial U).$$ Since $\underline{p}\colon \underline{U}\rightarrow \underline{Y}$ is strict étale and $V-\partial V$ is dense in $\underline{V}$, it follows that $$(V-\partial V)\times_{(Y-\partial Y)}(U-\partial U)$$ is dense in $\underline{V}\times_{\underline{Y}}\underline{U}$. Moreover, $W-\partial W$ is dense in $\underline{W}$ because $u$ is a closed immersion. We note that $\underline{V}$ is finite over $\underline{Y}$ by [@EGA IV.2.7.1(xv)] since $\underline{V}\times_{\underline{Y}}\underline{U}$ is finite over $\underline{U}$ and $\underline{p}\colon \underline{U}\rightarrow \underline{Y}$ is an étale cover. There is a naturally induced morphism $v\colon \underline{W}\rightarrow \underline{V}$. Since $u$ is an isomorphism, $v$ is a pullback of $p$, and hence $v$ is étale.
Let $V^N$ be the fs log scheme whose underlying scheme is the normalization of $\underline{V}$ and whose log structure is induced by $Y$. By [@EGA 11.3.13(ii)] the underlying fiber product $\underline{V^N}\times_{\underline{V}}\underline{W}$ is normal since $v$ is étale. Thus we have the cartesian square of schemes $$\begin{tikzcd}
\underline{W^N}\arrow[d]\arrow[r]&\underline{W}\arrow[d,"v"]\\
\underline{V^N}\arrow[r]&\underline{V}.
\end{tikzcd}$$ The log transfer structure gives rise to a morphism $r\colon W^N\rightarrow X$ of fs log schemes over $k$. By assumption, the two composite morphisms in the diagram $$W^N\times_{U}(U\times_Y U)
\rightrightarrows
W^N
\overset{r}{\rightarrow}
X$$ coincide. Since $p$ is strict étale there is a morphism $V^N\rightarrow X$ of fs log schemes over $k$ such that the composition $W^N\rightarrow V^N\rightarrow X$ is equal to $r$, see [@Ogu Corollary III.1.4.5]. The pair $(\underline{V},V^N\rightarrow X)$ gives a finite log correspondence $V$ from $X$ to $Y$ since $\underline{V}$ is finite over $\underline{Y}$, and the pullback of $V$ to $U$ is $W$.
\[A.5.11\] Let $t$ be one of the following topologies: strict Nisnevich, strict étale, and Kummer étale topologies. For every $t$-cover $f\colon U\rightarrow X$ of fs log schemes in $lSm/k$, the Čech complex $$\label{A.5.11.2}
\cdots\rightarrow a_t^*\Zltr(U\times_X U)\rightarrow a_t^*\Zltr(U)\rightarrow a_t^*\Zltr(X)\rightarrow 0$$ is exact as a complex of $t$-sheaves.
Arguing as in the proof of [@MVW Proposition 6.12] we check there is an exact sequence of $\Lambda$-modules $$\
\label{A.5.11.1}
\cdots\rightarrow \Zltr(U\times_X U)(T)\rightarrow \Zltr(U)(T)\rightarrow \Zltr(X)(T)\rightarrow 0,$$ where $\underline{T}$ is a hensel local scheme (resp. $\underline{T}$ is strictly local, resp. $T$ is log strictly local) if $t=sNis$ (resp. $t=s\acute{e}t$, resp. $t=k\acute{e}t$). Here $\Zltr(T)(T)$ denotes the colimit $\varinjlim \Zltr(X)(T_i)$ when $T$ is a limit of fs log schemes $T_i$ log smooth over $k$.
Let $Z$ be a saturated log scheme over $T$ such that $\underline{Z}$ quasi-finite over $\underline{T}$. We denote by $L(Z/T)$ the free abelian group generated by the pairs $$(\underline{W},W^N\rightarrow Z),$$ where $\underline{W}$ is an irreducible component of $\underline{Z}$ that is finite and surjective over a component of $T$, $W^N$ is the saturated log scheme whose underlying scheme is the normalization of $\underline{W}$ and whose log structure is induced by $T$, and $W^N\rightarrow Z$ is a morphism of saturated log schemes. We note that $L(Z/T)$ is covariantly functorial on $Z$. Now is the colimit of the complexes of the form $$\cdots\rightarrow L((Z_U)_Z^2/T)\rightarrow L(Z_U/T)\rightarrow L(Z/T)\rightarrow 0,$$ where $$Z_U:=Z\times_X U,
\;
(Z_U)_Z^n:=\underbrace{Z_U\times_Z \cdots \times_Z Z_U}_{n \text{ times}},$$ and the colimit runs over all strict closed subschemes $Z$ of $T\times X$ such that $Z$ is strict, finite, and surjective over $T$. Hence it suffices to check that the latter sequence is exact.
Note that $Z$ is henselian (resp. strictly henselian, resp. log strictly henselian) being finite over the henselian (resp. strictly henselian, resp. log strictly henselian) saturated log scheme $T$, see \[ket.2\] for the log strictly henselian case. Thus the strict Nisnevich (resp. strict étale, resp. Kummer étale) cover $Z_U\rightarrow Z$ splits, see Lemma \[ket.4\] for the Kummer étale case. Letting $s\colon Z\rightarrow Z_U$ be a splitting, the maps $$L(s\times_Z {\rm id})\colon L((Z_U)_Z^n)\rightarrow L((Z_U)_Z^{n+1})$$ furnish contracting homotopies, and the desired exactness follows.
Combining Propositions \[A.4.1\] and \[A.5.11\] we can remove $a_t^*$ in the equation if $t=sNis$ or $t=s\acute{e}t$.
\[A.5.22\] The strict Nisnevich topology, the strict étale topology, and the Kummer étale topology on $lSm/k$ are compatible with log transfers.
Follows from Propositions \[A.8.11\] and \[A.5.11\].
\[A.5.62\] Let $f\colon X'\rightarrow X$ be a log modification in $lSm/k$. For every $Y\in lSm/k$ and finite log correspondence $V\in \lCor(Y,X)$, there exists a dividing Zariski cover $g\colon Y'\rightarrow Y$ and a finite log correspondence $W\in \lCor(Y',X')$ fitting in a commutative diagram $$\begin{tikzcd}
Y'\arrow[d,"g"']\arrow[r,"W"]&X'\arrow[d,"f"]\\
Y\arrow[r,"V"]&X
\end{tikzcd}$$ in $lCor/k$.
The question is Zariski local on $Y$, so we may assume that $Y$ has an fs chart $P$. Let $p\colon V^N\times_X X'\rightarrow V^N$ be the projection, and let $q\colon V^N\rightarrow Y$ be the structure morphism. By Lemma \[A.9.21\], there is a subdivision of fans $M\rightarrow \Spec P$ such that the projection $V^N\times_{\A_P}\A_M\rightarrow V^N$ admits a factorization $$V^N\times_{\A_P}\A_M\rightarrow V^N\times_X X'\rightarrow V^N.$$ Setting $Y':=Y\times_{\A_P}\A_M$ and $V':=V\circ u$ where $u\colon Y'\rightarrow Y$ denotes the projection, the structure morphism $v\colon V'^N\rightarrow X$ factors through $V^N\times_Y Y'\cong V^N\times_{\A_P}\A_M$, so that $v$ factors through $V^N\times_X X'$. It follows that $v$ factors through $X'$.
Replacing $Y$ by $Y'$ and $V$ by $V'$, we may assume that the structure morphism $V^N\rightarrow X$ factors through $X'$. In this case there is a commutative diagram of fs log schemes $$\begin{tikzcd}
V^N\arrow[r,"w"]&Y\times X'\arrow[d]\arrow[r]&X'\arrow[d]\\
&Y\times X\arrow[r]&X.
\end{tikzcd}$$ Since $X'$ is proper over $X$, the morphism $w$ is proper. Let $\underline{W}$ be the image of $$\underline{V^N}\rightarrow \underline{Y}\times \underline{X'},$$ which we consider as a closed subscheme of $\underline{Y}\times \underline{X'}$ with the reduced scheme structure. Then $\underline{V^N}$ is the normalization of $\underline{W}$ since $\underline{V^N}$ is the normalization of the image of $$\underline{V^N}\rightarrow \underline{Y}\times \underline{X}.$$ Thus $w$ gives a correspondence $W$ from $Y$ to $X'$ with image $V$ in $\lCor(Y',X)$.
\[A.5.70\] For every log modification $f\colon Y\rightarrow X$ in $lSm/k$ the induced morphism $$a_{dNis}^*\gamma^*\Zltr(Y)\rightarrow a_{dNis}^*\gamma^*\Zltr(X)$$ of dividing Nisnevich sheaves is an isomorphism.
Let $T$ be a fs log scheme log smooth over $k$. In the commutative diagram $$\begin{tikzcd}
\lCor(T,Y)\arrow[r]\arrow[d,hook]&\lCor(T,X)\arrow[d,hook']\\
\Cor(T-\partial T,Y-\partial Y)\arrow[r,"\sim"]&\Cor(T-\partial T,X-\partial X)
\end{tikzcd}$$ the vertical morphisms are injections by Lemma \[A.5.10\], and the lower horizontal morphism is an isomorphism since $Y-\partial Y\cong X-\partial X$. Thus the upper horizontal morphism is an injection. Hence $\Zltr(Y)\rightarrow \Zltr(X)$ is a monomorphism, and likewise for $$a_{dNis}^*\gamma^*\Zltr(Y)\rightarrow a_{dNis}^*\gamma^*\Zltr(X)$$ since $a_{dNis}^*$ and $\gamma^*$ are exact, see and Proposition \[A.8.5\]. To show that it is an epimorphism, let $V\in \lCor(T,X)$ be a finite log correspondence. Due to Lemma \[A.5.62\] there exists a dividing Nisnevich cover $g\colon T'\rightarrow T$ and a finite log correspondence $W\in \lCor(T',Y)$ such that $f\circ W=V\circ g$. This finishes the proof.
\[A.5.23\] The dividing Nisnevich topology, the dividing étale topology, and the log étale topology on $lSm/k$ are compatible with log transfers.
We write the proof in the case of the log étale topology. Let $f\colon Y\rightarrow X$ be a log étale cover. We need to show that the Čech complex $$\cdots\rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(Y\times_X Y)\rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(Y)\rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(X)\rightarrow 0$$ is exact, see Proposition \[A.8.11\]. There exists a log modification $X'\rightarrow X$ such that the pullback $f'\colon Y\times_X X'\rightarrow X'$ of $f$ is a Kummer étale cover owing to Proposition \[A.5.44\]. Set $Y':=Y\times_X X'$. Then using Lemma \[A.5.70\], it suffices to show that the Čech complex $$\cdots \rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(Y'\times_{X'}Y')\rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(Y')\rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(X')\rightarrow 0$$ is exact. Since the sheafification functor $a_{l\acute{e}t}^*$ is exact, we are done by Proposition \[A.5.11\].
Dividing log correspondences {#subsec:divdinglogcor}
----------------------------
For $X,Y\in lSm/k$, we have used $\lCor(X,Y)$ to define dividing Nisnevich sheaves with log transfers. Since $\Zltr(Y)$ is not a dividing Nisnevich sheaf one cannot in general identify $\lCor(X,Y)$ with $$\hom_{\Shvltrkl}(a_{dNis}^*\Zltr(Y),a_{dNis}^*\Zltr(X)).$$ However, it turns out that $a_{dNis}^*\Zltr(Y)(X)$ can be described in terms of algebraic cycles. To that end we introduce the notion of dividing log correspondences.
\[A.5.47\] For $X,Y\in lSm/k$, a [*dividing elementary log correspondence*]{} $Z$ from $X$ to $Y$ is an integral closed subscheme $\underline{Z}$ of $\underline{X}\times \underline{Y}$ that is finite and surjective over a component of $\underline{X}$ together with a log modification $X'\rightarrow X$ and a morphism $$u'\colon Z'^N\rightarrow X'\times Y$$ satisfying the following properties:
1. The image of the composition $\underline{Z'^N}\stackrel{\underline{u'}}\rightarrow \underline{X'}\times \underline{Y}\rightarrow \underline{X}\times \underline{Y}$ is $\underline{Z}$.
2. $\underline{Z'^N}$ is the normalization of $\underline{Z}\times_{\underline{X}}\underline{X'}$.
3. The composition $Z'^N\stackrel{u'}\rightarrow X'\times Y\rightarrow X'$ is strict.
A [*dividing log correspondence*]{} from $X$ to $Y$ is a formal sum $\sum n_i Z_i$ of dividing elementary log correspondences $Z_i$ from $X$ to $Y$. Let $\lCor_k^{div}(X,Y)$ denote the free abelian group of dividing log correspondences from $X$ to $Y$. When no confusion seems likely to arise we shall omit the subscript $k$ and write $\lCor^{div}(X,Y)$.
\[A.5.48\] For $X,Y\in lSm/k$, a finite log correspondence $Z\in \lCor(X,Y)$ is determined by $Z-\partial Z\in \lCor(X-\partial X,Y-\partial Y)$ owing to Lemma \[A.5.10\]. Thus $Z$ is determined by its underlying scheme $\underline{Z}$. Hence we arrive at the following equivalent definition: An elementary log correspondence $Z$ from $X$ to $Y$ is an integral closed subscheme $\underline{Z}$ of $\underline{X}\times \underline{Y}$ that is finite and surjective over a component of $\underline{X}$ together with a morphism $$u\colon Z^N\rightarrow X\times Y$$ satisfying the following properties:
1. The image of $\underline{u}\colon \underline{Z^N}\rightarrow \underline{X}\times \underline{Y}$ is $\underline{Z}$.
2. $\underline{Z^N}$ is the normalization of $\underline{Z}$.
3. The composition $Z^N\stackrel{u}\rightarrow X\times Y\rightarrow X$ is strict.
In view of the latter description, we can interpret any dividing log correspondence as a finite log correspondence after replacing $X$ with a log modification of $X$.
\[A.5.49\] For every $X,Y\in lSm/k$ there are isomorphisms $$\lCor^{div}(X,Y)\otimes \Lambda\cong a_{dNis}^*\Zltr(Y)(X)\cong a_{d\acute{e}t}^*\Zltr(Y)(X).$$
With the description of elementary log correspondences in Remark \[A.5.48\], this follows from Lemma \[A.5.45\].
\[A.5.50\] For $X,Y,Z\in lSm/k$, let us construct a composition for dividing log correspondences $$\label{A.5.50.1}
\circ\colon \lCor^{div}(X,Y)\times \lCor^{div}(Y,Z)\rightarrow \lCor^{div}(X,Z).$$ Owing to Proposition \[A.5.49\] and the Yoneda lemma we have $$\label{A.5.50.2}
\lCor^{div}(X,Y)\cong \hom_{\Shvltrkl}(a_{dNis}^*\ZZltr(X),a_{dNis}^*\ZZltr(Y)).$$ We obtain because we have compositions in $\ShvltrkZ$. Moreover, the composition is associative, and for $g\in \lCor(X,Y)$ we have $${\rm id}\circ f=f=f\circ {\rm id}.$$ Thus we can form a category $lCor^{div}/k$ with the same objects as $lSm/k$ and with morphisms given by $\lCor^{div}(X,Y)$.
A [*presheaf of $\Lambda$-modules with dividing log transfers*]{} is an additive presheaf of $\Lambda$-modules on the category $lCor^{div}/k$. We let $\Pshdltrkl$ denote the category of presheaves of $\Lambda$-modules with dividing log transfers.
A [*dividing Nisnevich sheaf of $\Lambda$-modules with dividing log transfers*]{} is a presheaf of $\Lambda$-modules with dividing log transfers such that the restriction to $lSm/k$ is a dividing Nisnevich sheaf.
We let $\Shvdltrkl$ (resp. $\Shv_{d\acute{e}t}^{\rm dltr}(k,\Lambda)$) denote the category of dividing Nisnevich (resp. dividing étale) sheaves of $\Lambda$-modules with dividing log transfers.
\[A.5.56\] There are equivalences of categories $$\begin{gathered}
\Shvdltrkl\cong \Shvltrkl,
\\
\Shv_{d\acute{e}t}^{\rm dltr}(k,\Lambda)\cong \Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda),
\\
\Shv_{l\acute{e}t}^{\rm dltr}(k,\Lambda)\cong \Shv_{l\acute{e}t}^{\rm ltr}(k,\Lambda).\end{gathered}$$
We will only consider the dividing Nisnevich case since the proofs are similar. It suffices to show that any $\cF\in \Shvltrkl$ admits a unique dividing log transfer structure. For any $X,Y\in lSm/k$, owing to , we have that $$\hom_{\Shvltrkl}(a_{dNis}^*\Zltr(X),a_{dNis}^*\Zltr(Y))
\cong \
lCor^{div}(X,Y).$$ Thus for any $V\in \lCor^{div}(X,Y)$ the Yoneda lemma furnishes a canonical transfer map $$\begin{split}
\cF(X)\cong &\hom_{\Shvltrkl}(a_{dNis}^*\Zltr(X),\cF)\\
\rightarrow &\hom_{\Shvltrkl}(a_{dNis}^*\Zltr(Y),\cF)\cong \cF(Y).
\end{split}$$ Thus there exists a dividing log transfer structure on $\cF$.
For uniqueness, suppose that $\cG_0$ and $\cG_1$ are two sheaves with dividing log transfers whose restrictions to $\Shvltrkl$ are isomorphic to $\cF$. Since $\cG_0$ and $\cG_1$ agree on objects, it suffices to show that $\cG_0(V)=\cG_1(V)$ for every $V\in \lCor^{div}(X,Y)$. Owing to Proposition \[A.5.49\], there exists a dividing Nisnevich cover $p\colon X'\rightarrow X$ such that $V':=V\circ u\in \lCor^{div}(X',Y)$ is an element in $\lCor(X',Y)$. By the assumption that $\cF$ is the restriction of $\cG_0$ and $\cG_1$, we have an equality of homomorphisms $$\cG_0(V')=\cG_1(V')\colon \cF(Y)\rightarrow \cF(X').$$ Since $\cF$ is a dividing Nisnevich sheaf and $X'\rightarrow X$ is a dividing Nisnevich cover, $\cF(p)\colon \cF(X)\rightarrow \cF(X')$ is injective. Thus we can conclude $\cG_0(V')=\cG_1(V')$.
Let $lCor_{SmlSm}/k$ (resp. $lCor_{SmlSm}^{div}/k$) denote the full subcategory of $lCor/k$ (resp. $lCor^{div}/k$) consisting of all objects in $SmlSm/k$.
\[A.5.51\] Let $f\colon Y\rightarrow X$ be a log modification in $lSm/k$. Then $f$ is an isomorphism in $lCor^{div}/k$.
Consider the composition $$\Gamma_f^t\colon Y\stackrel{\Gamma_f}\longrightarrow Y\times X\rightarrow X\times Y,$$ where $\Gamma_f$ is the graph morphism and the second morphism switches the factors. The induced pullback $$Y\times_X Y\rightarrow (Y\times_X X)\times Y,$$ is isomorphic to the diagonal morphism $Y\rightarrow Y\times Y$. Thus $\Gamma_f^t$ is a dividing log correspondence from $X$ to $Y$, and it follows that $f$ is invertible in $lCor^{div}/k$.
Lemma \[A.5.51\] furnishes a class of non-finite morphisms in $lSm/k$ that become invertible in $lCor^{div}/k$, but not in $lCor/k$. Following the approach to motives with modulus in [@KSY2], this suggests that we should invert every proper birational morphism $f:Y\rightarrow X$ for which the naturally induced morphism $$Y-\partial Y\rightarrow X-\partial X$$ is an isomorphism. However, our strategy is different from [@KSY2]. Assuming resolution of singularities, in Theorem \[A.3.7\] we show that the naturally induced morphism of log motives $$M(f):M(Y)\rightarrow M(X)$$ is an isomorphism.
\[A.5.53\] There is an equivalence of categories $$lCor_{SmlSm}^{div}/k\simeq lCor^{div}/k.$$
Since $lCor_{SmlSm}^{div}/k$ is a full subcategory of $lCor^{div}/k$, it remains to show that any object $X$ of $lCor^{div}/k$ is isomorphic to an object of $lCor_{SmlSm}^{div}/k$. Owing to Proposition \[A.3.19\], there is a log modification $f\colon Y\rightarrow X$ such that $Y\in SmlSm/k$. Then we are done since $f$ is an isomorphism in $lCor^{div}/k$ by Lemma \[A.5.51\].
Sheaves with log transfers on $SmlSm/k$ {#Subsec::Sheaves.SmlSm}
---------------------------------------
There are many interesting examples of Nisnevich sheaves with transfers, e.g., $\mathbb{G}_m$ and $\Omega^i$. If $\cF$ is a Nisnevich sheaf with transfers on $Sm/k$, one may ask whether it extends to a dividing Nisnevich sheaf with log transfers on $lSm/k$, such that $\cF(X)$ makes sense for every $X\in lSm/k$. In practice one can divide this problem into two steps. The first step is to define $\cF(X)$ for every $X\in SmlSm/k$. And the second step is to define $\cF(X)$ for a general $X\in lSm/k$. In this subsection we explain this second step.
\[A.5.52\] We set $$\Pshltrklsm:=\Psh(lCor_{SmlSm}/k),$$ $$\Pshdltrklsm:=\Psh(lCor_{SmlSm}^{div}/k).$$ For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, we let $\Shv_t^{\rm ltr}(SmlSm/k,\Lambda)$ (resp. $\Shv_t^{\rm dltr}(SmlSm/k,\Lambda)$) denote the full subcategory of $\Pshltrklsm$ (resp. $\Pshdltrklsm$) consisting of $t$-sheaves on $SmlSm/k$.
\[A.5.54\] For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, there is an equivalence of categories $$\Shv_{t}(SmlSm/k,\Lambda)\simeq \Shv_t^{\rm log}(k,\Lambda).$$
We will only consider the dividing Nisnevich topology since the proofs are similar. For any $X\in lSm/k$, there exists a log modification $Y\rightarrow X$ such that $Y\in SmlSm/k$ owing to Proposition \[A.3.19\]. We conclude using the implication (i)$\Rightarrow$(ii) in [@SGA4 Théorème III.4.1].
Let $\iota$ denote the inclusion functor $lCor_{SmlSm}/k\rightarrow lCor/k$. For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, we have functors $$\begin{gathered}
\iota^*\colon \Psh^{\rm dltr}(k,\Lambda)
\rightarrow
\Psh^{\rm dltr}(SmlSm/k,\Lambda)
\\
\label{A.5.58.1}
\iota^*\colon
\Shv_{t}^{\rm ltr}(k,\Lambda)
\rightarrow
\Shv_{t}^{\rm ltr}(SmlSm/k,\Lambda)\end{gathered}$$ mapping $\cF$ to $\cF\circ \iota$. As in there are faithful functors $$\gamma\colon lSm/k\rightarrow lCor^{div}/k,\; \gamma\colon SmlSm/k\rightarrow lCor_{SmlSm}^{div}/k.$$ These induce functors $$\gamma^*\colon \Pshdltrkl\rightarrow \Pshlogkl,
\;
\gamma^*\colon \Pshdltrklsm\rightarrow \Psh(SmlSm/k,\Lambda)$$ mapping $\cF$ to $\cF\circ \gamma$.
\[A.5.55\] For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, there is an equivalence of categories $$\Shv_{t}^{\rm dltr}(SmlSm/k,\Lambda)\simeq \Shv_{t}^{\rm dltr}(k,\Lambda).$$
We will only consider the dividing Nisnevich topology since the proofs are similar. There is a commutative diagram $$\begin{tikzcd}
\Pshdltrkl\arrow[r,"\iota^*"]\arrow[d,"\gamma^*"']&\Pshdltrklsm\arrow[d,"\gamma^*"]\\
\Pshlogkl \arrow[r]&\Psh(SmlSm/k,\Lambda).
\end{tikzcd}$$ The restriction functor $\iota^*$ is an equivalence by Lemma \[A.5.53\]. Hence it suffices to show that for every $\cF\in \Pshdltrkl$, $\gamma^*(\cF)$ is a dividing Nisnevich sheaf if and only if $\theta(\gamma^*(\cF))$ is a dividing Nisnevich sheaf. This follows from Lemma \[A.5.54\].
\[A.5.57\] For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, there is an equivalence of categories $$\Shv_{t}^{\rm dltr}(SmlSm/k,\Lambda)\simeq \Shv_{t}^{\rm ltr}(SmlSm/k,\Lambda).$$
Parallel to the proof of Proposition \[A.5.56\].
\[A.5.58\] For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, the functor in is an equivalence.
Follows from Propositions \[A.5.56\], \[A.5.57\], and Lemma \[A.5.55\].
Thus the functor in admits a left adjoint $$\iota_\sharp\colon
\Shv_{t}^{\rm ltr}(SmlSm/k,\Lambda)
\rightarrow
\Shv_{t}^{\rm ltr}(k,\Lambda).$$
\[A.5.59\] Let $\cF$ be an object of $\Shv_{t}^{\rm ltr}(SmlSm/k,\Lambda)$, where $t$ is one of $dNis$, $d\acute{e}t$, and $l\acute{e}t$. For every $X\in lSm/k$ we have that $$\iota_\sharp \cF(X)
\cong
\colimit_{Y\rightarrow X}\cF(Y),$$ where the colimit runs over log modifications $Y\rightarrow X$ such that $Y\in SmlSm/k$.
We only discuss the dividing Nisnvich topology since the proofs are parallel. Since $\iota^*$ is an equivalence by Proposition \[A.5.58\], we have $\iota^*\iota_\sharp \cF\cong \cF$, so that $$\iota_\sharp \cF(Y)\cong \cF(Y)$$ for every $Y\in SmlSm/k$. Recall that $X_{div}^{Sm}$ is the full subcategory of $X_{div}$ consisting of $Y\rightarrow X$ such that $Y\in SmlSm/k$. According to Proposition \[A.9.81\], $X_{div}^{Sm}$ is cofinal in $X_{div}$, and hence $$\label{A.5.59.1}
\colimit_{Y\in X_{div}}\iota_\sharp \cF(Y)
\cong
\colimit_{Y\in X_{div}^{Sm}}\iota_\sharp \cF(Y)
\cong
\colimit_{Y\in X_{div}^{Sm}}\cF(Y).$$ Since $\iota_\sharp \cF(X)$ is a dividing Nisnevich sheaf, we deduce $$\label{A.5.59.2}
\iota_\sharp \cF(X)\cong \colimit_{Y\in X_{div}}\iota_\sharp \cF(Y)$$ Combining and gives the assertion.
Construction of triangulated categories of logarithmic motives {#sec.constructldmeff}
==============================================================
This section is dedicated to the construction and study of some basic properties of our categories of log motivic sheaves. First we explain how the dividing Nisnevich cohomology groups are related to the strict Nisnevich cohomology groups. Next we construct the triangulated categories $\ldmeff$ and $\ldaeff$ of effective log motives, with and without transfers, in the Nisnevich and étale topologies on log schemes over $k$. There are two equivalent ways of constructing $\ldmeff$, see Section \[ssecequivalence\]. One way is to use sheaves on $lSm/k$, and the other way is via sheaves on $SmlSm/k$. The advantage of having the latter category at our disposal is that it allows us to construct motivic sheaves that are, a priori, only defined and functorial on log schemes for which the underlying scheme is smooth over $k$ and with boundary a normal crossing divisor. In Section \[sec:Hodge\] we use this trick to prove representability of Hodge cohomology and cyclic homology.
Dividing Nisnevich cohomology groups
------------------------------------
A fundamental invariant of the dividing Nisnevich topology is the corresponding sheaf cohomology theory. In what follows we shall express the dividing Nisnevich cohomology groups in terms of strict Nisnevich cohomology groups by promoting Subsection \[subsec::structure\_dNis\] to cohomology. We also discuss the étale version of this result. As an application we discuss finite cohomological dimension property for our various topologies.
\[Div.8\] Let $S$ be a noetherian fs log scheme, and let $\cF$ be a strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaf on $lSm/S$. If $\cF$ is flabby, then $a_{dNis}^*\cF$ (resp. $a_{d\acute{e}t}^*\cF$, resp. $a_{l\acute{e}t}^*\cF$) is flabby.
We focus on the strict Nisnevich case since the proofs are similar. Owing to the implication $(b)\Rightarrow (a)$ in [@Milneetale Proposition III.2.12] it suffices to show that for every dividing Nisnevich cover $p:U\rightarrow X$ in $lSm/k$ and integer $i>0$, the corresponding Čech cohomology group vanishes $$H^i(a_{dNis}^*\cF(\cU))=0.$$ Here $\cU$ is the Čech nerve associated with $U\rightarrow X$.
According to Proposition \[A.5.44\], there exists a log modification $X'\rightarrow X$ such that the projection $U':=U\times_X X'\rightarrow X'$ is a strict Nisnevich cover. Since $a_{dNis}^*\cF$ is a complex of dividing Nisnevich sheaves, there is an isomorphism $$a_{dNis}^*\cF(\cU)\cong a_{dNis}^*\cF(\cU\times_X X').$$ Hence we can replace $U\rightarrow X$ by $U'\rightarrow X'$, so we reduce to the case when $p$ is a strict Nisnevich cover.
By Lemma \[A.5.45\] the complex $a_{dNis}^*\cF(\cU)$ is isomorphic to $$\colimit_{Y_0\in X_{div}}\cF(Y_0)
\rightarrow
\colimit_{Y_1\in U_{div}}\cF(Y_1)
\rightarrow
\colimit_{Y_2\in (U\times_X U)_{div}}\cF(Y_2)
\rightarrow
\cdots.$$ Applying Corollary \[Fan.14\] we see that the latter complex is isomorphic to $$\colimit_{Y\in X_{div}}\cF(Y)
\rightarrow
\colimit_{Y\in X_{div}}\cF(Y\times_X U)
\rightarrow
\colimit_{Y\in X_{div}}\cF(Y\times_X U\times_X U)
\rightarrow
\cdots.$$ This is a filtered colimit according to Proposition \[A.9.80\]. Since every filtered colimit in the category of $\Lambda$-modules is exact, it remains to check that $H^i$ of the complex $$\cF(Y)\rightarrow \cF(Y\times_X U)\rightarrow \cF(Y\times_X U\times_X U)\rightarrow \cdots$$ is trivial for every $Y\in X_{div}$ and $i>0$. To conclude we use the implication $(a)\Rightarrow (b)$ in [@Milneetale Proposition III.2.12] and the assumption that $\cF$ is flabby.
\[Div.3\] Let $S$ be a noetherian fs log scheme, and let $\cF$ be a bounded below complex of strict Nisnevich sheaves on $lSm/S$. Then for every $X\in lSm/S$ and $i\in \Z$ there is an isomorphism $$\begin{aligned}
\bH_{dNis}^i(X,a_{dNis}^*\cF)&\cong \colimit_{Y\in X_{div}}\bH_{sNis}^i(Y,\cF)
\\
\text{(resp.\ }\bH_{d\acute{e}t}^i(X,a_{d\acute{e}t}^*\cF)&\cong \colimit_{Y\in X_{div}}\bH_{s\acute{e}t}^i(Y,\cF),
\\
\text{resp.\ }\bH_{l\acute{e}t}^i(X,a_{l\acute{e}t}^*\cF)&\cong \colimit_{Y\in X_{div}}\bH_{k\acute{e}t}^i(Y,\cF)\text{)}.\end{aligned}$$
We only consider the strict Nisnevich case since the proofs are similar. We can replace $\cF$ by its injective resolution because the sheafification functor $a_{dNis}^*$ is exact. In particular, $\cF^i$ is flabby for every integer $i\in \Z$. Owing to Lemma \[Div.8\] we see that $a_{dNis}^*\cF^i$ is flabby. It follows that $$\bH_{dNis}^i(X,a_{dNis}^*\cF)\cong H^i(a_{dNis}^*\cF(X)).$$ Due to Lemma \[A.5.45\] there is an isomorphism $$a_{dNis}^*\cF(X)\cong \colimit_{Y\in X_{div}}\cF(Y).$$ Since this colimit is filtered by Proposition \[A.9.80\] and every filtered colimit in the category of $\Lambda$-modules is exact, we deduce that $$\bH_{dNis}^i(X,a_{dNis}^*\cF)\cong \colimit_{Y\in X_{div}}H^i(\cF(Y)).$$ To conclude the proof we observe that $$\bH_{sNis}^i(Y,\cF)
\cong
H^i(\cF(Y)).$$
Suppose $\cF$ is a torsion sheaf on the small log étale site $X_{l\acute{e}t}$. Then for every fs log scheme $X$ with an fs chart, the isomorphism $$H_{l\acute{e}t}^i(X,\cF)\cong H_{k\acute{e}t}^i(X,\cF)$$ is proven in [@MR3658728 Proposition 5.4(2)].
\[Div.5\] The category $lSm/k$ has finite $sNis$ and $dNis$-cohomological dimension for $\Lambda$-linear coefficients.
For every $X\in lSm/k$ and $\cF\in \Shv_{sNis}^{\rm log}(k,\Lambda)$, if $i>\dim X$, then due to [@MVW Example 12.2] we have the vanishing $$H_{sNis}^i(X,\cF)=H_{Nis}^i(\ul{X},\cF)=0.$$ If $Y$ is a log modification over $X$, then $\dim Y=\dim X$ by Lemma \[A.9.84\]. Thus we have the vanishing $$H_{sNis}^i(Y,\cF)=0.$$ Theorem \[Div.3\] finishes the proof.
\[ketcomp.13\] Let $X$ be an fs log scheme. Suppose $\cF$ is a sheaf of $\Q$-modules on the small strict étale site $X_{s\acute{e}t}$. Then, for every integer $i\geq 0$, there is an isomorphism of cohomology groups $$H_{sNis}^i(X,\cF)\cong H_{s\acute{e}t}^i(X,\cF).$$
Equivalently, we need to show there is an isomorphism $$H_{Nis}^i(\underline{X},\cF)\cong H_{\acute{e}t}^i(\underline{X},\cF).$$ This is the content of [@MVW Proposition 14.23].
\[ketcomp.12\] Let $X$ be an fs log scheme. Suppose $\cF$ is a sheaf of $\Q$-modules on the small Kummer étale site $X_{k\acute{e}t}$. Then, for every integer $i\geq 0$, there is an isomorphism of cohomology groups $$H_{s\acute{e}t}^i(X,\cF)\cong H_{k\acute{e}t}^i(X,\cF).$$
Let $\epsilon_*\colon \Shv(X_{k\acute{e}t})\rightarrow \Shv(X_{s\acute{e}t})$ be the functor sending $\cF$ to $\cF\vert_{X_{s\acute{e}t}}$. We need to show that $R^i\epsilon_*\cF=0$ for every integer $i>0$. Since $R^i\epsilon_*\cF$ is the strict étale sheaf associated with the presheaf $$(Y\in X_{s\acute{e}t})\mapsto H_{k\acute{e}t}^i(Y,\cF),$$ it suffices to show the vanishing $$H_{k\acute{e}t}^i(X,\cF)=0$$ for every $i>0$ whenever $\underline{X}$ is strictly local. According to [@MR1457738 Proposition 4.1], see also [@MR3658728 Proposition 1.2], the group $H_{k\acute{e}t}^i(X,\cF)$ is a torsion abelian group by [@weibel_1994 Corollary 6.11.4]. Since $\cF$ is a sheaf of $\Q$-modules, this implies the desired vanishing.
\[Div.6\] Let $t$ be one of the following topologies on $lSm/k$: $$s\acute{e}t,\; d\acute{e}t,\; k\acute{e}t,\text{ and }l\acute{e}t.$$ If $k$ has finite $\acute{e}t$-cohomological dimension for $\Lambda$-linear coefficients, then $lSm/k$ has finite $t$-cohomological dimension for $\Lambda$-linear coeffieicnts.
For any $\cF\in \Shv_t^{\rm log}(k,\Lambda)$, the kernel and cokernel of the natural morphism $$\cF\rightarrow \cF\otimes \Q$$ are torsion sheaves. Hence we only need to deal with the two separate cases: $\Lambda$ is a torsion ring, and $\Q\subset \Lambda$.
If $\Lambda$ is a torsion ring, then $lSm/k$ has finite $s\acute{e}t$ and $k\acute{e}t$-cohomological dimension for $\Lambda$-linear coefficients due to [@SGA4 Corollaries X.4.2, X.5.3] and [@MR3658728 Theorem 7.2]. To show that $lSm/k$ has finite $d\acute{e}t$ and $l\acute{e}t$-cohomological dimension for $\Lambda$-linear coefficients, apply Lemma \[A.9.84\] and Theorem \[Div.3\].
If $\Q\subset \Lambda$, then Lemmas \[ketcomp.13\] and \[ketcomp.12\] show that $lSm/k$ has finite $s\acute{e}t$ and $k\acute{e}t$-cohomological dimension for $\Lambda$-linear coefficients since $lSm/k$ has finite $sNis$-cohomological dimension due to Corollary \[Div.5\]. Then apply Lemma \[A.9.84\] and Theorem \[Div.3\] again to finish the proof.
\[Div.4\] Let $\cF$ be a bounded below complex of strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaves on $SmlSm/k$. Then for every $X\in SmlSm/k$ and integer $i\in \Z$ there is an isomorphism $$\begin{aligned}
\bH_{dNis}^i(X,a_{dNis}^*\cF)&\cong \colimit_{Y\in X_{div}^{Sm}}\bH_{sNis}^i(Y,\cF)
\\
\text{(resp.\ }\bH_{d\acute{e}t}^i(X,a_{d\acute{e}t}^*\cF)&\cong \colimit_{Y\in X_{div}^{Sm}}\bH_{s\acute{e}t}^i(Y,\cF),
\\
\text{resp.\ }\bH_{l\acute{e}t}^i(X,a_{l\acute{e}t}^*\cF)&\cong \colimit_{Y\in X_{div}^{Sm}}\bH_{k\acute{e}t}^i(Y,\cF)\text{)}.\end{aligned}$$ The same holds if we replace $X_{div}^{Sm}$ by $X_{divsc}^{Sm}$.
The proof is parallel to that of Theorem \[Div.3\] if we use Lemma \[A.5.46\] and Proposition \[A.9.82\] as replacements for Lemma \[A.5.45\] and Proposition \[A.9.80\]. To show the claim for $X_{divsc}^{Sm}$ use Corollary \[A.5.74\] and Lemma \[A.5.75\] instead.
Effective log motives {#subsection:cotcolm}
---------------------
Voevodsky’s category of effective motives $\dmeff$ (resp. effective motives without transfers $\daeff$) is the homotopy category with respect to the $\A^1$-local descent model structure on $\Co(\Shvtrkl)$ (resp. $\Co(\Shv_{Nis}(k,\Lambda))$). The motive $M(X)$ of $X\in Sm/k$ is defined as the object $\Ztr(X)$ (resp. $\Lambda(X)$) in $\dmeff$ (resp. $\daeff$). The category $\dmeff$ was initially constructed for bounded above complexes, see e.g., [@MVW Lecture 14]. The generalization to unbounded complexes was carried out in [@CD12 Definition 11.1.1].
By abuse of notation we let $\boxx$ be shorthand for the class of projections $X\times \boxx\rightarrow X$, where $X\in lSm/k$. We also consider $\boxx$ as the corresponding classes of morphisms of representable objects in $\Shvlogkl$ or $\Shvltrkl$.
The derived category of effective log motives $$\ldmeff$$ is the homotopy category of $\Co(\Shvltrkl)$ with respect to the $\boxx$-local descent model structure. The motive of $X\in lSm/k$ is the object $$M(X):=a_{dNis}^*\Zltr(X)\in\ldmeff.$$
Similarly, by abuse of notation, we write $\ldaeff$ for the homotopy category of $\Co(\Shvlogkl)$ with respect to the $\boxx$-local descent model structure and $M(X)$ for $a_{dNis}^*\Lambda(X)$.
By Proposition \[A.8.19\], $\ldmeff$ and $\ldaeff$ are symmetric monoidal triangulated categories such that for $X,Y\in lSm/k$, we have $$M(X)\otimes M(Y)=M(X\times Y).$$
For any dividing Nisnevich cover $Y\rightarrow X$ in $lSm/k$, we note that $$Y-\partial Y\rightarrow X-\partial X$$ is a Nisnevich cover in $Sm/k$. Moreover, we have that $$\omega(\boxx)=\A^1.$$ Thus by and , there are naturally induced adjunctions $$L\gamma_\sharp:\ldaeff\rightleftarrows \ldmeff:R\gamma^*,$$ and $$\label{eq::omegasharp}
\omega_\sharp:\ldmeff\rightleftarrows \dmeff:R\omega^*.$$
Similarly we have the naturally induced adjunctions $$L\gamma_\sharp:\daeff\rightleftarrows \dmeff:R\gamma^*,$$ and $$\omega_\sharp:\ldaeff\rightleftarrows \daeff:R\omega^*.$$
Further there is a diagram of adjunct functors $$\begin{tikzcd}
\ldaeff\arrow[d,shift left=0.75ex,leftarrow,"R\omega^*"]\arrow[d,shift right=0.75ex,"\omega_\sharp"']
\arrow[r,shift left=0.75ex,"L\gamma_\sharp"]\arrow[r,shift right=0.75ex,"R\gamma^*"',leftarrow]&\ldmeff\arrow[d,shift left=0.75ex,leftarrow,"R\omega^*"]\arrow[d,shift right=0.75ex,"\omega_\sharp"'] \\
\daeff\arrow[r,shift left=0.75ex,"L\gamma_\sharp"]\arrow[r,shift right=0.75ex,"R\gamma^*"',leftarrow] &\dmeff,
\end{tikzcd}$$ where $L\gamma_\sharp$ commutes with $\omega_\sharp$.
\[A.5.12\] A complex of dividing Nisnevich sheaves $\cF$ is [*strictly $\boxx$-invariant*]{} if for every fs log scheme $X$ log smooth over $k$ and integer $i\in \Z$, the projection $X\times \boxx\rightarrow X$ induces an isomorphism on sheaf cohomology groups $$\bH_{dNis}^i(X,\cF)\rightarrow \bH_{dNis}^i(X\times \boxx,\cF).$$
By definition of $\boxx$-local model structure, a complex of dividing Nisnevich sheaves $\cF$ is fibrant with respect to the $\boxx$-local descent model structure if and only if $\cF$ is fibrant with respect to the local model structure and $\cF$ is strictly $\boxx$-invariant.
\[A.5.13\] Let $\cF$ be a strictly $\boxx$-invariant complex of dividing Nisnevich sheaves with log transfers. Then, for every fs log scheme $X$ log smooth over $k$ and integer $i\in \Z$, there is an isomorphism $$\hom_{\ldmeff}(M(X),\cF[i])
\cong
\bH_{dNis}^i(X,\cF).$$ Similar statement holds for strictly $\boxx$-invariant complex of dividing Nisnevich sheaves $\cF$ and $\ldaeff$.
We focus on $\ldmeff$ since the proofs are similar. We may assume that $\cF$ is a fibrant object in $\Co(\Shvltrkl)$ with respect to the descent model structure given in §\[Subsection:derivedcategories\]. By Proposition \[A.8.19\](1), $\cF$ is a fibrant object in $\Co(\Shvltrkl)$ with respect to the $\boxx$-local descent model structure. Thus we have $$\hom_{\ldmeff}(M(X),\cF[i])
\cong
\hom_{\Deri(\Shvltrkl)}(a_{dNis}^*\Zltr(X),\cF[i]).$$ Proposition \[A.8.8\] finishes the proof.
\[A.5.73\] Let $\{\cF_i\}_{i\in I}$ be a set of strictly $\boxx$-invariant complexes of dividing Nisnevich sheaves. Then $\bigoplus_{i\in I}\cF_i$ is strictly $\boxx$-invariant too. The same holds for strictly $\boxx$-invariant complexes of dividing étale or log étale sheaves if $k$ has finite étale cohomological dimension for $\Lambda$-linear coefficients.
We first treat the dividing Nisnevich case. Due to Corollary \[Div.5\] and Proposition \[bigsmall.5\] for every $X\in lSm/k$ we have isomorphisms $$\begin{gathered}
\bH_{dNis}^i(X,\bigoplus_{i\in I}\cF)
\cong
\bigoplus_{i\in I}\bH_{dNis}^i(X,\cF),
\\
\bH_{dNis}^i(X\times \boxx,\bigoplus_{i\in I}\cF)
\cong
\bigoplus_{i\in I}\bH_{dNis}^i(X\times \boxx,\cF).\end{gathered}$$ Combining these isomorphisms finishes the proof. For the dividing étale and log étale cases, apply Corollary \[Div.6\] instead.
\[A.5.33\] The effective log motives $M(X)[n]$ indexed by $X\in lSm/k$ and $n\in \Z$ form a set of compact generators for the categories $\ldaeff$ and $\ldmeff$.
We focus on $\ldmeff$ since the proofs are similar. Let $\{\cF_i\}_{i\in I}$ be a set of strictly $\boxx$-invariant dividing Nisnevich complexes of sheaves with log transfers. Then $\bigoplus_{i\in I} \cF_i$ is strictly $\boxx$-invariant too by Lemma \[A.5.73\]. Owing to Proposition \[A.5.13\] we have $$\begin{gathered}
\hom_{\ldmeff}(M(X),\bigoplus_{i\in I}\cF_i[n])
\cong
\hom_{\mathbf{D}(\Shvlogkl)}(a_{dNis}^*\Lambda(X),\bigoplus_{i\in I}\cF_i[n]),
\\
\bigoplus_{i\in I}\hom_{\ldmeff}(M(X),\cF_i[n])
\cong
\bigoplus_{i\in I}\hom_{\mathbf{D}(\Shvlogkl)}(a_{dNis}^*\Lambda(X),\cF_i[n]).\end{gathered}$$ Proposition \[bigsmall.5\] and Corollary \[Div.5\] finish the proof.
\[A.5.34\] Similarly, the objects $M(X)[n]$, where $X\in Sm/k$ and $n\in \Z$, are compact in $\dmeff$ (resp. $\daeff$). The same objects generate $\dmeff$ (resp. $\daeff$) by the above or [@CD12 Theorem 11.1.13] (resp. [@CD12 Proposition 5.2.38]).
\[A.5.7\] If $f:Y\rightarrow X$ is a morphism in $lSm/k$, let $$M(Y\stackrel{f}\rightarrow X)$$ be the cone in $\ldmeff$ associated with the complex $$a_{dNis}^*\Zltr(Y)\rightarrow a_{dNis}^*\Zltr(X)$$ in $\Co(\Shvltrkl)$. We use the same notation for the object in $\ldaeff$ associated with the complex $a_{dNis}^*\Lambda(Y)\rightarrow a_{dNis}^*\Lambda(X)$ in $\Co(\Shvlogkl)$.
For a simplicial object $\mathscr{X}$ in $lSm/k$, let $M(\mathscr{X})$ denote the object in $\ldmeff$ associated with $\Zltr(\mathscr{X})$ in $\Co(\Shvltrkl)$ We use the same notation for the object in $\ldaeff$ associated with $\Lambda(\mathscr{X})$ in $\Co(\Shvlogkl)$.
We denote by $$\ldmeffprop$$ the smallest triangulated subcategory of $\ldmeff$ that is closed under small sums and shifts, and contains $M(X)$, where $X\in lSm/k$ and its underlying scheme $\underline{X}$ is proper over $\Spec{k}$.
We are interested in $\ldmeffprop$ because for a proper log smooth $X$, the log motive $M(X)$ is practically an object in $\dmeff$. Indeed, if $k$ admits resolution of singularities, Theorem \[thm::dmeff=ldmeffprop\] shows there is an equivalence of triangulated categories $$\ldmeffprop\simeq \dmeff.$$ Moreover, we note that the corresponding statement is false in the étale topology, see Remark \[Etalemot.5\].
\[A.5.42\]
1. For every strict Nisnevich distinguished square $$\begin{tikzcd}
Y'\arrow[d]\arrow[r]&Y\arrow[d]\\
X'\arrow[r]&X
\end{tikzcd}$$ in $lSm/k$, there is a naturally induced isomorphism $$M(Y'\rightarrow X')\rightarrow M(Y\rightarrow X)$$ in $\ldmeff$. The same holds true in $\ldaeff$.
2. For every log modification $f:Y\rightarrow X$, there is a naturally induced isomorphism $$M(Y)\rightarrow M(X)$$ in $\ldmeff$. The same holds true in $\ldaeff$.
Let us treat first the case of $\ldaeff$. To show (1), it suffices to show the homomorphism $$\hom_{\ldaeff}(M(Y\rightarrow X),\cF)\rightarrow \hom_{\ldaeff}(M(Y'\rightarrow X'),\cF)$$ is an isomorphism for any strictly $\boxx$-invariant complex of dividing Nisnevich sheaf $\cF$. For this it suffices to show there is an isomorphism $$\label{A.5.42.1}\hom_{\Deri(\Shvlogtkl)}(a_{dNis}^*\Lambda(Y\rightarrow X),\cF)
\rightarrow
\hom_{\Deri(\Shvlogtkl)}(a_{dNis}^*\Lambda(Y'\rightarrow X'),\cF).$$ This follows from the exactness of . Part (2) can be obtained similarly if we use a dividing distinguished square instead of a strict Nisnevich distinguished square.
To show (1) in the case of $\ldmeff$, it suffices to show the homomorphism $$\hom_{\ldaeff}(M(Y\rightarrow X),\cF)
\rightarrow
\hom_{\ldaeff}(M(Y'\rightarrow X'),\cF)$$ is an isomorphism for any strictly $\boxx$-invariant complex of dividing Nisnevich sheaf $\cF$ with log transfers. This follows from Proposition \[A.5.13\] since is an isomorphism. We omit the proof of (2) because it is similar to the one for (1).
\[A.5.43\] The commutative square $$\label{A.5.43.1}
\begin{tikzcd}
\A_\N\arrow[r]\arrow[d]&(\P^1,0)\cong \boxx\arrow[d]\\
\A^1\arrow[r]&\P^1
\end{tikzcd}$$ is a strict Nisnevich distinguished square, so that the induced morphism $$M(\A_\N\rightarrow \A^1)\rightarrow M(\boxx\rightarrow \P^1)\cong \Lambda(1)[2]$$ is an isomorphism in $\ldmeff$. This is analogous to the isomorphism $$M(\A^1/\mathbb{G}_m)\rightarrow M(\P^1/\A^1)\cong \Lambda(1)[2]$$ in $\dmeff$ and $\daeff$. Indeed, applying $\omega$ to the square we retain the Nisnevich distinguished square $$\begin{tikzcd}
\mathbb{G}_m\arrow[d]\arrow[r]&\P^1-\{0\}\arrow[d]
\\
\A^1\arrow[r]&\P^1
\end{tikzcd}$$ for the standard covering of the projective line. In the log situation, however, we note that $\Zltr(\A_\N)$ is not a subsheaf of $\Zltr(\A^1)$.
\[A.5.66\] Recall that $div$ is the class of morphisms in $lSm/k$ consisting of log modifications $Y\rightarrow X$ in $lSm/k$. We denote by $(\boxx,div)$ the class of morphisms in $lSm/k$ consisting of all the projections $X\times \boxx\rightarrow X$, where $X\in lSm/k$, and the log modifications $Y\rightarrow X$ in $lSm/k$.
\[A.4.28\] A useful property of $\A^1$-invariant presheaf $\cF$ with transfers is that for any Zariski cover $U_1$ and $U_2$ of an open subscheme $U$ of $\A^1$, there is a split exact sequence $$\label{A.4.28.1}
0\rightarrow \cF(U)\rightarrow \cF(U_1)\oplus \cF(U_2)\rightarrow \cF(U_1\cap U_2)\rightarrow 0,$$ see [@MVW Lemma 22.4]. This is no longer true for $\boxx$-invariant presheaves with log transfers. In what follows we check that $\Zltr(\A^1)$ provides a counterexample.
For $X\in lSm/k$, we have that $$\Zltr(\A^1)(X)=\Cor(\underline{X},\A^1).$$ Let $V$ be an elementary correspondence from $\underline{X}\times \P^1$ to $\A^1$. For any point $x$ of $\underline{X}$, consider the restriction $V_x$ of $V$ to $\{x\}\times \P^1$. Since $V_x$ is finite over $\{x\}\times \P^1$, $V_x$ is proper over $k$. Thus the image of $V'$ in $\A^1$ is again proper, so the image should be a union of points. Let $y$ be a rational point of $\P^1$, let $V'$ (resp. $V_x'$) be the restriction of $V$ to $\underline{X}\times \{y\}$ (resp. $\{x\}\times \{y\}$) and let $p:\underline{X}\times \P^1\rightarrow \underline{X}$ and $p_x:\{x\}\times \P^1\rightarrow \{x\}$ be the projections. From the above we have $p_x^*V_x'=V_x$ and hence $p^*V'=V$. Thus the morphism $$p^*:\Zltr(\A^1)(X)\rightarrow \Zltr(\A^1)(X\times \P^1)$$ is surjective. Since $p$ has a section, it follows that $p^*$ is injective. So far we have observed that $\Zltr(\A^1)$ is $\boxx$-invariant.
Next we turn to the split exactness of . Set $U:=\A^1=\Spec{k[x]}$, $U_1:=\A^1-\{0\}$, and $U_2:=\A^1-\{1\}$. Then there is an element $$[1/x+1/(x-1)]\in \Zltr(\A^1)(U_1\cap U_2),$$ where for $f:U_1\cap U_2\rightarrow \A^1$, we let $[f]$ denotes the associated correspondence. On the other hand, every element in $\Zltr(\A^1)(U_j)$ is of the form $$a_1[f_1]+\cdots +a_r[f_r],$$ where $a_i\in \Lambda$ and $f_i\in k[x,x^{-1}]$ if $j=1$ and $f_i\in k[x,(x-1)^{-1}]$ if $j=2$. There is no way to express $[1/x+1/(x-1)]$ as a sum of such elements, so the homomorphism $$\Zltr(U_1)\oplus \Zltr(U_2)\rightarrow \Zltr(U_1\cap U_2)$$ is not even surjective.
Let $\cF$ be an $\A^1$-invariant Nisnevich sheaf $\cF$ with transfers. The spilt exactness of is used to prove that also the presheaf $H_{Nis}^n(-,\cF)$ is $\A^1$-invariant, see e.g., [@MVW Theorem 24.1]. Due to Example \[A.4.28\], it is unclear whether for any $\boxx$-invariant strict Nisnevich sheaf $\cF$ with transfers, the presheaf $H_{sNis}^n(-,\cF)$ is $\boxx$-invariant.
Effective étale log motives
---------------------------
In this subsection we define the étale versions of $\ldmeff$ and $\ldaeff$.
\[Etalemot.1\] The derived category of effective dividing étale log motives $$\ldmeffet$$ is the homotopy category of $$\Co(\Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda))$$ with respect to the $\boxx$-local descent model structure. The motive of $X\in lSm/k$ is the object $$M(X):=a_{d\acute{e}t}^*\Zltr(X)\in\ldmeffet.$$
Similarly, we write $\ldaeffet$ for the homotopy category of $\Co(\Shv_{d\acute{e}t}^{\rm log}(k,\Lambda))$ with respect to the $\boxx$-local descent model structure and $M(X)$ for $a_{d\acute{e}t}^*\Lambda(X)$.
The derived category of effective log étale log motives $$\ldmefflet$$ is the homotopy category of $\Co(\Shv_{l\acute{e}t}^{\rm ltr}(k,\Lambda))$ with respect to the $\boxx$-local descent model structure. The effective log étale motive of $X\in lSm/k$ is the object $$M(X):=a_{l\acute{e}t}^*\Zltr(X)\in\ldmefflet.$$
Similarly, we write $\ldaefflet$ for the homotopy category of $\Co(\Shv_{l\acute{e}t}^{\rm log}(k,\Lambda))$ with respect to the $\boxx$-local descent model structure and $M(X)$ for $a_{l\acute{e}t}^*\Lambda(X)$.
By Proposition \[A.8.19\], $\ldmeffet$, $\ldaeffet$, $\ldmefflet$, and $\ldaefflet$ are symmetric monoidal triangulated categories such that for all $X,Y\in lSm/k$, $$M(X)\otimes M(Y)=M(X\times Y).$$
With respect to the descent model structures, there are Quillen adjunctions $$a_{d\acute{e}t}^*:\Co(\Shv_{dNis}^{\rm ltr}(k,\Lambda))\rightleftarrows \Co(\Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda)):a_{d\acute{e}t*}$$ and $$a_{l\acute{e}t}^*:\Co(\Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda))\rightleftarrows \Co(\Shv_{l\acute{e}t}^{\rm ltr}(k,\Lambda)):a_{l\acute{e}t*}.$$
Owing to [@MR1944041 Theorem 3.3.20(1)], this is again a Quillen adjunction with respect to the $\boxx$-local model structures. By passing to the homotopy categories we obtain the adjunctions $$La_{d\acute{e}t}^*:\ldmeff\rightleftarrows \ldmeffet:Ra_{d\acute{e}t*}$$ and $$La_{l\acute{e}t}^*:\ldmeffet\rightleftarrows \ldmefflet:Ra_{l\acute{e}t*}.$$
Let $Y\rightarrow X$ be a morphism in $lSm/k$, and let $\mathscr{X}$ be a simplicial object of $lSm/k$. As in Definition \[A.5.7\], we associate objects $$M(Y\stackrel{f}\rightarrow X)
\text{ and }
M(\mathscr{X})$$ in $\ldmeffet$, $\ldaeffet$, $\ldmefflet$, and $\ldaefflet$.
\[Etalemot.2\] A complex of dividing étale sheaves $\cF$ is [*strictly $\boxx$-invariant*]{} if for every fs log scheme $X$ log smooth over $k$ and integer $i\in \Z$, the projection $X\times \boxx\rightarrow X$ induces an isomorphism on sheaf cohomology groups $$\bH_{d\acute{e}t}^i(X,\cF)\rightarrow \bH_{d\acute{e}t}^i(X\times \boxx,\cF).$$
Similarly, a complex of log étale sheaves $\cF$ is [*strictly $\boxx$-invariant*]{} if for every fs log scheme $X$ log smooth over $k$ and integer $i\in \Z$, the projection $X\times \boxx\rightarrow X$ induces an isomorphism on sheaf cohomology groups $$\bH_{l\acute{e}t}^i(X,\cF)\rightarrow \bH_{l\acute{e}t}^i(X\times \boxx,\cF).$$
\[Etalemot.3\] Let $\cF$ be a strictly $\boxx$-invariant complex of dividing étale (resp. log étale) sheaves with log transfers. Then, for every fs log scheme $X$ log smooth over $k$ and integer $i\in \Z$, there is an isomorphism $$\hom_{\ldmeffet}(M(X),\cF)
\cong
\bH_{d\acute{e}t}^i(X,\cF)$$ $$\text{(resp.\ }
\hom_{\ldmefflet}(M(X),\cF)
\cong
\bH_{l\acute{e}t}^i(X,\cF)
\text{)}.$$
The proof is parallel to that of Proposition \[A.5.13\].
\[Etalemot.4\] We denote by $$\ldmeffetprop
\;
\text{(resp.\ }
\ldmeffletprop
\text{)}$$ the smallest triangulated subcategory of $\ldmeffet$ (resp. $\ldmefflet$) that is closed under small sums and shifts, and contains $M(X)$, where $X\in lSm/k$ and its underlying scheme is proper over $\Spec{k}$.
\[Etalemot.5\] Suppose $k$ has positive characteristic $p>0$. In contrast to Theorem \[thm::dmeff=ldmeffprop\], we do [*not*]{} have an equivalence of triangulated categories between $$\ldmeffetpropZ
\text{ and }
\dmeffetZ.$$
To wit, note that $\Z/p\cong 0$ in $\dmeffetZ$ by [@MVW Example 9.16]. Notice that since $\Z\cong M(k)\in \ldmeffetpropZ$, we have $\Z/p\in \ldmeffetpropZ$. In Example \[Nontriviality\] we will see that $\Z/p$ is nontrivial in $\ldmeffetpropZ$.
Suppose that $k$ has finite $\acute{e}t$-cohomological dimension for $\Lambda$-linear coefficients. Then the effective log motives $M(X)[n]$ indexed by $X\in lSm/k$ and $n\in \Z$ form a set of compact generators for the categories $$\ldaeffet,\;\ldmeffet,\; \ldaefflet,\text{ and }\ldmefflet.$$
Argue as in Proposition \[A.5.33\] by appealing to Corollary \[Div.6\].
Construction of motives using $SmlSm/k$ {#ssecequivalence}
---------------------------------------
We may use the results of Section \[Subsec::Sheaves.SmlSm\] to construct categories of effective log motives for objects in $SmlSm/k$.
\[SmlSm.1\] Let $$\ldmeffSmlSm$$ be the homotopy category of $$\Co(\Shvltrklsm)$$ with respect to the $\boxx$-local descent model structure.
Similarly, we let $$\ldmeffetSmlSm$$ be the homotopy category of $$\Co(\Shv_{d\acute{e}t}^{\rm ltr}(SmlSm/k,\Lambda))$$ with respect to the $\boxx$-local descent model structure.
\[SmlSm.2\] There are naturally induced equivalences of triangulated categories $$\begin{aligned}
\ldmeffSmlSm & \simeq \ldmeff, \\
\ldmeffetSmlSm & \simeq \ldmeffet, \\
\ldmeffletSmlSm & \simeq \ldmefflet.\end{aligned}$$
We will only consider the dividing Nisnevich topology since the proofs are similar. Owing to Proposition \[A.5.58\] there is an equivalence of categories $$\Shvltrklsm
\simeq
\Shvltrkl.$$
Let us consider the descent model structures on the categories $$\Co(\Shvltrklsm)$$ and $$\Co(\Shvltrkl).$$ Since the descent model structures is invariant under equivalences of categories, there is a Quillen equivalence with respect to the descent model structures $$\Co(\Shvltrklsm)\rightleftarrows \Co(\Shvltrkl).$$ Owing to [@MR1944041 Theorem 3.3.20(1)] this becomes a Quillen equivalence if we impose the $\boxx$-local descent model structure.
\[SmlSm.3\] Suppose $\cF$ is a complex of dividing Nisnevich sheaves with log transfers on $SmlSm/k$. Recall that $X_{div}^{Sm}$ denotes the category of log modifications $Y\rightarrow X$, where $Y\in SmlSm/k$. Owing to Lemma \[A.5.59\], we can naturally extend $\cF$ to a complex $i_\sharp \cF$ on $lSm/k$ via the formula $$\iota_\sharp \cF(X)
:=
\colimit_{Y\in X_{div}^{Sm}}\cF(Y).$$ By Lemma \[A.5.54\] there is an equivalence of categories $$\Shv_{dNis}(SmlSm/k,\Lambda)\simeq \Shvlogkl.$$ This implies that for every $X\in SmlSm/k$ and integer $i\in \Z$ we have an isomorphism of cohomology groups $$\label{SmlSm.3.1}
H_{dNis}^i(X,\cF)\cong H_{dNis}^i(X,\iota_\sharp \cF).$$
If $\cF$ is strictly $\boxx$-invariant, i.e., for every $X\in SmlSm/k$ $$H_{dNis}^i(X,\cF)\cong H_{dNis}^i(X\times \boxx,\cF),$$ shows $\iota_\sharp \cF$ is strictly $\boxx$-invariant too. Owing to Proposition \[A.5.13\], for every $X\in SmlSm/k$ and integer $i\in \Z$, we obtain an isomorphism $$\label{SmlSm.3.2}
\hom_{\ldmeff}(M(X),\iota_\sharp \cF[i])\cong H_{dNis}^i(X,\cF).$$
If $\cF$ is a complex of dividing étale (resp. log étale) sheaves with log transfers on $SmlSm/k$, then we can argue similarly to conclude the isomorphisms $$\label{SmlSm.3.3}
\hom_{\ldmeffet}(M(X),\iota_\sharp \cF[i]) \cong H_{d\acute{e}t}^i(X,\cF)$$ and $$\label{SmlSm.3.4}
\hom_{\ldmefflet}(M(X),\iota_\sharp \cF[i])\cong H_{l\acute{e}t}^i(X,\cF).$$
Calculus of fractions and homotopy theory of presheaves {#section:calculus}
=======================================================
Throughout this section, $T$ is a small category with finite limits. Let $\pt$ denote the final object in $T$. Suppose $\mathcal{A}$ is a set of morphisms in $T$ that admits a calculus of right fractions. We let $I$ be an object of $T$ admitting morphisms $$i_0,i_1:{\rm pt}\rightarrow I,\;\; p:I\rightarrow {\rm pt}$$ in $T$ and a morphism $$\mu:I\times I\rightarrow I$$ in $T[\mathcal{A}^{-1}]$ satisfying $$\mu(i_0\times {\rm id})=\mu({\rm id}\times i_0)=i_0p,$$ $$\mu(i_1\times {\rm id})=\mu({\rm id}\times i_1)={\rm id}.$$ Moreover, we assume $$i_0\amalg i_1:{\rm pt}\amalg {\rm pt}\rightarrow I$$ is a monomorphism. Note that $I$ is an interval of $T[\mathcal{A}^{-1}]$ in the sense of [@MV §2.3].
Recall that $\mathcal{A}$ admits a *calculus of right fractions* if the following conditions holds [@GZFractions dual of I.2.2].
1. The set $\mathcal{A}$ contains the isomorphisms and is closed under composition.
2. (Right Ore condition) For any diagram $Y\stackrel{f}\rightarrow X\stackrel{g}\leftarrow X'$ in $\cC$ with $f\in \cA$, there is a commutative diagram $$\begin{tikzcd}
Y'\arrow[d,"f'"']\arrow[r]&Y\arrow[d,"f"]\\
X'\arrow[r,"g"]&X
\end{tikzcd}$$ in $\cC$ such that $f'\in \cA$.
3. (Right cancellability condition) For any diagram $$\begin{tikzcd}
Z\arrow[r,shift left=0.5ex,"f"]\arrow[r,shift right=0.5ex,"g"']&Y\arrow[r,"u"]&X
\end{tikzcd}$$ in $\cC$ with $u\in \cA$ and $u\circ f=u\circ g$, there is a morphism $v:W\rightarrow Z$ in $\cA$ such that $f\circ v=g\circ v$.
In [@MV §2.3], Morel-Voevodsky constructs the singular functor ${\rm Sing}$ for an interval $I$ of $T$. Since our assumption is that $I$ is an interval of $T[\cA^{-1}]$, instead of $T$, we cannot apply the same arguments verbatim. For example, in the proof of [@MV Corollary 3.5, p. 89] it is essential that $\mu\in T$ when dealing with the multiplication map $\mu:I\times I\rightarrow I$. In this section, we construct a singular functor ${\rm Sing}$ for $\mu$ in $T[\cA^{-1}]$. We argue that there is another way of constructing ${\rm Sing}$ in [@Conservativity] without assuming the existence of $\mu$. The style of this construction is different from that in [@MV §2.3]. See [@AdditiveHomotopy Section 4.5] for a way of constructing the Sing functor obtained by forming a weak interval object [@AdditiveHomotopy Definition 2.19] in the category of simplicial presheaves on $T$.
Properties of localization functors
-----------------------------------
Let $\sPsh(T)$ denote the category of simplicial presheaves on $T$. A morphism $f:F\rightarrow G$ of simplicial presheaves is called
(i) a [*pointwise weak equivalence*]{} if $F(X)\rightarrow G(X)$ is a weak equivalence of simplicial sets for any object $X$ of $T$.
(ii) an [*injective cofibration*]{} if $F(X)\rightarrow G(X)$ is a cofibration of simplicial sets for any object $X$ of $T$.
(iii) an [*injective fibration*]{} if it has the right lifting property with respect to injective cofibrations that are pointwise weak equivalences.
(iv) a [*projective fibration*]{} if $F(X)\rightarrow G(X)$ is a fibration of simplicial sets for any object $X$ of $T$.
(v) a [*projective cofibration*]{} if it has the left lifting property with respect to projective fibrations that are pointwise weak equivalences.
Recall that the model structure on $\sPsh(T)$ with the pointwise weak equivalences, injective cofibrations, and injective fibrations is called the [*injective model structure*]{} (see [@JardineLocal Theorem 5.8]). The model structure on $\sPsh(T)$ with the pointwise weak equivalences, projective cofibrations, and projective fibrations is called the [*projective model structure*]{} (see [@JardineLocal §5.5]).
\[A.1.3\] Associated to the localization functor $$v:T\rightarrow T[\mathcal{A}^{-1}]$$ we let $$v^*:\sPsh(T[\mathcal{A}^{-1}])\rightarrow \sPsh(T)$$ denote the functor given by $$v^*F(X)
:=
F(X).$$ Here $F$ is a simplicial presheaf on $T[\mathcal{A}^{-1}]$ and $X$ is an object of $T$.
By [@SGA4 Proposition I.5.1], there exist adjoint functors $$\begin{tikzcd}
\sPsh(T)\arrow[rr,shift left=1.5ex,"v_\sharp "]\arrow[rr,"v^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"v_*"']&&\sPsh(T[\mathcal{A}^{-1}]),
\end{tikzcd}$$ where $v_\sharp$ is left adjoint to $v^*$, and $v^*$ is left adjoint to $v_*$.
\[A.5.1\] For any object $X$ of $T$, we denote by $\cA\downarrow X$ the category where an object is a morphism $Y\rightarrow X$ in $\cA$, and a morphism is a commutative diagram $$\begin{tikzcd}
Y'\arrow[rr,"g"]\arrow[rd]&&Y\arrow[ld]\\
&X
\end{tikzcd}$$ in $T$ such that $g\in \cA$.
Since $\mathcal{A}$ admits a calculus of right fractions, by the dual of [@GZFractions I.2.3], we have the formula $$\label{A.1.3.1}
\hom_{T[\mathcal{A}^{-1}]}(X,S)
\cong
\colimit_{Y\in \mathcal{A}\downarrow X}\hom_T(Y,S).$$
\[A.1.4\] The functor $v_\sharp $ is given by $$\label{A.1.4.1}
v_\sharp F(X)\cong \colimit_{Y\in \mathcal{A}\downarrow X} F(Y).$$
By [@SGA4 I.5.1.1], we reduce to the case when $F=Z\times \Delta[n]$, where $Z\in T$. In this case follows from .
\[A.2.1\] The adjunction $$v_\sharp :\sPsh(T)\rightleftarrows \sPsh(T[\mathcal{A}^{-1}]):v^*$$ is a Quillen pair with respect to the projective model structures.
The functor $v:T\rightarrow T[\mathcal{A}^{-1}]$ commutes with finite limits by using the dual of [@GZFractions Proposition I.3.1]. Thus $v$ is a morphism of sites with respect to the trivial topologies. We are done by [@MR2034012 Corollary 8.3].
\[A.2.7\] The functor $$v^*:\sPsh(T[\mathcal{A}^{-1}])\rightarrow \sPsh(T)$$ preserves pointwise weak equivalences and injective cofibrations. In particular, the adjunction $$v^*:\sPsh(T[\mathcal{A}^{-1}])\rightleftarrows \sPsh(T):v_*$$ is a Quillen pair with respect to the injective model structures.
Pointwise weak equivalences and injective cofibrations are defined objectwise, so the claim follows from the fact that $v:T\rightarrow T[\mathcal{A}^{-1}]$ is essentially surjective ([@GZFractions 1.1]).
For simplicial presheaves $F$ and $G$ on $T$, let ${\mathbf S}(F,G)$ denote the simplicial enrichment (or internal hom), which is given by the simplicial set $${\mathbf S}(F,G)_n:=\hom_{\sPsh(T)}(F\times \Delta[n],G).$$
\[A.2.3\] Let $\mathcal{A}$ be a set of morphisms in $T$.
(1) A simplicial presheaf $F$ on $T$ is [*$\mathcal{A}$-invariant*]{} if the morphism $$F(X\times Z)\rightarrow F(Y\times Z)$$ is a pointwise weak equivalence for any morphism $Y\rightarrow X$ in $\mathcal{A}$ and any $Z\in T$.
(2) A morphism $f:F\rightarrow G$ of simplicial presheaves on $T$ is a [*$\mathcal{A}$-weak equivalence*]{} if for any $\mathcal{A}$-invariant simplicial presheaf $H$ on $T$, $f$ induces a weak equivalence of simplicial sets $${\mathbf S}(G,H)\rightarrow {\mathbf S}(F,H).$$
(3) A morphism $f:F\rightarrow G$ of simplicial presheaves on $T$ is an [*injective $\mathcal{A}$-fibration*]{} if it has the right lifting property with respect to injective cofibrations that are $\mathcal{A}$-weak equivalences.
Recall that the left Bousfield localization of the injective model structure on $\sPsh(T)$ by $\mathcal{A}$-weak equivalences is called the [*injective $\mathcal{A}$-local model structure*]{}, which exists by [@MV Theorem 2.5, p. 71]. Note that the classes of $\mathcal{A}$-weak equivalences, injective cofibrations, and injective $\mathcal{A}$-fibrations form the injective $\mathcal{A}$-local model structure.
\[A.2.2\] The functor $v_\sharp :\sPsh(T)\rightarrow \sPsh(T[\mathcal{A}^{-1}])$ maps $\mathcal{A}$-weak equivalences to pointwise weak equivalences.
Let $\mathcal{C}$ be the class of morphisms $f:F\rightarrow G$ of simplicial presheaves on $T$ such that $v_\sharp f$ is a pointwise weak equivalence. By [@Ayo07 Proposition 4.2.74], it suffices to check the following conditions.
(i) $\mathcal{C}$ contains pointwise weak equivalences and morphisms of the form $$u\times {\rm id}:Y\times F\rightarrow X\times F,$$ where $F$ is a simplicial presheaf on $T$, and $u:Y\rightarrow X$ is a morphism in $\mathcal{A}$.
(ii) $\mathcal{C}$ has the 2 out of 3 property.
(iii) The class of morphisms in $\mathcal{C}$ that are projective cofibrations is stable by pushouts and transfinite compositions.
By [@GZFractions Proposition I.3.1] the functor $v$ commutes with finite limits. Therefore $v_\sharp (u\times {\rm id})$ is an isomorphism since $v_\sharp u$ is an isomorphism. If $f:F\rightarrow G$ is a pointwise weak equivalence, then $F(X)\rightarrow G(X)$ is a weak equivalence for any object $X$ of $T$. According to [@Dug Proposition 7.3], filtered colimits of pointwise weak equivalences are pointwise weak equivalences. Since $\mathcal{A}$ admits a calculus of right fractions, $\mathcal{A}\downarrow X$ is cofiltered. From the formula we deduce that $v_\sharp F(X)\rightarrow v_\sharp G(X)$ is a pointwise weak equivalence. This verifies condtion (i).
The conditions (ii) and (iii) follow from Proposition \[A.2.1\] and the fact that the class of pointwise weak equivalences and class of projective cofibrations satisfy (ii) and (iii).
\[A.1.5\] Let $F$ be a simplicial presheaf on $T$. If $F$ is $\mathcal{A}$-invariant, then the unit map $$F\rightarrow v^*v_\sharp F$$ is a pointwise weak equivalence.
By Proposition \[A.1.4\], we need to show that the induced morphism $$\label{A.1.5.1}
F(X)\rightarrow \colimit_{Y\in \mathcal{A}\downarrow X}F(Y)$$ is a pointwise weak equivalence for any $X\in T$. Each $F(X)\rightarrow F(Y)$ with $Y\in \mathcal{A}\downarrow X$ is a pointwise weak equivalence since $F$ is $\mathcal{A}$-invariant, and the category $\mathcal{A}\downarrow X$ is cofiltered since $\mathcal{A}$ admits a calculus of right fractions. Thus is a pointwise weak equivalence since any filtered colimit of pointwise weak equivalences is again a pointwise weak equivalence by [@Dug Proposition 7.3].
\[A.2.5\] The unit of the adjunction ${\rm id}\rightarrow v^*v_\sharp $ is an $\mathcal{A}$-weak equivalence.
Let $F$ be a simplicial presheaf on $T$ and choose an injective $\mathcal{A}$-fibrant resolution $f:F\rightarrow G$. Note that $\cG$ is $\cA$-invariant. Then $v^*v_\sharp f$ is a pointwise weak equivalence according to Propositions \[A.2.7\] and \[A.2.2\]. Hence to show that $F\rightarrow v^*v_\sharp F$ is an $\mathcal{A}$-weak equivalence, it suffices to show that $G\rightarrow v^*v_\sharp G$ is an $\mathcal{A}$-weak equivalence. For the latter see Proposition \[A.1.5\].
Let $(I,\mathcal{A})$ denote the set of morphisms $\{i_0:\pt\rightarrow I\}\cup \mathcal{A}$ in $T$. From Definition \[A.2.3\](2), we have the notions of $I$-weak equivalences and $(I,\cA)$-weak equivalences.
\[A.2.4\] The functor $$v^*:\sPsh(T[\mathcal{A}^{-1}])\rightarrow \sPsh(T)$$ maps $I$-weak equivalences to $(I,\mathcal{A})$-weak equivalences.
Let $\mathcal{C}$ be the class of morphisms $f:F\rightarrow G$ of simplicial presheaves on $T[\mathcal{A}^{-1}]$ such that $v^*f$ is an $(I,\mathcal{A})$-weak equivalence. By [@Ayo07 Proposition 4.2.74], it suffices to check the following conditions.
(i) $\mathcal{C}$ contains pointwise weak equivalences and morphisms of the form $$i_0\times {\rm id}:{\rm pt}\times F\rightarrow I\times F,$$ where $F$ is a simplicial presheaf on $T[\mathcal{A}^{-1}]$.
(ii) $\mathcal{C}$ has the 2 out of 3 property.
(iii) The class of morphisms in $\mathcal{C}$ that are injective cofibrations is stable by pushouts and transfinite compositions.
Since $v^*$ is a right adjoint of $v_\sharp $, it follows that $v^*$ commutes with limits. Thus to show $v^*(i_0\times {\rm id})$ is an $(I,\mathcal{A})$-weak equivalence, it suffices to show that $v^*i_0$ is an $(I,\mathcal{A})$-weak equivalence. Note that $v^*i_0\cong v^*v_\sharp i_0$. Consider the induced commutative diagram $$\begin{tikzcd}
{\rm pt}\arrow[d]\arrow[r,"i_0"]&I\arrow[d]\\
v^*v_\sharp {\rm pt}\arrow[r,"v^*v_\sharp i_0"]&v^*v_\sharp I
\end{tikzcd}$$ of simplicial presheaves on $T$. The upper horizontal morphism is an $I$-weak equivalence, and the vertical morphisms are $\mathcal{A}$-weak equivalences by Proposition \[A.2.5\]. Hence $v^*v_\sharp i_0$ is an $(I,\mathcal{A})$-weak equivalence. By Proposition \[A.2.7\], $\mathcal{C}$ contains the pointwise weak equivalences. This verifies condition (i).
The conditions (ii) and (iii) follow from Proposition \[A.2.7\] and the fact that the $(I,\mathcal{A})$-weak equivalences and the injective cofibrations satisfy (ii) and (iii).
Construction of the singular functor ${\rm Sing}$
-------------------------------------------------
Since $I$ is an interval in $T[\mathcal{A}^{-1}]$, we are entitled to the cosimplicial presheaf $$\Delta_I^\bullet\in T[\mathcal{A}^{-1}]$$ defined as follows, see [@MV p. 88]. On objects we set $\Delta_I^n:=I^n$. Let $p:I^n\rightarrow {\rm pt}$ denote the canonical morphism to the final object, and let $pr_k:I^k\rightarrow I$ denote the $k$-th projection. For any morphism $f:\{0,\ldots,n\}\rightarrow \{0,\ldots,m\}$ in $\Delta$, let $$\phi(f):\{1,\ldots,m\}\rightarrow \{0,\ldots,n+1\}$$ denote the function given by $$\phi(f)(i):=\left\{
\begin{array}{cl}
\min\{l\in \{0,\ldots,n\}:f(l)\geq i\}&\text{if this set is nonempty}\\
n+1&\text{otherwise}.
\end{array}
\right.$$ The morphism $\Delta_I(f):I^n\rightarrow I^m$ is given by $$pr_k\circ \Delta_I(f):=\left\{
\begin{array}{cl}
pr_{\phi(f)(k)}&\text{if }\phi(f)(k)\in \{1,\ldots,n\}\\
i_0\circ p&\text{if }\phi(f)(k)=n+1\\
i_1\circ p&\text{if }\phi(f)(k)=0.
\end{array}
\right.$$
We also have the simplicial presheaf $${\rm Sing}_\bullet^I(F)\in T[\mathcal{A}^{-1}]$$ defined in [@MV p. 88]. Recall that ${\rm Sing}_\bullet^I(F)$ is the diagonal simplicial presheaf of the bisimplicial presheaf $$(\Delta_I^\bullet,F_\bullet):(m,n)\mapsto \hom_{\Psh(T[\mathcal{A}^{-1}])}(\Delta_I^m,F_n).$$
\[A.1.7\] For any simplicial presheaf $F$ on $T$, we set $${\rm Sing}_\bullet^{(I,\mathcal{A})}(F):=v^*{\rm Sing}_\bullet^I(v_\sharp F).$$
For simplicity we shall often omit the superscript $(I,\mathcal{A})$ in ${\rm Sing}_\bullet^{(I,\mathcal{A})}(F)$.
\[A.1.10\] Let $f,g:F\rightarrow G$ be two morphisms of simplicial presheaves on $T$. An [*elementary $(I,\mathcal{A})$-homotopy*]{} is a morphism $$H:v_\sharp F\times I\rightarrow v_\sharp G$$ of simplicial presheaves on $T[\mathcal{A}^{-1}]$ such that $$H\circ i_0=v_\sharp f
\text{ and }
H\circ i_1=v_\sharp g.$$ Two morphisms of simplicial presheaves on $T$ are [*$(I,\mathcal{A})$-homotopic*]{} if they can be connected by a finite number of $(I,\mathcal{A})$-homotopies.
A morphism $f:F\rightarrow G$ of simplicial presheaves on $T$ is a [*strict $(I,\mathcal{A})$-homotopy equivalence*]{} if there is a morphism $g:G\rightarrow F$ of simplicial presheaves on $T$ such that $f\circ g$ and $g\circ f$ are $(I,\mathcal{A})$-homotopic to the identity morphism.
In the following results, we argue following [@MV Proposition 3.4 - Corollary 3.8, pp. 89-90] and [@AdditiveHomotopy Proposition 4.23 - Proposition 4.26].
\[A.1.11\] The canonical morphism $$i_0:{\rm pt}\rightarrow I$$ is a strict $(I,\mathcal{A})$-homotopy equivalence.
The diagram of simplicial presheaves on $\cT[\cA^{-1}]$ $$\begin{tikzcd}
I\arrow[r,shift left=0.5ex,"i_0\times {\rm id}"]\arrow[r,shift right=0.5ex,"i_1\times {\rm id}"']&I\times I\arrow[r,"\mu"]&I
\end{tikzcd}$$ displays an elementary $(I,\cA)$-homotopy from ${\rm id}=\mu\circ (i_0\times {\rm id})$ to $i_0p=\mu\circ(i_1\times {\rm id})$. Since $pi_0$ is the identity morphism on $\pt$, we are done.
\[A.1.12\] Suppose $f,g:F\rightarrow G$ are $(I,\mathcal{A})$-homotopic morphisms of simplicial presheaves on $T$. Then the induced morphisms $${\rm Sing}_\bullet(f),{\rm Sing}_\bullet(g):{\rm Sing}_\bullet(F)\rightarrow {\rm Sing}_\bullet(G)$$ are simplicially homotopic.
We may assume there is an elementary $(I,\mathcal{A})$-homotopy $H:v_\sharp F\times I\rightarrow v_\sharp G$ from $f$ to $g$. Then $H$ is an $I$-homotopy from $v_\sharp f$ to $v_\sharp g$, so ${\rm Sing}_\bullet(v_\sharp f)$ and ${\rm Sing}_\bullet(v_\sharp g)$ are simplicially homotopic by [@MV Proposition 3.4 in p. 89]. It remains to show that $v^*$ preserves simplicial homotopies.
Since $v^*$ is a right adjoint of $v_\sharp $, it follows that $v^*$ commutes with limits. Thus $$v^*(F\times \Delta[1])\cong v^*F\times \Delta[1]$$ for any simplicial presheaf $F$ on $T[\mathcal{A}^{-1}]$, so $v^*$ preserves simplicial homotopies.
\[A.1.13\] For any simplicial presheaf $F$ on $T$, the morphism $${\rm Sing}_\bullet(X)\rightarrow {\rm Sing}_\bullet(X\times I)$$ induced by $i_0:{\rm pt}\rightarrow I$ is a pointwise weak equivalence.
Follows from Propositions \[A.1.11\] and \[A.1.12\].
\[A.1.14\] For any simplicial presheaf $F$ on $T$, the canonical morphism $$F\rightarrow {\rm Sing}_\bullet(F)$$ is an $(I,\mathcal{A})$-weak equivalence.
We have the factorization $$F\rightarrow v^*v_\sharp F\rightarrow v^*{\rm Sing}_\bullet^I(v_\sharp F).$$ The first morphism is an $\mathcal{A}$-weak equivalence due to Proposition \[A.2.5\]. Thus, by Proposition \[A.2.4\], it suffices to show the canonical morphism $$v_\sharp F\rightarrow {\rm Sing}_\bullet^I(v_\sharp F)$$ is an $I$-weak equivalence. This follows from [@MV Corollary 3.8, p. 89].
Recall that $\Lambda$ is a commutative unital ring.
Let $\sPsh(T,\Lambda)$ denote the category of simplicial sheaves of $\Lambda$-modules on $T$, and let $\Co^{\leq 0}(\Psh(T,\Lambda))$ denote the category of complexes $\cF$ of sheaves of $\Lambda$-modules on $T$ such that $H^i(\cF)=0$ for any $i>0$.
By the Dold-Kan correspondence [@weibel_1994 Theorem 8.4.1], there are equivalences of categories $$N:\sPsh(T,\Lambda)\rightleftarrows \Co^{\leq 0}(\Psh(T,\Lambda)):\Gamma$$ and $$N:\sPsh(T[\cA^{-1}],\Lambda)\rightleftarrows \Co^{\leq 0}(\Psh(T[\cA^{-1}],\Lambda)):\Gamma$$
An object $F$ of $\Co^{\leq 0}(\Psh(T,\Lambda))$ is *$\cA$-invariant* if $\Gamma(F)$ is $\cA$-invariant. A morphism $f:F\rightarrow G$ in $\Co^{\leq 0}(\Psh(T,\Lambda))$ (resp. $\Co^{\leq 0}(\Psh(T[\cA^{-1}],\Lambda))$) is a $(I,\cA)$-weak equivalence (resp. $\cA$-weak equivalence) if $\Gamma(f):\Gamma(F)\rightarrow \Gamma(G)$ is a $(I,\cA)$-weak equivalence (resp. $\cA$-weak equivalence).
Let $\Co^{-}(\Psh(T,\Lambda))$ denote the category of bounded above complexes of sheaves of $\Lambda$-modules. Then an object $F\in \Co^{-}(\Psh(T,\Lambda))$ is called $\cA$-*invariant* if there exists an integer $n\in \Z$ such that $F[n]\in\Co^{\leq 0}(\Psh(T,\Lambda))$ is $\cA$-*invariant*.
The $(I,\cA)$-weak equivalences in $\Co^{-}(\Psh(T,\Lambda))$ and the $\cA$-weak equivalences in $\Co^{-}(\Psh(T[\cA^{-1}],\Lambda))$ are defined similarly.
Suppose $G$ is a presheaf of $\Lambda$-modules on $T$ and consider the simplicial presheaf $\Lambda$-modules $\Sing_{\bullet}^{(I,\cA)}(G)$ on $T$. Taking the alternating sum of the face maps yields the *Suslin complex* of presheaves $$C_*^{(I,\cA)}G.$$
We let $C_*G$ be shorthand for $C_*^{(I,\cA)}G$ when no confusion seems likely to arise, and define $$C_*^{DK}(G):=N({\rm Sing}_\bullet^{(I,\cA)}(G)).$$ Here $C_*^{DK}(G)$ is canonically a normalized subcomplex of $C_*(G)$. The morphism $$C_*^{DK}(G)\rightarrow C_*(G)$$ is a quasi-isomorphism but not an isomorphism in general.
If $F$ is a bounded above complex of presheaves on $T$, we can consider $C_*F$ and $C_*^{DK}F$ as double complexes and form their total complexes ${\rm Tot}(C_*F)$ and ${\rm Tot}(C_*^{DK}F)$.
Suppose $H\in \sPsh(T,\Lambda)$. The Eilenberg-Zilber theorem says that there is a quasi-isomorphism $$\label{A.1.15.1}
N({\rm Sing}_\bullet^{(I,\cA)}(H))
\cong
{\rm Tot}(C_*^{DK}(N(H))).$$
As in Definition \[A.1.3\] there exist adjoint functors $$\begin{tikzcd}
\Co^{\leq 0}(\Psh(T,\Lambda))\arrow[rr,shift left=1.5ex,"v_\sharp "]\arrow[rr,"v^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"v_*"']&&\Co^{\leq 0}(\Psh(T[\mathcal{A}^{-1}],\Lambda)),
\end{tikzcd}$$ where $v_\sharp$ is left adjoint to $v^*$, and $v^*$ is left adjoint to $v_*$. For $F\in \Co^{\leq 0}(\Psh(T[\mathcal{A}^{-1}],\Lambda))$ and $X\in T$, we have $$\Gamma(v^*F)(X)\cong \Gamma(v^*F(X))=\Gamma(F(v(X))=\Gamma(F)(v(X))\cong v^*(\Gamma(F))(X).$$ Thus $\Gamma$ commutes with $v^*$. Similarly, we can show that $N$ commutes with $v^*$. By adjunction we see that both $\Gamma$ and $N$ commute with $v_\sharp$.
\[A.1.15\] Let $F$ be a bounded above complex of presheaves of $\Lambda$-modules on $T$. Then the following properties hold.
1. If $F$ is $\cA$-invariant, then the unit of the adjunction $$F\rightarrow v^*v_\sharp F$$ is a quasi-isomorphism.
2. The unit of the adjunction $$F\rightarrow v^*v_\sharp F$$ is an $\mathcal{A}$-weak equivalence.
3. The morphism $${\rm Tot}(C_*F)\rightarrow {\rm Tot}(C_*(F\times I))$$ induced by $i_0:{\rm pt}\rightarrow I$ is a quasi-isomorphism.
4. The canonical morphism $$F\rightarrow {\rm Tot}(C_*F)$$ is an $(I,\mathcal{A})$-weak equivalence.
We observe that all the results in this section hold verbatim for simplicial presheaves of $\Lambda$-modules. It suffices to prove these assertions for the shifted complex $F[n]$, $n\in \Z$. Thus we may assume $F\in \Co^{\leq 0}(\Psh(T,\Lambda))$. As noted above both $\Gamma$ and $N$ commute with both $v_\sharp$ and $v^*$. Moreover, quasi-isomorphisms (resp. $\cA$-weak equivalences, resp. $(I,\cA)$-weak equivalences) in $\Co^{\leq 0}(\Psh(T,\Lambda))$ corresponds to pointwise weak equivalences (resp. $\cA$-weak equivalences, resp. $(I,\cA)$-weak equivalences) in $\sPsh(T,\Lambda)$ under $\Gamma$ and $N$.
Suppose $F$ is $\cA$-invariant. Then $\Gamma(F)$ is $\cA$-invariant, and the $\Lambda$-module version of Proposition \[A.1.5\] shows the morphism $$\Gamma(F)\rightarrow v^*v_\sharp \Gamma(F)\xrightarrow{\cong} \Gamma(v^*v_\sharp F)$$ is a pointwise weak equivalence. Thus $F\rightarrow v^*v_\sharp F$ is a quasi-isomorphism, which shows (1). A similar argument shows (2). Due to we have isomorphisms $$\begin{gathered}
{\rm Tot}(C_*F)\simeq {\rm Tot}(C_*^{DK}F)\simeq N({\rm Sing}_\bullet^{(I,\cA)}(\Gamma(F))),
\\
{\rm Tot}(C_*(F\times I))\simeq {\rm Tot}(C_*^{DK}(F\times I))\simeq N({\rm Sing}_\bullet^{(I,\cA)}(\Gamma(F\times I))).\end{gathered}$$ Owing to the $\Lambda$-module version of Corollary \[A.1.13\] the composite morphism $${\rm Sing}_\bullet^{(I,\cA)}(\Gamma(F))\rightarrow {\rm Sing}_\bullet^{(I,\cA)}(\Gamma(F)\times I)\xrightarrow{\cong} {\rm Sing}_\bullet^{(I,\cA)}(\Gamma(F\times I))$$ is an isomorphism. Combining the above we deduce (3). A similar argument shows (4).
Properties of logarithmic motives
=================================
In this section, we formulate and prove fundamental properties of our categories of logarithmic motives. Most of these properties can be stated for categories associated to a base scheme $S$, not necessarily a field, subject to a list of axioms listed below. This setup affords the case of $\ldmeff$ and its variants, see Proposition \[A.5.71\].
It can be quite interesting to observe that certain implications hold among the axioms. For example, $(\P^n, \P^{n-1})$-invariance can be deduced by assuming Nisnevich descent (Nis-desc), $\boxx$-invariance, and dividing-descent, see Proposition \[A.6.1\]. On the other hand, we can include $(\P^n, \P^{n-1})$-invariance as part of the axioms. In that case dividing-descent can be deduced from (Nis-desc) and $(\P^n, \P^{n-1})$-invariance for every $n\geq 1$. We refer to Section \[ssec:invariance-asuming-PnPn-1\] for further details.
After proving some general properties about $\P^n$-bundles and $\boxx^n$-bundles, we begin by showing an invariance property for certain admissible blow-ups. This is used for showing that log motives satisfies the $(\P^n, \P^{n-1})$-invariance property. We proceed by defining Thom motives, a.k.a. motivic Thom spaces, prove a log version of homotopy purity, and construct Gysin maps for closed embeddings.
Assuming that $k$ admits resolution of singularities allow us to prove a stronger invariance property for proper birational morphisms, see Theorem \[A.3.12\]. Finally we discuss strictly $(\P^\bullet,\P^{\bullet-1})$-invariant and strictly dividing invariant complexes of sheaves on $SmlSm/k$. This can be seen as a convenient tool for constructing objects of $\ldmeff$ without explicitly verifying the dividing-invariance property on the level of sheaf cohomology.
Throughout this section, we fix a base scheme $S$ and let $\mathscr{S}/S$ denote either $lSm/S$ or $SmlSm/S$. Moreover, we fix a symmetric monoidal triangulated category $\cT$, and we fix a monoidal functor $$M\colon \mathbf{Square}_{\mathscr{S}/S}\rightarrow \cT$$ satisfying (Sq) and (MSq). For the category $\mathbf{Square}_{\mathscr{S}/S}$, $n$-squares, and the conditions (Sq) and (MSq), we refer to Appendix \[appendix:cat\_tool\]. The reason why we choose to use $n$-squares is that this is a convenient way to construct certain cones canonically within triangulated categories without referring to model categories or $\infty$-categories. For example, $2$-squares are used in [@Deg08] to develop the theory of Gysin triangles in $\A^1$-motivic theory. In most of our argument we only need $n$-squares for $n\leq 2$, but we need $3$-squares in Subsection \[ssec:ThomMotives\].
We shall often assume some of the following properties.
1. *Zariski separation property.* If $\{X_1,\ldots,X_r\}$ is a finite Zariski cover of $X$ and $I=\{i_1,\ldots,i_s\}\subset \{1,\ldots,r\}$, we set $$X_I:=X_{i_1}\times_S \cdots \times_S X_{i_s}.$$ Then for any commutative square of the form $$\label{A.3.0.2}
\begin{tikzcd}
Y'\arrow[d]\arrow[r]&Y\arrow[d]
\\
X'\arrow[r]&X,
\end{tikzcd}$$ the morphism $$M(Y'\rightarrow X')\rightarrow M(Y\rightarrow X)$$ is an isomorphism if for every nonempty $I\subset \{1,\ldots,r\}$ there is an isomorphism $$M(Y_I'\rightarrow X_I')\rightarrow M(Y_I\rightarrow X_I)$$ where $$Y_I:=Y\times_X X_I, X_I':=X'\times_X X_I, \text{ and }Y_I':=Y'\times_Y Y_I.$$
2. *$\boxx$-invariance property.* For any $X\in \mathscr{S}/S$ the projection $X\times \boxx\rightarrow X$ induces an isomorphism $$M(X\times \boxx)\xrightarrow{\cong} M(X).$$
3. *$sNis$-descent property.* Every strict Nisnevich distinguished square of the form induces an isomorphism $$M(Y'\rightarrow X')\xrightarrow{\cong} M(Y\rightarrow X).$$
4. *$div$-descent property.* Every log modification $f:Y\rightarrow X$ in $\mathscr{S}/S$ induces an isomorphism $$M(Y)\xrightarrow{\cong} M(X).$$
5. *$(\P^\bullet,\P^{\bullet-1})$-invariance property.* For any $X\in \mathscr{S}/S$ and positive integer $n>0$ the projection $X\times (\P^n,\P^{n-1})\rightarrow X$ induces an isomorphism $$M(X\times (\P^n,\P^{n-1}))\xrightarrow{\cong} M(X).$$ Here we consider $\P^{n-1}$ as the hyperplane of $\P^n$ defined by the equation $x_n=0$ with respect to the coordinates $[x_0:\cdots:x_n]$.
6. *Admissible blow-up property.* For every proper birational morphism $f\colon Y\rightarrow X$ in $\mathscr{S}/S$ such that the naturally induced morphism $Y-\partial Y\rightarrow X-\partial X$ is an isomorphism, there is a naturally induced isomorphism $$M(Y)\xrightarrow{\cong} M(X).$$
The following figure illustrates how these properties are related. $$\begin{tikzpicture}
\node[draw,align=center] at (-5,0)
{($Zar$-sep)
\\
($\boxx$-inv)
\\
($sNis$-des)
\\
(adm-blow)
\\
$\mathscr{S}=lSm/k$};
\node[draw,align=center] at (0,0)
{($Zar$-sep)
\\
($\boxx$-inv)
\\
($sNis$-des)
\\
($div$-des)
\\
$\mathscr{S}=lSm/S$};
\node[draw,align=center] at (5,0)
{($Zar$-sep)
\\
($sNis$-des)
\\
($(\P^\bullet,\P^{\bullet-1})$-inv)
\\ $\mathscr{S}=SmlSm/S$};
\draw (-3.5,0.6)--(-1.5,0.6);
\draw (-3.5,0.5)--(-1.5,0.5);
\node at (-1.5,0.55) {$>$};
\node at (-2.5,0.8) {\footnotesize Trivial if $S=k$};
\draw (-3.5,-0.6)--(-1.5,-0.6);
\draw (-3.5,-0.5)--(-1.5,-0.5);
\node at (-3.5,-0.55) {$<$};
\node at (-2.5,-0.8) {\footnotesize Theorem \ref{A.3.12}};
\draw (1.35,0.6)--(3.35,0.6);
\draw (1.35,0.5)--(3.35,0.5);
\node at (3.35,0.55) {$>$};
\node at (2.35,0.8) {\footnotesize Proposition \ref{A.6.1}};
\draw (1.35,-0.6)--(3.35,-0.6);
\draw (1.35,-0.5)--(3.35,-0.5);
\node at (1.35,-0.55) {$<$};
\node at (2.35,-0.8) {\footnotesize Construction \ref{Pn.4}};
\end{tikzpicture}$$
\[A.5.60\] The Tate object in $\cT$ is defined by $$M(S)(1)
:=
M(S\stackrel{i_0}\rightarrow \P_S^1)[-2],$$ where $i_0:S \rightarrow \P_S^1$ is the $0$-section.
For $n\geq 0$, we set $$M(S)(n)
:=
\underbrace{M(S)(1)\otimes \cdots \otimes M(S)(1)}_{n\text{ times}},$$ where $M(S)(0):=M(S)$ by convention. The $n$th Tate twist of $K\in\cT$ is defined as $$K(n)
:=
K\otimes M(S)(n).$$
\[A.5.71\] The conditions [(Sq)]{} and [(MSq)]{} of Appendix \[appendix:cat\_tool\] and the properties [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and [($div$-des)]{} hold in the triangulated categories $$\begin{gathered}
\ldmeff,\;\ldaeff,\;\ldmeffet,\;\ldaeffet,
\\
\ldmefflet,\;\ldaefflet.\end{gathered}$$
We work in $\ldmeff$ since the proofs for the other categories are similar. For every $n$-square $C$, we set $$M(C):={\rm Tot}(\Zltr(C)).$$ For every $n$-square $C$ with $n\geq 1$, there exists a naturally induced sequence $${\rm Tot}(\Zltr(s_{i,0}^*C))\rightarrow {\rm Tot}(\Zltr(s_{i,1}^*C))\rightarrow {\rm Tot}(\Zltr(C))\rightarrow {\rm Tot}(\Zltr(s_{i,0}^*C))[1].$$ Moreover, for every $n$-square $C$ and $m$-square $D$, there exists a canonical isomorphism (not just a quasi-isomorphism) of complexes $${\rm Tot}(\Zltr(C))\otimes {\rm Tot}(\Zltr(D))\rightarrow {\rm Tot}(\Zltr(C\otimes D)).$$ The latter is reminiscent of an Eilenberg-Zilber map, but it is in fact much simpler since no shuffles are involved. We can readily deduce that the conditions (Sq) and (MSq) hold.
Property ($\boxx$-inv) holds by construction. Proposition \[A.5.42\] implies ($sNis$-des) and ($div$-des). To show ($Zar$-sep), let $\mathscr{X}$ be the Čech nerve of the covering $\{X_1,\ldots,X_r\}$, and set $$\mathscr{X}':=X'\times_X \mathscr{X}, \mathscr{Y}:=Y\times_X \mathscr{X}, \mathscr{Y}':=Y'\times_X \mathscr{X}.$$ Then, by assumption, we have the homotopy cartesian diagram $$\begin{tikzcd}
M(\mathscr{Y}')\arrow[d]\arrow[r]&M(\mathscr{Y})\arrow[d]\\
M(\mathscr{X}')\arrow[r]&M(\mathscr{X}).
\end{tikzcd}$$ This allows us to conclude that ($Zar$-sep) holds owing to the isomorphisms $$M(X)\cong M(\mathscr{X}), M(Y)\cong M(\mathscr{Y}), M(X')\cong M(\mathscr{X}'), \text{ and } M(Y')\cong M(\mathscr{Y}').$$
Some toric geometry and examples of bundles
-------------------------------------------
Let $d\geq 0$ and suppose $\sigma$ is a $d$-codimensional cone of a fan $\Sigma$. We can naturally associate a $d$-dimensional closed subscheme $V(\sigma)$ of $\ul{\A_\Sigma}$, which is the closure of the orbit space ${\rm orb}(\sigma)$ in $\ul{\A_\Sigma}$, see [@TOda Proposition 1.6, Corollary 1.7]. Let us explain the structure in the case that $\Sigma=\Spec{P}$ for some fs monoid $P$. Note that the only maximal cone of $\Sigma$ is $P^\vee$.
The cone $\sigma$ is associated with a face $F$ of $P^\vee$, which corresponds by duality to the face $$G:=F^\bot\cap P=\{x\in P:\langle x,y\rangle =0\text{ for all }y\in F\}.$$ Then ${\rm orb}(\sigma)$ is identified with $$\Spec{\Z[F^\bot \cap P^\gp]}\cong \Spec{\Z[G^\gp]},$$ see [@Fulton:1436535 p. 52]. Owing to [@Ogu Proposition I.3.3.4] the closed immersion $$V(\sigma)\rightarrow \ul{\A_\Sigma}$$ is induced by the natural epimorphism $$\Z[P]\rightarrow \Z[P]/\Z[P-G],$$ where we regard $\Z[P-G]$ as an ideal of $\Z[P]$.
If more specifically $P=\N^p\oplus \N^q$ and $F$ is the face $\N^p\oplus 0$ of $P^\vee$ for some integers $p,q\geq 0$, then $G=0\oplus \N^q$. Thus if we express $\Z[P]$ as $\Z[x_1,\ldots,x_{p+q}]$, then $$\Z[P]/\Z[P-G]=\Z[x_1,\ldots,x_{p+q}]/(x_1,\ldots,x_p).$$
Let $\Sigma$ be a fan. Suppose that $\Sigma'$ is a subfan of $\Sigma$, i.e., every cone of $\Sigma'$ is a cone of $\Sigma$. We denote by $$\A_{(\Sigma,\Sigma')}$$ the fs log scheme whose underlying scheme is $\ul{\A_\Sigma}$ and whose log structure is the compactifying log structure associated with the open immersion $$(\ul{\A_{\Sigma}}-\bigcup_{\sigma\in \Sigma'-\{0\}}V(\sigma))\rightarrow \ul{\A_\Sigma}.$$ For the convenience of the reader we will often draw figures illustrating the fans $\Sigma$ and $\Sigma'$. In our drawings of $2$-dimensional fans, thick (resp. thin) lines record the $1$-dimensional cones in $\Sigma'$ (resp. not in $\Sigma'$), and dark (resp. light) grey regions records the $2$-dimensional cones in $\Sigma'$ (resp. not in $\Sigma'$). For example, the figure $$\label{eq.examplefan}
\begin{tikzpicture}
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (0,0)--(1,0)--(1,1)--(0,1);
\fill[black!40] (0,0)--(0,1)--(-1,1)--(-1,0);
\draw (0,0)--(1,0);
\draw[ultra thick] (0,0)--(0,1);
\draw[ultra thick] (0,0)--(-1,0);
\end{tikzpicture}$$ depicts the pair of fans $$\Sigma=\{{\rm Cone}((1,0),(0,1)),{\rm Cone}((0,1),(-1,0))\},
\;
\Sigma'=\{{\rm Cone}((0,1),(-1,0))\}.$$ Here, the notation $\Sigma=\{\sigma,\ldots \}$ means that $\Sigma$ consists of all faces of the elements of the set $\{\sigma,\ldots\}$. The algebraic variety encoded in the figure can be described as follows. We set $\sigma_1 ={\rm Cone}((1,0),(0,1))$, $\sigma_2=\Sigma' = {\rm Cone}((0,1),(-1,0))$, $M=\Z^2$, and $N= {\rm Hom}_\Z(M,\Z)$. The toric variety $\ul{\A_\Sigma}$ is obtained by glueing the two affine schemes $$\mathbb{A}_{\sigma_1}=\Spec{\Z[\sigma_1^\vee \cap N]} = \Spec{\Z[x,y]}$$ and $$\mathbb{A}_{\sigma_2} = \Spec{\Z[\sigma_2^\vee \cap N]} = \Spec{\Z[x^{-1},y]}$$ along their common open subset $$\Spec{\Z[\tau^\vee\cap N]} = \Spec{\Z[x,x^{-1}, y]},$$ where $\tau$ is the 1-dimensional cone $(0,1)$. The underlying scheme is $$X = \P^1\times \A^1,$$ where $x$ is the coordinate on $\P^1$ and $y$ is the coordinate of $\A^1$. Consider the divisor $\partial X = \infty\times \A^1+\P^1\times 0$ on $X$. The corresponding compactifying log structure $(X,\partial X)$ can be identified with the fs log scheme $\A_{(\Sigma,\Sigma')}$.
In our drawings of $3$-dimensional fans, thick points records $1$-dimensional cones in $\Sigma'$, thick (resp. thin) lines record $2$-dimensional cones in $\Sigma'$ (resp. not in $\Sigma'$), and dark (resp. light) grey regions record $3$-dimensional cones in $\Sigma'$ (resp. not in $\Sigma'$).
Before going into the discussion of motives of general fs log schemes, we will first investigate some elementary examples of $\P^1$- and $\boxx$-bundles.
Let us define the geometric notion of vector bundles over fs log schemes.
\[df:logvectorbundle\] Let $X$ be an fs log scheme. A [*vector bundle*]{} $\xi:\cE\rightarrow X$ is a morphism of fs log schemes such that $\underline{\xi}:\underline{\cE}\rightarrow \underline{X}$ is a usual vector bundle and the diagram $$\begin{tikzcd}
\cE\arrow[r,"\xi"]\arrow[d]\arrow[r]&X\arrow[d]\\
\underline{\cE}\arrow[r,"\underline{\xi}"]&\underline{X}
\end{tikzcd}$$ is cartesian where the vertical arrows are the canonical morphisms removing the log structures. In other words, the log structure on $\cE$ is the pullback of the log structure on $X$ along $\xi$.
The projection $X\times_S \A_S^n\rightarrow X$ is a vector bundle. This is called the [*trivial*]{} vector bundle of rank $n$.
We say that “an fs log scheme $X$ satisfies Zariski locally a property $P$” if for any point $x$ of $X$ and Zariski neighborhood $V$ of $x$, there is a Zariski neighborhood $U$ of $x$ inside $V$ such that $U$ satisfies $P$.
\[df::projectivebundle\] Let $X$ be an fs log scheme. A $\P^n$-*bundle* $\xi:\cP\rightarrow X$ is a morphism of fs log schemes that Zariski locally on $X$ is isomorphic to the projection $X\times \P^n\rightarrow X$.
Similarly, a $(\P^n,\P^{n-1})$-*bundle* $\xi:\cP\rightarrow X$ is a morphism of fs log schemes that Zariski locally on $X$ is isomorphic to the projection $X\times (\P^n,\P^{n-1})\rightarrow X$. Note that we have $(\P^1,\P^{0})=\boxx$.
Recall that we view $\P^{n-1}$ as the hyperplane defined by $x_n=0$ where $[x_0:\cdots:x_n]$ is the coordinate on $\P^n$.
\[A.6.4\] The blow-up $X$ of $\P^n$ along the point $[0:\cdots:0:1]$ is a $\P^1$-bundle over $\P^{n-1}$.
While this should be a well-known fact, since we need to fix some notation for later use, we give a proof using the language of toric geometry. Let $e_1,\ldots,e_n$ be the standard coordinates of $\Z^n$. The blow-up $X$ in question is the toric variety associated to the fan with maximal cones $$\sigma_i:={\rm Cone}(e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_n,e_1+\cdots+e_n)\quad (1\leq i\leq n),$$ $$\sigma_i':={\rm Cone}(e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_n,-e_1-\cdots-e_n)\quad (1\leq i\leq n).$$ For a fixed $i$, the toric variety $U_i$ formed by $\sigma_i$ and $\sigma_i'$ is isomorphic to $\A^{n-1}\times \P^1$.
Recall that $\P^{n-1}$ is the toric variety associated to the fan with maximal cones $$\tau_i:={\rm Cone}(e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_{n-1},-e_1-\cdots-e_{n-1})\quad (1\leq i\leq n-1),$$ $$\tau_n:={\rm Cone}(e_1,\ldots,e_{n-1}).$$ For the homomorphism of lattices $$\varphi:\Z^n\rightarrow \Z^{n-1}$$ mapping $(x_1,\ldots,x_n)$ to $(x_1-x_n,\ldots,x_{n-1}-x_n)$, we have that $$\varphi(\sigma_i\cup \sigma_i')=\tau_i.$$ Thus the morphism $X\rightarrow \P^{n-1}$ induced by $\varphi$ is a $\P^1$-bundle.
\[A.6.5\] The blow-up of $(\P^n,\P^{n-1})$ along the point $[0:\cdots:0:1]$ is a $\boxx$-bundle over $\P^{n-1}$.
The cones $\sigma_i$ and $\sigma_i'$ appearing in the proof of Lemma \[A.6.4\] correspond to $$U_i:={\rm Spec}\left( \Z\left[\frac{x_0}{x_{i-1}},\ldots,\frac{x_{i-2}}{x_{i-1}},\frac{x_i}{x_{i-1}},\ldots,\frac{x_{n-1}}{x_{i-1}},\frac{x_{i-1}}{x_n}\right]\right)$$ and $$U_i':={\rm Spec}\left( \Z\left[\frac{x_0}{x_{i-1}},\ldots,\frac{x_{i-2}}{x_{i-1}},\frac{x_i}{x_{i-1}},\ldots,\frac{x_{n-1}}{x_{i-1}},\frac{x_{n}}{x_{i-1}}\right]\right),$$ respectively. Their log structures induced by $(\P^n,\P^{n-1})$ are given by $$\N \left(\frac{x_{n}}{x_{i-1}}\right)\rightarrow \Z\left[\frac{x_0}{x_{i-1}},\ldots,\frac{x_{i-2}}{x_{i-1}},\frac{x_i}{x_{i-1}},\ldots,\frac{x_{n-1}}{x_{i-1}},\frac{x_{n}}{x_{i-1}}\right].$$ In the proof of Lemma \[A.6.4\] we noted that $$U_i\cup U_i'\cong \A^{n-1}\times \P^1.$$ If we impose the above log structure, then this becomes $\A^{n-1}\times \boxx$, and the claim follows.
\[A.3.28\] For $n\geq 2$ we let $X$ be the blow-up of $\P^n$ at the point $[0:\cdots:0:1]$. Let $H$ be the divisor of $\P^n$ defined by $(x_0=0)$, let $E$ be the exceptional divisor on $X$, and let $H'$ be the strict transform of $H$ in $X$. Then $$(X,H'+E)$$ is a $\boxx$-bundle over $(\P^{n-1},\P^{n-2})$.
Recall the schemes $U_i$ and $U_i'$ appearing in the proof of Lemma \[A.6.5\]. We equip $U_i$ with the log structures
$$\N\left(\frac{x_0}{x_{i-1}}\right)\oplus \N\left(\frac{x_{i-1}}{x_{n}}\right)\rightarrow \Z\left[\frac{x_0}{x_{i-1}},\ldots,\frac{x_{i-2}}{x_{i-1}},\frac{x_i}{x_{i-1}},\ldots,\frac{x_{n-1}}{x_{i-1}},\frac{x_{n}}{x_{i-1}}\right]\text{ if }i>1,$$
$$\N\left(\frac{x_0}{x_{n}}\right)\rightarrow \Z\left[\frac{x_0}{x_{i-1}},\ldots,\frac{x_{i-2}}{x_{i-1}},\frac{x_i}{x_{i-1}},\ldots,\frac{x_{n-1}}{x_{i-1}},\frac{x_{n}}{x_{i-1}}\right]\text{ if }i=1,$$ and $U_i'$ with the log structures
$$\N\left(\frac{x_0}{x_{i-1}}\right)\rightarrow \Z\left[\frac{x_0}{x_{i-1}},\ldots,\frac{x_{i-2}}{x_{i-1}},\frac{x_i}{x_{i-1}},\ldots,\frac{x_{n-1}}{x_{i-1}},\frac{x_{i-1}}{x_{n}}\right]\text{ if }i>1,$$
$$0\rightarrow \Z\left[\frac{x_0}{x_{i-1}},\ldots,\frac{x_{i-2}}{x_{i-1}},\frac{x_i}{x_{i-1}},\ldots,\frac{x_{n-1}}{x_{i-1}},\frac{x_{i-1}}{x_{n}}\right]\text{ if }i=1.$$
Let $X_i$ and $X_i'$ denote the associated fs log schemes, and let $Y_i$ be the fs log scheme associated with $$\N\left(\frac{x_0}{x_{i-1}}\right)\rightarrow \Z\left[\frac{x_0}{x_{i-1}},\ldots,\frac{x_{i-2}}{x_{i-1}},\frac{x_i}{x_{i-1}},\ldots,\frac{x_{n-1}}{x_{i-1}}\right].$$ With these definitions we have $$X_i\cup X_i'\cong Y_i\times \boxx.$$ This finishes the proof because the $Y_i$’s glue together to form $(\P^{n-1},\P^{n-2})$. Here we consider $\P^{n-2}$ as the divisor on $\P^{n-1}$ defined by $(x_0=0)$.
\[A.3.33\] Let $X$ be the blow-up of $\P^n$ along the point $[0:\cdots:0:1]$, and let $E$ be the corresponding exceptional divisor on $X$. Then $(X,E)$ is a $\boxx$-bundle over $\P^{n-1}$.
Recall the schemes $U_i$ and $U_i'$ appearing in the proof of Lemma \[A.6.5\]. Their log structures induced by $(X,E)$ are given by
$$\N \left(\frac{x_{i-1}}{x_{n}}\right)\rightarrow \Z\left[\frac{x_0}{x_{i-1}},\ldots,\frac{x_{i-2}}{x_{i-1}},\frac{x_i}{x_{i-1}},\ldots,\frac{x_{n-1}}{x_{i-1}},\frac{x_{i-1}}{x_{n}}\right].$$ Our claim follows by noting there is an isomorphism $$U_i\cup U_i'\cong \boxx\times \A^{n-1}.$$
\[A.3.40\] Suppose $Y$ is a $\boxx$-bundle over $X\in\mathscr{S}/S$. If $\cT$ satisfies [($\boxx$-inv)]{} and [($sNis$-des)]{}, then the projection $Y\rightarrow X$ induces an isomorphism $$M(Y)\xrightarrow{\cong} M(X).$$
Use induction on the number of open subsets in a trivialization of $Y$, then from ($sNis$-des) we may assume $Y\cong X\times \boxx$. To conclude we apply ($\boxx$-inv).
Invariance under admissible blow-ups along smooth centers {#ssec:invariancesmoothblowups}
---------------------------------------------------------
In this subsection, assuming the properties ($Zar$-sep), ($\boxx$-inv), and ($sNis$-des) we will extend the invariance under log modifications ($div$-des) to admissible blow-ups along smooth centers, whose definition is given below.
\[A.3.17\] Suppose that $Z$ is an effective Cartier divisor on $X\in Sm/S$, and let $i\colon Z\rightarrow X$ be the corresponding closed immersion. Then $Z$ is a [*strict normal crossing divisor*]{} on $X$ over $S$ if Zariski locally on $X$, there is a cartesian square $$\begin{tikzcd}
Z\arrow[d,"i"']\arrow[r]&S\times \Spec {\Z[x_1,\ldots,x_n]/(x_1\cdots x_r)}\arrow[d]\\
X\arrow[r]& S\times \Spec{ \Z[x_1,\ldots,x_n]}
\end{tikzcd}$$ of schemes over $S$ for some $r\leq n$, where the horizontal morphisms are étale.
We say the divisors $Z_1,\ldots,Z_r$ on $X\in Sm/S$ form a [*strict normal crossing divisor*]{} if $$Z_1+\cdots+Z_r$$ is a strict normal crossing divisor on $X$ over $S$.
Suppose that $Z$ is a strict normal crossing divisor on $X$ over $S$, and let $v:W\rightarrow X$ be a closed immersion of schemes smooth over $S$. We say that $W$ has [*strict normal crossing*]{} with $Z$ over $S$ if Zariski locally on $X$, there is a commutative diagram of cartesian squares of schemes over $S$ $$\begin{tikzcd}
Z\arrow[d,"i"']\arrow[r]&S\times \Spec{ \Z[x_1,\ldots,x_n]/(x_1\cdots x_r)}\arrow[d]\\
X\arrow[d,leftarrow,"v"']\arrow[r]&S\times \Spec{ \Z[x_1,\ldots,x_n]}\arrow[d,leftarrow]\\
W\arrow[r]&S\times \Spec {\Z[x_1,\ldots,x_n]/(x_{i_1},\ldots,x_{i_s})}
\end{tikzcd}$$ for some $r\leq n$ and $1\leq i_1<\cdots<i_s\leq n$, where the right vertical morphisms are the evident closed immersions and the horizontal morphisms are étale. Note that, a priori, in this formulation $W$ can be contained in a component of $Z$. If $r\leq i_1< \cdots i_s\leq n$, we get, in particular, that the restriction of $Z$ to $W$ is again a Cartier divisor, and that it forms a strict normal crossing divisor on $W$.
\[def:associative-notation-log\] For divisors $Z_1,\ldots,Z_r$ forming a strict normal crossing divisor on $X\in Sm/S$, for $1\leq s\leq r$, we use the convenient “associative” notation $$\begin{split}
((X,Z_{s+1}+\cdots +Z_r),Z_1+\cdots+Z_s)&:=(X,Z_1+\cdots+Z_r)
\end{split}$$ where $Z_i'$ is any fs log scheme such that $\underline{Z_i'}\cong Z_i$ (we refer to Definition \[A.0.2\] for the notation $(X,Z_1+\cdots+Z_r)$).
To illustrate, suppose $X\in SmlSm/S$ has compactifying log structure given by $Z_{s+1}+\cdots +Z_r$, written $(X, Z_{s+1}+\cdots+Z_r)$. And let us consider a strict normal crossing divisor $Z_1+\cdots+Z_s$ on $X$ such that $Z_1+\cdots+Z_s + Z_{s+1}+\cdots +Z_r$ is again strict normal crossing. This determines another object of $SmlSm/S$, namely $(X, Z_{1}+\cdots+Z_r)$, equipped with a canonical map $$(X, Z_{1}+\cdots+Z_r) \to (X, Z_{s+1}+\cdots+Z_r)$$ obtained by adding the divisor $Z_{1}+\cdots+Z_s$ to the log structure.
For instance, if $Y:=(X,Z_{s+1},\cdots,Z_r)$, then we obtain $$(Y,Z_1+\cdots+Z_s)=(X,Z_1+\cdots +Z_r).$$ If $Y=\P^1\times \boxx=(\P^1\times \P^1,\P^1\times \{\infty\})$ and $Z_1'=\{\infty\}\times \boxx$, then $$(\P^1\times \boxx,\{\infty\}\times \boxx)=\boxx^2.$$
An *admissible blow-up along a smooth center over $S$* (or along a smooth center for short) is a proper birational morphism $$f:X'\rightarrow X$$ in $SmlSm/S$ for which the following properties hold.
1. $\underline{f}$ is the blow-up along a center $Z$ in $\underline{X}$ such that $Z$ is smooth over $S$.
2. $\partial X'$ is the union of $f^{-1}(Z)$ and the strict transform of $\partial X$ with respect to $f$.
3. $Z$ is contained in $\partial X$ and has strict normal crossing with $\partial X$ over $S$ in the sense of Definition \[A.3.17\].
Note that by Lemma \[lem::SmlSm\], we may assume that $\partial X$ is a strict normal crossing divisor on $X$, so that condition (3) makes sense. The open immersion $X-Z\rightarrow X$ lifts to $X'$. Hence the log structure on $X'$ is precisely the compactifying log structure associated with the composite open immersion $$X-\partial X\rightarrow X-Z\rightarrow X'.$$ If $Z$ has codimension $d$ in $X$, then we say that $f$ is an *admissible blow-up along a codimension $d$ smooth center*. Let $\cA\cB l_{Sm}/S$ denote the smallest class of morphisms in $\mathscr{S}/S$ containing all admissible blow-ups along a smooth center and closed under composition.
Let us give a fundamental example of admissible blow-ups along a smooth center.
\[A.3.41\] Let $X=\A_\N^p\times \A^q$ and let $e_1,\ldots,e_{p+q}$ denote the standard coordinates on $\Z^n$. We set $$\sigma:={\rm Cone}(e_1,\ldots,e_{p+q}),\;
\sigma':={\rm Cone}(e_1,\ldots,e_p),$$ and form the associated fans $$\Sigma:=\{\sigma\},\;\Sigma':=\{\sigma'\}.$$ Then $X$ can be written as $\A_{(\Sigma,\Sigma')}$.
Let $X'$ be the admissible blow-up of $X$ along the origin. Then $$\underline{X'}=\underline{\A_{\Gamma}},$$ where $\Gamma$ denotes the star subdivision $\Sigma^*(\sigma)$ of $\Sigma$ relative to $\sigma$. Let $\Gamma'$ be the fan consisting of all cones $\sigma\in \Gamma$ satisfying $$\sigma\subset {\rm Cone}(e_1,\ldots,e_p,e_1+\cdots+e_p).$$ Note that $$\partial X=V(e_1)\cap \cdots \cap V(e_p).$$ The strict transform of $V(e_i)\subset X$ for $1\leq i\leq p$ with respect to $f$ is again $V(e_i)\subset X'$. The exceptional divisor on $X'$ is $V(e_1+\cdots+e_p)$. Using these descriptions of the divisors we deduce that $$X'=\A_{(\Gamma,\Gamma')}.$$ For $p=q=1$, we are considering $\A^2$ with log structure given by the affine line $(x=0)=H \subset \A^2$, and $X'$ is $B_{(0,0)}(\A^2)$ with log structure $\tilde{H} + E$, where $\tilde{H}$ is the strict transform of $H$. Note that $X'\to X$ *is not* a log modification. In fact, since the log structure on $\A^2$ is given by $\N\to \Spec{\Z[x,y]}$, $n\mapsto x^n$, there are no non-trivial log modification.
We will first prove a special case of Theorem \[A.3.7\] (invariance under admissible blow-ups), which in fact is the key point of the proof in the general case. Roughly speaking, it says that the cone of an admissible blow-up morphism is insensitive to the addition of a smooth component of the boundary divisor, provided that it stays normal crossing with the centre. This is carried out, however, without constructing directly any map between the two cones. A posteriori, this invariance property could be re-interpreted using the Gysin triangle.
\[A.3.39\] Assume that $\cT$ satisfies [($\boxx$-inv)]{} and [($sNis$-des)]{}. For $X\in \mathscr{S}/S$, let $H_1$ and $H_2$ be the axes of $Y:=X\times \A^2$, let $H_1'$ and $H_2'$ be the strict transforms in the blow-up $Y':=X\times B_{\{(0,0)\}}\A^2$ at $(0,0)$, and let $E$ be the exceptional divisor on $Y'$. Then the naturally induced morphism $$M(Y',E+H_2')\rightarrow M(Y,H_2)$$ is an isomorphism if and only if the naturally induced morphism $$M(Y',E+H_1'+H_2')\rightarrow M(Y,H_1+H_2)$$ is an isomorphism.
It is enough to prove the claim for $X=\Spec{\Z}$. We consider the following cones $$\begin{gathered}
\sigma_1:={\rm Cone}((1,0),(1,1)),
\;
\sigma_2:={\rm Cone}((1,1),(0,1)),
\;
\sigma_3:={\rm Cone}((0,1),(-1,0)),
\\
\sigma_4:={\rm Cone}((-1,0),(0,-1)),
\;
\sigma_5:={\rm Cone}((0,-1),(1,0)).\end{gathered}$$ Form the following fans $$\begin{gathered}
\Sigma_1:=\{\sigma_1\cup \sigma_2,\sigma_3,\sigma_4,\sigma_5\},
\;
\Sigma_2:=\{\sigma_1,\sigma_2,\sigma_3,\sigma_4,\sigma_5\},
\;
\Sigma_3:=\{\sigma_1,\sigma_2\cup \sigma_3,\sigma_4,\sigma_5\},
\\
\Sigma_4:=\{\sigma_1\cup \sigma_2\},
\;
\Sigma_5:=\{\sigma_1,\sigma_2\},
\;
\Sigma_6:=\{\sigma_3,\sigma_4,\sigma_5\},
\;
\Sigma_7:=\{\sigma_2\cap \sigma_3,\sigma_1\cap \sigma_5\},
\\
\Sigma_8:=\{\sigma_2,\sigma_3\},
\;
\Sigma_9:=\{\sigma_2\cup \sigma_3\},
\;
\Sigma_{10}:=\{\sigma_1,\sigma_4,\sigma_5\},
\;
\Sigma_{11}:=\{\sigma_1\cap \sigma_2,\sigma_3\cap \sigma_4\},
\\
\Sigma_1'=\Sigma_6':=\{\sigma_3,\sigma_4\},
\;
\Sigma_2':=\{\sigma_2,\sigma_3,\sigma_4\},
\;
\Sigma_3':=\{\sigma_2\cup \sigma_3,\sigma_4\},
\\
\Sigma_4'=\Sigma_7':=\{\sigma_2\cap \sigma_3\},
\;
\Sigma_5':=\{\sigma_2\},
\;
\Sigma_8':=\{\sigma_2,\sigma_3\},
\\
\Sigma_9':=\{\sigma_2\cup\sigma_3\},
\;
\Sigma_{10}':=\{\sigma_1\cap \sigma_2,\sigma_4\},
\;
\Sigma_{11}':=\{\sigma_1\cap \sigma_2,\sigma_3\cap \sigma_4\}.\end{gathered}$$
For the convenience of the reader we include a figure illustrating the above fans. $$\begin{tikzpicture}[yscale=1, xscale=1]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (0,0)--(1,0)--(1,1)--(0,1);
\fill[black!40] (0,0)--(0,1)--(-1,1)--(-1,0);
\fill[black!40] (0,0)--(-1,0)--(-1,-1)--(0,-1);
\fill[black!20] (0,0)--(0,-1)--(1,-1)--(1,0);
\draw (0,0)--(1,0);
\draw[ultra thick] (0,0)--(0,1);
\draw[ultra thick] (0,0)--(-1,0);
\draw[ultra thick] (0,0)--(0,-1);
\node at (0,-1.5) {$(\Sigma_1,\Sigma_1')$};
\begin{scope}[shift={(3,0)}]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (0,0)--(1,0)--(1,1);
\fill[black!40] (0,0)--(1,1)--(0,1);
\fill[black!40] (0,0)--(0,1)--(-1,1)--(-1,0);
\fill[black!40] (0,0)--(-1,0)--(-1,-1)--(0,-1);
\fill[black!20] (0,0)--(0,-1)--(1,-1)--(1,0);
\draw (0,0)--(1,0);
\draw[ultra thick] (0,0)--(0,1);
\draw[ultra thick] (0,0)--(-1,0);
\draw[ultra thick] (0,0)--(0,-1);
\draw[ultra thick] (0,0)--(1,1);
\node at (0,-1.5) {$(\Sigma_2,\Sigma_2')$};
\end{scope}
\begin{scope}[shift={(6,0)}]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (0,0)--(1,0)--(1,1);
\fill[black!40] (0,0)--(1,1)--(-1,1)--(-1,0);
\fill[black!40] (0,0)--(-1,0)--(-1,-1)--(0,-1);
\fill[black!20] (0,0)--(0,-1)--(1,-1)--(1,0);
\draw (0,0)--(1,0);
\draw[ultra thick] (0,0)--(1,1);
\draw[ultra thick] (0,0)--(-1,0);
\draw[ultra thick] (0,0)--(0,-1);
\node at (0,-1.5) {$(\Sigma_3,\Sigma_3')$};
\end{scope}
\end{tikzpicture}$$ $$\begin{tikzpicture}[yscale=1, xscale=1]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (0,0)--(1,0)--(1,1)--(0,1);
\draw (0,0)--(1,0);
\draw[ultra thick] (0,0)--(0,1);
\node at (0,-1.5) {$(\Sigma_4,\Sigma_4')$};
\begin{scope}[shift={(3,0)}]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (0,0)--(1,0)--(1,1);
\fill[black!40] (0,0)--(1,1)--(0,1);
\draw (0,0)--(1,0);
\draw[ultra thick] (0,0)--(1,1);
\draw[ultra thick] (0,0)--(0,1);
\node at (0,-1.5) {$(\Sigma_5,\Sigma_5')$};
\end{scope}
\begin{scope}[shift={(6,0)}]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (0,0)--(0,-1)--(1,-1)--(1,0);
\fill[black!40] (0,1)--(-1,1)--(-1,-1)--(0,-1);
\draw (0,0)--(1,0);
\draw[ultra thick] (0,0)--(0,1);
\draw[ultra thick] (0,0)--(-1,0);
\draw[ultra thick] (0,0)--(0,-1);
\node at (0,-1.5) {$(\Sigma_6,\Sigma_6')$};
\end{scope}
\begin{scope}[shift={(9,0)}]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\draw (0,0)--(1,0);
\draw[ultra thick] (0,0)--(0,1);
\node at (0,-1.5) {$(\Sigma_7,\Sigma_7')$};
\end{scope}
\end{tikzpicture}$$ $$\begin{tikzpicture}[yscale=1, xscale=1]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!40] (0,0)--(1,1)--(-1,1)--(-1,0);
\draw[ultra thick] (0,0)--(1,1);
\draw[ultra thick] (0,0)--(-1,0);
\draw[ultra thick] (0,0)--(0,1);
\node at (0,-1.5) {$(\Sigma_8,\Sigma_8')$};
\begin{scope}[shift={(3,0)}]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!40] (0,0)--(1,1)--(-1,1)--(-1,0);
\draw[ultra thick] (0,0)--(1,1);
\draw[ultra thick] (0,0)--(-1,0);
\node at (0,-1.5) {$(\Sigma_9,\Sigma_9')$};
\end{scope}
\begin{scope}[shift={(6,0)}]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (0,0)--(0,-1)--(1,-1)--(1,1);
\fill[black!40] (0,0)--(-1,0)--(-1,-1)--(0,-1);
\draw (0,0)--(1,0);
\draw[ultra thick] (0,0)--(1,1);
\draw[ultra thick] (0,0)--(-1,0);
\draw[ultra thick] (0,0)--(0,-1);
\node at (0,-1.5) {$(\Sigma_{10},\Sigma_{10}')$};
\end{scope}
\begin{scope}[shift={(9,0)}]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\draw[ultra thick] (0,0)--(1,1);
\draw[ultra thick] (0,0)--(-1,0);
\node at (0,-1.5) {$(\Sigma_{11},\Sigma_{11}')$};
\end{scope}
\end{tikzpicture}$$
Here we have $\A_{(\Sigma_1,\Sigma_1')} = \boxx\times (\P^1, 0+\infty)$, and $\A_{(\Sigma_2,\Sigma_2')} = B_{(0,0)}(\P^1\times \P^1)$ with divisor the strict transform of the divisor on $\boxx\times (\P^1, 0+\infty)$ together with the exceptional divisor. Finally, $\A_{(\Sigma_3,\Sigma_3')}$ is (isomorphic to) the blow-up of $\P^2$ at the point $[1:0:0]$, with the coordinate lines $X_0=0$, $X_1=0$ and $X_2=0$ as divisors. Note that there is also a blow-up morphism $B_{(0,0)}(\P^1\times \P^1) \to \A_{(\Sigma_3,\Sigma_3')}$. We consider the following Zariski open coverings of these log schemes. $$\label{eq:diag-chart1}
\begin{tikzcd}
\A_{(\Sigma_7,\Sigma_7')}\arrow[d]\arrow[r]&
\A_{(\Sigma_6,\Sigma_6')}\arrow[d]
\\
\A_{(\Sigma_4,\Sigma_4')}\arrow[r]&
\A_{(\Sigma_1,\Sigma_1')},
\end{tikzcd}
\quad
\begin{tikzcd}
\A_{(\Sigma_7,\Sigma_7')}\arrow[d]\arrow[r]&
\A_{(\Sigma_6,\Sigma_6')}\arrow[d]
\\
\A_{(\Sigma_5,\Sigma_5')}\arrow[r]&
\A_{(\Sigma_2,\Sigma_2')},
\end{tikzcd}$$ $$\label{eq:diag-chart2}
\begin{tikzcd}
\A_{(\Sigma_{11},\Sigma_{11}')}\arrow[d]\arrow[r]&
\A_{(\Sigma_{10},\Sigma_{10}')}\arrow[d]
\\
\A_{(\Sigma_8,\Sigma_8')}\arrow[r]&
\A_{(\Sigma_2,\Sigma_2')},
\end{tikzcd}
\quad
\begin{tikzcd}
\A_{(\Sigma_{11},\Sigma_{11}')}\arrow[d]\arrow[r]&
\A_{(\Sigma_{10},\Sigma_{10}')}\arrow[d]
\\
\A_{(\Sigma_9,\Sigma_9')}\arrow[r]&
\A_{(\Sigma_3,\Sigma_3')}.
\end{tikzcd}$$ Here, the second diagram in and the first diagram in are two different open charts for the blow-up of $\P^1\times \P^1$ at $(0,0)$ (and the toric notation helps keeping track of the different log structures). Note that $\A_{(\Sigma_6,\Sigma_6')}$ is a common Zariski open subset of $\P^1\times \P^1$ and of the blow-up $B_{(0,0)}(\P^1\times \P^1)$, namely it is the open subset $\P^1\times \P^1- \{(0,0)\}$ (which does not contain the centre of the blow-up). Similarly, note that $\A_{(\Sigma_{10},\Sigma_{10'})}$ is a common Zariski open subset of $\A_{(\Sigma_3,\Sigma_3')}$ and of the blow-up of $\P^1\times \P^1$ at $(0,0)$ (this time containing the center of the blow-up).
Using ($sNis$-des) we have a zig-zag of isomorphisms $$\begin{gathered}
M(\A_{(\Sigma_5,\Sigma_5')}\rightarrow \A_{(\Sigma_2,\Sigma_2')})
\xrightarrow{\cong}
M(\A_{(\Sigma_4,\Sigma_4')}\rightarrow \A_{(\Sigma_1,\Sigma_1')}),
\\
M(\A_{(\Sigma_8,\Sigma_8')}\rightarrow \A_{(\Sigma_2,\Sigma_2')})
\xrightarrow{\cong}
M(\A_{(\Sigma_9,\Sigma_9')}\rightarrow \A_{(\Sigma_3,\Sigma_3')}).\end{gathered}$$
The squares in can be completed to a $3$-dimensional commutative diagram with back side $$\begin{tikzcd}
\A_{(\Sigma_5,\Sigma_5')}\arrow[d]\arrow[r]&
\A_{(\Sigma_2,\Sigma_2')}\arrow[d]
\\
\A_{(\Sigma_4,\Sigma_4')}\arrow[r]&
\A_{(\Sigma_1,\Sigma_1')},
\end{tikzcd}$$ where the morphisms $\A_{(\Sigma_2,\Sigma_2')}\to \A_{(\Sigma_1,\Sigma_1')}$ and $\A_{(\Sigma_5,\Sigma_5')}\to \A_{(\Sigma_4,\Sigma_4')}$ are induced by the blow-up morphisms $B_{(0,0)}(\P^1\times\P^1)\to \P^1\times \P^1$ and $B_{(0,0)}(\A^2)\to \A^2$, respectively. Taking into account the log structures, the morphism $(Y',E+H_2')\rightarrow (Y,H_2)$ is equal to the morphism $\A_{(\Sigma_5,\Sigma_5')}\rightarrow \A_{(\Sigma_4,\Sigma_4')}$. Similarly, we can complete the squares in to a 3-dimensional commutative diagram with back side $$\begin{tikzcd}
\A_{(\Sigma_8,\Sigma_8')}\arrow[d]\arrow[r]&
\A_{(\Sigma_2,\Sigma_2')}\arrow[d]
\\
\A_{(\Sigma_9,\Sigma_9')}\arrow[r]&
\A_{(\Sigma_3,\Sigma_3')},
\end{tikzcd}$$ where the vertical morphisms are again induced by the blow-ups. More precisely, via the linear transformation $$\Z^2\rightarrow \Z^2$$ mapping $(a,b)$ to $(a-b,a)$, the morphism $(Y',E+H_1'+H_2')\rightarrow (Y,H_1+H_2)$ can be identified with $\A_{(\Sigma_8,\Sigma_8')}\rightarrow \A_{(\Sigma_9,\Sigma_9')}$ (note that $\A_{\Sigma_9} = \Spec{\Z[x^{-1}, yx^{-1}]}$). Due to Proposition \[A.3.44\] there are induced isomorphisms $$\begin{gathered}
\label{A.3.39.7}
M(\A_{(\Sigma_5,\Sigma_5')}\rightarrow \A_{(\Sigma_4,\Sigma_4')})
\xrightarrow{\cong}
M(\A_{(\Sigma_2,\Sigma_2')}\rightarrow \A_{(\Sigma_1,\Sigma_1')}),
\\
\label{A.3.39.8}
M(\A_{(\Sigma_8,\Sigma_8')}\rightarrow \A_{(\Sigma_9,\Sigma_9')})
\xrightarrow{\cong}
M(\A_{(\Sigma_2,\Sigma_2')}\rightarrow \A_{(\Sigma_3,\Sigma_3')}).\end{gathered}$$ Thus from and we reduce to showing the equivalence of $$M(\A_{(\Sigma_2,\Sigma_2')}\rightarrow \A_{(\Sigma_1,\Sigma_1')})=0$$ and $$M(\A_{(\Sigma_2,\Sigma_2')}\rightarrow \A_{(\Sigma_3,\Sigma_3')})=0.$$
Both $\A_{(\Sigma_1,\Sigma_1')}$ and $\A_{(\Sigma_3,\Sigma_3')}$ are $\boxx$-bundles over $\boxx$, so that Proposition \[A.3.40\] gives isomorphisms $$M(\A_{(\Sigma_1,\Sigma_1')})\rightarrow M(\Spec \Z),
\;
M(\A_{(\Sigma_3,\Sigma_3')})\rightarrow M(\Spec \Z).$$ This finishes the proof.
Recall now some terminology from [@Deg].
A [*closed pair*]{} $(X,Z)$ [*smooth*]{} over $S$ is a closed immersion $Z\rightarrow X$ where both $X$ and $Z$ are object in $Sm/S$. A [*cartesian*]{} (resp. [*excisive*]{}) morphism $(f,f')$ of closed pairs $(X,Z)\rightarrow (X',Z')$ smooth over $S$ is a commutative diagram of $S$-schemes $$\begin{tikzcd}
Z\arrow[r]\arrow[d,"f'"']&X\arrow[d,"f"]\\
Z'\arrow[r]&X'
\end{tikzcd}$$ which is cartesian (resp. such that $f':Z\rightarrow Z'$ is an isomorphism).
Let $(X,Z)$ be a closed pair smooth over $S$, and consider the smooth closed pair $(\A_S^{r+s},\A_S^r)$ given by $$a_0\times {\rm id}:\A_S^s\rightarrow \A_S^{r+s},$$ where $a_0:S\rightarrow \A_S^r$ is the $0$-section. A [*parametrization*]{} of $(X,Z)$ is a cartesian morphism $(f,f'):(X,Z)\rightarrow (\A_S^{r+s},\A_S^s)$ such that $f$ is étale.
\[A.3.16\] We will use the following technique appearing in the proof of homotopy purity in [@MV Lemma 2.28, p. 117] — see also [@Deg §4.5.2] for another exposition, and see [@MR3431674 §7] for formulations in related settings. Let $(X,Z)$ be a closed pair smooth over $S$. Then by [@EGA IV.17.12.2], Zariski locally on $X$, there are étale morphisms $w$ and $w'$ making the diagram $$\begin{tikzcd}
Z\arrow[r,"w'"]\arrow[d]&\A_S^r\arrow[d]
\\
X\arrow[r,"w"]&\A_S^{r+s}.
\end{tikzcd}$$ cartesian. Setting $X_1:=\A_Z^s$ the morphism ${\rm id}\times w':X_1\rightarrow \A_S^{r+s}$ is étale. Let $\Delta$ be the diagonal of $Z$ over $\A_S^{r+s}$. This is a closed subscheme of $Z\times_{\A_S^{r+s}}Z$, and we define $$X_2
:=
X\times_{\A_S^{r+s}}X_1-(X\times_{\A_S^{r+s}}Z-\Delta)\cup (Z\times_{\A_S^{r+s}}X_1-\Delta).$$ By [@Deg Lemma 4.5.6], $\Delta$ is a closed subscheme of $X_2$, and $X_2$ is an open subscheme of $X\times_{\A_S^{r+s}}X_1$. Let $i_2:Z\cong \Delta\rightarrow X_2$ be the closed immersion. There are cartesian excisive morphisms of closed pairs smooth over $S$ $$\label{eq:excisivesquares}
(X,Z)\stackrel{(u,{\rm id})}\longleftarrow (X_2,Z)\stackrel{(v,{\rm id})}\longrightarrow (X_1,Z) = (\A^s_Z, Z),$$ where $u$ (resp. $v$) denotes the étale morphism induced by the projection $$X\times_{\A_S^{r+s}}X_1\rightarrow X
(\text{resp.}\ X\times_{\A_S^{r+s}}X_1\rightarrow X_1).$$ By working Zariski locally on $X$, the above morphisms of closed pairs smooth over $S$ allow us to reduce claims about $X$ and $Z$ to claims about $X_1$ and $Z$.
If $Z=S$, then $r=0$ and $X_1\rightarrow \A_S^{r+s} = \A^s_S$ is an isomorphism. Thus $X_2\cong X$, and the morphism $v:X_2\rightarrow X_1$ is the parametrization map $X\rightarrow \A_S^s$.
\[A.3.7\] Assume that $\cT$ satisfies [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and [($div$-des)]{}. Let $Y'\rightarrow Y$ be a morphism in $\cA\cB l_{Sm}/S$. Then $$\label{A.3.7.1}
M(Y')\rightarrow M(Y)$$ is an isomorphism.
We may assume that $Y'$ is an admissible blow-up of $Y$ along a connected codimension $d$ smooth center $Z$ for some integer $d\geq 2$. We set $X:=\underline{Y}$ and $X':=\underline{Y'}$. Write $Z'$ for the exceptional divisor, $Z':=Z\times_X X'$. Let $Z_1,\ldots,Z_r$ be divisors on $X$ smooth over $S$ such that $\partial Y=Z_1+\cdots+Z_r$. We write $Z_i'$ for the strict transform of $Z_i$ in the blow-up $X'$, and set $$Y:=(X,Z_1+\cdots+Z_r),
\;
Y':=(X',Z_1'+\cdots+Z_r'+Z').$$ Let $W_i:=\A^{i-1}\times \Spec \Z\times \A^{n-i}$ denote the smooth divisor of $\A^n$ given by $y_i=0$ for $1\leq i\leq n$, where $(y_1,\ldots, y_n)$ are coordinates on $\A^n$. Write $C_1, \ldots, C_l$ for the irreducible components of the intersections $Z\cap (Z_{i_1}\ldots \cap Z_{i_k})$, where $(i_1, \ldots, i_k)$ is a subset of $(1,\ldots, r)$ of length $k$, for $k=1,\ldots, r$. Finally we set $$\label{eq:definition-s}
X(s):= {\rm max}\{c_1, \ldots, c_l\},$$ where $c_i$ is the codimension of $C_i$ in $X$.
Next, observe the question is Zariski local on $X$, owing to ($Zar$-sep), so that we may assume there exists an étale morphism $u:X\rightarrow S\times \A^n$ in $Sm/S$ for which
1. $u^{-1}(S\times W_i)=Z_i$ for $1\leq i\leq r$ ,
2. $u^{-1}(S\times W)=Z$, where $$W:=\A^{r-p}\times (\Spec \Z)^p \times (\Spec \Z)^q\times \A^{n-r-q}$$ for some $1\leq p\leq r$ and $0\leq q\leq n-r$.
Note that $W$ is a smooth closed subscheme of codimension $p+q$ in $\A^n$ having strict normal crossing with $W_1+\ldots +W_i$ for every $i=1,\ldots, n$.
Assume that $S=\Spec{\Z}$ for simplicity of notation (the general case is deduced from this one by base change $S\to \Spec{\Z}$). For $r+1\leq i\leq r+q$ we set $$Z_i:=u^{-1}(W_i), \quad
T:=u^{-1}((\Spec \Z)^{r+q}\times \A^{n-r-q}).$$ Note that $T\subseteq Z$ and that $d=p+q$ is the codimension of $Z$ in $X$. Moreover, Zariski-locally, the codimension of $T$ in $X$ agrees with $X(s)$ defined in . We also observe that is trivially an isomorphism if $\dim X=q=r=0$. Similarly, if $X(s)=1$, then $Z=T$ is a divisor in $X$ and coincides with one of the components of $\partial X$. Thus the blow-up is trivially an isomorphism. We can now proceed by induction on $X(s)>1$.
**Step 1** [*Reduction to the case when $X=\A^{p+q}$ and $r=p$.*]{} Starting with a smooth pair $(X,T)$ and $(X,T)\rightarrow (\A^n,\A^{n-r-q})$, we may use the technique of Construction \[A.3.16\] with respect to the parametrization $u\colon (X,T)\to (\A^n, \A^{n-r-q})$ to form the schemes $X_1$ and $X_2$. In this case $X_1 = \A^{r+q}_T$, $X_2$ is open in $X\times_{\A^n} X_1$, and there exist étale morphisms $X_2\to X$, $X_2\to X_1$.
Let $X'_2$ be the pullback $X'\times_X X_2$. We equip $X_2$ (resp. $X_2'$) with the log structure induced by $Y$ (resp. $Y'$), setting $$Y_2:=Y\times_X X_2,\;\;Y_2':=Y'\times_X X_2.$$ Next we consider the open complement of $T$ in $X$ $$U:=(X-T,(Z_1-T)+\cdots+(Z_r-T)),$$ and the pullbacks $$U':=U\times_Y Y',\;\; U_2:=U\times_X X_2,\;\;U_2':=U'\times_X X_2.$$
We obtain strict Nisnevich distinguished squares in $\mathscr{S}/S$ $$\begin{tikzcd}
U_2\arrow[d]\arrow[r]&Y_2\arrow[d]\\
U\arrow[r]&Y
\end{tikzcd}\quad
\begin{tikzcd}
U_2'\arrow[d]\arrow[r]&Y_2'\arrow[d]\\
U'\arrow[r]&Y'.
\end{tikzcd}$$ By ($sNis$-des) there are isomorphisms $$\label{A.3.7.2}
M(U_2\rightarrow U)\xrightarrow{\cong} M(Y_2\rightarrow Y),
\;
M(U_2'\rightarrow U')\xrightarrow{\cong} M(Y_2'\rightarrow Y').$$ Observe that by removing $T$, we obtain strict inequalities $$U(s)<X(s), \quad U_2(s)< X(s).$$ Thus, by induction, there are isomorphisms $$M(U_2'\rightarrow U_2)\cong M(U'\rightarrow U)\cong 0,$$ which by Proposition \[A.3.44\] imply the isomorphism $$\label{A.3.7.3}
M(U_2'\rightarrow U')\xrightarrow{\cong} M(U_2\rightarrow U).$$ Combining and , we reduce to showing $$M(Y_2'\rightarrow Y_2)\cong 0.$$ Here we can replace $X$ by $X_2$. Similarly, we can repeat the argument using the right excisive square in and replace $X_2$ by $X_1$. We note that $X_1=\A^{r+q}\times T$, and that the subschemes $Z_i$ and $Z$ of $X$ correspond to the subschemes $$\A^{i-1}\times \Spec{\Z}\times \A^{r+q-i}\times T
\text{ and }
\A^{r-p}\times (\Spec{\Z})^p\times (\Spec{\Z})^q \times T,$$ respectively. Thus we have reduced to the case $$\label{eq:step1equation}
X=\A^{r+q}\times T,
\;\;
Z_i=\A^{i-1}\times \Spec \Z \times \A^{r+q-i}\times T,$$ $$Z=\A^{r-p}\times (\Spec \Z)^p\times (\Spec \Z)^q\times T.$$
Note that under , we have further an identification $$Y\cong \A_\N^{r-p}\times \A_\N^p\times \A^q\times T.$$ Using the monoidal structure in $\cT$, we may further assume that $T=\Spec{\Z}$ and $r=p$, i.e., $$X=\A^{p+q},\;\;Z_i=\A^{i-1}\times \Spec \Z \times \A^{p+q-i},\;\;Z=\Spec{\Z},$$ and $$Y\cong \A_\N^p\times \A^q.$$
**Step 2** [*Cases when $d=2$.*]{} If $p=2$ and $q=0$, then the morphism $$(X',Z_1'+\cdots+Z_r'+Z')\rightarrow (X,Z_1+\cdots+Z_r)$$ in $\cT$ is a log modification since this is $\A_M\rightarrow \A_\N^p$, where $M$ is the star subdivision of $\N^p$ relative to itself as in Definition \[Starsubdivision\]. Thus we are done by ($div$-des). The case when $p=q=1$ is already done in Proposition \[A.3.39\]. Together with Step 1, we see that $$M(Y')\rightarrow M(Y)$$ is an isomorphism for general $Y'\rightarrow Y$ if $d=2$.
**Step 3** [*General case.*]{} Recall that we may assume $$X=\A^{p+q},\;\;Z_i=\A^{i-1}\times \Spec \Z\times \A^{p+q-i},\;\;Z=(\Spec \Z)^p\times (\Spec \Z)^q.$$ We will construct $X'$ from $X$ by suitable blow-ups and blow-downs along codimension $2$ smooth centers.
Let $\{e_1,\ldots,e_{p+q}\}$ denote the standard basis of $\Z^{p+q}$. Starting with $\Sigma_0:=\N^{p+q}$, for $i=1,\ldots,p+q-1$ we inductively let $\Sigma_i$ be the star subdivision $\Sigma_{i-1}^*(\eta_{i+1})$, where $$\eta_i
:=
{\rm Cone}( e_1,\ldots,e_i).$$ Let $\Sigma_i'$ be the fan consists of cones $\sigma\in \Sigma_i$ such that $$\sigma\subset {\rm Cone}(e_1,e_2,\ldots,e_p,e_1,e_1+e_2,\ldots,e_1+\cdots+e_i).$$ Let $\Gamma_0'$ be shorthand for $\Sigma_0^*(\eta_{p+q})$. For $i=1,\ldots,p+q-2$, we inductively define $\Gamma_i'$ to be the star subdivision $\Gamma_i'^*(\eta_{i+1})$. We let $\Gamma_i'$ be the fan consisting of cones $\sigma\in \Gamma_i'$ such that $$\sigma\subset {\rm Cone}(e_1,e_2,\ldots,e_p,e_1,e_1+e_2,\ldots,e_1+\cdots+e_i,e_1+\cdots+e_{p+q}).$$ Note that $(\Sigma_{p+q-1},\Sigma_{p+q-1}')=(\Gamma_{p+q-2},\Gamma_{p+q-2}')$. We have naturally induced maps $$(\Sigma_{p+q-1},\Sigma_{p+q-1}')\rightarrow \cdots \rightarrow (\Sigma_{0},\Sigma_{0}'),
\;
(\Gamma_{p+q-2},\Gamma_{p+q-2}')\rightarrow \cdots \rightarrow (\Gamma_{0},\Gamma_{0}').$$
The following figure illustrates the fans in the case when $p=2$ and $q=1$. $$\begin{tikzpicture}[yscale=0.7, xscale=0.82]
\fill[black!20] (0,0)--(1,2)--(2,0)--(0,0);
\draw (0,0)--(1,2)--(2,0);
\draw[ultra thick] (0,0)--(2,0);
\fill[black!20] (4,0)--(5,2)--(6,0)--(4,0);
\filldraw (0,0) circle (3pt);
\filldraw (2,0) circle (3pt);
\draw (4,0)--(5,2)--(6,0);
\draw (5,2)--(5,0);
\draw[ultra thick] (4,0)--(6,0);
\fill[black!20] (8,0)--(9,2)--(10,0)--(8,0);
\fill[black!40] (8,0)--(10,0)--(9,1);
\filldraw (4,0) circle (3pt);
\filldraw (5,0) circle (3pt);
\filldraw (6,0) circle (3pt);
\draw (8,0)--(9,2)--(10,0);
\draw[ultra thick] (8,0)--(10,0);
\draw (9,2)--(9,1);
\draw[ultra thick] (9,1)--(9,0);
\draw[ultra thick] (8,0)--(9,1)--(10,0);
\filldraw (8,0) circle (3pt);
\filldraw (9,0) circle (3pt);
\filldraw (10,0) circle (3pt);
\filldraw (9,1) circle (3pt);
\fill[black!20] (12,0)--(13,2)--(14,0)--(12,0);
\fill[black!40] (12,0)--(14,0)--(13,1);
\draw[ultra thick] (12,0)--(14,0);
\draw (12,0)--(13,2)--(14,0)--(12,0);
\draw[ultra thick] (12,0)--(13,1)--(14,0);
\draw (13,1)--(13,2);
\filldraw (12,0) circle (3pt);
\filldraw (13,1) circle (3pt);
\filldraw (14,0) circle (3pt);
\node [below] at (1,-0.2) {$(\Sigma_0,\Sigma_0')$};
\node [below] at (5,-0.2) {$(\Sigma_1,\Sigma_1')$};
\node [below] at (9,-0.2) {$(\Sigma_2,\Sigma_2')=(\Gamma_1,\Gamma_1')$};
\node [below] at (13,-0.2) {$(\Gamma_0,\Gamma_0')$};
\node [left] at (0,0) {$e_1$};
\node [right] at (2,0) {$e_2$};
\node [above] at (1,2) {$e_3$};
\node at (3,1) {$\longleftarrow$};
\node at (7,1) {$\longleftarrow$};
\node at (11,1) {$\longrightarrow$};
\end{tikzpicture}$$
From the description of admissible blow-ups in Example \[A.3.41\], we see that all the induced morphisms $$\A_{(\Sigma_{p+q-1},\Sigma_{p+q-1}')}\rightarrow \cdots \rightarrow \A_{(\Sigma_{0},\Sigma_{0}')},
\;
\A_{(\Gamma_{p+q-2},\Gamma_{p+q-2}')}\rightarrow \cdots \rightarrow \A_{(\Gamma_{0},\Gamma_{0}')}$$ are admissible blow-ups along a codimension 2 smooth center. Step 2 shows that the induced morphisms $$\begin{gathered}
M(\A_{(\Sigma_{p+q-1},\Sigma_{p+q-1}')})\rightarrow \cdots \rightarrow M(\A_{(\Sigma_{0},\Sigma_{0}')}),
\\
M(\A_{(\Gamma_{p+q-2},\Gamma_{p+q-2}')})\rightarrow \cdots \rightarrow M(\A_{(\Gamma_{0},\Gamma_{0}')})\end{gathered}$$ are isomorphisms. This implies there is a naturally induced isomorphism $$M(\A_{(\Gamma_{0},\Gamma_{0}')})
\xrightarrow{\cong}
M(\A_{(\Sigma_{0},\Sigma_{0}')}),$$ which can be identified with $$M(Y')
\xrightarrow{\cong}
M(Y).$$
Let $\cA\cB l_{div}/S$ denote the smallest class of proper birational morphisms in $lSm/S$ that is closed under composition and contains $\cA\cB l_{Sm}/S$ together with all the dividing coverings.
\[A.3.11\] Assume that $\cT$ satisfies [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and [($div$-des)]{}. Let $f:Y\rightarrow X$ be a morphism in $\cA\cB l_{div}/S$. Then there is a naturally induced isomorphism $$M(Y)\xrightarrow{\cong} M(X).$$
If $f$ is a dividing cover, then apply ($div$-des). If $f$ is in $\cA\cB l_{Sm}/S$, then apply Theorem \[A.3.7\]. Since any morphism in $\cA\cB l_{div}/S$ is a composition of dividing covers and morphisms in $\cA\cB l_{Sm}/S$, we are done.
For $X\in SmlSm/S$ and $Y\rightarrow X\in \cA\cB l_{div}/S$, there exists a log modification $Z\rightarrow Y$ such that the composite morphism $Z\rightarrow X$ is in $\cA \cB l_{Sm}/S$.
Without loss of generality we may assume $f$ is a dividing cover. Proposition \[A.9.75\] shows that $f$ is a log modification. Owing to Theorem \[Fan.16\] the morphism $f$ is dominated by a sequence of log modifications $X_n\to \ldots \to X_1\to X$ along a smooth center, see Definition \[Fan.39\]. This suffices to conclude the proof.
Blow-up triangles and $(\mathbb{P}^n,\mathbb{P}^{n-1})$-invariance
------------------------------------------------------------------
Throughout this section we shall assume that the triangulated category $\cT$ satisfies the properties [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and [($div$-des)]{}. Our aim is to conclude that ($(\P^\bullet,\P^{\bullet-1})$-inv) holds. As in the formulation of ($(\P^\bullet,\P^{\bullet-1})$-inv), recall that we write $$[x_0:\cdots:x_n]$$ for the coordinates on $\P^n$ and view $\P^{n-1}=H_n$ as the hyperplane defined by $x_n=0$. The next result is a direct consequence of Theorem \[A.3.7\].
\[A.6.1\] For every $X\in\mathscr{S}/S$ the projection $X\times (\P^n,\P^{n-1})\rightarrow X$ induces an isomorphism $$M(X\times (\P^n,\P^{n-1}))\xrightarrow{\cong} M(X).$$
Owing to the isomorphism $$M(X\times (\P^n,\P^{n-1}))\cong M(X)\otimes M(\P^n\times S,\P^{n-1}\times S),$$ it suffices to consider the case $X=S$. We proceed by induction on $n$. The case $n=0$ is trivial, and the case $n=1$ is precisely the [($\boxx$-inv)]{} property, so we may assume $n>1$. For the standard coordinates $e_1,\ldots,e_n$ on $\Z^n$, recall that $\P^n$ is the toric variety associated to a fan $\Sigma_1$ with the maximal cones $$\Cone(e_1,\ldots,e_n),$$ $$\Cone(e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_n,-e_1-\cdots-e_n)\quad (1\leq i\leq n-1),$$ $$\tau:=\Cone(e_1,\ldots,e_{n-1},-e_1-\cdots-e_n).$$
We shall compare $\P^n$ with $\P^{n-1}\times \P^1$. Both are dominated by a common blow-up, which we describe next. Consider the star subdivision $$\Sigma_3:=(\Sigma_1^*(\tau'))^*(\tau),$$ where $$\tau':=\Cone(e_n,-e_1-\cdots-e_n).$$ The maximal cones of $\Sigma_3$ are given by $$\sigma_{1}:=\Cone(e_1,\ldots,e_n),$$ $$\sigma_{i2}:=\Cone(e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_n,-e_1-\cdots-e_{n-1})\quad (1\leq i\leq n-1),$$ $$\sigma_{i3}:=\Cone(e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_{n-1},-e_1-\cdots-e_n,-e_1-\cdots-e_{n-1})\quad (1\leq i\leq n-1),$$ $$\sigma_{i4}:=\Cone(e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_n,-e_1-\cdots-e_n,-e_n)\quad (1\leq i\leq n-1),$$ $$\sigma_{5}:=\Cone(e_1,\ldots,e_{n-1},-e_n).$$
Moreover, $\P^{n-1}\times \P^1$ is the toric variety corresponding to a fan $\Sigma_2$ with the maximal cones $$\sigma_{1},\;\sigma_{5},$$ $$\sigma_{i2}\quad (1\leq i\leq n-1),$$ $$\sigma_{i6}:=\Cone(e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_{n-1},-e_n,-e_1-\cdots-e_{n-1})\quad (1\leq i\leq n-1).$$ Since the maximal cones of the star subdivision $(\sigma_{i6})^*(\tau'')$ are $\sigma_{i3}$ and $\sigma_{i4}$, where $\tau'':=\Cone(-e_n,-e_1-\cdots-e_{n-1})$, we observe that $\Sigma_3=\Sigma_2^*(\tau'')$.
Form the following fans $$\begin{gathered}
\Sigma_1':=\{{\rm Cone}(-e_1-\cdots-e_n)\},
\;
\Sigma_2':=\{\tau''\},
\\
\Sigma_3':=\{{\rm Cone}(-e_1-\cdots-e_n,-e_1-\cdots-e_{n-1}),{\rm Cone}(-e_1-\cdots-e_n,-e_n)\}.\end{gathered}$$
When $n=2$, the following figure describes the above fans. $$\begin{tikzpicture}[yscale=1, xscale=1]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (1,1)--(1,-1)--(-1,-1)--(-1,1);
\draw (0,0)--(1,0);
\draw (0,0)--(0,1);
\draw[ultra thick] (0,0)--(-1,-1);
\node at (0,-1.5) {$(\Sigma_1,\Sigma_1')$};
\node at (2,0) {$\leftarrow$};
\begin{scope}[shift={(4,0)}]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (1,1)--(1,-1)--(-1,-1)--(-1,1);
\fill[black!40] (0,0)--(0,-1)--(-1,-1)--(-1,0);
\draw (0,0)--(1,0);
\draw (0,0)--(0,1);
\draw[ultra thick] (0,0)--(-1,-1);
\draw[ultra thick] (0,0)--(0,-1);
\draw[ultra thick] (0,0)--(-1,0);
\node at (2,0) {$\rightarrow$};
\node at (0,-1.5) {$(\Sigma_3,\Sigma_3')$};
\end{scope}
\begin{scope}[shift={(8,0)}]
\foreach \a in {-1,0,1}
\foreach \b in {-1,0,1}
\filldraw (\a,\b) circle (1pt);
\fill[black!20] (1,1)--(1,-1)--(-1,-1)--(-1,1);
\fill[black!40] (0,0)--(0,-1)--(-1,-1)--(-1,0);
\draw (0,0)--(1,0);
\draw (0,0)--(0,1);
\draw[ultra thick] (0,0)--(0,-1);
\draw[ultra thick] (0,0)--(-1,0);
\node at (0,-1.5) {$(\Sigma_2,\Sigma_2')$};
\end{scope}
\end{tikzpicture}$$ The scheme associated to the left hand drawing is $(\P^2, \P^1)$, where $\P^1$ is the line at infinity $x_2=0$. The middle drawing is the blow-up of $\P^1\times \P^1$ at $(\infty, \infty)$ with log structure given by the exceptional divisor, indicated by the diagonal line, together with the strict transforms of $\infty\times \P^1 + \P^1\times \infty$ (cf. the proof of Proposition \[A.3.39\]). Finally, the right hand drawing is $\boxx\times \boxx$. Note that the blow-up dominates both $\boxx\times \boxx$ and $(\P^2, \P^1)$. From the description of admissible blow-ups along smooth centers discussed in Example \[A.3.41\], we observe that the morphisms $$\A_{(\Sigma_1,\Sigma_1')}\leftarrow \A_{(\Sigma_3,\Sigma_3')}\rightarrow \A_{(\Sigma_2,\Sigma_2')}$$ are admissible blow-ups along smooth centers. More precisely, the morphism $$\A_{(\Sigma_3,\Sigma_3')} \to \A_{(\Sigma_2,\Sigma_2')}$$ is a dividing cover, while the morphism $$\A_{(\Sigma_3,\Sigma_3')} \to \A_{(\Sigma_1,\Sigma_1')}$$ is an admissible blow-up, albeit not a dividing cover.
Theorem \[A.3.7\] shows there are naturally induced isomorphisms $$M( \A_{(\Sigma_1,\Sigma_1')}\times S)\xleftarrow{\cong} M(\A_{(\Sigma_3,\Sigma_3')}\times S)\xrightarrow{\cong} M(\A_{(\Sigma_2,\Sigma_2')}\times S).$$ Since we have $$\A_{(\Sigma_1,\Sigma_1')}\cong (\P^n,\P^{n-1}) \text{ and }
\A_{(\Sigma_2,\Sigma_2')}\cong (\P^{n-1},\P^{n-2})\times \boxx,$$ there is a naturally induced isomorphism $$M((\P^n,\P^{n-1})\times S)\cong M((\P^{n-1},\P^{n-2})\times S)\otimes M(\boxx\times S)$$ Using induction the above implies there is an isomorphism $$M(\P^n\times S,\P^{n-1}\times S)\cong M(\boxx\times S)^{\otimes n}.$$ This finishes the proof by appealing to the property ($\boxx$-inv).
\[A.6.3\] Suppose $Y$ is a $(\P^n,\P^{n-1})$-bundle over $X\in\mathscr{S}/S$. Then the projection $Y\rightarrow X$ induces an isomorphism $$M(Y)\xrightarrow{\cong} M(X).$$
As in the proof of Proposition \[A.3.40\] by using induction on the number of open subsets in a trivialization of $Y$, we may assume $Y\cong X\times (\P^n,\P^{n-1})$. We conclude by reference to Proposition \[A.6.1\].
In the next theorem, we compare the relative motive $M(Z\to X)$ for a smooth closed pair $(X,Z)$ with the relative motive of the blow-up of $X$ along $Z$. The result would be automatic in presence of the localization property, which does not hold in our log setting (cf. the example in Section \[sec:Gysin\]).
\[A.3.13\] Let $(X,Z)$ be a closed pair smooth over $S$, and let $X'$ be the blow-up of $X$ along a smooth center $Z$. For $Z':=Z\times_X X'$ there is a naturally induced isomorphsm $$M(Z'\rightarrow X')\xrightarrow{\cong} M(Z\rightarrow X).$$
Note that both $X$ and $Z$ have trivial log structure. By Proposition \[A.3.44\], it suffices to show there is a naturally induced isomorphism $$M(Z'\rightarrow Z)\xrightarrow{\cong} M(X'\rightarrow X).$$ This question is Zariski local on $X$ by ($Zar$-sep), so that we may assume there exists an étale morphism $u:X\rightarrow \A^{r+s}\times S$ of schemes for which $u^{-1}(W\times S)=Z$, where $W$ denotes the subscheme $(\Spec Z)^r\times \A^s$ of $\A^{r+s}$. Using $u$ and $Z\hookrightarrow X$, we obtain schemes $X_1$ and $X_2$ over $k$ as in Construction \[A.3.16\], together with the corresponding excisive squares.
In $\mathscr{S}/S$ there are strict Nisnevich distinguished squares $$\begin{tikzcd}
X_2-Z\arrow[d]\arrow[r]&X_2\arrow[d]\\
X-Z\arrow[r]&X
\end{tikzcd}\quad
\begin{tikzcd}
X_2'-Z'\arrow[d]\arrow[r]&X_2'\arrow[d]\\
X'-Z'\arrow[r]&X'
\end{tikzcd}$$ where $X_2':=X_2\times_X X'$ (recall that $Z\cong \Delta \hookrightarrow X_2$). Property ($sNis$-des) implies there are isomorphisms $$M((X_2-Z)\rightarrow (X-Z))\xrightarrow{\cong} M(X_2\rightarrow X),$$ and $$M((X_2'-Z')\rightarrow (X'-Z'))\xrightarrow{\cong} M(X_2'\rightarrow X').$$
On account of the identifications $X_2-Z\cong X_2'-Z'$ and $X-Z\cong X'-Z'$, there is an isomorphism $$M(X_2'\rightarrow X')\xrightarrow{\cong} M(X_2\rightarrow X).$$ By Proposition \[A.3.44\] we find the isomorphism $$M(X_2'\rightarrow X_2)\xrightarrow{\cong} M(X'\rightarrow X).$$
Similarly, letting $X_1'$ denote the blow-up of $X_1$ along $Z$, there is a naturally induced isomorphism $$M(X_2'\rightarrow X_2)\xrightarrow{\cong} M(X_1'\rightarrow X_1).$$
By the above we may replace $X$ by $X_1$ and assume $X=\A^r\times Z$. Using the monoidal structure on $\cT$, we can further reduce to the case $X=\A^r$ and $Z=\{0\}$. Then $X'$ is $B_{0}(\A^r)$ and $Z'=E$ is the exceptional divisor. Let $Y$ be shorthand for $(\P^n\times S,H_n\times S)$, where $H_n=\P^{n-1}$ is the hyperplane at infinity. Let $Y'$ be the object of $\mathscr{S}/S$ with log structure induced by $Y$ and whose underlying scheme is the blow-up of $\P^n$ at the origin.
Using the open immersion $X\hookrightarrow Y$, we obtain strict Nisnevich distinguished squares in $\mathscr{S}/S$ $$\label{eq:blowup-reduction-nolog}
\begin{tikzcd}
X-Z\arrow[r]\arrow[d]&X\arrow[d]\\
Y-Z\arrow[r]&Y
\end{tikzcd}
\quad
\begin{tikzcd}
X'-Z'\arrow[r]\arrow[d]&X'\arrow[d]\\
Y'-Z'\arrow[r]&Y'.
\end{tikzcd}$$ By ($sNis$-des), there are isomorphisms $$\begin{aligned}
M((X-Z)\rightarrow (Y-Z))&\xrightarrow{\cong} M(X\rightarrow Y),\\
M((X'-Z')\rightarrow (Y'-Z'))&\xrightarrow{\cong} M(X'\rightarrow Y').\end{aligned}$$ Since $X'-Z'\cong X-Z$ and $Y'-Z'\cong Y-Z$, we obtain an isomorphism $$M(X'\rightarrow Y')\xrightarrow{\cong} M(X\rightarrow Y).$$ Using Proposition \[A.3.44\], and the relevant cube obtained by combining the two squares in , we obtain the isomorphism $$M(X'\rightarrow X)\xrightarrow{\cong} M(Y'\rightarrow Y).$$ Hence it suffices to show there is a naturally induced isomorphism $$M(Z'\rightarrow Z)\xrightarrow{\cong} M(Y'\rightarrow Y).$$
Due to Proposition \[A.6.1\] there is an isomorphism $$M(Z)\xrightarrow{\cong} M(Y),$$ given by the inverse of the projection $(\P^n, \P^{n-1})\to S$. Hence we are reduced to consider $M(Z')\rightarrow M(X')$. Lemma \[A.6.5\] shows that $X'$ is a $\boxx$-bundle over $\P_S^{n-1}$. Thus by Proposition \[A.6.3\], the projection $X'\rightarrow \P_S^{n-1}$ induces an isomorphism $$M(X')\xrightarrow{\cong} M(\P_S^{n-1}).$$ This concludes the proof since the composition $Z'\rightarrow X'\rightarrow \P_S^{n-1}$ is an isomorphism of schemes.
Thom motives {#ssec:ThomMotives}
------------
Throughout this section we shall assume that $\cT$ satisfies the properties [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and [($div$-des)]{}. Let $E$ be a vector bundle over a scheme $X\in Sm/k$ with the $0$-section $Z$. The Thom motive associated with $E$ in $\dmeff$ is defined as $$M((E-Z)\rightarrow E).$$ In the log setting, taking the open complement $E-Z$ of the $0$-section in $E$ does not give the “correct” homotopy type: we can see this by taking the Betti realization of the log scheme, as discussed in [@Ogu V.1]. See also Proposition \[prop::unitdiskbundle\] below. In the following, we will discuss an appropriate definition, and then prove some properties of our Thom motives.
\[A.3.15\] Suppose $Z_1,\ldots,Z_r$ form a strict normal crossing divisor on $X\in Sm/S$, and that a smooth closed subscheme $Z$ has strict normal crossing with $Z_1+\cdots+Z_r$ over $S$. For $Y:=(X,Z_1+\cdots+Z_r)$ the blow-up of $Y$ along $Z$ is defined as $$B_ZY:=(B_Z X,W_1+\cdots+W_r),$$ where $B_Z X$ is the blow-up of $X$ along $Z$ and $W_i$ is the strict transform of $Z_i$. If $E$ is the corresponding exceptional divisor on $X$, then there is a naturally induced morphism of schemes $$B_Z X-(E\cup W_1\cup \cdots\cup W_r)\rightarrow X-(Z_1\cup \cdots\cup Z_r).$$ Since the construction of compactifying log structure is functorial by [@Ogu §III.1.6], the above morphism induces a morphism of fs log schemes $$(B_Z Y,E)\rightarrow Y.$$ See Definition \[def:associative-notation-log\] for the notation. By definition, we say that $(B_Z Y,E)$ is the blow-up of $Y$ along $Z$. On the other hand, we do *not* have a naturally induced morphism of fs log schemes $B_ZY\rightarrow Y$ in general.
Regard $B_Z Y$ as the strict transform of $Y\times \{0\}$ in $B_{Z\times \{0\}}(Y\times \boxx)$. The deformation space of a pair $(Y,Z)$ is defined as $$D_Z Y:=B_{Z\times \{0\}}(Y\times \boxx)-B_Z Y.$$ The strict transform of $Z\times \boxx$ in $B_{Z\times \{0\}} (Y\times \boxx)$ does not intersect with $B_ZX$, so we may and will consider $Z\times \boxx$ as a smooth divisor on $D_Z Y$.
Consider the normal bundle $p:N_ZX\rightarrow X$ of $Z$ in $X$. The *normal bundle* of $Z$ in $Y$ is defined as $$N_Z Y:=(N_Z X,p^{-1}(Z_1)+\cdots+p^{-1}(Z_r)).$$ Note that $N_ZY\rightarrow Y$ is a vector bundle.
\[A.3.3\] Suppose $Z_1,\ldots,Z_r$ are divisors forming a strict normal crossing divisor on $X\in Sm/S$ over $S$, and that $Z$ has strict normal crossing with $Z_1+\cdots+Z_r$ over $S$. Let $f:X'\rightarrow X$ be an étale morphism in $Sm/S$ such that $Z':=f^{-1}(Z)\rightarrow Z$ is an isomorphism, and let $E$ (resp. $E'$) be the exceptional divisor on the blow-up $B_Z X$ (resp. $B_{Z'}X'$). Let $$Y:=(X,Z_1+\cdots+Z_r),
\;
Y':=(X',Z_1'+\cdots+Z_r'),$$ where $Z_i'$ denotes the pullback of $Z_i$ along $f$. Then $f$ induces an isomorphism $$M((B_{Z'}Y',E')\rightarrow Y')\xrightarrow{\cong} M((B_ZY,E)\rightarrow Y).$$
For notational convenience we set $$U:=(X-Z,(Z_1-Z)+\cdots+(Z_r-Z)),\; U':=U\times_Y Y'.$$ Since $X'\amalg X-Z\rightarrow X$ and $B_{Z'}X'\amalg X-Z\rightarrow B_Z X$ are Nisnevich covers by assumption, there are induced strict Nisnevich distinguished squares $$\begin{tikzcd}
U'\arrow[d]\arrow[r]&(B_{Z'}Y',E')\arrow[d]& U'\arrow[d]\arrow[r]&Y'\arrow[d]\\
U\arrow[r]&(B_Z Y, E)&U\arrow[r]&Y.
\end{tikzcd}$$ By passing to motives, property ($sNis$-des) shows there are naturally induced isomorhisms $$M(U'\rightarrow U)\xrightarrow{\cong} M((B_{Z'}Y',E')\rightarrow (B_Z Y,E)),$$ $$M(U'\rightarrow U)\xrightarrow{\cong} M(Y'\rightarrow Y).$$ This finishes the proof by appealing to Proposition \[A.3.44\] and the isomorphism $$M((B_{Z'}Y',E')\rightarrow (B_ZY,E))\xrightarrow{\cong} M(Y'\rightarrow Y).$$
Associated to vector bundles over fs log schemes we have the fundamental notion of a Thom motive; this plays a crucial role in the theory of log motives.
\[A.3.22\] Suppose $Z_1,\ldots,Z_r$ form a strict normal crossing divisor on $X\in Sm/S$. For $Y:=(X,Z_1+\cdots+Z_r)$ and a vector bundle $\xi:\cE\rightarrow Y$ of fixed rank, let $Z$ be the $0$-section of $\cE$. Then $$\xi^{-1}(Z_1),\ldots,\xi^{-1}(Z_r)$$ form a strict normal crossing divisor on $\underline{\cE}$, and $Z$ has strict normal crossing with $\xi^{-1}(Z_1)+\cdots+\xi^{-1}(Z_r)$ over $S$. Let $E$ be the exceptional divisor on the blow-up $B_Z(\cE)$. The [*Thom motive*]{} of $\cE$ over $X$ is defined as $$MTh_X(\cE)
:=
M((B_Z(\cE),E)\rightarrow \cE).$$ We often omit the subscript $X$ in the notation when no confusion seems likely to arise.
According to the following proposition, the Betti realization of the motivic Thom space is homotopy equivalent to the quotient of the unit disk bundle by the unit sphere bundle, which is a formulation of Thom spaces in algebraic topology.
\[prop::unitdiskbundle\] Suppose $X$ is a separated scheme over $\Spec\C$, and let $\xi:\cE\rightarrow X$ be a vector bundle. Then the Betti realization of $\cE$ (resp. $\widetilde{\cE}:=(B_Z(\cE),E)$) is homotopy equivalent to a unit disk bundle (resp. unit sphere bundle).
The question is Zariski local on $X$, so we may assume that $\xi$ is a trivial rank $n$ vector bundle. Consider the fs log scheme $Y:=(B_{\{0\}}\A_{\C}^n,D)$ where $D$ is the exceptional divisor. Then by Proposition \[A.9.74\], the Betti realization of $Y$ is homotopy equivalent to $$(Y-\partial Y)_{log}\cong (\A_{\C}^r-\{0\})_{an}\cong S^{2r-1},$$ where $S^i$ denotes the $i$-dimensional unit sphere. Recall the Betti realization of $\A_{\C}^r$ is homotopy equivalent to the $2r$-dimensional unit disk $D^{2r}$. Since $\cE\cong X\times_{\C}\A_\C^n$ and $\widetilde{\cE}\cong X\times_{\C} Y$, we are done.
We prove an analogue of [@MV Proposition 2.17(3), p. 112] as follows. Note that our proof, which involves the blow-up along the zero section of the vector bundle, is different from the proof given by Morel-Voevodsky [@MV].
\[A.3.34\] Let $\cE$ be a rank $n$ vector bundle over $X$ in $SmlSm/S$, and view $\P(\cE)$ as the closed subscheme of $\P(\cE\oplus \cO)$ at infinity. Then there is a canonical isomorphism $$MTh(\cE)\xrightarrow{\cong} M(\P(\cE)\rightarrow \P(\cE\oplus \cO)).$$ In particular, if $n=1$ and $\cE$ is trivial, then $$MTh(X\times \A^1)\cong M(X)(1)[2].$$
Let $Y$ (resp. $Y'$) be the blow-up of $\cE$ (resp. $\P(\cE\oplus \cO)$) along the zero section $Z_0$, and let $E$ (resp. $E'$) be the exceptional divisor on $Y$ (resp. $Y'$). View $\cE$ as the open subscheme $\P(\cE\oplus \cO)-\P(\cE)$ of $\P(\cE)$. Owing to ($sNis$-des) and Proposition \[A.3.44\], the cartesian square $$\begin{tikzcd}
(Y,E)\arrow[r]\arrow[d]& (Y',E')\arrow[d]\\
\cE\arrow[r]&\P(\cE\oplus \cO)
\end{tikzcd}$$ induces an isomorphism $$\label{A.3.34.1}
MTh(\cE)=M((Y,E)\rightarrow \cE)\xrightarrow{\cong} M((Y',E')\rightarrow \P(\cE\oplus \cO)).$$ Since $Z_0\cap \P(\cE)=\emptyset$, we can view $\P(\cE)$ as a divisor $Z$ of $Y'$ not intersecting with $E'$.
We claim the closed immersion $Z\rightarrow (Y',E')$ induces an isomorphism $$M(Z)\xrightarrow{\cong} M(Y',E').$$ Due to ($Zar$-sep), we reduce to the case when $\cE$ is trivial. In this case, owing to Lemma \[A.3.40\], $(Y',E')$ is a $\boxx$-bundle over $\P_X^{n-1}$. The composite morphism $Z\rightarrow (Y',E')\rightarrow \P_X^{n-1}$ is an isomorphism. Hence the composite morphism $$M(Z)\rightarrow M(Y',E')\rightarrow M(\P_X^{n-1})$$ is also an isomorphism. The second morphism is an isomorphism by Proposition \[A.6.3\]. Thus the first morphism is an isomorphism, which proves the claim.
From the above and Proposition \[A.3.48\], we deduce the isomorphism $$\label{A.3.34.2}
M((Y',E')\rightarrow \cP(\cE\oplus \cO))\cong M(\P(\cE)\rightarrow \P(\cE\oplus \cO)).$$ To conclude the proof we combine and .
For a smooth fan $\Sigma$ with a cone $\sigma$, we have used the notation $\Sigma^*(\sigma)$, which is the star subdivison (see Definition \[Starsubdivision\]) relative to $\sigma$. If $x$ is the center of $\sigma$, i.e., $x=(a_1+\cdots+a_n)$ whenever $\sigma={\rm Cone}(a_1,\ldots,a_n)$, we set $$\Sigma^*\langle x\rangle :=\Sigma^*(\sigma).$$
In $\A^1$-homotopy theory, the construction of the Thom space of a vector bundle $E$ on $X$ is compatible with the monoidal structure. That is, if $E_i$ a vector bundle on $X_i$ for $i=0,1$, there is a canonical isomorphism $$Th_{X_1}(E_1)\otimes Th_{X_2}(E_2) \cong Th_{X_1\times X_2}(E_1\times E_2).$$ This follows immediately from the construction, see [@MV Proposition 2.17 (1)]. Alas, in our setting this property is less evident. In the following we address this problem for double and triple products of vector bundles. This will be applied later in the proof of Proposition \[A.3.43\], see also Remark \[rmk:why-3-bundles-necessary\].
Assume that $k$ admits resolution of singularities as in Definition \[A.3.8\]. In this case, the compatibility between Thom motives and products can be deduced a posteriori from the one in $\dmeff$, provided the underlying scheme $\ul{X}$ is proper. This is a consequence of Proposition \[A.4.31\].
\[A.3.52\] Let $X$ be an irreducible separated scheme with two nowhere dense closed subschemes $Z_1$ and $Z_2$. We set $W_1:=Z_1\times_X B_{Z_2}X$ and $W_2:=B_{Z_2}X\times_X Z_2$. If $Y$ is the closure of the generic point of $B_{Z_1}X\times_X B_{Z_2}X$ whose image in $X$ is the generic point, then there are isomorphisms $$Y\cong B_{W_2}(B_{Z_1}X)\cong B_{W_1}(B_{Z_2}X).$$
Due to [@Fulton Section B.6.9], $B_{W_2}(B_{Z_1}X)$ is a closed subscheme of $B_{Z_1}X\times_X B_{Z_2}X$. Since $B_{W_2}(B_{Z_1}X)$ is proper and birational over $X$, we have $Y\cong B_{W_2}(B_{Z_1}X)$. We can similarly show $Y\cong B_{W_1}(B_{Z_2}X)$.
\[A.3.53\] Let $\Sigma$ be a fan in $N$ with two partial subdivisions $\Sigma_1$ and $\Sigma_2$. We set $\Sigma_{12}:=\Sigma_1\times_\Sigma \Sigma_2$, i.e., $$\Sigma_{12}=\{\sigma_1\cap \sigma_2:\sigma_1\in \Sigma_1,\sigma_2\in \Sigma_2\}.$$ If $Y$ is the closure of the generic point $\xi$ of $\ul{\A_{\Sigma_1}}\times_{\ul{\A_\Sigma}}\ul{\A_{\Sigma_2}}$ whose image in $\ul{\A_{\Sigma}}$ is the generic point, then there is an isomorphism $$Y\cong \ul{\A_{\Sigma_{12}}}.$$
We only need to show this for cones of $\Sigma$, $\Sigma_1$, and $\Sigma_2$. Hence we may assume $$\Sigma=\Spec{P},
\;
\Sigma_1=\Spec{P_1},
\;
\Sigma_2=\Spec{P_2},$$ where $P$, $P_1$, and $P_2$ are fs submonoids of the dual lattice $M$ of $N$ such that $$M\cong P^\gp\cong P_1^\gp\cong P_2^\gp.$$ There is an isomorphism $$\ul{\A_{P_1}}\times_{\ul{\A_P}}\ul{\A_{\P_2}}
\cong
\ul{\A_{P_1\oplus_P^{\rm mon}P_2}},$$ where $P_1\oplus_P^{\rm mon}P_2$ denotes the amalgamated sum in the category of monoids. The amalgamated sum $P_1\oplus_P^{\rm int}P_2$ in the category of integral monoids is the image of $P_1\oplus P_2$ in $M$, i.e., $$P_1\oplus_P^{\rm int}P_2=P_1+P_2.$$ There is a naturally induced homomorphism $$f\colon \Z[P_1\oplus_P^{\rm mon}P_2]\rightarrow \Z[M],$$ and since $Y$ is the closure of $\xi$ we observe that $Y=\Spec{\im{f}}$. Thus $$Y\cong \Spec{\Z[P_1+P_2]}$$ since $\im{f}$ agrees with $\Z[P_1+P_2]$. The fan $\Sigma_{12}$ has only one maximal cone $P_1^\vee\cap P_2^\vee$, which corresponds to $P_1+P_2$ by duality. This shows $Y\cong \ul{\A_{\Sigma_{12}}}$.
\[A.3.47\] For $X_1,X_2,X_3\in Sm/S$ and vector bundles $\cE_i\rightarrow X_i$ of fixed rank for $i=1,2,3$, we will for future usage construct 55 blow-ups of the triple product $$T:=\cE_1\times_S \cE_2\times_S \cE_3.$$
Let $Z_i$ be the $0$-section of $\cE_i$ for $i=1,2,3$, and we set $$\begin{gathered}
W_1:=Z_1\times_S \cE_2\times_S \cE_3,
\;
W_2:=\cE_1\times_S Z_2\times_S \cE_3,
\;
W_3:=\cE_1\times_S \cE_2\times_S Z_3,
\\
W_4:=Z_1\times_S Z_2\times_S \cE_3,
\;
W_5:=\cE_1\times_S Z_2\times_S Z_3,
\;
W_6:=Z_1\times_S Z_2\times_S Z_3.\end{gathered}$$ For every subset $I=\{i_1,\ldots,i_n\}\subset \{1,\ldots,6\}$ with $i_1<\ldots<i_n$, consider the fiber product $$B_{W_{i_1}}(T)\times_T \cdots \times_T B_{W_{i_n}}(T)$$ Collect its all generic points whose image in $T$ are also generic, and let $T_I$ denote the union of the closures of these generic points. Due to Lemma \[A.3.52\] if $j\notin I$, then $T_{I\amalg \{j\}}$ is the blow-up of $T_I$ along the preimage $W_{I,j}$ of $W_j$ in $T_I$.
We claim that $W_{I,j}$ is smooth if $$I\cap \{4,5,6\}=\{4,5\}.$$ This question is Zariski local on $X_1$, $X_2$, and $X_3$, so we may assume that $\cE_i$ is a trivial vector bundle of rank $p_i$ for $1\leq i\leq 3$. We set $n=p_1+p_2+p_3$. Let $e_1,\ldots,e_{n}$ be the standard coordinates in $\Z^{n}$, and let $\Sigma$ be the fan whose only maximal cone is ${\rm Cone}(e_1,\ldots,e_{n})$. We define the following sets $$\begin{gathered}
S_0:=\emptyset,
\;
S_1:=\{e_1,\ldots,e_{p_1}\},
\\
S_2:=\{e_{p_1+1},\ldots,e_{p_1+p_2}\},
\;
S_3:=\{e_{p_1+p_2},\ldots,e_{n}\},
\\
S_4:=S_1\cup S_2,
\;
S_5:=S_2\cup S_3,
\;
S_6:=S_1\cup S_2\cup S_3.\end{gathered}$$ We set $$\begin{gathered}
f_1:=e_1+\cdots+e_{p_1},
\;
f_2:=e_{p_1+1}+\cdots+e_{p_1+p_2},
\;
f_3:=e_{p_1+p_2+1}+\cdots+e_{n},
\\
f_4:=f_1+f_2,
\;
f_5:=f_2+f_3,
\;
f_6:=f_1+f_2+f_3.\end{gathered}$$ For every subset $I=\{i_1,\ldots,i_m\}\subset \{1,\ldots,6\}$ with $i_1<\ldots<i_m$ we set $$\Sigma_{I}:=\Sigma_{i_1}\times_{\Sigma}\cdots \times_{\Sigma}\Sigma_{i_m},$$ i.e., $$\Sigma_{I}=\{\sigma_1\cap \cdots \cap \sigma_m:\sigma_1\in \Sigma_{i_1},\ldots,\sigma_m\in \Sigma_m\}.$$ By convention $\Sigma_{\emptyset}:=\Sigma$. Due to Lemmas \[A.3.52\] and \[A.3.53\] there is an isomorphism $$T_I\cong \underline{\A_{\Sigma_I}}\times (X_1\times_S X_2\times_S X_3).$$ Hence it suffices to show that $\Sigma_{I\amalg \{j\}}$ is a star subdivision of $\Sigma_I$ relative to a smooth cone. This will be done in Lemma \[A.3.51\] after proving several preliminary results.
For $u\in S_1$, $v\in S_2$, $w\in S_3$, $p\in \{0,1,4,6\}$, $q\in \{0,2,4,5,6\}$, and $r\in \{0,3,5,6\}$, we set $$\sigma_{uvw}^{pqr}:=
\{
(a_1,\ldots,a_n)\in \N^n:
a_u\leq a_i\text{ if } i\in S_p,
\;
a_v\leq a_i\text{ if } i\in S_q,
\;
a_w\leq a_i\text{ if } i\in S_r
\}.$$ Note that if $p=0$ (resp. $q=0$, resp. $r=0$), then the condition $a_u\leq a_i$ (resp. $a_v\leq a_i$ resp. $a_w\leq a_i$) has no meaning. For notational convenience, in this case we also write $u=*$ (resp. $v=*$, resp. $w=*$).
If inequalities of the form $a_i\leq a_j$ and $a_j\leq a_i$ enter in the condition for some $i\neq j$ when forming $\sigma_{uvw}^{pqr}$, then $\dim \sigma_{uvw}^{pqr}<n$. By using this criterion, we can list all possible values of $(p,q,r)$ such that $\dim \sigma_{uvw}^{pqr}=n$, see Table \[pqrtable\].
$p$ $q$ $r$ $p$ $q$ $r$ $p$ $q$ $r$
------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------
$0$ or $1$ $0$ or $2$ $0$ or $3$ $6$ $0$ or $2$ $0$ or $3$ $4$ $5$ $0$ or $3$
$4$ $0$ or $2$ $0$ or $3$ $0$ or $1$ $6$ $0$ or $3$ $4$ $0$ or $2$ $5$
$0$ or $1$ $4$ $0$ or $3$ $0$ or $1$ $0$ or $2$ $6$ $0$ or $1$ $4$ $5$
$0$ or $1$ $5$ $0$ or $3$ $4$ $0$ or $2$ $6$
$0$ or $1$ $0$ or $2$ $5$ $6$ $0$ or $2$ $5$
: List of all concise $(p,q,r)$.[]{data-label="pqrtable"}
We say that $(p,q,r)$ is *concise* if it has a value as listed in the table. A concise triple $(p,q,r)$ is called *standard* if $\{4,5\}\not\subset \{p,q,r\}$. For $1\leq i\leq n$ and $1\leq t\leq 6$ to simplify notation we set $$f_0^i:=e_i,
\;
f_t^i:=f_t.$$ If $(p,q,r)$ is standard, then we can check by hand that $$\sigma_{uvw}^{pqr}={\rm Cone}(e_1,\ldots,e_{u-1},e_{u+1},\ldots,e_{v-1},e_{v+1},\ldots,e_{w-1},e_{w+1},\ldots,e_n,f_p^u,f_q^v,f_r^w).$$ We regard $\{0,\ldots,6\}$ as a partially ordered set whose order $\prec$ is generated by $$(0\prec 1,2,3),\; (1\prec4,5),\; (2\prec 4,5,6),\; (3 \prec 5,6),\; (4 \prec 6),\; (5\prec 6).$$ We use the following shorthand notation: $$\begin{gathered}
p^s:=\sup\{p,s\}\text{ (if }s\in \{1,4,6\}),
\;
q^s:=\sup\{q,s\}\text{ (if }s\in \{2,4,5,6\}),
\\
r^s:=\sup\{r,s\}\text{ (if }s\in \{3,5,6\}),\end{gathered}$$ where $\sup$ is taken with respect to the order $\prec$. Note that $$\label{A.3.47.7}
p^s=p\text{ or }s,
\;
q^s=q,\;s,\text{ or }6,
\;
r^s=r\text{ or }s.$$ If $p\neq 0$ and $s\in \{1,4,6\}$, then we have $$\label{A.3.47.2}
\begin{split}
&\sigma_{uvw}^{pqr}\cap \sigma_{u**}^{s00}
\\
=&
\{
(a_1,\ldots,a_n)\in \N^n:
a_u\leq a_i\text{ if } i\in S_p\cup S_s,
\;
a_v\leq a_i\text{ if } i\in S_q,
\;
a_w\leq a_i\text{ if } i\in S_r
\}
\\
=&
\sigma_{uvw}^{p^sqr}.
\end{split}$$ We similar have $$\begin{split}
\label{A.3.47.3}
\sigma_{uvw}^{pqr}\cap \sigma_{*v*}^{0s0}=\sigma_{uvw}^{pq^sr}\text{ (if }q\neq 0),&
\;
\sigma_{uvw}^{pqr}\cap \sigma_{**w}^{00s}=\sigma_{uvw}^{pqr^s}\text{ (if }r\neq 0),
\\
\sigma_{*vw}^{0qr}\cap \sigma_{u**}^{s00}=\sigma_{uvw}^{sqr},
\;
\sigma_{u*w}^{p0r}\cap \sigma_{*v*}^{0s0}&=\sigma_{uvw}^{psr},
\;
\sigma_{uv*}^{pq0}\cap \sigma_{**w}^{00s}=\sigma_{uvw}^{pqs}
\end{split}$$ for all appropriate $s$. Moreover, we have $$\label{A.3.47.4}
\dim\sigma_{uvw}^{pqr}\cap \sigma_{u'**}^{s00}<n\text{ if }u\neq u',\;p\neq 0,\;s\in \{1,4,6\}$$ since the inequalities $a_u\leq a_{u'}$ and $a_{u'}\leq a_u$ are appeared. Similarly, we have $$\begin{split}
\label{A.3.47.5}
\dim\sigma_{uvw}^{pqr}\cap \sigma_{*v'*}^{0s0}<n&\text{ if }v\neq v',\;q\neq 0,\;s\in \{2,4,5,6\},
\\
\dim\sigma_{uvw}^{pqr}\cap \sigma_{**w'}^{00s}<n&\text{ if }w\neq w',\;r\neq 0,\;s\in \{3,5,6\}.
\end{split}$$
\[A.3.50\] With notations as above, let $I$ be a subset of $\{1,\ldots,6\}$ such that $$I\cap \{4,5,6\}\neq \{4,5\}.$$ Then any $n$-dimensional cone of $\Sigma_I$ has can be written as $\sigma_{uvw}^{pqr}$ for some standard triple $(p,q,r)$.
Suppose $s\in \{1,\ldots,6\}-I$ and $(I\amalg \{s\})\cap \{4,5,6\}\neq \{4,5\}$. Any $n$-dimensional cone of $\Sigma_{I\amalg \{s\}}$ is of the form $\sigma\cap \sigma'$ for some $\sigma\in \Sigma_I$ and $\sigma'\in \Sigma_s$. Thus from , , , , and , we observe that any $n$-dimensional cone of $\Sigma_{I\amalg\{s\}}$ takes the form $\sigma_{uvw}^{pqr}$ with $p,q,r\in I\amalg \{0,s\}$ if any maximal cone of $\Sigma_I$ can be written as $\sigma_{uvw}^{pqr}$ with $p,q,r\in I\amalg\{0\}$. Now use induction on the number of elements of $I$ to deduce this holds for all $I$ satisfying $I\cap \{4,5,6\}\neq \{4,5\}$.
Let $\sigma_{uvw}^{pqr}$ be an $n$-dimensional cone of $\Sigma_I$. If $(p,q,r)$ is not concise, then $\sigma_{uvw}^{pqr}$ is not an $n$-dimensional cone. Thus $(p,q,r)$ is concise. If $(p,q,r)$ is not standard, then $\{4,5\}\in \{p,q,r\}$. We have shown in the above paragraph that this implies $\{4,5\}\in I$. From the condition $I\cap \{4,5,6\}\neq \{4,5\}$ we have $6\in I$. Then there are six cases according to Table \[pqrtable\]: $$\sigma_{uvw}^{pqr}=\sigma_{uvw}^{450},
\;
\sigma_{uvw}^{453},
\;
\sigma_{uvw}^{405},
\;
\sigma_{uvw}^{425},
\;
\sigma_{uvw}^{045},
\;
\sigma_{uvw}^{145}.$$ In the first and second (resp. third and fourth, resp. fifth and sixth) cases, $a_u,a_v\notin \sigma_{uvw}^{pqr}$ (resp. $a_u,a_w\notin \sigma_{uvw}^{pqr}$, resp. $a_v,a_w\notin \sigma_{uvw}^{pqr}$). Thus in all cases $\sigma_{uvw}^{pqr}$ is not contained in any of $$\sigma_{u'**}^{600},
\;
\sigma_{*v'*}^{060},
\;
\sigma_{**w'}^{060}.$$ This contradicts the fact that $6\in I$. Thus $(p,q,r)$ is standard.
\[A.3.51\] With notations as above, let $I$ be a subset of $\{1,\ldots,6\}$, and let $s$ be an element of $\{1,\ldots,6\}-I$. If $$\{4,5,6\}\cap I \neq \{4,5\}\text{ and }\{4,5,6\}\cap (I\amalg \{s\})\neq \{4,5\},$$ then $$\Sigma_{I\amalg \{s\}}=\Sigma_I^*(\tau),$$ where $\tau$ is the cone of $\Sigma_I$ generated by $f_s$.
Let $\sigma_{uvw}^{pqr}$ be a maximal cone of $\Sigma_I$. Lemma \[A.3.50\] shows that $(p,q,r)$ is standard. Suppose we have $$\{\sigma_{uvw}^{pqr}\}\times_{\Sigma}\Sigma_j
\neq
\{\sigma_{uvw}^{pqr}\}.$$ Due to , , , and , its $n$-dimensional cones are of the following forms: $$\begin{gathered}
\sigma_{uvw}^{sqr}\text{ (if }p\neq 0\text{ and }s\neq 2,3,5),
\;
\sigma_{uvw}^{psr}\text{ (if }q\neq 0\text{ and }s\neq 1,3),
\\
\sigma_{uvw}^{pqs}\text{ (if }q\neq 0\text{ and }s\neq 1,2,4),
\;
\sigma_{1vw}^{sqr},\ldots,\sigma_{p_1vw}^{sqr}\text{ (if }p= 0\text{ and }s\neq 2,3,5),
\\
\sigma_{u(p_1+1)w}^{psr},\ldots,\sigma_{u(p_1+p_2)w}^{psr}\text{ (if }q= 0\text{ and }s\neq 1,3),
\\
\sigma_{uv(p_1+p_2+1)}^{pqs}\ldots,\sigma_{uvn}^{pqs}\text{ (if }r=0\text{ and }s\neq 1,2,4).\end{gathered}$$ Note that some of these cones can have dimension $<n$.
Suppose $\sigma_{uvw}^{sqr}=\sigma_{uvw}^{pqr}\cap \sigma_{u**}^{s00}$ is a cone in $\{\sigma_{uvw}^{pqr}\}\times_\Sigma \Sigma_j$ and has dimension $n$ with $s\in \{1,4,6\}$ and $p\prec s$. Lemma \[A.3.50\] shows that $(s,q,r)$ is standard. Then $$\begin{gathered}
\sigma_{uvw}^{pqr}={\rm Cone}(e_1,\ldots,e_{u-1},e_{u+1},\ldots,e_{v-1},e_{v+1},\ldots,e_{w-1},e_{w+1},\ldots,e_n,f_{p}^u,f_q^v,f_r^w),
\\
\sigma_{uvw}^{sqr}={\rm Cone}(e_1,\ldots,e_{u-1},e_{u+1},\ldots,e_{v-1},e_{v+1},\ldots,e_{w-1},e_{w+1},\ldots,e_n,f_{s},f_q^v,f_r^w).\end{gathered}$$ As a special case we have $$\sigma_{*vw}^{0qr}={\rm Cone}(e_1,\ldots,e_{u-1},e_{u+1},\ldots,e_{v-1},e_{v+1},\ldots,e_{w-1},e_{w+1},\ldots,e_n,e_u,f_q^v,f_r^w).$$ Since $\sigma_{uvw}^{pqr}$ contains $\sigma_{uvw}^{sqr}$, we have $f_s\in \sigma_{uvw}^{pqr}$. Let $\tau$ be the subcone of $\sigma_{uvw}^{pqr}$ generated by $f_s$, and let $\{d_1,\ldots,d_l\}$ be the generator of $\tau$. Since $f_s\in \tau$, it follows that $\{f_s,d_1,\ldots,d_l\}$ is linearly independent. It follows that if $f_p^u\notin \tau$, then $\dim \sigma_{uvw}^{sqr}<n$. This contradicts the fact that $(s,q,r)$ is standard. In conclusion, we have:
1. To obtain $\sigma_{uvw}^{sqr}$ from $\sigma_{uvw}^{pqr}$ replace $f_p^u$ by $f_{s}$. To obtain $\sigma_{uvw}^{sqr}$ from $\sigma_{*vw}^{0qr}$ replace $e_u$ by $f_s$.
2. $f_p^u$ should be in $\tau$.
We have similar rules for $\sigma_{uvw}^{psr}$ and $\sigma_{uvw}^{pqs}$ also under similar assumptions.
If ${\rm Cone}(\epsilon_1,\ldots,\epsilon_t)$ is an arbitrary cone with the star subdivision $\Gamma$ relative to a subcone ${\rm Cone}(\epsilon_1,\ldots,\epsilon_{t'})$, then we have:
1. To obtain a maximal cone of $\Gamma$ replace $\epsilon_i$ by $\epsilon_1+\cdots+\epsilon_{t'}$.
2. $\epsilon_i$ should be in ${\rm Cone}(\epsilon_1,\ldots,\epsilon_{t'})$.
Compare (Rule 1) and (Rule 2) with (Rule 1’) and (Rule 2’) to deduce that $$\label{A.3.51.1}
\{\sigma_{uvw}^{pqr}\}\times_{\Sigma}\Sigma_s = \{\sigma_{uvw}^{pqr}\}^*(\tau),$$ which finishes the proof.
Let us illustrate with an example. If $(p,q,r,s)=(4,2,0,6)$, then $$\begin{gathered}
\sigma_{uv*}^{420}={\rm Cone}(e_1,\ldots,e_{u-1},e_{u+1},\ldots,e_{v-1},e_{v+1},\ldots,e_{w-1},e_{w+1},\ldots,e_n,f_{4},f_2,e_w),
\\
\sigma_{uv*}^{620}={\rm Cone}(e_1,\ldots,e_{u-1},e_{u+1},\ldots,e_{v-1},e_{v+1},\ldots,e_{w-1},e_{w+1},\ldots,e_n,f_6,f_2,e_w),
\\
\sigma_{uv*}^{460}={\rm Cone}(e_1,\ldots,e_{u-1},e_{u+1},\ldots,e_{v-1},e_{v+1},\ldots,e_{w-1},e_{w+1},\ldots,e_n,f_{4},f_6,e_w),
\\
\sigma_{uvw}^{426}={\rm Cone}(e_1,\ldots,e_{u-1},e_{u+1},\ldots,e_{v-1},e_{v+1},\ldots,e_{w-1},e_{w+1},\ldots,e_n,f_{4},f_2,f_6).\end{gathered}$$ The cone $\sigma_{uv*}^{460}$ has dimension $<n$, and $$\sigma_{uv*}^{620},\sigma_{uv(p_1+p_2+1)}^{426},\ldots,\sigma_{uvn}^{426}$$ are precisely all the maximal cones of $\{\sigma_{uv*}^{420}\}\times_{\Sigma}\Sigma_6$. In addition, $$\tau={\rm Cone}(f_4,e_{p_1+p_2+1},\ldots,e_n)$$ is the subcone of $\sigma_{uv*}^{420}$ generated by $f_6$, and $$\{\sigma_{uvw}^{420}\}\times_{\Sigma}\Sigma_6 = \{\sigma_{uvw}^{420}\}^*(\tau).$$
\[A.3.49\] For $X_1,X_2,X_3\in SmlSm/S$ and vector bundles $\cE_i\rightarrow X_i$ of fixed rank for $i=1,2,3$, we may apply Construction \[A.3.47\] to the vector bundles $\ul{\cE_1}\rightarrow \ul{X_1}$, $\ul{\cE_2}\rightarrow \ul{X_2}$, and $\ul{\cE_3}\rightarrow \ul{X_3}$. In this way we obtain the blow-up $\ul{T_I}$ of $$\ul{T}:=\ul{\cE_1}\times_S \ul{\cE_2}\times_S \ul{\cE_3}$$ for every subset $I$ of $\{1,\ldots,6\}$ such that $$I\cap\{4,5,6\}\neq \{4,5\}.$$ For $i=1,2,3$ let $Z_i$ be the $0$-section of $\cE_i$, and set $$\begin{gathered}
U:=(\cE_1-\partial \cE_1)\times_S (\cE_2-\partial \cE_2)\times_S (\cE_3-\partial \cE_3),
\;
U_1:=U-Z_1\times_S \cE_2\times_S \cE_3,
\\
U_2:=U-\cE_1\times_S Z_2\times_S \cE_3,
\;
U_3:=U-\cE_1\times_S \cE_2\times_S Z_3,
\\
U_4:=U-Z_1\times_S Z_2\times_S \cE_3,
\;
U_5:=U-\cE_1\times_S Z_2\times_S Z_3,
\;
U_6:=U-Z_1\times_S Z_2\times_S Z_3.\end{gathered}$$
If $I=\{i_1,\ldots,i_n\}$, then the open immersion $$j_I\colon U_I:=U_{i_1}\cap \cdots \cap U_{i_n}\rightarrow \underline{T}$$ can be lifted to $\ul{T_I}$ since the images of the exceptional divisors on $\ul{T_I}$ in $\ul{T}$ lie in the complement of $j_I$. Let $T_I$ denote the fs log scheme whose underlying scheme is $\ul{T_I}$ and whose log structure is the compactifying log structure associated with $j_I\colon U_I\rightarrow \underline{T}$. If $I=\{i_1,\ldots,i_n\}$, for simplicity of notation, we set $$T_{i_1\ldots i_n}:=T_I$$ In particular, we set $T:=T_\emptyset$.
We can divide all the $T_I$’s into the following groups such that if $T_I$ and $T_J$ belong to the same group, then $U_I=U_J$. $$\begin{gathered}
\label{A.3.49.1}
\{T\},
\;
\{T_1, T_{14}, T_{16}, T_{146}\},
\;
\{T_2, T_{24}, T_{25}, T_{26}, T_{246}, T_{256}, T_{2456}\},
\\
\{T_3, T_{35}, T_{36}, T_{356}\},
\;
\{T_{4}, T_{46}\},
\;
\{T_5, T_{56}\},
\;
\{T_6\},
\\
\{T_{12}, T_{124}, T_{125}, T_{126}, T_{1246}, T_{1256}, T_{12456}\},
\\
\{T_{13}, T_{134}, T_{135}, T_{136}, T_{1346}, T_{1356}, T_{13456}\},
\\
\{T_{23}, T_{234}, T_{235}, T_{236}, T_{2346}, T_{2356}, T_{23456}\},
\\
\{T_{15}, T_{156}, T_{1456}\},
\;
\{T_{34}, T_{346}, T_{3456}\},
\\
\{T_{123}, T_{1234}, T_{1235}, T_{1236}, T_{12346}, T_{12356}, T_{123456}\},
\;
\{T_{456}\}.
\end{gathered}$$ If $U_I=U_J$ and $\lvert J\rvert=\lvert I\rvert +1$, then $T_J$ is an admissible blow-up of $T_I$ along a smooth center since we have shown that $W_{I,j}$ in Construction \[A.3.47\] is smooth.
We can now turn to proving compatibility between Thom motives and products. As can be expected, since blow-ups are involved in our formulation, this requires some extra arguments than in the $\A^1$-setting.
\[A.3.42\] For $X_1,X_2\in SmlSm/S$ and vector bundles $\cE_1\rightarrow X_1$ and $\cE_2\rightarrow X_2$, there exists a canonical isomorphism $$\label{A.3.42.10}
MTh_{X_1}(\cE_1)\otimes MTh_{X_2}(\cE_2)\xrightarrow{\cong} MTh_{X_1\times_S X_2}(\cE_1\times_S \cE_2).$$
**Step 1** Apply Construction \[A.3.49\] to the vector bundles $\cE_1\rightarrow X_1$, $\cE_2\rightarrow X_2$, and $S\rightarrow S$ to obtain an fs log scheme $T_I\in SmlSm/S$ for every subset $I$ of $\{1,2,4\}$. According to Construction \[A.3.49\], the morphisms $$T_{124}\rightarrow T_{12},
\;
T_{14}\rightarrow T_1,
\;
T_{24}\rightarrow T_2$$ are admissible blow-ups along smooth centers. Theorem \[A.3.7\] shows the naturally induced morphisms $$\label{A.3.42.2}
M(T_{124})\rightarrow M(T_{12}),
\;
M(T_{14})\rightarrow M(T_1),
\;
M(T_{24})\rightarrow M(T_2)$$ are isomorphisms.
**Step 2** Form the naturally induced $3$-squares $$Q:=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{124}\arrow[dd]\arrow[rr]\arrow[ld]&&T_{14}\arrow[dd]\arrow[ld]\\
T_{24}\arrow[dd]\arrow[rr,crossing over]&&T\\
&T_{4}\arrow[rr]\arrow[ld]&&T_{4}\arrow[ld]\\
T_{4}\arrow[rr]&&T\arrow[uu,crossing over,leftarrow]
\end{tikzcd}
\quad
Q':=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{124}\arrow[dd]\arrow[rr]\arrow[ld]&&T_{14}\arrow[dd]\arrow[ld]\\
T_{24}\arrow[dd]\arrow[rr,crossing over]&&T\\
&T_{24}\arrow[rr]\arrow[ld]&&T_{4}\arrow[ld]\\
T_{24}\arrow[rr]&&T\arrow[uu,crossing over,leftarrow]
\end{tikzcd}$$ with $C_1:=\;$the bottom square of $Q$, $C_2:=\;$the top square of $Q$, $D_1:=\;$the front square of $Q$, $D_2:=\;$the back square of $Q$, $C_1':=\;$the bottom square of $Q'$, $C_2':=\;$the top square of $Q"$, $D_1':=\;$the front square of $Q'$, and $D_2':=\;$the back square of $Q'$. By (Sq), there exist canonical distinguished triangles $$\begin{gathered}
\label{A.3.42.17}
M(D_2')\rightarrow M(D_1')\rightarrow M(Q') \rightarrow M(D_2')[1],
\\
\label{A.3.42.18}
M(T_{24}\rightarrow T_{24})\rightarrow M(T\rightarrow T)\rightarrow M(D_1')\rightarrow M(T_{24}\rightarrow T_{24})[1].\end{gathered}$$ Due to Proposition \[A.3.45\] and we have $M(D_1')=0$. We will show in Step 3 that $M(D_2')=0$. Assuming this, from , we have $$M(Q')=0.$$ There exists a morphism of distinguished triangles due to (Sq) $$\label{A.3.42.19}
\begin{tikzcd}
M(C_2')\arrow[d]\arrow[r]&
M(C_1')\arrow[r]\arrow[d]&
M(Q')\arrow[d]\arrow[r]&
M(C_2')[1]\arrow[d]
\\
M(C_2)\arrow[r]&
M(C_1)\arrow[r]&
M(Q)\arrow[r]&
M(C_2)[1].
\end{tikzcd}$$ The first and fourth vertical morphisms are the identity, and the second vertical morphism is an isomorphism since we also have a morphism of distinguished triangles $$\begin{tikzcd}
M(T_{24}\rightarrow T_{24})\arrow[d]\arrow[r]&
M(T_4\rightarrow T)\arrow[r]\arrow[d]&
M(C_1')\arrow[d]\arrow[r]&
M(T_{24}\rightarrow T_{24})[1]\arrow[d]
\\
M(T_4\rightarrow T_4)\arrow[r]&
M(T_4\rightarrow T)\arrow[r]&
M(C_1)\arrow[r]&
M(T_4\rightarrow T_4)[1].
\end{tikzcd}$$ Thus the third vertical morphism in is an isomorphism. It follows that $$M(Q)=0.$$ Using (Sq) we deduce that the naturally induced morphism $$\label{A.3.42.21}
M(C_2)\rightarrow M(C_1)$$ is an isomorphism.
Let $C_3$ be the square $$\begin{tikzcd}
T_{12}\arrow[d]\arrow[r]&T_1\arrow[d]
\\
T_2\arrow[r]&T.
\end{tikzcd}$$ Since the morphisms in are isomorphism, by Proposition \[A.3.48\], the naturally induced morphism $$\label{A.3.42.20}
M(C_2)\rightarrow M(C_3)$$ is an isomorphism. By (MSq) there is a canonical isomorphism $$\label{A.3.42.9}
MTh_{X_1}(\cE_1)\otimes MTh_{X_2}(\cE_2)\cong M(C_3).$$ Combine , , and to obtain .
**Step 3** It remains to verify that $M(D_1')=0$, which is equivalent to showing the naturally induced morphism $$M(T_{124}\rightarrow T_{24})\rightarrow M(T_{14}\rightarrow T_4)$$ is an isomorphism due to (Sq). By ($Zar$-sep), this question is Zariski local on $X_1$ and $X_2$. Hence we may assume that $\cE_1$ and $\cE_2$ are trivial, say of rank $p_1$ and $p_2$. In this case, we can further reduce to the case when $X_1=X_2=S$ by applying (MSq).
For simplicity of notation we suppose that $S=\Spec{\Z}$. Recall the notations $f_1$, $f_2$, and $\Sigma_I$ for $I\subset \{1,2,4\}$ in Construction \[A.3.47\]. We set $$\begin{gathered}
\Sigma_4':=\{{\rm Cone}(f_1+f_2)\},
\;
\Sigma_{14}':=\{{\rm Cone}(f_1,f_1+f_2)\},
\;
\Sigma_{24}':=\{{\rm Cone}(f_2,f_1+f_2)\},
\\
\Sigma_{124}':=\{{\rm Cone}(f_1,f_1+f_2),{\rm Cone}(f_2,f_1+f_2)\}.\end{gathered}$$ With these definitions we have $$T_{124}=\A_{(\Sigma_{124},\Sigma_{124}')},
\;
T_{14}=\A_{(\Sigma_{14},\Sigma_{14}')},
\;
T_{24}=\A_{(\Sigma_{24},\Sigma_{24}')},
\;
T_4=\A_{(\Sigma_4,\Sigma_3')}.$$
When $p=2$ and $q=1$, we can describe the fans as follows. $$\begin{tikzpicture}[yscale=0.7, xscale=0.82]
\fill[black!20] (0,0)--(1,2)--(2,0)--(0,0);
\draw (0,0)--(1,2)--(2,0)--(0,0);
\draw (0,0)--(1,1)--(2,0);
\draw[ultra thick] (1,0)--(1,1)--(1,2);
\filldraw (1,0) circle (3pt);
\filldraw (1,1) circle (3pt);
\filldraw (1,2) circle (3pt);
\node [left] at (0,0) {$e_1$};
\node [right] at (2,0) {$e_2$};
\node [left] at (1,2) {$e_3$};
\node [below] at (1,-0.1) {$(\Sigma_{124},\Sigma_{124}')$};
\begin{scope}[shift={(4,0)}]
\fill[black!20] (0,0)--(1,2)--(2,0)--(0,0);
\draw (0,0)--(1,2)--(2,0)--(0,0);
\draw (0,0)--(1,1)--(2,0);
\draw[ultra thick] (1,0)--(1,1);
\draw (1,1)--(1,2);
\filldraw (1,0) circle (3pt);
\filldraw (1,1) circle (3pt);
\node [below] at (1,-0.1) {$(\Sigma_{14},\Sigma_{14}')$};
\end{scope}
\begin{scope}[shift={(8,0)}]
\fill[black!20] (0,0)--(1,2)--(2,0)--(0,0);
\draw (0,0)--(1,2)--(2,0)--(0,0);
\draw (0,0)--(1,1)--(2,0);
\draw[ultra thick] (1,1)--(1,2);
\filldraw (1,1) circle (3pt);
\filldraw (1,2) circle (3pt);
\node [below] at (1,-0.1) {$(\Sigma_{24},\Sigma_{24}')$};
\end{scope}
\begin{scope}[shift={(12,0)}]
\fill[black!20] (0,0)--(1,2)--(2,0)--(0,0);
\draw (0,0)--(1,2)--(2,0)--(0,0);
\draw (0,0)--(1,1)--(2,0);
\draw (1,1)--(1,2);
\filldraw (1,1) circle (3pt);
\node [below] at (1,-0.1) {$(\Sigma_{4},\Sigma_{4}')$};
\end{scope}
\end{tikzpicture}$$
For any cone $\sigma$, we write $\{\sigma\}$ for the fan whose only maximal cone is $\sigma$. Due to ($Zar$-sep) it suffices to check that the naturally induced morphism $$\label{A.3.42.11}
M(T_{124}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}}
\rightarrow
T_{24}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}})
\rightarrow
M(T_{14}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}}
\rightarrow
T_4\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}})$$ is an isomorphism for every cone $\sigma$ of $\Sigma_4$. Equivalently, owing to Proposition \[A.3.44\], it suffices to check that the naturally induced morphism $$\label{A.3.42.12}
M(T_{124}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}}
\rightarrow
T_{14}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}})
\rightarrow
M(T_{24}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}}
\rightarrow
T_4\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}})$$ is an isomorphism. If $$\sigma\subset {\rm Cone}(e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_{p_1+p_2},f_1+f_2)$$ for some $1\leq i\leq p_1$, then the naturally induced morphisms $$T_{124}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}}
\rightarrow
T_{24}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}},
\;
T_{14}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}}
\rightarrow
T_4\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}}$$ are isomorphisms. Thus the morphism is also an isomorphism. On the other hand, if $$\sigma\subset {\rm Cone}(e_1,\ldots,e_{p_1+j-1},e_{p_1+j+1},\ldots,e_{p_1+p_2},f_1+f_2)$$ for some $1\leq j\leq p_2$, then the naturally induced morphisms $$T_{124}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}}
\rightarrow
T_{14}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}},
\;
T_{24}\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}}
\rightarrow
T_4\times_{\underline{\A_{\Sigma_4}}}\underline{\A_{\{\sigma\}}}$$ are isomorphisms. Thus the morphism is also an isomorphism.
\[A.3.46\] For $X_1,X_2,X_3\in SmlSm/S$ and vector bundles $\cE_i\rightarrow X_i$ for $i=1,2,3$, there are naturally induced isomorphisms rendering the diagram $$\label{A.3.46.1}
\begin{tikzcd}[column sep=tiny]
MTh_{X_1}(\cE_1)\otimes MTh_{X_2}(\cE_1)\otimes MTh_{X_3}(\cE_1)\arrow[d, "\simeq"']\arrow[r, "\simeq"]&
MTh_{X_1}(\cE_1)\otimes MTh_{X_2\times_S X_3}(\cE_2\times_S \cE_3)\arrow[d, "\simeq"]
\\
MTh_{X_1\times_S X_2}(\cE_1\times_S \cE_2)\otimes MTh_{X_3}(\cE_3)\arrow[r, "\simeq"]&
MTh_{X_1\times_S X_2\times_S X_3}(\cE_1\times_S \cE_2\times_S \cE_3)
\end{tikzcd}$$ commutative.
We note the isomorphisms are obtained from Proposition \[A.3.42\]. Applying Construction \[A.3.49\] to the vector bundles $\cE_1\rightarrow X_1$, $\cE_2\rightarrow X_2$, and $\cE_3\rightarrow X_3$ we obtain an fs log scheme $T_I\in SmlSm/S$ for every subset $I$ of $\{1,\ldots,6\}$ such that $$I\cap \{4,5,6\}\neq \{4,5\}.$$ For $i=1,2,3$, let $Z_i$ be the $0$-section of $\cE_i$, and let $E_i$ be the exceptional divisor on the blow-up $B_{Z_i}(\cE_i)$. All of the blow-ups $\ul{T_I}$ contains $$(\cE_1-\partial \cE_1)\times_S (\cE_2-\partial \cE_2)\times_S (\cE_3-\partial \cE_3)$$ as an open subscheme. Let $T_I$ denote the fs log scheme whose underlying scheme is $\ul{T_I}$ and whose log structure is the compactifying log structure associated with the above open immersion.
Form the $3$-squares $$Q_1:=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{123}\arrow[dd]\arrow[rr]\arrow[ld]&&T_{13}\arrow[dd]\arrow[ld]\\
T_{23}\arrow[dd]\arrow[rr,crossing over]&&T_3\\
&T_{12}\arrow[rr]\arrow[ld]&&T_1\arrow[ld]\\
T_2\arrow[rr]&&T\arrow[uu,crossing over,leftarrow]
\end{tikzcd}
\quad
Q_2:=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{1234}\arrow[dd]\arrow[rr]\arrow[ld]&&T_{134}\arrow[dd]\arrow[ld]\\
T_{234}\arrow[dd]\arrow[rr,crossing over]&&T_{3}\\
&T_{124}\arrow[rr]\arrow[ld]&&T_{14}\arrow[ld]\\
T_{24}\arrow[rr]&&T\arrow[uu,crossing over,leftarrow]
\end{tikzcd}$$ $$Q_3:=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{34}\arrow[dd]\arrow[rr]\arrow[ld]&&T_{34}\arrow[dd]\arrow[ld]\\
T_{34}\arrow[dd]\arrow[rr,crossing over]&&T_{3}\\
&T_{4}\arrow[rr]\arrow[ld]&&T_4\arrow[ld]\\
T_4\arrow[rr]&&T\arrow[uu,crossing over,leftarrow]
\end{tikzcd}
\quad
Q_2':=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{1235}\arrow[dd]\arrow[rr]\arrow[ld]&&T_{135}\arrow[dd]\arrow[ld]\\
T_{235}\arrow[dd]\arrow[rr,crossing over]&&T_{35}\\
&T_{125}\arrow[rr]\arrow[ld]&&T_{1}\arrow[ld]\\
T_{25}\arrow[rr]&&T\arrow[uu,crossing over,leftarrow]
\end{tikzcd}$$ $$Q_3':=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{15}\arrow[dd]\arrow[rr]\arrow[ld]&&T_{15}\arrow[dd]\arrow[ld]\\
T_{5}\arrow[dd]\arrow[rr,crossing over]&&T_{5}\\
&T_{15}\arrow[rr]\arrow[ld]&&T_{1}\arrow[ld]\\
T_{5}\arrow[rr]&&T\arrow[uu,crossing over,leftarrow]
\end{tikzcd}
\quad
Q:=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{123456}\arrow[dd]\arrow[rr]\arrow[ld]&&T_{13456}\arrow[dd]\arrow[ld]\\
T_{23456}\arrow[dd]\arrow[rr,crossing over]&&T_{356}\\
&T_{12456}\arrow[rr]\arrow[ld]&&T_{1456}\arrow[ld]\\
T_{2456}\arrow[rr]&&T\arrow[uu,crossing over,leftarrow]
\end{tikzcd}$$ $$Q':=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{6}\arrow[dd]\arrow[rr]\arrow[ld]&&T_{6}\arrow[dd]\arrow[ld]\\
T_{6}\arrow[dd]\arrow[rr,crossing over]&&T_{6}\\
&T_{6}\arrow[rr]\arrow[ld]&&T_{6}\arrow[ld]\\
T_{6}\arrow[rr]&&T.\arrow[uu,crossing over,leftarrow]
\end{tikzcd}$$ Then form the $2$-squares $$C_1:=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&
T_{34}\arrow[ld]\arrow[dd]
\\
T_3\arrow[dd]
\\
&
T_4\arrow[ld]
\\
T
\end{tikzcd}
\quad
C_2:=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&
T_{346}\arrow[ld]\arrow[dd]
\\
T_{36}\arrow[dd]
\\
&
T_{46}\arrow[ld]
\\
T
\end{tikzcd}
\quad
C_3:=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&
T_{6}\arrow[ld]\arrow[dd]
\\
T_6\arrow[dd]
\\
&
T_6\arrow[ld]
\\
T
\end{tikzcd}
\quad
C:=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&
T_{13456}\arrow[ld]\arrow[dd]
\\
T_{356}\arrow[dd]
\\
&
T_{1456}\arrow[ld]
\\
T
\end{tikzcd}$$ $$C_1':=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{15}\arrow[ld]\arrow[rr]&&T_1\arrow[ld]
\\
T_5\arrow[rr]&&T
\end{tikzcd}
\quad
C_2':=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{156}\arrow[ld]\arrow[rr]&&T_{16}\arrow[ld]
\\
T_{56}\arrow[rr]&&T
\end{tikzcd}$$ $$C_3':=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{6}\arrow[ld]\arrow[rr]&&T_{6}\arrow[ld]
\\
T_{6}\arrow[rr]&&T
\end{tikzcd}
\quad
C':=
\begin{tikzcd}[row sep=tiny, column sep=tiny]
&T_{12456}\arrow[ld]\arrow[rr]&&T_{1456}\arrow[ld]
\\
T_{2456}\arrow[rr]&&T.
\end{tikzcd}$$
From the above there is a naturally induced commutative diagram $$\begin{tikzcd}
M(Q_1)\arrow[r,leftarrow,"(a)"]\arrow[d,leftarrow,"(a)"']&
M(Q_2)\arrow[d,leftarrow,"(c)"]\arrow[r,"(b)"]&
M(Q_3)\arrow[r,leftarrow,"(a)"]&
M(C_1)\arrow[d,leftarrow,"(a)"]
\\
M(Q_2')\arrow[d,"(b)"']\arrow[r,leftarrow,"(c)"]&
M(Q)\arrow[d,leftarrow,"(e)"]\arrow[r,leftarrow,"(e)"]\arrow[rd]&
M(C)\arrow[r,"(d)"]&
M(C_2)\arrow[d,"(b)"]
\\
M(Q_2')\arrow[d,leftarrow,"(a)"']&
M(C')\arrow[d,"(d)"]&
M(Q')\arrow[d,leftarrow,"(f)"]\arrow[r,leftarrow,"(f)"]&
M(C_3)\arrow[d,leftarrow,"(a)"]
\\
M(C_1')\arrow[r,leftarrow,"(a)"]&
M(C_2')\arrow[r,"(b)"]&
M(C_3')\arrow[r,leftarrow,"(a)"]&
M(T_6\rightarrow T).
\end{tikzcd}$$ The morphisms labeled $(a)$ and $(b)$ are isomorphisms, see the proof of Proposition \[A.3.42\]. Moreover, after inverting all the morphisms with label $(b)$, the outer diagram can be identified with by means of (MSq). The morphisms with labels $(c)$ and $(d)$ are isomorphisms by Proposition \[A.3.48\], Theorem \[A.3.7\], and the discussion about admissible blow-ups along smooth centers . It follows that the morphisms labeled $(e)$ are also isomorphisms.
Ley $C''$ denote the square $$C'':=
\begin{tikzcd}
T_6\arrow[d]\arrow[r]&
T_6\arrow[d]
\\
T_6\arrow[r]&
T_6.
\end{tikzcd}$$ By (Sq) there are distinguished triangles $$\begin{gathered}
M(C'')\rightarrow M(C_3)\rightarrow M(Q')\rightarrow M(C'')[1],
\;
M(C'')\rightarrow M(C_3')\rightarrow M(Q')\rightarrow M(C'')[1],
\\
M(T_6\rightarrow T_6)\rightarrow M(T_6\rightarrow T_6)\rightarrow M(C'')\rightarrow M(T_6\rightarrow T_6)[1].\end{gathered}$$ Since $M(T_6\rightarrow T_6)=0$ by Proposition \[A.3.45\], it follows that $M(C'')=0$. Thus the morphisms labeled $(f)$ are isomorphisms. The diagram remains commutative after inverting all morphisms with labels $(a)$, $(c)$, $(e)$, and $(f)$. This means that also the diagram commutes.
\[A.3.43\] For every $X\in SmlSm/S$ and integer $n\geq 1$, there is a canonical isomorphism $$MTh_X(X\times \A^n)\cong M(X)(n)[2n].$$
Owing to Proposition \[A.3.34\] there are isomorphisms $$MTh_X(X\times \A^1)\cong M(X)(1)[2],
\;
MTh_S(S\times \A^1)\cong M(S)(1)[2].$$ Then apply Proposition \[A.3.42\] to obtain an isomorphism $$MTh_X(X\times \A^r)\otimes MTh_S(S\times \A^1)\cong MTh_X(X\times \A^{r+1})$$ for every integer $r\geq 1$. Induction on $n$ concludes the proof.
\[rmk:why-3-bundles-necessary\] We can also produce an isomorphism $$MTh_X(X\times \A^n)\xrightarrow{\cong} M(X)(n)[2n]$$ by applying Proposition \[A.3.42\] in a different order, e.g., as in $$\begin{aligned}
& MTh(X\times \A^1)\otimes( MTh_S(S\times \A^1)\otimes MTh_S(S\times \A^1) )
\\
\rightarrow &
MTh(X\times \A^1)\otimes MTh_S(S\times \A^2)
\rightarrow
MTh_X(X\times \A^3).\end{aligned}$$ Proposition \[A.3.46\] ensures that the isomorphism $MTh_X(X\times \A^n)\cong M(X)(n)[2n]$ is indeed independent of choosing such an order.
For every $X\in SmlSm/S$ and integer $n\geq 1$, there is a canonical isomorphism $$M(\P_X^{n-1}\rightarrow \P_X^{n})\cong M(X)(n)[2n].$$
By (MSq) there is a canonical isomorphism $$M(X)\otimes M(\P_S^{n-1}\rightarrow \P_S^n)\xrightarrow{\cong} M(\P_X^{n-1}\rightarrow \P_X^n).$$ Thus we may reduce to the case when $X=S$, and conclude by applying Propositions \[A.3.34\] and \[A.3.43\].
This corollary is analogous to the isomorphism $$\P^n/\P^{n-1}\simeq T^n$$ in $\A^1$-homotopy theory of schemes [@MV Corollary 2.18, p. 112].
Gysin triangles {#sec:Gysin}
---------------
Throughout this section we shall assume $\cT$ satisfies the properties [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and [($div$-des)]{}. In $\dmeff$, recall from [@MVW Theorem 15.15] that for any smooth scheme $X$ with a smooth closed subscheme $Z$ with codimension $c$, there is the Gysin isomorphism $$M((X-Z)\rightarrow X)\cong M(Z)(c)[2c].$$ This does not hold in $\ldmeff$. For example, if $X=\P^1$ and $Z=\{\infty\}$, then the said Gysin isomorphism would imply $$M(\A^1\rightarrow \P^1)\cong \Lambda(1)[2].$$ It follows that $M(\A^1)\cong \Lambda$, so that $\A^1$ is invertible in $\ldmeff$. This is evidently a contradiction since the object $\mathbb{G}_a\otimes \Lambda$ is in $\ldmeff$, but $$\hom_{\ldmeff}(M(\A^1),\mathbb{G}_a\otimes \Lambda)\not\cong \hom_{\ldmeff}(M(k),\mathbb{G}_a\otimes \Lambda),$$ see Theorem \[thmHodge\].
Our goal in this section is to formulate a Gysin isomorphism in $\ldmeff$. As for the construction of Thom motives performed in the previous section, we need to “compactify” $X-Z$ without changing the “homotopy type” of $X-Z$. This can be done by taking the blow-up $B_Z X$ of $X$ along $Z$ and add the log structure given by the exceptional divisor $E$ on $B_Z X$. In this case, our formulation of the Gysin isomorphism should take the form $$\mathfrak{g}_{X,Z}:M((B_Z X,E)\rightarrow X)\stackrel{\cong}\rightarrow MTh(N_ZY).$$ Here $MTh(N_Z Y)$ is the Thom motive associated with the normal bundle $N_ZY$. We shall formulate the Gysin isomorphism more generally in Construction \[A.3.37\] allowing nontrivial log structures on $X$.
Recall that the technique of deformation to the normal cone is used in the proof of purity for $\A^1$-homotopy theory of schemes [@MV Theorem 2.23, p. 115]. We will develope a log version of the said technique based on our unit interval $\boxx$.
Let $X\in Sm/S$, and let $Z$ be a smooth closed subscheme of $X$. We have the commutative diagram of deformation to the normal cone $$\label{A.3.15.1}
\begin{tikzcd}
Z\arrow[d]\arrow[r]&Z\times \boxx\arrow[d]\arrow[r,leftarrow]&Z\arrow[d]\\
X\arrow[d]\arrow[r]&D_Z X\arrow[d]\arrow[r,leftarrow]&N_ZX\arrow[d]\\
\{1\}\arrow[r,"i_1"]&\boxx\arrow[r,"i_0",leftarrow]&\{0\}
\end{tikzcd}$$ where
1. each square is cartesian,
2. $i_0$ is the $0$-section, and $i_1$ is the $1$-section,
3. $Z\rightarrow N_ZX$ is the $0$-section,
4. $D_ZX\rightarrow \boxx$ is the composition $D_ZX\rightarrow B_Z(X\times \boxx)\rightarrow X\times \boxx\rightarrow \boxx$ where the third arrow is the projection.
Suppose $Z_1,\ldots,Z_r$ form a strict normal crossing divisor on $X$, and that $Z$ has strict normal crossing with $Z_1+\cdots+Z_r$ over $S$. For $Y:=(X,Z_1+\cdots+Z_r)$ we obtain the commutative diagram $$\label{A.3.15.2}
\begin{tikzcd}
B_Z Y\arrow[d]\arrow[r]&B_{Z\times \boxx}(D_Z Y)\arrow[d]\arrow[r,leftarrow]&B_Z(N_ZY)\arrow[d]\\
Y\arrow[d]\arrow[r]&D_Z Y\arrow[d]\arrow[r,leftarrow]&N_ZY\arrow[d]\\
\{1\}\arrow[r,"i_1"]&\boxx\arrow[r,"i_0",leftarrow]&\{0\}.
\end{tikzcd}$$
A difference between the technique of deformation to the normal cone used in the proof of [@MV Theorem 2.23, p. 115] and ours is that we use $X\times \boxx$ instead of $X\times \A^1$.
\[A.3.37\] We are ready to formulate and prove our purity theorem for log motives. With the above definitions, let $E$ (resp. $E^D$, $E^N$) be the exceptional divisor on $B_ZY$ (resp. $B_{Z\times \boxx}(D_Z Y)$, resp. $B_Z(N_ZY)$). In the case $Z$ has codimension $1$ in $Y$, we have $E=Z$ (resp. $E^D=Z\times \boxx$, resp. $E^N=Z$) by convention. From the diagram , we have the naturally induced morphisms $$M((B_ZY,E)\rightarrow Y)\rightarrow M((B_{Z\times \boxx}(D_Z Y),E^D)\rightarrow D_Z Y),$$ $$M((B_Z(N_Z Y),E^N)\rightarrow N_Z Y)\rightarrow M((B_{Z\times \boxx}(D_Z Y),E^D)\rightarrow D_Z Y).$$ If these morphisms are isomorphisms, then via a zig-zag we obtain the isomorphism $$\mathfrak{g}_{Y,Z}:M((B_Z Y,E)\rightarrow Y)\rightarrow MTh(N_Z Y)$$ that is refered to as the Gysin isomorphism for the closed pair $(X,Z)$. This will be the log analogue to [@MV Theorem 2.23].
\[A.3.36\] Suppose that $Z_1,Z_2,\ldots,Z_r$ are smooth divisors forming a strict normal crossing divisor on $X\in Sm/S$ and that $Z$ is a smooth closed subscheme of $X$ having strict normal crossing with $Z_1+ \cdots + Z_r$ over $S$ such that $Z$ is not contained in any component of $Z_1\cup \cdots\cup Z_r$. Let $E$ (resp. $E^D$, resp. $E^N$) be the exceptional divisor on $B_Z X$ (resp. $B_{Z\times \boxx}(D_Z X)$, resp. $B_Z(N_Z X)$). For $Y:=(X,Z_1+\cdots+Z_r)$ the morphisms $$M((B_Z Y,E)\rightarrow Y)\rightarrow M(B_{Z\times \boxx}(D_ZY),E^D)\rightarrow D_ZY),$$ $$M((B_Z(N_ZY),E^N)\rightarrow N_ZY)\rightarrow M(B_{Z\times \boxx}(D_ZY),E^D)\rightarrow D_ZY).$$ induced by are isomorphisms and functorial in the pair $(Y,Z)$.
[**Step 1**]{} We proceed as in the proof of Theorem \[A.3.7\], using a local parametrization of $(X,Z)$. For $i=1, \ldots, r$, consider the divisor $$W_i:= \A^{i-1}\times \Spec{\Z}\times \A^{r+s-i}$$ on $\A^{r+s}$, and let $W$ denote the subscheme $\A^r\times (\Spec{\Z})^p \times \A^{s-p}$. The question is Zariski local on $X$ by ($Zar$-sep), and $Z$ is not contained in any component of $Z_1\cup \cdots\cup Z_r$, so we may assume there exists an étale morphism $u:X\rightarrow \A_S^{r+s}$ of schemes over $S$ for which $u^{-1}(S\times W_i)=Z_i$ for $1\leq i\leq r$ and $u^{-1}(S\times W)=Z$.
[**Step 2**]{} Using $u$ and $Z\rightarrow X$, Construction \[A.3.16\] produces schemes $X_j$ over $S$ for $j=1,2$. Form $Y_j$ from $X_j$ in the same way as we obtained $Y$ from $X$, and let $E_j$ (resp. $E$) be the exceptional divisor on $B_Z Y_j$ (resp. $B_ZY$).
[**Step 3**]{} Since blow-ups commute with flat base change, we have the cartesian square $$\begin{tikzcd}
D_Z X_2\arrow[d]\arrow[r]&X_2\times\boxx\arrow[d]\\
D_Z X\arrow[r]&X\times \boxx
\end{tikzcd}$$ of fs log schemes over $S$. It follows that $D_ZX_2\rightarrow D_ZX$ is étale. Let $E_2^D$ (resp. $E_2^N$) be the exceptional divisor on $B_{Z\times \boxx}(D_Z Y_2)$ (resp. $B_{Z}(N_Z Y_2)$). Since $X_2\rightarrow X$ and $N_ZX_2\rightarrow N_ZX$ are also étale, Proposition \[A.3.3\] implies there are isomorphisms $$M((B_{Z}Y_2,E_2)\rightarrow Y_2)\xrightarrow{\cong} M((B_ZY,E)\rightarrow Y),$$ $$M((B_{Z\times \boxx}(D_{Z}Y_2),E_2^D)\rightarrow D_ZY_2)\xrightarrow{\cong} M((B_{Z\times \boxx}(D_ZY),E^D)\rightarrow D_ZY),$$ and $$M((B_Z(N_{Z}Y_2),E_2^N)\rightarrow N_ZY_2)\xrightarrow{\cong} M((B_Z(N_{Z}Y),E^N)\rightarrow N_ZY).$$ Hence we can replace $X$ by $X_2$. Similarly, we can replace $X_2$ by $X_1$. This reduces to the situation when $X=T\times \A^{r+s},$ $Z_i=T\times W_i$ for $1\leq i\leq r$, and $Z=T\times W$ for some $T\in Sm/S$.
[**Step 4**]{} Let $F$ (resp. $F^D$, resp. $F^N$) be the exceptional divisor on $B_{\{0\}}(\A^p)$ (resp. $B_{\{0\}\times \boxx}(D_{\{0\}}(\A_S^p))$, resp. $B_{\{0\}\times \boxx}(N_{\{0\}}(\A_S^p))$). Setting $V:=(Z,Z\cap Z_1+\cdots+Z\cap Z_r)$ we observe that $$V=(T\times W,T\times (W_1\cap W)+\cdots+T\times (W_r\cap W))\cong T\times \A_{\N}^r\times (\Spec {\Z})^{p}\times \A^{s-p},$$ $$(B_ZY,E)=(T\times B_{W}(\A^{r+s}),T\times W_1+\cdots +T\times W_{r})\cong V\times (B_{\{0\}}(\A^p),F).$$ and $$Y=(T \times \A^{r+s},T\times W_1+\cdots+T\times W_r)\cong V\times \A^p.$$ By (MSq), it remains to show that induces isomorphisms $$\begin{split}
&M(V)\otimes M((B_{\{0\}}(\A_S^p),F)\rightarrow \A_S^p)\\
\xrightarrow{\cong} &M(V)\otimes M((B_{\{0\}\times \boxx}(D_{\{0\}}(\A_S^p),F^D)\rightarrow D_{\{0\}}(\A_S^p),
\end{split}$$ $$\begin{split}
& M(V)\otimes M((B_{\{0\}}(N_{\{0\}}(\A_S^p)),F^N)\rightarrow N_{\{0\}}(\A_S^p))\\
\xrightarrow{\cong} & M(V)\otimes M((B_{\{0\}}(D_{\{0\}}(\A_S^p)),F^D)\rightarrow D_{\{0\}}(\A_S^p).
\end{split}$$
[**Step 5**]{} Due to Step 4 we may assume that $X=\A_S^p$, $r=0$, and $Z=\{0\}\times S$. We set $S=\Spec{\Z}$ for simplicity. Let $e_1,\ldots,e_p$ denote the standard coordinates in $\Z^p$. There are corresponding cones $$\begin{gathered}
\tau:={\rm Cone}(e_1,\ldots,e_{p-1}),
\;
\sigma_1:={\rm Cone}(e_1,\ldots,e_{p-1},e_1+\cdots+e_p),
\\
\sigma_2:={\rm Cone}(e_1,\ldots,e_{p-1},-e_p),
\;
\sigma_3:={\rm Cone}(e_1,\ldots,e_{p-1},-e_1-\cdots-e_p),\end{gathered}$$ and their fans $$\Sigma_1:=\{\sigma_1,\sigma_2\},
\;
\Sigma_2:=\{\sigma_1,\sigma_3\},
\;
\Sigma_1':=\{{\rm Cone}(-e_p)\},
\;
\Sigma_2':=\{{\rm Cone}(-e_1-\cdots-e_p)\}.$$ Moreover, we shall cosider the star subdivisions $$\Sigma_3:=\Sigma_2^*(\sigma_3),
\;
\Sigma_4:=\Sigma_1^*(\tau),
\;
\Sigma_5:=\Sigma_2^*(\tau),
\;
\Sigma_6:=\Sigma_3^*(\tau),$$ and the subfans $$\begin{gathered}
\Sigma_3':=\{{\rm Cone}(-e_p,-e_1-\cdots-e_p)\},
\;
\Sigma_4':=\{{\rm Cone}(e_1+\cdots+e_{p-1},-e_p)\},
\\
\Sigma_5':=\{{\rm Cone}(e_1+\cdots+e_{p-1},-e_1-\cdots-e_p)\},
\\
\Sigma_6':=\{{\rm Cone}(e_1+\cdots+e_{p-1},-e_p),{\rm Cone}(-e_p,-e_1-\cdots-e_p)\}.\end{gathered}$$
For $1\leq i\leq p-1$, we set $$\eta_i:={\rm Cone}(e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_{p-1},-e_p,-e_1-\cdots-e_p),$$ and form the fans $$\Sigma_7:=\{\eta_1,\ldots,\eta_{p-1}\},
\;
\Sigma_7':=\Sigma_3'.$$
When $p=3$, the above fans take the following forms. $$\begin{tikzpicture}[yscale=0.7, xscale=0.82]
\fill[black!20] (0,0)--(1,1)--(2,0)--(0,0);
\fill[black!20] (0,0)--(2,0)--(1,-1)--(0,0);
\draw (0,0)--(1,1)--(2,0)--(0,0);
\draw (0,0)--(1,-1)--(2,0);
\node [left] at (0,0) {$e_1$};
\node [right] at (2,0) {$e_2$};
\node [above] at (1,1) {$e_1+e_2+e_3$};
\node [below] at (1,-1) {$-e_3$};
\node [below] at (1,-2.1) {$(\Sigma_1,\Sigma_1')$};
\filldraw (1,-1) circle (3pt);
\begin{scope}[shift={(5,0)}]
\fill[black!20] (0,0)--(1,1)--(2,0)--(0,0);
\fill[black!20] (0,0)--(2,0)--(1,-2)--(0,0);
\draw (0,0)--(1,1)--(2,0)--(0,0);
\draw (0,0)--(1,-2)--(2,0);
\filldraw (1,-2) circle (3pt);
\node [right] at (1,-2) {$-e_1-e_2-e_3$};
\node [below] at (1,-2.1) {$(\Sigma_2,\Sigma_2')$};
\end{scope}
\begin{scope}[shift={(10,0)}]
\fill[black!20] (0,0)--(1,1)--(2,0)--(0,0);
\fill[black!20] (0,0)--(2,0)--(1,-2)--(0,0);
\draw (0,0)--(1,1)--(2,0)--(0,0);
\draw (0,0)--(1,-1)--(2,0);
\draw[ultra thick] (1,-1)--(1,-2);
\draw (0,0)--(1,-2)--(2,0);
\filldraw (1,-2) circle (3pt);
\filldraw (1,-1) circle (3pt);
\node [below] at (1,-2.1) {$(\Sigma_3,\Sigma_3')$};
\end{scope}
\end{tikzpicture}$$ $$\begin{tikzpicture}[yscale=0.7, xscale=0.82]
\fill[black!20] (0,0)--(1,1)--(2,0)--(0,0);
\fill[black!20] (0,0)--(2,0)--(1,-1)--(0,0);
\draw (0,0)--(1,1)--(2,0)--(0,0);
\draw (0,0)--(1,-1)--(2,0);
\draw[ultra thick] (1,0)--(1,-1);
\filldraw (1,0) circle (3pt);
\filldraw (1,-1) circle (3pt);
\node [below] at (1,-2.1) {$(\Sigma_4,\Sigma_4')$};
\begin{scope}[shift={(4,0)}]
\fill[black!20] (0,0)--(1,1)--(2,0)--(0,0);
\fill[black!20] (0,0)--(2,0)--(1,-2)--(0,0);
\draw (0,0)--(1,1)--(2,0)--(0,0);
\draw (0,0)--(1,-2)--(2,0);
\draw[ultra thick] (1,0)--(1,-2);
\filldraw (1,0) circle (3pt);
\filldraw (1,-2) circle (3pt);
\node [below] at (1,-2.1) {$(\Sigma_5,\Sigma_5')$};
\end{scope}
\begin{scope}[shift={(8,0)}]
\fill[black!20] (0,0)--(1,1)--(2,0)--(0,0);
\fill[black!20] (0,0)--(2,0)--(1,-2)--(0,0);
\draw (0,0)--(1,1)--(2,0)--(0,0);
\draw (0,0)--(1,-1)--(2,0);
\draw[ultra thick] (1,0)--(1,-2);
\draw (0,0)--(1,-2)--(2,0);
\filldraw (1,-2) circle (3pt);
\filldraw (1,-1) circle (3pt);
\filldraw (1,0) circle (3pt);
\node [below] at (1,-2.1) {$(\Sigma_6,\Sigma_6')$};
\end{scope}
\begin{scope}[shift={(12,0)}]
\fill[black!20] (0,0)--(1,-1)--(2,0)--(1,-2);
\draw (0,0)--(1,-1)--(2,0);
\draw[ultra thick] (1,-1)--(1,-2);
\draw (0,0)--(1,-2)--(2,0);
\filldraw (1,-2) circle (3pt);
\filldraw (1,-1) circle (3pt);
\node [below] at (1,-2.1) {$(\Sigma_7,\Sigma_7')$};
\end{scope}
\end{tikzpicture}$$
Observe that $$D_ZY=\A_{(\Sigma_1,\Sigma_1')},
\;
(B_{Z\times \boxx}(D_Z Y),E^D)=\A_{(\Sigma_4,\Sigma_4')}.$$ The fan $\{\sigma_1\}$ corresponds to the trivial line bundle $Y\times \A^1$, and the $0$-section (resp. $1$-section) of the bundle is $Y$ (resp. $N_ZY$). Hence there are naturally induced commutative diagrams $$\begin{tikzcd}
&
\A_{(\Sigma_1,\Sigma_1')}\arrow[d]&
\\
Y\arrow[r]\arrow[ru]\arrow[rd]&
\A_{(\Sigma_3,\Sigma_3')}\arrow[d]&
N_ZY\arrow[l]\arrow[lu]\arrow[ld]
\\
&
\A_{(\Sigma_2,\Sigma_2')},
\end{tikzcd}
\;\;
\begin{tikzcd}
&
\A_{(\Sigma_4,\Sigma_4')}\arrow[d]&
\\
(B_Z Y,E)\arrow[r]\arrow[ru]\arrow[rd]&
\A_{(\Sigma_6,\Sigma_6')}\arrow[d]&
(B_Z(N_Z Y),E^N)\arrow[l]\arrow[lu]\arrow[ld]
\\
&
\A_{(\Sigma_5,\Sigma_5')}.
\end{tikzcd}$$
Due to these diagrams we are reduced to showing the naturally induced morphisms $$\begin{gathered}
\label{A.3.36.1}
M(\A_{(\Sigma_4,\Sigma_4')}\rightarrow \A_{(\Sigma_1,\Sigma_1')})\rightarrow M(\A_{(\Sigma_6,\Sigma_6')}\rightarrow \A_{(\Sigma_3,\Sigma_3')}),
\\
\label{A.3.36.2}
M(\A_{(\Sigma_6,\Sigma_6')}\rightarrow \A_{(\Sigma_3,\Sigma_3')})\rightarrow M(\A_{(\Sigma_5,\Sigma_5')}\rightarrow \A_{(\Sigma_2,\Sigma_2')}),
\\
\label{A.3.36.3}
M((B_Y Z,E)\rightarrow Y)\rightarrow M(\A_{(\Sigma_5,\Sigma_5')}\rightarrow \A_{(\Sigma_2,\Sigma_2')}),
\\
\label{A.3.36.4}
M((B_Z(N_Z Y),E^N)\rightarrow N_ZY)\rightarrow M(\A_{(\Sigma_5,\Sigma_5')}\rightarrow \A_{(\Sigma_2,\Sigma_2')})\end{gathered}$$ are isomorphisms.
The naturally induced morphisms $$\A_{(\Sigma_6,\Sigma_6')}\rightarrow \A_{(\Sigma_5,\Sigma_5')},
\;
\A_{(\Sigma_3,\Sigma_3')}\rightarrow \A_{(\Sigma_2,\Sigma_2')}$$ are admissible blow-ups along $$V({\rm Cone}(e_1+\cdots+e_{p-1},-e_1-\cdots-e_p)),
\;
V(\sigma_3).$$ Theorem \[A.3.7\] shows that $$M(\A_{(\Sigma_3,\Sigma_3')}\rightarrow \A_{(\Sigma_2,\Sigma_2')})=0,
\;
M(\A_{(\Sigma_6,\Sigma_6')}\rightarrow \A_{(\Sigma_5,\Sigma_5')})=0.$$ Proposition \[A.3.44\] allows us to deduce that is an isomorphism.
We can identify $\A_{(\Sigma_2,\Sigma_2')}$ and $\A_{(\Sigma_5,\Sigma_5')}$ with $$Y\times \boxx,
\;
(B_Z Y,E)\times \boxx.$$ The fibers at $0\in \boxx$ (resp. $1\in \boxx$) correspond to $N_ZY$ and $(B_Z(N_ZY),E^N)$ (resp. $Y$ and $(B_ZY,E)$). Thus ($\boxx$-inv) shows that and are isomorphisms.
There are strict Zariski distinguished squares $$\begin{tikzcd}
\A_{(\Sigma_1,\Sigma_1')}\cap \A_{(\Sigma_7,\Sigma_7')}\arrow[r]\arrow[d]& \A_{(\Sigma_7,\Sigma_7')}\arrow[d]
\\
\A_{(\Sigma_1,\Sigma_1')}\arrow[r]& \A_{(\Sigma_3,\Sigma_3')},
\end{tikzcd}
\quad
\begin{tikzcd}
\A_{(\Sigma_4,\Sigma_4')}\cap \A_{(\Sigma_7,\Sigma_7')}\arrow[r]\arrow[d]& \A_{(\Sigma_7,\Sigma_7')}\arrow[d]
\\
\A_{(\Sigma_4,\Sigma_4')}\arrow[r]& \A_{(\Sigma_6,\Sigma_6')}.
\end{tikzcd}$$ Since $$\A_{(\Sigma_1,\Sigma_1')}\cap \A_{(\Sigma_7,\Sigma_7')}=\A_{(\Sigma_4,\Sigma_4')}\cap \A_{(\Sigma_7,\Sigma_7')},$$ we can use ($sNis$-des) to obtain the isomorphism $$M(\A_{(\Sigma_4,\Sigma_4')}\rightarrow \A_{(\Sigma_6,\Sigma_6')})\xrightarrow{\cong} M(\A_{(\Sigma_1,\Sigma_1')}\rightarrow \A_{(\Sigma_3,\Sigma_3')}).$$ Combined with Proposition \[A.3.44\] we deduce that is an isomorphism, which finishes the proof. Moreover, functoriality follows from the construction.
\[A.3.38\] With reference to Theorem \[A.3.36\], the [*Gysin isomorphism*]{} is the isomorphism $$\mathfrak{g}_{Y,Z}:M((B_ZY,E)\rightarrow Y)\rightarrow MTh(N_ZY),$$
Invariance under admissible blow-ups {#ssec:invadmblowrefined}
------------------------------------
In this section, we assume that $\cT$ satisfies the properties [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and [($div$-des)]{}, $S=\Spec k$, and $k$ admits resolution of singularities. Under these hypothesis we prove that motives are invariant under admissible blow-ups. This invariance will be used for comparing $\ldmeff$ with $\dmeff$. We start by introducing the following (weaker) version of the notion of admissible blow-up in Definition \[A.3.17\].
\[A.3.1\] An [*admissible blow-up*]{} $Y\rightarrow X$ in $\mathscr{S}/S$ is a proper birational morphism of fs log schemes such that the induced morphism $$Y-\partial Y\rightarrow X-\partial X$$ of schemes is an isomorphism. We denote by $\cA\cB l/S$ the class of admissible blow-ups.
Note that $\cA\cB l_{Sm}/S$ and $\cA\cB l_{div}/S$ are subclasses of $\cA\cB l/S$.
The class $\cA\cB l/S$ is equivalent to the class $\cS_b$ in [@AdditiveHomotopy] if $S=\Spec k$ and $\mathscr{S}/S=SmlSm/k$.
\[A.3.8\] The field $k$ admits [*resolution of singularities*]{} if the following conditions hold.
1. For any integral scheme $X$ of finite type over $k$, there is a proper birational morphism $Y\rightarrow X$ of schemes over $k$ such that $Y$ is smooth over $k$.
2. Let $f:Y\rightarrow X$ be a proper birational morphism of integral schemes over $k$ such that $X$ is smooth over $k$, and let $Z_1,\ldots,Z_r$ be smooth divisors forming a strict normal crossing divisor on $X$. Assume that $$f^{-1}(X-Z_1\cup \cdots\cup Z_r)\rightarrow X-Z_1\cup \cdots\cup Z_r$$ is an isomorphism. Then there is a sequence of blow-ups $$X_n\stackrel{f_{n-1}}\rightarrow X_{n-1}\stackrel{f_{n-2}}\rightarrow \cdots\stackrel{f_0}\rightarrow X_0=X$$ along smooth centers $W_i\subset X_i$ such that
1. the composition $X_n\rightarrow X$ factors through $f$,
2. $W_i$ is contained in the preimage of $Z_1\cup\cdots\cup Z_r$ in $X_i$,
3. $W_i$ has strict normal crossing with the sum of the strict transforms of $$Z_1,\ldots,Z_r,f_0^{-1}(W_0),\ldots,f_{i-1}^{-1}(W_{i-1})$$ in $X_i$.
Note that $k$ must be perfect in order to satisfy (1). If $k$ has characteristic $0$, then (1) (resp. (2)) is satisfied by [@Hir Main theorem I] (resp. [@Hir Main theorem II]).
\[A.3.2\] Assume that $k$ admits resolution of singularities. Then the class $\cA\cB l_{div}/k$ admits a calculus of right fractions.
By definition, the class in question contains isomorphisms and is closed under composition. If $f,g:X\rightarrow Y$ and $s:Y\rightarrow Z$ are morphisms of fs log schemes log smooth over $k$ such that $s\circ f=s\circ g$ and $s\in \cA\cB l_{div}/k$, then $f$ and $g$ agree on the dense open subscheme $X-\partial X$ of $X$ since the morphism $Y-\partial Y\rightarrow Z-\partial Z$ induced by $s$ is an isomorphism. Thus $f=g$ by Lemma \[A.5.9\] and the right cancellabilty condition holds.
It remains to check the right Ore condition. Suppose $f:Y\rightarrow X$ and $g:X'\rightarrow X$ are morphisms where $f\in \cA\cB l_{div}/k$. By Proposition \[A.3.19\], we can choose a dividing cover $Y_1\rightarrow Y$ such that $\partial Y_1$ is a strict normal crossing divisor. Then by resolution of singularities, there is a morphism $Y'\rightarrow Y_1$ in $\cA\cB l_{div}/k$ that factors through the projection $Y_1\times_X X'$. Thus we have the commutative diagram $$\begin{tikzcd}
Y'\arrow[d]\arrow[r]&Y\arrow[d,"f"]\\
Y\arrow[r,"g"]&X.
\end{tikzcd}$$ Here the left vertical arrow is in $\cA\cB l_{div}/k$, so the right Ore condition is satisfied.
\[A.3.9\] Assume that $k$ admits resolution of singularities. For any morphism $f:Y\rightarrow X$ in $\cA\cB l/k$, there is a morphism $Y'\rightarrow X$ in $\cA\cB l_{div}/k$ that factors through $f$.
There exists a dividing cover $g:X_1\rightarrow X$ such that $\underline{X_1}$ is smooth over $k$ and $\partial X_1$ is a strict normal crossing divisor, see Proposition \[A.3.19\]. Applying part (2) of Definition \[A.3.8\] to the projection $p:X_1\times_X Y\rightarrow X_1$, we obtain a morphism $h:Y'\rightarrow X_1$ in $\cA \cB l_{Sm}/k$ that factors through $p$. Then the composition $gh$ is in $\cA \cB l_{div}/k$ and factors through $f$.
\[A.3.10\] Assume that $k$ admits resolution of singularities. Then $\cA\cB l/k$ admits a calculus of right fractions.
The right Ore condition follows from Propositions \[A.3.2\] and \[A.3.9\]. The proof of the right cancellability condition is similar to that in Proposition \[A.3.2\].
\[A.3.12\] Assume that $k$ admits resolution of singularities. Then any morphism $f:Y\rightarrow X$ in $\cA\cB l/k$ induces an isomorphism $$M(Y)\xrightarrow{\cong} M(X).$$
Propositions \[A.3.2\] and \[A.3.10\] show that $\cA=\cA\cB l/k$ and $\cB=\cA\cB l_{div}/k$ satisfy a calculus of right fractions. All the conditions in Lemma \[A.1.16\] are met by Propositions \[A.3.11\] and \[A.3.9\], so we are done.
\[A.3.20\] By Theorem \[A.3.12\], we can apply the results in \[section:calculus\] to $$\cA=\cA Bl/k;
\,
I=\boxx;
\,
T=lSm/k, Cor/k.$$
For example, for any $(I,\cA)$-weak equivalence $f:\cF\rightarrow \cG$ of bounded above complexes of presheaves (resp. presheaves with transfers) on $lSm/k$, the image of $f$ in $\ldaeff$ (resp. $\ldmeff$) is an isomorphism.
Invariance under log modifications {#ssec:invariance-asuming-PnPn-1}
----------------------------------
In this section we assume that $\mathscr{S}=SmlSm/S$ and $\cT$ satisfies the properties [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and ($(\P^\bullet,\P^{\bullet-1})$-inv). Under these hypothesis we show that the property ($div$-des) holds.
\[A.3.26\] Let $H_1$ and $H_2$ be the closed subschemes $\{0\}\times \A^1$ and $\A^1\times \{0\}$ of $\A^2$, let $H_1'$ and $H_2'$ be the strict transforms of $H_1$ and $H_2$ in ${\rm Bl}_{\{(0,0)\}}(\A^2)$, and let $E$ be the exceptional divisor on ${\rm Bl}_{\{(0,0)\}}(\A^2)$. Then for any $X\in \mathscr{S}/S$ there is a naturally induced isomorphism $$M(X\times ({\rm Bl}_{\{(0,0)\}}\A^2,H_1'+H_2'+E))\xrightarrow{\cong} M(X\times (\A^2,H_1+H_2)).$$
Assume that $S=X=\Spec{\Z}$ for simplicity of notation as usual. Due to Proposition \[A.3.39\] it suffices to show there is a naturally induced isomorphism $$M({\rm Bl}_{\{(0,0)\}}\A^2,H_1'+E)\xrightarrow{\cong} M(\A^2,H_1).$$ Let $\overline{H_1}$ and $\overline{H_2}$ be the closure of $H_1$ and $H_2$ in $\P^2$, and let $\overline{H_1'}$ and $\overline{H_2'}$ be the closures of $H_1'$ and $H_2'$ in ${\rm Bl}_{\{O\}}(\P^2)$, where $O:=[0:0:1]$. There are Zariski distinguished squares $$\begin{tikzcd}[column sep=small]
(\A^2,H_1)-\{O\}\arrow[d]\arrow[r]&
(\A^2,H_1)\arrow[d]
\\
(\P^2,\overline{H_1})-\{O\}\arrow[r]&
(\P^2,\overline{H_1})
\end{tikzcd}
\quad
\begin{tikzcd}[column sep=small]
(\A^2,H_1)-\{O\}\arrow[d]\arrow[r]&
({\rm Bl}_{\{(0,0)\}}\A^2,H_1'+E)\arrow[d]
\\
(\P^2,\overline{H_1})-\{O\}\arrow[r]&
({\rm Bl}_{\{O\}}\P^2,\overline{H_1'}+E).
\end{tikzcd}$$ Applying ($sNis$-des) to these squares yields an isomorphism $$M(({\rm Bl}_{\{(0,0)\}}\A^2,H_1'+E)\rightarrow ({\rm Bl}_{\{O\}}\P^2,\overline{H_1'}+E))
\xrightarrow{\cong}
M((\A^2,H_1)\rightarrow (\P^2,\overline{H_1}))$$ By Proposition \[A.3.44\] we find the isomorphism $$M(({\rm Bl}_{\{(0,0)\}}\A^2,H_1'+E)\rightarrow (\A^2,H_1))
\xrightarrow{\cong}
M(({\rm Bl}_{\{O\}}\P^2,\overline{H_1'}+E)\rightarrow (\P^2,\overline{H_1})).$$ Hence we only need to show that the morphism $$M({\rm Bl}_{\{O\}}\P^2,\overline{H_1'}+E)\rightarrow M(\P^2,\overline{H_1})$$ is an isomorphism. Since $({\rm Bl}_{\{O\}}\P^2,\overline{H_1'}+E))$ is a $\boxx$-bundle over $\boxx$ by Lemma \[A.3.28\], ($(\P^\bullet,\P^{\bullet-1})$-inv) gives $$M({\rm Bl}_{\{O\}}\P^2,\overline{H_1'}+E)\cong M(\Z)\cong M(\P^2,\overline{H_1}).$$
\[A.3.32\] For a number $r\in \N^+$, let ${\rm Fan}_r$ denote the category where objects are $r$-dimensional fans and morphisms are subdivisions. For an $r$-dimensional smooth fan $\Sigma$, let ${\rm Fan}_r^{sm}/\Sigma$ denote the full category of ${\rm Fan}_r/\Sigma$ consisting of smooth $r$-dimensions fans that are subdivisions of $\Sigma$. Let $\cS \cD/\Sigma$ be the class of morphisms in ${\rm Fan}_r/\Sigma$, and let $\cS \cD_2/\Sigma$ denote the class of morphisms in ${\rm Fan}_r^{sm}/\Sigma$ that are obtained by finite successions of star subdivisions relative to two-dimensional cones.
\[A.3.29\] Let $\Sigma$ be a fan. The class $\cS \cD/\Sigma$ admits a calculus of right fractions. If in addition $\Sigma$ is smooth, then the class $\cS \cD_2/\Sigma$ also admits a calculus of right fractions.
Let $r$ be the dimension of $\Sigma$. By definition, $\cS \cD_2/\Sigma$ and $\cS \cD_2/\Sigma$ are closed under compositions. For any objects $\Sigma_1$ and $\Sigma_2$ of ${\rm Fan}_r/\Sigma$ or ${\rm Fan}_r^{sm}/\Sigma$, there is at most one morphism from $\Sigma_2$ to $\Sigma_1$. Thus the right cancellabilty condition holds for $\cS \cD/\Sigma$ and $\cS \cD_2/\Sigma$. It remains to check the right Ore condition. Suppose that $f:\Sigma_2\rightarrow \Sigma_1$ and $g:\Sigma_1'\rightarrow \Sigma_1$ are in $\cS \cD/\Sigma$. Let $\Sigma_2'$ be the fan generated by $\Sigma_2$ and $\Sigma_1'$. We have the naturally induced commutative diagram of fans $$\begin{tikzcd}
\Sigma_2'\arrow[d,"f'"]\arrow[r]&\Sigma_2\arrow[d,"f"]\\
\Sigma_1'\arrow[r,"g"]&\Sigma_1.
\end{tikzcd}$$ This shows the Ore condition for $\cS \cD/\Sigma$.
Suppose that $\Sigma$, $\Sigma_1$, $\Sigma_1'$, and $\Sigma_2$ are smooth. The fan $\Sigma_2'$ is not necessarily smooth, but $\Sigma_2'$ has a subdivision $\Sigma_2''$ that is smooth by toric resolution of singularities, see [@CLStoric Theorem 11.1.9]. By Lemma \[A.3.31\], there exists a morphism $f':\Sigma_2'''\rightarrow \Sigma_1$ in $\cS \cD_2/\Sigma$ that factors through $$\Sigma_2''\rightarrow \Sigma_2'\stackrel{f'}\rightarrow \Sigma_1'.$$ This shows the right Ore condition for $\cS \cD_2/\Sigma$.
\[A.3.27\] Every log modification $f:Y\rightarrow X$ in $SmlSm/S$ induces an isomorphism $$M(f):M(Y)\xrightarrow{\cong} M(X).$$
By ($Zar$-sep), the question is Zariski local on $X$. Hence by definition, we may assume that there exists a morphism $g:Z\rightarrow Y$ such that both $fg$ and $g$ are log blow-ups. It suffices to show that $M(fg):M(Z)\rightarrow M(X)$ and $M(g):M(Z)\rightarrow M(X)$ are isomorphisms, so we reduce to the case when $f$ is a log blow-up.
The question is Zariski local on $X$, so we may assume that $X$ has an fs chart $P$ with $P\cong \N^r$ for some $r$ such that $X\rightarrow \A_P$ is smooth by Lemma \[lem::Smchart\]. By [@Ogu Lemma III.2.6.4], we may further assume that $f$ is the projection $X\times_{\A_P}\A_M\rightarrow X$ for some subdivision of fans $M\rightarrow \Spec{P}$. Consider the dual monoid $P^\vee$ as a fan, and consider $M$ as a fan that is a subdivision of $P^\vee$.
We have the functors $\cS \cD/P^\vee\rightarrow \cT$ and $\cS \cD_2/P^\vee\rightarrow \cT$ given by $$(N\in {\rm Fan}_r/P^{\vee}) \mapsto M(X\times_{\A_P}\A_{N}).$$ By Lemma \[A.3.29\], both $\cS \cD/P^\vee$ and $\cS \cD_2/P^\vee$ admit a calculus of right fractions. Owing to Lemma \[A.1.16\], to have the isomorphism $$M(f):M(X\times_{\A_P}\A_M)\xrightarrow{\cong} M(X).$$ Hence we are reduced to showing that $$M(g):M(X\times_{\A_P}\A_{N'})\rightarrow M(X\times_{\A_P}\A_N)$$ is an isomorphism for any morphism $N'\rightarrow N$ in $\cS \cD_2/P^\vee$. It suffices to consider the case when $N'$ is the star subdivision of $N$ relative to a $2$-dimensional cone. Using Zariski locality, we reduce to the case when $N$ is a $2$-dimensional cone and $N'$ is its star subdivision. Since $X$ is smooth over $\A_P$, we see that $X\times_{\A_P}\A_N$ is smooth over $\A_N$. Hence we can replace $X$ by $X\times_{\A_P}\A_N$ and $N'$ by $M$, so that it suffices to show $$M(f):M(X\times_{\A_P}\A_M)\rightarrow M(X)$$ is an isomorphism when $P=\N^2$ and $M$ is the star subdivision of $P^\vee$.
Next we set $$Z:=\underline{X}\times_{\underline{\A_{P}}}\underline{{\rm pt}_{P}}.$$ Since $X$ is smooth over its chart $\A_{P}$, Zariski locally on $X$, there is a strict étale morphism $X\rightarrow \A^n \times \A_{P}$ such that the composition with the projection $\A^n\times \A_{P}\rightarrow \A_{P}$ is equal to $X\rightarrow \A_P$. In this case, $(\underline{X},Z)$ has a parametrization $(\A^{n+2},\A^n)$ and Construction \[A.3.16\] produces schemes $\underline{X_j}$ for $j=1,2$. Give $\underline{X_2}$ the log structure induced by that of $X$, and we obtain an fs log scheme $X_2$. Set $X_1:=Z\times \A_P$, which gives the log structure on $\underline{X_1}$. Now define $Y_1:=X_1\times_{\A_P}\A_M$ and $Y_2:=X_2\times_{\A_P}\A_M$. Let $E$, $E_1$, and $E_2$ be the exceptional divisors on $Y$, $Y_1$, and $Y_2$. Note that $E_2\rightarrow E$ and $E_2\rightarrow E_1$ are isomorphisms since $Z$ is unchanged in $X_2\rightarrow X$ and $X_2\rightarrow X_1$.
Owing to Lemma \[A.3.26\], $M(Y_1)\rightarrow M(X_1)$ is an isomorphism, i.e., we have $$M(Y_1\rightarrow X_1)\simeq 0.$$ We obtain strict Nisnevich distinguished squares in $SmlSm/S$ $$\begin{tikzcd}
X_2-Z\arrow[r]\arrow[d]&X_2\arrow[d]\\
X-Z\arrow[r]&X
\end{tikzcd}
\begin{tikzcd}
Y_2-E_2\arrow[r]\arrow[d]&Y_2\arrow[d]\\
Y-E\arrow[r]&Y.
\end{tikzcd}$$
By ($sNis$-des), there are isomorphisms $$M((Y_2-E_2)\rightarrow (Y-E))\stackrel{\sim}\rightarrow M(Y_2\rightarrow Y),\;
M((X_2-Z)\rightarrow (X-Z))\stackrel{\sim}\rightarrow M(X_2\rightarrow X).$$ Since $Y_2-E_2'\cong X_2-Z$ and $Y-E\cong X-Z$, there is an isomorphism $$M(Y_2\rightarrow Y)\cong M(X_2\rightarrow X_2)$$ Using Proposition \[A.3.44\], we obtain an isomorphism $$M(Y_2\rightarrow X_2)\cong M(Y\rightarrow X).$$ Similarly, there is an isomorphism $$M(Y_2\rightarrow X_2)\cong M(Y_1\rightarrow X_1).$$ Thus $M(Y\rightarrow X)\cong 0$ since $M(Y_1\rightarrow X_1)\cong 0$, so $M(Y)\rightarrow M(X)$ is an isomorphism.
Strictly $(\mathbb{P}^\bullet,\mathbb{P}^{\bullet-1})$-invariant complexes
--------------------------------------------------------------------------
In this section we introduce the notions of strictly $(\P^\bullet,\P^{\bullet-1})$-invariant and strictly dividing invariant complexes of sheaves on $SmlSm/k$. We show that these two notions are equivalent. As a consequence, we provide another way of constructing objects of $\ldmeff$ and $\ldmeffet$.
\[Pn.1\] We denote by $(\P^{\bullet},\P^{\bullet-1})$ the class of projections $$(\P^{n},\P^{n-1})\times X\rightarrow X$$ for all $X\in SmlSm/k$ and positive integers $n>0$.
\[Pn.2\] Let $\cF$ be a complex of $t$-sheaves on $SmlSm/k$, where $t$ is one of $sNis$, $s\acute{e}t$, or $k\acute{e}t$. We say that $\cF$ is *strictly dividing invariant* if for every log modification $Y\rightarrow X$ in $SmlSm/k$ and integer $i\in \Z$ there is an isomorphism $$\bH_{t}^i(X,\cF)\cong \bH_{t}^i(Y,\cF).$$
We say that $\cF$ is *strictly $(\P^{\bullet},\P^{\bullet-1})$-invariant* if for every $X\in SmlSm/k$, $i\in \Z$, and integer $n>0$ there is an isomorphism $$\bH_{t}^i(X,\cF)\cong \bH_{t}^i(X\times (\P^n,\P^{n-1}),\cF).$$
\[Pn.3\] Let $\cF$ be a complex of $t$-sheaves on $SmlSm/k$, where $t$ is one of $sNis$, $s\acute{e}t$, or $k\acute{e}t$. Then $\cF$ is strictly dividing invariant if and only if $\cF$ is strictly $(\P^\bullet,\P^{\bullet-1})$-invariant.
We only consider the strict Nisnevich topology since the proofs are similar. Without loss of generality we may assume that $\cF$ is strict Nisnevich local.
On $\Co(\Shv_{sNis}(SmlSm/k,\Lambda))$ we have the $div$-local descent model structure and the $(\P^\bullet,\P^{\bullet-1})$-local descent model structure. Let $\mathbf{Ho}_{div}$ and $\mathbf{Ho}_{(\P^\bullet,\P^{\bullet-1})}$ denote the corresponding homotopy categories. Proposition \[A.5.71\] shows the properties [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and [($div$-des)]{} hold in $\ldaeff$. We can similarly prove that [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and [($div$-des)]{} hold in $\mathbf{Ho}_{div}$, while [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, and ($(\P^\bullet,\P^{\bullet-1})$-inv) hold in $\mathbf{Ho}_{(\P^\bullet,\P^{\bullet-1})}$. Owing to Proposition \[A.6.1\] and Theorem \[A.3.27\], we see that [($Zar$-sep)]{}, [($\boxx$-inv)]{}, [($sNis$-des)]{}, [($div$-des)]{}, and ($(\P^\bullet,\P^{\bullet-1})$-inv) hold in both $\mathbf{Ho}_{div}$ and $\mathbf{Ho}_{(\P^\bullet,\P^{\bullet-1})}$.
Suppose now that $\cF$ is strictly dividing invariant. Then $\cF$ is a fibrant object with respect to the $div$-local descent model structure on $\Co(\Shv_{sNis}(SmlSm/k,\Lambda))$. Thus for every $X\in SmlSm/k$ and integer $i\in \Z$ there is an isomorphism $$\hom_{\mathbf{Ho}_{div}}(\Lambda(X),\cF[i])\cong \bH_{sNis}^i(X,\cF).$$ Since ($(\P^\bullet,\P^{\bullet-1})$-inv) holds in $\mathbf{Ho}_{div}$ this shows $\cF$ is strictly $(\P^{\bullet},\P^{\bullet-1})$-invariant.
Conversely, if $\cF$ is strictly $(\P^\bullet,\P^{\bullet-1})$-invariant, then $\cF$ is a fibrant object with respect to the $(\P^\bullet,\P^{\bullet-1})$-local descent model structure on $\Co(\Shv_{sNis}(SmlSm/k,\Lambda))$. Thus for every $X\in SmlSm/k$ and integer $i\in \Z$ there is an isomorphism $$\hom_{\mathbf{Ho}_{(\P^\bullet,\P^{\bullet-1})}}(\Lambda(X),\cF[i])\cong \bH_{sNis}^i(X,\cF).$$ Since ($div$-des) holds in $\mathbf{Ho}_{(\P^\bullet,\P^{\bullet-1})}$, this shows $\cF$ is strictly dividing invariant.
\[Pn.4\] Let us begin with a strictly $(\P^\bullet,\P^{\bullet-1})$-invariant bounded below complex $\cF$ of strict Nisnevich sheaves with log transfers on $SmlSm/k$. Owing to Theorem \[Pn.3\] we see that $\cF$ is strictly dividing invariant. Applying Theorem \[Div.4\], this means that for every $X\in SmlSm/k$ and integer $i\in \Z$ there are isomorphisms $$\label{Pn.4.1}
H_{dNis}^i(X,a_{dNis}^*\cF)\cong \colimit_{Y\in X_{div}^{Sm}} H_{sNis}^i(Y,\cF)\cong H_{sNis}^i(X,\cF).$$ This shows that $a_{dNis}^*\cF$ is strictly $\boxx$-invariant. Then using Construction \[SmlSm.3\], we see that $\iota_\sharp a_{dNis}^*\cF$ is strictly $\boxx$-invariant too. In this way to obtain an object $\iota_\sharp a_{dNis}^*\cF$ of $\ldmeff$ from $\cF$. Combining and we have that $$\hom_{\ldmeff}(M(X),\iota_\sharp a_{dNis}^*\cF)\cong H_{dNis}^i(X,a_{dNis}^*\cF)\cong H_{sNis}^i(X,\cF).$$
Similarly, if $\cF$ is a strictly $(\P^\bullet,\P^{\bullet-1})$-invariant bounded below complex of strict étale (resp. Kummer étale) sheaves with log transfers, then there is an isomorphism $$\begin{gathered}
\label{Pn.4.2}
\hom_{\ldmeffet}(M(X),\iota_\sharp a_{d\acute{e}t}^*\cF)\cong H_{s\acute{e}t}^i(X,\cF),
\\
\label{Pn.4.3}
\text{(resp.\ }\hom_{\ldmefflet}(M(X),\iota_\sharp a_{l\acute{e}t}^*\cF)\cong H_{k\acute{e}t}^i(X,\cF).\text{)}\end{gathered}$$
Comparison with Voevodsky’s motives {#section:cvvmotives}
===================================
Our main results in this section are the projective bundle theorem and a comparison result showing that Voevodsky’s category of derived motives $\dmeff$ identifies with the full subcategory of $\A^1$-local objects in our derived category of log motives $\ldmeff$.
Presheaves with transfers and direct image functors
---------------------------------------------------
\[A.4.4\] Let $\cF$ be an $\A^1$-invariant Nisnevich sheaf with transfer on $Sm/k$. Let $X$ be a scheme log smooth over $k$ and let $j\colon U\rightarrow X$ be an open immersion whose complement is a strict normal crossing divisor on $X$. If $k$ is perfect, then for any $i\in \Z$, there is an isomorphism $$H_{Nis}^i(X,j_*j^*\cF)
\cong H_{Nis}^i(U,\cF).$$
Let $Z_1,\ldots,Z_r$ be the divisors forming the complement of $j$, and set $$U_i:=X-(Z_{i+1}\cup \cdots \cup Z_r).$$ Now let $j_i\colon U_i\rightarrow U_{i+1}$ denote the induced open immersion. By the Leray spectral sequence $$E_2^{pq}=H^p(U_{i+1},R^qj_{i*}j_i^*\cF) \Rightarrow H^{p+q}(U_i,\cF),$$ it suffices to show that $$R^qj_{i*}j_i^*\cF=0$$ for any $q>0$. Thus we reduce to the case when $X-U$ is a smooth divisor on $X$. Let $c\colon X-U\rightarrow X$ denote the complement of $j\colon U\rightarrow X$.
For every $q\in \Z$, let $H_{Nis}^q(\cF)$ denote the presheaf given by $$X\mapsto H_{Nis}^q(X,\cF).$$ Owing to [@MVW Theorem 24.1] the presheaf $H_{Nis}^q(\cF)$ is $\A^1$-invariant. By [@MVW Proposition 23.5, Example 23.8, Theorem 23.12] we obtain the vanishing $$R^qj_*j^*\cF
\cong
c_*H_{Nis}^q(\cF)_{(X\times \A^1,Z\times \A^1)}
\cong
c_*( H_{Nis}^q(\cF)_{-1})_{Nis}
=
0$$ for $q>0$. This allows us to proceed as in the proof of [@MVW Proposition 24.3].
\[A.4.19\] Suppose $\cF$ is a presheaf on a category $\cC$ equipped with a topology $t$. We denote the $t$-sheafification of $\cF$ by $$\cF_t
:=
a_t\cF.$$
\[A.4.20\] Let $X$ be a smooth scheme over $k$ with a strict normal crossing divisor $W$. Let $Z$ be an irreducible component of $W$ with corresponding closed immersion $i\colon Z\rightarrow X$. For a presheaf $\cF$ on $X$, we set $$\cF_{(X,W,Z)}:=i^*(\coker(u_*u^*\cF\rightarrow v_*v^*\cF))_{Zar}.$$ Here $u\colon X-\overline{(W-Z)}\rightarrow X$ and $v\colon X-W\rightarrow X$ are the naturally induced open immersions.
There is an exact sequence $$\label{A.4.20.1}
(u_*u^*\cF)_{Zar}\rightarrow (v_*v^*\cF)_{Zar}\rightarrow i_*\cF_{(X,W,Z)}\rightarrow 0$$ of Zariski sheaves on $X$ as in [@MR1764200 Remark 4.10].
\[A.4.25\] With the above notations, let $f\colon X'\rightarrow X$ be an étale morphism of smooth schemes over $k$ such that $Z':=f^{-1}(Z)\rightarrow Z$ is an isomorphism. For $W':=f^{-1}(W)$ there is a naturally induced isomorphism $$\cF_{(X,W,Z)}\xrightarrow{\cong} \cF_{(X',W',Z')}$$ of sheaves on $Z$.
First note that $W'$ is a strict normal crossing divisor on $X'$ and $Z'$ is an irreducible compoenent of $Z'$ since $f$ is étale, so we can use the notation $\cF_{(X',W',Z')}$. Let $$u'\colon X'-\overline{(W'-Z')}\rightarrow X'$$ and $$v'\colon X'-W'\rightarrow X'$$ be the naturally induced open immersions, and let $U$ be an open subscheme of $X$. For $U':=U\times_X X'$ we have the commutative diagram with exact rows $$\begin{tikzcd}[column sep=tiny]
\cF(U\times_X (X-Z))\arrow[d]\arrow[r]&\cF(U\times_X (X-W))\arrow[d]\arrow[r]& \coker(u_*u^*\cF\rightarrow v_*v^*\cF)(U)\arrow[r]\arrow[d,"\varphi"]&0\\
\cF(U'\times_{X'}(X'-Z'))\arrow[r]&\cF(U'\times_{X'}(X'-W'))\arrow[r]&\coker(u_*'u'^*\cF\rightarrow v_*'v'^*\cF)(U')\arrow[r]&0.
\end{tikzcd}$$ Let $a$ be an element of $\coker(u_*u^*\cF\rightarrow v_*v^*\cF)(U)$. If $\varphi(a)=0$, then Zariski locally on $U$, we have $a=0$ by [@MR1764200 Proposition 4.12(1)]. On the other hand, suppose $b$ is an element of $\coker(u_*'u'^*\cF\rightarrow v_*'v'^*\cF)(U')$. Then, Zariski locally on $U$, $b$ is in the image of $\varphi$ by [@MR1764200 Proposition 4.12(2)].
\[A.4.26\] Let $Z$ be a smooth scheme over $k$ with a strict normal crossing divisor $Y$, and let $j\colon Z-Y\rightarrow Z$ be the obvious open immersion. For any presheaf $\cF$ with transfers, there is a canonical isomorphism $$(j_*j^*\cF_{-1})_{Zar}\cong \cF_{(Z\times \A^1,(Z\times \{0\})\cup (Y\times \A^1),Z\times \{0\})}$$ of Zariski sheaves on $Z$.
We shall argue as in the proof of [@MR1764200 Proposition 4.11] (see also the proof of [@MVW Proposition 23.10]). Fix a point $z$ of $Z$. Let $U$ be an affine open neighborhood of $z$ in $Z$, and let $V$ be an affine open neighborhood of $U\times \{0\}$ in $U\times \A^1$. Following the proof of [@MVW Proposition 23.10], there exists a neighborhood $U'$ of $z$ in $U$ and an affine open subscheme $T$ of $\P_k^1$ such that $U'\times T$ contains both $\P_{U'}^1-V$ and $U'\times \{0\}$ where $V':=(U'\times \A^1)\times_{U\times \A^1}V$. We form the fiber product $Y_{U'}:=Y\times_Z U'$.
Since $Y$ is given by a Cartier divisor, we may further assume that $U'-Y_{U'}$ is affine, and hence $(U'-Y_{U'})\times T$ is affine. Setting $V'':=V'\times_{U'}(U'-Y_{U'})$ we deduce the standard triple (in the sense of [@MVW Definition 11.5]) $$T_{U',V'}:=(\P_{U'-Y_{U'}}^1,\P_{U'-Y_{U'}}^1-V'',(U'-Y_{U'})\times \{0\}),$$ see the second paragraph in the proof of [@MVW Proposition 23.10]. Next we form $$T_{U'}:=(\P_{U'-Y_{U'}}^1,(U'-Y_{U'})\times \{\infty\},(U'-Y_{U'})\times \{0\}) \text{ and } V_0'':=V''-(U'\times \{0\}).$$
Following the proof of [@MVW Proposition 23.10] we see the identity on $\P_{U'}^1$ is a finite morphism of standard triples $T_{U',V'}\rightarrow T_{U'}$ in the sense of [@MVW Definition 21.1]. Using [@MVW Theorem 21.6], we obtain a split exact sequence $$0 \rightarrow \cF((U'-Y_{U'})\times \A^1)\rightarrow \cF((U'-Y_{U'})\times (\A^1-\{0\}))\oplus \cF(V'')\rightarrow \cF(V_0'')\rightarrow 0.$$ Since $\cF$ is homotopy invariant, the morphism $$\cF((U'-Y_{U'})\times \A^1)\rightarrow \cF((U'-Y_{U'})$$ is injective. This implies that the morphism $\cF(V'')\rightarrow \cF(V_0'')$ is injective and $$(j_*j^*\cF_{-1})(U')\cong \cF(V_0'')/\cF(V'').$$ The right side is $\coker(u_*u^*\cF\rightarrow v_*v^*\cF)(V')$, where $$u\colon (Z\times \A^1)-(Y\times \A^1)\rightarrow Z\times \A^1,
\;
v\colon Z\times \A^1-((Z\times \{0\}\cup (Y\times \A^1))\rightarrow Z\times \A^1$$ are the naturally induced open immersions. We are done by passing to the limit over all $U'$ and $V'$.
\[A.4.27\] Let $X$ be a scheme smooth over $k$ with a strict normal crossing divisor $W$, let $Z$ be an irreducible component of $W$, and let $$j\colon (X-\overline{(W-Z)})\times_X Z\rightarrow Z$$ be the naturally induced open immersion. For any presheaf $\cF$ with transfers, Zariski locally on $X$, there is an isomorphism $$(j_*j^*\cF_{-1})_{Zar}\cong \cF_{(X,W,Z)}$$ of Zariski shaves on $X$.
Since the question is Zariski local on $X$, we may assume there exists an étale morphism $f\colon X\rightarrow \A_k^n$ of schemes over $k$ such that $W$ is the preimage of the union of axes in $\A_k^n$ and $Z=f^{-1}(\{0\}\times \A_k^{n-1})$, see Definition \[A.3.17\].
Consider the morphism of closed pairs $$(Z,X)\leftarrow (Z,X_2)\rightarrow (Z,X_1)$$ in Construction \[A.3.16\]. Note that $X_1=Z\times \A^m$ where $m$ is the codimension of $Z$ in $X$. Using Lemma \[A.4.25\] twice, we can replace $(Z,X)$ by $(Z,X_1)$. In this case, $W$ becomes the strict normal crossing divisor $$(Z\times \{0\})\cup ((Z\cap (\overline{(W-Z)})\times \A^1)$$ on $X_1=X\times \A^1$. We are done by Lemma \[A.4.26\].
If $W=Z$, then Lemma \[A.4.27\] is a special case of [@MR1764200 Theorem 4.14].
\[A.4.17\] Let $\cF$ be an $\A^1$-invariant presheaf with transfer on $Sm/k$. Let $X$ be a smooth scheme over $k$ and let $j\colon U\rightarrow X$ be an open immersion whose complement is a strict normal crossing divisor $Z$ on $X$. If $k$ is perfect, then the sheafification morphism $\cF\rightarrow \cF_{Nis}$ induces an isomorphism $$\label{A.4.17.1}
(j_*j^*\cF)_{Nis}\xrightarrow{\cong} (j_*j^*\cF_{Nis})_{Nis}\cong j_*j^*(\cF_{Nis}).$$
Assume the sheafification morphism $\cF\rightarrow \cF_{Zar}$ induces an isomorphism $$\label{A.4.17.2}
(j_*j^*\cF)_{Zar}\xrightarrow{\cong} (j_*j^*\cF_{Zar})_{Zar}\cong j_*j^*(\cF_{Zar}).$$ The functors $j^*$ and $j_*$ map Nisnevich sheaves to Nisnevich sheaves, and $\cF_{Zar}\cong \cF_{Nis}$ by [@MR1764200 Proposition 5.5]. Thus $j_*j^*(\cF_{Zar})$ is a Nisnevich sheaf on $Sm/X$, so $(j_*j^*\cF)_{Zar}$ is a Nisnevich sheaf on $Sm/X$ due to . It follows that is an isomorphism. Therefore it suffices to show that is an isomorphism.
We proceed by induction on the number $r$ of connected components of $Z$. If $r=0$, then $j$ is an isomorphism, and there is nothing to prove. Suppose $r\geq 1$, and let $Z_1,\ldots,Z_r$ be the irreducible components of $Z$. For each $i$, set $S_i:=Z_1\cup \cdots \cup Z_i$, and let $$u_i\colon X-S_i\rightarrow X,\;j_i\colon X-S_{i+1}\rightarrow X-S_i.\; z_i\colon Z_i\rightarrow X$$ be the naturally induced immersions.
For any open immersion $v\colon V'\rightarrow V$ of schemes, the functor $v^*$ commutes with Zariski sheafification and satisfies smooth base change with respect to $w_*$ for any open immersion $w\colon W\rightarrow V$. The latter means that $v^*w_*\cong w_*'v'^*$ where $v'\colon V'\times_V W\rightarrow W$ and $w\colon V'\times_V W\rightarrow W$ are the projections. Hence the problem is Zariski local on $X$. Moreover, by [@MR1764200 Corollary 4.19, Proposition 4.34], Lemma \[A.4.27\] and induction, we may assume there are isomorphisms $$\cF_{(X,S_{i+1},Z_{i+1})}\cong (j_{i*}j_i^*\cF_{-1})_{Zar}\cong j_{i*}j_i^*((\cF_{Zar})_{-1})$$ and an exact sequence $$0\rightarrow u_{i*}u_i^*\cF_{Zar}\rightarrow u_{i+1,*}u_{i+1}^*(\cF_{Zar})\rightarrow {z_i}_*(j_{i*}j_i^*(\cF_{Zar})_{-1})\rightarrow 0$$ of Zariski sheaves on $X$. Using , there is an induced commutative diagram with exact rows $$\begin{tikzcd}
&(u_{i*}u_i^*\cF)_{Zar}\arrow[d]\arrow[r]&(u_{i+1,*}u_{i+1}^*\cF)_{Zar}\arrow[d]\arrow[r]&z_{i*}((j_{i*}j_i^*\cF_{-1})_{Zar})\arrow[r]\arrow[d]&0\\
0\arrow[r]&u_{i*}u_i^*(\cF_{Zar})\arrow[r]&u_{i+1,*}u_{i+1}^*(\cF_{Zar})\arrow[r]&z_{i*}(j_{i*}j_i^*(\cF_{Zar})_{-1})\arrow[r]&0,
\end{tikzcd}$$ where the vertical morphisms are induced by the Zariski sheafification functor. Here the left vertical morphism is an isomorphism by induction, and the right vertical morphism is an isomorphism since $(j_{i*}j_i^*\cF_{-1})_{Zar}\cong j_{i*}j_i^*((F_{-1})_{Zar})$ by induction on $r$ and $(\cF_{-1})_{Zar}\cong (\cF_{Zar})_{-1}$. The snake lemma implies that also the middle vertical morphism is an isomorphism.
The above paragraph shows that $(u_{i*}u_i^*\cF)_{Zar}\rightarrow u_{i*}u_i^*(\cF_{Zar})$ is an isomorphism by induction on $i$. When $i=0$, this is trivial since $u_0$ is the identity. Specializing to the case $i=r$ finishes the proof because $u_r=j$.
We refer to the definition of the functor $$\omega^*\colon \Shvtrkl\rightarrow \Shvltrkl.$$
\[A.4.18\] Let $\cF$ be a strictly $\A^1$-invariant complex of Nisnevich sheaves with transfer on $Sm/k$. Let $X$ be a smooth scheme over $k$ and let $j\colon U\rightarrow X$ be an open immersion whose complement is a strict normal crossing divisor on $X$. If $k$ is perfect, then for every $i\in \Z$, there is an isomorphism $$\bH_{Nis}^i(X,j_*j^*\cF)\cong \bH_{Nis}^i(U,\cF).$$ In particular, for every $Y\in SmlSm/k$ and $i\in \Z$, there is an isomorphism $$\bH_{sNis}^i(Y,\omega^*\cF)\cong \bH_{Nis}^i(Y-\partial Y,\cF).$$
For any complex $\cG$ of presheaves and $i\in Z$, we let $H^i(\cG)$ denote the associated cohomology presheaf. The Nisnevich cohomological dimensions of $X$ and $U$ are finite, see e.g., [@MVW Example 11.2]. Hence by [@weibel_1994 5.7.9], there are two strongly convergent hypercohomology spectral sequences $E$ and $E'$ given by $$E_2^{pq}=H_{Nis}^p(X,(H^q(j_*j^*\cF))_{Nis})\Rightarrow \bH^{p+q}(X,j_*j^*\cF),$$ and $$E_2'^{pq}=H_{Nis}^p(U,(H^q(\cF))_{Nis})\Rightarrow \bH^{p+q}(U,\cF).$$ There is a naturally induced morphism of spectral sequences $E\rightarrow E'$.
Since $j_*$ and $j^*$ are both exact functors between presheaf categories there is an isomorphism $$H^q(j_*j^*\cF)\cong j_*j^*(H^q(\cF)).$$ Thus $(H^q(j_*j^*\cF))_{Nis}\cong j_*j^*(H^q(\cF)_{Nis})$ by Lemma \[A.4.17\]. We deduce the maps on $E_2$ terms are isomorphisms by Lemma \[A.4.4\] since $(H^q(\cF))_{Nis}$ is an $\A^1$-invariant Nisnevich sheaf by [@MVW Proposition 14.8]. Owing to the strong convergence of both the spectral sequences above the induced maps between the filtered target groups is also an isomorphism.
\[A.4.7\] Let $\cF$ be a strictly $\A^1$-invariant complex of Nisnevich sheaves with transfers on $Sm/k$. For every $X\in lSm/k$ and integer $i\in \Z$ there is a naturally induced isomorphism $$\label{A.4.7.7}
\bH_{dNis}^i(X,\omega^*\cF)\cong \bH_{Nis}^i(X-\partial X,\cF).$$
If $Y\rightarrow X$ is a log modification there is a naturally induced isomorphism $$Y-\partial Y\xrightarrow{\cong} X-\partial X.$$ Due to Lemma \[A.5.70\] we deduce the isomorphism $$a_{dNis}^*\Lambda(Y)\xrightarrow{\cong} a_{dNis}^*\Lambda(X).$$ Moreover, we conclude there are isomorphisms $$\begin{aligned}
\bH_{dNis}^i(X,\omega^*\cF)
& \cong
\hom_{\mathbf{D}(\Shvltrkl)}(a_{dNis}^*\Lambda(X),\omega^*\cF) \\
& \cong
\hom_{\mathbf{D}(\Shvltrkl)}(a_{dNis}^*\Lambda(Y),\omega^*\cF) \\
& \cong
\bH_{dNis}^i(Y,\omega^*\cF).\end{aligned}$$ This shows both sides of are invariant under log modifications. By Proposition \[A.3.19\], we may assume that $\underline{X}$ is smooth and $\partial X$ is a strict normal crossing divisor.
Choose a fibrant resolution $f\colon \cF\rightarrow \cG$ in the model category $\Co(\Shvtrkl)$ with respect to the descent model structure. This implies that $f$ is a quasi-isomorphism and $$\label{A.4.7.4}
\bH_{Nis}^i(X-\partial X,\cF)\cong H^i(\cG(X-\partial X)).$$ Let us show that $\omega^*\cG$ is a fibrant object in the model category $\Co(\Shvltrkl)$ with respect to the descent model structure. By definition of $\omega^*$, for every integer $i\in \Z$, there is an isomorphism $$\label{A.4.7.1}
H^i(\omega^*\cG(X))\cong H^i(\cG(X-\partial X)).$$ We need to show that for every dividing Nisnevich hypercover $\mathscr{X}\rightarrow X$ we have an isomorphism $$H^i(\omega^*\cG(X))\rightarrow H^i({\rm Tot}^\pi(\omega^*\cG(\mathscr{X}))),$$ see .
Since $\mathscr{X}-\partial \mathscr{X}\rightarrow X-\partial X$ is a Nisnevich hypercover, the assumption that $\cG$ is fibrant in $\Co(\Shvtrkl)$ implies $$\label{A.4.7.2}
H^i(\cG(X-\partial X))\cong H^i({\rm Tot}^\pi \cG(\mathscr{X}-\partial \mathscr{X})).$$ Using we have that $$\label{A.4.7.3}
H^i({\rm Tot}^\pi(\cG(\mathscr{X}-\partial \mathscr{X})))
\cong
H^i({\rm Tot}^\pi(\omega^*\cG(\mathscr{X}))).$$ By combining , , and we deduce that $\omega^*\cG$ is fibrant. Hence there is an isomorphism $$\label{A.4.7.5}
\bH_{dNis}^i(X,\omega^*\cG)\cong H^i(\omega^*\cG(X)).$$
Due to Lemma \[A.4.18\] there are isomorphisms $$\bH_{sNis}^i(X,\omega^*\cF)\cong \bH_{Nis}^i(X-\partial X,\cF)$$ and $$\bH_{sNis}^i(X,\omega^*\cG)\cong \bH_{Nis}^i(X-\partial X,\cG).$$ Since $f$ is a quasi-isomorphism, we deduce that $\omega^*f$ is a quasi-isomorphism for the strict Nisnevich topology. Thus $\omega^*f$ is a quasi-isomorphism for the dividing Nisnevich topology. It follows that $$\label{A.4.7.6}
\bH_{dNis}^i(X,\omega^*\cF)\cong \bH_{dNis}^i(X,\omega^*\cG).$$ To conclude the proof we combine , , , and .
\[ketcomp.9\] Let $\cF$ be a Nisnevich (resp. an étale) sheaf on $Sm/k$. Then $\omega^*\cF$ is a dividing Nisnevich (resp. log étale) sheaf on $lSm/k$.
Let $p\colon U\rightarrow X$ be a dividing Nisnevich (resp. log étale) cover. We need to show that the naturally induced sequence $$\omega^*\cF(X)\rightarrow \omega^*\cF(U)\rightrightarrows \omega^*\cF(U\times_X U)$$ is exact. This sequence is isomorphic to $$\cF(X-\partial X)\rightarrow \cF(U-\partial U)\rightrightarrows\cF(U-\partial U\times_{X-\partial X} U-\partial U),$$ which is exact since $U-\partial U\rightarrow X-\partial X$ is a Nisnevich (resp. an étale) cover and $\cF$ is a Nisnevich (resp. an étale) sheaf.
\[A.4.13\] Let $\cF$ be a strictly $\A^1$-invariant complex of Nisnevich sheaves with transfers on $Sm/k$. Then $\omega^*\cF$ is a both strictly $(\P^\bullet,\P^{\bullet-1})$-invariant and strictly $\A^1$-invariant complex of dividing Nisnevich sheaves with log transfers.
Lemma \[ketcomp.11\] shows that $\omega^*\cF$ is a complex of dividing Nisnevich sheaves with transfers. For $X\in lSm/k$, owing to Lemma \[A.4.7\] there are isomorphisms $$\begin{gathered}
\bH_{dNis}^i(X,\omega^*\cF)\cong \bH_{Nis}^i(X-\partial X,\cF),\\
\bH_{dNis}^i(X\times (\P^n,\P^{n-1}),\omega^*\cF)\cong \bH_{Nis}^i((X-\partial X)\times \A^{n},\cF),
\\
\bH_{dNis}^i(X\times \A^1,\omega^*\cF)\cong \bH_{Nis}^i((X-\partial X)\times \A^1,\cF).\end{gathered}$$ To conclude the proof we use the assumption that $\cF$ is strictly $\A^1$-invariant.
$\A^1$-local objects in $\ldmeff$
---------------------------------
We denote by $$v\colon \LCor\rightarrow \LCor[(\cA\cB l/k)^{-1}]$$ the evident localization functor, where $\cA\cB l/k$ is the class of admissible blow-ups in Definition \[A.3.1\]. As observed in Definition \[A.1.16\] there exist adjoint functors $$\begin{tikzcd}
\Co^{\leq 0}(\Psh(\LCor,\Lambda))\arrow[rr,shift left=1.5ex,"v_\sharp "]
\arrow[rr,"v^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"v_*"']&&\Co^{\leq 0}(\Psh(\LCor[(\cA\cB l/k)^{-1}],\Lambda)).
\end{tikzcd}$$ Here $v_\sharp$ is left adjoint to $v^*$ and $v^*$ is left adjoint to $v_*$.
\[A.4.2\] Assume that $k$ admits resolution of singularities. Let $X$ be a smooth and proper scheme over $k$, and let $Y$ be an fs log scheme log smooth over $k$. Then there is a naturally induced isomorphism $$v^*v_\sharp \Zltr(X)(Y)\cong \Zltr(X)(Y-\partial Y).$$
Let $Y'\rightarrow Y$ be an admissible blow-up. The induced morphism $$Y'-\partial Y\rightarrow Y-\partial Y$$ of schemes over $k$ is an isomorphism. Thus we can form the composite homomorphism of $\Lambda$-modules $$\varphi_{Y'}\colon \Zltr(X)(Y')\rightarrow \Zltr(X)(Y'-\partial Y')\stackrel{\sim}\rightarrow \Zltr(X)(Y-\partial Y).$$ From the formula we have the isomorphism $$v^*v_\sharp \Zltr(X)(Y)\cong \colimit_{Y'\in (\cA\cB l/k) \downarrow Y} \Zltr(X)(Y').$$ Collecting the $\varphi_{Y'}$’s, we get the induced homomorphism of $\Lambda$-modules $$\varphi\colon v^*v_\sharp \Zltr(X)(Y)\rightarrow \Zltr(X)(Y-\partial Y).$$ Let $V\in \lCor(Y',X)$ be a finite log correspondence. If $\varphi_{Y'}(V)=0$, then $V=0$ since $V$ is the closure of $\varphi_{Y'}(V)$ in $Y'\times X$. Thus $\varphi$ is injective. It remains to show that $\varphi$ is surjective.
Let $W\in \Cor(Y-\partial Y,X)$ be an elementary correspondence. By Raynaud-Gruson [@RaynaudGruson Théorème 5.7.9], there is an algebraic space $\underline{Y'}$ and a blow-up $\underline{f}\colon\underline{Y'}\rightarrow \underline{Y}$ such that $\underline{f}$ is an isomorphism on $Y-\partial Y$ and the closure $\underline{W'}$ of $\underline{W}$ in $\underline{Y'}\times \underline{X}$ is flat over $\underline{Y'}$. The algebraic space $\underline{Y'}$ is a scheme because it is a blow-up of a scheme. Further, $\underline{W'}$ is finite over $\underline{Y'}$ since $\underline{Y'}\times \underline{X}$ is proper over $\underline{Y'}$.
By resolution of singularities, there is a blow-up $g\colon \ul{Y''}\rightarrow \ul{Y'}$ such that $\ul{Y''}$ is smooth over $k$ and the complement of $(\underline{f}\circ \underline{g})^{-1}(Y-\partial Y)$ in $\underline{Y''}$ consists of strict normal crossing divisors $Z_1',\ldots,Z_r'$. Setting $Y'':=(\underline{Y''},Z_1'+\cdots+Z_r')$ the induced morphism $Y''\rightarrow Y$ of fs log schemes log smooth over $k$ is an admissible blow-up. Since the closure $\underline{W''}$ of $\underline{W}$ in $\underline{Y''}\times \underline{X}$ is a closed subscheme of $\underline{W'}\times_{\underline{Y'}}\underline{Y''}$, $\underline{W''}$ is finite over $\underline{Y''}$. It follows that $W$ can be extended to a finite correspondence from $\underline{Y''}$ to $X$. Since $X$ has the trivial log structure, such a finite correspondence gives a finite log correspondence from $Y''$ to $X$ by Example \[A.5.31\](2). Thus $\varphi$ is surjective.
\[A.4.30\] Assume that $k$ admits resolution of singularities. For every smooth and proper scheme $X$ over $k$ there is an isomorphism $$a_{dNis}^*\Zltr(X)\cong \omega^*\Ztr(X)$$ in $\ldmeff$. Likewise, there are isomorphisms $$a_{d\acute{e}t}^*\Zltr(X)\cong \omega^*\Ztr(X)
\text{ and }
a_{l\acute{e}t}^*\Zltr(X)\cong \omega^*\Ztr(X)$$ in $\ldmeffet$ and $\ldmefflet$, respectively.
We only treat the case of dividing Nisnevich coverings since the proofs are similar. Owing to Lemma \[A.5.45\] we have an isomorphism $$a_{dNis}^*v^*v_\sharp \Zltr(X) \cong v^*v_\sharp a_{dNis}^*\Zltr(X).$$ Moreover, Proposition \[A.4.2\] implies $$a_{dNis}^*v^*v_\sharp \Zltr(X)\cong a_{dNis}^*\omega^*\Ztr(X),$$ and from Remark \[A.3.20\] we deduce $$v^*v_\sharp a_{dNis}^*\Zltr(X)\cong a_{dNis}^*\Zltr(X).$$ To conclude, note that $\omega^*\Ztr(X)$ is a dividing Nisnevich sheaf, see Lemma \[ketcomp.11\].
For a bounded above complex $\cF$ of presheaves with log transfers, recall the Suslin double complex $C_*\cF$ from Definition \[A.1.16\]. If $\cF$ is a presheaf with log transfers, then for $U\in lSm/k$, we have $$C_*\cF(U):= (\cdots \rightarrow \colimit_{Y\in (\cA \cB l/k)\downarrow (U\times \Delta_{\boxx}^1)}\cF(Y)\rightarrow \colimit_{Y\in (\cA \cB l/k)\downarrow U}\cF(Y)).$$ For a bounded above complex $\cF$ of presheaves with log transfers, we also have the Suslin double complex $C_*^{\A^1}\cF$. If $\cF$ is a presheaf with transfers, then for $U\in Sm/k$, $$C_*^{\A^1}\cF(U):=(\cdots \rightarrow \cF(U\times \Delta_{\A^1}^1)\rightarrow \cF(U)).$$
\[A.4.12\] Suppose $\cF$ is a bounded above complex of presheaves with log transfers on $Sm/k$. Then there is a quasi-isomorphism of total complexes $${\rm Tot}(C_*\omega^*\cF)\cong {\rm Tot}(\omega^* C_*^{\A^1} \cF).$$
Using the technique in [@MVW Lemma 9.12], we reduce to the case when $\cF$ is a presheaf with log transfers. For every $X\in lSm/k$, admissible blow-up $Y\rightarrow \Delta_{\boxx}^n\times X$, $m\in \Z$, and $n\in \N$, we have $$\omega^*\cF(Y)=\cF(Y-\partial Y)\cong \cF(\Delta_{\A^1}^n\times (X-\partial X)).$$ Thus for the internal hom object $$\underline{\hom}(\Delta_{\A^1}^n,\cF)$$ mapping $U\in Sm/k$ to $\cF(U\times \Delta_{\A^1}^n)$, we have that $$\colimit_{Y\in (\cA\cB l/k) \downarrow(\Delta_{\boxx}^n\times X)}\omega^*\cF(Y)
=
\cF(\Delta_{\A^1}^n\times (X-\partial X))
\cong
(\omega^* \underline{\hom}(\Delta_{\A^1}^n,\cF))(X).$$ Since $$C_*\omega^*\cF(U):= (\cdots \rightarrow \colimit_{Y\in (\cA \cB l/k)\downarrow (U\times \Delta_{\boxx}^1)}\omega^*\cF(Y)\rightarrow \colimit_{Y\in (\cA \cB l/k)\downarrow U}\omega^*\cF(Y)),$$ and $$\omega^*C_*^{\A^1}\cF(U):=(\cdots \rightarrow \omega^*\underline{\hom}(\Delta_{\A^1}^1,\cF)(X)\rightarrow \omega^*\underline{\hom}(\Delta_{\A^1}^0,\cF)(X)),$$ we deduce the desired isomorphism.
\[A.4.29\] Assume that $k$ admits resolution of singularities. Let $X$ be a scheme smooth over $k$. Then there are isomorphisms $$\omega^*\Ztr(X)\cong
C_*\omega^*\Ztr(X)\cong
\omega^*C_*^{\A^1}\Ztr(X)$$ in $\ldmeff$, $\ldmeffet$, and $\ldmefflet$.
For the first isomorphism, apply Proposition \[A.1.15\](4) and Remark \[A.3.20\]. For the second isomorphism, apply Proposition \[A.4.12\].
\[A.4.9\] Assume that $k$ admits resolution of singularities. Let $X$ be a scheme smooth over $k$, and let $Y$ be an fs log schemes log smooth over $k$. Then for every integer $i\in \Z$ there is a naturally induced isomorphism $$\hom_{\ldmeff}(M(Y),\omega^* \Ztr(X)[i])
\cong
\hom_{\dmeff}(M(Y-\partial Y),M(X)[i]).$$
Due to Proposition \[A.4.29\] we have an isomorphism $$\hom_{\ldmeff}(M(Y),\omega^*\Ztr(X)[i])\\
\cong \hom_{\ldmeff}(M(Y),\omega^*C_*^{\A^1}\Ztr(X)[i]).$$ Since the Suslin complex $C_*^{\A^1}\Ztr(X)$ is strictly $\A^1$-invariant for the Nisnevich topology by [@MVW Corollary 14.9], we deduce $\omega^*C_*^{\A^1}\Ztr(X)$ is strictly $\boxx$-invariant for the dividing Nisnevich topology by Proposition \[A.4.13\]. Thus owing to Proposition \[A.5.13\] there is an isomorphism $$\hom_{\ldmeff}(M(Y),\omega^*C_*^{\A^1}\Ztr(X)[i])\cong \bH_{dNis}^i(Y,\omega^*C_*^{\A^1}\Ztr(X)).$$ From Lemma \[A.4.7\] we obtain the isomorphism $$\bH_{dNis}^i(Y,\omega^*C_*^{\A^1}\Ztr(X))\cong \bH_{Nis}^i(Y-\partial Y,C_*^{\A^1}\Ztr(X)).$$ We also have an isomorphism due to [@MVW Proposition 14.16] $$\bH_{Nis}^i(Y-\partial Y,C_*^{\A^1}\Ztr(X))\cong \hom_{\dmeff}(M(Y-\partial Y),M(X)[i]).$$ To conclude we combine the above isomorphisms.
\[A.4.8\] Assume that $k$ admits resolution of singularities. Let $X$ be a smooth and proper scheme over $k$, and let $Y$ be an fs log scheme log smooth over $k$. Then for every $i\in \Z$ there is a natural isomorphism $$\hom_{\ldmeff}(M(Y)[i],M(X))
\cong
\hom_{\dmeff}(M(Y-\partial Y)[i],M(X)).$$
Immediate from Propositions \[A.4.30\] and \[A.4.9\].
We do not expect that Proposition \[A.4.8\] holds for non-proper schemes.
Recall from the adjunction $$\omega_\sharp :\ldmeff \rightleftarrows \dmeff:R\omega^*.$$
\[A.4.15\] Assume that $k$ admits resolution of singularities. Let $X$ be a proper and smooth scheme over $k$. Then the unit of the adjunction ${\rm id}\rightarrow R\omega^*\omega_\sharp$ induces an isomorphism $$M(X)\rightarrow R\omega^*\omega_\sharp M(X)$$ in $\ldmeff$.
Appealing to the generators $M(Y)[i]$ of $\ldmeff$, where $Y\in lSm/k$ and $i\in \Z$, it suffices to show there is an isomorphism $$\hom_{\ldmeff}(M(Y)[i],M(X))
\cong
\hom_{\ldmeff}(M(Y)[i],R\omega^*M(X)).$$
This follows from Proposition \[A.4.8\] and the isomorphisms $$\begin{split}&\hom_{\ldmeff}(M(Y)[i],R\omega^*M(X))\\
\cong &\hom_{\dmeff}(\omega_\sharp M(Y)[i],M(X))\\
\cong &\hom_{\dmeff}(M(Y-\partial Y)[i],M(X)).
\end{split}$$
\[A.4.11\] Let $Z_1,\ldots,Z_r$ be divisors forming a strict normal crossing divisor on a smooth scheme $X$ over $k$. Suppose the smooth subscheme $Z$ of $X$ has strict normal crossing with and is not contained in $Z_1\cup\cdots\cup Z_r$. Let $E$ be the exceptional divisor of the blow-up $B_ZX$. Then for $$Y:=(X,Z_1+\cdots+Z_r)$$ there is a commutative diagram in $\ldmeff$ $$\begin{tikzcd}
M((B_Z Y,E)\rightarrow Y)\arrow[d,"\sim"']\arrow[r]&R\omega^*M((Y-\partial Y\up Z\rightarrow Y-\partial Y)\arrow[d,"\sim"]\\
MTh(N_ZY)\arrow[r]&R\omega^*(MTh(N_{Z-\partial Y\cap Z}(Y-\partial Y))).
\end{tikzcd}$$ The vertical morphisms are the Gysin isomorphisms, and the horizontal morphisms are induced by the unit of the adjunction ${\rm id}\rightarrow R\omega^*\omega_\sharp$.
With the same notations in the proof of Theorem \[A.3.36\], we set $$\begin{gathered}
U:=(B_Z Y,E),
\;
V:=Y,
\;
U':=(B_{Z\times \boxx}(D_ZY),E^D),
\;
V':=B_{Z\times \boxx}(D_ZY),
\\
U'':=(B_Z(N_ZY),E^N),
\;
V'':=N_Z Y.\end{gathered}$$ The diagram $$\begin{tikzcd}
M(U\rightarrow V)\arrow[d]\arrow[r]&R\omega^*M((U-\partial U)\rightarrow (V-\partial V))\arrow[d]\\
M(U'\rightarrow V')\arrow[d,leftarrow]\arrow[r]&R\omega^*M((U-\partial U)\rightarrow (V'-\partial V'))\arrow[d,leftarrow]\\
M(U''\rightarrow V'')\arrow[r]&R\omega^*M((U''-\partial U'')\rightarrow (V''-\partial V''))
\end{tikzcd}$$ commutes because the horizontal morphisms are obtained from an adjunction. Our assertion follows from this.
\[A.4.31\] Let $X$ be an fs log scheme in $SmlSm/k$ with a rank $n$ vector bundle $\cE$. View $\P(\cE)$ as the closed subscheme of $\P(\cE\oplus \cO)$ at infinity. Then $\cF:=\cE-\partial \cE$ is a vector bundle over $X-\partial X$, and there is a commutative diagram $$\label{A.4.31.1}
\begin{tikzcd}
MTh(\cE)\arrow[d]\arrow[r]&
R\omega^*MTh(\cF)\arrow[d]
\\
M(\P(\cE)\rightarrow \P(\cE\oplus \cO))\arrow[r]&
R\omega^* M(\P(\cF)\rightarrow \P(\cF\oplus \cO)).
\end{tikzcd}$$ Here the vertical morphisms are obtained from and [@MV Proposition 2.17, p. 112], and the horizontal morphisms are induced by the unit of the adjunction ${\rm id}\rightarrow R\omega^*\omega_\sharp$.
We form the fs log schemes $Y$, $Y'$ and the divisors $E$, $E'$ as in the proof of Proposition \[A.3.34\], and set $$V:=(Y,E)-\partial (Y,E),
\;
V':=(Y',E')-\partial (Y',E').$$ Then there is a commutative diagram $$\begin{tikzcd}
M((Y,E)\rightarrow \cE)\arrow[d]\arrow[r]&
M((Y',E')\rightarrow \P(\cE\oplus \cO))\arrow[r,leftarrow]\arrow[d]&
M(\P(\cE)\rightarrow \P(\cE\oplus \cO))\arrow[d]
\\
R\omega^*M(V\rightarrow \cF)\arrow[r]&
R\omega^*M(V'\rightarrow \P(\cF\oplus \cO))\arrow[r,leftarrow]&
R\omega^*M(\P(\cF)\rightarrow \P(\cF\oplus \cO)).
\end{tikzcd}$$ In the proof of Proposition \[A.3.34\] and [@MV Proposition 2.17, p. 112] it is shown that the horizontal morphisms in the right hand side square are isomorphisms. After inverting these, we obtain the required commutative diagram.
\[A.4.10\] Assume that $k$ admits resolution of singularities. Let $X$ be an fs log scheme log smooth over $k$. If the underlying scheme $\underline{X}$ is proper over $k$, then the unit of the adjunction $\id\rightarrow R\omega^*\omega_\sharp$ induces an isomorphism $$M(X)\rightarrow R\omega^*\omega_\sharp M(X).$$ Therefore, for every $Y\in lSm/k$ and integer $i\in \Z$, there is a natural isomorphism $$\hom_{\ldmeff}(M(Y)[i],M(X))
\cong
\hom_{\dmeff}(M(Y-\partial Y)[i],M(X-\partial X)).$$
The question is dividing local on $X$, so we may assume that $X\in SmlSm/k$ by Proposition \[A.3.19\]. Then $\partial X$ is a strict normal crossing divisior formed by smooth divisors $Z_1,\ldots,Z_r$ on $X$, and we can form $$\begin{gathered}
X_1:=(\underline{X},Z_2+\cdots+Z_r),
\;
U:=X-\partial X,
\;
U_1:=X_1-\partial X_1,
\\
V:=Y-\partial Y,
\;
W_1:=Z_1-\partial X\cap Z_1.\end{gathered}$$
The proof proceeds by induction on $r$. Proposition \[A.4.15\] implies the claim for $r=0$. Assuming $r>0$, by Proposition \[A.4.31\] there is a commutative diagram $$\begin{tikzcd}
MTh(N_{Z_1}X)\arrow[d]\arrow[r]&
R\omega^*MTh(N_{W_1}U)\arrow[d]
\\
M(\P(N_{Z_1}X)\rightarrow \P(N_{Z_1}X\oplus \cO))\arrow[r]&
R\omega^* M(\P(N_{W_1}U)\rightarrow \P(N_{W_1}U\oplus \cO)).
\end{tikzcd}$$ The vertical morphisms are isomorphisms by Proposition \[A.3.34\] and [@MV Proposition 2.17, p. 112]. By induction there are isomorphisms $$M(\P(N_{Z_1}X))\xrightarrow{\cong} R\omega^*M(\P(N_{W_1}U))$$ and $$M(\P(N_{Z_1}X\oplus \cO))\xrightarrow{\cong} R\omega^*M(\P(N_{W_1}U\oplus \cO)).$$ It follows that the lower horizontal morphism is an isomorphism, and hence the upper horizontal morphism is an isomorphism.
Due to Proposition \[A.4.11\] there is a commutative diagram $$\begin{tikzcd}
M(X\rightarrow X_1)\arrow[d]\arrow[r]& R\omega^*M(U\rightarrow U_1)
\arrow[d]\\
MTh(N_{Z_1}X) \arrow[r]&R\omega^*MTh(N_{W_1}U).
\end{tikzcd}$$ The above shows the lower horizontal morphism is an isomorphism. The vertical morphisms are isomorphisms by Theorem \[A.3.36\] and [@MVW Theorem 15.15], respectively. This allows us to conclude for the upper horizontal morphism.
By induction $M(X_1)\rightarrow R\omega^*M(U_1)$ is an isomorphism. The five lemma implies $M(X)\rightarrow R\omega^*M(U_1)$ is an isomorphism.
\[A.4.16\] Assume that $k$ admits resolution of singularities. Then there is an adjunction $$R\omega^*: \dmeff\rightleftarrows \ldmeff:R\omega_*$$
By [@CD12 Proposition 1.3.19] we need to show that $\dmeff$ admits a set of objects $\cG$ with the following properties.
1. For each $K\in \cG$, $K$ (resp. $R\omega^*K$) is compact in $\dmeff$ (resp. $\ldmeff$).
2. $\dmeff$ (resp. $\ldmeff$) is generated by $\cG$ (resp. $R\omega^*\cG$).
3. $\ldmeff$ is compactly generated.
By Remark \[A.5.34\] (resp. Proposition \[A.5.33\]), $\dmeff$ (resp. $\ldmeff$) is generated by compact objects $M(X)[i]$ for $X\in Sm/k$ (resp. $X\in lSm/k$). Hence it remains to check that $R\omega^*M(X)$ is a compact object in $\ldmeff$.
By resolution of singularities, there is a projective fs log scheme $\overline{X}$ in $SmlSm/k$ with $X\cong \overline{X}-\partial \overline{X}$. Theorem \[A.4.10\] shows there is an isomorphism $$M(\overline{X})\xrightarrow{\cong} R\omega^*M(\overline{X}-\partial \overline{X})\cong R\omega^*M(X).$$ Thus $R\omega^*M(X)$ is compact in $\ldmeff$ since the motive $M(\overline{X})$ is compact in $\ldmeff$.
To summarize the above we note there are adjunctions $$\begin{tikzcd}
\ldmeff\arrow[rr,shift left=1.5ex,"\omega_\sharp "]\arrow[rr,"R\omega^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"R\omega_*"']&&\dmeff.
\end{tikzcd}$$
\[A.4.14\] Assume that $k$ admits resolution of singularities. Then the functor $$R\omega^*\colon \dmeff\rightarrow \ldmeff$$ is fully faithful.
We need to show that the unit of the adjunction ${\rm id}\rightarrow R\omega_*R\omega^*$ is an isomorphism. By invoking the generators $M(X)[i]$ of $\dmeff$, where $X\in Sm/k$ and $i\in \Z$, we only need to show there is an isomorphism $$M(X)\rightarrow R\omega_*R\omega^*M(X).$$ Again by invoking the generators, we are reduced to showing there is an isomorphism $$\hom_{\dmeff}(M(Y)[i],M(X))\cong \hom_{\dmeff}(M(Y)[i],R\omega_*R\omega^*M(X))$$ for any $Y\in Sm/k$ and $i\in \Z$.
By resolution of singularities, there are projective fs log schemes $\overline{X}$ and $\overline{Y}$ in $SmlSm/k$ such that $X\cong \overline{X}-\partial \overline{X}$ and $Y\cong \overline{Y}-\partial \overline{Y}$. Owing to Theorem \[A.4.10\] we have that $$R\omega^*M(X)\cong M(\overline{X}),
\;
R\omega^*M(Y)\cong M(\overline{Y}).$$ Applying Theorem \[A.4.10\] we deduce the isomorphisms $$\begin{aligned}
\hom_{\dmeff}(M(Y)[i],M(X))\cong &\hom_{\ldmeff}(M(\overline{Y})[i],M(\overline{X}))
\\
\cong &\hom_{\ldmeff}(R\omega^*M(Y)[i],R\omega^*M(X)).\end{aligned}$$ By adjunction, this concludes the proof.
\[A.4.24\] An object $\cF$ of $\ldmeff$ is called [*$\A^1$-local*]{} if for any $X\in lSm/k$ and $i\in \Z$ the projection $X\times \A^1\rightarrow X$ induces an isomorphism $$\hom_{\ldmeff}(M(X)[i],\cF)\rightarrow \hom_{\ldmeff}(M(X\times \A^1)[i],\cF).$$
We note that if a complex $\cF$ of dividing Nisnevich sheaves with log transfers is both strictly $\A^1$-invariant and strictly $\boxx$-invariant, then the associated object in $\ldmeff$ is $\A^1$-local.
\[A.4.21\] Assume that $k$ admits resolution of singularities. Let $\cF$ be an $\A^1$-local object of $\ldmeff$. Then, for every $Y\in lSm/k$ and $i\in \Z$, there is a natural isomorphism $$\hom_{\ldmeff}(M(Y)[i],\cF)
\cong
\hom_{\ldmeff}(M(Y-\partial Y)[i],\cF).$$
By taking a fibrant replacement, we may assume $\cF$ is a strictly $\boxx$-invariant complex of dividing Nisnevich sheaves with log transfers. Then, by Proposition \[A.5.13\], there is an isomorphism $$\hom_{\ldmeff}(M(Y)[i],\cF)
\cong
\bH_{dNis}^i(Y,\cF).$$ If $Y$ has the trivial log structure, then the dividing Nisnevich topology on $Y$ agrees with the Nisnevich topology on $Y$, and we obtain $$\label{A.4.21.1}
\hom_{\ldmeff}(M(Y)[i],\cF)
\cong
\bH_{Nis}^i(Y,\cF).$$ Thus the restriction $\cG$ of $\cF$ on $Sm/k$ is a strictly $\A^1$-invariant complex of Nisnevich sheaves with transfers.
The claim is dividing Nisnevich local on $Y$, so we may assume $\partial Y$ is a strict normal crossing divisor formed by smooth divisors $Z_1,\ldots,Z_r$ on $Y$. We set $$Y_1:=(\underline{Y},Z_2+\cdots+Z_r),$$ $$W=(Z_1,Z_1\cap Z_2+\cdots+Z_1\cap Z_r),$$ and proceed by induction on $r$. The claim is evident for $r=0$.
In the following we assume $r\geq 1$ and set $$Y':=(D_{Z_1}Y_1,Z_1\times \boxx),
\;
Y_1':=D_{Z_1}Y,
\;
Y'':=(N_{Z_1} Y_1,Z_1),
\;
Y_1'':=N_{Z_1}Y_1.$$ There is a naturally induced commutative diagram $$\begin{tikzpicture}[baseline= (a).base]
\node[scale=.94] (a) at (0,0)
{
\begin{tikzcd}[column sep=tiny]
\hom_{\ldmeff}(M(Y\rightarrow Y_1)[i],\cF)\arrow[d]\arrow[r]
&\hom_{\dmeff}(M(Y-\partial Y\rightarrow Y_1-\partial Y_1)[i],\cG)\arrow[d]
\\
\hom_{\ldmeff}(M(Y'\rightarrow Y_1')[i],\cF)\arrow[d,leftarrow]\arrow[r]
&\hom_{\dmeff}(M(Y'-\partial Y'\rightarrow Y_1'-\partial Y_1')[i],\cG)\arrow[d,leftarrow]
\\
\hom_{\ldmeff}(M(Y''\rightarrow Y_1'')[i],\cF)\arrow[r]&\hom_{\dmeff}(M(Y-\partial Y\rightarrow Y_1''-\partial Y_1'')[i],\cG).
\end{tikzcd}
};
\end{tikzpicture}$$ The left vertical morphisms are isomorphisms by Theorem \[A.3.36\], and the right vertical morphisms are isomorphisms by and the proof of [@MR1764202 Proposition 3.5.4]. The lower horizontal morphism is an isomorphism by induction, so the upper horizontal morphism is also an isomorphism. By induction there is an isomorphism $$\hom_{\ldmeff}(M(Y_1)[i],\cF)\rightarrow \hom_{\dmeff}(M(Y_1-\partial Y_1)[i],\cG)$$ This finishes the proof on account of the five lemma.
\[A.4.22\] Assume that $k$ admits resolution of singularities. Then the essential image of the functor $$R\omega^*\colon \dmeff\rightarrow \ldmeff$$ is the full subcategory of $\ldmeff$ consisting of $\A^1$-local objects.
Let $\cG$ be an object of $\dmeff$. Then, for every $Y\in lSm/k$ and $i\in \Z$, the isomorphisms $$\begin{split}
\hom_{\ldmeff}(M(Y)[i],R\omega^*\cG) & \cong \hom_{\dmeff}(\omega_\sharp M(Y)[i],\cG) \\
& \cong \hom_{\dmeff}(M(Y-\partial Y)[i],\cG) \\
& \cong \hom_{\dmeff}(M((Y-\partial Y)\times \A^1)[i],\cG) \\
& \cong \hom_{\dmeff}(\omega_\sharp M(Y\times \A^1)[i],\cG) \\
& \cong \hom_{\ldmeff}(M(Y\times \A^1)[i],R\omega^*\cG)
\end{split}$$ show that $R\omega^*\cG$ is $\A^1$-local.
Now let $\cF$ be an $\A^1$-local object of $\ldmeff$. To show that $\cF$ is in the essential image of $R\omega^*$, it suffices to show that the unit of the adjunction $\cF\rightarrow R\omega^*\omega_\sharp \cF$ is an isomorphism. We show this by checking that, for any $Y\in lSm/k$ and $i\in \Z$, there is a naturally induced isomorphism $$\hom_{\ldmeff}(M(Y)[i],\cF)\rightarrow \hom_{\dmeff}(\omega_\sharp M(Y)[i],\omega_\sharp \cF).$$
The open immersion $Y-\partial Y\rightarrow Y$ induces a commutative diagram $$\begin{tikzcd}
\hom_{\ldmeff}(M(Y)[i],\cF)\arrow[d]\arrow[r]&
\hom_{\dmeff}(\omega_\sharp M(Y)[i],\omega_\sharp \cF)\arrow[d]
\\
\hom_{\ldmeff}(M(Y-\partial Y)[i],\cF)\arrow[r]&
\hom_{\dmeff}(\omega_\sharp M(Y-\partial Y)[i],\omega_\sharp \cF).
\end{tikzcd}$$ The left vertical homomorphism is an isomorphism by Proposition \[A.4.21\], and the right vertical homomorphism is an isomorphism since $$\omega_\sharp M(Y)\cong M(Y-\partial Y)\cong \omega_\sharp M(Y-\partial Y).$$ Hence to show the upper horizontal homomorphism is an isomorphism, it suffices to show the lower horizontal homomorphism is an isomorphism, i.e., we are reduced to the case when $Y\in Sm/k$.
We may view $\cF$ as a strictly $\boxx$-invariant object of $\Co(\Shvltrkl)$. Recall from the adjunction $$L\lambda_\sharp\colon \Deri(\Shvtrkl)\rightleftarrows \Deri(\Shvltrkl)\colon \lambda^*\cong R\lambda^*\cong \omega_\sharp.$$ Thus, for every $Y\in Sm/k$ and $i\in \Z$, there are isomorphisms $$\begin{aligned}
\hom_{\Deri(\Shvtrkl)}(\Ztr(Y)[i],\omega_\sharp \cF)
& \cong
\hom_{\Deri(\Shvtrkl)}(\Ztr(Y)[i],\lambda^* \cF) \\
& \cong
\hom_{\Deri(\Shvltrkl)}(L\lambda_\sharp \Ztr(Y)[i], \cF) \\
& \cong
\hom_{\Deri(\Shvltrkl)}(\Zltr(Y)[i],\cF) \\
& \cong
\hom_{\ldmeff}(M(Y)[i],\cF)\end{aligned}$$ Together with the assumption that $\cF$ is $\A^1$-local, we deduce $\omega_\sharp F$ is $\A^1$-local, and $$\hom_{\Deri(\Shvtrkl)}(\Ztr(Y)[i],\omega_\sharp \cF)\cong \hom_{\dmeff}(M(Y)[i],\omega_\sharp \cF).$$ This shows the isomorphism $$\hom_{\ldmeff}(M(Y)[i],\cF)\cong \hom_{\dmeff}(M(Y)[i],\omega_\sharp \cF),$$ To conclude the proof we note that $$\omega_\sharp M(Y)\cong M(Y).$$
\[thm::dmeff=ldmeffprop\] Assume that $k$ admits resolution of singularities. Then there is an equivalence of triangulated categories $$\ldmeffprop\simeq \dmeff.$$
Owing to Theorem \[A.4.22\] it suffices to identify $\ldmeffprop$ with the essential image of the functor $$R\omega^*\colon \dmeff\rightarrow \ldmeff.$$ The essential image of $R\omega^*$ is the smallest triangulated subcategory of $\ldmeff$ that is closed under small sums and contains $R\omega^*M(X)$ for every $X\in Sm/k$. On the other hand, $\ldmeffprop$ is the smallest triangulated subcategory of $\ldmeff$ that is closed under small sums and contains the motive $M(Y)$ for every $Y\in lSm/k$, where $Y$ is proper over $k$.
Owing to Theorem \[A.4.10\], there is an isomorphism $$M(Y)\cong R\omega^*M(Y-\partial Y).$$ Thus $\ldmeffprop\subset \im R\omega^*$. By resolution of singularities, there exists an object $X'\in SmlSm/k$ such that $X'-\partial X'\cong X$. Moreover, by Theorem \[A.4.10\], there is an isomorphism $$M(X')\cong R\omega^*M(X).$$ Thus we have the inclusion $$\im R\omega^*\subset \ldmeffprop.$$
\[A.4.23\] Assume that $k$ admits resolution of singularities. Then the functor $$R\omega^*\colon \dmeff\rightarrow \ldmeff$$ is [*not*]{} essentially surjective. Indeed, in Theorem \[thmHodge\] we show that for every $X\in SmlSm/k$ and $i,j\geq 0$ there is an isomorphism $$\hom_{\ldmeff}(M(X), \Omega^j_{-/k})
\cong
H^i_{Zar}(\ul{X}, \Omega^j_{X/k}).$$ However, the groups $H^i_{Zar}(\ul{X}, \Omega^j_{X/k})$ and $H^i_{Zar}(\ul{X}\times \A^1, \Omega^j_{X\times \A^1/k})$ are non-isomorphic, so that Theorem \[A.4.22\] implies $\Omega^j_{X/k}$ is not in the essential image of $R\omega^*$.
Projective bundle formula
-------------------------
In this subsection we formulate and prove the projective bundle formula under the assumption of resolution of singularities. We begin by discussing orientations on $\ldmeff$.
\[A.7.1\] We say that $\ldmeff$ admits an [*orientation*]{} if for any $X\in lSm/k$, there is a morphism $$c_1\colon {\rm Pic}(\underline{X})\rightarrow \hom_{\ldmeff}(M(X),\Lambda(1)[2])$$ of sets, functorial in $X$, such that the class of the canonical line bundle in ${\rm Pic}(\P^1)$ corresponds to the canonical morphism $M(\P^1)\rightarrow \Lambda(1)[2]$.
\[A.7.2\] Assume that $k$ admits resolution of singularities. Recall that $\Lambda(1)[2]=M(\pt\rightarrow \P^1)$. Since $\pt$ and $\P^1$ are smooth and proper, by Proposition \[A.4.8\], we have $${\rm Pic}(X-\partial X)
\cong
\hom_{\ldmeff}(M(X),\Lambda(1)[2]).$$ Thus we are entitled to the composition $${\rm Pic}(\underline{X})\stackrel{u^*}
\rightarrow
{\rm Pic}(X-\partial X)\stackrel{\sim}\rightarrow \hom_{\ldmeff}(M(X),\Lambda(1)[2]),$$ where $u\colon X-\partial X\rightarrow \underline{X}$ is the induced open immersion. Hence $\ldmeff$ admits an orientation.
\[A.7.3\] Let $X$ be a fs log scheme log smooth over $k$, and let $\cP$ be a $\P^n$-bundle over $X$, i.e., Zariski locally on $X$, the morphism $\cP\rightarrow X$ is isomorphic to the projection $\P^n\times X\rightarrow X$. If $\ldmeff$ admits an orientation, then the canonical line bundle over $\cP$ induces a morphism $$\tau_1\colon M(\cP)\rightarrow \Lambda(1)[2]$$ in $\ldmeff$. More generally, for $1\leq i\leq n$, there is a morphism $$\label{A.7.3.1}
\tau_i\colon M(\cP)\rightarrow \Lambda(i)[2i]$$ given by the composition $$M(\cP)
\stackrel{\Delta}\longrightarrow
M(\cP)^{\otimes i}
\stackrel{\tau_1^{\otimes i}}\longrightarrow
(\Lambda(1)[2])^{\otimes i}
\cong
\Lambda(i)[2i].$$ We also have a morphism $$\sigma_i\colon M(\cP)\rightarrow M(X)(i)[2i]$$ given by the composition $$M(\cP)\stackrel{\Delta}\longrightarrow M(\cP)\otimes M(\cP)\stackrel{\id \times \tau_i}\longrightarrow M(\cP)\otimes \Lambda(i)[2i]\longrightarrow M(X)\otimes \Lambda(i)[2i]\cong M(X)(i)[2i].$$ This defines an induced morphism $$\sigma\colon M(\cP)\rightarrow \bigoplus_{i=0}^n M(X)(i)[2i].$$
\[A.6.6\] Assume that $k$ admits resolution of singularities. For every $X\in lSm/k$ and integer $n\geq 0$, the morphism $$\sigma\colon M(\P_X^n)\rightarrow \bigoplus_{i=0}^n M(X)(i)[2i]$$ is an isomorphism.
Using the monoidal structure, we have that $$M(\cP)\cong M(X)\otimes M(\P^n)\in \ldmeff.$$ Hence we may assume that $X=\Spec k$, i.e., for our purposes it suffices to show there is an isomorphism $$M(\P^n)
\cong
\bigoplus_{i=0}^n \Lambda(i)[2i].$$ It suffices to show that for $Y\in lSm/k$ and $i\in \Z$ there is an isomorphism $$\hom_{\ldmeff}(M(Y)[i],M(\P^n))\cong \hom_{\ldmeff}(M(Y)[i],\bigoplus_{i=0}^n \Lambda(i)[2i]).$$ By Proposition \[A.4.8\], there is a commutative diagram with vertical isomorphisms $$\begin{tikzcd}[column sep=tiny]
\hom_{\ldmeff}(M(Y)[i],M(\P^n))\arrow[d,"\sim"']\arrow[r]&\hom_{\ldmeff}(M(Y)[i],\bigoplus_{i=0}^n \Lambda(i)[2i])\arrow[d,"\sim"]\\
\hom_{\dmeff}(M(Y-\partial Y)[i],M(\P^n))\arrow[r]&\hom_{\dmeff}(M(Y-\partial Y)[i],\bigoplus_{i=0}^n \Lambda(i)[2i]).
\end{tikzcd}$$ Hence it suffices to show there is an isomorphism $$M(\P^n)
\cong
\bigoplus_{i=0}^n \Lambda(i)[2i]$$ in $\dmeff$, which is noted in [@MVW Corollary 15.5, Exercise 15.11].
\[A.6.2\] Assume that $k$ admits resolution of singularities. Let $\mathcal{E}$ be a vector bundle of rank $n+1$ over an fs log scheme $X$ log smooth over $k$. Then the naturally induced morphism $$\sigma\colon M(\P(\cE))\rightarrow \bigoplus_{i=0}^n M(X)(i)[2i]$$ is an isomorphism in $\ldmeff$.
We argue as in [@Deg08 Theorem 3.2]. For $\cP:=\P(\cE)$ and an open immersion $U\rightarrow X$ of fs log schemes over $k$, consider the induced commutative diagram $$\begin{tikzcd}
\cP_U\arrow[d]\arrow[r]& \cP_U\times \cP\arrow[d]\\
\cP\arrow[r]&\cP\times \cP
\end{tikzcd}$$ of fs log schemes over $k$, where $\cP_U:=\cP\times_X U$ and the horizontal morphisms are the graph morphisms. There are induced morphisms $$M(\cP_U\rightarrow \cP)
\rightarrow
M(\cP_U\times \cP\rightarrow \cP\times \cP)
\stackrel{\sim}
\rightarrow M(\cP_U\rightarrow \cP)\otimes M(\cP)\rightarrow M(U\rightarrow X)\otimes M(\cP).$$
Using the morphism $M(\cP)\rightarrow \bigoplus_{i=0}^n \Lambda(i)[2i]$ from Construction \[A.7.3\], we obtain $$M(\cP_U\rightarrow \cP)\rightarrow \bigoplus_{i=0}^n M(U\rightarrow X)(i)[2i].$$
We use induction on the finite number $m$ of Zariski open subsets appearing in a trivialization of $\cE$. The case $m=1$ is already done in Proposition \[A.6.6\]. Hence suppose $m>1$. Then there exists a Zariski cover $\{V_1,\ldots,V_m\}$ such that $\cE$ is trivial on $V_i$ for every $i$. Set $$U_1:=V_1,
\;
U_2:=V_2\cup \cdots\cup V_m,
\;
\text{and }
U_{12}:=U_1\cap U_2.$$
There is a naturally induced commutative diagram in $\ldmeff$ $$\begin{tikzcd}
M(\cP_{U_{12}}\rightarrow \cP_{U_2})\arrow[r]\arrow[d]&\bigoplus_{i=0}^n M(U_{12}\rightarrow U_{2})(i)[2i]\arrow[d]\\
M(\cP_{U_1}\rightarrow \cP)\arrow[r]&\bigoplus_{i=0}^n M(U_1\rightarrow X)(i)[2i].
\end{tikzcd}$$ The vertical morphisms are isomorphisms by Mayer-Vietoris since $\{U_1,U_2\}$ is a Zariski cover of $X$. By induction the claim holds for $U_1$, $U_2$, and $U_{12}$. This implies the upper horizontal morphism is an isomorphism, so that the lower horizontal arrow is also an isomorphism.
As in [@Deg08 Lemma 3.1] there is a commutative diagram $$\begin{tikzcd}[column sep=small]
M(\cP_{U_1})\arrow[d]\arrow[r]&M(\cP)\arrow[d]\arrow[r]&M(\cP_{U_1}\rightarrow \cP)\arrow[d]\arrow[r]&M(\cP_{U_1})[1]\arrow[d]\\
\bigoplus_i M(U_1)(i)[2i]\arrow[r]&\bigoplus_i M(X)(i)[2i]\arrow[r]&\bigoplus_i M(U_1\rightarrow X)(i)[2i]\arrow[r]&\bigoplus_i M(U_1)(i)[2i+1]
\end{tikzcd}$$ The first and third vertical morphisms are isomorphisms, so that the second vertical morphism is also an isomorphism.
\[A.6.10\] In the proof of [@MVW Theorem 15.12] an extra argument is needed to justify the induction argument, see the proofs of [@Deg08 Lemma 3.1, Theorem 3.2].
\[A.6.7\] Assume that $k$ admits resolution of singularities. Let $\cE$ be a rank $n$ vector bundle over $X\in SmlSm/k$. Then there is a canonical isomorphism $$MTh(\cE)\cong M(X)(n)[2n].$$
Recall that we have the morphisms $$\tau_i\colon M(\P(\cE))\rightarrow \Lambda(i)[2i]
\text{ and }
\tau_i\colon M(\P(\cE\oplus \cO))\rightarrow \Lambda(i)[2i]$$ The pullback of the canonical line bundle over $\P(\cE\oplus \cO)$ by the closed immersion $u\colon \P(\cE)\rightarrow \P(\cE\oplus \cO)$ is isomorphic to the canonical line bundle over $\P(\cE)$. This means that there is a commutative diagram $$\begin{tikzcd}
M(\P(\cE))\arrow[r,"\tau_1"]\arrow[d,"M(u)"']&
\Lambda(1)[2]\arrow[d,"{\rm id}"]
\\
M(\P(\cE\oplus \cO))\arrow[r,"\tau_1"]&
\Lambda(1)[2].
\end{tikzcd}$$ Thus for every integer $i\geq 0$, there is a commutative diagram $$\begin{tikzcd}
M(\P(\cE))\arrow[d]\arrow[r,"\Delta"]&
M(\P(\cE))^{\otimes i}\arrow[r,"\tau_1^{\otimes i}"]\arrow[r]\arrow[d,"{\rm id}"]&
\Lambda(1)[2]^{\otimes i}\arrow[d]
\\
M(\P(\cE\oplus \cO))\arrow[r,"\Delta"]&
M(\P(\cE\oplus \cO))^{\otimes i}\arrow[r,"\tau_1^{\otimes i}"]\arrow[r]&
\Lambda(1)[2]^{\otimes i}.
\end{tikzcd}$$
Applying the above for $1\leq i\leq n-1$ we obtain the commutative diagram $$\begin{tikzcd}
M(\P(\cE))\arrow[d,"M(u)"']\arrow[r,"\sigma"]&
\bigoplus_{i=0}^{n-1} M(X)(i)[2i]\arrow[d,"\beta:=\sigma\circ M(u)\circ \sigma^{-1}"']\arrow[rd,"{\rm id}"]
\\
M(\P(\cE\oplus \cO))\arrow[r,"\sigma"]&
\bigoplus_{i=0}^n M(X)(i)[2i]\arrow[r,"p"]&
\bigoplus_{i=0}^{n-1} M(X)(i)[2i],
\end{tikzcd}$$ where $p$ is the canonical projection. It follows that there is a canonical commutative diagram with horizontal split distinguished triangles and vertical isomorphisms $$\begin{tikzcd}[column sep=small]
\bigoplus_{i=0}^{n-1}M(X)(i)[2i]\arrow[d,"\cong"']\arrow[r,"\beta"]&
\bigoplus_{i=0}^{n}M(X)(i)[2i]\arrow[d,"\cong"]\arrow[r]&
M(X)(n)[2n]\arrow[r,"0"]&
\bigoplus_{i=0}^{n-1}M(X)(i)[2i+1]\arrow[d,"\cong"]
\\
M(\P(\cE))\arrow[r]&
M(\P(\cE\oplus \cO))\arrow[r]&
MTh(\cE)\arrow[r,"0"]&
M(\P(\cE))[1].
\end{tikzcd}$$ The canonical composite morphism $$\alpha\colon M(X)(n)[2n]\rightarrow \bigoplus_{i=0}^n M(X)(i)[2i]\xrightarrow{\cong}M(\P(\cE\oplus \cO))\rightarrow MTh(\cE)$$ makes the above diagram into a morphism of distinguished triangles, where the first morphism is the canonical embedding. By the five lemma $\alpha$ is an isomorphism.
\[A.6.8\] Assume that $k$ admits resolution of singularities. For $X\in SmlSm$ and $Z$ a codimension $n$ smooth subscheme of $\underline{X}$ such that $Z$ has strict normal crossing with $\partial X$ and is not contained in $\partial X$, there is a canonical isomorphism $$M((B_Z X,E)\rightarrow X)\rightarrow M(Z)(n)[2n].$$ Here $E$ is the exceptional divisor of the blow-up $B_Z X$.
Immediate from Theorems \[A.3.36\] and \[A.6.7\].
Motives with rational coefficients
----------------------------------
If $\Lambda$ is a $\Q$-algebra, then according to [@MVW Theorem 14.30], there is an equivalence of Voevodsky’s triangulated categories of Nisnevich motives and étale motives $$\dmeff\cong \dmeffet.$$ In what follows, we argue as in [@MVW Lecture 14] and prove an analogue in the log setting: If $\Q\subset \Lambda$, then there is an equivalence of triangulated categories $$\ldmeff\cong \ldmeffet.$$
\[MotRat.1\] Let $\cF$ be a strict Nisnevich sheaf of $\Q$-modules with log transfers. Then $\cF$ is a strict étale sheaf.
Owing to Proposition \[A.5.22\] the strict étale topology is compatible with log transfers. Thus $a_{s\acute{e}t}^*\cF$ is a strict étale sheaf of $\Q$-modules with log transfers. We need to show there is a naturally induced isomorphism $$p\colon \cF\rightarrow a_{s\acute{e}t*}a_{s\acute{e}t}^*\cF.$$ To that end it suffices to show that $$a_{s\acute{e}t}^*(\ker p)=a_{s\acute{e}t}^*(\coker p)=0.$$ Hence we are reduced to consider the case when $a_{s\acute{e}t}^*\cF=0$.
We may assume that $\cF$ is nontrivial. Then there exists a henselization $T$ of an fs log scheme in $lSm/k$ such that $\cF(T)\neq 0$, i.e., $\ul{T}$ is a henseliization of $\ul{Z}$ for some $Z\in lSm/k$ and $T=\ul{T}\times_{\ul{Z}}Z$. Choose a nonzero element $c\in \cF(T)$. Since $a_{s\acute{e}t}^*\cF=0$, there exists a strict étale cover $f\colon X\rightarrow T$ such that $f^*(c)\in \cF(X)$ is zero. Owing to the implication $(a)\Rightarrow (c'')$ in the proof of [@EGA Théorème IV.18.5.11] there exists a factorization $g\colon Y\rightarrow T$ of $f$, where $g$ is a finite étale morphism. It follows that $g^*(c)=0\in cF(Y)$.
The graph of $g\colon Y\rightarrow T$ gives an elementary log correspondence $h\in \lCor(T,Y)$. If $d$ is the degree of $g$, then the composition $g\circ h\in \lCor(T,T)$ is equal to $d$ times the identity correspondence. Thus the composition $$\cF(T)\stackrel{g^*}\rightarrow \cF(Y)\stackrel{h^*}\rightarrow \cF(T)$$ is an isomorphism since $\cF$ is a sheaf of $\Q$-modules. This implies $c=0$ since $g^*(c)=0$, which is a contradiction.
\[MotRat.2\] The proof of Lemma \[MotRat.1\] does not work for the Kummer étale topology. Indeed, in this case, $Y$ is not necessarily strict over $X$, and hence the graph of $g$ is not an elementary log correspondence from $S$ to $Y$.
\[MotRat.3\] Suppose $\cF$ be a dividing Nisnevich sheaf of $\Q$-modules with log transfers. Then $\cF$ is a dividing étale sheaf. Thus if $\Q\subset \Lambda$, there is an equivalence of categories $$\Shv_{dNis}^{\rm ltr}(k,\Lambda)\cong \Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda).$$
Owing to Lemma \[MotRat.1\] $\cF$ is a strict étale sheaf. For any log modification $Y\rightarrow X$ in $lSm/k$ there is an isomorphism $\cF(X)\rightarrow \cF(Y)$ since $\cF$ is a dividing sheaf. To conclude we apply Lemma \[A.5.45\].
\[MotRat.4\] If $\Q\subset \Lambda$, there is an equivalence of triangulated categories $$\ldmeff\cong \ldmeffet.$$
Let $\cG$ be the set of objects of the form $\Zltr(X)$ for $X\in lSm/k$. With respect to this set, we are entitled to the corresponding descent model structures and the equivalent categories $$\Co(\Shv_{dNis}^{\rm ltr}(k,\Lambda))\text{ and }\Co(\Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda)),$$ and hence a Quillen equivalence $$\Co(\Shv_{dNis}^{\rm ltr}(k,\Lambda))\rightleftarrows \Co(\Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda)).$$ Due to [@MR1944041 Theorem 3.3.20(1)] the above is also a Quillen equivalence with respect to the $\boxx$-local descent model structures, and we are done by passing to the homotopy categories.
Locally constant log étale sheaves
----------------------------------
If $\Lambda$ is a torsion ring coprime to the exponential characteristic of $k$, then [@MVW Theorem 9.35] shows there is an equivalence of triangulated categories $$\mathbf{D}^{-}(k_{\acute{e}t},\Lambda)\rightarrow \dmeffetminus.$$ The proof relies on Suslin’s rigidity theorem [@MVW Theorem 7.20]. We refer to [@CDEtale] for a way to remove boundedness. In this subsection, we discuss the log étale version of the above equivalence.
Let $\eta\colon k_{\acute{e}t}\rightarrow lSm/k$ denote the inclusion functor. Then $\eta$ is a continuous and cocontinuous functor with respect to the étale topology on $k_{\acute{e}t}$ and the log étale topology on $lSm/k$. Since $\eta$ preserves finite limits, $\eta$ is a morphism of sites due to [@SGA4 IV.4.9.2]. As a consequence there exist adjunct functors $$\label{equation:ketlog}
\begin{tikzcd}
\Shv(k_{\acute{e}t},\Lambda)
\arrow[rr,shift left=1.5ex,"\eta_\sharp "]
\arrow[rr,"\eta^*" description,leftarrow]
\arrow[rr,shift right=1.5ex,"\eta_*"']&&
\Shv_{l\acute{e}t}^{\rm log}(k,\Lambda),
\end{tikzcd}$$ and $\eta_\sharp$ is exact.
\[ketcomp.3\] A log étale sheaf $\cF$ on $lSm/k$ is *locally constant* if the counit of the adjunction $\eta_\sharp \eta^*\cF\rightarrow \cF$ in is an isomorphism.
\[ketcomp.11\] The functor $\eta_\sharp\colon \Shv(k_{\acute{e}t},\Lambda)\rightarrow \Shv_{l\acute{e}t}^{\rm log}(k,\Lambda)$ is fully faithful.
The functor $\eta\colon k_{\acute{e}t}\rightarrow lSm/k$ is fully faithful, so for every $X\in k_{\acute{e}t}$ the unit $$\Lambda(X)\rightarrow \eta^*\eta_\sharp \Lambda(X)$$ is an isomorphism. Since $\eta_\sharp$ and $\eta^*$ commute with colimits, for every $\cF\in \Shv(k_{\acute{e}t},\Lambda)$ we have $$\eta^*\eta_\sharp\cF
\cong
\eta^*\eta_\sharp \colimit_{X\rightarrow \cF}\Lambda(X)
\cong
\colimit_{X\rightarrow \cF} \eta^*\eta_\sharp\Lambda(X)
\cong
\colimit_{X\rightarrow \cF} \Lambda(X)
\cong
\cF.$$ It follows that $\eta_\sharp$ is fully faithful.
\[ketcomp.14\] Suppose that there is a cartesian square of locally noetherian fs log schemes $$\begin{tikzcd}
Y'\arrow[d,"f'"']\arrow[r,"g'"]&Y\arrow[d,"f"]
\\
X'\arrow[r,"g"]&X
\end{tikzcd}$$ with the following properties.
1. $X$ and $X'$ have trivial log structure, and $X'\cong \limit X_i$ for some projective system of étale schemes of finite type over $X$ such that every transition morphism $X_j\rightarrow X_i$ is affine.
2. $f$ is a morphism of finite type.
Then for every $\cF\in \mathbf{D}(X_{l\acute{e}t},\Lambda)$ there is a base change isomorphism $$g^*Rf_* \cF\xrightarrow{\cong} Rf_*'g'^*\cF.$$
As in the proof of [@CDEtale Theorem 1.1.14], this can be reduced to showing that for every log étale sheaf $\cF$ of $\Lambda$-modules on $X_{l\acute{e}t}$ there is a base change isomorphism $$\label{ketcomp.14.1}
g^*f_*\cF\xrightarrow{\cong} f_*'g'^*\cF.$$ We can also replace $X'$ by $X_i$ in the projective system and $Y'$ by $Y\times_X X_i$ and then apply the limit. Hence we reduce to the case when $g$ is étale. In this case, can be proved directly.
\[ketcomp.1\] Suppose $\Lambda$ is a torsion ring coprime to the exponential characteristic of $k$. Let $X$ be an fs log scheme in $lSm/k$ and $q\in\Z$. For every complex $\cF$ of locally constant sheaves of $\Lambda$-modules on $lSm/k$, the projection morphism $p\colon X\times \boxx\rightarrow X$ induces an isomorphism of log étale cohomolgy groups $$p^*\colon \bH_{l\acute{e}t}^q(X,\cF)\xrightarrow{\cong} \bH_{l\acute{e}t}^q(X\times \boxx,\cF).$$
By applying Proposition \[ketcomp.14.1\] to the case when $g$ is $\Spec{k^s}\rightarrow \Spec{k}$, we reduce to the case when $k$ is separably closed. Then $X$ and $X\times \boxx$ have finite cohomological dimension by virtue of [@MR3658728 Theorem 7.2(2)]. Apply [@weibel_1994 5.7.9] to obtain two strongly convergent hypercohomology spectral sequences $$E_{pq}^2=H^p(X,a_{l\acute{e}t}^* H^q(\cF))\Rightarrow \bH_{l\acute{e}t}^{p+q}(X,\cF).$$ and $$E_{pq}'^2=H^p(X\times \boxx,a_{l\acute{e}t}^* H^q(\cF))\Rightarrow \bH_{l\acute{e}t}^{p+q}(X\times \boxx,\cF).$$ Since there is a naturally induced morphism of spectral sequences $E_{pq}\rightarrow E_{pq}'$, we reduce to case when $\cF$ is a locally constant sheaf of $\Lambda$-modules on $X_{l\acute{e}t}$.
Since $k$ has become separably closed, $\cF$ is a constant torsion sheaf. We can further reduce to the case when $\cF=\Z/n\Z$ for some integer $n>0$ invertible on $k$. Thanks to [@MR3658728 Theorem 5.17] there are isomorphisms $$H_{k\acute{e}t}^q(X,\Z/n\Z)\xrightarrow{\cong} H_{l\acute{e}t}^q(X,\Z/n\Z),
\;
H_{k\acute{e}t}^q(X\times \boxx,\Z/n\Z)\xrightarrow{\cong} H_{l\acute{e}t}^q(X\times \boxx,\Z/n\Z).$$ For this reason it suffices to show that there is an isomorphism of Kummer étale cohomology groups $$p^*\colon H_{k\acute{e}t}^q(X,\Z/n\Z)\xrightarrow{\cong} H_{k\acute{e}t}^q(X\times \boxx,\Z/n\Z).$$
Owing to [@MR1922832 Corollary 7.5] there are isomorphisms $$\begin{aligned}
H_{k\acute{e}t}^q(X,\Z/n\Z)&\xrightarrow{\cong} H_{\acute{e}t}^q(X-\partial X,\Z/n\Z),
\\
H_{k\acute{e}t}^q(X\times \boxx,\Z/n\Z)&\xrightarrow{\cong} H_{\acute{e}t}^q((X-\partial X)\times \A^1,\Z/n\Z).\end{aligned}$$ By $\A^1$-homotopy invariance of étale cohomology [@SGA4 Corollaire XV.2.2] there is an isomorphism $$H_{\acute{e}t}^q(X-\partial X,\Z/n\Z)\xrightarrow{\cong} H_{\acute{e}t}^q((X-\partial X)\times \A^1,\Z/n\Z).$$ Combining these isomorphisms concludes the proof.
\[ketcomp.2\] For every integer $n>1$ coprime to the exponential characteristic of $k$, the sheaf of $n$th roots of unity $\mu_n$ is *not* strictly $\boxx$-invariant in the strict étale topology. Indeed, for every integer $i\in \Z$, we have $$H_{s\acute{e}t}^i(\boxx,\mu_n)
\cong
H_{\acute{e}t}^i(\P^1,\mu_n).$$ In general, the latter group is non-isomorphic to $H_{\acute{e}t}^i(\Spec{k},\mu_n)$. In particular, if $k$ contains all $n$th roots of unity, then $\Z/n$ is *not* strictly $\boxx$-invariant in the strict étale topology.
\[ketcomp.4\] With $\Lambda$ as above, every locally constant log étale sheaf $\cF$ on $lSm/k$ admits a unique log transfer structure.
Owing to [@MVW Lemma 6.11] $\cF$ has a unique transfer structure on $Sm/k$. For every $X\in lSm/k$, apply Proposition \[ketcomp.1\] to have an isomorphism $$\cF(X)\cong \cF(X-\partial X).$$ Thus for every finite log correspondence $V\in \lCor(X,Y)$, where $Y\in lSm/k$, the transfer map $$(V-\partial V)^*\colon \cF(X-\partial X)\rightarrow \cF(Y-\partial Y)$$ extends uniquely to a transfer map $$V^*\colon \cF(X)\rightarrow \cF(Y).$$
With $\Lambda$ as above, as a consequence we have an adjunction $$\label{equation:ketloglambda}
\eta_\sharp: \Shv(k_{\acute{e}t},\Lambda)\rightleftarrows \Shv_{l\acute{e}t}^{\rm ltr}(k,\Lambda):\eta^*,$$ and $\eta_\sharp$ is exact and fully faithful. From we deduce the canonical Quillen adjunction $$\eta_\sharp:\Co(\Shv(k_{\acute{e}t},\Lambda))\rightleftarrows \Co(\Shv_{l\acute{e}t}^{\rm ltr}(k,\Lambda)):\eta^*$$ with respect to the descent model structure on the left side and the $\boxx$-local descent model structure on the right side. On the corresponding homotopy categories, we obtain the canonical adjunction: $$L\eta_\sharp:\mathbf{D}(\Shv(k_{\acute{e}t},\Lambda))\rightleftarrows \ldmefflet:R\eta^*$$ Since $\eta_\sharp$ is exact, $L\eta_\sharp\simeq \eta_\sharp$.
With $\Lambda$ as above, the functor $$\eta_\sharp\colon \mathbf{D}(\Shv(k_{\acute{e}t},\Lambda))\rightarrow \ldmefflet$$ is fully faithful.
Let $\cF$ be an object of $\Co(\Shv(k_{\acute{e}t},\Lambda))$. By Proposition \[ketcomp.1\], $\eta_\sharp \cF$ is strictly $\boxx$-invariant, and hence there is an isomorphism $$R\eta^*\eta_\sharp \cF\simeq \eta^*\eta_\sharp\cF.$$ To conclude we observe that the functor $$\eta_\sharp\colon \mathbf{C}(\Shv(k_{l\acute{e}t},\Lambda))\rightarrow \mathbf{C}(\Shv_{l\acute{e}t}^{\rm ltr}(k,\Lambda))$$ is fully faithful since the functor $\eta_\sharp\colon \Shv(k_{l\acute{e}t},\Lambda)\rightarrow \Shv_{l\acute{e}t}^{\rm log}(k,\Lambda)$ is fully faithful.
Next we state a rigidity conjecture which is a log analogue of [@MVW Theorem 9.35].
\[conjrigidity\] With $\Lambda$ as above, the functor $$\eta_\sharp\colon \mathbf{D}(\Shv(k_{l\acute{e}t},\Lambda))\rightarrow \ldmefflet$$ is an equivalence.
Related to Conjecture \[conjrigidity\] we have the following log analogue of Suslin’s rigidity theorem [@MVW Theorem 7.20].
\[ketcomp.5\] Every strictly $\boxx$-invariant log étale sheaf $\cF$ equipped with log transfers and torsion coprime to the exponential characteristic of $k$ is locally constant.
It is unclear whether one can expect that every $\boxx$-invariant log étale sheaf with log transfers is locally constant. The sheaf $a_{l\acute{e}t}^*\Zltr(\A^1)$ is a potential counterexample, see Example \[A.4.28\].
Comparison with Voevodsky’s étale motives
-----------------------------------------
In this section we state our expectations regarding an étale version of Theorem \[thm::dmeff=ldmeffprop\]. Since Lemma \[A.4.4\] fails in the étale topology, the proof of Theorem \[thm::dmeff=ldmeffprop\] cannot be directly adopted to étale log motives.
\[ketcomp.6\] There is an equivalence of triangulated categories $$\ldmeffet\cong \ldmefflet.$$
Theorem \[thmHodge\] on Hodge sheaves provides some first evidence for Conjecture \[ketcomp.6\] in form of the isomorphism $$\hom_{\ldmeffet}(M(X),\Omega_{/k}^j[i])\xrightarrow{\cong} \hom_{\ldmefflet}(M(X),\Omega_{/k}^j[i]).$$ On the other hand, since $\Z/n$ is not strictly $\boxx$-invariant for the strict étale topology, it is unclear whether the cohomology groups $$\hom_{\ldmeffet}(M(X),\Z/n[i])\text{ and } \hom_{\ldmefflet}(M(X),\Z/n[i])$$ are isomorphic, see Example \[ketcomp.2\].
\[ketcomp.7\] Suppose that $\Lambda=\Z/n$, where $n$ is invertible in $k$. Then there are equivalences of triangulated categories $$\ldmeffet\leftarrow \ldmeffetprop\rightarrow \dmeffet,$$ and $$\ldmefflet\leftarrow \ldmeffletprop\rightarrow \dmeffet,$$
The Hodge sheaves $\Omega_{/k}^j\in\ldmeffet$ by Theorem \[thmHodge\]. We expect the same holds for the de Rham-Witt sheaves $W_m\Omega_{/k}^j$. Moreover, one may guess that all the objects of $\ldmeffet$ can be build from $W_m\Omega_{/k}^j$ and objects of $\dmeffet$. Due to the vanishing $$W_m\Omega_{X/k}^j(X)\otimes \Z[1/p]=0,$$ where $p$ denotes the characteristic of the base field $k$, the de Rham-Witt sheaves may become trivial in $\ldmeffet$ under our assumption. This indicates that $\ldmeffet$ and $\dmeffet$ are comprised of the same objects.
Hodge sheaves {#sec:Hodge}
=============
The main goal of this section is to show that Hodge cohomology for (log) schemes defined over a perfect field $k$ is representable in the category of effective log motives $\ldmeff$. This demonstrates in clear terms a fundamental difference between our category and Voevodsky’s category of motives $\dmeff$. Hodge cohomology is not representable in the latter category.
In the main step towards representability of Hodge cohomology, see Theorem \[thm:Omega\_log\_transfer\], we show that the presheaf $\Omega^j$ on $lSm/k$ given by the assignment $$X\mapsto \Omega^j_{X/k}(X)$$ extends to a presheaf with log transfers on $lSm/k$ in the sense of Section \[sec:logtransfers\]. Analogous to an argument in [@BS §6.2], showing $(\P^{\bullet},\P^{\bullet-1})$-invariance is then fairly straightforward by using a suitable interpretation of the projective bundle formula.
The action of finite correspondences on Hodge sheaves (without log structure) is well-known, and follows for example from the computations in [@lecomtewach] (at least in characteristic zero). A more conceptual approach, based on coherent duality and cohomology with support, was developed by Chatzistamatiou and Rülling in [@ChatzistamatiouRullingANT] and [@ChatzistamatiouRullingDocumenta]. We will basically adapt the same script to the logarithmic setting.
Logarithmic derivations and differentials
-----------------------------------------
For every fs log scheme $X$ over $k$ and $i\geq 0$ the sheaf of logarithmic differentials on $X$ is defined by $$\Omega^i_{X/k}
:=
\begin{cases}
\cO_X & i=0 \\
\wedge^i \Omega^1_{X/k} & i>0.
\end{cases}$$ We shall often omit the subscript $/k$ and write $\Omega_X^i$ when no confusion seems likely to arise. Recall that $\Omega^1_{\underline{X}}$ is the sheaf of $\cO_{X}$-modules generated by universal log derivations. We refer to [@Ogu Chapter IV] and Section \[log-differential\] for details.
If $X\in SmlSm/k$, and $(t_1,\ldots, t_n)$ are local coordinates, we recall that $\Omega^1_{X}$ is the free $\cO_{\underline{X}}$-module generated by the symbols $dt_1, \ldots dt_n$. For the open immersion $$j\colon \underline{X}- \partial X \to \underline{X},$$ the sheaf $$\Omega^1_{X}=\Omega^1_{\underline{X}}(\log \partial X)$$ of differentials with log poles along $\partial X$ is the subsheaf of $j_*j^* \Omega^1_{\underline{X}}$ generated by $$d\log t_1, \ldots, d\log t_r, dt_{r+1}, \ldots dt_n.$$ Here, the ideal sheaf of $\partial X$ in $\underline{X}$ is locally defined by $I=(t_1\cdots t_r)$.
A *quasi-coherent* sheaf $\cF$ on an fs log scheme $X$ is a quasi-coherent sheaf on $\underline{X}$. We similarly define *coherent* sheaves and *locally free* coherent sheaves on $X$. For example, $\Omega_{X}^i$ is a coherent sheaf for every fs log scheme $X$ of finite type over $k$. Indeed, if $i=1$ this is [@Ogu Lemma IV. 1.2.16] and $\Omega_{X}^i$ is the $i$th exterior power of $\Omega_{X}^1$ for $i>1$. Moreover, if $X\in SmlSm/k$, then $\Omega_X^i$ is locally free.
By generalizing [@Milneetale Proposition III 3.7, Remark III 3.8] to the Nisnevich case we obtain isomorphisms $$\label{eqn:coherentcohomology}
H_{Zar}^i(X,\cF)\cong H_{sNis}^i(X,\cF)\cong H_{s\acute{e}t}^i(X,\cF)$$ for every quasi-coherent sheaf $\cF$ on $X$ and integer $i\geq 0$.
\[Omega\_dividing\_sheaf\] Let $X$ be an fs log scheme. Then every quasi-coherent sheaf $\cF$ on $X$ is a log étale sheaf.
The question is Zariski local on $X$. Hence we may assume that there exists a presentation of the form $$\bigoplus_{j\in J}\cO_X\rightarrow \bigoplus_{i\in I}\cO_X\rightarrow \cF\rightarrow 0.$$ Since the sheafification functor is exact, we only need to show that $\cO_X$ is a log étale sheaf. This follows from [@KatoLog2 §3.3].
$(\P^n, \P^{n-1})$-invariance of logarithmic differentials {#PnPn-1invariancediff}
----------------------------------------------------------
For every $n\geq 1$, let $(\P^n, \P^{n-1})$ be the log smooth log scheme $\P^n = \P^n_k$ with compactifying log structure induced by the open immersion $$j\colon \A^n =\P^n- H\hookrightarrow \P^n,$$ where $H\cong\P^{n-1} \subset \P^n$ is a $k$-rational hyperplane. If $X$ is an fs log scheme in $lSm/k$, we form the product $X\times (\P^n, \P^{n-1})$ and the projection $$\pi\colon X\times (\P^n, \P^{n-1}) \to X.$$
\[prop:boxinvdiff\] Suppose $X\in SmlSm/k$. For every $i\geq 1$ the naturally induced map $$\pi^*\colon \Omega^i_{X} \to {R}\pi_*\Omega^i_{X\times (\P^n, \P^{n-1})}$$ is an isomorphism in the bounded derived category of coherent sheaves ${\bf D}^b(\underline{X}_{\rm Zar})$.
(Cf. [@BS §6.2]). We begin by setting our notation. Let $X = (\underline{X}, \partial X)$, where $\partial X$ is a strict normal crossing divisor in $\underline{X}$. Let $\iota_H\colon H\to \P^n$ be the closed embedding of the rational hyperplane $H\cong \P^{n-1}$ in $\P^n$. We need to prove there is an isomorphism $$\label{eq:eqboxinvdiff}
\pi^* \colon \Omega^i_{\underline{X}}(\log \partial X) \to R\pi_*\Omega^i_{\underline{X} \times \P^n}(\log H+\partial X)$$ in $D^b(\underline{X}_{\rm Zar})$. On the right hand side of , we write $H+\partial X$ for the strict normal crossing divisor on $\underline{X}\times \P^n$ given by $\underline{X} \times H + \partial X \times \P^n$. We shall prove the claim by induction on the number of components of $\partial X$.
Let us write $$\partial X = \partial X_1 + \ldots + \partial X_m,$$ where the $\partial X_i$’s are the irreducible components of $\partial X$. When $\partial X = \emptyset$, the statement follows from the projective bundle formula for the sheaves of logarithmic differential forms. This should be well known, but for the lack of a reference we give the argument.
Write $Res_H$ for the residue map along $\ul{X}\times H \subset \ul{X}\times \P^n$ that fits into a short exact sequence (see e.g., [@EVbuch §2.3]) $$\label{eq:eqboxinvdiff2}
0\to \Omega^i_{\ul{X}\times \P^n} \to \Omega^i_{\ul{X}\times \P^n}(\log H) \xrightarrow{Res_H} \iota_{H,*}\Omega^{i-1}_{\ul{X}\times H}\to 0$$ of $\cO_{\underline{X}\times \P^n}$-modules. Pushing forward to $\ul{X}$ we get a distinguished triangle $$\label{eq:eqboxinvdiff3}
R\pi_{H,*} \Omega^{i-1}_{\ul{X}\times H}[-1] \xrightarrow{f} R\pi_{ *}\Omega^i_{\ul{X}\times \P^n} \to R\pi_* \Omega^i_{\ul{X}\times \P^n}(\log H)
\rightarrow R\pi_{H,*} \Omega^{i-1}_{\ul{X}\times H}$$ in ${\bf D}^b(\ul{X}_{\rm Zar})$, where $\pi_H\colon \ul{X}\times H\to \ul{X}$ is the projection. Taking cup products with the powers of the first Chern class $$\xi = c_1(\cO_{\P^n}(H)) \in H_{Zar}^1(\P^n, \Omega^1_{\P^n})$$ of $H$ determines an isomorphism in the derived category of bounded complexes of $\cO_{\ul{X}}$-modules $$\label{eq:eqboxinvdiff4}
\bigoplus_{0\leq j\leq n} \Omega^{i-j}_{\ul{X}} [-j] \xrightarrow{\simeq} R\pi_* \Omega^i_{\ul{X}\times \P^n}, \quad (a_0, \ldots, a_n)\mapsto \sum_{0\leq j\leq n} \pi^*(a_j) \cup \xi^j.$$ This implies the projective bundle formula in Hodge cohomology for the trivial bundle. The existence of this map is clear once one interprets $\xi$ as a section of $R^1\pi_* \Omega^1_{\ul{X}\times \P^n}$, and the quasi-isomorphism is well-known to follow from a direct local computation. See for example [@ArapuraKang Lemma 3.2] but the result is much older (see [@MR265370]).
Similarly, there is an isomorphism $$\label{eq:eqboxinvdiff5}
\bigoplus_{0\leq j\leq n-1} \Omega^{i-1-j}_{\ul{X}}[-j] \xrightarrow{\simeq} R (\pi_H)_* \Omega^{i-1}_{\ul{X}\times H}, \quad (a_0, \ldots, a_{n-1})\mapsto \sum_{0\leq j\leq n-1} \pi_H^*(a_j) \cup \xi^j_H,$$ where $$\xi_H\in H^1(H, \Omega^1_{H/k})$$ is the restriction $\iota_H^*(\xi)$ of $\xi$ to $H$. For every local section $a\in \Omega^{i-1-j}_{\ul{X}}$ a direct local computation and the definition of the residue map imply $$\label{eq:eqboxinvdiff9}
f( \pi_H^*(a)\cup \xi_H^j) = \pi^*(a)\cup \xi^{j+1}.$$
Combining , , , and we obtain the commutative diagram $$\label{eq:eqboxinvdiff6}
\begin{tikzcd} \bigoplus_{0\leq j\leq n-1} \Omega^{i-1-j}_{\ul{X}}[- j-1] \arrow[r, "\simeq"] \arrow[d, swap, "\nu"] & R\pi_{H,*} \Omega^{i-1}_{\ul{X}\times H}[-1] \arrow[d, "f"]\\
\bigoplus_{0\leq j\leq n} \Omega^{i-j}_{\ul{X}} [-j] \arrow[r, "\simeq"] & R\pi_{ *}\Omega^i_{\ul{X}\times \P^n}
\end{tikzcd}$$ where the left vertical morphism $\nu$ is given by the (split) inclusion into the direct sum $$\bigoplus_{0\leq j\leq n} \Omega^{i-j}_{\ul{X}} [-j]$$ and the horizontal morphisms are quasi-isomorphisms. Using and the equality $\pi^*(-) = \pi^*(-)\cup \xi^0$. We deduce the isomorphism by noting that the cokernel of $\nu$ is exactly $\Omega^i_{\ul{X}}$.
Next, we assume that $\partial X \neq \emptyset$, so that $m\geq 1$. We proceed following the script of Sato [@SatoLogHW §2] (a sketch for relative differentials can be found in [@BS §6.2]).
Write $I = \{1, \ldots, m \}$ and for $1\leq a\leq m$, we define the disjoint union $$\partial X^{[a]} = \coprod_{1\leq i_1<\cdots< i_a\leq m} \partial X_{i_1}\cap \ldots \cap \partial X_{i_a}.$$ Note that each $\partial X^{[a]}$ is regular (since $\partial X$ is a strict normal crossing divisor on $\ul{X}$), and that there is a canonical morphism $$\nu_a\colon \partial X^{[a]}\to \ul{X}$$ for each $a\geq 1$. The log structure on $X$ induces the compactifiying log structure on each $\partial X^{[a]}$. More explicitely, we have $$E_a
=
\coprod_{1\leq i_1< \ldots < i_a\leq m} \big( (\partial X_{i_1}\cap \ldots \cap \partial X_{i_a}) \cap \sum_{j\notin \{i_1, \ldots, i_a\} } \partial X_j \big).$$ This is a normal crossing divisor on $\partial X^{[a]}$ (with strictly less then $m$ components on each irreducible component of $\partial X^{[a]}$). The $cdh$-covering of $\partial X$ given by the $\partial X^{[a]}$’s gives rise to the exact sequence of sheaves $$\begin{aligned}
\label{eq:exboxinvdiff7}
0\to \Omega^{i}_{\ul{X}} \xrightarrow{\epsilon_X} \Omega^i_{\ul{X}}(\log \partial X) \to
& \nu_{1, *} \Omega^{i-1}_{\partial X^{[1]}} \xrightarrow{\rho_2} \nu_{2, *}\Omega^{i-2}_{\partial X^{[2]}} \to\ldots \\ \nonumber
\ldots& \to \nu_{a, *} \Omega^{i-a}_{\partial X^{[a]}} \xrightarrow{\rho_a} \nu_{a+1, *} \Omega^{i-a-1}_{\partial X^{[a+1]}}\to \ldots,\end{aligned}$$ on $\ul{X}$, where each morphism $\rho_{a}$ is induced by the alternating sum on residues, see e.g., [@SatoLogHW Proposition 2.2.1].
We may repeat the construction above for $X\times (\P^n, \P^{n-1})$. First note that there is an evident canonical morphism of log schemes $$X\times (\P^n, \P^{n-1}) \to \ul{X}\times (\P^n, \P^{n-1}).$$ Letting $\nu_a$ denote the natural map $\partial X^{[a]} \times \P^n \to \ul{X}\times \P^n$, we get the following exact sequence of sheaves on $\ul{X}\times \P^n$ (using the above convention that $H$ stands for $\ul{X}\times H$ as a divisor on $\ul{X}\times \P^n$) $$\begin{aligned}
\label{eq:exboxinvdiff8}
0\to \Omega^{i}_{\ul{X}\times \P^n}(\log H)& \xrightarrow{\epsilon_{X\times \P^n}} \Omega^i_{\underline{X} \times \P^n}(\log H+\partial X) \to
\nu_{1, *} \Omega^{i-1}_{\partial X^{[1]}\times \P^n} \xrightarrow{\rho_2} \ldots \\
\nonumber \ldots & \to \nu_{a, *} \Omega^{i-a}_{\partial X^{[a]}\times \P^n} \xrightarrow{\rho_a}
\nu_{a+1, *} \Omega^{i-a-1}_{\partial X^{[a+1]}\times \P^n} \to \ldots \end{aligned}$$ Here, on each scheme $\partial X^{[a]}\times \P^n$ we consider the compactifying log structure given by the inclusion of the complement of the normal crossing divisor $E^{[a]}\times \P^n + \partial X^{[a]} \times H$. To simplify the notation, we set $$A_a:= \Omega^{i-a}_{\partial X^{[a]}\times \P^n}
\text{ and }
B_a:=\Omega^{i-a}_{\partial{X}^{[a]}}.$$ Let $F_A$ be the cokernel of $\epsilon_{X\times \P^n}$ and let $F_B$ be the cokernel of $\epsilon_X$. Then using we obtain the spectral sequence $$R^{a+q}\pi_* (\nu_{a+1, *} A_{a+1}) \Rightarrow R^{a+q}\pi_* F_A.$$
Now, by induction, there is an isomorphism $$\pi^*\colon \nu_{a+1, *} B_{a+1}\xrightarrow{\simeq} R^{a+q}\pi_* (\nu_{a+1, *} A_{a+1}),$$ and hence also $$F_B \xrightarrow\simeq R \pi_* F_A$$ using the above spectral sequence. The proposition follows now from the 5-lemma and the case $\partial X = \emptyset$ by comparing the two distinguished triangles $$\Omega^i_{\ul{X}} \xrightarrow{\epsilon_X} \Omega^i_{\ul{X}}(\log \partial X) \to F_B\to \Omega^i_{\ul{X}}[1]$$ and $$R\pi_* \Omega^{i}_{\ul{X}\times \P^n}(\log H)
\xrightarrow{\epsilon_{X\times \P^n}} R\pi_*\Omega^i_{\underline{X} \times \P^n}(\log H+\partial X) \to R\pi_* F_A \to R\pi_* \Omega^{i}_{\ul{X}\times \P^n}(\log H) [1]$$ in the bounded derived category ${\bf D}^b(\ul{X}_{\rm Zar})$.
\[Pn-invariance of Log differentials-Zar\] Let $X$ be an fs log scheme in $SmlSm/k$. Then for every $i$, $j$, and $n$, there is an isomorphism $$H_{Zar}^i(X\times (\P^n,\P^{n-1}),\Omega^j)\cong H_{Zar}^i(X,\Omega^j).$$
This is immediate from Proposition \[prop:boxinvdiff\].
\[prop::set=ket\] Let $X$ be an fs log scheme, and let $\cF$ be a quasi-coherent sheaf on $X$. Then for every $i\geq 0$ there is an isomorphism $$H_{s\acute{e}t}^i(X,\cF)\xrightarrow{\cong} H_{k\acute{e}t}^i(X,\cF).$$
The question is strictly étale local on $X$. Hence we may assume that $X$ is affine. Our claim follows now from Kato’s [@KatoLog2 Proposition 6.5] with a proof given by Niziol in [@MR2452875 Proposition 3.27].
\[Invariance of coherent cohomology under log blow-ups\] Let $f\colon Y\rightarrow X$ be a log modification in $lSm/k$, and let $\cF$ be a locally free coherent sheaf on $X$. Then for every $i\geq 0$ and $j\geq 0$ there is an isomorphism $$H_{s\acute{e}t}^i(X,\cF)\xrightarrow{\cong} H_{s\acute{e}t}^i(Y,f^*\cF).$$
The question is strict étale local on $X$, so we may assume that $X$ has an fs chart $P$. Owing to Proposition \[A.9.21\] there is a subdivision $\Sigma$ of $\Spec{P}$ such that there is an isomorphism $$Y\times_{\A_P}\A_\Sigma\xrightarrow{\cong} X\times_{\A_P}\A_\Sigma.$$ Let $p:X\times_{\A_P}\A_\Sigma\rightarrow X$ and $q:Y\times_{\A_P}\A_\Sigma\rightarrow X$ be the projections. There is a naturally induced commutative diagram $$\begin{tikzcd}
H_{s\acute{e}t}^i(X,\cF)\arrow[d]\arrow[r]&H_{s\acute{e}t}^i(X\times_{\A_P}\A_\Sigma,p^*\cF)\arrow[d,"\cong"]
\\
H_{s\acute{e}t}^i(Y,f^*\cF)\arrow[r]&X_{s\acute{e}t}^i(Y\times_{\A_P}\A_\Sigma,q^*\cF).
\end{tikzcd}$$ The horizontal morphisms are isomorphisms due to [@Vetere Remark 5.19], which is a consequence of [@MR1296725 Theorem 11.3] and the projection formula [@stacks-project Tag 01E8]. Thus also the left vertical morphism is an isomorphism.
\[Invariance of Log differentials under log blow-ups\] Let $f\colon Y\rightarrow X$ be a log modification in $lSm/k$. Then for every $i\geq 0$ and $j\geq 0$ there is an isomorphism $$f^*\colon H_{s\acute{e}t}^i(X,\Omega^j)\xrightarrow{\cong} H_{s\acute{e}t}^i(Y,\Omega^j).$$
Since $f$ is log étale we have $f^*\Omega_X^j\cong \Omega_Y$. Proposition \[Invariance of coherent cohomology under log blow-ups\] allows us to conclude the proof.
\[prop::hodgelet\] Let $X$ be an fs log scheme in $lSm/k$. For every $i\geq 0$ and $j\geq 0$ there exist functorial isomorphisms $$\begin{split}
\label{eqn::hodgelet1}
H_{Zar}^i(X,\Omega^j)
\cong^{(1)} &
H_{sNis}^i(X,\Omega^j)
\cong^{(3)}
H_{dNis}^i(X,\Omega^j)
\\
\cong^{(1)} &
H_{s\acute{e}t}^i(X,\Omega^j)
\cong^{(3)}
H_{d\acute{e}t}^i(X,\Omega^j)
\\
\cong^{(2)} &
H_{k\acute{e}t}^i(X,\Omega^j)
\cong^{(3)}
H_{l\acute{e}t}^i(X,\Omega^j)
\end{split}$$
The isomorphisms labeled (1) follow from the fact that the cohomology of coherent sheaves with respect to the Zariski, Nisnevich, and étale topologies coincides, see . Proposition \[prop::set=ket\] implies (2). Finally, the isomorphisms labeled (3) follow from Theorem \[Div.3\] and Corollary \[Invariance of Log differentials under log blow-ups\].
The comparison of strict étale, Kummer étale, and log étale Hodge cohomology groups are also carried out in [@Vetere Theorems 6.13, 6.14].
\[Pn-invariance of Log differentials\] Let $X$ be an fs log scheme in $lSm/k$. Suppose that $t$ is one of following topologies: $Zar$, $sNis$, $s\acute{e}t$, $k\acute{e}t$, $dNis$, $d\acute{e}t$, and $l\acute{e}t$. Then for every $i$, $j$, and $n$, there is an isomorphism $$H_{t}^i(X\times (\P^n,\P^{n-1}),\Omega^j)\cong H_{t}^i(X,\Omega^j).$$
Owing to Corollary \[Invariance of Log differentials under log blow-ups\] and Propositions \[prop::hodgelet\] and \[A.3.19\], we reduce to the case when $X\in SmlSm/k$. Then apply Corollary \[Pn-invariance of Log differentials-Zar\].
A recollection on Grothendieck duality
--------------------------------------
We recall some material from Grothendieck-Verdier duality theory for coherent sheaves [@HartshorneRD]. If $\ul{X}$ is a separated scheme of finite type over $k$, and $\pi_{\ul{X}}\colon \ul{X}\to \Spec{k}$ is the structural morphism, we have a pair of adjoint functors $(R \pi_{\underline{X}, *}, \pi_{\ul{X}}^!)$ between ${\bf D}^b(k)$ and ${\bf D}^b(\ul{X}_{\rm Zar}) = {\bf D}^b(\ul{X})$ (since we only consider the Zariski topology we often omit the subscript). More generally, if $f\colon \ul{X}\to \ul{Y}$ is any morphism of finite type, there is an adjoint pair $$Rf_*\colon {\bf D}^b(\ul{X})\leftrightarrows {\bf D}^b(\ul{Y}) \colon f^!$$ between the bounded derived categories of coherent sheaves. The adjunction induces a natural transformation of functors $$Rf_* R \Hom_{\ul{X}}(-, f^{!}(-)) \xrightarrow{\simeq} R \Hom_{\ul{Y}}(Rf_*(-), -)$$ called the standard Grothendieck duality isomorphism ($\Hom$ refers to the inner Hom of sheaves on $\ul{X}_{\rm Zar}$ to avoid confusion with the inner Hom in the categories of motives). We denote by $D_{\ul{X}}$ the dualizing functor $$D_{\ul{X}}
=
R \Hom_{\ul{X}}(-, \pi_{\ul{X}}^{!}(k))
\colon
{\bf D}^b(\ul{X})\to {\bf D}^{b}(\ul{X})$$
If $\ul{X}$ is smooth of dimension $d$ over $k$, then there is a canonical isomorphism $$\pi_{\ul{X}}^! k \cong \Omega^d_{\ul{X}}[d].$$ Moreover, for every $j\geq 0$ and $n\in \Z$, there is an isomorphism (canonical up to the choice of a sign) $$\Omega_{\ul{X}}^j[n]\xrightarrow{\simeq} D_{\ul{X}}(\Omega^{d-j}_{\ul{X}})[n-d].$$ The latter isomorphism is induced by taking exterior products of forms $$\Omega^j_{\ul{X}}
\xrightarrow{\simeq}
\Hom_{\ul{X}}(\Omega^{d-j}_{\ul{X}}, \Omega^d_{\ul{X}}), \quad \omega
\mapsto
(\eta\mapsto \omega\wedge \eta).$$
Our next goal is to extend the above results to the log setting. For the analytic setting, see Esnualt-Viehweg [@EsnaultViehweglogDR], which in turn was inspired by analogous results in the theory of $\mathcal{D}$-modules (see e.g., Bernstein [@BBDModules]). We begin with the following
Let $X$ be an fs log scheme in $SmlSm/k$. By Lemma \[lem::SmlSm\], we have that $X = (\ul{X}, \partial X)$, where $\ul{X}$ is a smooth $k$-scheme and $\partial X =\partial X_1+\ldots \partial X_r$ is a strict normal crossing divisor on $\ul{X}$. We let $I_{\partial X}$ be the invertible sheaf of ideals defining $\partial X$.
Let $X$ be an integral fs log scheme in $SmlSm/k$ of dimension $d_X$. Then, for any $j\geq 0$ and every $n$, there is an isomorphism $$\label{eq:dualitylog}
\theta_X\colon \Omega^j_X[n] = \Omega^j_{\ul{X}}(\log \partial X)[n]
\xrightarrow{\simeq}
D_{\ul{X}} (\Omega^{d_X-j}_{X} \tensor_{\cO_{\ul{X}}} I_{\partial X})[n-d_X]$$ in ${\bf D}^b(\ul{X})$.
To prove , we note there is an isomorphism of sheaves $$\label{eq:dualitylogbis}
\Omega^j_{\ul{X}}(\log \partial X)
\xrightarrow{\simeq}
\Hom_{\ul{X}}(\Omega^{d_X-j}_{\ul{X}}(\log \partial X) \tensor_{\cO_{\ul{X}}} I_{\partial X}, \Omega^{d_X}_{\ul{X}} )$$ induced by the exterior product of forms $$\omega\mapsto (\beta \tensor u \mapsto \omega\wedge (\beta \tensor u)),$$ where $u$ is a local section of $I_{\partial X}$. Here we are implicitly making a choice of sign since the assignment $\beta\tensor u \mapsto \beta\wedge (\omega \tensor u)$ is equally valid. We note the inclusion $$\Omega^{d_X-j}_{\ul{X}}(\log \partial X) \tensor_{\cO_{\ul{X}}} I_{\partial X} \subset \Omega^{d_X-j}_{\ul{X}}$$ (every section of the sheaf is a regular differential).
To show is an isomorphism we resort to a computation in local coordinates. We have $$\Omega^{d_X}_{\ul{X}}(\log \partial X) \tensor I_{\partial X} \cong \Omega^{d_X}_{\ul{X}},$$ since $$(d\log x_1 \wedge \ldots d\log x_r \wedge dx_{r+1} \wedge \ldots dx_{d_X})\, x_1\cdots x_r = dx_1\wedge \ldots dx_{d_X},$$ where $(x_1\cdots x_r) = I_{\partial X}$ locally around any point of $\partial X$. Next, note that for every integer $n\in \Z$, we have $$\begin{aligned}
\label{eq:dualitylog2}
&\Hom_{\ul{X}}(\Omega^{d_X-j}_{\ul{X}}(\log \partial X) \tensor_{\cO_{\ul{X}}} I_{\partial X}, \Omega^{d_X}_{\ul{X}} )[n] && \\
\nonumber&= \Hom^{\bullet}_{\ul{X}}(\Omega^{d_X-j}_{\ul{X}}(\log \partial X) \tensor_{\cO_{\ul{X}}} I_{\partial X}, \Omega^{d_X}_{\ul{X}}[d_X])[n-d_X] &&
\text{(sheaf ${\rm Hom}$ complex)}\\\nonumber
& =R {\Hom}_{\ul{X}}( \Omega^{d_X-j}_{\ul{X}}(\log \partial X) \tensor_{\cO_{\ul{X}}} I_{\partial X} , \pi_X^{!}(k))[n-d_X] &&
\text{($\Omega^{d_X-j}_{\ul{X}}(\log \partial X)$ is locally free) }\end{aligned}$$ Here, we have used the notation $\Hom_{\ul{X}}^\bullet$ for the inner Hom of complexes of sheaves on $\ul{X}$, and $R\Hom$ for the derived functor of $\Hom_{\ul{X}}$. By combining with we get the desired isomorphism.
1. One can use the twisted dualizing functor $$D_X = D_{\ul{X}}( - \tensor I_{\partial X})$$ to prove that holds more generally for the logarithmic de Rham complex $DR_{\partial X} (\mathcal{V})$ associated to a coherent $\cO_{\ul{X}}$-module $\mathcal{V}$ equipped with a meromorphic connection with regular singularities along $\partial X$. For details see [@EsnaultViehweglogDR (A.2)].
2. Let $j\colon U=\ul{X}- \partial X\hookrightarrow \ul{X}$ be the open immersion of the complement of the boundary divisor on $\ul{X}$. Then, for every $j\in \Z$, we have $$\Omega^j_U
=
(\Omega^j_X)_{| U} \cong D_{\ul{X}}(\Omega^{d_X-j}_X\tensor I_{\partial X})_{|U}[-d_X] = D_U(\Omega^{d_X-j}_U)[-d_X].$$
Cohomology with support and cycle classes
-----------------------------------------
Following [@ChatzistamatiouRullingANT], in order to define the action of finite log correspondences on logarithmic differentials, we will need to construct a cycle class in Hodge cohomology associated to a closed subscheme. A priori this is potentially a subtle point, since we might need to deal with closed immersions of schemes which are not strict as morphisms of log schemes (this is the case for an arbitrary finite log correspondence). As it turns out, this can be avoided as discussed later in the section, and the following discussion of cohomology with support in the log setting suffices for our purposes.
Let $X$ be an integral fs log scheme of finite type over $k$, and let $v\colon Z\hookrightarrow X$ be a strict closed immersion of log schemes. For $\cF$ a Zariski sheaf of abelian groups on $X_{Zar}$ we set $$\Gamma_Z(X,\cF):=\ker (\cF(X) \to \cF(X-Z)).$$ It is elementary to see that the functor $\Gamma_Z(X,-)$ is left exact and there exists an induced derived functor $$R\Gamma_Z(X,-):\Deri^b(X_{Zar})\rightarrow \Deri^b({\bf Ab}).$$
We define cohomology with support by setting $$H_Z^i(X,\cF):=R\Gamma_Z^i(X,\cF).$$ Note that we have $R\Gamma_X(X,-)=R\Gamma(X, -)$. For $Z\subset X$ strict as above, we define the *Hodge cohomology of $X$ with support in $Z$* (cf. [@ChatzistamatiouRullingANT §2]) as the sum $$H_{\rm Hodge}^*(X, Z) = \bigoplus_{i,j \geq 0}H^i_{Z}(X, \Omega^j_X).$$ If $Z$ is empty, we write $H_{\rm Hodge}^*(X)$ for $H_{\rm Hodge}^*(X, \emptyset)$.
\[constr:pullback\] For $f\colon X\to Y$ a morphism and $Z$ a strict closed subscheme of $Y$, we write $f^{-1}(Z)$ for $X\times_Y Z$ seen as a strict closed subscheme of $X$. There is an isomorphism of functors $$R\Gamma_{f^{-1}(Z)} \xrightarrow{\cong} R\Gamma_{Z} Rf_*.$$
We say that a strict closed subscheme $W$ of $X$ contains $f^{-1}(Z)$ if the strict closed immersion $f^{-1}(Z)\hookrightarrow X$ admits a factorization $f^{-1}(Z)\subset W \hookrightarrow X$. In this situation the latter isomorphism furnishes a natural transformation $$\label{eq:pullbacksupport}
f^*\colon
R\Gamma_Z \to R\Gamma_{Z} Rf_* Lf^*
\xrightarrow{\cong}
R\Gamma_{f^{-1}(Z)}Lf^*
\to
R\Gamma_W Lf^*.$$ See [@ChatzistamatiouRullingANT §2.1.5] for more details. The displayed natural transformations are morphisms in the functor category from $\Deri^b({Y}_{Zar})$ to $\Deri^b({\bf Ab})$. Here the log structures on $X$ and $Y$ are irrelevant since only involves standard operations on sheaves. The pullback $f^*$ in (\[eq:pullbacksupport\]) is compatible with composition, see [@ChatzistamatiouRullingANT (2.1.8)].
\[rmk:pullback\]
1. Every object and every morphism involved in the natural transformation is defined inside the usual bounded derived category of coherent sheaves on $\ul{X}$ or $\ul{Y}$. In particular, the sheaf $$\Omega^j_{X} = \Omega^j_{\ul{X}}(\log \partial X)$$ is considered as a coherent sheaf on $\ul{X}$ (and likewise for $\Omega^j_{Y}$ and $\ul{Y}$). In other words, the log structures on $X$ and $Y$ do not play any role in the definition of the derived categories that we consider, and the log geometric information is completely captured by the sheaves of log differentials.
Since $\Omega^j_Y$ is locally free when $Y\in SmlSm/k$, we have $Lf^* \Omega^j_Y=f^*\Omega^j_Y$ and composing with the morphism $f^*\Omega^j_Y\to \Omega^j_X$ from [@Ogu Proposition IV.1.2.15] we obtain $$f^*\colon R\Gamma_Z \Omega^j_Y \to R\Gamma_W \Omega^j_X.$$ In particular, this yields the pullback morphism in Hodge cohomology $$f^*\colon H^*_{\rm Hodge}(Y, Z)\to H^*_{\rm Hodge}(X, W).$$ In the above $Y\in SmlSm/k$ for simplicity but we do not impose the same assumption on $X$.
2. If $X\in SmlSm/k$ there is a canonical morphism to the underlying scheme $p_X\colon X\to \ul{X}$. For every (irreducible) closed subscheme $\ul{Z}$ of $\ul{X}$, the morphism $$Z=\ul{Z}\times_{\ul{X}} X \to X$$ is a strict closed immersion (here the log structure on $Z$ is the restriction of the log structure of $X$). Applying Construction \[constr:pullback\] to $p_X$ yields the morphism $$p_X^*\colon H^*_{\rm Hodge}(\ul{X}, \ul{Z})
=
\bigoplus_{i,j}H^i_{\ul{Z}}(\ul{X}, \Omega_{\ul{X}}^j)
\to
\bigoplus_{i,j} H^i_{Z}(X, \Omega^j_X) = H^{*}_{\rm Hodge}(X, Z).$$ Here $H^i_{Z}(X, \Omega^j_X)$ is the cohomology group with support $H^i_{\ul{Z}}(\ul{X}, \Omega^j_{\ul{X}}(\log \partial X))$. The morphism $p_X^*$ is induced by the inclusion of locally free sheaves $$\Omega^j_{\ul{X}}\to \Omega^j_{\ul{X}}(\log \partial X)$$ on $\ul{X}$ or, intrinsically using log geometry, is the canonical morphism discussed in [@Ogu Example IV 1.2.17].
3. We can further compose the morphism in (2) with the morphism induced by the open immersion $j\colon X- \partial X \to X$ to obtain $$H^*_{\rm Hodge}(\ul{X}, \ul{Z}) \xrightarrow{p_X^*} H^*_{\rm Hodge}( X, Z )\xrightarrow{j^*} H^*_{\rm Hodge}(X- \partial X, Z- \partial Z),$$ where $Z- \partial Z = Z\cap (X- \partial X)$. The composite morphism is induced by $$\Omega^j_{\ul{X}} \to j_* \Omega^j_{X- \partial X},$$ which clearly factors through the sheaf $\Omega^j_{\ul{X}}(\log \partial X)$.
Next, we discuss a derived pushforward functor following the script in [@ChatzistamatiouRullingANT §2.2]. Owing to Remark \[rmk:pullback\](3) it suffices to perform our constructions in the bounded derived category $\Deri^b(\ul{X})$.
\[const:push\] Let $f\colon X\to Y$ be a proper map between integral fs log schemes. We will assume $Y\in SmlSm/k$ and that $X$ is normal with compactifying Deligne-Faltings log structure $\partial X\to X$, where $\partial X$ is an effective Cartier divisor on $X$. A special case is the boundary of $X\in lSm/k$. Recall that $D_{\ul{X}}(-)$ is the dualizing functor on $\Deri^b(\ul{X})$. Using standard coherent duality theory [@HartshorneRD] we obtain a morphism in $\Deri^b(\ul{Y})$ $$\label{eq:push1}
f_*\colon Rf_* D_{\ul{X}}(\Omega^q_{X}) \to D_{\ul{Y}}(\Omega^q_Y),$$ defined as the composition $$\begin{aligned}
Rf_* R\Hom_{\ul{X}}(\Omega^q_X, \pi_{\ul{X}}^!(k)) &\xrightarrow{\cong} Rf_* R\Hom_{\ul{X}}(\Omega^q_X, f^!\pi_{\ul{Y}}^!(k)) \\
& \to R\Hom_{\ul{Y}}( Rf_*\Omega^q_X, Rf_* f^! \pi_{\ul{Y}}^!(k)) \\
& \xrightarrow{Tr_f} R\Hom_{\ul{Y}}( Rf_*\Omega^q_X, \pi_{\ul{Y}}^!(k)) \\
& \xrightarrow{f^*} R\Hom_{\ul{Y}}( \Omega^q_Y, \pi_{\ul{Y}}^!(k)).\end{aligned}$$ Here $Tr_f$ is gotten from the trace morphism $Rf_* f^!\to \id$, and $f^*\colon \Omega_Y^q\to Rf_* \Omega^q_X$ is the pullback of forms with log poles. If we further assume that $f$ is a strict morphism of log schemes, then we have $$\Omega^q_X \tensor I_{\partial X}
=
\Omega^q_X\tensor f^* I_{\partial Y}$$ where $f^*$ denotes the pullback of coherent sheaves (locally free, in this case). Further the strictness assumption implies $f^{-1}(\partial Y) = \partial X$ as effective Cartier divisors on $X$. Thus the construction of furnishes a morphism $$\label{eq:push2}
f_*\colon Rf_* D_{\ul{X}}(\Omega^q_X\tensor I_{\partial X}) \to D_{\ul{Y}}(\Omega^q_Y\tensor I_{\partial Y}).$$
The assumption that $Y\in SmlSm/k$ and $X$ has a compactifying Deligne-Faltings log structure is needed to incorporate the ideal sheaves $I_{\partial X}$ and $I_{\partial Y}$ in Construction \[const:push\], but it is not used to define . In Construction \[Construction of pushforward with support\] we use the assumption on $X$ to define pushforward with proper support.
For the proof of the following result we refer the reader to [@ChatzistamatiouRullingANT Proposition 2.2.7].
\[lem:obviousproperties\] For the pushforward morphism in (\[eq:push1\]) we have $(\id)_*=\id$. If $f\colon X\to Y$ and $g\colon Y\to Z$ are proper maps between fs log schemes in $SmlSm/k$, then $$(g\circ f)_* = g_* \circ Rg_*(f_*)\colon Rg_*Rf_*D_{\ul{X}}(\Omega^q_X)\to D_{\ul{Z}}(\Omega^q_Z)$$ for every $q\geq 0$. Moreover, if $f$ and $g$ are strict, the same holds for the pushforward morphism in .
Let $i\colon X\hookrightarrow Y$ be a strict closed immersion of fs log schemes in $SmlSm/k$, and assume that $\ul{X}$ has pure codimension $c$ in $\ul{Y}$, so that $\ul{X}\hookrightarrow \ul{Y}$ is a closed immersion of smooth schemes of pure codimension $c$. If we combine with Lemma \[eq:dualitylog\] then for every $g\geq0$ we have the composite $$\label{eq:push3closedimm}
i_* \Omega^q_X
\xrightarrow{i_*\theta_X} i_*D_{\ul{X}} (\Omega_{X}^{d_X-q}\tensor I_{\partial X})[-d_X]
\xrightarrow{i_*} D_{\ul{Y}}(\Omega_Y^{d_X-q}\tensor I_{\partial Y})[-d_X]
\xrightarrow{\theta_Y} \Omega^{c+q}_{Y}[c].$$ Here $\theta_X$ is the duality isomorphism for $X$, $\theta_Y$ is the duality isomorphism for $Y$, and $d_X$ is the dimension of $X$.
\[functorialitypush\] Let $i\colon X\hookrightarrow Y$ be a strict closed immersion of fs log schemes in $SmlSm/k$. Let $f\colon Y'\to Y$ be a morphism in $SmlSm/k$ and assume there is a cartesian square $$\begin{tikzcd}
X'\arrow[d,"f'"']\arrow[r,"i'"]&Y'\arrow[d,"f"]\\
X\arrow[r,"i"]&Y
\end{tikzcd}$$ where $X'\in SmlSm/k$. Assume that $\ul{X}$ has pure codimension $c=d_Y-d_X=d_{Y'}-d_{X'}$, where $d_{(-)}$ denotes the dimension of a scheme.
Then for all $q\geq 0$ there is a commutative diagram $$\begin{tikzcd}
i_* Rf'_* \Omega_X'^q \arrow[r,"\eqref{eq:push3closedimm}'"]& Rf_* \Omega^{c+q}_{Y'}[c] \\
i_* \Omega^q_X \arrow[u,"(f')^*"] \arrow[r,"\eqref{eq:push3closedimm}"]& \Omega^{c+q}_{Y}[c]. \arrow[u, "f^* "']
\end{tikzcd}$$
The proof requires an explicit description of the map using local cohomology sheaves. Since the log structure does not cause any complications, the argument in [@ChatzistamatiouRullingANT Corollary 2.2.22] applies, *mutatis mutandis*, to our situation.
Assume that $\ul{X}$ is integral. Applying $R\Gamma_X(Y,-)$ to , we get a morphism $$R\Gamma(X,\Omega^q_X) = R\Gamma_X(Y,Ri_* \Omega^q_X) = R\Gamma_X(Y, i_* \Omega^q_X )
\to
R\Gamma_X(Y,\Omega^{c+q}_{Y})[c]$$ in $\Deri^b(\mathbf{Ab})$. Furthermore, on cohomology, we obtain $$H(i_*)\colon H^*(X, \Omega^q_X) \to H^{*+c}_X(Y, \Omega^{q+c}_Y).$$ In particular, for $q=*=0$, there is a morphism $$\label{eq:cycleclasssmooth}
H(i_*)\colon
H^0(X, \cO_X)
\to
H^c_X(Y, \Omega_Y^c).$$ The image of $1\in H^0(X, \cO_X)$ along the map $H(i_*)$ of is called the *cycle class* of the strict closed subscheme $X$ in $Y$. In order to extend the definition to an arbitrary strict closed subscheme, we first establish some auxiliary results.
\[vanishing of hodge cohomology with support\] Let $\ul{X}$ be a smooth and separated scheme over $k$, and let $\ul{Z}$ be a closed subscheme of $\ul{X}$ of pure codimension $c$. Let $D$ be a strict normal crossing divisor on $\ul{X}$. Then, for any $i<c$ and $j$, we have the vanishing $$H_{\ul{Z}}^i(\ul{X},\Omega_{\ul{X}}^j(\log D))=0.$$
By the localization sequence one is reduced to the case where $X$ is local and $Z$ is the closed point. In this case the assertion follows from the vanishing of local cohomology, using that $\Omega_{\ul{X}}^j(\log D)$ is locally free and that a regular local ring is Gorenstein, see e.g., [@localcohomology Theorem 6.3].
Lemma \[vanishing of hodge cohomology with support\] immediately implies the following result.
\[injectivity of hodge cohomology\] Let $\ul{X}$ be a smooth and separated scheme over $k$, and let $$\ul{W}\stackrel{\mu}\rightarrow \ul{Z}\stackrel{\nu}\rightarrow \ul{X}$$ be closed immersions such that the codimension of $\nu$ (resp. $\mu$) is $c$ (resp. $\geq 1$). Then the naturally induced homomorphism $$H_{\ul{Z}}^c(\ul{X},\Omega_{\ul{X}}^c)
\rightarrow
H_{\ul{Z}-\ul{W}}^c(\ul{X}-\ul{W},\Omega_{\ul{X}-\ul{W}}^c)$$ is injective.
We can now recall the definition of Grothendieck’s “fundamental class” for a closed subscheme of a smooth $k$-scheme.
\[lem:cycleclassGroth\] Let $\ul{X}$ be a smooth and separated scheme over $k$, and let $\ul{Z}$ be a closed subscheme of $\ul{X}$ of pure codimension $c$. Consider the open immersion $j\colon U=\ul{X}-\ul{Z}_{sing}\rightarrow \ul{X}$, and consider the induced homomorphism $$H(i_*)
\colon
H_{\ul{Z}_{\rm reg}}^0(\ul{Z}_{\rm reg},\cO_{\ul{Z}_{\rm reg}})
\rightarrow H_{\ul{Z}_{\rm reg}}^c(U,\Omega_{U}^c),$$ where $\ul{Z}_{\rm reg}$ is the regular (equivalenty, smooth, since $k$ is perfect) locus of $\ul{Z}$. Then the following statements hold.
1. The naturally induced homomorphism $$j^*:H_{\ul{Z}}^c(\ul{X},\Omega_{\ul{X}}^c)\rightarrow H_{\ul{Z}_{reg}}^c(U,\Omega_{U}^c).$$ is injective.
2. There exists a unique element $cl(\ul{Z})$ in $H_{\ul{Z}}^c(\ul{X},\Omega_{\ul{X}}^c)$ such that $$j^*(cl(\ul{Z}))=H(i_*)(1).$$
Assertion (1) follows from Lemma \[injectivity of hodge cohomology\] and assertion (2) is explained in [@ChatzistamatiouRullingANT Proposition 3.1.1]
Putting the above together, we arrive at the following construction of a cycle class.
\[Construction of cycle class\] Let $X$ be an integral fs log scheme in $SmlSm/k$, and let $\nu:Z\rightarrow X$ be a strict closed immersion of fs log schemes such that $\ul{Z}$ has pure codimension $c$ in $\ul{X}$. In this setting, we define the *cycle class* of $Z$ $$cl(Z)\in H_Z^c(X,\Omega_X^c)$$ as follows: Let $j\colon X-\partial X\rightarrow X$ and $p_X\colon X\rightarrow \underline{X}$ denote the canonical morphisms to the underlying scheme of $X$. In degree $c$ we have the homomorphisms $p_X^*$ and $j^*$ $$H_{\underline{Z}}^c(\underline{X},\Omega_{\underline{X}}^c)\stackrel{p_X^*}
\rightarrow H_Z^c(X,\Omega_X^c)\stackrel{j^*}
\rightarrow H_{Z-\partial Z}^c(X-\partial X,\Omega_{X-\partial X}^c)$$ from Remark \[rmk:pullback\](3). Since pullbacks are compatible with compositions, the composition in question is the restriction map $\underline{j}^*$ for cohomology with support on the underlying schemes (both $\ul{X}$ and $X-\partial X$ have trivial log structures). Lemma \[injectivity of hodge cohomology\] shows $\underline{j}^*$ is injective. It follows that also $p_X^*$ is injective, and we set $$cl(Z):=p^*_X(cl(\underline{Z})).$$ Note that $j^*cl(Z)\cong cl(Z-\partial Z)$ by Lemma \[lem:cycleclassGroth\] and Lemma \[functorialitypush\]. By the same token, when $Z\in SmlSm/k$, the class $cl(Z)$ agrees with $H(i_*)(1)$ of .
Pushforward with proper support
-------------------------------
Let $f\colon X\to Y$ be a strict morphism of integral fs log schemes in $SmlSm/k$, and let $u\colon Z\hookrightarrow X$ be a strict closed immersion of fs log schemes. We will assume that the restriction of $f$ to $Z$ is proper and surjective onto $Y$.
\[df:goodcomp\] A good compactification of $f$ as above is a factorization $$f=\ol{f}\circ j\colon X\hookrightarrow \ol{X}\xrightarrow{\ol{f}} Y,$$ where $\ol{X}\to Y$ is a proper and strict map, $\ol{X}$ is an integral normal fs log scheme with compactifying Deligne-Faltings log structure $\partial X\to X$, where $\partial X$ an effective Cartier divisor on $\ul{X}$.
\[rmk:goodcomp2\] A good compactification exists for every $f$ as above. Indeed, Nagata’s compactification theorem [@ConradNagata] shows that every separated morphism of finite type $\ul{f}\colon \ul{X}\to \ul{Y}$ between Noetherian schemes admits a compactification, i.e., there exists a factorization into an open immersion followed by a proper map $\ol{f}\colon \ul{X}'\to Y$. Letting $\ol{X}$ denote the fs log scheme with log structure $(\ol{f}^*\cM_Y)_{\rm log}$ on $\ul{X}'$ we obtain a morphism $\ol{f}\colon\ol{X}\to Y$. By a standard argument, we may assume that $\ul{X}'$ is normal (simply take the normalization of $\ul{X}$ which does not alter the interior) and that the subscheme $\partial X$, where the log structure is nontrivial, is the support of an effective Cartier divisor. If we further assume that resolution of singularities holds over $k$, we may assume $\ol{X}\in SmlSm/k$.
\[Construction of pushforward with support\] Let $f:X\rightarrow Y$ be a strict morphism of integral schemes in $SmlSm/k$. Let $u:Z\rightarrow X$ be a strict closed immersion of fs log schemes, and assume that the restriction of $f$ to $Z$ is proper and surjective. Set $d_X:=\dim X$, $d_Y:=\dim Y$, and $r:=d_X-d_Y$. In this setting, for every $i$ and $j$, we will construct the pushforward morphism $$\label{pushforward for differentials with supports}
H(f_*):H_Z^{i+r}(X,\Omega_X^{j+r})\rightarrow H^i(Y,\Omega_Y^j).$$ First recall from Lemma \[eq:dualitylog\] the duality isomorphisms $$\begin{aligned}
\label{eq:properpush1}
\theta_X
\colon
H_Z^{i+r}(X,\Omega_X^{j+r})
& \xrightarrow{\cong} H_Z^{i+r-d_X}(X,D_{\underline{X}}(\Omega_X^{d_X-j-r}\otimes I_{\partial X}))\\
\nonumber
& =H_Z^{i-d_Y}(X,D_{\underline{X}}(\Omega_X^{d_Y-j}\otimes I_{\partial X})),\end{aligned}$$ and $$\label{eq:properpush1bis}
\theta_Y\colon H^{i+r}(Y,\Omega_Y^{j+r})\xrightarrow{\cong} H^{i-d_Y}(Y,D_{\underline{Y}}(\Omega_Y^{d_Y-j}\otimes I_{\partial Y})).$$ Here $\theta_X$ is obtained from by applying $R\Gamma_Z(X,-)$. Thus we are left to construct $$H_Z^{i-d_Y}(X,D_{\underline{X}}(\Omega_X^{d_Y-j}\otimes I_{\partial X}))
\rightarrow
H^{i-d_Y}(Y,D_{\underline{Y}}(\Omega_Y^{d_Y-j}\otimes I_{\partial Y})).$$
If $f$ is proper, this is precisely the content of after applying the cohomology with support functor (here we use the surjectivity of the map from $Z$ to $Y$, otherwise we would have to introduce a support in the target). When $f$ is not proper, choose a good compactification $\ol{f}$ in the sense of Definition \[df:goodcomp\], which is possible by Remark \[rmk:goodcomp2\]. Then we have a natural isomorphism $$\label{eq:properpush2}
H_Z^{i-d_Y}(X,D_{\underline{X}}(\Omega_X^{d_Y-j}\otimes I_{\partial X}))
\xrightarrow{\cong}
H_Z^{i-d_Y}(\ol{X},D_{\underline{\ol{X}}}(\Omega_{\ol{X}}^{d_Y-j}\otimes I_{\partial {\ol{X}}}))$$ by excision since $Z\subset X\hookrightarrow\ol{X}$ is a strict closed immersion. Applying to $\ol{f}$ we get $$\label{eq:properpush3}
H_Z^{i-d_Y}(\ol{X},D_{\underline{\ol{X}}}(\Omega_{\ol{X}}^{d_Y-j}\otimes I_{\partial {\ol{X}}}))
\to
H^{i-d_Y}(Y,D_{\underline{Y}}(\Omega_Y^{d_Y-j}\otimes I_{\partial Y})).$$ Composing , , and the inverse of , we obtain .
The next lemma is a routine check, see [@ChatzistamatiouRullingANT Definition 2.3.2, Proposition 2.3.3].
The map is independent of the choice of compactification. Let $f\colon X\to Y$ and $g\colon Y\to V$ be two strict morphisms of integral fs log schemes in $SmlSm/k$ such that the restrictions $f\circ u$ and $g\circ f\circ u$ are proper and surjective. If we set $l=\dim X-\dim V$, then $$H((g\circ f)_*) = H(g_*)\circ H(f_*) \colon H^{i+l}_Z(X, \Omega^{j+l}_X)\to H^i(V, \Omega^j_V).$$
The action of log correspondences
---------------------------------
Suppose $Z$ is an elementary log correspondence from $X$ to $Y$ in the sense of Definition \[A.5.2\], where $X,Y\in SmlSm/k$. Our aim in this subsection to construct a pullback morphism $$Z^*
\colon
\Omega^j(Y)
\rightarrow
\Omega^j(X).$$ If we extend by linearity to all finite log correspondences and verify compatibility with the composition, this equips the logarithmic differential $\Omega^j$ with the structure of a presheaf with log transfers.
\[Exactness and strictness\] Let $$\begin{tikzcd}
Y'\arrow[d,"f'"']\arrow[r,"g'"]&Y\arrow[d,"f"]\\
X'\arrow[r,"g"]&X
\end{tikzcd}$$ be a cartesian square of fs log schemes. Suppose that $g'$ is exact. If there exists a morphism $h:X\rightarrow T$ such that $hg$ is strict, then $g'$ is strict.
Owing to [@Ogu Definition III.1.2.1] there are naturally induced morphisms $$g_{log}^*\cM_X\rightarrow \cM_{X'},\;h_{log}^*\cM_T\rightarrow \cM_X,\;(hg)_{log}^*\cM_T\rightarrow \cM_{X'}.$$ Since $hg$ is strict, the third morphism is an isomorphism. This implies that the first morphism is an epimorphism, so that $$\cM_{X'/X}:=\coker(g_{log}^*\cM_X\rightarrow \cM_{X'})=0.$$ The question is Zariski local on $X'$ and $X$, so owing to [@Ogu Theorem III.1.2.7], we may assume that $g$ has a neat chart $\theta:P\rightarrow P'$ ([@Ogu Definition II.2.4.4]). This shows $\theta^{\rm gp}$ is an isomorphism since $\cM_{X'/X}=0$. The question is also Zariski local on $Y$, so we may assume $f$ admits an fs chart $P\rightarrow Q$.
By the construction of fiber products in the category of fs log schemes, $Y'$ has an fs chart $$Q':=P'\oplus_P Q,$$ which is an amalgamated sum in the category of fs monoids. By [@Ogu Theorem I.1.3.4] we have an isomorphism $$Q'^{\rm gp}\cong P'^{\rm gp}\oplus_{P^{\rm gp}}Q^{\rm gp}.$$ Since $\theta^{\rm gp}:P^{\rm gp}\rightarrow P'^{\rm gp}$ is an isomorphism, we see that $Q^{\rm gp}\rightarrow Q'^{\rm gp}$ is an isomorphism. This implies that $Q\rightarrow Q'$ is an isomorphism since $g'$ is exact. Thus $g'$ is strict.
\[constructionAction\] Let $Z$ be an elementary log correspondence from $X$ to $Y$ where $X,Y\in SmlSm/k$. We will first construct $$Z^*:\Omega^j(Y)\rightarrow \Omega^j(X)$$ in the case when $X$ is a quasi-projective scheme and has an fs chart $P$. By definition, there is a strict finite surjective morphism $Z^N\rightarrow X$ and a morphism $Z^N\rightarrow Y$. By Theorem \[FKatoThm\], there exists a log blow-up $V\rightarrow X\times Y$ such that the pullback $f':Z^N\times_{(X\times Y)}V\rightarrow V$ of $f$ is exact. Using Lemma \[Exactness and strictness\], we deduce that $f'$ is strict.
By Proposition \[Fan.12\], there is a log modification $X'\rightarrow X$ such that the pullback $$(Z^N\times_{X\times Y}V)\times_X X'\rightarrow Z^N\times_X X'$$ is an isomorphism. Next, note that by Proposition \[A.9.75\], every log modification is a surjective proper log étale monomorphism. By [@Ogu Corollary 3.2.4], if $f$ is log étale then $f^* \Omega^1_{X} \xrightarrow{\simeq} \Omega^1_{Y}$, since both $X$ and $Y$ are coherent log schemes. Thus we get an isomorphism on sections $$\Omega^j(X)\cong \Omega^j(X').$$ Let $Z'$ be the correspondence from $X'$ to $Y$ defined by $Z^N\times_X X'$. If we can construct a morphism $$Z'^*:\Omega^j(Y)\rightarrow \Omega^j(X'),$$ then we obtain $Z^*:\Omega^j(Y)\rightarrow \Omega^j(X)$ via the composite $$\Omega^j(Y)\stackrel{Z'^*}
\rightarrow
\Omega^j(X')
\stackrel{\sim}\rightarrow
\Omega^j(X).$$
Hence we may replace $(X,Y,Z,V)$ by $(X',Y',Z,V')$ where $V':=V\times_X X'$. In this case, the projection $$Z^N\times_{(X\times Y)}V\rightarrow Z^N$$ is an isomorphism, and the projection $Z^N\times_{(X\times Y)}V\rightarrow V$ gives a morphism $$\nu:Z^N\rightarrow V.$$ Let $U$ be the maximal open subscheme of $V$ that is strict over $X$, which exists by Lemma \[lem::maximal strict open\]. Since $Z^N$ is strict over $X$, $\nu$ factors through $\mu:Z^N\rightarrow U$, and there is a commutative diagram $$\begin{tikzcd}
Z^N\arrow[r,"\mu"]\arrow[rd]\arrow[rdd]&U\arrow[rd,"p_{U,Y}"]\arrow[d]\\
&X\times Y\arrow[d]\arrow[r]&Y\\
&X.
\end{tikzcd}$$
Since $Z^N$ is quasi-projective, $\mu$ admits a factorization $$Z^N\rightarrow U'\rightarrow \P_U^m\rightarrow U$$ into a closed immersion followed by an open immersion and the projection morphism for some $m\geq 0$. We let $p_{U'}:U'\rightarrow U$ (resp. $q_{U'}$) be the composition $U'\rightarrow \P_U^m\rightarrow U$ (resp. $U'\stackrel{p_{U'}}\rightarrow U\rightarrow X\times Y\rightarrow X$). Now using Constructions \[Construction of cycle class\] and \[Construction of pushforward with support\], we have the homomorphisms $$\begin{split}
&H^i(Y,\Omega^j)\stackrel{p_{U,Y}^*}\rightarrow H^i(U,\Omega^j)
\stackrel{p_U^*}\rightarrow H^i(U',\Omega^j)\\
\stackrel{\cup cl(Z^N,U')}\longrightarrow &H_Z^{i+m+d_Y}(U',\Omega^{j+m+d_Y})
\stackrel{q_{U*}}\rightarrow H^i(X,\Omega^j).
\end{split}$$ When $i=0$ we deduce the desired correspondence action $Z^*:\Omega^j(Y)\rightarrow \Omega^j(X)$.
For a general $X$, we choose a Zariski cover $\{X_a\}_{a\in I}$ of $X$ such that each $X_a$ is quasi-projective and has an fs chart. Set $X_{ab}:=X_a\times_X X_b$, and for each $a$ and $b$ we choose a Zariski cover $\{X_{abc}\}_{c\in J_{ab}}$ of $X_{ab}$ such that each $X_{abc}$ is quasi-projective and has an fs chart. Since $\Omega^j_X$ is a Zariski sheaf, there is an exact sequence $$0\rightarrow \Omega^j(X)\rightarrow \bigoplus_{a\in I}\Omega^j(X_a)\rightarrow \bigoplus_{a,b\in I,c\in J_{ab}} \Omega^j(X_{abc}).$$ This induces an exact sequence $$\begin{split}
0\rightarrow &\hom(\Omega^j(Y),\Omega^j(X))\rightarrow \hom(\Omega^j(Y),\oplus_{a\in I}\Omega^j(X_a))\\
\rightarrow &\hom(\Omega^j(Y),\oplus_{a,b\in I,c\in J_{ab}}\Omega^j(X_{abc})).
\end{split}$$
Letting $Z_a\in \lCor(X_a,Y)$ and $Z_{abc}\in \lCor(X_{abc},Y)$ denote the restrictions of $Z\in \lCor(X,Y)$, we have constructed $$Z_a^*\in \hom(\Omega^j(Y),\Omega^j(X_a)),
\;
Z_{abc}^*\in \hom(\Omega^j(Y),\Omega^j(X_{abc})).$$ By Lemma \[Correspondence, open immersion, and differentials\] below, we obtain the desired element $$\label{eq:corconst}
Z^*\in \hom(\Omega^j(Y),\Omega^j(X))$$ using the above exact sequence. Our construction extends to the log setting the correspondence action given by Rülling in [@KSY §A.4].
\[Correspondence, open immersion, and differentials\] Let $Z\in lCor(X,Y)$ be a log correspondence where $X,Y\in SmlSm/k$, let $j:U\rightarrow X$ be an open immersion, and let $Z_U\in lCor(U,Y)$ be the restriction of $Z$. Consider the naturally induced map $$j^*:\Omega^j(X)\rightarrow \Omega^j(X).$$ If $X$ is quasi-projective and has an fs chart $P$, then we have $$j^*\circ Z^*=Z_U^*.$$
This is immediate from the construction.
\[thm:Omega\_log\_transfer\] The presheaf $\Omega^j$ on $lSm/k$ admits a log transfer structure.
We will first show that $\Omega^j$ on $SmlSm/k$ admits a log transfer structure. Let $V\in lCor(X,Y)$ and $W\in lCor(Y,Z)$ be log correspondences. From Construction \[constructionAction\] we have $$V^*\in \hom(\Omega^j(Y),\Omega^j(X)),\;
W^*\in \hom(\Omega^j(Z),\Omega^j(Y)),$$ and $$(W\circ V)^*\in \hom(\Omega^j(Z),\Omega^j(X)).$$ It remains to check that $$(W\circ V)^* = V^*\circ W^*.$$
Due to Construction \[constructionAction\], we may assume that $X$, $Y$, and $Z$ are quasi-projective schemes equipped with fs charts. Consider the diagram $$\begin{tikzcd}
\Omega^j(Z) \arrow[r, "W^*"] \arrow[d, hook] & \Omega^j(Y) \arrow[r, "V^*"]\arrow[d, hook] & \Omega^j(X)\arrow[d, hook] \\
\Omega^j(Z- \partial Z) \arrow[r, "(W^\circ)^*"] & \Omega^j(Y- \partial Y) \arrow[r, "(V^\circ)^*"] & \Omega^j(X- \partial X).
\end{tikzcd}$$ The correspondences $V^\circ$ and $W^\circ$ are as in Lemma \[A.5.10\].
The vertical morphisms are injective by definition of the differential forms with log poles as subsheaf of the pushforward of the sheaf of differential forms from the open subset where the log structure is trivial. By Lemma \[Correspondence, open immersion, and differentials\] and by the construction [@KSY §A.4], the pullbacks $(V^\circ)^*$ and $(W^\circ)^*$ render the two squares commutative. By [@KSY Theorem A.4.1], the lower horizontal composition $(W^\circ \circ V^\circ)^*$ agrees with $(V^\circ)^*\circ (W^\circ)^*$. Since the vertical maps are injective and $(W\circ V)^\circ = W^\circ \circ V^\circ$ by Lemma \[lem:composition\], we conclude that $(W\circ V)^*= V^*\circ W^*$ by a diagram chase. This completes the proof that $\Omega^j$ admits a log transfer structure on $SmlSm/k$.
Thanks to Lemma \[A.5.59\], the presheaf on $lSm/k$ given by $$X\mapsto \colimit_{Y\in X_{div}^{Sm}}\Omega^j(Y)$$ admits a log transfer structure. To conclude observe that this presheaf is isomorphic to $\Omega^j$ since $\Omega^j$ is a dividing sheaf.
Let $X$ be a smooth $k$-scheme. By [@HuberJorder Theorem 3.6] the sections of the $h$-sheafification $a_h (\Omega^j)$ of $\Omega^j$ on $X$ agree with $\Omega^j(X)$. This implies that the sheaves $\Omega^j$ on $Sm/k$ admit a unique transfer structure. Thus the transfer structure defined by Rülling and extended in this section to log schemes agrees with the transfer structure defined in [@lecomtewach], at least with rational coefficients and in characteristic zero.
\[prop::commutativity\_differential\] For every integer $j\geq 0$, the differential $$d:\Omega^j\rightarrow \Omega^{j+1}$$ is a morphism of presheaves with log transfers.
We need to show that for every finite log correspondence $V\in \lCor(X,Y)$, with $X,Y\in lSm/k$, the back square in the diagram $$\begin{tikzcd}[row sep=tiny, column sep=tiny]
\Omega^j(Y)\arrow[dd]\arrow[rr]\arrow[rd]&
&
\Omega^{j+1}(Y)\arrow[dd]\arrow[rd]
\\
&
\Omega^{j}(Y-\partial Y)\arrow[rr,crossing over]&
&
\Omega^{j+1}(Y-\partial Y)\arrow[dd]
\\
\Omega^j(X)\arrow[rr]\arrow[rd]&
&
\Omega^{j+1}(X)\arrow[rd]
\\
&
\Omega^j(X-\partial X)\arrow[rr]\arrow[uu,crossing over,leftarrow]&
&
\Omega^{j+1}(X-\partial X).
\end{tikzcd}$$ commutes. The top and bottom squares commute since $d:\Omega^j\rightarrow \Omega^{j+1}$ is a morphism of presheaves. The left and right squares commute since $\Omega^j$ is a presheaf with log transfers by Theorem \[thm:Omega\_log\_transfer\]. In the proof of Theorem \[thm:Omega\_log\_transfer\] we have observed that the homomorphisms $$\Omega^j(X)\rightarrow \Omega^j(X-\partial X)
\text{ and }
\Omega^j(Y)\rightarrow \Omega^j(Y-\partial Y)$$ are injective. Thus we are reduced to showing that the front square commutes, i.e., we may assume that $X,Y\in Sm/k$. This case is checked in [@KSY Theorem B.2.1].
Hodge cohomology and cyclic homology {#ssec-Hodgecyclic}
------------------------------------
After having established the action of log correspondences on the sheaves of logarithmic differentials, we are ready to establish representability of Hodge cohomology and cyclic homology in $\ldmeff$.
\[thmHodge\] For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, the sheaves $\Omega^j$ are strictly $\boxx$-invariant $t$-sheaves with log transfers. Thus for every $X\in lSm/k$ and $i\in \Z$, there is a natural isomorphism $$\begin{aligned}
\hom_{\ldmeff}(M(X), \Omega^j[i])
\cong
H^i_{Zar}(X, \Omega^j).\end{aligned}$$
Lemma \[Omega\_dividing\_sheaf\], Proposition \[prop::hodgelet\], Corollary \[Pn-invariance of Log differentials\], and Theorem \[thm:Omega\_log\_transfer\] imply that $\Omega^j$ is a strictly $\boxx$-invariant $t$-sheaves with log transfers on $SmlSm/k$. For the isomorphisms $$\begin{aligned}
\hom_{\ldmeff}(M(X), \Omega^j[i])
&\cong
\hom_{\ldmeffet}(M(X), \Omega^j[i]) \\
&\cong
\hom_{\ldmefflet}(M(X),\Omega^j[i]) \\
&\cong
H^i_{Zar}(X, \Omega^j), \end{aligned}$$ see and .
\[Nontriviality\] Suppose $k$ is a field of positive characteristic $p>0$ and let $\Lambda$ be an $\mathbb{F}_p$-algebra. Then the Artin-Schreier sequence $$0\rightarrow \mathbb{G}_a\rightarrow \mathbb{G}_a\rightarrow \Z/p\rightarrow 0$$ is exact on the small étale site of $k$. By Theorem \[thmHodge\] the sheaf $\mathbb{G}_a$ on $SmlSm/k$ is a strictly $\boxx$-invariant dividing étale sheaf with log transfers. As a consequence, $\Z/p$ is a strictly $\boxx$-invariant dividing étale sheaf with log transfers. Applying we see that for every $X\in SmlSm/k$ there is an isomorphism $$\hom_{\ldmeffetZ}(M(X), \Z/p)
\cong
H^0_{Zar}(\ul{X}, \Z/p).$$ In particular, if $X$ is nonempty, this is a nontrivial group. It follows that also $\Z/p$ is nontrivial in $\ldmeffetZ$. Similarly, $\Z/p$ is nontrivial in $\ldmeffletZ$.
Assume now that $k$ has characteristic zero and let $\Lambda=\mathbb{Q}$. Let $HC_n(X)$ denote the cyclic homology groups of a smooth $k$-scheme $X$. The Zariski descent method of [@weibelcyclic] and the Hochschild-Kostant-Rosenberg Theorem [@Loday Theorem 3.4.12] provide a natural isomorphism $$\label{equation:eqcyclic}
HC_n(X)
\cong
\bigoplus_{p\in \Z}\bH_{Zar}^{2p-n}(X, \Omega^{\leq 2p}).$$ Proposition \[prop::commutativity\_differential\] shows that the differentials in the complex $\Omega^\bullet$ on $SmlSm/k$ are compatible with the log transfer structure, so that $\Omega^{\leq 2p}$ is a complex of Nisnevich sheaves with (log) transfers. Theorem \[thmHodge\] and imply representability of cyclic homology in ${\mathbf{logDM}^{\rm eff}(k, \mathbb{Q})}$.
\[thmHC\] Suppose $k$ has characteristic zero and let $\Lambda=\mathbb{Q}$. For every ${X}\in Sm/k$ and $n\geq 0$, there is a natural isomorphism $$\label{equation:cyclichomology}
\hom_{{\mathbf{logDM}^{\rm eff}(k, \mathbb{Q})}}(M({X}), \bigoplus_{p\in \Z} \Omega^{\leq 2p}[2p-n])
\cong
HC_n({X}).$$
Note that for $X\in lSm/k$, the left hand side of can be taken as definition of cyclic homology for fs log smooth log schemes over a field of characteristic zero.
Logarithmic geometry
====================
In this section, we provide a minimal reminder of the basic definitions regarding logarithmic geometry and state results that are used in the main body of our work. One can intuitively view the category of log schemes as an extension of the category of schemes given by essentially allowing “schemes with boundary” meaning pairs $(X,\partial X)$, where $X$ is a scheme and $\partial X\subset X$ is a normal crossing divisor, In many ways, the log geometry of $(X, \partial X)$ models the geometry on the complement $X\setminus \partial X$: for example, they have the same de Rham cohomology and étale sites. Many classical invariants and properties of schemes extend naturally to log schemes. There are notions of smooth and étale maps, pullbacks, forms and differentials, and of coherent sheaves. The Kato-Nakayama or Betti analytification functor provides an important bridge between log schemes and topological spaces; in our motivic setting, this is clearly expressed in the construction of Thom spaces.
The structure sheaf $\mathcal{O}_X$ of $X$ is a sheaf of rings in the Zariski topology. Its subsheaf of invertible elements is a Zariski sheaf of commutative monoids under multiplication $\mathcal{O}_X^{\times}$. Moreover, it is related to the sheaf of differentials via the logarithmic differential $$\label{equation:differential}
\text{dlog}
\colon
\mathcal{O}_{X}^{\times}\to \Omega_{X}$$ defined locally by $$f\mapsto \frac{df}{f}.$$ A foundational insight of log geometry is that for a scheme with boundary, extends to a morphism $$\label{equation:logdifferential}
\text{dlog}
\colon
\mathcal{O}_{X}^{\times}(\partial X)\to \Omega_{X}(\partial X)$$ with values in the sheaf of logarithmic differentials with respect to the normal crossing divisor $\partial X$. Here, $\mathcal{O}_{X}^{\times}(\partial X)$ denotes the monoid of functions which are invertible outside of $\partial X$ (and may have arbitrary order of vanishing at $\partial X$). Much of the geometry of the pair $(X, \partial X)$ is captured by the enhanced logarithmic differential map in . In log geometry one generalizes the above to the “logarithmic boundary data” of a scheme together a submonoid sheaf.
A textbook reference for logarithmic geometry is [@Ogu]. Some of the first accounts of the subjects include [@MR1463703] and [@MR1922832].
Monoids
-------
\[monoid\] A monoid is a commutative semigroup equipped with a unit. A homomorphism of monoids is required to preserve the unit element. Let $\Mon$ denote the category of monoids.
Given a monoid $P$, we can associate the group completion $$P^\gp
:=
\{(a,b) |(a,b)\sim(c,d) \ \mbox{if} \ \exists \ p \in P \ \mbox{such that} \ p+a+d=p+b+c\}.$$ Note the canonical morphism $P\to P^\gp$ is universal among the homomorphisms of monoids $P\to G$ such that the target $G$ is an abelian group.
Let $P^\ast$ denote the set of all $p\in P$ such that $p+q=0$ for some $q\in P$. It is called the *unit group* of $P$.
\[df:integral\] For a monoid $P$ the image of the homomorphism $P\rightarrow P^\gp$ sending $a$ to $(a,0)$ is denoted by $P^{\rm int}$. A monoid $P$ is said to be *integral* if $P\rightarrow P^{\rm int}$ is an isomorphism and finitely generated if there exists a surjective homomorphism $\N^{k}\to P$ for some $k$. It is called *fine* if it is integral and finitely generated. For an integral monoid $P$ we denote by $P^{\rm sat}$ the submonoid of $P^\gp$ such that whenever $p\in P^\gp$ and $n$ is a positive integer such that $np \in P$ then $p \in P^{\rm sat}$. An integral monoid $P$ is said to be *saturated* if $P\rightarrow P^{\rm sat}$ is an isomorphism. A monoid $P$ is *sharp* if its unit group $P^*$ is trivial.
We abbreviate the combined condition “fine and saturated" to “fs."
For homomorphisms of monoids (resp. integral monoids, resp. saturated monoids) $P\rightarrow Q$ and $P\rightarrow P'$, the amalgamated sum $Q\oplus_P^{\rm mon} P'$ (resp. $Q\oplus_P^{\rm int} P'$, resp. $Q\oplus_P P'$) in the category of monoids (resp. integral monoids, resp. saturated monoids) is the colimit of the diagram $Q\leftarrow P\rightarrow P'$.
In general, the monoids $Q\oplus_P^{\rm mon} P'$, $Q\oplus_P^{\rm int} P'$, and $Q\oplus_P P'$ are not isomorphic to each other even if $P$, $P'$, and $Q$ are saturated. The monoid $Q\oplus_P^{\rm int} P'$ is isomorphic to the submonoid of the abelian group $Q^\gp\oplus_{P^\gp}P'^\gp$ generated by the images of $Q$ and $P'$. There is an isomorphism $$Q\oplus_P P'\cong (Q\oplus_P^{\rm int}P')^{\rm sat}.$$ For an explicit description of $Q\oplus_P^{\rm mon} P'$ we refer to [@Ogu Proposition I.1.1.5].
For a monoid $P$, we set $$\overline{P}:=P\oplus_{P^\ast}^{\rm mon}0.$$ Note that $\overline{P}$ is always sharp.
\[KummerDef\] A homomorphism of integral monoids $\theta:P\rightarrow Q$ is called Kummer if it is injective and $\theta\otimes \Q:P\otimes \Q\rightarrow Q\otimes \Q$ is surjective.
A *face* $F$ of a monoid $P$ is a submonoid of $P$ such that for every $p,q\in P$ $$p+q\in F\Rightarrow p,q\in F.$$ An *ideal* $I$ of a monoid $P$ is a subset of $P$ satisfying the condition $$p\in I,q\in P\Rightarrow p+q\in I.$$ For an element $p$ of $P$, let $\langle p\rangle$ denote the smallest face containing $p$, and let $(p)$ denote the smallest ideal containing $p$.
For a face $F$ of a monoid $P$ we set $$P_F
:=
\{(a,b)\in P\times F|(a,b)\sim(c,d) \ \mbox{if} \ \exists \ p \in F \ \mbox{such that} \ p+a+d=p+b+c\}.$$ With this definition we have $P_P=P^\gp$. We also set $$P/F:=P\oplus_F^{\rm mon} 0.$$ There is an isomorphism $\overline{P_F}\cong P/F$. Finally, for an element $f\in P$, we set $$P_f:=P_{\langle f\rangle}.$$
Logarithmic structures
----------------------
For a scheme $X$, let $\cO_X$ be the structure sheaf. When we refer to it as a sheaf of monoids, we consider always its multplicative structure.
\[log-str\] Let $X$ be a scheme. A pre-logarithmic structure on $X$ is a sheaf of monoids $\cM_{X}$ on the small Zariski site of $X$ combined with a morphism of sheaves of monoids $$\alpha\colon \cM_{X} \longrightarrow \mathcal{O}_{X}$$ called the structure morphism. A pre-logarithmic structure is called a logarithmic structure if $\alpha$ induces an isomorphism $$\alpha^{-1}(\mathcal{O}_{X}^{*})\cong \mathcal{O}^{*}_{X}.$$
The pair $(X,\cM_{X})$ is called a log scheme. For notational convenience, whenever we abbreviate $(X,\cM_X)$ to $Y$ we set $\underline{Y}:=X$ and $\cM_Y:=\cM_X$. The scheme $\underline{Y}$ is called the underlying scheme of $Y$.
Note that the last condition in Definition \[log-str\] gives us a canonical embedding $\mathcal{O}_{X}^{*}\hookrightarrow \cM_{X}$ as a subsheaf of groups. The quotient sheaf of monoids $$\overline{\cM}_{X}
\colon=
\cM_{X}/\mathcal{O}_{X}^{*}$$ is called the characteristic monoid of the log structure $\cM_{X}$.
The morphism $\alpha$ is sometimes denoted $\exp$ for “exponential" and an inverse $\mathcal{O}_X^\ast \rightarrow \cM_{X}$ is denoted $\log$ for “logarithmic." Any scheme $X$ is a log scheme with $\cM_{X}=\mathcal{O}_{X}^{\times}$ and $\alpha$ the inclusion. This is the trivial log structure on $X$.
\[A.9.15\] Let ${\bf Log}_X$ denote the category of log structures on a scheme $X$.
\[A.9.13\] Let $f:X\rightarrow Y$ be a morphism of schemes. For a log structure $\cM_{X}\rightarrow \cO_X$ on $X$ we set $$f_*^{log}\cM_{X}
:=
f_*\cM_{X}\times_{f_*\cO_X}\cO_Y.$$ For a log structure $\cM_{Y}\rightarrow \cO_Y$ on $Y$, let $f_{log}^*\cM_{Y}$ be the log structure induced by the prelog structure $$f^{-1}(\cM_{Y})\rightarrow f^{-1}(\cO_Y)\rightarrow \cO_X.$$
With these definitions there exists an adjoint pair of functors $$f_{log}^*
:
{\bf Log}_X
\rightleftarrows
{\bf Log}_Y:f_*^{log}.$$
A morphism of log schemes $(X,\cM_{X})\to (Y,\cM_{Y})$ is a pair $(f,\phi)$, where $f\colon X\to Y$ is a morphism of schemes and $\phi\colon f^{-1}\cM_{Y}\to \cM_{X}$ is a homomorphism of sheaves of monoids such that the diagram $$\label{diagram:normalcrossing}
\vcenter{\xymatrix{
f^{-1}\cM_{Y} \ar[r]^-{\phi} \ar[d] & \cM_{X} \ar[d]\\
f^{-1}\mathcal{O}_{Y} \ar[r]^-{f^{\ast}} & \mathcal{O}_{X}
}}$$ commutes. Note that giving a morphism $\varphi:f^{-1}\cM_{Y}\rightarrow \cM_{X}$ is equivalent to giving a morphism $f_{log}^*\cM_{Y}\rightarrow \cM_{X}$.
We write $\LogSch$ for the category of log schemes.
A log morphism between schemes equipped with their trivial log structures is just a morphism of schemes.
\[ex:logpoints\] Let $X$ be a smooth scheme with an effective divisor $D\subset X$. Then we have a standard logarithmic structure on $X$ associated to the pair $(X, D)$ or the divisorial log structure induced by $D$, where $$\cM_{X} := \{f \in \mathcal{O}_{X} \ | \ f|_{X\setminus D} \in \mathcal{O}^*_{X}\}$$ with the structure morphism $\cM_{X} \to \mathcal{O}_{X}$ given by the canonical inclusion. Note that any section of $\mathcal{O}^*_{X}$ is already in $\cM_{X}$. In this case the log structure encodes the intuition that $D$ as a sort of “boundary” in $X$. With reference to Section \[subsection:cols\], picking local equations for the branches of $D$ passing through a point $x\in X$ gives local charts for the log structure with monoid $\N^{n}$, where $n$ is the number of branches.
\[ex:affine-toric-log\] To a monoid $P$ one associates the monoid ring $\Z[P]$ of $P$ over the integers and the affine toric variety $\underline{\A_{P}} := \text{Spec}(\Z[P])$. The standard logarithmic structure $\cM_{\A_P}$ on $\underline{\A_P}$ is induced from the pre-logarithmic structure $$P \to \Z[P] = \Gamma(\mathcal{O}_{\A_{P}})$$ defined by the obvious inclusion. This is called the canonical log structure on $\underline{\A_{P}}$, and we set $\A_P:=(\underline{\A_P},\cM_{\A_P})$. For example, we can consider $\A_{\N}$ as $\Spec{\Z[\N]}$ with the log structure associated to the origin. More generally, we may replace $\Z$ with a ring $R$. In this way obtain a contravariant functor from $\Mon$ to $\LogSch$.
Of particular interest is the case when $P$ is a toric monoid.
Any open subscheme of a log scheme can be equipped with the restriction of the log structure,
\[ex:logpoints2\] Let $Q$ be a sharp monoid. Define the homomorphism $\alpha\colon Q\to k$ by $\alpha(0)=1$ and $\alpha(q)=0$ if $q\neq 0$. The associated log scheme $(\Spec{k},Q)$ is a called a log point in general and a standard log point when $Q=\N$. This log structure is equivalent to that of $\alpha \colon Q\oplus k^{\ast}\to k$ given by $\alpha(0,x)=x$ and $\alpha(q,x)=0$ if $q\neq 0$. When $Q=\{0\}$ we obtain the trivial log structure. If $k$ is algebraically closed, then every log structure on $\Spec k$ is equivalent to a log point.
A log structure isolates the “singular” information carried by a pre-log structure, i.e., it provides logarithmic behaviour for functions which are not already invertible. Any pre-log structure can be turned into a log structure. The following provides a left adjoint of the inclusion functor from pre-log structures into log structures.
Let $\alpha\colon \cM_{X} \rightarrow \mathcal{O}_{X}$ be a pre-logarithmic structure. Then the associated logarithmic structure $M^{a}_{X}$ is the pushout of the diagram $$\xymatrix{
\mathcal{O}_{X}^{*} & \alpha^{-1}(\mathcal{O}_{X}^{*}) \ar[l] \ar[r] & \cM_{X}
}$$ in the category of sheaves of monoids on the small Zariski site of $X$, endowed with $$M^{a}_{X} \rightarrow \mathcal{O}_{X} \qquad (a,b)\mapsto \alpha(a)b \qquad\qquad (a \in M, b \in \mathcal{O}_{X}^{*}).$$
Any morphism of schemes $f\colon X\to \Spec {\Z[P]}$ induces a log structure on $X$ as follows: $f$ gives a morphism of monoids $P\to \mathcal{O}_X(X)$ that induces $\widetilde{\alpha}\colon\underline{P}\to \mathcal{O}_X$, where $\underline{P}$ is the constant sheaf. It is typically not true that $\widetilde{\alpha}$ induces an isomorphism between $\widetilde{\alpha}^{-1}\mathcal{O}^\times_X$ and $\mathcal{O}^\times_X$, but the procedure above fixes the behaviour of the units, and this produces a log structure $\alpha\colon M^{a}_{X}\to \mathcal{O}_X$. The quotient $M^{a}_{X}/\mathcal{O}_X^\times$ is obtained from $\underline{P}$ by locally killing the sections of $\underline{P}$ that become invertible in $\mathcal{O}_X$. In particular, all the stalks of $M^{a}_{X}/\mathcal{O}_X^\times$ are quotients of the monoid $P$.
Monoschemes and fans
--------------------
In this section we review the definition of fans and the relation between monoschemes and fans.
A *monoidal space* is a topological space $X$ equipped with a sheaf of monoids $\cM_{X}$.
A monoidal space $X$ is *quasi-compact* (resp. *quasi-separated*, resp. *connected*) if the underlying topological space is quasi-compact (resp. quasi-separated, resp. connected).
Let $P$ be a monoid. Its monoidal space $\Spec{P}$ is defined as follows. The underlying set of $\Spec{P}$ consists of the faces of $P$. For an element $f\in P$, we set $$D(f):=\{F\in \Spec{P}:f\in F\}.$$ The *Zariski* topology on $\Spec{P}$ is the topology generated by $$\mathscr{B}:=\{D(f):f\in P\},$$ i.e., $\mathscr{B}$ is a base for $\Spec{P}$. The assignment $$D(f)\mapsto P_f$$ defines a presheaf on $\mathscr{B}$, and its sheafification defines a sheaf $\cM_{\Spec{P}}$ on $\Spec{P}$. The pair $$\Spec{P}:=(\Spec{P},\cM_{\Spec{P}})$$ is a monoidal space.
The meaning of the notation $\Spec{\Z}$ in this text, either as the affine scheme associated with the ring $\Z$ or as the monoidal space associated with the monoid $\Z$, should always be clear from the context.
A monoidal space $X$ is an fs *monoscheme* if there is an open cover $\{X_i\}$ of $X$ such that for some fs monoid $P_i$ there is an isomorphism $$X_i\cong \Spec {P_i}.$$ If $\cM_{X,x}^{\rm gp}$ is torsion free for every point $x\in X$, we say that $X$ is *toric*. If $X\cong \Spec{P}$ for some fs monoid $P$, we say that $X$ is *affine*.
The category of fs monoschemes has fiber products, see [@Ogu Proposition II.1.3.5].
A quasi-compact and quasi-separated fs monoscheme $X$ is *separated* (resp. *proper*) if the naturally induced homomorphism $$\hom(\Spec \N, X)\rightarrow \hom(\Spec \Z, X)$$ is injective (resp. bijective).
\[df:toricvarieties\] We follow standard notation for toric varieties and set $$N:=\Z^{d},
\;
M:=\hom_{\Z}(N,\Z),
\;
N_{\R}:=N\otimes_{\Z}\R,
\;
M_{\R}:=M\otimes_{\Z}\R.$$ We also often use the notation $N^\vee:=M$. If $\sigma\subset N_{\R}$ is a strictly convex rational polyhedral cone, we denote its dual cone by $$\sigma^{\vee}
:=
\{
m\in M_{\R} \vert \langle m,n \rangle = 0, \ \forall \ n\in\sigma
\}.$$ Gordan’s lemma tells us that $M\cap \sigma^{\vee}$ is a finitely generated monoid.
A *fan* $\Sigma$ in $N$ is a finite set of cones in $N$ satisfying the following conditions:
1. If $\sigma\in \Sigma$, then every face of $\sigma$ is in $\Sigma$.
2. If $\sigma,\tau\in \Sigma$, then $\sigma\cap \tau\in \Sigma$.
Let $\underline{\A_{\Sigma}}$ be the toric variety associated to a fan $\Sigma$ in $M$. By gluing the log structures associated to the homomorphisms $$M\cap \sigma^{\vee}
\to
\Z[M\cap \sigma^{\vee}]$$ for every cone $\sigma$ in $\Sigma$, we obtain an fs log scheme $\A_{\Sigma}$ over $\Spec{\Z}$. This is in fact the divisorial log structure on $\A_{\Sigma}$ coming form the toric boundary $\partial \A_{\Sigma}\subseteq \A_{\Sigma}$, i.e., the complement of the torus orbit in $X$.
For a fan $\Sigma$ in a lattice $N$, the *support* $\lvert \Sigma \rvert$ of $\Sigma$ is the union of all cones of $\Sigma$.
For fans $\Sigma$ and $\Sigma'$ in lattices $N$ and $N'$, a morphism $\Sigma'\rightarrow \Sigma$ of fans is a homomorphism of lattices $f:N'\rightarrow N$ such that for every cone $\sigma'$ of $N'$, $f(\sigma')\subset \sigma$ for some cone $\sigma$ of $N$. A morphism $\Sigma'\rightarrow \Sigma$ of fans is a *partial subdivision* if the associated homomorphism $f:N'\rightarrow N$ of lattices is an isomorphism. A partial subdivision $\Sigma'\rightarrow \Sigma$ is a *subdivision* if $\lvert \Sigma'\rvert=\lvert\Sigma\rvert$. If $\Sigma'\rightarrow \Sigma$ is a morphism of fans, then we can naturally associate a morphism $\A_{\Sigma'}\rightarrow \A_{\Sigma}$ of fs log schemes.
Let $\mathbf{Fan}$ denote the category of fans, and let $\mathbf{MSch}^{fs}$ denote the category of fs monoschemes.
\[fan=monoscheme2\] There is a fully faithful functor $$\label{fan=monoscheme2.1}
\mathbf{Fan}\rightarrow \mathbf{MSch}^{fs}.$$ The essential image consists of all separated, connected, and toric fs monoschemes whose underlying topological spaces are finite sets.
We refer to [@Ogu Theorem II.1.9.3].
\[fan=monoscheme\] Here we give an explicit description of the functor . Let $\Sigma$ be a fan with the cones $\sigma_1,\ldots,\sigma_r$ in a lattice $N$. Consider the dual cones $\sigma_1^\vee,\ldots,\sigma_r^\vee$. We can glue $\Spec{\sigma_1^\vee},\ldots,\Spec{\sigma_r^\vee}$ altogether to canonically obtain an fs monoscheme. Thus we can regard fans as fs monoschemes and morphisms of fans as morphisms of monoschemes.
If $P$ is an fs monoid such that $P^\gp$ is a lattice $N$, then $\Spec{P}$ corresponds to the fan with a single maximal cone $P^\vee$ in a lattice $N^\vee$.
\[Fiberproduct\_fans\] Let $f\colon\Theta\rightarrow \Sigma$ be a subdivision (resp. partial subdivision) of fans in a lattice $N$, and let $g\colon \Sigma'\rightarrow \Sigma$ be a morphism of fans with a lattice homomorphism $\theta\colon N'\rightarrow N$. Then the fiber product $\Theta':=\Theta\times_\Sigma \Sigma'$ in the category of fs monoschemes is a fan in $N'$, and the projection $f'\colon \Theta'\rightarrow \Sigma'$ is a subdivision (resp. partial subdivision).
Let $\sigma=\Spec{P}$ be a cone of $\Sigma$, where $P$ is an fs monoid, and let $\tau=\Spec{Q}$ and $\sigma'=\Spec{P'}$ be cones of $\Theta$ and $\Sigma'$ where $P'$ and $Q$ are fs monoids. Suppose that $f(\tau)=\sigma$ and $g(\sigma')=\sigma$. Then $Q':=Q\oplus_P P'$ is the submonoid of $N'^\vee$ generated by $\theta^\vee(Q)$ and $P'$, i.e., $Q'=\theta^\vee(Q)+P'$. Thus we have $Q'^\vee=(\theta^\vee(Q))^\vee\cap P'^\vee$. From this local description of $\Theta'$, we see that $\Theta'$ is the fan in $N'$ whose cones are of the form $$\label{Fiberproduct_fans.1}
\{\theta^\vee(\tau) \cap \sigma':\sigma'\in \Sigma'\text{ and }\tau\in \Theta\}.$$
Since $\Sigma'$ and $\Theta'$ are in the same lattice $N'$, $f'\colon\Theta'\rightarrow \Sigma'$ is a partial subdivision. If $f$ is a subdivision, then $\lvert \Theta\rvert = \lvert \Sigma \rvert$. Using we deduce that $\lvert \Theta'\rvert=\lvert \Sigma'\rvert$.
\[monomorphismoffans\] Let $f:\Sigma'\rightarrow \Sigma$ be a subdivision of fans in a lattice $N$. Then $f$ is a monomorphism in the category of fans.
Let $\xi$ and $\xi'$ are the generic points of the monoschemes $\Sigma$ and $\Sigma'$. There are isomorphisms $$N\cong \cM_{\Sigma,\xi}\cong \cM_{\Sigma',\xi'}.$$ Here $\Sigma$ and $\Sigma'$ can be viewed as separated fs monoschemes, and we may apply [@Ogu Corollary II.1.6.4] to conclude that $f$ is a monomorphism in the category of fs monoschemes. In particular, $f$ is a monomorphism in the category of fans.
\[Propersubdivision\] Let $\Sigma$ be a fan in a lattice $N$. Then there is a canonical bijection $$\hom(\Spec{\N},\Sigma)\cong \lvert \Sigma \rvert.$$
A homomorphism $f:\Z\rightarrow N$ gives a morphism $\Spec{\N}\rightarrow \Sigma$ precisely when $f(\N)$ lies in a cone of $\Sigma$, i.e., when $f(\N)$ lies in $\lvert \Sigma \rvert$. This implies the bijection.
\[Starsubdivision\] Suppose $\Sigma$ is a fan in $\Z^n$, and let $\tau$ be a cone of $\Sigma$ such that for any cone $\sigma$ containing $\tau$, $\sigma$ is smooth. Express $\tau$ as an $r$-dimensional cone ${\rm Cone}(a_1,\ldots,a_r)$. Then for each maximal cone $\sigma={\rm Cone}(a_1,\ldots,a_r,a_{r+1},\ldots,a_n)$ of $\Sigma$ containing $\tau$, replace $\sigma$ by the fan with maximal cones $$\begin{split}
&{\rm Cone}(a_1,\ldots,a_{r-1},a_{r+1},\ldots,a_n,a_1+\cdots+a_r),\\
&{\rm Cone}(a_1,\ldots,a_{r-2},a_r,\ldots,a_n,a_1+\cdots+a_r),\cdots,{\rm Cone}(a_2,\ldots,a_n,a_1+\cdots+a_r).
\end{split}$$ The resulting new fan $\Sigma^*(\tau)$ is called the [*star subdivision*]{} of $\Sigma$ relative to $\tau$.
\[A.3.31\] Let $\Sigma$ and $\Sigma'$ be two smooth fans with $|\Sigma|=|\Sigma'|$. Then there is a subdivision $\Sigma''$ of $\Sigma$ obtained by a finite succession of star subdivisions relative to two-dimensional cones such that $\Sigma''$ is a subdivision of $\Sigma'$.
See [@MR803344] and [@TOda pp. 39-40].
\[Compactification\_fan\] Let $\Sigma$ be a fan in a lattice $N$. Then there exists a fan $\Sigma'$ in $N$ containing $\Sigma$ such that $\lvert \Sigma' \rvert=N$.
See [@TOda p. 18].
Strict morphisms
----------------
Let $f:X \rightarrow Y$ be a morphism of schemes. Given a logarithmic structure $\cM_{Y}$ on $Y$, we have defined a logarithmic structure $f_{log}^*\cM_{Y}$ on $X$, called the inverse image of $\cM_{Y}$ or the pullback log structure on $X$. Using the inverse image of logarithmic structures, we arrive at the following definition which is incorporated in our notion of finite log correspondences.
A morphism of log schemes $(X,\cM_{X}) \to (Y,\cM_{Y})$ is called strict if the induced homomorphism $f_{log}^{*}\cM_{Y}\to \cM_{X}$ is an isomorphism, and it is said to be a strict closed immersion if it is strict and $X\to Y$ is a closed immersion.
In Example \[ex:affine-toric-log\] the log structure on $\Spec{R[P]}$ can be viewed as the pullback log structure on $\Spec{\Z[P]}$ along the morphism induced by $\Z\to R$.
Let $f\colon X\to \A^{1}$ be a smooth morphism. Then $f$ is a strict morphism of log schemes with respect to the divisorial log structures given by $0\in\A^{1}$ and $f^{-1}(0)\subseteq X$.
\[lem::maximal strict open\] Let $f:X\rightarrow Y$ be a morphism of fs log schemes. Then there exists a maximal open subscheme $U$ of $X$ such that the restriction $f_{\vert U}$ of $f$ to $U$ is strict over $Y$.
In [@MR1754621 Lemma 1.5] it is shown that if $X$ is strict at a point $x$, then $f$ restricts to a strict morphism over an étale neighborhood. The same argument applies to fs log schemes equipped with Zariski log structures,
Charts of logarithmic structures {#subsection:cols}
--------------------------------
Let $X$ be a log scheme and $P$ be a monoid. A [*chart*]{} for $\cM_{X}$ is a morphism $P\rightarrow \Gamma(X,\cM_{X})$ such that the induced map of logarithmic structures $P^{a}\to \cM_{X}$ is an isomorphism. Here, $P^{a}$ is the logarithmic structure associated to the pre-logarithmic structure given by $$P\rightarrow \Gamma(X,\cM_{X})\rightarrow \Gamma(X,\mathcal{O}_{X}).$$ If $P$ is an fs monoid, the chart is said to be an fs chart.
A chart for $\cM_{X}$ is equivalent to a strict morphism $$f
\colon
X
\to
\A_P.$$ In fact, by [@Ogu III.1.2.9], there is a bijection $$\hom_{\LogSch}(X,\A_P)
\rightarrow
\hom_{\Mon}(P,\Gamma(X,\cM_{X}))$$ associating to $f$ the composition $$\xymatrix{
P \ar[r]&\Gamma(X, P_X)\ar[r] & \Gamma(X, \cM_{X}).
}$$ The morphism $f\colon X\rightarrow \A_P$ also gives a morphism of monoidal spaces $$X=(X,\cM_X)\rightarrow \Spec{P}.$$
A chart for a morphism of log schemes $(X,\cM_{X})\to (Y,\cM_{Y})$ is a triple $(P\to \cM_{X},Q\to \cM_{Y},Q\to P)$, where $P\to \cM_{X}$ and $Q\to \cM_{Y}$ are charts, and $Q\to P$ is is a homomorphism such that the diagram $$\label{diagram:chartmaps}
\vcenter{\xymatrix{
Q \ar[r] \ar[d] & P \ar[d]\\
f_{log}^{\ast}\cM_{Y} \ar[r] & \cM_{X}
}}$$ commutes.
A log scheme $X$ is said to be *coherent* if étale locally there is a chart $P\rightarrow \cM_{X}$ with $P$ a monoid. If moreover $P$ can be chosen to be integral (resp. fine, resp. saturated, resp. fs), then $X$ is called an *integral* (resp. *fine*, resp. *saturated*, resp. *fs*) log scheme. If such a chart exists Zariski locally we say that $X$ has a *Zariski log structure*. Finally, if $P$ can be chosen isomorphic to $\N^k$ then the logarithmic structure is called locally free.
We are particularly interested in Zariski log structures, which locally arise from pullbacks of monoid algebras. The property of being fine and saturated is a logarithmic analogue of the property of being normal and of finite type [@Ogu Chapter IV.1.2].
Suppose that $Y\rightarrow X$ and $X'\rightarrow X$ are morphisms of coherent (resp. integral, resp. saturated) log schemes. The fiber product in the category of coherent (resp. integral, resp. saturated) log schemes is denoted by $$Y\times_X^{\rm coh}X'
\text{ (resp.\ }Y\times_X^{\rm int}X',
\text{ resp.\ }Y\times_X X'\text{)}.$$
Suppose that $Q\rightarrow P$ and $P'\rightarrow P$ are homomorphisms of monoids (resp. integral monoids, resp. saturated monoids). Then there exists an isomorphism $$\begin{gathered}
\A_Q\times_{\A_P}^{\rm coh}\A_{P'}\cong \A_{Q\oplus_P^{\rm mon} P'}
\\
\text{ (resp.\ } \A_Q\times_{\A_P}^{\rm int}\A_{P'}\cong \A_{Q\oplus_P^{\rm int} P'},
\text{ resp.\ } \A_Q\times_{\A_P}\A_{P'}\cong \A_{Q\oplus_P P'}\text{)}.\end{gathered}$$
\[Fiber.product.strict.morphism\] Let $f:X\rightarrow S$ be a strict morphism of fs log schemes. Then for any morphism $S'\rightarrow S$ of fs log schemes, the induced morphism $$\underline{X\times_S S'}\rightarrow \underline{X}\times_{\underline{S}}\underline{S'}$$ is an isomorphism.
From the construction of fiber products in the category of fs log schemes, see [@Ogu Corollary III.2.1.6], it suffices to show that the fiber product $X\times_S^{\rm coh}S'$ in the category of coherent log schemes is an fs log scheme. This question is Zariski local on $S'$, so we may assume that $S'$ has an fs chart $S'\rightarrow \A_P$. Since $f$ is strict, the projection $X\times_S^{\rm coh}S'\rightarrow X$ is also strict. Then $X\times_S^{\rm coh}S'$ has an fs chart. Thus $X\times_S^{\rm coh}S'$ is an fs log scheme.
\[A.9.8\] Suppose $X$ is a coherent log scheme with a chart $\beta:P\rightarrow \cM_{X}$, and let $x\in X$ be a point. Then $\beta$ is called exact at $x$ if $$\overline{\beta_x}:\overline{P}\rightarrow \overline{\cM}_{X,x}$$ is an isomorphism, and $\beta$ is called neat at $x$ if it is exact at $x$ and $P$ is sharp.
\[A.9.10\] (1) Let $X$ be a fine log scheme with a fine chart $\beta:P\rightarrow \cM_{X}$, and let $x\in X$ be a point. Then, Zariski locally on $X$, $\beta$ factors through an exact chart at $x$ by [@Ogu Remark III.2.3.2].
\(2) Let $X$ is an fs log scheme, and let $x\in X$ be a point. Then Zariski locally on $X$, there exists a neat chart at $x$ by [@Ogu Proposition III.2.3.7].
\[lem::Smchart\] Suppose $X$ is an fs log scheme log smooth over a scheme $S$. Then Zariski locally on $X$, there is a neat fs chart $P$ such that the induced morphism $X\rightarrow S\times \A_P$ is strict smooth. If $\underline{X}$ is smooth over $S$, then $P\cong \N^r$ for some integer $r\in \N$.
Since the question is Zariski local on $X$ we may assume that $X$ admits a neat chart $P$ at a point $x\in X$, see Remark \[A.9.10\](2). In particular, since $P$ is sharp, $P^{\rm gp}$ is torsion-free by [@Ogu Proposition I.1.3.5(2)]. Owing to [@Ogu Proposition III.2.3.7] there is an isomorphism $$\cM_{X,x}^{\rm gp}\cong \cO_{X,x}^*\oplus P.$$ This implies that the order of the torsion subgroup of $\cM_{X,x}^{\rm gp}$ is invertible in $k$. Hence we can apply [@Ogu Theorem IV.3.3.1(3)] to conclude that $X\rightarrow S\times \A_P$ is strict smooth.
If $\underline{X}$ is smooth over $S$, then $S\times \underline{\A_P}$ is smooth over $S$ by [@EGA Proposition IV.17.7.7]. This shows that $P\cong \N^r$ for some $r$ by [@CLStoric Theorem 1.3.12] because $P$ is sharp.
We refer to [Definition \[A.3.17\]]{} for the notion of a strict normal crossing divisor on a scheme.
\[lem::SmlSm\] Suppose that $X\in SmlSm/S$. Then there exists a strict normal crossing divisor $Z$ on $\ul{X}$ and an isomorphism $$X\cong (\underline{X},Z).$$
The question is Zariski local on $\ul{X}$. Owing to Lemma \[lem::Smchart\] there is an fs chart $P\cong \N^r$ of $X$ such that the induced morphism $f:X\rightarrow S\times \A_P$ is strict smooth. That is, $\A_P\cong (\underline{\A_P},W)$, where $W$ is the strict normal crossing divisor induced by the log structure on $\A_P$. Using the strict smooth morphism $f:X\rightarrow S\times \A_P$, we see the claimed isomorphism follows for the strict normal crossing divisor $Z:=f^*W$.
\[df::closedimmersion\] A morphism of fs log schemes $f:X\rightarrow Y$ is a *closed immersion* if the following two properties are satisfied:
1. The underlying morphism of schemes $\underline{f}:\underline{Z}\rightarrow \underline{X}$ is a closed immersion.
2. The homomorphism $f^\flat:f_{log}^*(\cM_Y)\rightarrow \cM_X$ is surjective.
Logarithmic smoothness {#subsec::logsmooth}
----------------------
Log smoothness is one of the central concepts in log geometry. Kato [@MR1463703] showed that every log smooth morphism factors étale locally as a usual smooth morphism composed with a morphism induced by a homomorphism of monoids, which essentially determines the log structure.
Consider the following commutative diagram of log schemes illustrated with solid arrows:
$$\label{diagram:smooth}
\vcenter{\xymatrix{
T_{0} \ar[r]^{\phi} \ar[d]_{i}& X \ar[d]^{f}\\
T_{1} \ar[r]^{\psi} \ar@{-->}[ur] & Y
}}$$
Here $i$ is a strict closed immersion defined by a square zero ideal (note that $T_{0}$ and $T_{1}$ have the same underlying topological spaces). Logarithmic smoothness is defined by an infinitesimal lifting property.
\[df:logsmooth\] A morphism $f:X\rightarrow Y$ of fine log schemes is logarithmically smooth (resp. logarithmically étale) if the underlying morphism $\ul{X}\rightarrow \ul{Y}$ is locally of finite presentation and for any commutative diagram (\[diagram:smooth\]), étale locally on $T_{1}$ there exists a (resp. there exists a unique) morphism $g:T_{1}\rightarrow X$ such that $\phi=g\circ i$ and $\psi=f\circ g$.
We have the following useful criterion for smoothness from [@MR1463703 Theorem 3.5].
\[KatoStrThm\] Let $f:X\rightarrow Y$ be a morphism of fine log schemes. Assume there is a chart $Q\rightarrow \cM_{Y}$, where $Q$ is a finitely generated integral monoid. Then the following are equivalent:
1. $f$ is logarithmically smooth (resp. logarithmically étale).
2. Étale locally on $X$, there is a chart $(P\rightarrow \cM_{X},Q\rightarrow \cM_{Y},Q\rightarrow P)$ extending $Q\rightarrow \cM_{Y}$ and satisfying the following properties.
1. The kernel and the torsion part of the cokernel (resp. the kernel and the cokernel) of $Q^{\gp}\rightarrow P^{\gp}$ are finite groups of orders invertible on $X$.
2. The induced morphism $X\rightarrow Y\times_{\text{Spec}(\Z[Q])} \text{Spec}(\Z[P])$ is étale in the usual sense.
In the second part of Theorem \[KatoStrThm\], the étale locality condition is used to ensure that $X$ admits a chart. Thus, since we are working with Zariski log structures, we may assume the required chart exists Zariski locally on $X$.
\[ex:toriclogsmooth\] Suppose $P$ is a fs monoid such that $P^\gp$ is torsion free. Let $X=\A_P$ and $Y=\Spec{\Z}$ equipped with the trivial logarithmic structure. Then, according to the theorem, $X$ is logarithmically smooth relative to $Y$, though the underlying toric variety might be singular. This is Kato’s “magic of log."
\[ex:normalcrossinglogsmooth\] Let $\sigma$ be the cone of $M_{\R}=\R^{d}$ generated by the standard basis vectors $e_{i}$, $1\leq i\leq d$. For integers $a_{j}>0$, $1\leq i\leq d$, let $\tau$ be the subcone of $\sigma$ generated by $a_{1}e_{1}+\dots+a_{d}e_{d}$. We set $P:=M\cap\sigma$ and $Q:=M\cap\tau$. Assuming the greatest common divisor $N:=\text{gcd}(a_{1},\dots,a_{d})$ is invertible in the field $k$, we can identify the fiber product $$\Spec{k}\times_{\Spec{\Z[1/N][Q]}}\Spec{\Z[1/N][P]}$$ with the normal crossing variety $$X
:=
\Spec{
k[x_{1},\dots,x_{d}]/(x_{1}^{a_{1}}\cdots x_{d}^{a_{d}})
}.$$ There are morphisms $$\label{diagram:normalcrossing2}
\vcenter{\xymatrix{
\N \ar[r] \ar[d]_{\phi} & k \ar[d]\\
\N^{d} \ar[r]^-{\psi} & k[x_{1},\dots,x_{d}]/(x_{1}^{a_{1}}\cdots x_{d}^{a_{d}}),
}}$$ where $\phi(1)=a_{1}e_{1}+\dots+a_{d}e_{d}$ and $\psi(e_{i})=x_{i}$. The induced morphism $$(X,\N^{d})
\to (\Spec{k},\N)$$ is log smooth.
Suppose $(X,\cM_{X})$ is an fs log scheme which is log smooth over $\Spec{k}$. Then $X$ is covered étale locally by schemes which are smooth over toric varieties, and there exists a divisor $D$ of $X$ such that the log structure is equivalent to that of $j^{\ast}\mathcal{O}_{X\setminus D}\cap \mathcal{O}_{X}$, where $j\colon X\setminus D\to X$ is the inclusion.
The endomorphism of $\A_\N$ induced by $t\mapsto t^{n}$ is a Kummer morphism of log schemes, and a log étale morphism in case $n$ and $\text{char}(k)$ are coprime.
Logarithmic differentials
-------------------------
One basic idea in log geometry is that the sheaves of monoids specify where $d\log$ makes sense. To form sheaves of logarithmic differentials, one adds to the sheaf of differentials symbols of the form $d\log \alpha(m)$ for all elements $m\in \cM_{X}$.
\[log-differential\] Let $f: X \to Y$ be a morphism of fine log schemes. The sheaf of relative logarithmic differentials $\Omega_{X/Y}^{1}$ is given by $$\Omega_{X/Y}^{1}
\colon=
\Bigl(\Omega_{X/Y}\oplus(\mathcal{O}_{X}\otimes_{\Z} M^{\gp}_{X}) \Bigr)\big/\mathcal{K},$$ where $\mathcal{K}$ is the $\mathcal{O}_{X}$-module generated by local sections of the following forms:
1. $(d\alpha(a),0)-(0,\alpha(a)\otimes a)$ with $a\in \cM_{X}$;
2. $(0,1\otimes a)$ with $a\in \im(f^{-1}\cM_{Y}\rightarrow \cM_{X})$.
The universal derivation $(\partial,D)$ is given by $$\partial
\colon
\mathcal{O}_{X}\stackrel{d}\rightarrow\Omega_{X/Y}\rightarrow\Omega_{X/Y}^{1}$$ and $$D
\colon
\cM_{X}\rightarrow \mathcal{O}_{X}\otimes_{\Z} \cM_{X}^{\gp}\rightarrow\Omega_{X/Y}^{1}.$$
Let $h: Q \rightarrow P$ be a morphism of fine monoids. For the induced morphism $f:\A_P\rightarrow\A_Q$ we have $$\Omega_{f}^{1}=\mathcal{O}_{\A_P}\otimes_{\Z} \coker{h^{\gp}}.$$ The generators correspond to the logarithmic differentials $d\log\alpha(p)$ for $p\in P$, which are regular on the torus $\text{Spec}(\Z[P^{\gp}])$, modulo those coming from $Q$.
Logarithmic differentials behave somewhat analogously to differentials. Let $$X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z$$ be a sequence of morphisms of fine log schemes.
1. There is a natural exact sequence $$f^{*}\Omega_{Y/Z}^{1} \rightarrow \Omega_{X/Z}^{1}\rightarrow \Omega_{Z/Y}^{1}\rightarrow 0.$$
2. If $f$ is log smooth, then $\Omega_{f}^{1}$ is a locally free $\mathcal{O}_{X}$-module of finite type, and there is an exact sequence $$0\rightarrow f^{*}\Omega_{Y/Z}^{1} \rightarrow \Omega_{X/Z}^{1}\rightarrow \Omega_{X/Y}^{1}\rightarrow 0.$$ In particular, if $f$ is log étale, then $\Omega_{f}^1=0$.
3. If $g f$ is log smooth and the sequence in (2) is exact and splits locally, then $f$ is log smooth.
A proof can be found in [@Ogu Chapter IV].
Let $\A_\Sigma$ be the log scheme introduced in Definition \[df:toricvarieties\]. We have $$\Omega_{\A_{\Sigma}}^{1}
\cong
\mathcal{O}_{\A_{\Sigma}}\otimes_{\Z} N.$$
The normal crossing variety $$X
:=
\Spec{
k[x_{1},\dots,x_{n}]/(x_{1}\cdots x_{d}})$$ defines a log structure of “semistable type” over the standard log point $(\Spec{k},\N)$. This is a special case of Example \[ex:logpoints\] with $X=\Spec{k[\N^{n}]}$ and $D=V(x_{1}\cdots x_{d})$, corresponding to the submonoid of $\mathcal{O}_{X}$ generated by $\mathcal{O}_{X}^{\times}$ and $\{x_{1},\dots,x_{d}\}$. We have that $\Omega_{X}^{1}$ is a free $\mathcal{O}_{X}$-module generated by the logarithmic differentials $$\frac{dx_{1}}{x_{1}}
=
d\log x_{1},
\cdots,
\frac{dx_{d}}{x_{d}}
=
d\log x_{d},
dx_{d+1},
\cdots,
dx_{n}$$ subject to the relation $$d\log x_{1}
+
\cdots
+
d\log x_{d}
=
0.$$
Exact morphisms
---------------
A homomorphism $\theta:P\rightarrow Q$ of monoids is *exact* if the commutative diagram $$\begin{tikzcd}
P\arrow[d,"\theta"']\arrow[r]&
P^\gp\arrow[d,"\theta^\gp"]
\\
Q\arrow[r]&
Q^\gp
\end{tikzcd}$$ is cartesian.
A morphism $f:Y\rightarrow X$ of log schemes is *exact* if for every point $y\in Y$ the naturally induced homomorphism $$\cM_{X,f(x)}\rightarrow \cM_{Y,y}$$ is exact.
Let $f$ be an exact morphism of integral (resp. saturated) log schemes. Then every pullback of $f$ in the category of integral (resp. saturated) log schemes is also exact.
See [@Ogu Proposition III.2.2.1(2)]. This reference states only the integral case. However the saturated case can be proved similarly, see [@Ogu Proposition I.4.2.1(6)].
\[kummer-is-logetandstrict\] Let $f:X\rightarrow Y$ be a log étale morphism of fs log schemes. Then $f$ is Kummer if and only if $f$ is exact.
We refer to [@MR1922832 1.6].
Kato-Nakayama spaces of fs log schemes
--------------------------------------
If $X$ is a fine and saturated log scheme that is locally of finite type over $\text{Spec}({\C})$ we are entitled to the log analytic space $X_{\text{an}}$. The idea of Kato-Nakayama is to associate to $X_{\text{an}}$ a topological space $X_{\log}$ that embodies the log structure of $X$ in a topological way by using points instead of sheaves of monoids. By construction there exists a continuous map $\tau\colon X_{\log}\to X_{\text{an}}$ that is proper and surjective. Let $U\subseteq X_{\text{an}}$ be the trivial locus of the log structure, i.e., the largest open subset over which $\mathcal{O}_{X_{\text{an}}}^{\times}\hookrightarrow M_{X_{\text{an}}}$ is an isomorphism. Then the open immersion $i\colon U\to X_{\text{an}}$ factors through $\tau$, so that $X_{\log}$ can be considered as a “relative compactification” of $i$. We give a short review of Kato–Nakayama spaces and refer to [@Ogu Section V.1] for more details.
Let $\mathrm{pt}^{\dagger}$ be the log analytic point $\mathrm{pt}=\text{Spec}(\C)_{\text{an}}$ with monoid $M_{\mathrm{pt}}=\R_{\geq 0}\times S^1$, where each factor is given the monoidal multiplicative structure so that $\R_{\geq 0}^{\gp}=0$, and map $\alpha\colon M_{\mathrm{pt}}\to \C$ given by $(r,a)\mapsto ra$. As a set, $$X_{\log}:=
\hom(\mathrm{pt}^{\dagger},X_{\text{an}})$$ is comprised of morphisms of log analytic spaces from $\mathrm{pt}^{\dagger}$ to $X_{\text{an}}$. One identifies $X_{\log}$ with the set of pairs $(x,\phi)$, where $x\in X_{\text{an}}$ and $\phi\colon M_{X_{\text{an}},x}^{\gp} \rightarrow S^{1}$ is a homomorphism of abelian groups such that $\phi(f)={f(x)}/{|f(x)|}\in S^{1}$ for every $f\in \mathcal{O}_{X_{\text{an}},x}^\times$. With this description, $\tau\colon X_{\log}\to X_{\text{an}}$ is the projection to the first coordinate $(x,\phi)\mapsto x$. For every open subset $U\subseteq X_{\text{an}}$ and section $m\in M_{X_{\text{an}}}(U)$, we obtain a function $f_{U,m}\colon U_{\log}\rightarrow S^{1}$, defined by $(x,\phi)\mapsto \phi(m_x)\in S^{1}$. Here $U_{\log}$ denotes the subset of $X_{\log}$ of points $(x,\phi)$ with $x\in U$, i.e., the Kato–Nakayama space of the log analytic space $(U,M_{X_{\text{an}}}|_U)$. The topology on $X_{\log}$ is the coarsest topology that makes the projection $\tau\colon X_{\log}\to X_{\text{an}}$ and all the functions $f_{U,m}\colon U_{\log}\rightarrow S^1$ continuous. For every $x\in X_{\text{an}}$, the fiber $\tau^{-1}(x)$ is a torsor under the space of homomorphisms of abelian groups $\hom(M_{X_{\text{an}},x}^{\gp},S^{1})$. If $X_{\text{an}}$ is fs (or more generally if $M_{X_{\text{an}},x}$ is torsion-free), then $\tau^{-1}(x)$ is non-canonically isomorphic to a real torus $(S^{1})^r$ where $r$ is the rank of the free abelian group $\overline{M}^{\gp}_{X_{\text{an}},x}=M^{\gp}_{X_{\text{an}},x}/\mathcal{O}_{X,x}$.
The construction of Kato-Nakayama spaces is functorial and compatible with strict morphisms of fine saturated log analytic spaces $X_{\text{an}}\rightarrow Y_{\text{an}}$, i.e., there is a naturally induced cartesian square of topological spaces: $$\xymatrix{
X_{\log}\ar[r] \ar[d]& Y_{\log}\ar[d]\\
X_{\text{an}}\ar[r] & Y_{\text{an}}
}$$
\[example:kn\]
- For $\A^1$ with its toric log structure $\N\to \Spec{\N}=\A^1$ we have $(\A^1)_{\log}=\R_{\geq 0}\times S^{1}$. The projection map $\tau\colon \R_{\geq 0}\times S^{1}\rightarrow \C$ sends $(r,a)$ to $ra$. More generally, let $P$ be a fine monoid and $X$ the affine toric scheme $\Spec{\C[P]}$ with its toric log structure. Then $$X_{\log}
=
\hom(P,\R_{\geq 0}\times S^{1})
=
\hom(P,\R_{\geq 0})\times \hom(P,S^{1})$$ is locally compact with topology induced by the one on $\R_{\geq 0}\times S^{1}$.
- If $X$ is smooth and $M_D$ is the compactifying log structure coming from a normal crossing divisor $D\subseteq X$, then $(X,M_D)_{\log}$ can be identified with the real oriented blowup of $X$ along $D$. This is a “smooth manifold with corners”.
The last example suggests that $X_{\log}$ should be thought of as the complement of an “open tubular neighbourhood of the log structure.” Our next observation is used in Proposition \[prop::unitdiskbundle\].
\[A.9.74\] Let $X$ be an fs log scheme log smooth over $\C$. Then $X_{\log}$ is homotopy equivalent to $(X-\partial X)_{\text{an}}$.
As observed in [@MR1700591 Remark 1.5.1], $X$ is a manifold with boundary. Our assertion follows readily from this.
Let $D\subseteq X_{\text{an}}$ be a simple normal crossings divisor. Then $(X,M_D)_{\log}$ is a manifold with boundary; its interior is $X_{\text{an}}-D$ [@Ogu Theorem V.1.3.1]. The inclusion $X_{\text{an}}-D \rightarrow (X_{\text{an}},M_D)_{\rm log}$ is a homotopy equivalence.
Log blow-ups
------------
In this section we briefly recall Niziol’s work on log blow-ups [@Niz]. Recall that an ideal of a monoid $P$ is a subset $I\subseteq P$ such that $P+I\subseteq I$.
A subsheaf $\cI\subseteq \cM_{X}$ of ideals of the monoid $\cM_{X}$ is coherent if locally for the étale topology of $X$ there is a chart $P\rightarrow \cM_{X}$ for the log structure and a finitely generated ideal $I\subseteq P$ of the monoid $P$, such that $\cI$ coincides with the subsheaf of $\cM_{X}$ generated by the image of $I$.
A subsheaf $\cI \subseteq \cM_{X}$ of ideals is invertible if it can be locally generated by a single element, or equivalently if it is induced locally by a principal ideal of a chart $P\rightarrow \cM_{X}$. If $\cI$ is coherent, then it is invertible if and only if $\cI_x \subseteq M_{X,x}$ is a principal ideal for every $x\in X$. For a coherent sheaf of ideals $\cI$ of the log structure on $X$, the log blow-up of $X$ along $\cI$ is a log scheme $(X_\cI,M_\cI)$ equipped with a map $f_\cI\colon X_\cI \rightarrow X$ satisfying a universal property: the ideal generated by $f_\cI^{-1}\cI$ in $M_{\cI}$ is invertible and every morphism of log schemes $f\colon Y\rightarrow X$ such that the ideal generated by $f^{-1}\cI$ on $Y$ is invertible factors uniquely through $f_\cI$. Locally on $X$, for a chart $P\rightarrow \cM_{X}$ and a finitely generated ideal $I\subseteq P$ inducing $\cI$, the log blow-up is given by the pullback along $X\rightarrow \text{Spec}(k[P])$ of the blow-up $\text{Proj}( \bigoplus_n I^n)\rightarrow \text{Spec}(k[P])$.
The following result due to Niziol strengthens Kato’s work [@MR1296725 (10.4)].
\[A.3.19\] For $X\in lSm/k$, there is a log blow-up $Y\rightarrow X$ such that $Y\in SmlSm/k$.
This is a special case of [@Niz Theorem 5.10].
When forming triangulated categories of logarithmic motives, for every log blow-ups $Y\rightarrow X$ in $lSm/k$ we invert the naturally induced morphism of motives $$M(Y)\rightarrow M(X).$$ Since every fs log scheme $X$ admits a log blow-up equipped with a Zariski log structure, see [@Niz Theorem 5.4], we may assume that the log structure of $X$ — and thus of every log blow-up [@Niz Proposition 4.5] — is Zariski.
Log blow-ups performed in the category of fine log schemes or in the category of fs log schemes differ by a “normalization” or “saturation” on the level of monoids. See [@Niz Section 4] for details.
Let $X$ be a (normal) toric variety equipped with its natural log structure. Every (saturated) log blow-up of $X$ turns out to be a “toric blow-up” given by subdivisions of the fan $\Sigma$ of $X$. Indeed, by [@Niz Proposition 4.3], the log blow-up along a coherent sheaf of ideals $\cI\subseteq \cM_{X}$ is the same as the normalization of the blow-up of $X$ along the coherent sheaf of ideals $\langle \alpha(\cI)\rangle\subseteq \mathcal{O}_X$ for $\alpha\colon \cM_{X}\rightarrow \mathcal{O}_X$. When $X=\text{Spec}(k[P])$, the global sections of this image form a homogeneous ideal of $k[P]$, and the blow-up along this ideal affords a description via a subdivision of the corresponding cone. Patching together the subdivisions on the affine pieces yields a subdivision of $\Sigma$ realizing the log blow-up.
Set $X=\A_{\N\oplus \N}$, and let $K$ be the zero ideal of $\N\oplus \N$. Now let $\cK$ be the coherent sheaf of ideals in $\cM_{X}$ associated with $K$, and let $P_1$ (resp. $P_2$) be the submonoid of $\Z\oplus \Z$ generated by $(-1,1)$ and $(1,0)$ (resp. $(1,-1)$ and $(0,1)$). From the construction of log blow-ups in [@Ogu Lemma II.1.7.3, Remark II.1.7.4], we see that the log blow-up $Y$ of $X$ along $\cK$ is the gluing of $\A_{P_1}$ and $\A_{P_2}$ along $\A_{\Z\oplus \Z}$. On the underlying schemes, we note that $\underline{Y}$ is the usual blow-up of $\underline{X}=\A^2_{k}$ along the origin.
Log étale monomorphisms {#subsec::mono}
-----------------------
A morphism of schemes is a closed (resp. an open) immersion if and only if it is a proper (resp. an étale) monomorphism by [@EGA Corollaire IV.18.12.6, Théorème IV.17.9.1]. Thus a morphism of schemes is an isomorphism if and only if it is a surjective proper étale monomorphism.
In contrast to this, a proper (resp. a log étale) monomorphism need not be a closed (resp. an open) immersion, and a surjective proper log étale monomorphism not need be an isomorphism. On the level of log motives, however, we will turn surjective proper log étale monomorphisms into isomorphisms. For this purpose we shall study log étale monomorphisms.
\[A.5.15\] By [@Ogu Proposition III.2.6.3(3)], log blow-ups are universally surjective proper monomorphisms. Zariski locally on $X$ and $Y$, $f$ has a chart $P\rightarrow Q$ such that $P^{\rm gp}\rightarrow Q^{\rm gp}$ is an isomorphism, which can be checked by appealing to [@Ogu Lemma II.1.7.3, Remark II.1.7.4]. Thus $f$ is log étale by [@Ogu Theorem IV.3.3.1]. In conclusion, $f$ is a universally surjective proper log étale monomorphism.
\[A.9.71\] Let $f:X\rightarrow S$ be a log étale morphism of fs log schemes, and let $i:S\rightarrow X$ be a section of $f$. Then $i$ is an open immersion.
According to the assumptions we can form the commutative diagram of fs log schemes $$\begin{tikzcd}
S\arrow[r,"i"]\arrow[d,"i"]&X\arrow[d,"i'"]\arrow[r,"f"]&S\arrow[d,"i"]\\
X\arrow[r,"a"]&X\times_S X\arrow[r,"p_2"]&X,
\end{tikzcd}$$ where
(i) $a$ denotes the diagonal morphism.
(ii) $p_2$ denotes the projection on the second factor.
(iii) Each square is cartesian.
It suffices to show that $a:X\rightarrow X\times_S X$ is an open immersion. Since the diagonal morphism $$\underline{X}
\rightarrow
\underline{X}\times_{\underline{S}}\underline{X}$$ is radicial, we are reduced to showing that $a$ is strict étale by [@EGA Théorème IV.17.9.1]. As in [@EGA Corollaire IV.17.3.5], the morphism $a$ is log étale. Thus it remains to show $a$ is strict. We divide the proof into three steps.
\(I) [*Locality on $S$.*]{} Let $g:S'\rightarrow S$ be a strict étale cover of fs log schemes, and set $X'=X\times_S S'$. Then the commutative cartesian diagram of fs log schemes $$\begin{tikzcd}
X'\arrow[r]\arrow[d]&X'\times_{S'}X'\arrow[d]\\
X\arrow[r]&X\times_S X
\end{tikzcd}$$ shows the question is strict étale local on $S$.
\(II) [*Locality on $X$.*]{} Let $h:X'\rightarrow X$ be a strict étale cover of fs log schemes. Then we have the commutative diagram of fs log schemes $$\begin{tikzcd}
&X'\times_S X'\arrow[d,"h''"]\\
X'\arrow[d,"h"']\arrow[r,"a'"]\arrow[ru,"a''"]&X\times_S X'\arrow[d,"h'"]\\
X\arrow[r,"a"]&X\times_S X,
\end{tikzcd}$$ where
(i) The lower square is cartesian.
(ii) $a''$ denotes the diagonal morphism.
(iii) $h':={\rm id}\times h$ and $h'':=h\times {\rm id}$.
Assume that $a''$ is a strict morphism. Then $a'$ is also strict since $h''$ is strict, while $a$ is also strict since $h$ is a strict étale cover. Conversely, if $a$ is a strict morphism, then $a'$ is strict and therefore also $a''$ is strict. Thus the question is strict étale local on $X$.
\(III) [*Final step of the proof.*]{} By [@Ogu Theorem IV.3.3.1], Zariski locally on $X$ and $S$ we have an fs chart $\theta:P\rightarrow Q$ of $f$ such that the following holds.
(i) $\theta$ is injective, and the cokernel of $\theta^{\rm gp}$ is finite,
(ii) The induced morphism $X\rightarrow S\times_{\mathbb{A}_P}\mathbb{A}_Q$ is strict étale.
Hence by parts (I) and (II), we may assume that $(X,S)=(\mathbb{A}_Q,\mathbb{A}_P)$. Then it suffices to show that the diagonal homomorphism $$\mathbb{A}_Q
\rightarrow
\mathbb{A}_{Q}\oplus_{\mathbb{A}_P}\mathbb{A}_Q$$ is strict. In effect, it suffices to show $a\oplus (-a)\in (Q\oplus_P Q)^*$ for every $a\in Q$. Choose $n\in \mathbb{N}^+$ such that $na\in P^{\rm gp}$. Because the summation homomorphism $$P^{\rm gp}\oplus_{P^{\rm gp}}P^{\rm gp}
\rightarrow
P^{\rm gp}$$ is an isomorphism, the two elements $(na)\oplus 0$ and $0\oplus (na)$ of $Q\oplus_P Q$ are equal. Thus we have $n(a\oplus (-a))=0$. Since $Q\oplus_P Q$ is an fs monoid, it follows that $a\oplus (-a)\in Q\oplus_P Q$. This means that $a\oplus (-a)\in (Q\oplus_P Q)^*$ since $n(a\oplus (-a))=0$.
\[A.9.19\] Let $f:X\rightarrow S$ be a log étale monomorphism of fs log schemes, and let $P$ be an fs chart of $S$. Then Zariski locally on $X$, there exists a chart $\theta:P\rightarrow Q$ of $f$ with the following properties.
1. The induced morphism $X\rightarrow S\times_{\mathbb{A}_P} \mathbb{A}_Q$ is an open immersion,
2. $\theta^{\rm gp}:P^{\rm gp}\rightarrow Q^{\rm gp}$ is an isomorphism.
The question is Zariski local on $X$. Let $x$ be a point on $X$. By appealing to [@Ogu Theorem IV.3.3.1], we may assume there exists an fs chart $\theta':P\rightarrow Q'$ with the following properties.
1. The induced morphism $X\rightarrow S\times_{\A_P}\A_{Q'}$ is strict étale.
2. $\theta'$ is injective, and the cokernel of $\theta'^{\rm gp}$ is finite of order invertibile on $\cO_S$.
3. The chart $Q'$ of $X$ is exact at $x$.
Note that $Q:=P^{\rm gp}\cap Q'$ is an fs monoid by Gordon’s lemma [@Ogu Theorem I.2.3.19]. Then the induced homomorphism $P^{\rm gp}\rightarrow Q^{\rm gp}$ is an isomorphism, i.e., we have (ii). Thus the projection $$S\times_{\A_P}\A_{Q}\rightarrow S$$ is a log étale monomorphism. Consider $f$ as the composition $X\rightarrow S\times_{\A_P}\A_Q\rightarrow S$. Then the first arrow is a monomorphism since $f$ is a monomorphism. According to [@Ogu Remark IV.3.1.2], the first arrow is also log étale. Since the inclusion $Q\rightarrow Q'$ is Kummer, see Definition \[KummerDef\], replacing $X\rightarrow S$ by $X\rightarrow S\times_{\mathbb{A}_P}\mathbb{A}_Q$, we may also assume that $\theta'$ is Kummer.
Let $X_1$ (resp. $S_1$) be the strict closed subscheme of $X$ (resp. $S$) whose underlying scheme is $\{x\}$ (resp. $\{f(x)\}$) with the reduced scheme structure. Since $f$ is a monomorphism, the induced pullback morphism $X_1\rightarrow S_1$ is also a monomorphism. Since this is a morphism of log points, it can be written as $$\Spec{Q'\oplus k''^*\rightarrow k''}\rightarrow \Spec{P\oplus k'^*\rightarrow k'},$$ where $\Spec {k''} =\underline{X_1}$ and $\Spec {k'}=\underline{S_1}$.
Suppose for contradiction that $\theta'$ is not an isomorphism. Then the fiber product $X_1\times_{S_1}X_1$ has more than one point, and hence the diagonal $X_1\rightarrow X_1\times_{S_1}X_1$ is not an isomorphism. This contradicts the fact that $X_1\rightarrow S_1$ is a monomorphism. Thus $\theta'$ is an isomorphism.
Then by (i)’, $f$ is strict étale. Since $f$ is a monomorphism, $f$ is an open immersion by [@EGA Théorème IV.17.9.1]. Thus we have (i).
\[A.9.67\] Suppose $P$ is an fs monoid such that $P^\gp$ is torsion free and $u:\Sigma\rightarrow \Spec P$ is subdivision of fans. Then the morphism $\A_u:\A_\Sigma\rightarrow \A_P$ is universally surjective.
Let $p:S\rightarrow \A_P$ be a morphism of fs log schemes, and set $X:=S\times_{\A_P}\A_\Sigma$. We need to show that the projection morphism $f:X\rightarrow S$ is surjective. This question is Zariski local on $S$, so we may assume that $p$ has a factorization $$S\rightarrow \A_Q\stackrel{\theta}\rightarrow \A_P,$$ where the first morphism is strict and $\theta:P\rightarrow Q$ is a homomorphism of fs monoids. The projection $$\Sigma\times_{\Spec P}\Spec Q\rightarrow \Spec Q$$ is also a subdivision of fans due to Lemma \[Fiberproduct\_fans\]. Hence, by replacing $P$ by $Q$, we may assume that $p$ is strict. In this case $\underline{f}$ is a pullback of $\underline{\A_u}$ by Lemma \[Fiber.product.strict.morphism\]. Since $\underline{\A_u}$ is surjective and any surjective morphism in the category of schemes is universally surjective, $\underline{f}$ is also surjective.
\[A.9.21\] Let $(f_i:X_i\rightarrow S)_{i\in I}$ be a finite family of log étale monomorphisms where $X_i$ is quasi-compact for every $i\in I$. Suppose that there is a strict morphism $S\rightarrow \A_\Sigma$ where $\Sigma$ is a fan. Then there exists a subdivision of fans $$u:\Sigma'\rightarrow \Sigma$$ such that for each $i\in I$, the pullback $$f_i'
\colon
X_i\times_{\mathbb{A}_\Sigma}\mathbb{A}_{\Sigma'}\rightarrow S\times_{\mathbb{A}_\Sigma}\mathbb{A}_{\Sigma'}$$ is an open immersion. If $f_i$ is surjective and proper, then $f_i'$ is an isomorphism.
Let us start with a log étale monomorphism $f:X\rightarrow S$, where $X$ is quasi-compact. Suppose that $\Sigma$ is in a lattice $N$. Choose fs submonoids $\{P_j\}_{j\in J}$ of $\Z^r$ such that the dual monoids $P_j^{\vee}$ corresponds the maximal cones of $\Sigma$. Then $$S_j:=X\times_{\A_\Sigma}\A_{P_j}$$ affords an fs chart $P_j$, and $\{S_j\rightarrow S\}_{j\in J}$ is a finite Zariski cover. By Lemma \[A.9.19\], there exists a finite Zariski cover $\{U_{jk}\rightarrow S_j\}_{k\in K_j}$ such that for each $k$, there is a homomorphism $\theta_k:P_j\rightarrow Q_{jk}$ of fs monoids with the properties that $\theta_k^{\rm gp}$ is an isomorphism and the induced morphism $$U_{jk}\rightarrow S_j\times_{\mathbb{A}_{P_j}}\mathbb{A}_{Q_{jk}}$$ is an open immersion.
Lemma \[Compactification\_fan\] yields a fan $\Theta_{jk}$ in $N$ such that $Q_{jk}^\vee\in \Theta_{jk}$ and $\lvert \Theta_{jk}\rvert=N$. Let $\Sigma_0$ be the fan in $N$ whose only cone is $N$, and let $\Sigma'$ be the fiber product of $\Sigma$ and all $\Theta_{jk}$ over $\Sigma_0$. Owing to Lemma \[Fiberproduct\_fans\] we get that $\Sigma'$ is a subdivision of $\Sigma$. Moreover, for each cone $\sigma$ of $\Sigma'$ there is $j\in J$ and $k\in K_j$ such that $\sigma\subset Q_{jk}^{\vee}$. This implies that the induced morphism $$u_j':U_{jk}\times_{\mathbb{A}_\Sigma}\mathbb{A}_{\Sigma'}\rightarrow S_j\times_{\mathbb{A}_\Sigma}\mathbb{A}_{\Sigma'}$$ is an open immersion. In particular, $u_j'$ is strict étale. Thus the pullback $$f':X\times_{\A_\Sigma}\A_{\Sigma'}\rightarrow S\times_{\A_\Sigma}\A_{\Sigma'}$$ is strict étale. Since $f'$ is a monomorphism, it follows that $f'$ is an open immersion by [@EGA Théorème IV.17.9.1].
Assuming that $f$ is surjective and proper, we consider the cartesian square $$\begin{tikzcd}\label{A.9.21.1}
X\times_{\A_\Sigma}\A_{\Sigma'}\arrow[r,"f'"]\arrow[d,"g'"']&S\times_{\A_\Sigma}\A_{\Sigma'}\arrow[d,"g"]\\
X\arrow[r,"f"]&S.
\end{tikzcd}$$ As a result of Lemma \[A.9.67\], we get that $\A_u$ is universally surjective, and hence $g$ and $g'$ are universally surjective. Thus $(fg')^{-1}(s)$ is nonempty for every point $s\in S$. The induced open immersion $(fg')^{-1}(s)\rightarrow g^{-1}(s)$ is proper, and it is an isomorphism according to Zariski’s main theorem since $g^{-1}(s)$ is connected. Thus $f'$ is surjective, and hence an isomorphism.
In the general case, choose a subdivision of fans $\Sigma_i'\rightarrow \Sigma$ for each $i\in I$ such that the pullback $$X_i\times_{\A_\Sigma}\A_{\Sigma_i'}\rightarrow S\times_{\A_\Sigma}\A_{\Sigma_i'}$$ is an open immersion. Letting $\Sigma'$ be the fiber product $\Sigma_{i_1}'\times_{\Sigma}\cdots \times_{\Sigma}\Sigma_{i_r}'$ where $I=\{i_1,\ldots,i_r\}$, we have that the pullback $$X_i\times_{\A_\Sigma}\A_{\Sigma'}\rightarrow S\times_{\A_\Sigma}\A_{\Sigma'}$$ is an open immersion for any $i$. To conclude we note that the induced morphism $\Sigma'\rightarrow \Sigma$ is a subdivision of fans owing to Lemma \[Fiberproduct\_fans\].
\[A.9.22\] Suppose $f:X\rightarrow S$ is a surjective proper log étale monomorphism of fs log schemes. Then any pullback along $f$ is a surjective proper log étale monomorphism. In particular, $f$ is universally surjective.
The question is Zariski local on $S$, so we may assume that $S$ is quasi-compact and has an fs chart $P$. Then $X$ is quasi-compact since $f$ is proper. By Proposition \[A.9.21\], there exists a subdivision of fans $u:M\rightarrow \Spec P$ such that the morphism $f'$ in is an isomorphism. In the proof of Proposition \[A.9.21\], we have checked that the morphism $g$ in is universally surjective. Thus $f$ is universally surjective. To conclude we use that a pullback along a proper log étale monomorphism is a proper log étale monomorphism.
\[A.9.70\] Recall from [@FKato] that a morphism $f:X\rightarrow S$ of fs log schemes is called a [*log modification*]{} if Zariski locally on $S$, there exists a log blow-up $g:Y\rightarrow X$ such that the composition $fg$ is also a log blow-up. [^3]
\[A.9.76\] As observed in [@FKato], a morphism of fs log schemes induced by a subdivision of toric fans in the sense of [@Ogu §I.1.9] is an example of a log modification, see also [@TOda pp. 39-40].
\[A.9.75\] Let $f:X\rightarrow S$ be a morphism of fs log schemes. Then $f$ is a surjective proper log étale monomorphism if and only if $f$ is a log modification.
Assume that $f$ is a log modification. The question is Zariski local on $S$, so we may assume that there exists a log blow-up $g:Y\rightarrow S$ such that $fg$ is also a log blow-up. Since $g$ and $fg$ are surjective proper log étale monomorphisms as observed in Example \[A.5.15\], $f$ is a surjective proper log étale morphism. Consider the induced commutative diagram of fs log schemes $$\begin{tikzcd}
Y\times_X Y\arrow[d]\arrow[r]&Y\times_S Y\arrow[d]\\
X\arrow[r,"\Delta"]&X\times_S X.
\end{tikzcd}$$ Since $g$ and $fg$ are monomorphisms, the diagonal morphisms $Y\rightarrow Y\times_S Y$ and $Y\rightarrow Y\times_X Y$ are isomorphisms. Thus the upper horizontal morphism is an isomorphism. The right vertical morphism is surjective since $g$ is universally surjective. Thus $\Delta$ is surjective. By Lemma \[A.9.71\], $\Delta$ is an open immersion, so $\Delta$ is an isomorphism. Thus $f$ is a monomorphism, and hence a surjective proper log étale monomorphism.
Conversely, assume that $f$ is a surjective proper log étale monomorphism. The question is Zariski local on $S$, so we may assume that $S$ is quasi-compact and has a neat fs chart $P$. Then $X$ is quasi-compact since $f$ is proper. By Proposition \[A.9.21\], there exists a subdivision of fans $\Sigma\rightarrow \Spec P$ such that the pullback $$X\times_{\A_P}\A_\Sigma\rightarrow S\times_{\A_P}\A_\Sigma$$ is an isomorphism. As observed in Example \[A.9.76\], there exists a log blow-up $T\rightarrow \A_\Sigma$ such that the composition $T\rightarrow \A_P$ is also a log blow-up. Then the morphisms $$X\times_{\A_P}T\rightarrow X \text{ and } X\times_{\A_P}T\rightarrow S$$ are log blow-ups, so $f$ is a log modification.
\[A.9.77\] Any composition of log modifications is a log modification. Any pullback of a log modification is a log modification.
See [@FKato Lemma 3.15(2),(3)]. Alternatively one can use Proposition \[A.9.75\].
\[FKatoThm\] Let $f:X\rightarrow Y$ be a morphism of noetherian fs log schemes. Then there exists a log modification $Y'\rightarrow Y$ such that the projection $$f':X\times_Y Y'\rightarrow Y'$$ is integral. In particular, $f'$ is exact.
See [@FKato Theorem (1)].
\[A.9.78\] Let $f:X\rightarrow S$ and $g:Y\rightarrow S$ be morphisms of fs log schemes. Suppose that $g$ is a log modification. Then there are at most one morphism $h:X\rightarrow Y$ such that $gh=f$. Moreover, if $f$ is a log modification, then $h$ is a log modification.
Suppose that such an $h$ exists. The graph morphism $u:X\rightarrow X\times_S Y$ is a pullback of the diagonal morphism $d:Y\rightarrow Y\times_S Y$, and $d$ is an isomorphism since $Y$ is a log modification. Thus $u$ is an isomorphism. This means that $h$ is isomorphic to the projection $X\times_S Y\rightarrow Y$. If $f$ is a log modification, then the projection is also a modification by Proposition \[A.9.77\].
\[A.9.79\] Let $X$ be an fs log scheme. We denote by $X_{div}$ the category of log modifications over $X$, and we denote by $X_{div}^{Sm}$ the full subcategory of $X_{div}$ consisting of log modifications $Y\rightarrow X$ such that $Y\in SmlSm/k$.
\[A.9.83\] Every exact log étale monomorphism $f:X\rightarrow S$ of fs log schemes is an open immersion.
The question is Zariski local on $S$ and $X$, so we may assume that $S$ has s neat fs chart $P$ at $s\in S$. Moreover, due to Lemma \[A.9.19\], we may assume there exists a chart $\theta:P\rightarrow Q$ of $f$ such that $X\rightarrow S\times_{\A_P}\A_Q$ is an open immersion and $\theta^{\rm gp}:P^{\rm gp}\rightarrow Q^{\rm gp}$ is an isomorphism.
Let $x$ be a point of $X$ mapping to $s$. Then $\overline{\cM}_{X,x}\cong Q/F$ for some face $F$ of $Q$. By the assumption that $f$ is exact we see that $$P\cong \overline{\cM}_{S,s}\rightarrow\overline{\cM}_{X,x}\cong Q/F$$ is exact. Thus $P\rightarrow Q_F$ is exact owing to [@Ogu Proposition I.4.2.1(3)], i.e., $$P\cong P^{\rm gp}\times_{Q^{\rm gp}}Q_F.$$ Since $\theta^{\rm gp}$ is an isomorphism, this implies $P\cong Q_F$. This shows the morphism $f$ is strict. Thus $f$ is a strict étale monomorphism, and hence an isomorphism according to [@EGA Théorème IV.17.9.1].
\[A.9.80\] Suppose $X$ is an fs log scheme. Then $X_{div}$ is a filtered category.
For any $Y,Y'\in X_{div}$, we have that $Y\times_X X'\in X_{div}$ owing to Proposition \[A.9.77\]. Thus $X_{div}$ is connected. Owing to Lemma \[A.9.78\] there exists at most one morphism $Y\rightarrow Y'$ of fs log schemes over $X$. Thus $X_{div}$ is filtered.
\[A.9.81\] For $X\in lSm/k$, the category $X_{div}^{Sm}$ is cofinal in $X_{div}$.
Follows from Proposition \[A.3.19\].
\[A.9.82\] For $X\in lSm/k$, the category $X_{div}^{Sm}$ is a filtered category.
Follows from Propositions \[A.9.80\] and \[A.9.81\].
\[Fan.12\] Let $f:X\rightarrow S$ be an exact morphism of noetherian fs log schemes. Then for every log modification $g:Y\rightarrow X$, there is a log modification $S'\rightarrow S$ such that the pullback $$r:Y\times_S S'\rightarrow X\times_S S'$$ is an isomorphism.
There exists a log modification $S'\rightarrow S$ such that the projection $$q:Y\times_S S'\rightarrow S'$$ is exact owing to Theorem \[FKatoThm\] Since $f$ is exact, the projection $p:X\times_S S'\rightarrow S'$ is exact. It follows that $$r:Y\times_S S'\rightarrow X\times_S S'$$ is exact owing to [@Ogu Proposition III.3.2.1(1)]. We deduce that $r$ is an open immersion owing to Proposition \[A.9.83\]. Since $r$ is surjective and proper, it is an isomorphism.
\[Fan.14\] Let $f:X\rightarrow S$ be an exact morphism of noetherian fs log schemes. Then the functor $$f^*:S_{div}\rightarrow X_{div}$$ mapping $S'\in S_{div}$ to $S'\times_S X$ is cofinal.
Immediate from Proposition \[A.9.82\].
Small Kummer étale and small log étale sites
--------------------------------------------
In this subsection we review the small Kummer étale site $X_{k\acute{e}t}$ and the small log étale site $X_{l\acute{e}t}$ for a saturated log scheme $X$. We also prove some basic properties.
\[ket.1\] A saturated log scheme $X$ is *log strictly local* (resp. a *log geometric point*) if $\underline{X}$ is the spectrum of a strictly henselian local ring (resp. a separably closed field), and for every integer $n>0$ prime to the characteristic of $k$ the multiplication by $n$ morphism on $\overline{\cM}(X)$ is bijective.
A saturated log scheme $X$ is *log strictly henselian* if $X$ is a finite disjoint union of log strictly local schemes.
For a saturated log scheme $X$, let $X_{k\acute{e}t}$ (resp. $X_{l\acute{e}t}$) denote the category of Kummer étale (resp. log étale) morphisms of saturated log schemes whose target is $X$. The topology on $X_{k\acute{e}t}$ (resp. $X_{l\acute{e}t}$) is generated by the pretopology with coverings comprised of Kummer étale (resp. log étale) morphisms $\{U_i\rightarrow U\}_{i\in I}$ for which $\amalg_{i\in I} U_i\rightarrow U$ is universally surjective in the category of *saturated* log schemes.
Suppose that $X$ is an fs log scheme. For $X_{k\acute{e}t}$, a family of morphisms $\{U_i\rightarrow U\}_{i\in I}$ is a covering if and only if $\amalg_{i\in I} U_i\rightarrow U$ is surjective, see e.g., [@MR3658728 Proposition 3.2]. In particular, a family of morphisms $\{U_i\rightarrow U\}_{i\in I}$ is a covering if and only if $\amalg_{i\in I} U_i\rightarrow U$ is universally surjective in the category of *fs* log schemes.
However, we really need to take the category of *saturated* log schemes in the definition of coverings in $X_{l\acute{e}t}$. For example, in the proof [@MR3658728 Proposition 3.14(1)], a surjectivity of $(\amalg_{i\in I}U_i)\times_U V\rightarrow V$ is used for some saturated but non fs log scheme over $U$.
\[ket.2\] Let $X$ be a log strictly henselian scheme. Then every saturated log scheme $Y$ that is finite and strict over $X$ is also strictly henselian.
Since $\underline{X}$ is strictly henselian, $\underline{Y}$ is strictly henselian too. Let $y$ be a closed point of $Y$ with image $x$ in $X$. Then $\overline{\cM}_{Y,y}\cong \overline{\cM}_{X,x}$ since $Y$ is strict over $X$. Thus for every integer $n>0$ prime to the characteristic of $k$ the multiplication by $n$ on $\overline{\cM}_{Y,y}$ is bijective.
For an integral monoid $P$ and $n\in \N^+$, we let $P^{1/n}$ denote the submonoid of $P^{\rm gp}\otimes \Q$ given by $$\label{ket.0.1}
P^{1/n}:=\{p\in P^{\rm gp}\otimes \Q:np\in P\}.$$
\[ket.3\] Let $S$ be an fs log scheme with an fs chart $P$ whose underlying scheme $\underline{S}$ is strictly local. Then every Kummer étale cover $f:X\rightarrow S$ admits a refinement of the form of a projection $$S\times_{\A_P} \A_{P^{1/n}}\rightarrow S$$ where $n\in N^+$ is invertible in $T$.
Any open subscheme of $X$ is again a Kummer étale cover since $\underline{S}$ is strictly local. Hence, owing to Theorem \[KatoStrThm\], we may assume that $f$ admits an fs chart $\theta:P\rightarrow Q$ with the following properties.
1. A naturally induced strict étale morphism $$p:X\rightarrow S\times_{\A_P}\A_Q.$$
2. The kernel and cokernel of $\theta^{\rm gp}:P^{\rm gp}\rightarrow Q^{\rm gp}$ is finite, and the orders are invertible in $S$.
Since $f$ is Kummer, (ii) implies that $Q\subset P^{1/n}$ for some $n$ invertible in $S$. Replacing $X$ by $X\times_{\A_Q}\A_{P^{1/n}}$, we may assume that $Q=P^{1/n}$. Then $k[Q]$ is a locally free $k[P]$-module, so that $\A_Q$ is finite over $\A_P$. Thus $S\times_{\A_P}\A_Q$ is finite over $\A_P$. This shows that the underlying scheme of $S\times_{\A_P}\A_Q$ is strictly henselian, since $\underline{S}$ is strictly henselian, and $p$ admits a section. It follows that $S\times_{\A_P}\A_Q$ is a refinement of $X$.
\[ket.4\] Suppose that $S$ is an fs log scheme. Let $f:X\rightarrow S$ be a morphism of saturated log schemes, and let $g:S'\rightarrow S$ be a Kummer étale morphism of fs log schemes. If $X$ is log strictly henselian, then the projection $X\times_S S'\rightarrow X$ has a section.
We may assume that $X$ is local. Replacing $\underline{S}$ by its strict henselization at the image of the closed point of $X$, we may assume that $\underline{S}$ is strictly local. Then since $S$ is local, $f$ admits an fs chart $\theta:P\rightarrow Q$. Owing to Lemma \[ket.3\] we may assume that $S'=S\times_{\A_P}\A_{P^{1/n}}$ where $n>0$ is an integer invertible in $k$. There is a naturally induced commutative diagram of saturated monoids $$\begin{tikzcd}
P^{1/n}\arrow[r]\arrow[d]&P\arrow[d,"\theta"]
\\
Q^{1/n}\arrow[r]&Q.
\end{tikzcd}$$ By the assumption on $Q$ the upper horizontal morphism is an isomorphism. This means that the homomorphism $Q\rightarrow Q\oplus_P P^{1/n}$ has a retraction, so the projection $X\times_S S'\rightarrow X$ has a section.
Sharpened fans
--------------
In this subsection we review the notion of Kato’s fan in [@MR1296725], which we call sharpened fan, and we study basic properties. Sharpened fan contains less information and is more flexible than fs monoscheme.
For a monoidal space $\Sigma=(\Sigma,\cM_\Sigma)$, we set $$\overline{\Sigma}:=(\Sigma,\overline{\cM}_\Sigma).$$ For a morphism of monoidal spaces $f:\Sigma'\rightarrow \Sigma$, let $$\overline{f}:\overline{\Sigma'}\rightarrow \overline{\Sigma}$$ denote the naturally induced morphism of sharpened fans.
For an fs monoid $P$, we set $$\overline{\rm Spec}(P):=\overline{\Spec{P}}.$$ Note that $\overline{\rm Spec}(P)\cong \overline{\rm Spec}(\overline{P})$.
\[Fan.1\] A *sharpened fan* $\Delta$ is a monoidal space such that there exists an open cover $\{U_i\}$ with an isomorphism $$U_i\cong \overline{\rm Spec} (P_i),$$ where $P_i$ is a sharp fs monoid. A morphism of sharpened fans is a morphism of monoidal spaces. A *cone* of $\Delta$ is an open subset of $\Delta$ that is isomorphic to $\overline{\rm Spec} (P)$ for some sharp fs monoid $P$. Note that there is a one-to-one correspondence between cones of $\Delta$ and points of $\Delta$.
We say that $\Delta$ is *affine* if there is an isomorphism $\Delta\cong \uSpec(P)$ for some sharp fs monoid $P$. We say that $\Delta$ is *smooth* if every cone of $\Delta$ is isomorphic to $\overline{\rm Spec} (\N^r)$ for some $r$.
For any property $\bP$ of maps of topological spaces, we say that a morphism of sharpened fan satisfies $\bP$ if the underlying map of topological spaces satisfies $\bP$.
\[Fan.7\] The reader may want to compare the above with the related notions of a conical polyhedral complex with an integral structures, a fan satisfying ($S_{fan}$), and an s-fan, see [@MR0335518 Definitions II.5 and II.6], [@MR1296725 9.4], and [@Ogu Remark II.1.9.4].
In the following we establish some basic properties for sharpened fans. We refer to [@Ogu Sections II.1.2, II.1.3] for the monoscheme versions.
\[Fan.21\] Let $P$ be a sharp fs monoid, and let $\Delta$ be a sharpened fan. Then there is a canonical bijection $$\alpha:\hom(\Delta,\uSpec(P))\xrightarrow{\cong} \hom(P,\Gamma(\Delta,\cM_{\Delta})).$$
We define $\alpha$ by the morphism sending $f:\Delta\rightarrow \uSpec(P)$ to its associated homomorphism on the global sections $$P\cong \Gamma(\uSpec(P),\cM_{\uSpec(P)})\rightarrow \Gamma(\Delta,\cM_{\Delta}).$$
Suppose that $\Delta=\uSpec(Q)$ for some sharp monoid $Q$. Then $\alpha$ has an inverse $$\hom(P,Q)\rightarrow \hom(\uSpec(Q),\uSpec(P))$$ sending $g:P\rightarrow Q$ to $\uSpec(g):\uSpec(Q)\rightarrow \uSpec(P)$. Thus $\alpha$ is bijective in this case.
In the general case, let $\{\sigma_i\}_{i\in I}$ be the set of cones of $\Delta$, and let $\{\sigma_{ijk}\}_{k\in I_{ij}}$ be the set of cones of $\sigma_i\cap \sigma_j$ where $i,j\in I$. There is a naturally induced diagram $$\begin{tikzcd}[column sep=small]
\hom(\Delta,\uSpec(P))\arrow[d]\arrow[r]&
\amalg_{i}\hom(\sigma_i,\uSpec(P))\arrow[d]\arrow[r,shift left=0.5ex]\arrow[r,shift right=0.5ex]&
\amalg_{i,j,k}\hom(\sigma_{ijk},\uSpec(P))\arrow[d]
\\
\hom(P,\Gamma(\Delta,\cM_{\Delta}))\arrow[r]&
\amalg_{i}\hom(P,\Gamma(\sigma_i,\cM_{\sigma_i}))\arrow[r,shift left=0.5ex]\arrow[r,shift right=0.5ex]&
\amalg_{i,j,k}\hom(P,\Gamma(\sigma_i,\cM_{\sigma_i})).
\end{tikzcd}$$ Here the horizontal diagrams are equalizers. The middle and right vertical morphisms are bijections by the above. Thus the left vertical morphism is a bijection.
On the category of sharpened fans the *Zariski topology* is the smallest Grothendieck topology containing all open covers. The Yoneda lemma allows us to identify any sharpened fan with its representable sheaf.
\[Fan.23\] Let $\cF$ be a sheaf of sets on the category of sharpened fans, and suppose that $\{\Delta_i\rightarrow \cF\}_{i\in I}$ is a set of morphisms from sharpened fans $\Delta_i$ satisfying the following conditions for every morphism $\Psi\rightarrow \cF$, where $\Psi$ is a sharpened fan.
1. For every $i\in I$, the sheaf $\Delta_i\times_{\cF}\Psi$ is representable, and the projection $$\Delta_i\times_{\cF}\Psi\rightarrow \Psi$$ is an open immersion.
2. The set $\{\Delta_i\times_{\cF}\Psi\rightarrow \Psi\}_{i\in I}$ is a Zariski cover.
Then $\cF$ is representable.
For $\Delta_{ij}:=\Delta_i\times_{\cF}\Delta_j$ for $i,j\in I$, the projections $\Delta_{ij}\rightarrow \Delta_i,\Delta_j$ are open immersions by the condition (i). Hence we can glue the morphisms $\Delta_i\rightarrow \cF$ for $i\in I$ to obtain a morphism $$f\colon\Delta\rightarrow \cF$$ from a sharpened fan $\Delta$.
It remains to show that $f$ is an isomorphism. For this purpose we need to show that for every morphism $\Psi\rightarrow \cF$ from a sharpened fan, the projection $$f_\Psi\colon\Delta\times_{\cF} \Psi \rightarrow \Psi$$ is an isomorphism. The sharpened fan $\Delta\times_\cF\Psi$ is again the gluing of $\Delta_i\times_\cF\Psi$ for $i\in I$ along $\Delta_{ij}\times_{\cF}\Psi$ for $i,j\in I$. The condition (ii) shows that $f_\Psi$ is an isomorphism.
\[Fan.24\] Let $P\rightarrow Q$ and $P\rightarrow P'$ be homomorphisms of sharp saturated monoids. Then the fiber product $$\uSpec(Q)\times_{\uSpec(P)}\uSpec(P')$$ is representable by $\uSpec(\overline{Q'})$, where $Q'$ is the amalgamated sum $Q\oplus_P P'$ in the category of saturated monoids.
Since $Q'$ is the amalgamated sum, for every saturated monoid $M$, there is a bijection $$\hom(Q',M)\xrightarrow{\cong} \hom(Q,M)\times_{\hom(P,M)}\hom(Q,M).$$ If $M$ is sharp, then $\hom(Q',M)\cong \hom(\overline{Q'},M)$, and hence there is a bijection $$\hom(\overline{Q'},M)\xrightarrow{\cong} \hom(Q,M)\times_{\hom(P,M)}\hom(Q,M).$$ Specializing the above to $M=\Gamma(\Delta,\cM_{\Delta})$, where $\Delta$ is any sharpened fan, and applying Proposition \[Fan.21\] concludes the proof.
\[Fan.25\] The category of sharpened fans has fiber products.
Let $f:\Psi\rightarrow \Delta$ and $g:\Delta'\rightarrow \Delta$ be morphisms of sharpened fans. Take the fiber product $\Psi':=\Psi\times_{\Delta}\Delta'$ in the category of sheaves. We need to show that $\Psi'$ is representable. Let $\{\sigma_i\}_{i\in I}$ be the set of cones of $\Delta$. For every $i\in I$, let $$\{\tau_{ij}\}_{j\in J_i}\text{ and }\{\sigma_{ik}'\}_{k\in K_i}$$ be the set of cones of $\Psi$ and $\Delta'$. Owing to Lemma \[Fan.24\], we see that $$\tau_{ijk}':=\tau_{ij}\times_{\sigma_i}\sigma_{ik}'$$ is representable for every $i$, $j$, and $k$. Moreover, for every morphism $\Theta\rightarrow \Psi'$ of sheaves, where $\Theta$ is a sharpened fan, there is an isomorphism $$(\amalg_{i,j,k}\tau_{ijk}')\times_{\Psi'}\Theta \cong \amalg_{i,j,k} (\tau_{ij}\times_\Psi \Theta)\times_{\sigma_i\times_\Delta \Theta}(\sigma_{ik}'\times_{\Delta'}\Theta).$$ The latter is representable and also a cover of $\Theta$. Thus Lemma \[Fan.23\] shows that $\Psi'$ is representable.
\[Fan.29\] Let $\alpha$ be the functor from the category of fs monoschemes to the category of sharpened fans sending any monoscheme $\Sigma$ to $\overline{\Sigma}$. Then $\alpha$ preserves fiber products.
Suppose that $\theta:P\rightarrow Q$ and $\varphi:P\rightarrow P'$ are homomorphisms of fs monoids. Let $Q''$ the amalgamated sum $\ol{P'}\oplus_{\ol{P}}\ol{Q}$ in the category of saturated monoids. Observe that $$\Spec{P'}\times_{\Spec{P}}\Spec{Q}\cong \Spec{P'\oplus_P Q},$$ and $$\uSpec(\overline{P})\times_{\uSpec(\overline{P})}\uSpec(\overline{Q})\cong \uSpec(\overline{Q''}).$$
There is a naturally induced commutative diagram $$\begin{tikzcd}
P'\oplus_P Q\arrow[r]\arrow[rd]&\overline{P'}\oplus_{\overline{P}}\overline{Q}\arrow[d]
\\
&\overline{P'\oplus_P Q},
\end{tikzcd}$$ which induces a commutative diagram $$\begin{tikzcd}
\overline{P'\oplus_P Q}\arrow[r,"\mu"]\arrow[rd,"{\rm id}"']&\overline{Q''}\arrow[d]
\\
&\overline{P'\oplus_P Q}.
\end{tikzcd}$$ The homomorphism $\overline{P'\oplus_P Q}\rightarrow \overline{Q''}$ is surjective since the homomorphism $P'\oplus_P Q\rightarrow Q''$ is surjective. It follows that the homomorphism $\overline{P'\oplus_P Q}\rightarrow \overline{Q''}$ is an isomorphism. We deduce that there is a canonical isomorphism $$\label{Fan.29.1}
\alpha(\Spec{P'}\times_{\Spec{P}}\Spec{Q})\rightarrow \alpha(\Spec{P'})\times_{\alpha(\Spec{P})}\alpha(\Spec{Q}).$$
Suppose that $\Theta\rightarrow \Sigma$ and $\Sigma'\rightarrow \Sigma$ be morphisms of fs monoschemes. There is an open cover $\{\sigma_i\}_{i\in I}$ of $\Sigma$, an open cover $\{\tau_{ij}\}_{j\in J_i}$ for every $i\in I$, and an open cover $\{\sigma_{ik}'\}_{k\in K_i}$ for every $i\in I$ such that $\sigma_i$, $\tau_{ij}$, and $\sigma_{ik}'$ are affine for every $i$, $j$, and $k$. Due to we have isomorphisms $$\alpha(\tau_{ij}\times_{\sigma_i}\sigma_{ik}')
\cong
\alpha(\tau_{ij})\times_{\alpha(\sigma_i)}\alpha(\sigma_{ik}').$$ Gluing these isomorphisms concludes the proof.
Next we explain how to construct an fs monoscheme from the data of a sharpened fan. We begin with proving the following lemma.
\[Fan.19\] Let $\Sigma$ be an fs monoscheme. Then the map $$\label{Fan.19.1}
\hom(\Spec{\N},\Sigma)\rightarrow \hom(\uSpec(\N),\overline{\Sigma})$$ sending $f$ to $\overline{f}$ is bijective.
If $g:\uSpec(\N)\rightarrow \overline{\Sigma}$ is a morphism, then the image of $g$ lies in a cone of $\overline{\Sigma}$ containing the image of the closed point of $\uSpec(\N)$. Hence we are reduced to the case $\Sigma=\Spec{P}$, where $P$ is an fs monoid.
There is a commutative diagram of sets with vertical bijections $$\begin{tikzcd}
\hom(\Spec{\N},\Spec{P})\arrow[d]\arrow[r]&
\hom(\uSpec(\N),\uSpec(\overline{P}))\arrow[d]
\\
\hom(P,\N)\arrow[r,"\mu"]&
\hom(\overline{P},\N),
\end{tikzcd}$$ where $\mu$ sends $g:P\rightarrow \N$ to $\overline{g}:\overline{P}\rightarrow \N$. Every homomorphism $g:P\rightarrow \N$ maps the unit group $P^*$ maps to $0$. This shows that $\mu$ is bijective.
\[Fan.10\] Let $\Delta=(\Delta,\cM_{\Delta})$ be a sharpened fan. Suppose that $N^\vee\rightarrow \cM_{\Delta}^{\rm gp}$ is a surjective morphism of sheaves, where $N$ is a lattice with dual $N^\vee$. Then $$\Delta_N:=(\Delta,N^\vee\times_{\cM_{\Delta}^{\rm gp}}\cM_{\Delta})$$ is a monoidal space. Moreover, $\Delta_N$ is toric and there is an isomorphism $\overline{\Delta_N}\cong \Delta$.
Let us check that $\Delta_N$ is an fs monoscheme. For this purpose, we may assume that $$\Delta=\overline{\rm Spec}(P),$$ where $P$ is a sharp fs monoid. Let $x$ be a point of $\Delta$ corresponding to a face $F$ of $P$. Since $\cM_{\Delta,x}\cong P/F$, there is an isomorphism $$\cM_{\Delta_N,x}\cong N^\vee\times_{(P/F)^{\rm gp}}(P/F).$$ Applying [@Ogu Proposition I.4.2.1(1)] to $P_F$ we obtain the cartesian square $$\begin{tikzcd}
P_F\arrow[d]\arrow[r]&
P/F\arrow[d]
\\
(P_F)^\gp\arrow[r]&
(P/F)^\gp.
\end{tikzcd}$$ It follows that $$\cM_{\Delta_N,x}\cong N^\vee\times_{(P_F)^\gp}P_F\cong N^\vee\times_{P^\gp}P_F,$$ which is a localization of the monoid $$P_N:=N^\vee\times_{P^{\rm gp}}P$$ with respect to the face $F_N:=N^\vee\times_{P^{\rm gp}}F$. Thus there is an isomorphism $$\Delta_N\cong \uSpec(P_N).$$ This proves the claim.
Suppose that $\Delta$ is connected. Then $\Delta_N$ is also connected, so that it has a unique generic point $\xi$, see [@Ogu p. 198]. It follows that $$\hom(\Spec{\Z},\Delta_N)\cong \hom(\cM_{\Delta_N,\xi}^\gp,\Z)\cong \hom(N^\vee,\Z)\cong N.$$ We obtain the composite map $$\label{Fan.20.1}
\varphi_{\Delta,N}:\hom(\uSpec(\N),\Delta)\rightarrow \hom(\Spec{\N},\Delta_N)\rightarrow \hom(\Spec{\Z},\Delta_N)\cong N,$$ where the first map is the inverse of , and the second map is induced by the homomorphism $\N\rightarrow \Z$.
\[Fan.20\] Let $\Delta$ be a connected sharpened fan whose underlying topological space is a finite set. Suppose that $N^\vee\rightarrow \cM_{\Delta}^\gp$ is a surjective morphism of sheaves, where $N$ is a lattice with dual $N^\vee$. If the morphism $$\varphi_{\Delta,N}:\hom(\uSpec(\N),\Delta)\rightarrow N$$ is injective, then the associated fs monoscheme $\Delta_N$ is a fan.
Owing to Theorem \[fan=monoscheme2\] we need to show that $\Delta_N$ is toric and separated. In Construction \[Fan.10\] we noted that $\Delta_N$ is toric. If $\varphi_{\Delta,N}$ is injective, then the homomorphism $$\hom(\Spec{\N},\Delta_N)\rightarrow \hom(\Spec{\Z},\Delta_N)$$ is injective. This means that $\Delta_N$ is separated.
\[Fan.18\] Suppose that $\Sigma$ is an fs monoscheme. Then $\Delta:=\overline{\Sigma}$ is a sharpened fan. If $\Sigma$ is a fan in a lattice $N$, then $\cM_{\Sigma}^\gp\cong N^\vee$. Thus the naturally induced morphism of sheaves $\cM_{\Sigma}^\gp\rightarrow \cM_{\Delta}^\gp$ gives a surjective morphism of sheaves $$N^\vee\rightarrow \cM_{\Delta}^\gp.$$ Applying [@Ogu Proposition I.4.2.1] to $\cM_\Sigma$ we deduce the isomorphism $$\cM_{\Sigma}\cong N^\vee\times_{\cM_\Delta^\gp}\cM_\Delta.$$ This shows $\Sigma\cong \Delta_N$, where $\Delta_N$ is the fs monoscheme in Construction \[Fan.10\].
As in the case of fans, there are notions of subdivision and partial subdivision of sharpened fans as follows.
\[Fan.8\] Let $f:\Delta'\rightarrow \Delta$ be a morphism of sharpened fans. We say that $f$ is a *subdivision* (resp. *partial subdivision*) if the following conditions are satisfied.
1. For any $x'\in \Delta'$, $\cM_{\Delta,f(x')}^{\rm gp}\rightarrow \cM_{\Delta',x'}^{\rm gp}$ is surjective.
2. For any $x\in \Delta$, $f^{-1}(x)$ is a finite set.
3. The map $$\hom(\uSpec(\N),\Delta')\rightarrow \hom(\uSpec(\N),\Delta)$$ is bijective (resp. injective).
In [@MR1296725 Definition 9.7] a subdivision in our sense is called a proper subdivision.
We note that compositions of subdivisions (resp. proper subdivisions) are again subdivisions (resp. proper subdivisions).
\[Fan.31\] Any subdivision $f:\Delta'\rightarrow \Delta$ of sharpened fans is surjective.
Let $x$ be a point of $\Delta$, which is the generic point of a cone $\uSpec(P)$ of $\Delta$, where $P$ is a sharp fs monoid. By appeal to [@Ogu Proposition I.2.2.1] there exists a homomorphism $p:P\rightarrow \N$ such that $p^{-1}(0)=0$. Let $y$ be the generic point of $\uSpec(\N)$, and set $h:=\uSpec(p)$. Then we have that $h(y)=x$. By condition (iii) in Definition \[Fan.8\] there is a morphism $g:\uSpec(\N)\rightarrow \Delta'$ such that $f\circ g=h$. It follows that $f(g(y))=h(y)=x$, and hence $f$ is surjective.
\[Fan.32\] Let $f:\Delta'\rightarrow \Delta$ and $g:\Delta''\rightarrow \Delta'$ be morphisms of sharpened fans. If $g$ and $f\circ g$ are subdivisions, then $f$ is a subdivision.
Since $g$ is surjective by Lemma \[Fan.31\], the condition (i) (resp. (ii)) in Definition \[Fan.8\] for $f\circ g$ implies the condition (i) (resp. (ii)) for $f$. Condition (iii) for $g$ and $f\circ g$ implies the condition (iii) for $f$.
\[Fan.34\] If $f\colon\Sigma'\rightarrow \Sigma$ is a subdivision (resp. partial subdivision) of fans, then $\overline{f}:\overline{\Sigma'}\rightarrow \overline{\Sigma}$ is a subdivision (resp. partial subdivision) of sharpened fans.
Suppose that $f$ is a $\Sigma$ is in a lattice $N$. Let $x$ be a point of $\Sigma$ corresponding to a cone $\sigma$. If $x'$ is a point of $\Sigma'$ such that $f(x')=x$, then there are isomorphisms $$\cM_{\Sigma',x'}^\gp\cong \cM_{\Sigma,x}^\gp\cong N^\vee.$$ It follows that $\overline{\cM}_{\Sigma',x'}^\gp\rightarrow \overline{\cM}_{\Sigma,x}^\gp$ is surjective, which is condition (i) in Definition \[Fan.8\]. There are only finitely many cones in $\Sigma'$. This shows that condition (ii) holds. Since $\lvert \Delta' \rvert\rightarrow \lvert \Delta \rvert$ is bijective (resp. injective), we deduce condition (iii).
\[Fan.9\] Let $\Sigma$ be a fan. Then for every subdivision (resp. partial subdivision) $f:\Delta'\rightarrow \Delta:=\overline{\Sigma}$ of sharpened fans, there exists a subdivision (resp. partial subdivision) of fans $g:\Sigma'\rightarrow \Sigma$ such that $\overline{g}\cong f$.
Suppose that $f$ is a partial subdivision. We may suppose that $\Sigma$ lies in a lattice $N$. As observed in Example \[Fan.18\], there is a surjective morphism of sheaves $N^\vee\rightarrow \cM_\Delta^\gp$ such that $\Sigma\cong \Delta_N$. Then we have a composite morphism of sheaves $$N^\vee\cong f^*N^\vee\rightarrow f^*\cM_{\Delta}^{\rm gp}\rightarrow \cM_{\Delta'}^{\rm gp},$$ which is surjective because of condition (i). This gives a morphism of fs monoschemes $g:\Delta_N'\rightarrow \Delta_N$ such that $\overline{g}\cong f$.
Let us check that $\Delta_{N}'$ is a fan. Condition (ii) implies that the underlying topological space of $\Delta'$ is a finite set. Moreover, condition (iii) implies that the map in $$\varphi_{\Delta',N}:\hom(\uSpec(\N),\Delta')\rightarrow N$$ is injective. Proposition \[Fan.20\] shows that $\Delta_N'$ is a fan. Since $\Delta$ and $\Delta'$ are in the same lattice $N$, we deduce that $g$ is a partial subdivision.
If $f$ is a subdivision, then condition (iii) implies that the map $$\hom(\Spec{\N},\Delta_N')\rightarrow \hom(\Spec{\N},\Delta_N)$$ is bijective by Lemma \[Fan.19\]. Proposition \[Propersubdivision\] shows that $g$ is a subdivision.
\[Fan.6\] There is a notion of star subdivision for smooth sharpened fans as in the case of toric fans, see Definition \[Starsubdivision\]. Let $\tau=\uSpec(Q)$ be a cone of a smooth sharpened fan $\Delta$ where $Q$ is a sharp fs monoid. For any cone $\sigma=\uSpec(P)$ of $\Delta$ containing $\tau$ where $P$ is a sharp fs monoid, consider the star subdivision $\Theta$ of $\Spec{P}$ relative to $\Spec{Q}$. Replace $\sigma$ by $\overline{\Theta}$ for every $\sigma$, then we obtain a new smooth sharpened fan $\Delta^*(\tau)$. This is called the *star subdivision* of $\Delta$ relative to $\tau$.
\[Fan.27\] Every partial subdivision $f\colon\Delta'\rightarrow \Delta$ of sharpened fans is a monomorphism in the category of sharpened fans.
We show that the diagonal morphism $\Delta'\rightarrow \Delta'\times_\Delta \Delta'$ is an isomorphism. To that end, it suffices to check that for every cone $\sigma$ of $\Delta$ the induced morphism $$\Delta'\times_\Delta \sigma\rightarrow \Delta'\times_\Delta \Delta'\times_\Delta \sigma$$ is an isomorphism. Hence we reduce to the case when $\Delta=\uSpec(P)$ for some sharp fs monoid $P$. In this case, due to Proposition \[Fan.9\], there exists a partial subdivision $g\colon \Sigma'\rightarrow \Sigma:=\Spec{P}$ of fans together with an isomorphism $\overline{g}\cong f$. Lemma \[monomorphismoffans\] implies $g$ is a monomorphism. By appeal to Lemma \[Fan.29\] we deduce that $f$ is a monomorphism.
\[Fan.33\] Let $f\colon\Psi\rightarrow \Delta$ be a subdivision (resp. partial subdivision) of sharpened fans, and $g\colon \Delta'\rightarrow \Delta$ a partial subdivision of sharpened fans. Then the projection $f'\colon \Psi':=\Psi\times_{\Delta}\Delta'\rightarrow \Delta'$ is a subdivision (resp. partial subdivision).
The question is local on $\Delta$, so we may assume that $\Delta$ is affine. Owing to Proposition \[Fan.9\] there exists a subdivision (resp. partial subdivision) of sharpened fans $u\colon\Theta\rightarrow\Sigma$ and a partial subdivision $v\colon \Sigma'\rightarrow \Sigma$ of sharpened fans such that $\overline{u}\cong f$ and $\overline{v}\cong g$. Let $u'\colon \Theta':=\Theta\times_\Sigma \Sigma'\rightarrow \Sigma'$ be the projection. Lemma \[Fiberproduct\_fans\] shows that $u'$ is a subdivision (resp. partial subdivision). By Lemma \[Fan.29\] we have $\overline{u'}\cong f'$. Due to Lemma \[Fan.34\] we can conclude that $f'$ is a subdivision (resp. partial subdivision).
\[Fan.26\] Let $f\colon Y\rightarrow X$ and $g\colon Z\rightarrow Y$ be monomorphisms in a category $\cC$ with fiber products. Then the projection $$f'\colon Y\times_X Z\rightarrow Z$$ is an isomorphism in $\cC$.
There is a naturally induced commutative diagram with cartesian squares $$\begin{tikzcd}
Z\times_X Z\arrow[d]\arrow[r,"g'"]&
Y\times_X Z\arrow[d]\arrow[r,"f'"]&
Z\arrow[d,"fg"]
\\
Z\arrow[r,"g"]&
Y\arrow[r,"f"]&
X.
\end{tikzcd}$$ Since the composite $fg$ is a monomorphism, it follows that $f'g'$ is an isomorphism. This shows that $f'$ is an epimorphism. Similarly, again since the pullback of a monomorphism is a monomorphism, it follows that $f'$ is a monomorphism. Thus $f'$ is an isomorphism.
\[Fan.17\] Let $p\colon\Delta'\rightarrow \Delta$ be a subdivision of smooth sharpened fans. If the underlying topological space of $\Delta$ is a finite set, then there exists a sharpened fan $\Delta''$ obtained by a finite succession of star subdivisions from $\Delta$ such that the morphism $\Delta''\rightarrow \Delta$ factors through $p$.
Let $\{\sigma_1=\uSpec(P_1),\ldots,\sigma_n=\uSpec(P_n)\}$ be all the cones of $\Delta$, where $P_1,\ldots,P_n$ are sharp fs monoids. We will inductively construct subdivisions $$\Delta_n\rightarrow \cdots \rightarrow \Delta_0:=\Delta$$ such that for every $1\leq i\leq n$ the projection $$\label{Fan.17.1}
\Delta'\times_\Delta \Delta_i\times_\Delta \sigma_i\rightarrow \Delta_i\times_\Delta \sigma_i$$ is an isomorphism.
Suppose that we have constructed $\Delta_{i-1}$. By Lemma \[Fan.33\] the projection $$\Delta'\times_\Delta \Delta_{i-1}\times_\Delta \sigma_i\rightarrow \Delta_{i-1}\times_\Delta \sigma_i$$ is a subdivision of sharpened fans. Proposition \[Fan.20\] shows there exist subdivisions $g:\Sigma\rightarrow \Spec{P_i}$ and $h:\Sigma'\rightarrow \Sigma$ of fans such that $\overline{g}$ and $\overline{h}$ are isomorphic to the projections $$\Delta_{i-1}\times_{\Delta}\sigma_i\rightarrow \sigma_i
\text{ and }
\Delta'\times_\Delta \Delta_{i-1}\times_{\Delta}\sigma_i\rightarrow \Delta_{i-1}\times_\Delta \sigma_i.$$ Due to Lemma \[A.3.31\] there is a subdivision $\Sigma''$ of $\Sigma$ obtained by a finite succession of star subdivisions from $\Sigma$ such that $\Sigma''\rightarrow \Sigma$ factors through $\Sigma'$. Since the underlying topological space of $\Sigma$ is an open subset of the underlying topological space of $\Delta_{i-1}$, we can take the corresponding finite succession of star subdivisions from $\Delta_{i-1}$ to obtain a subdivision $\Delta_i$ of $\Delta_{i-1}$. Then the morphism $$\Delta_i\times_{\Delta}\sigma_i\rightarrow \Delta_{i-1}\times_{\Delta}\sigma_i$$ is a subdivision and factors through $\Delta'\times_\Delta\Delta_{i-1}\times_\Delta \sigma_i$. The projection $$\Delta'\times_\Delta \Delta_{i-1}\times_\Delta \sigma_i\rightarrow \Delta_{i-1}\times_\Delta \sigma_i$$ is a subdivision by Lemma \[Fan.33\]. Now use Lemmas \[Fan.27\] and \[Fan.26\] to deduce that the morphism is an isomorphism. This completes the construction of $\Delta_i$ and hence, by induction, $\Delta_n$.
From we see that for every $1\leq i\leq n$ the projection $$\Delta_n\times_\Delta \Delta'\times_\Delta \sigma_i\rightarrow \Delta_n\times_\Delta \sigma_i$$ is an isomorphism. It follows that the projection $\Delta_n\times_\Delta \Delta'\rightarrow \Delta_n$ is an isomorphism since $\{\sigma_1,\ldots,\sigma_n\}$ comprises all the cones of $\Delta$. Hence the morphism $\Delta_n\rightarrow \Delta$ factors through $\Delta'$.
Frames of fs log schemes
------------------------
Let $X$ be an fs log scheme. If there exists a fan $\Sigma$ with a strict morphism $$X\rightarrow \A_\Sigma,$$ then any partial subdivision $\Sigma'$ of $\Sigma$ produces a new fs log scheme $X\times_{\A_\Sigma}\A_{\Sigma'}$. Alas, it is unclear whether such a fan $\Sigma$ always exists in the category of fs log schemes. For fs log schemes in $lSm/k$ we shall proceed by means of sharpened fans. As an application we will show that every log modification $Y\rightarrow X$ in $SmlSm/k$ admits a factorization into log modifications $$Y_n\rightarrow \cdots \rightarrow Y_1\rightarrow Y_0:=X$$ such that for every $1\leq i\leq n$ the morphism $$\underline{Y_i}\rightarrow \ul{Y_{i-1}}$$ is a blow-up along along a smooth center.
\[Fan.2\] Let $X$ be an fs log scheme. A *preframe* of $X$ is a sharpened fan $\Delta$ with a morphism of monoidal spaces $s\colon(X,\overline{\cM}_X)\rightarrow \Delta$.
A preframe $\Delta$ of $X$ is called a *frame* of $X$ if the following condition is satisfied: Locally on $X$ and $\Delta$, there exists a chart $t\colon (X,\cM_X)\rightarrow \Spec{P}$ with an fs monoid $P$ such that $\ol{t}$ is isomorphic to $s\colon(X,\overline{\cM}_X)\rightarrow \Delta$. Such a frame is an $s$-frame in the sense of [@Ogu Definition III.1.12.1].
\[Fan.3\] For every $X\in lSm/k$ we can associate a frame $$s_X:(X,\overline{\cM}_X)\rightarrow \Delta_X$$ satisfying the following properties.
1. Every preframe of $X$ uniquely factors through $s_X$.
2. The morphism of monoidal spaces $s_X$ is open and surjective.
3. For every morphism $f:X\rightarrow Y$ in $lSm/k$, there exists a unique morphism of sharpened fans $\Delta_f:\Delta_X\rightarrow \Delta_Y$ such that the diagram $$\begin{tikzcd}
(X,\overline{\cM}_X)\arrow[r,"f"]\arrow[d,"s_X"']&
(Y,\overline{\cM}_Y)\arrow[d,"s_Y"]
\\
\Delta_X\arrow[r,"\Delta_f"]&
\Delta_Y
\end{tikzcd}$$ commutes.
Owing to [@Ogu Theorems III.1.11.1, IV.3.5.1], $X$ is very solid in the sense of [@Ogu Definition III.1.10.1]. Then apply [@Ogu Proposition III.1.12.3] and its following paragraph.
\[Fan.15\] Let $X$ be a quasi-compact fs log scheme in $lSm/k$. Then the underlying space of the associated sharpened fan $\Delta_X$ is a finite set.
There exists a finite Zariski cover $\{U_i\}$ of $X$ with the property that there exists a chart $t_i\colon(U_i,\cM_{U_i})\rightarrow \Spec{P_i}$ such that the composite morphism $$(U_i,\ul{\cM}_{U_i})\rightarrow (X,\ul{\cM}_X)\stackrel{s_X}\rightarrow \Delta_X$$ factors through $\ol{t_i}$. Since the underlying topological space of $\Spec{P_i}$ is a finite set, we can use property (ii) of Proposition \[Fan.3\] to conclude.
\[Fan.5\] Suppose $X$ is an fs log scheme with a frame $$\pi:(X,\overline{\cM}_X)\rightarrow \Delta,$$ and let $p:\Delta'\rightarrow \Delta$ be a partial subdivision of sharpened fans. Let $\cC_{X,\Delta',\Delta}$ be the category of pairs $(f,\pi')$, where $f\colon X'\rightarrow X$ is a morphism of fs log schemes and $$\pi':(X',\overline{\cM}_{X'})\rightarrow \Delta'$$ is a morphism of monoidal spaces such that the diagram $$\begin{tikzcd}
(X',\overline{\cM}_{X'})\arrow[d]\arrow[r,"\pi'"]&
\Delta'\arrow[d,"p"]
\\
(X,\overline{\cM}_{X})\arrow[r,"\pi"]&
\Delta
\end{tikzcd}$$ commutes. In [@MR1296725 Proposition 9.9], it is shown that $\cC_{X,\Delta',\Delta}$ admits a final object $$X\times_{\Delta}\Delta'.$$ Here, the fiber product notation is used for convenience; technically, it is not a fiber product. Moreover, the morphism $\pi'$ gives a framing of $X'$.
Let us explain the local description of $X'$ in [@MR1296725 Proposition 9.9]. Suppose that $P$ is an fs chart of $X$ with a homomorphism $\theta:P\rightarrow Q$ of fs monoids such that $\theta^{\rm gp}$ is an isomorphism, $\Delta=\uSpec(\overline{P})$, and $\Delta'=\uSpec(\overline{Q})$. In this case we have $$X\times_{\Delta}\Delta'\cong X\times_{\A_P}\A_{Q}.$$ More generally, suppose that $\Delta=\uSpec(\overline{P})$ and $\Delta'=\overline{\Sigma}$, where $P$ is an fs chart of $X$ and $\Sigma$ is a partial subdivision of $\Spec{P}$. Then the local description gives an isomorphism $$\label{Fan.5.1}
X\times_{\Delta}\Delta'\cong X\times_{\A_P}\A_\Sigma.$$
\[Fan.36\] Let $X$ be an fs log scheme with a frame $\Delta$, and let $\Delta'\rightarrow \Delta$ and $\Delta''\rightarrow \Delta'$ be partial subdivisions of sharpened fans. Then there is a canonical isomorphism $$(X\times_\Delta \Delta')\times_{\Delta'}\Delta''
\cong
X\times_\Delta \Delta''.$$
We need to show that $(X\times_\Delta \Delta')\times_{\Delta'}\Delta''$ is a final object in the category $\cC_{X,\Delta'',\Delta}$. Suppose that $f:X'\rightarrow X$ is a morphism of fs log schemes with a morphism of monoidal spaces $(X',\overline{\cM}_{X'})\rightarrow \Delta''$ such that the diagram $$\begin{tikzcd}
(X',\overline{\cM}_{X'})\arrow[d]\arrow[r]&
\Delta''\arrow[d]
\\
(X,\overline{\cM}_{X})\arrow[r]&
\Delta
\end{tikzcd}$$ commutes. The universal property of $X\times_\Delta \Delta'$ means that there is a unique morphism $X'\rightarrow X\times_\Delta \Delta'$ such that the diagram $$\begin{tikzcd}
(X',\overline{\cM}_{X'})\arrow[d]\arrow[r]&
\Delta''\arrow[d]
\\
(X\times_\Delta \Delta',\overline{\cM}_{X\times_\Delta \Delta'})\arrow[r]&
\Delta'
\end{tikzcd}$$ commutes. Similarly, the universal property of $(X\times_\Delta \Delta')\times_{\Delta'}\Delta''$ means that there is a unique morphism $$X'\rightarrow (X\times_\Delta \Delta')\times_{\Delta'}\Delta''$$ in $\cC_{X,\Delta'',\Delta}$. This shows that $(X\times_\Delta \Delta')\times_{\Delta'}\Delta''$ is a final object in $\cC_{X,\Delta'',\Delta}$.
\[Fan.11\] Let $X$ be an fs log scheme in $lSm/k$. Then for every subdivision of fans $p:\Delta'\rightarrow \Delta_X$, the natural morphism $$f:X\times_{\Delta_X}\Delta'\rightarrow X$$ is a log modification.
The question is Zariski local on $X$, so we may assume that $X$ admits an fs chart $t\colon X\rightarrow \Spec{P}$ such that $\ol{t}$ is isomorphic to $s_X\colon (X,\ol{\cM}_X)\rightarrow \Delta_X$. Owing to Proposition \[Fan.9\] there exists a subdivision of fans $q:\Sigma'\rightarrow \Spec{P}$ such that $\ol{q}\cong p$. From the construction of $X\times_{\Delta_X}\Delta'$ we see that $$X\times_{\Delta_X}\Delta'\cong X\times_{\A_{\Spec{P}}}\A_{\Sigma'}= X\times_{\A_{P}}\A_{\Sigma'}.$$ This concludes the proof since the projection $X\times_{\A_{P}}\A_{\Sigma'}\rightarrow X$ is a log modification due to Example \[A.9.76\].
\[Fan.35\] Suppose $X$ is an fs log scheme in $lSm/k$ with an fs chart $t\colon X\rightarrow \Spec{P}$ where $P$ is an fs monoid. For every subdivision $\Sigma\rightarrow \Spec{P}$ of fans, the associated morphism of sharpened fans $$\Delta_{X\times_{\A_P} \A_\Sigma}\rightarrow \Delta_X$$ is a subdivision.
The question is Zariski local on $X$, so we may assume that $\overline{t}$ is isomorphic to $s_X\colon (X,\overline{\cM}_X)\rightarrow \Delta_X$. Due to there is an isomorphism $$X\times_{\A_P}\A_{\Sigma}\cong X\times_{\Delta_X}\overline{\Sigma}.$$ It follows that $\Delta_{X\times_{\A_P}\A_\Sigma}\cong \overline{\Sigma}$. Since $\Sigma\rightarrow \Spec{P}$ is a subdivision, $\overline{\Sigma}\rightarrow \Delta_X$ is a subdivision by Lemma \[Fan.34\].
\[Fan.30\] For any log modification $f\colon Y\rightarrow X$ in $lSm/k$, the associated morphism $u\colon\Delta_Y\rightarrow \Delta_X$ of sharpened fans is a subdivision.
The question is Zariski local on $X$, so we may assume that $X$ is quasi-compact and admits an fs chart $t\colon X\rightarrow \Spec{P}$ such that $\ol{t}$ is isomorphic to $s_X\colon (X,\ol{\cM}_X)\rightarrow \Delta_X$. Owing to Proposition \[A.9.21\] there is a subdivision of fans $p:\Sigma\rightarrow \Spec{P}$ such that the pullback $$Y\times_{\A_P}\A_\Sigma\rightarrow X\times_{\A_P}\A_\Sigma =:X'$$ is an isomorphism. Therefore we have associated morphisms $\Delta_{X'}\stackrel{v}\rightarrow \Delta_{Y}\stackrel{u}\rightarrow \Delta_X$ of sharpened fans.
We claim that $v$ and $u\circ v$ are subdivisions. Owing to Lemma \[A.9.19\] there is a Zariski cover $\{U_i\}$ of $Y$ such that for every $i$ the morphism $U_i\rightarrow X$ admits an fs chart $\theta\colon P\rightarrow Q_i$ such that $\theta^\gp\colon P^\gp\rightarrow Q_i^\gp$ is an isomorphism. This means that $\Spec{Q_i}$ is a partial subdivision of $\Spec{P}$. Lemma \[monomorphismoffans\] shows that the projection $$\Sigma_i:=\Spec{Q_i}\times_{\Spec{P}}\Sigma\rightarrow \Spec{Q_i}$$ is a subdivision of fans. Thus by Lemma \[Fan.35\] the associated morphism of sharpened fans $$\Delta_{U_i\times_{\A_P}\A_{\Sigma}}
\cong
\Delta_{U_i\times_{\A_{Q_i}}\A_{\Sigma_i}}
\rightarrow
\Delta_{U_i}$$ is a subdivision. It follows that $v$ is a subdivision. Moreover, Lemma \[Fan.35\] also shows that $u\circ v$ is a subdivision. As a consequence, $u$ is a subdivision by Lemma \[Fan.32\].
\[Fan.38\] Let $X$ be an fs log scheme in $SmlSm/k$. Then the associated sharpened fan $\Delta_X$ is smooth.
The question is Zariski local on $X$, so we may assume that $X$ admits an fs chart $t\colon X\rightarrow \Spec{P}$ such that $\ol{t}$ is isomorphic to $s_X\colon (X,\ol{\cM}_X)\rightarrow \Delta_X$. Lemma \[lem::Smchart\] shows that $P\cong \N^r$ for some integer $r\geq 0$. Thus $\Delta_X\cong \uSpec(\N^r)$ is smooth.
\[Fan.39\] For $X\in SmlSm/k$, a *log modification along a smooth center* is a log modification $p:Y\rightarrow X$ such that Zariski locally on $X$ there exists a chart $X\rightarrow \A_{\N^r}$ such that $p$ is isomorphic to the projection $$X\times_{\A_{\N^r}}\A_{\Sigma}
\to
X.$$ Here $\Sigma$ is the star subdivision of the dual of $\N^r$.
\[Fan.37\] Let $X$ be an fs log scheme in $SmlSm/k$. Then for every star subdivision of fans $p:\Delta'\rightarrow \Delta_X$, the natural morphism $$f\colon X\times_{\Delta_X}\Delta'\rightarrow X$$ is a log modification along a smooth center.
The question is Zariski local on $X$, so we may assume that $X$ admits an fs chart $t\colon X\rightarrow \Spec{P}$ such that $\ol{t}$ is isomorphic to $s_X\colon (X,\ol{\cM}_X)\rightarrow \Delta_X$. Due to Lemma \[Fan.38\], $P\cong \N^r$ for some integer $r\geq 0$. By the definition of star subdivisions of sharpened fans, there exists a face $Q$ of $P$ such that $\Delta'\cong \overline{\Sigma}$, where $\Sigma$ is the star subdivision of $\Spec{P}$ relative to $\Spec{Q}$. Then gives an isomorphism $$X\times_{\Delta_X}\Delta'\cong X\times_{\A_P}\A_{\Sigma}.$$ This shows that $f$ is a log modification along a smooth center.
\[Fan.16\] Let $f:Y\rightarrow X$ be a log modification of quasi-compact fs log schemes in $SmlSm/k$. Then there exists a sequence of log modifications along a smooth center $$X_n\rightarrow \cdots \rightarrow X_1\rightarrow X$$ such that the projection $X_n\times_X Y\rightarrow X_n$ is an isomorphism.
Lemma \[Fan.30\] shows that the associated morphism $\Delta_Y\rightarrow \Delta_X$ is a subdivision of sharpened fans. By Lemma \[Fan.37\] the associated sharpened fan $\Sigma_X$ is smooth. Lemma \[Fan.17\] yields star subdivisions of smooth sharpened fans $$\Delta_n\rightarrow \cdots \rightarrow \Delta_1\rightarrow \Delta_X$$ such that the subdivision $\Delta_n\rightarrow \Delta_X$ factors through $\Delta_Y\rightarrow \Delta_X$. We set $$X_i:=X\times_{\Delta_X}\Delta_i$$ for every $1\leq i\leq n$. By definition $X_n\rightarrow X$ factors through $f$, and by Lemma \[Fan.36\] there is an isomorphism $$X_n\cong X_{n-1}\times_{\Delta_{n-1}}\Delta_n.$$ Lemma \[Fan.37\] shows that the morphism $X_i\rightarrow X_{i-1}$ is a log modification along a smooth center for every $1\leq i\leq n$. To show that the projection $X_n\times_X Y\rightarrow X_n$ is an isomorphism, use Lemma \[Fan.26\].
Model structures on the category of complexes {#AppendixB}
=============================================
Descent structures
------------------
In this appendix we recall the construction of model structures on chain complexes. We follow the approach in [@CD09] using descent structures.
Let $\cT$ be a triangulated category, and let $\cA$ be an abelian category.
1. An object $X$ of $\cT$ is called [*compact*]{} if the functor $$\hom_{\cT}(X,-)$$ commutes with small sums.
2. Let $\cF$ be a family of objects of $\cA$. We say that $\cF$ [*generates*]{} $\cA$ if for any object $F$ of $\cA$, there is an epimorphism $\oplus X\rightarrow F$ from a small sum of objects of $\cF$.
3. Let $\cF$ be a family of objects of $\cT$. We say that $\cF$ [*generates*]{} $\cT$ if the family of functors $$\hom_{\cT}(X[n],-)$$ for $X\in \cF$ and $n\in \Z$ is conservative.
4. We say that $\cT$ is [*compactly generated*]{} if there is a family that generates $\cT$ and consists of compact objects.
5. A [*localizing subcategory*]{} of $\cT$ is a triangulated subcategory of $\cT$ that is stable by small sums. For a family $\cF$ of objects of $\cA$, we write $\langle \cF\rangle$ for the smallest localizing subcategory of $\cT$ containing $\cF$.
\[A.8.14\] Let $\cA$ be a Grothendieck abelian category. For $\cG$ an essentially small set of objects of $\cA$, and $\cH$ an essentially small set of complexes of $\cA$ we make the following definitions.
1. The class of $\cG$-[*cofibrations*]{} (or simply [*cofibrations*]{}) is the smallest class of morphisms in $\Co(\cA)$ that contains the morphisms $$\begin{tikzcd}
{[}0\arrow[r]\arrow[d]&F{]}\arrow[d,"{\rm id}"]\\
{[}F\arrow[r,"{\rm id}"]&F{]},
\end{tikzcd}$$ where $F\in \cG$ and $0$ sits in degree $-1$, and is closed under shifts, pushouts, retracts, and transfinite compositions.
2. A complex $K$ of $\cA$ is called $\cG$-[*cofibrant*]{} (or simply [*cofibrant*]{}) if $0\rightarrow K$ is a $\cG$-cofibration.
3. A complex $K$ of $\cA$ is called $\cG$-[*local*]{} (or simply [*local*]{}) if for any $\cG$-cofibrant complex $L$ and $n\in \Z$ the induced homomorphism $$\hom_{{\bf K}(\cA)}(L[n],K)\rightarrow \hom_{\Deri(\cA)}(L[n],K)$$ is an isomorphism of abelian groups.
4. A complex $K$ of $\cA$ is called $\cH$-[*flasque*]{} (or simply [*flasque*]{}) if for any $n\in \Z$ and $L\in \cH$ we have $$\hom_{{\bf K}(\cA)}(L[n],K)=0.$$
We say that $(\cG,\cH)$ is a [*descent structure*]{} for $\cA$ if $\cG$ generates $\cA$ and any $\cH$-flasque complex is $\cG$-local.
Assume that $\cA$ has a closed symmetric monoidal structure. Then $(\cG,\cH)$ is called [*weakly flat*]{} if $\cG$ is closed under tensor products up to isomorphism, the unit object of $\cA$ is in $\cG$, and $K\otimes L$ is acyclic for any $K\in \cG$ and $L\in \cH$.
Next we recall the [*injective model structure*]{} on chain complexes in Grothendieck abelian categories.
\[A.8.26\] Let $\cA$ be a Grothendieck abelian category. A monomorphism in $\Co(\cA)$ is called an [*injective cofibration*]{}. By [@CD09 Theorem 2.1], $\Co(\cA)$ is a proper cellular model category with quasi-isomorphisms as weak equivalences and injective cofibrations as cofibrations. A fibration with respect to the injective model structure is called an [*injective fibration*]{}.
We shall refer to the following model structure as the *descent model structure*.
\[A.8.15\] Let $\cA$ be a Grothendieck abelian category with a descent structure $(\cG,\cH)$. Then the following properties hold.
1. $\Co(\cA)$ is a proper cellular model category with quasi-isomorphisms as weak equivalences and $\cG$-cofibrations as cofibrations.
2. A complex $K$ is $\cG$-local if and only if it is $\cH$-flasque.
3. A morphism of complexes is a fibration if and only if it is a degreewise surjection with a $\cG$-local kernel.
4. Assume that $\cA$ has a closed symmetric monoidal structure. If $(\cG,\cH)$ is weakly flat, then $\Co(\cA)$ is a symmetric monoidal model category.
By [@CD09 Theorem 2.5] (resp. [@CD09 Corollary 5.5], resp. [@CD09 Proposition 3.2]), we have (1) and (2) (resp. (3), resp. (4)).
\[A.8.32\] Let $\cA$ be a Grothendieck abelian category, and let $\cG$ be an essentially small set of objects of $\cA$ generating $\cA$. Then the following properties hold.
1. $\cG$ generates $\Deri(\cA)$.
2. $\Deri(\cA)=\langle \cG\rangle$.
\(1) Let $f:F\rightarrow G$ be a morphism in $\Deri(\cA)$. Assume that the induced map $$\hom_{\Deri(\cA)}(X[n],F)\rightarrow \hom_{\Deri(\cA)}(X[n],G)$$ is an isomorphism for any $X\in \cG$ and $n\in Z$. For a cone $H$ of $f$ we have $$\hom_{\Deri(\cA)}(X[n],H)=0.$$ Choose an injective fibrant resolution $H\rightarrow I$. If $I$ is not quasi-isomorpic to $0$, then $H^n(I)\neq 0$ for some $n\in \Z$. Choose an epimorphism $J\rightarrow \ker d^n$ from a small sum $J$ of objects of $\cG$, where $d^i$ is the differential map $I^i\rightarrow I^{i+1}$. Then the $J\rightarrow \ker d^n$ does not factor through $\im d^{n-1}$, and we reach the contradiction $$\hom_{{\bf D}(\cA)}(J[-n],H)\cong \hom_{{\bf K}(\cA)}(J[-n],I) \neq 0.$$ Thus $I$ is quasi-isomorphic to $0$, and we conclude that $f$ is an isomorphism.
\(2) Since $\cG$ generates $\cA$, for any object $F$ of $\cA$, there is a quasi-isomorphism $P\rightarrow F$ where $P$ is a bounded above complex of objects that are small sums of objects of $\cG$. Thus we have an inclusion $\cA\subset \langle \cG\rangle$, which readily implies $\Deri(\cA)=\langle \cG\rangle$.
\[A.8.18\] Let $\cA$ be a Grothendieck abelian category with a descent structure $(\cG,\cH)$, and let $\cW$ be an essentially small set of $\cG$-cofibrant complexes of $\cA$. Recall the following definitions from [@CD09 §4].
1. A complex $G$ is called [*$\cW$-local*]{} if for any $n\in \Z$ and $F\in \cW$ we have $$\hom_{\Deri(\cA)}(F[n],G)=0.$$
2. A morphism $F\rightarrow F'$ of complexes is called a [*$\cW$-equivalence*]{} if for any $\cW$-local complex $G$ the induced homomorphism $$\hom_{\Deri(\cA)}(F'[n],G)\rightarrow \hom_{\Deri(\cA)}(F[n],G)$$ is an isomorphism of abelian groups.
3. A morphism of complexes is called a [*$\cW$-fibration*]{} if it is a fibration with respect to the descent model structure with a $\cW$-local kernel.
\[A.8.28\] Let $\cC$ be a category with a topology $t$ and sheafification functor $a_t^*$. The category of sheaves $\Shvckl$ of $\Lambda$-modules has a descent structure $(\cG,\cH)$ given as follows. Let $\Lambda(X)$ denote the free presheaf of $\Lambda$-modules generated by $X\in \cC$. For a hypercover $\mathscr{X}\rightarrow X$ we let $\Lambda(\mathscr{X})$ denote the associated complex of sheaves. Now let $\cG$ be the family of objects $a_t^*\Lambda(X)$ for all $X\in \cC$, and let $\cH$ be the family of cones of the morphisms $a_t^*\Lambda(\mathscr{X})\rightarrow a_t^*\Lambda(X)$ for all hypercovers $\mathscr{X}\rightarrow X$. With these definitions $(\cG,\cH)$ gives a descent structure of $\Shvckl$.
Note that a complex $\cF\in \Co(\Shvckl)$ is $\cH$-flasque if and only if for every hypercover $\mathscr{X}\rightarrow X$ we have that $$\label{A.8.28.1}
\cF(X)\simeq {\rm Tot}^\pi\cF(\mathscr{X}).$$ This is equivalent to saying that for every hypercover $\mathscr{X}\rightarrow X$ and integer $i\in \Z$ we have that $$\label{A.8.28.2}
\hom_{\mathbf{K}(\Shvckl)}(a_t^*\Lambda(X),\cF[i])
\cong
\hom_{\mathbf{K}(\Shvckl)}(a_t^*\Lambda(\mathscr{X}),\cF[i]).$$
\[A.8.19\] Let $\cA$ be a Grothendieck abelian category with a descent structure $(\cG,\cH)$, and let $\cW$ be an essentially small set of $\cG$-cofibrant complexes of $\cA$.
1. The category $\Co(\cA)$ is a proper cellular model category with $\cW$-equivalences as weak equivalences, $\cG$-cofibrations as cofibrations, and $\cW$-fibrations as fibrations. This model structure is the left Bousfield localization of the descent model structure by the maps $0\rightarrow T[n]$ for $T\in \cW$ and $n\in \Z$. Its homotopy category $\Deri_\cW(\cA)$ is equivalent to the Verdier quotient $\Deri(\cA)/\langle\cW\rangle$, where $\langle\cW\rangle$ is the smallest localizing subcategory of $\Deri(\cA)$ containing $\cW$.
2. Assume that $\cA$ has a closed symmetric monoidal structure and that $(\cG,\cH)$ is weakly flat. If $T\otimes G\in \cW$ for any $T\in \cW$ and $G\in \cG$, then $\Co(\cA)$ is a symmetric monoidal model category and the localization functor $\Deri(\cA)\rightarrow \Deri_\cW(\cA)$ is a symmetric monoidal functor. This model structure is called the [$\cW$-local descent model structure]{}.
According to [@CD09 Propositions 4.3, 4.7] (resp. [@CD09 Proposition 4.11]), we have (1) (resp. (2)).
\[A.8.16\] Let $\cA$ (resp. $\cA'$) be a Grothendieck abelian category with a descent structure $(\cG,\cH)$ (resp. $(\cG',\cH')$), and let $$f^*:\cA\rightleftarrows \cA':f_*$$ be a pair of adjoint additive functors. Assume that
1. for any $K$ in $\cG$, $f^*K$ is a direct sum of elements of $\cG'$,
2. for any $K$ in $\cH$, $f^*K$ is in $\cH'$.
Then the induced pair of adjunctions $$f^*:\Co(\cA)\rightleftarrows \Co(\cA'):f_*$$ is a Quillen pair with respect to the descent model structures.
This is the content of [@CD09 Proposition 2.14].
\[A.8.17\] Let $\cA$ be a Grothendieck abelian category with a descent structure $(\cG,\cH)$, let $\cA'$ be another Grothendieck abelian category, and let $$f^*:\cA\rightleftarrows \cA':f_*$$ be a pair of adjoint additive functors. Assume that
1. $f_*$ is exact,
2. for any complex $H$ in $\cH$, $f^*H$ is quasi-isomorphic to $0$.
Then $(f^*\cG,f^*\cH)$ is a descent structure for $\cA'$, and the induced pair of adjunctions $$f^*:\Co(\cA)\rightleftarrows \Co(\cA'):f_*$$ is a Quillen pair with respect to the descent model structures.
Since $f^*$ and $f_*$ are additive functors there is an induced adjoint functor pair $$f^*:{\bf K}(\cA)\rightleftarrows {\bf K}(\cA'):f_*$$
Let us consider the injective model structures on $\cA$ and $\cA'$. For a $f^*\cH$-flasque complex $L$ we choose an injective fibrant replacement $p:L\rightarrow L'$ in $\Co(\cA')$. For any $\cG$-cofibrant object $K$ in $\cA$, we have the induced commutative diagram of abelian groups: $$\begin{tikzcd}
\hom_{\Deri(\cA')}(f^*K,L)\arrow[d,leftarrow,"u"']\arrow[r,"w"]&\hom_{\Deri(\cA')}(f^*K,L')\arrow[d,leftarrow,"u'"]\\
\hom_{{\bf K}(\cA')}(f^*K,L)\arrow[d,"\sim"']\arrow[r]&\hom_{{\bf K}(\cA')}(f^*K,L')\arrow[d,"\sim"]\\
\hom_{{\bf K}(\cA)}(K,f_*L)\arrow[d,"v"']\arrow[r]&\hom_{{\bf K}(\cA)}(K,f_*L')\arrow[d,"v'"]\\
\hom_{\Deri(\cA)}(K,f_*L)\arrow[r,"w'"]&\hom_{\Deri(\cA)}(K,f_*L')\
\end{tikzcd}$$
In the following we show that $u$ is an isomorphism. This implies $L$ is a $f^*\cG$-local complex.
Since $p$ is a quasi-isomorphism, $w$ is an isomorphism. Moreover, $f_*p$ is also an isomorphism since $f_*$ is exact. It follows that $w'$ is an isomorphism. Since $L'$ is a injectively fibrant object, we deduce $u'$ is an isomorphism. For any $H\in \cH$, $f^*H$ is quasi-isomorphic to $0$, so $\hom_{\Deri(\cA')}(f^*H,L')=0$. Using that $u'$ is an isomorphism we obtain $$\hom_{{\bf K}(\cA')}(f^*H,L')=0.$$ Thus we have $$\hom_{{\bf K}(\cA')}(H,f_*L')=0.$$ That is, $f_*L'$ is $\cH$-flasque and hence $f_*L'$ is $\cG$-local since $(\cG,\cH)$ is a descent structure for $\cA$. This implies $v'$ is an isomorphism. Since $L$ is a $f^*\cH$-flasque complex we have $$\hom_{{\bf K}(\cA')}(f^*H,L)=0.$$ That is, $\hom_{{\bf K}(\cA)}(H,f_*L)=0$ and hence $f_*L$ is a $\cH$-flasque complex. This implies $v$ is an isomorphism.
To summarize, we have shown that $u'$, $v$, $v'$, $w$, and $w'$ are isomorphisms. This implies $u$ is an isomorphism and that $(f^*\cG,f^*\cH)$ is a descent structure for $\cA'$. The remaining assertion follows from Proposition \[A.8.16\].
\[A.8.20\] Let $\cA$ (resp. $\cA'$) be a Grothendieck abelian category with a descent structure $(\cG,\cH)$ (resp. $(\cG',\cH')$), let $\cW$ (resp. $\cW'$) be an essentially small set of $\cG$-cofibrant (resp. $\cG'$-cofibrant) complexes of $\cA$ (resp. $\cA'$), and let $$f^*:\cA\rightleftarrows \cA':f_*$$ be a pair of adjoint additive functors. Assume that
1. for any $K$ in $\cG$, $f^*K$ is a direct sum of elements of $\cG'$,
2. for any $K$ in $\cH$, $f^*K$ is in $\cH'$,
3. for any $K$ in $\cW$, $f^*$ is in $\cW'$.
Then the induced pair of adjunctions $$f^*:\Co(\cA)\rightleftarrows \Co(\cA'):f_*$$ is a Quillen pair with respect to the $\cW$-local descent model structures.
This is the content of [@CD09 Theorem 4.9].
\[A.8.21\] Let $\cA$ be a Grothendieck abelian category with a descent structure $(\cG,\cH)$ and let $\cA'$ be a Grothendieck abelian category. Suppose $\cW$ is an essentially small set of $\cG$-cofibrant complexes of $\cA$, and let $$f^*:\cA\rightleftarrows \cA':f_*$$ be a pair of adjoint additive functors. Assume that
1. $f_*$ is exact,
2. for any complex $H$ in $\cH$, $f^*H$ is quasi-isomorphic to $0$.
Then the induced pair of adjunctions $$f^*:\Co(\cA)\rightleftarrows \Co(\cA'):f_*$$ is a Quillen pair with respect to the $\cW$-local descent model structure on $\Co(\cA)$ and the $f^*\cW$-local model structure on $\cO(\cA')$.
This follows from Propositions \[A.8.17\] and \[A.8.20\].
A descent structure $(\cG,\cH)$ on a Grothendieck abelian category is called [*bounded*]{} if every object of $\cH$ is a bounded complexes of objects that are finite sums of objects of $\cG$.
\[A.8.29\] Let $\cA$ be a Grothendieck abelian category with a bounded descent structure $(\cG,\cH)$. If $\hom_{\cA}(X,-)$ commutes with small sums for any $X\in \cG$, then $X[n]$ is a compact object in $\Deri(\cA)$ for any $n\in \Z$ and $X\in \cG$.
See [@CD09 Theorem 6.2].
\[A.8.31\] Let $\cA$ be a Grothendieck abelian category with a bounded descent structure $(\cG,\cH)$, and let $\cW$ be an essentially small set of bounded complexes of objects that are finite sums of objects of $\cG$. If $\hom_{\cA}(X,-)$ commutes with small sums for any $X\in \cG$, then $\cG$ generates $\Deri_\cW(\cA)$ and $X[n]$ is a compact object in $\Deri_\cW(\cA)$ for any $n\in \Z$ and $X\in \cG$.
For any family $\cF$ of objects of $\Deri(\cA)$, let $\langle \cF \rangle$ denote the smallest localizing subcategory of $\Deri(\cA)$ containing $\cF$. Note that $\langle \cG\rangle = \Deri(\cA)$ by Proposition \[A.8.32\](2). Owing to Proposition \[A.8.19\](1), we have $\Deri_\cW(\cA)\cong \langle \cG\rangle / \langle \cW\rangle$. Proposition \[A.8.29\] shows $X[n]$ is a compact object in $\Deri_\cW(\cA)$ for any $n\in \cZ$ and $X\in \cG$. Thus any bounded complex of objects that are finite sums of objects of $\cG$ is compact in $\Deri(\cA)$. In particular, any object of $\cW$ is compact in $\Deri(\cA)$. To conclude we apply Thomason’s localization theorem [@Neeman Theorem 4.4.9] — in loc. cit., $\cT^{\aleph_0}$ denotes the class of compact objects in a triangulated category $\cT$.
Categorical toolbox {#appendix:cat_tool}
===================
In this appendix we collect some useful yoga of triangulated categories.
Category of squares
-------------------
We fix a category $\cA$ and a symmetric monoidal triangulated category $\cT$.
Non-canonicity of the cone construction is well-known in triangulated categories. To impose canonicality for cones of morphisms in $\cA$, we first introduce the notion of $n$-squares which helps us dealing with iterated cones.
For $n\geq 0$, let $[n]$ denote the $n$th cubical category, i.e., the (nonfull) subcategory of the category $\mathsf{Fin}$ of finite sets whose objects are $n+1$-uples $\ul{a} = (a_0, \ldots, a_n)\in \{0,1\}^n$. The set of morphisms ${\rm hom}_{[n]}(\ul{a}, \ul{b})$ for $\ul{a}\neq\ul{b}$ consists of a single element if $a_i\leq b_i$ for every index $i$, and it is empty otherwise. We note that all morphisms in $[n]$ are generated by morphisms of the form $a\rightarrow b$ such that there exists an index $1\leq i\leq n$ such that $$b_j=\left\{
\begin{array}{ll}
a_j+1&\text{if }j=i
\\
a_j&\text{if }j\neq i.
\end{array}
\right.$$ We define the face functors $$s_{i,0}, s_{i,1}\colon [n-1]\to [n], \quad i=1, \ldots, n$$ by inserting $0$ (resp. $1$) in the $i$-th coordinate.
An $n$-square in a category $\mathcal{A}$ is a functor $C\colon [n]\to \cA$, and a morphism of $n$-squares is a natural transformation of functors $C_1\Rightarrow C_2$ between $n$-squares. Note that a $0$-square is simply an object of $\cA$, a $1$-square is an arrow $C_0\to C_1$ where $C_0$ and $C_1$ are objects of $\cA$, and so on. We write $C_{(a_1,\ldots,a_n)} \in {\rm Ob}(\cA)$ for $C((a_1,\ldots,a_n))$. We denote by $$\mathbf{Square}_\cA$$ the coproduct of the categories of $n$-squares for all integer $n\geq 0$.
For an $n$-square $C$ with $n\geq 1$, let $s_{i,0}^*C$ (resp. $s_{i,1}^*$) denote the $(n-1)$-square that is the restriction of $C$ to all the coordinates of the form $$\begin{gathered}
(a_1,\ldots,a_{i-1},0,a_{i+1},\ldots,a_n)\\
\text{(resp.\ }(a_1,\ldots,a_{i-1},1,a_{i+1},\ldots,a_n)).\end{gathered}$$ or, equivalently, the composition $[n-1]\xrightarrow{s_{i, \epsilon}}[n]\xrightarrow{C}\cA$ for $\epsilon=0,1$. Note that if $C$ is an $(n+1)$-square, then there is a natural morphism of $n$-squares $$s_{i,0}^*C\rightarrow s_{i,1}^*C.$$
If $\cA$ is monoidal, for an $n$-square $C$ and $m$-square $D$ we denote by $$C\otimes D$$ the $m+n$ square such that the object at the coordinate $(a_1,\ldots,a_n,b_1,\ldots,b_m)$ is given by the tensor product of the objects at the coordinates $(a_1,\ldots,a_n)$ in $C$ and $(b_1,\ldots,b_m)$ in $D$. The morphisms in $C\otimes D$ are naturally induced by the morphisms in $C$ and $D$. This defines a monoidal structure on $\mathbf{Square}_\cA$.
We shall often assume that there exists a monoidal functor $$M\colon \mathbf{Square}_{\cA}\rightarrow \cT$$ satisfying the following conditions.
1. *Square condition.* For every integer $n\geq 1$ and $n$-square $C$, there exist canonical morphisms $$d_{i,C}\colon M(s_{i,1}^*C)\rightarrow M(C),
\;
\partial_{i,C}\colon M(C)\rightarrow M(s_{i,0}^*C)[1]$$ such that $$M(s_{i,0}^*C)\stackrel{M(\theta_{C,i})}\longrightarrow M(s_{i,1}^*C)\stackrel{d_{i,C}}\longrightarrow M(C)\stackrel{\partial_{i,C}}\longrightarrow M(s_{i,0}^*C)[1]$$ is a distinguished triangle, where $\theta_{C,i}\colon s_{i,0}^*C\rightarrow s_{i,1}^*C$ is the canonical morphism induced by $C$.
For every integer $n\geq 2$ with $1\leq i<j\leq n$ and $n$-square $C$, we require that the diagram $$\begin{tikzcd}[column sep=large]
M(s_{i,1}^*s_{j,1}^*C)\arrow[d,"d_{s_{j,1}^*C,i}"']\arrow[r,"s_{i,1}^*d_{j,C}"]&
M(s_{i,1}^*C)\arrow[d,"d_{i,C}"]\arrow[r,"s_{i,1}^*\partial_{i,C}"]&
M({s_{i,1}^*s_{j,0}^*C[1]})\arrow[d,"{d_{s_{j,0}^*C,i}[1]}"]
\\
M(s_{j,1}^*C)\arrow[r,"d_{j,C}"]\arrow[d,"\partial_{s_{j,1}^*C,i}"']&
M(C)\arrow[d,"\partial_{i,C}"]\arrow[r,"\partial_{i,C}"]&
M({s_{j,0}^*C[1]})\arrow[d,"-\partial_{s_{j,0}^*C,i}{[1]}"]
\\
M({s_{i,0}^*s_{j,1}^*C[1]})\arrow[r,"s_{i,0}^*d_{j,C}{[1]}"]&
M({s_{i,0}^*C[1]})\arrow[r,"-s_{i,1}^*\partial_{i,C}{[1]}"]&
M(s_{i,0}^*s_{j,0}^*C[2])
\end{tikzcd}$$ commutes.
2. *Monoidal square condition.* For every $n$-square $C$ and $m$-square $D$, let $$T_{C,D}:M(C)\otimes M(D)\rightarrow M(C\otimes D)$$ be the canonical morphism obtained by the monoidality of $M$. If $n\geq 1$, then for every $1\leq i\leq n$ we require that the diagram $$\begin{tikzpicture}[baseline= (a).base]
\node[scale=0.738] (a) at (0,0){
\begin{tikzcd}[column sep=huge]
M(s_{i,0}^*C)\otimes M(D)\arrow[d,"T_{s_{i,0}^*C,D}"']\arrow[r,"M(\theta_{i,C})\otimes M(D)"]&
M(s_{i,1}^*C)\otimes M(D)\arrow[r,"d_{i,C}\otimes M(D)"]\arrow[d,"T_{s_{i,1}^*C,D}"]&
M(C)\otimes M(D)\arrow[r,"\partial_{i,C}\otimes M(D)"]\arrow[d,"T_{C,D}"]&
M(s_{i,0}^*C)\otimes M(D)[1]\arrow[d,"T_{s_{i,0}^*C,D}{[1]}"]
\\
M(s_{i,0}^*C\otimes D)\arrow[r,"M(\theta_{i,C\otimes D})"]&
M(s_{i,1}^*C\otimes D)\arrow[r,"d_{i,C\otimes D}"]&
M(C\otimes D)\arrow[r,"\partial_{i,C\otimes D}"]&
M(s_{i,0}^*C\otimes D)[1]
\end{tikzcd}};
\end{tikzpicture}$$ commutes. Similarly, if $m\geq 1$, then for every $1\leq j\leq m$ we require that the diagram $$\begin{tikzpicture}[baseline= (a).base]
\node[scale=0.738] (a) at (0,0){
\begin{tikzcd}[column sep=huge]
M(C)\otimes M(s_{j,0}^*D)\arrow[d,"T_{C,s_{j,0}^*D}"']\arrow[r,"M(C)\otimes M(\theta_{i,D})"]&
M(C)\otimes M(s_{j,1}^*D)\arrow[r,"M(C)\otimes d_{j,D}"]\arrow[d,"T_{C,s_{j,1}^*D}"]&
M(C)\otimes M(D)\arrow[r,"M(C)\otimes \partial_{j,D}"]\arrow[d,"T_{C,D}"]&
M(C)\otimes M(s_{j,0}^*D)[1]\arrow[d,"T_{C,s_{j,0}^*D}{[1]}"]
\\
M(C\otimes s_{j,0}^*D)\arrow[r,"M(\theta_{n+j,C\otimes D})"]&
M(C\otimes s_{j,1}^*D)\arrow[r,"d_{n+j,C\otimes D}"]&
M(C\otimes D)\arrow[r,"\partial_{n+j,C\otimes D}"]&
M(C\otimes s_{j,0}^*D)[1]
\end{tikzcd}};
\end{tikzpicture}$$ commutes.
The conditions (Sq) and (MSq) are often automatically satisfied if $\cT$ arises from a simplicial, cofibrantly generated and locally presentable monoidal model category (such as the category of sheaves of complexes, that we consider in this text).
Let us give some basic consequences of (Sq).
\[A.3.45\] Let $f:Y\rightarrow X$ be a morphism in $\mathscr{S}/S$. Then the morphism $$M(f):M(Y)\rightarrow M(X)$$ is an isomorphism if and only if $M(Y\rightarrow X)=0$.
By (Sq) there is a canonical distinguished triangle $$M(Y)\rightarrow M(X)\rightarrow M(Y\rightarrow X)\rightarrow M(Y)[1],$$ which implies the claim.
\[A.3.48\] Let $C\rightarrow D$ be a morphism of $n$-squares. If the naturally induced morphism $$M(C_{(a_1,\ldots,a_n)})\rightarrow M(D_{(a_1,\ldots,a_n)})$$ is an isomorphism for every coordinate $(a_1,\ldots,a_n)\in \{0,1\}^n$, then the naturally induced morphism $$M(C)\rightarrow M(D)$$ is an isomorphism.
We use induction on $n$. The case $n=1$ is clear, so suppose that $n>1$. By (Sq) there is a morphism of distinguished triangles $$\begin{tikzcd}
M(s_{1,0}^*C)\arrow[d]\arrow[r]&
M(s_{i,1}^*C)\arrow[d]\arrow[r]&
M(C)\arrow[d]\arrow[r]&
M(s_{1,0}^*C)[1]\arrow[d]
\\
M(s_{1,0}^*D)\arrow[r]&
M(s_{1,1}^*D)\arrow[r]&
M(D)\arrow[r]&
M(s_{1,0}^*D)[1].
\end{tikzcd}$$ By induction, all the vertical morphisms except for $M(C)\rightarrow M(D)$ are isomorphisms. We conclude using the five lemma.
\[A.3.44\] For any commutative square in $\mathscr{S}/S$ $$C=\begin{tikzcd}
Y'\arrow[d]\arrow[r]&Y\arrow[d]\\
X'\arrow[r]&X
\end{tikzcd}$$ such that $$M(Y'\rightarrow X')\rightarrow M(Y\rightarrow X)$$ is an isomorphism, also $$M(Y'\rightarrow Y)\rightarrow M(X'\rightarrow X)$$ is an isomorphism.
The condition (Sq) implies that there are canonical distinguished triangles $$\begin{gathered}
\label{A.3.44.1}
M(Y'\rightarrow X')\rightarrow M(Y\rightarrow X)\rightarrow M(C)\rightarrow M(Y'\rightarrow X')[1],
\\
\label{A.3.44.2}
M(Y'\rightarrow Y)\rightarrow M(X'\rightarrow X)\rightarrow M(C)\rightarrow M(Y'\rightarrow Y)[1].\end{gathered}$$ By assumption, gives $M(C)=0$. Now use to conclude.
A result on calculus of fractions
---------------------------------
We record a result on calculus of fractions which is used repeatedly in this text. In applications, we are usually interested in the case when $S$ is a base scheme, $\cC=lSm/S$ or $SmlSm/S$, $\cD=\cT$, and $F=M$.
\[A.1.16\] Let $\cB\subset \cA$ be classes of morphisms in a category $\cC$ such that both classes admit a calculus of right fractions in $\cC$, and let $F:\cC\rightarrow \cD$ be a functor. Assume that for any morphism $f:Y\rightarrow X$ in $\cA$, there is a commutative diagram $$\begin{tikzcd}
&Y'\arrow[d,"g"]\arrow[ld,"h"']\\
Y\arrow[r,"f"']&X
\end{tikzcd}$$ in $\cC$, where $g\in \cB$. If $$F(g):F(Y')\rightarrow F(X)$$ is an isomorphism for every $g$ in $\cB$, then $$F(f):F(Y)\rightarrow F(X)$$ is an isomorphism for every $f$ in $\cA$.
The right Ore condition (dual to [@GZFractions I.2.2.c]) for $\cB$ implies there is a commutative diagram in $\cC$ $$\begin{tikzcd}
Y''\arrow[d,"g'"']\arrow[r,"f'"]&Y'\arrow[d,"g"]\\
Y\arrow[r,"f"']&X,
\end{tikzcd}$$ where $g'\in \cB$. Since $fhf'=gf'=fg'$ and $f\in \cA$, there is a morphism $r:Z\rightarrow Y''$ in $\mathcal{A}$ such that $hf'r=g'r$ by the right cancellation condition for $\cA$. By assumption, there is a morphism $s:Z'\rightarrow Y''$ in $\cB$ that factors through $r$. Thus $hf's=g's$, and there is a commutative diagram in $\cC$ $$\begin{tikzcd}
Z'\arrow[d,"g's"']\arrow[r,"f's"]&Y'\arrow[ld,"h"']\arrow[d,"g"]\\
Y\arrow[r,"f"']&X.
\end{tikzcd}$$ Now by assumption, $F(g)$ and $F(g's)$ are isomorphisms since $g$ and $g's$ are in $\cB$. Since $F(g)F(f's)=F(f)F(g's)$, we have that $$F(f's)F(g's)^{-1}=F(g)^{-1}F(f),$$ and therefore $${\rm id}=F(g's)F(g's)^{-1}=F(h)F(f's)F(g's)^{-1}=F(h)F(g)^{-1}F(f),$$ $${\rm id}=F(g)F(g)^{-1}=F(f)F(h)F(g)^{-1}.$$ This shows that $F(f)$ is an isomorphism.
[^1]: An alternate notion of finite correspondences between log schemes, free of any normalization constraints, was considered by K. Rülling in [@Rulling-logcor]. We thank him for sharing his note with us.
[^2]: Some authors opted for the French expression *quarrable* from SGA 4.
[^3]: In [@FKato], the condition is étale local on $S$. Since we are only interested in fs log schemes with Zariski log structures, we may restrict to the Zariski case.
|
---
abstract: 'Linear time-distance helioseismic inversions are carried out for vector flow velocities using travel times measured from two $\sim 100^2\,{\rm Mm^2}\times 20\,{\rm Mm}$ realistic magnetohydrodynamic quiet-Sun simulations of about 20 hr. The goal is to test current seismic methods on these state-of-the-art simulations. Using recent three-dimensional inversion schemes, we find that inverted horizontal flow maps correlate well with the simulations in the upper $\sim 3$ Mm of the domains for several filtering schemes, including phase-speed, ridge, and combined phase-speed and ridge measurements. In several cases, however, the velocity amplitudes from the inversions severely underestimate those of the simulations, possibly indicating nonlinearity of the forward problem. We also find that, while near-surface inversions of the vertical velocites are best using phase-speed filters, in almost all other example cases these flows are irretrievable due to noise, suggesting a need for statistical averaging to obtain better inferences.'
author:
- 'K. DeGrave, J. Jackiewicz, M. Rempel'
bibliography:
- 'mybib.bib'
title: 'Validating Time-Distance Helioseismology With Realistic Quiet Sun Simulations'
---
Introduction
============
Time-distance helioseismology [@duvall1993] is one of the few tools available that allows us to peer beneath the photosphere to better understand the Sun’s internal structure and dynamics. Through helioseismic inversions of wave travel-times measured at the solar surface, we can, in principle, better understand and characterize different kinds of subsurface perturbations (i.e. magnetic fields, density anomalies, flows, etc.) that may be present in the upper convection zone. This method has been used to study the near-surface structure of supergranulation [@gizon2003; @zhao2003; @jackiewicz2008; @duvall2010; @svanda2012] and sunspots [@zhao2001; @couvidat2006; @kosovichev2011a].
One way to test the accuracy of helioseismology is through comparisons with other helioseismic methods (i.e. comparing time-distance inversion results with those of ring-diagram analysis or helioseismic holography) [e.g., @gizon2009; @kosovichev2011b]. However, these types of studies can be misleading as agreement among methods does not necessarily guarantee correctness (which may be especially true around active regions). Perhaps the best way to to test the validity of time-distance inversions is by inverting for specific features present in solar simulations. This approach is advantageous in that the inversion results can easily be compared directly with the known values taken from the simulation. Synthetic data also allow us to carry out additional testing that would otherwise be impossible if only real solar data were available (i.e. testing the effects of data filter choice on results, tests of kernel performance through forward-modeling, etc.).
A small number of studies have been carried out previously to validate time-distance helioseismology by applying the method to simulated quiet-Sun data possessing varying degrees of realism. @zhao2007 performed time-distance inversions of realistic convection simulations [@benson2006] in the ray approximation and found that the recovered horizontal flows agree well (correlation coefficients of $\sim$0.5–0.9) with those of the simulation in the upper 5 Mm of the domain, though the vertical velocities were anticorrelated at nearly every depth. @svanda2011 used a snapshot of a convective simulation flow-field to produce forward-modeled travel-time maps, inverting them using the first Born approximation methods [@gb02] to validate their time-distance inversions. They found high correlation over the same range of depths for the horizontal flow components, and improved correlation for the vertical component.
In some sense, however, both cases may be too idealized. In the case of @zhao2007, the work was based upon a simulation which contained no magnetic fields. The work of @svanda2011 did indeed validate the inversion procedure, yet it was based on simplified measurements, as forward-modeled travel times of a frozen flow field may be too idealized in comparison to a full time-distance analysis of a Doppler time series.
$
\begin{array}{c}
\includegraphics[width=1\linewidth,clip=]{f1a.eps} \\
\includegraphics[width=1\linewidth,clip=]{f1b.eps}
\end{array}$
The goal of this work is to test current time-distance methods using the most realistic solar simulations available to us today. These magnetohydrodynamic simulations are fully convective and nonlinear in nature and contain large-scale supergranular-type flows. Such data, described in Section \[sec:sim\], will allow us to test methods in a regime that is as close as possible to the real Sun. In Section \[sec:filt\] the data filtering is described where two methods are considered. The filtered data are used to measure a large set of travel times in two different ways as discussed in Section \[sec:tt\]. Sections \[sec:kernels\] and \[sec:inv\] briefly review the forward and inversion problems in the first Born approximation. Also shown are comparisons of measured and forward-modeled travel times. Inversion results for the three vector flow components are given in Section \[sec:results\]. We discuss the results in the context of validation of these time-distance methods in Section \[sec:dis\].
This initial work is carried out with the more difficult goal of testing time-distance inversions near active regions in mind. The inversion procedure described here will be applied to a similarly realistic sunspot simulations to assess how capable or limited current linear time-distance inversions may be at recovering subsurface flows in these strong perturbation regimes. As sunspot helioseismology is an active area of research, such work should prove useful.
Quiet-Sun Simulation Data {#sec:sim}
=========================
The two quiet-Sun simulations (referred to as QS1 and QS2 hereafter) used for this study are based on the work of @rempel2009a [@rempel2009b]. They are publicly available for use in testing helioseismic methods[^1]. Both Cartesian simulation domains are 98.304 $\times$ 98.304 $\times$ 18.432 Mm in the horizontal and vertical directions. QS1 was computed with horizontal and vertical grid spacings of 0.064 Mm and 0.032 Mm respectively, while QS2 was computed with grid spacings of 0.128 Mm and 0.048 Mm. Boundary conditions for both are periodic in the horizontal directions. The top boundary of the domains is closed and located about $700~\rm{km}$ above the $\tau=1$ level. The bottom boundary is open (i.e. it allows convective flows to cross the boundary). This is implemented by imposing a symmetric boundary condition on all mass flux components (i.e. these quantities are mirrored into the ghost cells across the boundary). In QS1 the mean pressure is extrapolated into the ghost cells such that the value right at the boundary (between first ghost and domain cell) is fixed, while fluctuations are damped ($5\%$ reduction in the first ghost cell, $10\%$ reduction in the second). In QS2, the pressure fluctuations are set to zero at the boundary. Fixing the (mean) pressure at the boundary ensures that the total mass in the simulation domain remains constant apart from small fluctuations. The entropy is specified in inflow regions such that the resulting radiative losses in the photosphere lead to a solar-like energy flux, while down flows use a symmetric boundary condition.
The quiet-Sun simulations presented here are magnetic. We use a setup with zero net flux in which a turbulent magnetic field is maintained through a small-scale dynamo. We use a LES approach, relying only on numerical diffusivities similar to those introduced in [@rempel2009b] in order to minimize diffusivities. At the top boundary the magnetic field is vertical in QS2, while it is matched to a potential field extrapolation in QS1. We use a symmetric boundary condition for all three field components at the bottom. For details of the dynamo setup, we refer the reader to a forthcoming paper (Rempel 2014 in preparation).
Both simulations show large-scale flow structure that extends deeply through their domains (Fig. \[fig:QS\]). The near-surface horizontal flow magnitudes exhibit a root-mean-square (RMS) average of the order $350-400~\rm{m\,s^{-1}}$, while the average vertical velocity is $\sim 230~\rm{m\,s^{-1}}$, where we define the root-means-square (RMS) value for $n$ measurements of a quantity, $x$, as
$$x_{\rm{RMS}} = \sqrt{\frac{1}{n} \sum_i^n x_i^2}.$$
In general, the near-surface flows of QS1 are somewhat weaker than those of QS2. The simulations contain surface-gravity and acoustic waves that compare well to solar modes (see Fig. \[fig:filters\]), except where the boundaries of the domain have influence.
From each of these simulation runs we extracted a vertical velocity Doppler time series at the $\tau=0.01$ layer. We define this data cube as $v_z (\bm{r}, z=0, t)$, where $\bm{r}=(x,y)$ is the horizontal coordinate and $z$ is the vertical coordinate. The QS1 time series spans 25 hr, while QS2 spans 19.2 hr, and both are sampled with a time cadence $h_t= 45$ s. For the helioseismic analysis, the simulations were interpolated onto grids with $h_x=h_y=1$ Mm horizontal spacing, which is sufficiently small to capture the spatial variations due to acoustic waves.
$
\begin{array}{c}
\includegraphics[width=1.0\linewidth,clip=]{f2.eps}
\end{array}$
Filtering {#sec:filt}
=========
The first step in the analysis is to filter the data to obtain signal only from those waves whose travel times we would ultimately like to measure. For this work we chose to study the effects on time-distance inversion results of two filtering types – phase-speed filters, that isolate waves of similar phase speed over a range of radial orders, and ridge filters, that isolate wave packets that have the same radial order. Phase-speed filters extract the signal from waves that propagate to approximately the same depth in the solar interior. [@couvidat2006] showed that there exists an optimal set of such filters that effectively maximizes the signal-to-noise (S/N) of measured travel times. For this work, we selected the first five lowest phase-speed filters defined in @couvidat2006, referred to as ${\rm td_1}$–${\rm td_5}$ hereafter. Each of these filters is centered upon a line of constant phase-speed and is constructed according to the following analytic form: $$F(\omega, k) = e^{[- (\omega/k - v_{\rm ph})^2/ 2\delta v_{\rm ph}^2]}.
\label{phase}$$ Here, $v_{\rm{ph}}$ is the value of the central phase-speed; $\delta v_{\rm{ph}}=\rm{FWHM}/[2(\log 2)^{1/2}]$, where FWHM is the full width at half-maximum of the squared filter; $k = |\bm{k}|$ is the horizontal wavenumer, and $\omega$ is the angular frequency. Figure \[fig:filters\] shows the central phase-speed of the five phase-speed filters. These particular filters were chosen to keep only signal from those waves whose lower turning points are above a depth of $\sim$12 Mm (i.e. those waves whose phase-speeds are less than roughly 40 $\rm{km\,s^{-1}}$) to avoid any signal contamination from wave reflections at the bottom boundary of the simulation domain. When phase-speed filtering is used, the $f$-mode is first removed, as it follows a dispersion relation different from that of the acoustic $p$-modes.
As an alternative to phase-speed filtering, ridge filtering has been used to isolate wave signal by selecting those modes possessing the same radial order [@duvall2000; @jackiewicz2008; @gizon2009]. Here we use filters for the modes along ridges $f$, $p_1$–$p_3$ in the analysis. Higher radial order modes were not selected due to the fact that the amount of ridge power between the 40 $\rm{km\,s^{-1}}$ phase-speed line and the acoustic cutoff frequency was quite small, and we have seen some evidence of problems with our travel-time sensitivity kernels for these higher-order modes. The solid red lines in Fig. \[phase\] show each of the centers of the filters along the ridges considered here.
Each ridge filter, centered on its corresponding mode, was designed to keep full wave power extending 40% of the way in frequency to the next adjacent ridge on either side. This “flat gap" of full power transmission is then terminated by cosine wings that smoothly decrease to zero at 60% of the way to the next adjacent ridge. All filters, both ridge and phase-speed, were confined within the frequency range of 2.5–5.3 mHz.
With our initial Doppler time series, $\phi(x, y, t)$, we apply filters by multiplying the Fourier transform of the data cube with the square root of each filter $F_m (\bm{k},\omega)$ as
$$\phi_m (\bm{k},\omega) = \phi (\bm{k},\omega) \sqrt{F_m (\bm{k},\omega)}.$$
Here, $\phi_m (\bm{k},\omega)$ is the filtered cube in Fourier space containing wave signal isolated using filter $m$. The last step is to transform the filtered data back to real space, giving $\phi_m(x,y,t)$.
Travel-Time Measurements {#sec:tt}
========================
Temporal cross-covariances are computed from the filtered data $\phi_m (\bm{r},t)$ in the center-to-annulus and quadrant configurations [e.g., @duvall1997]. This is done by cross-correlating the signal at a point on the surface of the simulation domain with the signal averaged over a concentric annulus of some radius $\Delta$, and over 90$^\circ$ quadrants centered at the four discrete north, south, east, and west directions. The cross-covariances are computed for a range of $\Delta$ values depending on filter type. For the ridge-filtered data, $\Delta = 11-27$ Mm in increments of 4 Mm, totaling five radii per ridge. For the phase-speed filtered data, the valid $\Delta$ values over which the cross-correlations can be computed depend on central phase-speed as discussed in @couvidat2006. For this work, the ranges of $\Delta$ values used are 5–9, 7–11, 9–15, 15–19, 19–29 Mm in increments of 2 Mm for filters $\rm{td_1}$–$\rm{td_5}$ respectively. In total, we have $M=117$ measurements.
Comparison of Two Travel-Time Measurement Methods
-------------------------------------------------
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Travel-time differences were computed from each cross-covariance in the ‘oi’ (out minus in), ‘we’ (west - east), and ‘ns’ (north - south) geometries according to the two methods defined in @gb02 [@gb04] (hereafter referred to as GB02 and GB04, respectively). Both GB02 and GB04 measurement methods rely on calculating the difference between the cross-covariances at each point and a time-symmetric reference cross-covariance function, obtained either from a model or from an appropriate averaging of the data. The GB04 method is a linearization of the least-squares minimization prescribed in GB02. Example $p_1$ and ${\rm td}_3$ oi travel-time difference maps computed under the GB04 definition for both simulations are shown in Fig. \[ttoi\] for $\Delta = 15$ Mm. The travel times suggest that both simulations possess large ($\sim 20$ Mm scale) regions of convergent (positive travel-time difference) and divergent (negative travel-time difference) flows.
GB04 assumes that the travel times are linearly related to the small changes in the cross-covariances due to small subsurface perturbations, and will eventually become inaccurate in regions where strong perturbations dominate. GB02 has been shown to be more robust in a larger range of perturbation regimes [@couvidat2012]. Inspection of the data in Fig. \[fig:QS\] shows that both simulations possess large ($>$ 500 $\rm{m\,s^{-1}}$) flows that could potentially cause a problem for linear measurements and inversions [@jackiewicz2007a]. Therefore, travel times were computed under both definitions for use in the inversions to see if recovered flows vary in any significant way.
Figure \[gb0204\] shows the Pearson correlation coefficient between QS1 GB02 and GB04 travel-time maps computed for each filter and $\Delta$ in the oi geometry. We find a very high correlation between the two and there seems to be no trend with distance or filter type, except in the case of the $f$-mode. Even here, though, the correlation between the methods is greater than 0.99. However, we find that travel times measured with GB04 are on average 10% larger than those of GB02 in terms of RMS average.
A comparison of MDI travel times carried out by @couvidat2012 found a similarly high correlation between GB02 and GB04 travel time definitions. In their analysis, however, they found that at large ($\sim 30$ Mm) distances, GB02 and GB04 are highly correlated only at small travel-time differences between $\pm 5$ seconds. Outside this range, GB02 travel-times were generally found to be larger than GB04.
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Travel-Time Noise
-----------------
Solar oscillations are driven by stochastic convective motions, and measured travel times contain some level of realization noise depending on the observation length. This noise is very important to characterize as it induces correlations in the travel-time measurements and propagates through the inversion [@jensen2003; @couvidat2005; @couvidat2006; @jackiewicz2008]. For this work, we estimate the noise covariances for travel times computed from the simulation data by employing the Monte Carlo technique outlined in @gb04. The basic idea is to study small segments (90 min in this case) of the Doppler data and compute a series of noise realizations by adding complex Gaussian noise to its average power spectrum. Travel-times are measured from this noisy data, and the covariance between every travel-time map is measured for each available filter-$\Delta$-geometry combination. Rather than simply using the RMS variation in the travel times to estimate noise [@zhao2012], the noise covariance matrices should give a more accurate representation of the resulting error in the flow inversions.
Sensitivity Kernels {#sec:kernels}
===================
To infer flows, we relate our travel-time measurements to subsurface conditions present in the simulations. The relationship between travel-time perturbations and flow velocity, $\bm{v}$, developed in the GB02 and GB04 formulation, is given in the form of a linear integral equation
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$$\begin{aligned}
\delta\tau^{a}(\bm{r}) = h_rh_z\sum_{ij} \bm{K}^{a} (\bm{r}_i - \bm{r},z_j) \cdot \bm{v} (\bm{r}_i,z_j) + N^{a} (\bm{r})
\label{eq:dt}\end{aligned}$$
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where $h_r=h_xh_y$ and $h_z$ is the vertical grid spacing, $\bm{K}^a$ are three-dimensional vector-valued kernels describing the sensitivity of wave travel times to flows for each particular measurement geometry, filter, and $\Delta$, captured in the $a$ index. $N^a$ represents the noise for travel-time measurement $a$. The summation over $i$ represents a horizontal spatial convolution.
A set of kernels for flows $\{K_{v_x}, K_{v_y}, K_{v_z}\}$ in the $+\hat{\bm{x}}$ direction was computed in the single scattering Born approximation [@birch2007]. The model power spectra that are needed for the kernel computation (as prescribed in @gb02) are filtered exactly as the initial data power spectra used to derive the travel times for which we are modeling. These “point-to-point” kernels are then combined in an appropriate way to be consistent with the ‘oi’, ‘we’, and ‘ns’ travel-time geometries. Figure \[fig:kerns\] shows the approximate depth sensitivity of a selection of kernels computed using ridge and phase-speed filtering. All the kernels are strongly peaked at the top of the box with lower-amplitude peaks slightly below this region. Only one set of kernels and covariance matrices is subsequently used in the inversions, as the power spectra of the simulations are indistinguishable for the modes of interest.
Because we are working with synthetic rather than real solar data, we have an advantage in that we can directly asses the abilities of our kernels through forward modeling. To do this, we computed a set of forward-modeled travel-time maps by convolving our sensitivity kernels with the known flow fields taken directly from the simulations (Eq. \[eq:dt\], without the noise term). In the case of no measurement noise and perfect kernels, each modeled travel-time map would match its measured counterpart exactly. Of course this is not the case, so we expect instead some mismatch. Figure \[fig:forward\] shows the correlation between measured and forward-modeled travel times for every filter over the appropriate range of $\Delta$ values for QS1. In the ridge-filter case, we find that the correlation is high for filters $f$, $p_1$, and $p_2$, but drops significantly for $p_3$. This trend of decreasing correlation has also been observed for $p_4$. This was ultimately the basis for using only $f$, $p_1$–$p_3$ in our analysis. The phase-speed filters show something similar with the correlation starting high when phase-speed is low, but dropping sharply when when it becomes large. QS2 shows the same phenomenon, with the $p_3$ and td$_5$ correlations actually being slightly worse than the QS1 case. It is not entirely obvious why such a decrease in correlation should be observed. As the computation of sensitivity kernels is heavily dependent upon accurately modeling the data power spectra, a mismatch between data and model is likely contributing to the poorer agreement in these cases. We do indeed find that it is often more difficult to model the acoustic power of higher-order modes and the power at higher phase-speeds mostly because of the weaker mode power at longer wavelengths due to the finite simulation domain size. Another consideration is that the kernels are computed using a standard solar model, and while the simulation and model stratification generally agree, deviation does occur in near-surfaces layers where the energy transport in the simulation is treated more realistically. The effects of this have not been studied in any quantitative manner.
SOLA Inversion Method {#sec:inv}
=====================
The goal of the time-distance inversions is to recover subsurface vector flows ($\bm{v}$ in Eq. \[eq:dt\]) in the upper layers of QS1 and QS2 given a set of measured travel times, sensitivity kernels, and the appropriate noise-covariance matrices. To do this, we employ the Subtractive Optimally Localized Averaging (SOLA) method [@pijpers1992], modified for computations in the Fourier spatial domain [@jackiewicz2012]. The general idea behind a SOLA inversion is to find a set of ‘inversion weights’ that average the set of travel times in a linear fashion. Once obtained, the weights in this case are spatially convolved with the travel times to give an estimate of a chosen component $\alpha=\{x,y,z\}$ of the flow at a targeted location centered at a depth $z_0$ within the simulation domain $$v_{\alpha}^{\rm inv} (\bm{r}; z_0) = \sum_i \sum_{a=1}^M w_a^{\alpha} (\bm{r}_i - \bm{r}; z_0) \delta \tau^a(\bm{r}_i),
\label{vinv}$$ where we recall that $M$ is the number of travel-time maps considered in a particular inversion. As described in other work [@svanda2011; @jackiewicz2012], the inversion weights are used to construct linear combinations of the sensitivity kernels, an averaging kernel, that is localized in 3D space and acceptably matches a pre-defined (usually Gaussian) ‘target function’ $T$. In practice the match is usually far from perfect due to noise and a limited number of travel times. Another consideration is the extent to which the two flow components not being inverted for “leak” into the inferred velocity, known as the ‘cross-talk.’ This is a major factor only in inversions for the vertical velocity.
The SOLA method attempts to minimize the misfit between averaging kernel and target function while at the same time minimizing inversion noise, cross-talk, and the spatial localization of the weights [@jackiewicz2007b; @svanda2011]. This amounts to an optimization problem with the balance of these quantities being determined by a set of regularization parameters, denoted here by $\mu$, $\nu$, and $\epsilon$ respectively . One is free to choose regularization values, though when inverting for horizontal flow components, priority is often placed first on reaching an acceptable noise level rather than focusing on any of the other parameters. For example, when inverting for supergranular flows with a typical RMS velocity $\sim$250 $\rm{m\,s^{-1}}$, one might choose to accept a noise level of $\sim 30 - 40~\rm{m\,s^{-1}}$. On the other hand, inversions for the vertical flow component will likely place a greater importance on minimizing cross-talk and the localization of the weights. In practice, when inversions are performed, a series of regularization values are tested to find which combination gives the best results.
Inversion Results {#sec:results}
=================
We carry out inversions using each of the three filtering schemes (i.e. ridge, phase-speed, and ridge+phase-speed) for all three velocity components $(v_x, v_y, v_z)$ at depths of 1, 3, and 5 Mm below the surface using the GB04 travel-time definition. To our knowledge, only one previous study [@svanda2013] has employed and tested a combination of ridge+phase-speed filtered data in helioseismic inversions.
In what follows, inversions for flows, denoted by $v_\alpha^{\rm inv}$, are directly compared to the flows from the numerical simulations. To make direct comparisons on the appropriate spatial scales at each targeted depth, the artificial data are smoothed to the expected resolution of the inverted flows. This is carried out by a convolution of the inversion target function $T$, with the raw simulation flow field to obtain the “targeted” answer $v_\alpha^{\rm tgt}$ for flow component $\alpha$ $$v_\alpha^{\rm tgt}(\bm{r};z_0) = h_r h_z\sum_{ij} T_\alpha(\bm{r}_i - \bm{r},z_j;z_0)v_\alpha(\bm{r}_i,z_j),
\label{valph}$$ similar to the notation in @svanda2011. The sum over index $i$ represents a horizontal convolution, while the sum of the products over $j$ takes place at the same depth slices. In the case of an “ideal” inversion when the averaging kernel matches the target function with minimal noise variance, the quantities $v_\alpha^{\rm tgt}$ and $v_\alpha^{\rm inv}$ will be similar. As is often the case in many of the example inversions presented here, however, the averaging kernel does not sufficiently match the target, particularly for the vertical flow inversions. When this is the case, we will also be interested in studying the quantities $$v_\alpha^{(\beta)}(\bm{r};z_0) = h_r h_z\sum_{ij} \mathcal{K}_\alpha^{(\beta)}(\bm{r}_i - \bm{r},z_j;z_0)v_\beta(\bm{r}_i,z_j),
\label{vbet}$$ where $\mathcal{K}_\alpha^\beta$ is the $\beta$ component of the averaging kernel from an inversion for $\alpha$. Again, in the ideal case $v_\alpha^{(\beta)}$ will be zero when $\alpha\neq\beta$. More likely, however, the individual components obtained from this expression are nonzero and can be used to quantify the cross-talk from any inversion. In addition, the component $v_\alpha^{(\alpha)}\equiv v_\alpha$ is the targeted flow, equivalent to Eq. (\[valph\]) when the target and averaging kernel are identical.
Horizontal Flow Inversions
--------------------------
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Figure \[fig:qs1vx\] shows the resulting $v_{x,y}^{\rm inv}$ horizontal flow maps for QS1 obtained at each depth for the separate ridge and phase-speed filter inversions (rows 1 and 2 respectively), and for the combined ridge+phase-speed filter inversions (row 3). The noise for each inversion is $\sim35\,\rm{m\,s^{-1}}$. For comparison, the simulation flows, $v_{x,y}^{\rm tgt}$ (see Eq. \[valph\]), at each corresponding depth are shown in row 4. These represent the best case scenario that we can hope to accomplish with our inversions. The approximate horizontal and vertical resolutions of each inversion (i.e. the horizontal and vertical FWHM of the target function) along with the regularization parameters used are given in Table \[tab1\], inversion set 1. We find that these inversions capture the large-scale supergranule-sized flows present in the simulation, as well as the smaller features throughout the domain at the two shallowest depths. We therefore measure high spatial correlation between $v_{x,y}^{\rm inv}$ and $v_{x,y}^{\rm tgt}$ flow maps down to a depth of 3 Mm for each of the three filtering scenarios (Table \[tab2\]). At a depth of 5 Mm, the acoustic wave coverage is small in comparison to the shallower layers, and so it is quite difficult to accurately recover flow structure there. However, the flow correlation at the largest depth is not that small, $\approx 0.6$, and there is visible similarity in the maps. We note here that the $v_x$ correlation is higher than that of $v_y$ in every single case. The same phenomena has been documented previously by @zhao2007, though it is not clear why there should be a preferential bias of one flow component over the other.
Figure \[fig:qs2vx\] shows the equivalent QS2 $v_{x,y}^{\rm inv}$ horizontal flow maps obtained at each depth for both ridge and phase-speed filter types (rows 1 and 2 respectively), for the combined ridge+phase-speed filters (row 3), and for the simulation at each corresponding depth (row 4). The horizontal and vertical resolutions of each inversion along with the regularization parameters used are identical to the QS1 case and are also given in Table \[tab1\], inversion set 1. Again, each of the three filtering scenarios show strong correlation between $v_{x,y}^{\rm inv}$ and $v_{x,y}^{\rm tgt}$ in the upper 3 Mm, dropping substantially as we reach a depth of 5 Mm (Table \[tab2\]).
The divergence correlation coefficients presented for the QS2 simulation in Table \[tab2\] tend to show a slightly higher correlation than for the QS1 case. This could be attributed to the fact that QS1 contains more small-scale structure that the inversions have a hard time resolving, even after smoothing. To support this conjecture, Fig. \[fig:vpow\] shows the $v_x$ spatial power averaged over wavevector for both simulations at a depth of 1 Mm. This was computed using the simulation data both before and after convolution with the 1 Mm depth target function. We see in both cases that QS2 has slightly more power over large spatial scales and less at smaller ones. Interestingly, the correlation differences between $v_x$ and $v_y$ found for QS1 are not present in the results for QS2, as seen in Table \[tab2\].
Additionally, forward-modeled travel times were inverted to compare with the results from the measured ones to see if there are significant differences in the results. Figure \[forward\_vx\] shows the correlation between $v_x^{\rm inv}$ and $v_x^{\rm tgt}$ for both QS1 and QS2 at each depth for each filtering scheme when either the measured or modeled travel times were used. In both cases, the $v_x^{\rm inv}$ flows obtained using the forward-modeled travel times are more consistent with $v_x^{\rm tgt}$ at every depth.
### Flow amplitudes
What is not captured reasonably well by the horizontal inversions, particularly for QS2, is the amplitude of the flows, as is clearly seen in the vector velocity plots as well as in Table \[tab2\] where the slopes of the correlation scatter plots are recorded. Here, even the best inversions underestimate those of $v_{x,y}^{\rm tgt}$ by up to 45% in some cases, well above the standard deviation of the noise. We attribute much of the discrepancy to poor misfit between the inversion averaging kernels and the target function. To explore this further, Figure \[smoothed\] shows the raw QS2 simulation $v_x$ flow component at many depths, smoothed with a variety of wider and wider Gaussian target functions. At a given depth and smoothing value, a perfect inversion would fall near to the appropriate curve within the noise level. The starred points are the phase-speed inversion values, clearly well below the expected ones. The filled circles represent the raw simulation data instead smoothed by the averaging kernels ($v_x^{(x)}$ from Eq. \[vbet\]) rather than the proper Gaussian functions. As these points fall much closer to the inversion values, especially for the 2 deeper inversions, it is evident that the misfit is the source of the low amplitudes. The shallowest inversion mismatch is farther out of the range of the noise deviation, and is instead likely due to the large near-surface flows in the simulation that we cannot retrieve in a linear inversion. @jackiewicz2007a first showed a nonlinear response of travel-time differences for steady flows that the first Born approximation does not capture. The break from linearity typically occurs when flows reach $\sim$300-400 $\rm{m\,s^{-1}}$ at nominal travel distances. The depth slices through the simulations presented in Fig. \[fig:QS\] show that both simulations clearly possess flows that exceed this linearity threshold near the surface which remain even after smoothing.
$v_x^{\rm inv}$$v_x^{(x)}$$v_x^{(y)}$$v_x^{(z)}$$v_x^{\rm inv} - \sum_\beta v_x^{(\beta)}$$v_x^{\rm tgt} - \sum_\beta v_x^{(\beta)}$
Further explanation for the poor amplitude inferences could be cross-talk effects from the vertical flow field. Figure \[crosstalkx\] shows the components $v_x^{(x)}$, $v_x^{(y)}$, $v_x^{(z)}$ of the near-surface QS2 simulation computed from our $v_x$ inversion averaging kernels. However, we find that little of the off-diagonal terms are “leaking” into the targeted component, as inspection of the averaging kernels themselves further verifies in Fig. \[fig:avex\] (as a note, averaging kernel plots like Fig. \[fig:avex\] have been computed for every inversion listed in Table \[tab1\] and can be viewed in the online supplement). The first two panels of Fig. \[crosstalkx\] indeed agree rather well. This is expected because of the weaker vertical flows [@svanda2011], and shows that the cross-talk minimization is working well in this case and not contributing to this problem.
The fifth panel in Fig. \[crosstalkx\] illustrates a map of the random noise in the inversion whose magnitude is consistent within a factor of 2 with the value predicted from the inversion machinery. The last panel in Fig. \[crosstalkx\] reinforces the result that the amplitudes recovered in the horizontal inversions are significantly different (smaller) than those in the inversion, and that this difference is due to the mismatch between the averaging kernels and the target.
### Effects of filtering
Examining the calculated correlations in Table \[tab2\], there is a slight advantage when using the phase-speed filtering scheme. This is true particularly in the shallow layers of QS1. For QS2 we find that the correlations between phase-speed $v_{x,y}^{\rm inv}$ and $v_{x,y}^{\rm tgt}$ are consistently better than the ridge filtering and combined ridge+phase-speed filtering cases at every depth.
It is important to note here that no clear overall improvement is found in either simulation when performing inversions in the upper 5 Mm using the combined ridge+phase-speed filtering scheme. In terms of both magnitude and spatial correlation, the combined filtering generally showed no significant advantage over phase-speed filtering alone. However, for the same fixed inversion noise level, the combined filtering can significantly improve the misfit between target and averaging kernel (or conversely, improve the level of inversion noise for the same fixed misfit). This improvement in inversion noise level for the combined filtering scheme was also seen by @svanda2013, and is likely due to having more kernels. In the case of the near-surface inversions presented here, though, the improved misfit does not seem to translate into any improvement in our inversion results.
We note than in nearly every horizontal inversion performed, for the same noise level, the level of misfit was the worst when ridge filtering alone was used. This is in contrast to @svanda2013 where ridge filtering showed a substantial advantage over phase-speed filtering at a depth of 2.2 Mm when the two methods were compared directly. This was possibly explained by the fact that their ridge filter inversions contained many more independent measurements than the phase-speed filter inversions. For this work, we tried to avoid this potential bias, and the ridge and phase-speed filter inversions were all carried out with nearly identical numbers of independent measurements.
A Note on Converging Flow
-------------------------
There have been several studies in which time-distance helioseismology has been used to examine the flow field around supergranulation within the first few Mm of the upper convection zone [@gizon2003; @zhao2003; @woodard2007; @jackiewicz2008; @duvall2010; @svanda2012]. A major focus of these works is not only to characterize the structure of these features, but also to determine the depth at which supergranules terminate, as this would shed some light on their mechanism of formation. @duvall1998 [@zhao2003] found evidence of a supergranule return flow below some 10 Mm below the photosphere. These conclusions are drawn based on the correlation of near-surface flow divergence with the divergence of flows recovered at increasing depths. The correlation decreases quickly as one moves deeper through the convection zone and has been shown to eventually reverse sign.
Recently, @svanda2013 suggested that the detection of this return flow could in fact be spurious, simply arising due to the loss of supergranule coherence combined with increasing noise as one probes larger depths. This idea is testable with our simulation data as each domain actually possesses supergranule-sized flows whose large-scale structure stays coherent well throughout the upper half of each simulation domain. Inversions at depths larger than 5 Mm, however, are difficult with our current kernel selection because there is little sensitivity in these regions. The idea here, though, is not to show that we can accurately recover flows at these depths (because we cannot) but to show that a spurious “return flow” detection is certainly possible when carrying out inversions where there is little wave signal and where subsurface flows are weak.
Additional horizontal inversions for each of the three filtering schemes were performed at depths of 7 and 9 Mm using QS2. QS2 was chosen simply based on the fact that it contains more well-defined supergranule structure than QS1. Target FWHM values for these inversions are given in Table \[tab1\], inversion set 2. In contrast with the shallow inversion results, we find a marked advantage for the case of ridge+phase-speed filtering over ridge or phase-speed filtering alone. As before, the combined filtering scheme gives a much better misfit between target and averaging kernel for the same amount of noise. This is especially important at large depths when getting a reasonable error generally means having to allow the misfit to decrease drastically. This can become problematic because the averaging kernel becomes increasingly non-localized around the target depth, and begins including unwanted signal from shallower regions. For the same misfit at these depths, the inversion error for the combined filtering case was 20–40 $\rm{m\,s^{-1}}$ less than for the ridge or phase-speed filtering cases alone.
Using the combined filtering scheme, each inversion flow map at depth was correlated with the near-surface one at 1 Mm. Figure \[sg\] shows the correlation as a function of depth when both the horizontal divergence, $\nabla_{\rm h}\cdot \bm{v}_{\rm h}^{\rm inv}$, and the separate $v_x^{\rm inv}$ and $v_y^{\rm inv}$ flow component are used as proxies. We see for the inverted quantities that the correlation decreases rapidly at larger depths, where a reversal in sign occurs around 7 Mm. In fact, $v_x^{\rm inv}$ and $v_y^{\rm inv}$ give different crossover depths as indicated by the divergence correlation profile. However, performing the same exercise using the targeted flows (i.e. those of $v_{x,y}^{\rm tgt}$) shows that no reversal exists at any depth. This suggests that the observation of a correlation sign reversal seen in real solar inversions could in fact be spurious, merely arising as a consequence of the current “standard inversion” shortcomings.
Vertical Flow Inversions
------------------------
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\begin{array}{c c c}
\includegraphics[width=0.3\linewidth,clip=]{f16a.eps} &
\includegraphics[width=0.3\linewidth,clip=]{f16b.eps} &
\includegraphics[width=0.3\linewidth,clip=]{f16c.eps}\\
\includegraphics[width=0.3\linewidth,clip=]{f16d.eps} &
\includegraphics[width=0.3\linewidth,clip=]{f16e.eps} &
\includegraphics[width=0.3\linewidth,clip=]{f16f.eps}
\end{array}$
Inversions for the vertical velocities in the two simulations were also carried out, the parameters of which are presented in Table \[tab1\], inversion set 3. As pointed out in @svanda2011, the structure of the vertical velocity sensitivity kernels only allows for the retrieval of the fluctuations of $v_z$ from the mean value $\langle v_z \rangle$. In everything that follows, our comparisons are made between the inverted vertical flows and the mean-subtracted simulation vertical flows. Figure \[fig:qs1vz\] shows the resulting QS1 $v_z^{\rm inv}$ flow maps obtained from inverting the phase-speed travel times at each depth (top row). For comparison, the target flows from the simulation, $v_z^{\rm tgt}$, are also shown (bottom row). The strategy for the shallowest inversion was to obtain a noise level in the $10-20~\rm{m\,s^{-1}}$ range. This is motivated by the RMS values of $v_z^{\rm tgt}$ being on the order of 35 $\rm{m\,s^{-1}}$. This gives reasonable averaging kernels here, while at larger depths this constraint on the noise limit has to be considerably relaxed to obtain a reasonable averaging kernel. Consequently, this leads to very inaccurate flow amplitude determinations at larger depth. We find from Table \[tab2\] that the correlation between $v_z^{\rm inv}$ and $v_z^{\rm tgt}$ for QS1 is rather poor in every case. The artificial flows ($v_z^{\rm tgt}$) show a pattern of many small-scale up flows and down flows present over the full range of depths which is not recovered in $v_z^{\rm inv}$. Phase-speed inversions generally resulted in higher correlations than the combined filtering scheme, but the correlation values are very low. The ridge inversions were significantly worse, even showing a relatively strong negative correlation at a depth of 1 Mm. Figure \[fig:qs2vz\] shows the equivalent $v_z$ flow maps for QS2. In this case, the near-surface correlations are significantly higher than those of QS1, in the $0.5-0.7$ range at a depth of 1 Mm. Again, the ridge filter correlations are substantially lower than the other filtering schemes, especially in the near-surface layers. At larger depths, the correlations rapidly decrease and the signal is dominated by noise, rendering the recovered amplitudes meaningless.
To ensure that there were no mistakes in the QS1 analysis that led to such poor correlation between $v_z^{\rm inv}$ and $v_z^{\rm tgt}$, a test was carried out by exchanging the inversion weights. While by eye the sets of weights from $v_z^{\rm inv}$ for QS1 and QS2 are very similar, we computed flows according to Eq. \[vinv\] with the weights from QS1 and the travel-time measurements from QS2, and vice versa. The results showed no quantitative differences.
$v_z^{(x)}$$v_z^{(y)}$$v_z^{(z)}$$v_z^{\rm inv}$$v_z^{\rm inv} - \sum_\beta v_z^{(\beta)}$$v_z^{\rm tgt} - \sum_\beta v_z^{(\beta)}$
Cross-talk effects in standard helioseismic inversions of numerical simulations can lead to the retrieval of vertical flows with the wrong sign [@zhao2007; @dombroski2013]. As already mentioned, the inversion procedure here uses a constraint that attempts to limit such effects, while taking into account the associated trade-offs in the misfit and noise. To understand better the poor results for the vertical inversions, we explored the cross-talk contribution.
Representative averaging kernels from the vertical inversions are shown in Fig. \[fig:avez\]. We see that the cross-talk minimization in the inversion algorithm appears to be effective, in that the $x, y$ components are relatively small in magnitude, by about a factor of 50 in this example. However, this evidently does not translate into well constrained flows, as Fig. \[crosstalkz\] shows the components of the diagonal and off-diagonal flows, $v_z^{(\beta)}$, for QS1. The cross-talk here can reach up to 35% of the overall maximal flow amplitudes, indicating that the strong horizontal flows play a detrimental role in the retrieval of the vertical velocities because of their larger strength.
The vertical velocity inversions are impacted strongly by noise. It is therefore challenging to construct averaging kernels that are a close match to the target function. The first statement is demonstrated in Fig. \[crosstalkz\] for QS1, whereby the proxy noise component (panel 5) is very similar to the inverted flow component itself (panel 4). The fact that the last panel in Fig. \[crosstalkz\] shows a signal that is of the order of the simulated velocities themselves proves the second statement. It should be emphasized that although the vertical-velocity inversions do not perform as well as in the tests of @svanda2011, the differences should mainly be attributed to the simulation data and overall procedure, as we have not started from a single snapshot with random noise added and tried to recover the input velocities.
Summary and Discussion {#sec:dis}
======================
Current time-distance analysis has been tested using two realistic magnetohydrodynamic quiet-Sun simulations. Measurements made using the GB02 and GB04 travel-time definitions are found to correlate very well with one another, varying linearly over a large range of distances. The travel-times computed using GB04 are on average $10\%$ larger than those of GB02 in terms of RMS variation. Correlation between measured and forward-modeled travel-times computed in the first Born approximation is generally high for filters $f, p_1-p_2$ and $\rm{td_1}-\rm{td_4}$, but is found to decrease rapidly for filters $p_3$ and $\rm{td_5}$. We possibly attribute this to inadequate modeling of the simulation power at higher phase-speeds and for higher-order modes.
SOLA inversions were carried out using both travel-time definitions for several filtering schemes, including phase-speed, ridge, and combined phase-speed and ridge measurements to recover flows in the upper layers of both simulations. We find that horizontal flow maps correlate well ($\sim0.8$) with the simulations in the upper 3 Mm of the domains. At a depth of 5 Mm, correlation deteriorates significantly ($\sim0.6$), though some large-scale flow structure is still visible. Simply increasing the number of measurements used in the inversions would likely help to improve wave coverage at larger depths, but this is not a trivial undertaking due to current computational constraints. We find that even for our best inversions, we severely underestimate the flow velocities at every depth, possibly indicating non-linearity of the forward problem caused by the very strong ($>500~\rm{m\,s^{-1}}$) near-surface flows present in both simulation domains.
Inversions employing phase-speed filtering alone seem to show an advantage in the upper 5 Mm when compared to the other filtering schemes. At larger depths, however, the combined ridge+phase-speed filtering produced a better match between averaging kernel and target function for a fixed inversion noise level. Ridge filtering was generally found to give the worst correlation values.
Vertical flow inversions show poor correlation with the simulation over the full range of depths for QS1, but noticeably better results for QS2. Amplitude determination of these vertical velocities fails everywhere but the nearest surface layer, as noise dominates these inferences. While the inversion procedure to minimize the important cross-talk terms appears to work effectively in the inversion procedure itself, we do not accurately retrieve the vertical flows as was the case in [@svanda2011]. This is likely due to the differences in the simulation data we used, such as overall flow amplitudes, time-series length, and utilization of forward-modeled travel times.
In summary, the large-scale flows present in these very sophisticated solar-like simulations are not adequately retrievable with current time-distance techniques, and these results cause us to hesitate to invert for real solar features. Improvements to forward and inverse modeling may need to be made for studying individual structures over short time scales. Longer data sets can be used, but this eliminates the possibility of determining individual supergranule vertical flow profiles as their lifetimes are on average only $1-2$ days. However, these findings suggest that perhaps the most promising way to proceed is the statistical averaging scheme developed in @duvall2010 and @svanda2011 for supergranule-type flows. We have quantified the level of discrepancy between the seismic inferences and the known answer for these simulations, and a forthcoming study attempts a similar analysis for the flows in sunspot simulations.
------------------------------ ---------- -------- ------- ----------------- ----------------- ---------------------------- ------------------- ------------------------- ---------------------- --
inversion sim filter depth FWHM$_{\rm{h}}$ FWHM$_{\rm{z}}$ $\mu$ $\nu$ $\epsilon$ noise
\[Mm\] \[Mm\] \[$\rm{s^2\, m^{-2}}$\] \[$\rm{Mm^{3}}$\] \[$\rm{s^4 \,m^{-2}}$\] \[$\rm{m\,s^{-1}}$\]
\[0.5ex\] set 1 $(v_x, v_y)$ QS1, QS2 all 1 Mm 10 2 (0.4–1.8)$\times$10$^{-8}$ 1 1.0$\times$10$^{-6}$ $\sim$35
3 Mm 12 2 (0.4–1.8)$\times$10$^{-8}$ 1 1.0$\times$10$^{-6}$ $\sim$35
5 Mm 14 2 (0.4–1.8)$\times$10$^{-8}$ 1 1.0$\times$10$^{-6}$ $\sim$35
set 2 $(v_x, v_y)$ QS2 comb 7 Mm 16 3 4.6$\times$10$^{-10}$ 1 1.0$\times$10$^{-6}$ 64
9 Mm 18 3 7.5$\times$10$^{-12}$ 1 1.0$\times$10$^{-6}$ 86
QS1 ridge 1 Mm 12 2 7.2$\times$10$^{-9}$ 80 3.2$\times$10$^{-6}$ 18
3 Mm 14 2 2.3$\times$10$^{-11}$ 80 1.0$\times$10$^{-9}$ 814
5 Mm 16 2 1.0$\times$10$^{-11}$ 80 3.2$\times$10$^{-9}$ 567
phase 1 Mm 12 2 3.7$\times$10$^{-8}$ 80 1.0$\times$10$^{-5}$ 10
3 Mm 14 2 2.7$\times$10$^{-10}$ 80 1.0$\times$10$^{-8}$ 210
5 Mm 16 2 2.3$\times$10$^{-11}$ 80 1.0$\times$10$^{-9}$ 445
comb 1 Mm 12 2 3.7$\times$10$^{-8}$ 80 1.0$\times$10$^{-5}$ 12
3 Mm 14 2 6.1$\times$10$^{-10}$ 80 3.2$\times$10$^{-8}$ 152
5 Mm 16 2 2.3$\times$10$^{-11}$ 80 1.0$\times$10$^{-8}$ 236
QS2 ridge 1 Mm 12 2 7.2$\times$10$^{-9}$ 80 3.2$\times$10$^{-6}$ 18
3 Mm 14 2 1.0$\times$10$^{-11}$ 80 1.0$\times$10$^{-9}$ 814
5 Mm 16 2 1.0$\times$10$^{-11}$ 80 1.0$\times$10$^{-9}$ 567
phase 1 Mm 12 2 1.9$\times$10$^{-8}$ 80 3.2$\times$10$^{-5}$ 18
3 Mm 14 2 1.4$\times$10$^{-10}$ 80 1.0$\times$10$^{-8}$ 210
5 Mm 16 2 1.2$\times$10$^{-11}$ 80 3.2$\times$10$^{-9}$ 445
comb 1 Mm 12 2 3.7$\times$10$^{-8}$ 80 1.0$\times$10$^{-5}$ 12
3 Mm 14 2 2.7$\times$10$^{-10}$ 80 3.2$\times$10$^{-8}$ 152
5 Mm 16 2 5.2$\times$10$^{-11}$ 80 1.0$\times$10$^{-8}$ 236
\[1ex\]
------------------------------ ---------- -------- ------- ----------------- ----------------- ---------------------------- ------------------- ------------------------- ---------------------- --
Columns from left to right are the inversion set number, the specific simulation, the type of filtering, target inversion depth, the horizontal target width, the vertical target size, the noise, cross-talk, and weight spread trade-off parameters, respectively, and the inversion noise level.
--------------- ------- ------- ------- ------- ------- -------------- ------ ----------- ----------- ----------- ------------------ ----------------
sim fiter depth $v_x$ $v_y$ $v_z$ $\rm{mag_h}$ div s$_{v_x}$ s$_{v_y}$ s$_{v_z}$ s$_{\rm{mag_h}}$ s$_{\rm{div}}$
\[0.5ex\] QS1 ridge 1 0.87 0.74 -0.37 0.57 0.87 0.90 0.93 -0.29 0.64 1.20
ridge 3 0.80 0.66 0.01 0.45 0.78 0.63 0.67 0.00 0.41 1.08
ridge 5 0.58 0.53 -0.03 0.04 0.57 0.40 0.59 0.00 0.03 1.04
phase 1 0.92 0.70 0.36 0.58 0.86 0.97 0.82 0.46 0.68 0.95
phase 3 0.86 0.74 0.34 0.62 0.85 0.65 0.76 0.02 0.53 0.89
phase 5 0.69 0.62 0.21 0.23 0.69 0.73 0.88 0.01 0.32 1.46
comb 1 0.89 0.70 0.28 0.56 0.85 1.05 0.95 0.33 0.77 1.14
comb 3 0.85 0.72 0.25 0.57 0.83 0.57 0.67 0.04 0.44 0.80
comb 5 0.70 0.59 0.12 0.21 0.65 0.46 0.55 0.01 0.17 0.80
QS2 ridge 1 0.86 0.90 0.50 0.63 0.92 1.07 1.21 0.65 0.85 1.39
ridge 3 0.82 0.75 0.39 0.63 0.73 1.31 1.27 0.01 1.09 1.78
ridge 5 0.45 0.50 -0.02 0.26 0.33 0.99 0.98 0.00 0.51 1.30
phase 1 0.91 0.89 0.81 0.68 0.93 1.57 1.55 0.67 1.27 1.84
phase 3 0.86 0.84 0.53 0.65 0.88 0.92 0.97 0.04 0.77 1.27
phase 5 0.71 0.78 -0.11 0.54 0.81 0.97 1.09 -0.01 0.76 1.71
comb 1 0.89 0.90 0.79 0.65 0.92 1.25 1.20 0.58 0.92 1.30
comb 3 0.85 0.77 0.49 0.63 0.82 0.95 0.90 0.06 0.75 1.30
comb 5 0.65 0.71 0.04 0.43 0.74 0.83 0.83 0.00 0.53 1.53
\[1ex\]
--------------- ------- ------- ------- ------- ------- -------------- ------ ----------- ----------- ----------- ------------------ ----------------
Columns 4-6 are the correlation coefficients for the given flow component. ‘${\rm mag_h}$’ denotes $\sqrt{v_x^2+v_y^2}\equiv |\bm{v}_h|$, and ‘div’ the horizontal divergence $\nabla_{\rm h}\cdot\bm{v}_{\rm h}$. The last 5 columns give the slope of the best-fit lines through the correlation plots for the same quantities.
The authors gratefully acknowledge support by the NASA SDO Science Center through contract NNH09CE41C awarded to NWRA and very fruitful discussions with Doug Braun. K.D. and J.J. also acknowledge funding from a NASA EPSCoR award to NMSU under contract NNX09AP76A. M. Rempel is partially supported through NASA contracts NNH09AK02I, NNH12CF68C and NASA grant NNX12AB35G. The National Center for Atmospheric Research is sponsored by the National Science Foundation. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center under project s0925.
[^1]: Pleave visit <http://www2.hao.ucar.edu/>, then “Observations/Data” and then “Numerical Sunspot Models.”
|
---
abstract: 'In this work, we revisited the ZGB model in order to study the behavior of its phase diagram when two well-known random networks play the role of the catalytic surfaces: the Random Geometric Graph and the Erdös-Rényi network. The connectivity and, therefore, the average number of neighbors of the nodes of these networks can vary according to their control parameters, the neighborhood radius $\alpha$ and the linking probability $p$, respectively. In addition, the catalytic reactions of the ZGB model is governed by the parameter $y$, the adsorption rate of carbon monoxide molecules on the catalytic surface. So, to study the phase diagrams of the model on both random networks, we carried out extensive steady-state Monte Carlo simulations in the space parameters ($y,\alpha$) and ($y,p$) and showed that the continuous phase transition is greatly affected by the number of neighbors per node while the discontinuous one remains present in the diagram throughout the interval of study.'
author:
- 'E. B. Vilela'
- 'H. A. Fernandes'
- 'F. L. P. Costa'
- 'P. F. Gomes'
title: Phase diagrams of the ZGB model on random networks
---
Introduction
============
Phase transitions and critical phenomena of equilibrium systems have, for a long time, been studied mainly through Monte Carlo simulations, making it one of the most important methods in Statistical Mechanics. More recently, nonequilibrium systems have also attracted a lot of attention, since it was discovered that they can exhibit similar behavior to those found in equilibrium systems. One of those systems which has been extensively studied and has become a prototype in nonequilibrium Monte Carlo simulations is the surface reaction model known as ZGB model. In their pioneering work, R.M. Ziff, E. Gulari, and Y. Barshad [@ziff1986] proposed a model that describes some nonequilibrium aspects of the production of carbon dioxide ($\textrm{CO}_2$) molecules through the reaction of carbon monoxide ($\textrm{CO}$) molecules with oxygen (O) atoms, both adsorbed on a catalytic surface represented by square lattices. Despite its simplicity, this model presents a rich phase diagram with two irreversible phase transitions, one continuous and another discontinuous, separating the reactive phase from two absorbing phases [@ziff1986; @meakin1987; @fisher1989]. In addition, this model had shown to be capable of reproducing some aspects of transition-metal catalysts being, therefore, of interest for possible technological applications [@bond1987; @zhdanov1994; @marro1999]. For instance, discontinuous phase transitions are observed in some experimental works on platinum [@golchet1978; @matsushima1979; @ehsasi1989; @christmann1991; @block1993; @berdau1999] even though there are no experimental evidences of the continuous phase transition presented in the ZGB model. In fact, reaction processes in catalytic surfaces have also gained much attention due to their importance on the petrochemical and automotive industries [@Hagen2006].
Since its discovery, several modified versions of the original model have been proposed. Similarly, some works have considered the desorption and/or diffusion [@fischer1989; @dumont1990; @albano1992; @brosilow1992; @tome1993; @kaukonen1989; @jensen1990; @buendia2009; @grandi2002; @buendia2013; @chan2015; @buendia2015; @dasilva2018; @fernandes2018] of CO molecules. Other works included the presence of impurities on the surface [@hoenicke2000; @buendia2012; @buendia2013; @buendia2015; @hoenicke2014; @fernandes2019] or took into account the attractive and/or repulsive interactions between the adsorbed molecules [@buendia2015; @satulovsky1992]. Notably, most numerical works on the ZGB model has considered regular square lattices (SL) as the catalytic surfaces where the reactions occur. On the other hand, only few works took into account different network structures, such as hexagonal [@meakin1987; @provata2005; @noussiou2007; @provata2007], fractal [@albano1990; @mai1992; @albano1994; @gao1999a; @gao1999b] and Voronoi-Delaunay random lattices [@Oliveira2016]. Khan and Yaldram [@khan2000] have also considered a two-layer structure built through the interaction among the sites of two square lattices. A simple way to create a flexibility on the pattern of the sites communication is to arrange them in a more sophisticated manner and a natural suggestion is a simple random network. Unlike regular networks, e.g. square or hexagonal lattices, in which there exists only a single node degree (each site, or node, has a fix number of connections or neighbors), the random networks present a tractable degree distribution thus creating a whole new set of topological properties.
In this work, we have revisited the ZGB model in order to study the behavior of its phase transitions through extensive equilibrium Monte Carlo simulations on two different and well-known random networks: the Random Geometric Graph (RGG) and the Erdös-Rényi network (ERN). These two networks are used as prototype for communication patterns and the connections between their sites follow precise rules defined by their control parameters.
This paper is organized as follows: in Sec. \[model\] we describe the ZGB model and its phase diagram, as well as the two random networks considered in this study. In Sec. \[results\] we show our main results and present the discussions about our findings. Finally, the conclusions are presented in Sec. \[sec:conclusions\].
The model {#model}
=========
The Ziff-Gulari-Barshad model
-----------------------------
The Ziff-Gulari-Barshad (ZGB) model [@ziff1986] is a catalytic surface model that mimics the production of carbon dioxide molecules (CO$_2$) when carbon monoxide (CO) and oxygen (O$_2$) molecules, both in the gas phase, impinge the surface with rates $y$ and $1-y$, respectively. In the adsorption process, the O$_2$ molecules dissociate into two oxygen (O) atoms and, therefore, the sites of the surface can be filled with O atoms, CO molecules, or be vacant ($V$). Once adsorbed on the surface, the CO molecules and O atoms can react producing CO$_2$ molecules. The set of reactions follows the Langmuir-Hinshelwood mechanism [@evans1991] and can be represented by the following reaction equations: $$\begin{aligned}
\textrm{CO}(g)+V &\longrightarrow \textrm{CO}(a), \label{reaction1} \\
\textrm{O}_2(g)+2V &\longrightarrow 2\textrm{O}(a), \label{reaction2} \\
\textrm{CO}(a)+\textrm{O}(a) &\longrightarrow \textrm{CO}_2(g)+2V. \label{eq:reaction}\end{aligned}$$ Eq. \[reaction1\] is related to the CO molecule adsorption process. The molecule in the gas ($g$) phase impinges the surface with a rate $y$ being adsorbed ($a$) on it only if the randomly chosen site is vacant. On the other hand, as stated by Eq. \[reaction2\], if the O$_2$ molecule is chosen (with a rate $1-y$), it dissociates into two O atoms and both are adsorbed on the surface only if the two neighbor sites, also chosen at random, are empty. If any of the adsorption sites is occupied, the adsorption processes do not occur and the O$_2$ molecule returns to the gas phase. The production of CO$_2$ molecules, which is represented by Eq. \[eq:reaction\], occurs whenever O atoms and CO molecules are neighbors on the surface. In this reaction, the CO$_2$ molecule desorbs from the surface leaving two vacant sites behind. As shown above, $y$ is the sole parameter that controls the kinetics of the ZGB model.
Regardless its simplicity, the model presents two irreversible phase transitions when implemented on the regular square lattice (SL). The first one is a continuous phase transition which takes place at $y_1 \simeq 0.3874$ [@voigt1997] and separates the absorbing phase where system is poisoned by O atoms ($0 \leq y < y_1$) and the active phase ($y_1 \leq y < y_2$) in which there exists the production of CO$_2$ molecules. This active phase persists until $y_2 \simeq 0.5256$ [@ziff1992] when a discontinuous phase transition occurs and the system is trapped in the CO absorbing state ($y_2 < y \leq 1$).
It is notable that this simple model is capable to describe the discontinuous phase transition experimentally verified by several authors, as stated above. In contempt of its existence in theoretical studies, there are no experimental evidences of the continuous phase transition. This last transition has also being studied by several authors with results supporting that the critical point belongs to the directed percolation (DP) universality class [@jensen1990a; @grinstein1989; @Fernandes2016].
Random Networks
---------------
In this work, we study the ZGB model in which the catalytic surface is represented by two possible random networks: the Random Geometric Graph [@penrose2003; @spodarev2013] and the Erdös-Rényi network [@erdos1959; @erdos1960; @Gilbert1959]. A network (or a graph) is a set of vertices (or nodes) connected by a list of links (or edges) [@Newman2010; @Barabasi2016]. Here, the vertices are the sites of the catalytic surfaces and the links representing the connectivity among the vertices provide the neighbors of each site. Each node and its connectivity are randomly defined on the catalytic surface according to the criterion for each considered random network. This criterion creates a distribution of all network properties such as degree (number of neighbors of each node), centrality, number of components and others. On the contrary, for the regular square and hexagonal lattices, each site has exactly four and six neighbors, respectively. Some of the questions we will try to answer in this work are the following: Are the continuous and discontinuous phase transitions present on both random networks? What is the effect of the connectivity on the properties of the ZGB model? Before answering these questions, we recapitulate the main properties of the random networks considered in this study and compare them to the square lattice (SL).
### The Random Geometric Graph (RGG)
The RGG is constructed by placing $M$ nodes (sites) at random in a square box of linear size $L$. We maintain the superficial density of sites $d = M/L^2$ constant in order to discard its effect when the size $M$ of the network is adjusted. In this manner, for a given $M$ we set $L = \sqrt{M/ d}$. Since the effective area of one site is $1/d$, we adopt the linear size $\ell = 1/\sqrt{d}$ as the spatial scale to measure all distances in our study [@Gomes2019]. The neighborhood of a given site, as well as the network properties, is defined by a single control parameter $r = \alpha \ell$ with $\alpha>0$, the neighborhood radius. Figure 1 shows a snapshot of a network with $M=256$.
![Example of a RGG with $M=256$ nodes, $d=1.0$ and $r=\alpha = 1.0$. The square box has a linear size $L = \sqrt{M} = 16$. Only a part of the area is represented in the figure in order to emphasize the neighborhood radius. A complete square is shown on the Fig. \[RGG\_comps\]. Each color represents a different component and the isolated nodes (without neighbors) are in black. The circle with radius $\alpha$ defines the neighborhood of influence of the node in its center.[]{data-label="RGG"}](fig_RGG_inicial_v7.pdf){width="3.0"}
The criterion to define the connectivity is the following: two sites are connected, i.e., they are neighbors, if and only if the euclidean distance between them is less than $\alpha$. An isolated set of nodes connected to each other defines a component [@Newman2010] as represented in Fig. \[RGG\] (each color is a different component). In other words, a component of a graph is a group of nodes such that it is possible to go from any node to any other node through the connections among them. A graph with only one component is called connected, which means there is no isolated nodes. Therefore, the spatial positions of the nodes are the criteria to define the network connectivity of the RGG.
The number $k_i$ of neighbors of a given site $i$ is called the degree and the average network degree is given by $$\bar{k}=\sum_i^M \frac{k_i}{M}. \nonumber$$ The value of $\bar{k}$ for the RGG can vary from 0, when $\alpha = 0$ and the system is totally disconnected, to $ M-1$ when $\alpha > \sqrt{2}L/2$ (half the diagonal of the square) and the system is fully connected, i.e., everyone is everyone’s neighbor. This particularity brings an important consequence: for small values of $\alpha$, the network is not connected and there exist many components. The number of components $\mathcal{N}$ depends on the radius $\alpha$ since when it increases, $\bar{k}$ also increases and, consequently, the components grow up in size decreasing the value of $\mathcal{N}$. So, $\mathcal{N}$ can vary from $M$ when $\bar{k} = 0$ meaning a totally disconnected system, to 1 when $\bar{k}=M-1$ and the network is fully connected.
Another important quantity to be considered is the size of the largest component $\mathcal{S}$, i.e., the number of sites belonging to the largest component of the network. This quantity can vary from 1 meaning that all nodes are isolated ($\bar{k}=0$) to $M$ when the network is fully connected (only one component, which is the size of the network) [@Reia2020]. Therefore, for a network with a certain set of components, the size of its largest component can bring relevant information about the connectivity of the nodes. For instance, there exists a radius threshold in which the largest component $\mathcal{S}$ percolates and the network gets connected [@dall2002]. In order to study the phase transitions of the ZGB model on this random network, the radius threshold becomes an important parameter from which the phase transitions of the system can occur.
### The Erdös Rényi network (ERN)
The Erdös Rényi network (ERN) is also constructed by placing $M$ nodes in the network at random. However, the connections are chosen according to the probability $p$, i.e., each node has a probability $p$ to be connected to another one [@Newman2010; @Barabasi2016]. Hence, there is no spatial position in the definition of the ERN. This is the main difference between the RGG and the ERN: for a given number $M$ of nodes, while the neighborhood radius $\alpha$ is the control parameter for the RGG, the probability $p$ is the control parameter for the ERN. The other quantities such as $\mathcal{N}$, $\mathcal{S}$ and $\bar{k}$ have the same meaning for the two networks. Therefore, the probability $p$ is responsible for producing a network ranging from totally disconnected ($p = 0$ and $\bar{k} = 0$) to fully connected ($p=1$ and $\bar{k} = M-1$). In the general case we have $\bar{k} = p(M-1)$. As it was shown above for the RGG, there is also a critical value of $p$ and $\bar{k}$ that makes the largest component large enough to allow the percolation of the system [@Bolobas2001; @erdos1959]: $p_c = 1/M$ which implies $\bar{k}_c=1$. If $p>p_c$ the network is percolated [@Achlioptas2009]. Indeed, it is needed at least one neighbor for each site in order to have a percolated graph.
For regular SL, the edges are the nearest-neighbors pairs of sites so that $k_i=4$ for all site $i$. Hence, the lattice is always connected (only one component): $\mathcal{N}=1$, $\mathcal{S}=M$.
Results and discussion {#results}
======================
Monte Carlo simulations and topological properties of the networks
------------------------------------------------------------------
Before presenting our main results, we make a few general considerations about the Monte Carlo simulations carried out in this work, as well as, present the main topological properties of the RGG and ERN.
The model was simulated on catalytic surfaces with $M=128^2$ sites with periodic boundary conditions. We used the density $d=1.0$ throughout this paper, which gives $\ell=1.0$ and $r=\alpha$. During the simulations, we compute the densities of CO molecules, $\rho_c$, and O atoms, $\rho_o$, both adsorbed on the catalytic surfaces, as well as the density of vacant sites, $\rho_v$, and the density of CO$_2$ molecules, $\rho_2$, released from the surface during the catalytic reaction. These densities are given by: $$\begin{aligned}
\rho_c &=& \dfrac{n_c}{M}, \qquad \qquad \rho_o = \dfrac{n_o}{M}, \nonumber \\
\rho_2 &=& \dfrac{n_2}{M}, \qquad \qquad \rho_v = \dfrac{n_v}{M}, \nonumber\end{aligned}$$ where $n_c$, $n_o$, $n_2$ and $n_v$ are, respectively, the number of sites occupied by CO molecules, O atoms, the number of CO$_2$ molecules produced in the active phase and number of empty sites.
The simulations started with all sites empty (initial condition) and we discarded the first $8.0 \times 10^3$ Monte Carlo steps (MCS) in order to reach the steady state. The results (the average values of the densities) were obtained by considering the following $1.5\times 10^4$ MCS. All the simulations were carried out using Fortran and all the graphics were created using Python.
To study the two random networks, we considered three topological properties: the average degree distribution $\bar{k}$, as well as the normalized number of components $n$ and largest component $s$ given by $$n = \dfrac{\mathcal{N}}{M}, \qquad \qquad \textrm{and} \qquad \qquad s = \dfrac{\mathcal{S}}{M}. \nonumber$$ So, for a totally disconnected network we have $\mathcal{N}=M$ and $\mathcal{S}=1$ producing $n=1$ and $s=1/M$. On the other hand, for a fully connected network $\mathcal{N}=1$ and $\mathcal{S}=M$ which gives $n=1/M$ and $s=1$.
Figures \[topological\](a) and (b) show these three quantities for the RGG and ERN, respectively. These results were obtained in the interval $0.1 \leq \alpha \leq 10$ and $10^{-5} \leq p \leq 10^{-3}$ which are centered on the percolation point. As can be seen in this figure, $\bar{k}$ has linear behavior in log scale for both networks.
![Topological properties as function of the control parameters for (a) RGG and (b) ERN. The curves show the average degree distribution $\bar{k}$ (in blue triangles), the number of components $n$ (in pink circles), and the larges component $s$ (in red squares). The dashed lines are just a guide to the eye.[]{data-label="topological"}](topological_RGG_v2.pdf "fig:"){width="3.05"} ![Topological properties as function of the control parameters for (a) RGG and (b) ERN. The curves show the average degree distribution $\bar{k}$ (in blue triangles), the number of components $n$ (in pink circles), and the larges component $s$ (in red squares). The dashed lines are just a guide to the eye.[]{data-label="topological"}](topological_ERN_v2.pdf "fig:"){width="3.0"}
Another consequence of the degree distributions is that the random networks do not percolate for all values of the control parameters $\alpha$ and $p$. The percolation begins when the system is close to be connected meaning that there are only a few components.
As shown in Fig. \[topological\](a), the percolation starts for $\alpha \gtrsim 1.0$ since $s$ goes from $\simeq 0$ to $\simeq 1.0$ suddenly around this point meaning that even a small variation of $\alpha$ can produce huge transformations on the network from disconnected to connected. The normalized number of components $n$ can also provide important information about the graph percolation even though it decreases smoothly from one to zero and. At $\alpha \simeq 1.0$ its value is $n \sim 0.1$ meaning that only 10% of the nodes are not connected to the largest component.
A similar behavior is found for the ERN (see Fig. \[topological\] (b)) although the curve of $s$ is not as abrupt as for the RGG. We found that the percolation process is centered around $p \simeq 10^{-4}$ when 70% of the nodes ($n=0.3$) are connected to the largest component. It starts at $p \sim 6 \times 10^{-5}$, which is in accordance with the predicted value $p_c = 1/M \sim 6.1 \times 10^{-5}$ [@Barabasi2016]. The average degree in this point is also in agreement with the prediction $\bar{k}_c=1$. These topological features will be important for the description of the phase diagrams, shown in the following sections.
Densities and phase transitions
-------------------------------
At first, we compare the behavior of the densities as function of $y$ for the ZGB model on the RGG and ERN with the one on the SL. The curves for the two random networks are very similar and, therefore, we decided to present only the results for RGG. In Fig. \[densidadesRGG\], we present our findings for three different values of $\alpha$.
{width="2.3"} {width="2.3"} {width="2.3"}
As shown in Fig. \[densidadesRGG\](a), for low values of $\alpha$ ($\alpha = 1.0$), the system is in a regime without production of CO$_2$ molecules for any value of $y>0$. In this case, the catalytic surface is composed by more than two thousand components, each one with a unique particle species: or O atoms or CO molecules. For $y=0$, there exist only O atoms (when the network has components with two or more nodes) and vacant sites. For low values of $y$, most components are filled with O atoms but some isolated nodes or even larger components may have CO molecules. As $y$ increases, the number of components with CO molecules also increases, as it should, and for $y \simeq 0.5$ the number of CO molecules grows rapidly while $\rho_o$ decreases. This phase diagram shows that the network is in an absorbing state for all values of $y$. However, this state is not as the two absorbing ones of the original ZGB model since there are no O- or CO-poisoned states. Instead, this absorbing phase is composed from O atoms, CO molecules and vacant sites. For this reason, we name this phase state as inactive state.
Figure \[densidadesRGG\](b) shows that the framework is completely different for $\alpha=2.0$ since the phase diagram now resembles the original one of the ZGB model: two poisoned states separated by an active phase with production of CO$_2$ molecules. However, the window of the active phase is larger than that of the standard model. The point separating the O-poisoned phase from the active state is $y \lesssim 0.24$ and the point separating the active phase from the CO-poisoned phase is $y \gtrsim 0.58$. It is worth to mention that, for this value of $\alpha$, the number of components $n$ decreases substantially (see Fig. \[topological\](a)) approaching zero meaning that the network is already connected and percolated. That figure also shows that for this value of $\alpha$ the average degree is $\bar{k}\simeq 12.6$ and $s\simeq 1$.
The phase diagram remains with three well-defined phases and the reactive window grows up until $ \alpha \lesssim 12$. From this point on, the O-poisoned state disappears for $y>0$ and the only present phase transition is the discontinuous one. Figure \[densidadesRGG\](c) shows that for $\alpha=12.0$ there is no O-poisoned state for any value of $y$ greater than zero preventing the model to have the continuous phase transition. On the other hand, the discontinuous phase transitions occurs at $y \simeq 0.65$. For this value of $\alpha$, $\bar{k} \simeq 452$, such that the most $O_2$ molecules which impinge the surface are adsorbed on it. In addition, the catalytic reaction, i.e., the CO$_2$ production, is greatly increased even for low values of $y$ by the amount of neighbors of each O atom adsorbed after the dissociation process.
The behaviour of these densities in the ERN (not shown) are completely analogous to the ones described above for RGG. The inactive state in the ERN can be observed for $p \lesssim 10^{-4}$ although a small production of CO$_2$ starts when the system percolates at $p =6.1 \times 10^{-5}$. The two poisoned states and the active phase can be observed at $p=5\times 10^{-4}$ . In this configuration the active phase range is $0.19 \lesssim y \lesssim 0.61$. The last configuration where the O poisoned states is absent occurs at $p \gtrsim 10^{-2}$, such that at $p = 10^{-1}$ we obtain an almost identical behaviour as the one presented at Fig. \[densidadesRGG\](c).
For each random network there is a control parameter ($\alpha$ for RGG and $p$ for ERN) which defines its average degree $\bar{k}$, the distribution of components ($n$ and $s$) and all other topological features. On the model side, the only control parameter is the CO adsorption rate $y$, therefore, only two parameters define the phase of the system in each case. For this reason, we are able to construct color maps showing the densities as function of ($\alpha,y$) for the RGG and ($p,y$) for the ERN. However, we also suppress the results for the ERN since they are very similar to the RGG. Figure \[mapa\_rho\] presents the density color maps for the RGG in the range $0<y<0.8$ and $0<\alpha<12$.
{width="3.1"} {width="3.1"}\
{width="3.1"} {width="3.1"}
In Fig. \[mapa\_rho\](a), we present the color map of $\rho_o$ in order to show the regions of the parameter space in which there exist O atoms. As can be seen the O-poisoned phase, which occurs whenever $\rho_o=1.0$, is indicated by the dark red color at the left side of the figure, which holds on only for low values of $\alpha$ and $y \lesssim 0.5$. For $\alpha \longrightarrow 0$ the only possible molecule to be adsorbed on the surface is the CO molecule (blue color) since the network has only isolated components. On the other hand, as stated above, for large values of $\alpha$, the chance of an O atom adsorbed on the surface to react with one CO molecule is high enough to prevent the poisoning of the network and so, $\rho_o < 1.0$. This figure also shows that for $y \gtrsim 0.65$ the density of O atoms drops to zero for any value of $\alpha$ and, as presented in Fig. \[mapa\_rho\](b), this is the region where the system reaches the CO-poisoned state defined by $\rho_c=1.0$. This state, indicated by the dark red color, can be observed for $y \simeq 0.5$ and low values of $\alpha$ as well. Figure \[mapa\_rho\](c) presents the density of vacant sites $\rho_v$. In this plot we can see that the dark blue color is related to the poisoned states in which there is no empty sites and $\rho_v=0$. On the other hand, one can notice a red colored region for $\alpha \gtrsim 0.3$ and $y$ around 0.6, which in turn indicates that each site has a large number of neighbors as well as there are more CO molecules impinging the surface than O$_2$ molecules. So, almost every O$_2$ molecule which reaches the surface is absorbed on it, interacting with CO molecules and creating two CO$_2$ molecules. Finally, in Fig. \[mapa\_rho\](d) one can see that the region of greatest production of CO$_2$ molecules is also the region of higher number of vacant sites. We can also observe the system approaching a stable regime at $\alpha \gtrsim 10$ in which there is no more continuous phase transition and the point $y$ of the discontinuous one becomes constant.
CO$_2$ production
-----------------
In order to compare the production of CO$_2$ molecules between the two random networks considered in this study with the traditional SL, we chose the parameters $\alpha$ and $p$ so that $\bar{k} \simeq 4$, thus creating networks whose sites have a comparable number of neighbors to those of the SL ($k=4$). Figure \[comparacao\_RN\_SQ\](a) shows the CO$_2$ density $\rho_2$ as function of $y$ for the three networks.
![(a) Density $\rho_2$ vs $y$ with $M=128^2$ and $d=1.0$ for the two random networks with $k \sim 4$ compared to the standard square lattice. The dashed line is just a guide to the eye. (b) Volume of total production of CO$_2$ $V_2=\int \rho_2 dy$ as function of $\bar{k}$ for the two random networks and for the square lattice. RGG: red circle. ERN: blue square. SL: yellow triangle.[]{data-label="comparacao_RN_SQ"}](fig_redes_k4_b.pdf "fig:"){width="3.0"} ![(a) Density $\rho_2$ vs $y$ with $M=128^2$ and $d=1.0$ for the two random networks with $k \sim 4$ compared to the standard square lattice. The dashed line is just a guide to the eye. (b) Volume of total production of CO$_2$ $V_2=\int \rho_2 dy$ as function of $\bar{k}$ for the two random networks and for the square lattice. RGG: red circle. ERN: blue square. SL: yellow triangle.[]{data-label="comparacao_RN_SQ"}](fig_redes_V2_b.pdf "fig:"){width="3.0"}
Although the connectivity of both random networks is almost the same, Fig. \[comparacao\_RN\_SQ\](a) shows a striking discrepancy in the behavior of $\rho_2$. While ERN produces a lot of CO$_2$ molecules for a large window of $y-$values, the active phase of RGG is restricted to a narrow window and with little production of CO$_2$. In addition, the production of CO$_2$ molecules on the ERN resembles the one on the regular SL even though the continuous phase transition is not observed on the ERN with $\bar{k}\simeq 4$. This result shows that although the connectivity is the first parameter considered when using different networks it is not the only important one to determine their properties. The effect of the topological properties of the networks can be more explicitly seen in the integrated intensity of $\rho_2$ as function of the average degree $\bar{k}$, which is proportional to the total volume of CO$_2$ produced in the range $0<y<1.0$. Figure \[comparacao\_RN\_SQ\](b) displays the greater efficiency in CO$_2$ production for the ERN when compared to the RGG at $\bar{k} < 100$ while at $\bar{k}>100$ both networks have similar productions. The SL is represented by the yellow triangle at $\bar{k}=4$. At this point, ERN is twice as more efficient than SL and the production of CO$_2$ through RGG is close to zero. This is a dramatic evidence of the importance of the network topological properties on the system.
Phase diagrams
--------------
As shown above, the ZGB model considered in this work presents one active phase with production of CO$_2$ molecules, an O-poisoned state for low values of $y$ and $\alpha$ (RGG) or $p$ (ERN) and a CO-poisoned phase for large values of $y$. These three phases are also observed for the standard ZGB model on regular SL. In addition, the random networks produce another absorbing phase for low values of $\alpha$ and $p$ (see Fig. \[densidadesRGG\](a)) with the presence of both CO molecules, O atoms and vacant sites in a large number of components, however without production of CO$_2$ molecules. Here, for simplicity, we named this absorbing phase as the inactive phase. Figures \[phase\_diagram\](a) and (b) show the phase diagram for the RGG and ERN, respectively, for $0 \leq y \leq 0.8$, $0 \leq \alpha \leq 13$, and $10^{-5} \leq p \leq 10^{-1}$.
![Phase diagrams of the ZGB model on the (a) RGG and (b) ERN. Blue region represents the active phase ($\rho_2>0$) where CO$_2$ is produced. The red region is the CO-poisoned phase ($\rho_c>0.9999$) and the green one is the O-poisoned state where $\rho_o >0.9999$). The black region is the inactive phase. The horizontal white dashed lines indicate $\alpha = 1.0$ (RGG) and $p=6.0 \times 10^{-5}$ (ERN). The steps for the RGG are $\delta y = 0.005$ and $\Delta \alpha = 0.1$. There are 74 points in the vertical axis for the phase diagram of the ERN and $\Delta y = 0.01$. The results for $y > 0.8$, $\alpha > 13$ and $p > 10^{-1}$ are not shown since there are no further relevant information on these regions.[]{data-label="phase_diagram"}](diagrama_RGG_v2.pdf "fig:"){width="3.1"} ![Phase diagrams of the ZGB model on the (a) RGG and (b) ERN. Blue region represents the active phase ($\rho_2>0$) where CO$_2$ is produced. The red region is the CO-poisoned phase ($\rho_c>0.9999$) and the green one is the O-poisoned state where $\rho_o >0.9999$). The black region is the inactive phase. The horizontal white dashed lines indicate $\alpha = 1.0$ (RGG) and $p=6.0 \times 10^{-5}$ (ERN). The steps for the RGG are $\delta y = 0.005$ and $\Delta \alpha = 0.1$. There are 74 points in the vertical axis for the phase diagram of the ERN and $\Delta y = 0.01$. The results for $y > 0.8$, $\alpha > 13$ and $p > 10^{-1}$ are not shown since there are no further relevant information on these regions.[]{data-label="phase_diagram"}](diagrama_ERN_v2.pdf "fig:"){width="3.2"}
In the green colored region on both diagrams one finds the O-poisoned state and the red one represents the CO-poisoned state. The blue colored region shows the phase where there are sustainable production of CO$_2$ molecules, i.e., where the catalytic reaction occurs. As can be seen on both phase diagrams, this active region occupies the majority of the area indicating that the CO$_2$ production is largely enhanced with the increase of $\alpha$ and $p$ (which, in turn, increases the average degree $\bar{k}$). Finally, the black colored region is the inactive region of the phase diagrams where the networks possess many components preventing the systems to percolate. For $\alpha < 1.0$ (for RGG) and $p < 5.0 \times 10^{-5}$ (for ERN), there are many components isolated from each other such that CO molecules can be trapped in one component, O atoms can be trapped in another one with two or more nodes, and there may be some vacant sites thus preventing the formation of any of the three phases present in the original model. For $\alpha=1.0$, $p=6.0 \times 10^{-5}$ (see the white dashed lines) and $0.4 < y < 0.6$, each phase diagram presents an appendix (intersection between the blue region and the white dashed line), i.e., a small region belonging to the active phase. As can be seen in Fig. \[topological\](a) and (b), these are the points where the networks start the processes of percolation even though there are many components yet. However, at these points the largest component is already large enough to start the catalytic reaction and the active phase is initiated. For increasing values of $\alpha$ and $p$ the other phases appear as well as the continuous and discontinuous phase transition points.
Another important result we obtained in this study is that with increasing $\alpha$ and $p$, the continuous transition is eliminated for $y>0$ while the discontinuous transition continues to exist and stabilizes at $y \simeq 0.65$ for both random networks. Therefore, the ZGB model on random networks is capable to reproduce, at least qualitatively, the existence of only one phase transition, the discontinuous one, as predicted by experimental works [@golchet1978; @matsushima1979; @ehsasi1989; @christmann1991; @block1993].
Visualization of the network
----------------------------
In order to better understand and make an intuitive idea of the network topological properties, crucial to understand the phase diagrams, we present in this section snapshots of the graphs for different control parameters. To visualize the RGG and ERN for different values of $\alpha$ and $p$, respectively, we plotted a figure in which each component is represented by specific color. Figure \[RGG\_comps\] shows the snapshots of the RGG for three different values of $\alpha$ and the largest component is represented by the red dots.
{width="2.3"} {width="2.3"} {width="2.3"}
In the Fig. \[RGG\_comps\](a), we show the graph for $\alpha = 0.5$, which gives $\bar{k} \sim 0.79$ and creates a large number of components $\mathcal{N}=nM\simeq 0.663 \times 128^2\simeq 1.09\times 10^4$ (see Fig \[topological\]). As the sites of different components are not connected, there can be O atoms isolated in some components, CO molecules isolated in another components, or even vacant sites, creating what we call the inactive phase. In the Fig. \[RGG\_comps\](b), we show a network with $\alpha = 1.0$ which produces $n \sim 0.1$. At this point, the largest component (in red) starts to be visible and its size allows the production of CO$_2$ molecules for some values of $y$ (the intersection between the blue region and the white dashed line on the Fig. \[phase\_diagram\](a)), although there is still no percolation. At $\alpha=2.0$ (Fig. \[RGG\_comps\](c)), the network is connected (composed of only one component): $n \sim 1/M$ and $s \sim 1.0$, and the three phases of the original model can be observed. Fig. \[ERN\_comps\] shows the snapshots of the ERN for three different values of $p$ and the largest component is also represented by the red dots. As the spatial position of the nodes has no meaning, we constructed the graphs in such a way that the isolated nodes are spread on the outermost part of the figure, the connected ones stay on the innermost part. The larger the degree closer to the center the node stays, so that the largest component is placed at the center of the box.
Although the ERN and RGG are conceived in a very different way, the analysis of both networks are similar. In the Fig. \[componentes\_ERN\_left\], we plot a snapshot with $p=5\times 10^{-5}$ which produces $ n \sim 0.59$ such that most of the sites are isolated preventing the formation of active and poisoned phases (black region of Fig. \[phase\_diagram\](b)). This behaviour is the expected one as the critical value for percolation is $p_c=1/M \sim 6.1 \times 10^{-5}$ as described above. For $p=10^{-4}$ (Fig. \[componentes\_ERN\_middle\_v2\]), the largest component grows up considerably ($s \sim 0.661$) as the system just started the percolation, enabling the appearance of the active phase and of a small appendix at the inactive phase of the Fig. \[phase\_diagram\]. The graph for $p= 5 \times 10^{-4}$ presented in Fig. \[componentes\_ERN\_right\_v2\] shows that the percolated network is almost connected (a few sites are still isolated) and the active and poisoned phases appear.
Conclusions {#sec:conclusions}
===========
In summary, we have performed steady-state Monte Carlo simulations in order to investigate the behavior of the phase transitions of the ZGB model when two different kinds of random networks are used as the surfaces where the catalytic reactions of the model occur. For these networks, the number of neighbors of a given site can be controlled leading to important changes in the phase diagram. For low values of $\bar{k}$ the system is not percolated and all three original phases of the ZGB model on the SL are absent. After the network percolates the system enters in the regime where all ZGB original ZGB three phases are present. A remarkable feature is the absence of the O-poisoning phase in both networks for large average degrees. This finding entails the absence of the continuous transition in agreement with experimental works.
Our work sets a new precedent by showing that the communication pattern among the sites on the catalytic surface plays an important role to the point of theoretically reproduce only the phase transition observed experimentally. Furthermore, the crucial properties of the network go far beyond the simple number of neighbors, as evidenced by the calculation of the volume of CO$_2$ molecules in the three evaluated networks. Models like ZGB and others, that have been traditionally implemented in the square lattice, can have their understanding improved when others networks are used. The interdisciplinary association of a catalytic surface model with the Network Science greatly enriches the mathematical tools used in the scope of Statistical Mechanics.
P.F. Gomes would like to thank S. M. Reia and J. F. Fontanari for helpful discussions. This work was partially supported by the Brazilian Agencies CNPq (H.A. Fernandes under the grant 408163/2018-6) and FAPEG (P.F. Gomes under the grant 17833 - 03/2015).
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---
abstract: 'We formulate an affine invariant implementation of the algorithm in [@Nest83]. We show that the complexity bound is then proportional to an affine invariant regularity constant defined with respect to the Minkowski gauge of the feasible set.'
address:
- 'CMAP, UMR CNRS 7641, Ecole Polytechnique, Palaiseau, France.'
- 'CMAP, UMR CNRS 7641, Ecole Polytechnique, Palaiseau, France.'
author:
- 'Alexandre d’Aspremont'
- Martin Jaggi
title: An Optimal Affine Invariant Smooth Minimization Algorithm
---
Introduction {#s:intro}
============
In this short note, we show how to implement the smooth minimization algorithm described in [@Nest83; @Nest03] so that both its iterations and its complexity bound are invariant by a change of coordinates in the problem. We focus on the minimization problem \[eq:min-pb\] & f(x)\
& x Q,\
where $f$ is a convex function with Lipschitz continuous gradient and $Q$ is a compact convex set. Without too much loss of generality, we will assume that the interior of $Q$ is nonempty and contains zero. When $Q$ is sufficiently simple, in a sense that will be made precise later, @Nest83 showed that this problem could be solved with a complexity of $O(1/\sqrt{\epsilon})$, where $\epsilon$ is the precision target. Furthermore, it can be shown that this complexity bound is optimal for the class of smooth problems [@Nest03a].
While the dependence in $O(1/\sqrt{\epsilon})$ of the complexity bound in @Nest83 is optimal, the constant in front of that bound still depends on a few parameters which vary with implementation: the choice of norm and prox regularization function. This means in particular that, everything else being equal, this bound is not invariant with respect to an affine change of coordinates, so the complexity bound varies while the intrinsic complexity of problem remains unchanged. Here, we show one possible fix for this inconsistency, by choosing a norm and a prox term for the algorithm in [@Nest83; @Nest03] which make its iterations and complexity invariant by a change of coordinates.
Smooth Optimization Algorithm {#s:algo}
=============================
We first recall the basic structure of the algorithm in [@Nest83]. While many variants of this method have been derived, we use the formulation in [@Nest03]. We [*choose a norm $\|\cdot\|$*]{} and assume that the function $f$ in problem is convex with Lipschitz continuous gradient, so \[eq:lip-ineq\] f(y) f(x) +f(x),y-x+Ly-x\^2, x,yQ, for some $L>0$. We also [*choose a [prox]{} function*]{} $d(x)$ for the set $Q$, i.e. a continuous, strongly convex function on $Q$ with parameter $\sigma$ (see [@Nest03a] or [@Hiri96] for a discussion of regularization techniques using strongly convex functions). We let $x_0$ be the center of $Q$ for the prox-function $d(x)$ so that $$x_0\triangleq\argmin_{x\in Q}d(x),$$ assuming w.l.o.g. that $d(x_0)=0$, we then get in particular \[eq:d-strong-convex\] d(x)x-x\_0\^2. We write $T_Q(x)$ a solution to the following subproblem \[eq:tq\] T\_Q(x) \_[yQ]{}{f(x),y-x +Ly-x\^2} We let $y_0\triangleq T_Q(x_0)$ where $x_0$ is defined above. We recursively define three sequences of points: the current iterate $x_k$, the corresponding $y_k=T_Q(x_k)$, and the points \[eq:zk-def\] z\_k \_[x Q]{}{d(x)+\_[i=0]{}\^k\_i\[ f(x\_i)+f(x\_i),x-x\_i\]} and a step size sequence $\alpha_k\geq 0$ with $\alpha_0\in(0,1]$ so that \[eq:xyz-update\] x\_[k+1]{}=\_k z\_k+(1-\_k)y\_k\
y\_[k+1]{}=T\_Q(x\_[k+1]{})\
where $\tau_k=\alpha_{k+1}/A_{k+1}$ with $A_k=\sum_{i=0}^k\alpha_i$. We implicitly assume here that $Q$ is simple enough so that the two subproblems defining $y_k$ and $z_k$ can be solved very efficiently. We have the following convergence result.
\[th:conv\] Suppose $\alpha_k=(k+1)/2$ with the iterates $x_k$, $y_k$ and $z_k$ defined in (\[eq:zk-def\]) and (\[eq:xyz-update\]), then for any $k\geq 0$ we have $$f(y_k)-f(x^{\star})\leq \frac{4Ld(x^{\star})}{\sigma(k+1)^2 }$$ where $x^{\star}$ is an optimal solution to problem (\[eq:min-pb\]).
See [@Nest03].
If $\epsilon>0$ is the target precision, Theorem \[th:conv\] ensures that Algorithm \[alg:smooth\] will converge to an $\epsilon$-accurate solution in no more than \[eq:maxiter\] iterations. In practice of course, $d(x^\star)$ needs to be bounded a priori and $L$ and $\sigma$ are often hard to evaluate.
\[1\] $x_0$, the prox center of the set $Q$. Compute $\nabla f(x_k)$. Compute $y_k=T_Q(x_k)$. Compute $z_k = \argmin_{x \in Q}\left\{\frac{L}{\sigma}d(x)+\sum_{i=0}^k\alpha_i[ f(x_i)+\langle \nabla f(x_i),x-x_i\rangle]\right\}$. Set $x_{k+1}=\tau_k z_k + (1-\tau_k)y_k$. $x_N,y_N \in Q$.
While most of the parameters in Algorithm \[alg:smooth\] are set explicitly, the norm $\|\cdot\|$ and the prox function $d(x)$ are chosen arbitrarily. In what follows, we will see that a natural choice for both makes the algorithm affine invariant.
Affine invariant implementation {#s:aff}
===============================
We can define an affine change of coordinates $x=Ay$ where $A\in\reals^{n \times n}$ is a nonsingular matrix, for which the original optimization problem in is transformed so \[eq:aff-pb\] & f(x)\
& x Q,\
& f(y)\
& y Q,\
in the variable $y\in\reals^n$, where \[eq:aff-chg\] f(y) f(Ay) Q A\^[-1]{}Q. Unless $A$ is pathologically ill-conditioned, both problems are equivalent and should have invariant complexity bounds and iterations. In fact, the complexity analysis of Newton’s method based on the self-concordance argument developed in [@Nest94] produces affine invariant complexity bounds and the iterates themselves are invariant. Here we will show how to choose the norm $\|\cdot\|$ and the prox function $d(x)$ to get a similar behavior for Algorithm \[alg:smooth\].
Choosing the norm {#ss:norm}
-----------------
We start by a few classical results and definitions. Recall that the [*Minkowski gauge*]{} of a set $Q$ is defined as follows.
\[def:mink\] Let $Q\subset \reals^n$ containing zero, we define the Minkowski gauge of $Q$ as $$\gamma_Q(x) \triangleq \inf\{ \lambda \geq 0 : x \in \lambda Q\}$$ with $\gamma_Q(x)=0$ when $Q$ is unbounded in the direction $x$.
When $Q$ is a compact convex, centrally symmetric set with respect to the origin and has nonempty interior, the Minkowski gauge defines a [*norm*]{}. We write this norm $\|\cdot\|_Q\triangleq \gamma_Q(\cdot)$. From now on, we will assume that the set $Q$ is centrally symmetric or use for example $\bar Q=Q - Q$ (in the Minkowski sense) for the gauge when it is not (this can be improved and extending these results to the nonsymmetric case is a classical topic in functional analysis). Note that any affine transform of a centrally symmetric convex set remains centrally symmetric. The following simple result shows why $\|\cdot\|_Q$ is potentially a good choice of norm for Algorithm \[alg:smooth\].
\[lem-lip-aff\] Suppose $f: \reals^n \rightarrow \reals$, $Q$ is a centrally symmetric convex set with nonempty interior and let $A\in\reals^{n \times n}$ be a nonsingular matrix. Then $f$ has Lipschitz continuous gradient with respect to the norm $\|\cdot\|_Q$ with constant $L>0$, i.e. $$f(y) \leq f(x) +\langle \nabla f(x),y-x\rangle +\frac{1}{2}L\|y-x\|_Q^2, \quad x,y\in Q,$$ if and only if the function $f(Aw)$ has Lipschitz continuous gradient with respect to the norm $\|\cdot\|_{A^{-1}Q}$ with the same constant $L$.
Let $y=Az$ and $x=Aw$, then $$f(y) \leq f(x) +\langle \nabla f(x),y-x\rangle +\frac{1}{2}L\|y-x\|_Q^2, \quad x,y\in Q,$$ is equivalent to $$f(Az) \leq f(Aw) +\left\langle A^{-T} \nabla_{\!w} f(Aw), Az-Aw\right\rangle +\frac{1}{2}L\|Az-Aw\|_Q^2, \quad z,w\in A^{-1}Q,$$ and, using the fact that $\|Ax\|_Q=\|x\|_{A^{-1}Q}$, this is also $$f(Az) \leq f(Aw) +\left\langle \nabla_{\!w} f(Aw),A^{-1}(Az-Aw)\right\rangle +\frac{1}{2}L\|z-w\|_{A^{-1}Q}^2, \quad z,w\in A^{-1}Q,$$ hence the desired result.
An almost identical argument shows the following analogous result for the property of *strong convexity* with respect to the norm $\|\cdot\|_Q$ and affine changes of coordinates.
\[lem-strong-aff\] Suppose $f: \reals^n \rightarrow \reals$, $Q$ is a centrally symmetric convex set with nonempty interior and let $A\in\reals^{n \times n}$ be a nonsingular matrix. Suppose $f$ is strongly convex with respect to the norm $\|\cdot\|_Q$ with parameter $\sigma>0$, i.e. $$f(y) \geq f(x) +\langle \nabla f(x),y-x\rangle +\frac{1}{2}\sigma\|y-x\|_Q^2, \quad x,y\in Q,$$ if and only if the function $f(Ax)$ is strongly convex with respect to the norm $\|\cdot\|_{A^{-1}Q}$ with the same parameter $\sigma$.
We now turn our attention to the choice of prox function in Algorithm \[alg:smooth\].
Choosing the prox {#ss:prox}
-----------------
Choosing the norm as $\|\cdot\|_Q$ allows us to define a norm without introducing an arbitrary geometry in the algorithm, since the norm is extracted directly from the problem definition. When $Q$ is smooth, a similar reasoning allows us to choose the prox term in Algorithm \[alg:smooth\], and we can set $d(x)=\|x\|_Q^2$. The immediate impact of this choice is that the term $d(x^*)$ in is bounded by one, by construction. This choice has other natural benefits which are highlighted below. We first recall a result showing that the conjugate of a squared norm is the squared dual norm.
\[lem:conj-sqr\] Let $\|\cdot\|$ be a norm and $\|\cdot\|^*$ its dual norm, then $$\frac{1}{2}\left(\|y\|^*\right)^2=\sup_x ~y^Tx - \frac{1}{2} \|x\|^2.$$
We recall the proof in [@Boyd03 Example3.27] as it will prove useful in what follows. By definition, $x^Ty\leq \|y\|^* \|x\|$, hence $$y^Tx - \frac{1}{2} \|x\|^2 \leq \|y\|^* \|x\| - \frac{1}{2} \|x\|^2 \leq \frac{1}{2} \left(\|y\|^*\right)^2$$ because the second term is a quadratic function of $\|x\|^2$, with maximum $\left(\|y\|^*\right)^2/2$. This maximum is attained by any $x$ such that $x^Ty= \|y\|^* \|x\|$ (there must be one by construction of the dual norm), normalized so $\|x\|=\|y\|^*$, which yields the desired result.
This last result (and its proof) shows that solving the prox mapping is equivalent to finding a vector aligned with the gradient, with respect to the Minkowski norm $\|\cdot\|_Q$. We now recall another simple result showing that the dual of the norm $\|\cdot\|_Q$ is given by $\|\cdot\|_{Q^\circ}$ where $Q^\circ$ is the polar of $Q$.
\[lem:dual-mink\] Let $Q$ be a centrally symmetric convex set with nonempty interior, then $\|\cdot\|_Q^* = \|\cdot\|_{Q^\circ}$.
We write x\_[Q\^]{} & = & { 0 : x Q\^}\
& = & { 0 : x\^Tyt, y Q}\
& = & { 0 : \_[yQ]{} x\^Tyt}\
& = & \_[yQ]{} x\^Ty\
& = & x\_Q\^\* which is the desired result.
The last remaining issue to settle is the strong convexity of the squared Minkowski norm. Fortunately, this too is a classical result in functional analysis, as a squared norm is strongly convex with respect to itself if and only if its dual norm has a smoothness modulus of power 2.
However, this does not cover the case where the norm $\|.\|_Q$ is not smooth. In that scenario, we need to pick the norm based on $Q$ but find a smooth prox function not too different from $\|.\|_Q$. This is exactly the problem studied by [@Judi08b] who define the regularity of a Banach space $(E,\|.\|_E)$ in terms of the smoothness of the best smooth approximation of the norm $\|.\|_E$. We first recall a few more definitions, and we will then show that the regularity constant defined by [@Judi08b] produces an affine invariant bound on the term $d(x^*)/\sigma$ in the complexity of the smooth algorithm in [@Nest83].
\[def:bm\] Suppose $\|\cdot\|_X$ and $\|\cdot\|_Y$ are two norms on a space $E$, the *distortion* $d(\|\cdot\|_X,\|\cdot\|_Y)$ between these two norms is equal to the smallest product $ab>0$ such that $$\frac{1}{b} \|x\|_Y \leq \|x\|_X \leq a \|x\|_Y$$ over all $x\in E$.
Note that $\log d(\|\cdot\|_X,\|\cdot\|_Y)$ defines a metric on the set of all symmetric convex bodies in $\reals^n$, called the [*Banach-Mazur distance*]{}. We then recall the regularity definition in [@Judi08b].
\[def:reg\] The regularity constant of a Banach space $(E,\|.\|)$ is the smallest constant $\Delta>0$ for which there exists a smooth norm $p(x)$ such that
1. $p(x)^2/2$ has a Lipschitz continuous gradient with constant $\mu$ w.r.t. the norm $p(x)$, with $1\leq\mu\leq \Delta$,
2. the norm $p(x)$ satisfies \[eq:disto-reg\] x\^2 p(x)\^2 x\^2, hence $d(p(x),\|.\|)\leq \sqrt{\Delta/\mu}$.
Note that in finite dimension, since all norms are equivalent to the Euclidean norm with distortion at most $\sqrt{\dim E}$, we know that all finite dimensional Banach spaces are at least $(\dim E)$-regular. Furthermore, the regularity constant is invariant with respect to an affine change of coordinates since both the distortion and the smoothness bounds are. We are now ready to prove the main result of this section.
\[prop:reg-bound\] Let $\epsilon>0$ be the target precision, suppose that the function $f$ has a Lipschitz continuous gradient with constant $L_Q$ with respect to the norm $\|\cdot\|_Q$ and that the space $(\reals^n,\|\cdot\|_Q^*)$ is $D_Q$-regular, then Algorithm \[alg:smooth\] will produce an $\epsilon$-solution to problem in at most \[eq:aff-bnd\] iterations. The constants $L_Q$ and $D_Q$ are affine invariant.
If $(\reals^n,\|\cdot\|_Q^*)$ is $D_Q$-regular, then by Definition \[def:reg\], there exists a norm $p(x)$ such that $p(x)^2/2$ has a Lipschitz continuous gradient with constant $\mu$ with respect to the norm $p(x)$, and [@Judi08b Prop.3.2] shows by conjugacy that the prox function $d(x)\triangleq p^*(x)^2/2$ is strongly convex with respect to the norm $p^*(x)$ with constant $1/\mu$. Now means that $$\sqrt{\frac{\mu}{D_Q}}~ \|x\|_Q \leq p^*(x) \leq \|x\|_Q, \quad \mbox{for all $x\in Q$}$$ since $\|\cdot\|^{**}=\|\cdot\|$, hence d(x+y) & & d(x) + d(x), y+ p\^\*(y)\^2\
& & d(x) + d(x), y+ y\_Q\^2\
so $d(x)$ is strongly convex with respect to $\|\cdot\|_Q$ with constant $\sigma=1/D_Q$, and using as above $$\frac{d(x^*)}{\sigma} = \frac{p^*(x^*)^2 D_Q}{2} \leq \frac{\|x^*\|_Q^2 D_Q}{2} \leq \frac{D_Q}{2}$$ by definition of $\|\cdot\|_Q$, if $x^*$ is an optimal (hence feasible) solution of problem . The bound in then follows from and its affine invariance follows directly from affine invariance of the distortion and Lemmas \[lem-lip-aff\] and \[lem-strong-aff\].
Examples
========
To illustrate our results, consider the problem of minimizing a smooth convex function over the unit simplex, written \[eq:min-simplex\] & f(x)\
& \^T x1, x0,\
in the variable $x\in\reals^n$. As discussed in [@Judi07 §3.3], choosing $\|\cdot\|_1$ as the norm and $d(x)=\log n+\sum_{i=1}^n x_i \log x_i$ as the prox function, we have $\sigma=1$ and $d(x^*)\leq \log n$, which means the complexity of solving using Algorithm \[alg:smooth\] is bounded by \[eq:l1-complex\] 8 where $L_1$ is the Lipschitz constant of $\nabla f$ with respect to the $\ell_1$ norm. This choice of norm and prox has a double advantage here. First, the prox term $d(x^*)$ grows only as $\log n$ with the dimension. Second, the $\ell_\infty$ norm being the smallest among all $\ell_p$ norms, the smoothness bound $L_1$ is also minimal among all choices of $\ell_p$ norms.
Let us now follow the construction of Section \[s:aff\]. The simplex $C=\{x\in\reals^n: \ones^Tx\leq 1, x \geq 0\}$ is not centrally symmetric, but we can symmetrize it as the $\ell_1$ ball. The Minkowski norm associated with that set is then equal to the $\ell_1$-norm, so $\|\cdot\|_Q=\|\cdot\|_1$ here. The space $(\reals^n,\|\cdot\|_\infty)$ is $2 \log n$ regular [@Judi08b Example 3.2] with the prox function chosen here as $\|\cdot\|_{(2 \log n)}^2/2$. Proposition \[prop:reg-bound\] then shows that the complexity bound we obtain using this procedure is identical to that in . A similar result holds in the matrix case.
Conclusion
==========
From a practical point of view, the results above offer guidance in the choice of a prox. function depending on the geometry of the feasible set $Q$. On the theoretical side, these results provide affine invariant descriptions of the complexity of the feasible set and of the smoothness of the objective function, written in terms of the regularity constant of the polar of the feasible set and the Lipschitz constant of $\nabla f$ with respect to the Minkowski norm. However, while we show that it is possible to formulate an affine invariant implementation of the optimal algorithm in [@Nest83], we do not yet show that this is always a good idea... In particular, given our choice of norm the constants $L_Q$ and $D_Q$ are both affine invariant, with $L_Q$ optimal by construction and our choice of prox function minimizing $D_Q$ over all smooth square norms, but this does not mean that our choice of norm (Minkowski) minimizes the product $L_Q \min\{D_Q/2,n\}$, hence that we achieve the best possible bound for the complexity of the smooth algorithm in [@Nest83].
Acknowledgments {#acknowledgments .unnumbered}
===============
AA and MJ would like to acknowledge support from the European Research Council (project SIPA). MJ also acknowledges support from the Swiss National Science Foundation (SNSF).
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\#1[[\[hep-th/\#1\]]{}]{} \#1(\#2)\#3 [Nucl. Phys. [**B\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Phys. Rept.[**\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3[Phys. Lett. [**\#1B**]{} (\#2) \#3]{} \#1(\#2)\#3[Phys. Rev. Lett. [**\#1**]{} (\#2) \#3]{} \#1(\#2)\#3[Phys. Rev. [**D\#1**]{} (\#2) \#3]{} \#1(\#2)\#3[Ann. Phys. [**\#1**]{} (\#2) \#3]{} \#1(\#2)\#3[Rev. Mod. Phys. [**\#1**]{} (\#2) \#3]{} \#1(\#2)\#3[Commun. Math. Phys. [**\#1**]{} (\#2) \#3]{} \#1(\#2)\#3[Mod. Phys. Lett. [**\#1**]{} (\#2) \#3]{} \#1(\#2)\#3[Int. J. Mod. Phys. [**A\#1**]{} (\#2) \#3]{} \#1(\#2)\#3[JHEP [**\#1**]{} (\#2) \#3]{} \#1(\#2)\#3[J. Math. Phys. [**\#1**]{} (\#2) \#3]{}
Ivan K. Kostov[^1][Member of CNRS]{}[^2][Permanent address: C.E.A. - Saclay, Service de physique th[é]{}orique, F-91191 Gif-sur-Yvette, France]{}${}^\dagger$ [*and*]{} Pierre Vanhove[^3][Address after 1 October 1998: DAMTP, Cambridge University, Cambridge CB3 9EW, UK]{}[^4][ivan.kostov, pierre.vanhove@cern.ch]{}
[*Theory Division, CERN*]{}
[*1211 Geneva 23, Switzerland*]{}
.3in
.3in
plus 2pt minus 2pt plus 2pt minus 2pt
The interpretation of the dimensional reductions of ten-dimensional supersymmetric Yang–Mills (SYM) theory as effective theories for the dynamics of $p$-dimensional extended objects (D$p$-branes) initiated a new wave of interest in these theories. It culminated in the BFSS conjecture that a system of interacting D0-branes, described by the $D=10$ SYM theory reduced to one dimension, provides, in the large $N$ limit, a constructive definition of M-theory, the hypothetical theory encompassing all known string theories and eleven-dimensional supergravity. It also led to Matrix string theory , which describes non-perturbatively type IIA string theory by $D=10$ SYM theory reduced to two dimensions. Finally, an interpretation of the completely reduced SYM theory as type IIB string theory has been advanced in . All three reduced SYM theories are closely related and, in the large $N$ limits each one contains, in a certain sense, the other two. We will refer to the SYM theory reduced to $2, 1$ and $ 0$ dimensions as the DVV, BFSS and IKKT model, correspondingly.
The most basic information about these theories, namely concerning their vacuum excitations, can be obtained by studying their partition functions. The only partition function computed at present is that of the completely reduced theory, which we will denote by $\CZ_{ \rm IKKT}$. It was studied by several groups in order to prove the existence of bound states in the BFSS model . It was conjectured by Green and Gutperle that where the numerical factor $\CF_N$ was computed later by Krauth, Nicolai and Staudacher . Recently, this conjecture was rigorously proved by Moore, Nekrasov and Shatashvili .
In this paper we present the quasi-classical calculation of the partition function of the Euclidean matrix string theory (the DVV model) compactified on a rectangular torus $\CT^2$ with periods $R$ and $T$, with Ramond-Ramond boundary conditions. Replacing the fermionic and bosonic potentials by constraints, we obtain Further we argue that this expression which, by its definition, is valid only in the limit of large area $RT$, is actually exact. Our argument is based on the quasi-classical calculation of the DVV partition function in which the constant mode $A_\s^{(0)}$ of the $U(1)$-component of the gauge field subtracted by inserting a delta-function, and which we denote by $\langle \delta(A_\s^{(0)}/\sqrt{2\pi})\rangle_{_{\rm DVV}}$. In the limit $RT\to 0$ the modified partition function coincides with the partition function of the completely reduced theory Comparing our quasi-classical result extrapolated to the $RT\to 0$ limit with the exact expression we find perfect agreement, including the numerical factor computed in . It is therefore very plausible that our quasi-classical results are valid everywhere, [*i.e.*]{}, that these models possess the property of having [*exact quasi-classics*]{}.
The excitations that contribute to the DVV partition function can be interpreted as free type IIA Green-Schwarz strings wrapping the torus, with additional abelian gauge degrees of freedom on the world-sheet. The sum over all possible wrappings reproduces, in the large $N$ limit, the integration over the string moduli.
The paper is organized as follows. In Section 2, we define the DVV partition function and make a quantitative description of the dimensional reductions that give the BFSS and IKKT partition functions. In Sections 3 and 4, we present the derivation of and . Section 5 contains discussions of our results.
The Euclidean DVV model is ten-dimensional $U(N)$ SYM theory dimensionally reduced to a two-dimensional cylinder . The field content of the theory includes the two-dimensional $U(N)$ gauge field $\{ A_a\} _{a=1}^{2}$, the bosonic Hermitian Higgs fields $\{X^I\}_{I=1}^{ 8}$, and the Majorana–Weyl fermions $\{\Psi_\alpha\}_{\alpha=1}^{16}$. In order to study the partition function, we compactify the theory on a rectangular two-torus $\CT^2$ with periods $R$ and $T$,
$$\CT^2 =\{ \vec \s = (\s, \tau) \ | \ \s \in [0, R], \tau \in [0,T]\}.$$ The DVV partition function is defined by the functional integral where the action is $D_a = \partial_a - i A_a$ is the covariant derivative, and a minimal set of pairs of fermionic and bosonic zero-modes: is subtracted from the integration measure of the matter fields. The integration measure over the gauge fields is divided as usual by the volume $ {\rm Vol} (\CG)$ of the gauge group.
After shrinking one of the periods of the torus, the theory degenerates to a SYM theory reduced to one dimension. In the limit $R\to 0$, the component $A_\s$ of the gauge field enters in the action in the same way as the eight Higgs fields, and the $O(8)$ symmetry of the action is enhanced to $O(9)$. The functional integral describes, in this limit, the partition function of a one-dimensional reduction of the SYM theory, with $A_\s$ playing the role of $X_9$. However, this is not yet the partition function of the BFSS model because the [*integration measure*]{} of the field $A_\s$ is not identical to that of the eight remaining Higgs fields. In order to obtain the same integration measure, we have to introduce in the original functional integral a delta-function of the constant mode $\s$-component of the gauge field. Let us denote by $\langle \delta(A_\s
^{(0)}/\sqrt{2\pi})\rangle _{_{\rm DVV}} $ the DVV functional integral with the inserted delta-function. In the limit $R\to 0$ it indeed reduces to the partition function $\CZ_{_{\rm BFSS}}$ of the BFSS model at finite temperature $T$, defined by the action
The IKKT matrix integral with the action depends on a single parameter, the coupling constant $g= g_{\rm YM}^2$. In refs. the integral was understood as an integral over traceless matrices. Here we use the same normalizations as in but, in order to facilitate the comparison with the previous integrals, we define the integration measures by where $dX $ and $d\Psi$ are the flat measures with normalization The partition function of the IKKT model, with the above normalization of the measure, is equal to where $\CF_N$ is the “group factor"
The $T\to 0$ limit of the BFSS model has been analysed by Sethi and Stern in connection with the computation of the Witten index. The BFSS action reduces, in the limit $T\to 0$ and after identifying the constant mode of the gauge field with the Higgs field $X_{10}$, to the $O(10)$-symmetric IKKT action multiplied by the area $RT$ of the torus. The limit for the measure is less trivial. Let us first consider the measure of the gauge field.
The one-dimensional gauge field $A$ corresponds, by the exponential map, to a generic element of the local gauge group $U (\tau)= \hat T
e^{i\int_0^\tau A(t) dt}$. It is therefore convenient to parametrize the constant mode $A$ of the gauge field in terms of the group element $U =
e^{iTA} $ and integrate over the Haar measure on $SU(N)$ (normalized as $\int_{SU(N)} dU =1$). In the vicinity of any of the $N$ central elements of $SU(N)$ the group element can be parametrized by $U= e^{2\pi i k/N}
e^{iTA} $ and the Haar measure becomes where $\CF_N$ is given by and the measure $[dA]$ is identical to the measure $[dX]$ defined in . (The details of the calculation can be found in .) As was explained by Sethi and Stern , in the limit $T\to 0$ the integral over the gauge field is saturated by the vicinity of the $N$ central elements of $SU(N)$ and therefore by eq. For the measures of the matter fields we find, from the kinetic part of the action and the normalization , Taking into account the rescaling of the zero-modes and combining all factors of $T$ and $R$, we finally obtain
As argued by Dijkgraaf, Verlinde and Verlinde , in the infrared limit $ RT\to\infty$ the non-diagonal components of the gauge and matter fields become infinitely massive, and the bosonic and fermionic potentials turn into constraints. Under the constraint that all matrices $\Phi = \{ X_I, \Psi_\a, iD_a= i\p_a +A_a\} $ are simultaneously diagonalizable, for each field configuration there exists a unitary matrix $V(\s,\tau)$ such that $$\Phi(\s, \tau) = V^{-1} (\s, \tau ) \Phi^D(\s, \tau )V(\s, \tau ),$$ where $\Phi^D = {\rm diag}\{\Phi_1,...,\Phi_N\}$. We have therefore where $\hat S= V(0, \tau) V^{-1}(R, \tau )$ and $\hat T= V(\s, 0)V^{-1} (\s,
T )$. By construction, $\hat S\hat T = \hat T \hat S $. Assuming that all the eigenvalues are distinct, the only unitary transformations relating two diagonal matrices represent permutations of their diagonal elements. Therefore the matrices $\hat S$ and $\hat T$ act as two commuting permutations $\hat s:
i\to s_i $ and $ \hat t: i \to t_i $ of the symmetric group $S_N$
and, in particular, do not depend on the coordinates $\s$ and $\tau$.
Each pair of permutations describes an $N$-covering of the target-space torus consisting, in general, of several connected components. Each connected component can be interpreted as the world-sheet of a string wrapping several times the torus in both directions. A $q$-component covering corresponds to a decomposition $$\hat s = \prod_{k=1}^q \hat s^{(k)},\ \ \hat t = \prod_{k=1}^q \hat
t^{(k)},$$ where all factors commute with each other and satisfy The $k$-th world-sheet torus is the complex plane $\omega =\s + i\tau $ factored by the periods $\omega_1 = n_k R +i \ (l_k T) $ and $ \omega_2 =
j_k R +i\ (m_k T)$; it covers the target torus $N_k = n_km_k - l_kj_k$ times. In this way the orbifold structure of the target space generates the sum over all twisted boundary conditions, which becomes, in the limit $N\to\infty$ and for the “long" strings only, the integral over all complex structure of the world-sheet tori. Owing to the periodicity condition on the fermions, each connected component has 16 fermionic zero modes. In our problem we are only interested in contributions with exactly 16 fermionic zero-modes; therefore only one-sheet $(q=1)$ coverings will be relevant. A covering with periods $\omega_1 = n R +i (l T) $ and $\omega_2 =j R+i (m T)$ wrapping the target torus $N = n m -jl$ times is defined by two permutations, $\hat s$ and $\hat t$, satisfying $$\hat s \hat
t=\hat t \hat s, \ \ \hat s^n \hat t^l=\hat s^j \hat t^m=1.$$ By a mapping class transformation we can reduce this to The explicit solution of is, up to an internal homomorphism, After having identified the distinct topological sectors, we can write the partition function as a sum over all pairs $m$ and $n$ such that $mn=N$ and $j=0,1,..., n-1$. The diagonal matrix variables $\Phi^D(\s,\tau)$, with boundary conditions belonging to the equivalence classes $[m, n; j]$, can be considered as scalar variables defined on the torus with periods $\omega_1= nR$ and $\omega_2
= jR +i (mT)$ in the $\omega = \s + i \tau$ plane.
Now let us proceed to the evaluation of the partition function in the infrared limit. In the limit $RT\to\infty$ the measure over the gauge field reduces to the integral over the diagonal components of the gauge field, normalized by the volume of the diagonal gauge group $ \CG^D$ which is the localized $U(1)^{N}$. Taking into account the overall factor $1/N!$ because the eigenvalues are determined up to a permutation, we get where the boundary conditions in each term are twisted according to eq. . The number of permutations in each class $[m,n;j]$ is equal to the number of combinations of $m$ and $n$ elements times the number of cyclic permutations of order $m$ and $n$, $${N!\over n! m!} (m-1)! (n-1)! = (N-1)! .$$ Summing the contributions of all equivalence classes we get where $\CZ_{[m, n; j]}$ is the partition function of the Abelian $(N=1)$ $DVV$ model defined on the torus of area $NRT $ with periods $ \omega_1 =nR$ and $\omega_2 =jR + i( mT)$. The latter is a product of the partition function of the Abelian gauge field and that associated with the diagonal components of the matter field The contribution of the $U(1)$ gauge field to the partition function is given, in the gauge $A^D_\tau =0$, to the functional integral with respect to the angular variable Here $A_\s$ denotes the Abelian gauge field on the “world-sheet" torus that corresponds to the $N$-component field of $A^D_\s$ on the space-time torus.
The gauge-field partition function is The result of the integration depends only on the area $RT$ and not on the modular parameter of the torus, which reflects the symmetry of the two-dimensional gauge theory with respect to area-preserving diffeomorphisms. Further, with our conventions for the zero modes, the integral of the matter fields is exactly 1 because of supersymmetry This gives the result which can be written, after a Poisson resummation, as
The evaluation of the modified DVV can be done in the same way as in the previous section. In the topological sector $[m,n;j]$ the constant mode $A_\s^{(0)}$ is expressed through the angular variable as The combinatorics is the same as in the previous section and the only difference is in the expression of the gauge-field partition function in each topological sector. The latter is given by the one-dimensional functional integral with respect to the field $\theta(\tau)$ taking its values in the unit circle and is evaluated as
Summing over all topological sectors we find, instead of , Comparing this expression, whose derivation implies the infrared limit $RT\to \infty$, with Eq. , we see that it is equally true in the ultraviolet limit $RT\to 0$.
We have computed the quasi-classical partition function of the Euclidean Matrix string theory compactified on a two-dimensional torus $\CT^2$ , with doubly periodic boundary conditions and a minimal set of zero-modes removed. We have shown that the relevant degrees of freedom are described, in the limit of large area of the torus, by an Abelian supersymmetric sigma-model accompanied by a $U(1)$ gauge theory defined on the orbifold space $S^N \CT^2 =
(\CT^2)^{\otimes N}/S_N$, where $S_N$ is the symmetric group of $N$ elements. The sigma-model and the gauge field are coupled through the boundary conditions, which are described by a pair of commuting permutations of $S_N$.
Each such configuration can be interpreted as a set of non-interacting strings in a light-cone gauge, with additional gauge degrees of freedom on the world-sheet. The world-sheet of each of such string defines a multiple covering of the space-time torus $\CT^2$, characterized by a modular parameter $\omega_2/\omega_1$, with $\omega_{1,2} $ sweeping the lattice $R\ZZ
+i T\ZZ$. In the large-$N$ limit, the sum over coverings converges to an integral with a correct modular-invariant measure, under the condition that we are far from the boundary of the moduli space. This means that the partition function of a “long" string in the DVV model is indeed that of the Green-Schwarz string in light-cone gauge, with additional Abelian gauge degree of freedom on the world-sheet. To complete the proof of the (perturbative) matrix-string correspondence, one has to consider the matrix-field configurations that define branched coverings of the space-time torus which, according to the original suggestion in , should describe strings in interaction. Some steps in this direction was made by the authors .
On the other hand, near the boundary of the moduli space, that is when one of the periods is kept finite while $N\to\infty$, the sum keeps its discrete character and the corresponding excitations (“short" strings) describe particles. Thus the distinction between particle and string excitations in the DVV model can be made only after performing the large-$N$ limit. The latter is not yet completely understood, in spite of the recent progress made in . With the convention that a minimal set of zero-modes is deleted from the functional measure, which is the case considered in the present paper, the allowed field configurations of the orbifold theory are described by a single string whose world-sheet wraps $N$ times the space-time torus $\CT^2$. Our computation gives an intuitive understanding of the sum over the divisors of $N$ in the expression for the partition function of the IKKT model . It is quite analogous to the argument presented by Green and Gutperle in which this last partition function was compared with the partition function of the matrix quantum mechanics at infinite temperature. Their calculation was based on the assumption of the existence of bound states of D0-branes, which lead to the sum over the divisors of $N$. In our case the sum over the divisors comes from the sum over the conjugacy classes of permutations defining the different topological sectors of the $S_N$-orbifold theory. This is a trivial example of the Hecke operator, defined by its action on a modular-invariant form $\CA$ (actually a constant in our case) which takes care of the inequivalent $N$-fold wrappings. (For more general discussion see .) Comparing our calculation with the argument of Green and Gutperle, we see that the bound states of D0-branes can indeed be interpreted as strings winding several times around the spatial ($\s$-)dimension of the two-torus.
The partition functions of the DVV and the IKKT models are related to the half-BPS saturated amplitudes, such as the $t_8t_8\CR^4$ term in type IIB theory . Our computation confirms the rules for counting wrapped D-branes used in . Moreover, we have shown that the high-temperature limit of the DVV model reproduces the D-instantons contributions predicted in , which confirms the dual interpretation of the DVV model as describing wrapped D1-branes.
We consider as the principal result of this paper not the computation of the IR partition function itself, which is rather trivial, but its comparison with the known result for the partition function of the completely reduced gauge theory (the IKKT model). For this purpose we have established the exact relations between the partition functions of the SYM theories reduced to 2, 1 and 0 dimensions, which we presented in Section 2. Then we were able to check that when extrapolated to the small-area limit, the quasi-classical result reproduces exactly the known expression of the IKKT partition function. Therefore we conjecture that expressions and are actually [*exact*]{}, which means, that the theory has the property of having exact quasi-classics. We expect that this can be proved by extending to the two-dimensional case the calculation of Moore, Nekrasov and Shatashvili based on Witten’s description of the two-dimensional gauge theory as cohomological field theory .
Finally, let us remark that it is possible to give a very simple explanation of the pre-exponential factors in expression if the partition function of the DVV model is considered as a certain limit of the partition function of a supersymmetric Schild string defined on the orbifold $S^N\CT^2$. We intend to report on this subject in the near future.
[**Acknowledgements**]{}
We thank Costas Bachas, Volodya Kazakov, Matthias Staudacher and Tom Wynter for many useful discussions. Partial support was received by I.K. from the European contract TMR ERBFMRXCT960012. P.V. was supported by a short-term visiting fellowship from CERN.
[^1]: $^\ast$
[^2]: $
^\diamond $
[^3]: $^\bullet$
[^4]: $
^\dagger$
|
---
abstract: |
We present here near-infrared spectroscopy in the H and K bands of a selection of nearly 80 stars that belong to various AGB types, namely S type, M type and SR type. This sample also includes 16 Post-AGB (PAGB) stars. [**From these spectra, we seek correlations between the equivalent widths of some important spectral signatures and the infrared colors that are indicative of mass loss. Repeated spectroscopic observations were made on some PAGB stars to look for spectral variations. We also analyse archival SPITZER mid-infrared spectra on a few PAGB stars to identify spectral features due to PAH molecules providing confirmation of the advanced stage of their evolution. Further, we model the SEDs of the stars (compiled from archival data) and compare circumstellar dust parameters and mass loss rates in different types.**]{}
[**Our near-infrared spectra show that in the case of M and S type stars, the equivalent widths of the CO(3-0) band are moderately correlated with infrared colors, suggesting a possible relationship with mass loss processes. A few PAGB stars revealed short term variability in their spectra, indicating episodic mass loss: the cooler stars showed in CO first overtone bands and the hotter ones showed in HI Brackett lines. Our spectra on IRAS 19399+2312 suggest that it is a transition object. From the SPITZER spectra, there seems to be a dependence between the spectral type of the PAGB stars and the strength of the PAH features. Modelling of SEDs showed among the M and PAGB stars that the higher the mass loss rates, the higher the \[K-12\] colour in our sample.**]{}
author:
- |
V. Venkata Raman${^1}$[^1] and B.G. Anandarao${^1}$ [^2]\
$^1$Physical Research Laboratory (PRL), Navrangpura, Ahmedabad - 380009, India
title: 'Infrared Spectroscopic Study of a Selection of AGB and Post-AGB Stars'
---
\[firstpage\]
[Stars: AGB and Post-AGB – Techniques: spectroscopic – Stars: circumstellar matter – Stars: mass-loss – Stars: evolution – ISM: dust, extinction]{}
Introduction
============
During their asymptotic giant branch (AGB) stage, intermediate mass stars undergo substantial mass loss triggered by pulsation shocks and radiation pressure (e.g., Willson [@will00]; van Winckel [@vanw03]). Near-infrared spectroscopy is one of the recognized tools to study the mass loss process (e.g., Kleinmann & Hall [@klei86]; Volk, Kwok & Hrivnak [@volk99]; Lancon & Wood [@lanc00]; Bieging, Rieke & Rieke [@beig02]; Winters et al. [@wint03]). The near-infrared JHK bands, contain several important diagnostic lines that can be used as signatures to probe the atmospheres of these stars (Hinkle, Hall & Ridgway [@hink82]).
One of the most dramatic manifestations of the AGB stage is the high rate of mass loss that is mainly attributed to pulsational levitation of atmosphere followed by expulsion of matter by the radiation pressure (Wood [@wood79]; Bowen [@bowe88]; Gail & Sedlmayr [@gail88]; Vassiliadis & Wood [@vass93]; Habing [@habi96]; Feast [@feas00]; Winters et al. [@wint00]; Willson [@will00]; Woitke [@woit03]; Garcia-Lario [@garc06]). One of the issues that we would address is the possible signatures of pulsational mass loss in the spectra of AGB stars. We would also like to study the infrared spectral signatures that the intermediate mass stars leave while evolving beyond AGB stage to become Planetary Nebulae (e.g., Kwok [@kwok00])
In this work we report near-infrared (H and K bands) spectroscopy on a selection of about 80 AGB stars of different types, namely M type, S type, Semi-regular variables (SR) and post-AGB (PAGB) candidates (Section 2). In Section 3.1, we try to seek significant differences among these types and correlate spectral signature strengths with pulsation parameters or mass loss indicators (such as the infrared color indices) in case of M and S and SR types.
Further, we present the SPITZER archival spectra in the mid-infrared region on a few post-AGB stars in order to find out the evolutionary stage of these stars in terms of the spectral features seen in the spectra (Section 3.2). Following this, using the DUSTY code, we model the infrared spectral energy distributions of these sample stars, constructed from archival data in the visible, near and far-infrared regions to investigate basic differences in the photospheric and circumstellar parameters among the various types of AGB stars (Section 3.3). Some interesting aspects on variation of spectral lines in a few PAGB stars are discussed in Section 4. Section 5 lists important conclusions of the present work.
Star Sp. Type $\phi$ H-K K-12 CO(3-0) CO(4-1) CO(H) CO(2-0) CO(3-1) CO(K) $\dot{M}$
------------ ---------- -------- ------ ------- --------- --------- ------- --------- --------- ------- ---------------------
02234-0024 M4e 0.34 0.44 2.31 16.1 7.8 51.5 21.1 6.9 61.9 $2.6(-6)^{\dagger}$
03082+1436 M5.5e 0.33 0.45 1.99 11.3 9.2 64.2 25.1 13.3 62.3 $9.8(-7)^{\dagger}$
03489-0131 M4III ... 0.31 1.86 4.8 5.3 61.9 18.4 20.2 45.9 1.3(-6)
04352+6602 Sev 0.44 0.50 1.22 ... ... ... 31.8 29.8 101.2 $1.4(-7)^{\dagger}$
05265-0443 M7e 0.57 0.49 1.32 13.6 10.1 53.1 15.1 7.8 22.9 $4.6(-7)^{\dagger}$
05495+1547 S7.5 0.49 0.59 2.79 11.6 8.4 47.4 11.9 16.0 47.6 3.5(-6)
06468-1342 S ... 0.52 1.03 14.5 16.6 85.0 32.9 31.2 110.6 ...
06571+5524 Se 0.59 0.64 1.67 20.0 22.1 105.6 18.3 25.6 52.8 $1.7(-6)^{\dagger}$
07043+2246 S6.9e 0.36 0.18 1.17 11.6 13.9 77.0 34.3 28.8 98.8 $3.1(-7)^{\dagger}$
07092+0735 Se 0.30 0.71 3.02 16.4 15.4 72.1 20.5 25.8 81.0 2.9(-6)
07197-1451 S 0.26 0.55 2.32 11.8 8.0 65.0 22.3 9.7 43.5 4.6(-6)
07299+0825 M7e 0.57 0.36 1.90 16.2 10.5 64.1 26.8 14.1 65.3 $8.5(-7)^{\dagger}$
07462+2351 SeV 0.96 0.52 -0.90 12.3 13.8 68.8 28.1 33.9 108.0 ...
07537+3118 0.09 0.53 1.40 13.7 9.9 76.5 22.8 25.0 77.5 ...
07584-2051 S2,4e 0.19 0.44 1.00 12.6 11.0 78.4 21.2 17.9 39.1 ...
08188+1726 SeV ... 0.41 2.28 14.3 13.6 66.1 22.2 24.8 71.3 ...
08308-1748 S ... 0.43 0.72 10.1 15.9 72.6 25.7 19.1 92.9 1.7(-7)
08416-3220 ... 0.41 0.97 12.2 14.2 75.8 33.0 35.0 105.3 ...
08430-2548 ... 0.46 1.51 12.2 13.0 68.3 19.6 16.6 80.2 ...
09411-1820 Me 0.31 0.49 1.63 5.1 7.3 33.2 8.5 5.7 22.1 $7.9(-7)^{\dagger}$
12372-2623 M4 0.92 0.40 1.39 9.9 14.8 64.2 20.2 23.8 50.8 ...
17001-3651 M 0.70 0.58 2.17 9.6 8.9 57.0 11.3 8.9 35.8 $1.9(-6)^{\dagger}$
17186-2914 S ... 0.70 1.76 14.5 13.2 80.7 23.9 26.1 89.2 2.9(-6)
17490-3502 M 0.60 0.54 2.26 16.4 16.1 78.1 24.6 17.6 95.8 ...
17521-2907 Se 0.11 0.56 1.64 18.9 19.5 88.4 20.1 21.9 71.8 4.6(-6)
18508-1415 SeV 0.44 0.38 1.00 3.4 5.1 64.3 8.8 5.5 31.1 ...
19111+2555 S 0.57 0.80 4.36 19.1 17.6 66.8 13.3 11.2 24.5 $4.6(-6)^{\dagger}$
19126-0708 S.. 0.90 0.79 3.81 21.6 19.1 102.0 21.5 23.5 59.4 $2.9(-6)^{\dagger}$
19354+5005 S.. 0.24 0.52 2.28 26.6 22.8 108.0 30.2 33.8 98.2 $1.5(-6)^{\dagger}$
23041+1016 M7e 0.87 0.52 2.40 7.3 9.0 50.7 12.4 6.3 18.7 $1.5(-6)^{\dagger}$
Star Sp. Type H-K K-12 CO(3-0) CO(4-1) CO(H) CO(2-0) CO(3-1) CO(K) $\dot{M}$
------------ ---------- ------ ------- --------- --------- ------- --------- --------- ------- ---------------------
04599+1514 M0 0.49 0.96 12.8 17.3 80.8 24.2 28.6 99.4 $1.4(-7)^{\dagger}$
05036+1222 0.60 1.09 12.7 15.1 80.9 26.2 28.1 54.3 ...
05374+3153 M2Iab 0.16 1.44 13.7 12.2 77.5 21.6 26.6 88.3 $7.3(-7)^{\dagger}$
07247-1137 Sv.. 0.34 0.85 8.6 10.7 57.8 25.8 19.1 74.5 ...
07392+1419 M3II-III 0.09 -1.44 5.3 8.4 38.9 16.7 14.7 55.3 7.3(-7)
09070-2847 S 0.25 0.91 10.7 12.8 74.1 22.1 37.1 59.2 ...
09152-3023 0.47 1.21 4.0 9.4 44.2 18.1 14.4 39.1 ...
10538-1033 M... 0.29 0.74 8.1 4.5 44.6 22.9 9.6 64.9 ...
11046+6838 M... 0.40 0.39 3.7 9.5 41.1 15.0 10.4 54.7 ...
12219-2802 S 0.30 0.72 9.2 9.4 57.0 22.1 17.8 69.8 ...
12417+6121 SeV 0.41 0.95 16.7 18.0 82.4 28.2 29.5 92.2 $1.0(-7)^{\dagger}$
13421-0316 K5 0.22 0.57 ... 8.3 40.9 20.0 14.9 59.8 ...
13494-0313 M... 0.42 0.56 3.0 9.2 55.5 17.4 20.0 47.4 ...
14251-0251 S... 0.34 0.71 3.7 9.1 45.8 20.2 16.7 60.0 ...
15494-2312 S 0.38 0.62 5.1 8.1 46.1 19.8 17.2 59.3 ...
16205+5659 M... 0.18 0.78 5.4 7.1 45.3 13.0 19.1 61.6 ...
16209-2808 S 0.54 0.92 10.7 13.6 73.9 21.9 27.2 79.5 1.0(-7)
Star Sp. Type $\phi$ H-K K-12 CO(3-0) CO(4-1) CO(H) CO(2-0) CO(3-1) CO(K) $\dot{M}$
------------ ---------- -------- ------ ------ --------- --------- ------- --------- --------- ------- ---------------------
04030+2435 S4.2V ... 0.40 0.63 4.1 8.3 45.0 16.2 17.9 34.1 ...
04483+2826 CII ... 0.38 1.04 7.0 10.9 60.3 24.5 17.6 58.3 1.4(-7)
05213+0615 M4 0.41 0.39 1.37 9.3 11.1 64.8 19.3 21.4 56.9 ...
05440+1753 S0.V 0.33 0.66 1.89 22.7 19.5 99.5 33.2 30.3 88.1 1.4(-6)
06333-0520 M6 0.85 0.38 1.20 19.7 10.1 77.8 19.1 24.8 61.5 1.7(-7)
08272-0609 M6e ... 0.40 1.47 11.3 7.8 51.7 19.1 15.1 81.4 5.6(-7)
08372-0924 M5II ... 0.36 1.57 5.5 6.5 52.3 18.5 12.4 49.0 1.7(-7)
08555+1102 M5III ... 0.26 1.28 7.4 7.9 61.8 15.5 13.9 43.7 2.4(-7)
09076+3110 M6IIIaSe ... 0.31 1.20 13.5 11.6 65.9 18.5 18.8 56.7 $3.8(-7)^{\dagger}$
10436-3459 S ... 0.37 1.61 10.5 15.4 61.2 30.3 23.9 79.0 $9.5(-7)^{\dagger}$
15492+4837 M6S 0.77 0.40 1.58 12.7 13.0 63.8 19.9 24.5 44.4 $9.5(-7)^{\dagger}$
16334-3107 S4.7:.V ... 0.43 1.13 15.0 22.9 93.4 22.5 29.7 104.8 $1.0(-7)^{\dagger}$
17206-2826 M.. ... 0.42 0.82 10.3 11.4 68.1 15.2 19.6 63.8 $1.0(-7)^{\dagger}$
17390+0645 SV... ... 0.36 0.92 12.1 7.6 56.7 29.8 14.2 70.9 ...
19497+4327 ... 0.60 0.82 6.7 11.5 59.2 25.2 18.1 73.8 ...
Star Sp. Type H-K K-12 Pa$\beta$ Br$\gamma$ CO(K) 21$\mu$m $T_{d}$ $\dot{M}$
------------- ---------- ------ ------ ----------- ------------ ------- ---------- ---------- ---------------------
Z02229+6208 G9a 0.44 6.38 ... 9.2 8.4 ... 300 2.9(-5)
04296+3429 G0Ia 0.49 7.34 ... 4.1 17.7 -2.11 270 $4.5(-5)^{\dagger}$
05113+1347 G8Ia 0.25 5.98 ... 6.5 12.9 ... 350 2.1(-5)
06556+1623 Bpe 1.11 5.10 ... -52.4 ... ... 1200/150 1.0(-5)
07134+1005 F5Ia 0.10 6.45 ... 5.1 ... -1.73 280 $2.3(-5)^{\dagger}$
07284-0940 K0Ibpv 0.23 5.64 ... ... 10.9 ... 450 1.0(-5)
07430+1115 G5Ia 0.44 6.35 ... 3.9 28.9 ... 320 3.3(-5)
17423-1755 Be 1.48 5.30 ... -5.6 -23.6 ... 125 $4.9(-6)^{\dagger}$
18237-0715 Be 0.55 2.29 -25.6 -14.9 ... ... 70 5.8(-6)
19114+0002 G5Ia 0.27 4.84 ... 5.4 ... ... 120 $4.9(-5)^{\dagger}$
19157-0247 B1III 1.19 5.76 ... ... -10.8 ... 270 $1.1(-6)^{\dagger}$
19399+2312 B1e 0.11 4.80 -6.2 -15.6 ... ... ... ...
22223+4327 F9Ia 0.24 4.44 7.5 5.8 ... -1.31 125 $1.0(-5)^{\dagger}$
22272+5435 G5Ia 0.38 5.55 4.4 3.4 30.2 -0.83 250 $1.6(-5)^{\dagger}$
22327-1731 A0III 0.90 4.95 3.8 ... 28.7 ... 800 5.0(-6)
23304+6147 G2Ia 0.36 6.48 3.4 5.1 12.8 -2.53 250 $3.5(-5)^{\dagger}$
Near-Infrared Spectroscopic Observations:
=========================================
Near-infrared spectroscopic observations were made in the J(only for some of the sample PAGB stars), H and K bands on the selection of stars, using the NICMOS-3 near-infrared camera/spectrograph at the Cassegrain focus of the 1.2 m telescope at Mt. Abu, Western India, during the period 2003-07. The observations were made at a spectral resolution of $\sim$ 1000. Spectrophotometric standard stars of A0V type were used for each programme star at similar air mass to remove the telluric absorptions and for relative flux calibration (cf. Lancon & Rocca-Volmerange [@lanc92]). For sky background subtraction, two traces were taken for each star for each cycle of integration at two spatially separated positions along the slit within 30 arcsec. Typical overall integration times varied between 0.6 to 160 sec for the brightest and faintest stars in our sample (in each of the three bands). The atmospheric OH vibration-rotation lines were used to calibrate wavelength. The HI absorption lines from the standard star spectra are removed by interpolation. The resulting spectra of the standard star were used for the removal of telluric lines from the program star by ratioing. The ratioed spectra were then multiplied by a blackbody spectrum at the effective temperature corresponding to the standard star to obtain the final spectra. Typically the S/N ratios at the strongest and the weakest spectral features varied between 100 and 3 respectively. Data processing was done using standard spectroscopic tasks (e.g., APALL) inside IRAF.
The Sample Stars:
-----------------
The programme stars were selected from published literature/catalogues (Fouque et al. [@fouq92], Kerschbaum & Hron [@kers94], Chen et al. [@chen95], Wang & Chen [@wang02], and Stasinska et al. [@stas06]). Our sample contains about 80 stars with the selection based on the observability at Mt. Abu during the period November-May and JHK band magnitudes brighter than 8-9. Among these stars, 30 are M type; 17 S type; 15 SR type and 16 are PAGB stars (some of which are known to be transition/proto-PNe, see Ueta et al. [@ueta03] and Kelly & Hrivnak [@kell05]). The greater bias towards M types in the sample is due to the fact that they are brighter in K band than the others. Tables 1-4 list all the stars in the four categories along with the \[H-K\] and \[K-12\] colors computed from 2MASS and IRAS photometric archival data. The phases ($\phi$) of the variable stars of M and SR types, corresponding to the dates of our observations, are also listed in the tables (epochs taken from Kholopov et al. [@khol88]). The spectral types are taken from SIMBAD data base.
Results and Discussion:
=======================
Fig 1 shows some typical spectra for different types considered here. In general, we find in the AGB stars, the photospheric absorption lines of Na I doublet at 2.21 $\mu$m, Ca I triplet at $\sim$ 2.26 $\mu$m, as well as Mg I at 1.708 $\mu$m in all the sample stars. Brackett series lines of HI were found mostly among the PAGB stars where a hotter central source is believed to be present. In addition, the CO vibration-rotation lines of first and second overtone bands ($\Delta v = 2$ and $\Delta v = 3$ respectively) were seen in a large number of M, S and SR type stars and in a few PAGB stars. Some PAGB stars showed HI Brackett emission lines (IRAS 06556+1623, 17423-1755, 18237-0715 and 19399+2312 for the last one of which our spectra are new) and an indication of fainter emission in CO. This could be due to the fact that as the central star hots up, the CO lines become progressively weaker and finally disappear when CO dissociates (at 11.2 eV). Around this time the HI recombination lines start becoming prominent in emission. Our spectra on IRAS 06556+1623 and 17423-1755 are quite comparable to those of Garcia-Hernandez et al. [@garc02], with a faint detection of H${_2}$ S(1) 1-0 line at 2.12 $\mu$m. For the PAGB object IRAS 22327-1731, our spectra showed for the first time absorption in HI lines and CO bands. Some of the PAGB objects that showed variability are discussed in section 4. Equivalent widths (EWs) were computed for all the spectral features that are detected in our sample stars. The errors in the equivalent widths are mainly from the S/N ratio of the lines. To that extent the errors are estimated to be about 3 Å, mostly applicable to the low S/N ($\sim$ 3) line detections, such as the metallic lines; but for high S/N ($\geq$ 5) line/bands the errors are $\leq$ 1 Å. Tables 1-4 list the EWs (in Å, positive for absorption and negative for emission) of the CO(3-0, 4-1) second overtone bands in H band, CO(2-0, 3-1) first overtone bands in K band, and HI Brackett series lines (only for PAGB stars), for the programme stars of each category.
Correlations and their implications:
------------------------------------
Based on physical processes, very interesting correlations were found among spectral features observed in AGB stars (Kleinmann & Hall [@klei86]; Lancon & Wood [@lanc00]; Bieging, Rieke & Rieke [@beig02]). We tried to see if our sample too shows these. Kleinmann & Hall [@klei86] and Bieging, Rieke & Rieke [@beig02] have found that the CO(2-0) and CO(3-1) band head strengths are correlated. From the tables 1-3, we can see this trend; in addition, one can see a correlation between CO(3-0) & CO(4-1) bands as well.
The first overtone bands of CO ($\Delta v = 2$) in the K band arise at T $\sim$ 800 K partly in the photosphere and partly in the circumstellar envelopes; while the second overtone bands ($\Delta v = 3$) in the H band arise at T $\sim$ 3000-4000 K entirely in the photospheric layers (e.g., Hinkle, Hall & Ridgway [@hink82]; Emerson [@emer96]). As a result, the pulsational effects may be more pronounced in the $\Delta v = 3$ bands than in the $\Delta v = 2$ bands; while the former will reflect the circumstellar matter properties better. Thus we would expect a better correlation between EW of $\Delta v = 3$ bands with pulsation period P than the $\Delta v = 2$ bands. It should be noted here that for the CO bands the S/N ratio is high(in many cases $\geq$ 5) and hence the errors in EW are $\leq$ 1 Å. Fig 2 shows equivalent widths of CO(3-0) and CO(2-0) plotted against the color \[H-K\] for M type stars. The phases of these variable stars are not necessarily the same between our observations and 2MASS or IRAS data acquisition (see Tables 1 for phases during our observations). Also shown in Fig 2 are the EWs of the CO lines from the S type stars (against the color \[K-12\]). The trend of correlation for CO(3-0) band is quite clear. For SR stars, we did not find any such trend.
In order to quantify the degree of correlation between two parameters, we have used the Spearman’s rank correlation method that does not assume any functional relationship between them (e.g., used by Loidl, Lancon & Jorgensen [@loid01]). Fig 2 shows only those correlations with CO(3-0) for which the rank coefficient ($\rho$) $\geq$ 0.50 and its significance ($s$) $\leq$ 0.01 (i.e., a chance occurrence of 1 in 100). We find that for M stars, for the CO(3-0) vs \[H-K\], $\rho$) is 0.50 with $s$ of 0.008 (a chance occurrence of 8 in 1000); while for S stars, for CO(3-0) vs \[K-12\] $\rho$ is 0.74 with $s$ of 0.003. These are moderately significant correlations as may also be visually seen from the plots. In comparison, the coefficients and their significance are quite poor for the CO(2-0) plots in Fig 2. For M stars, one can also see a trend in the CO ($\Delta v = 3$) vs log P better than CO($\Delta v = 2$) vs log P(not shown). Usually the mass loss is correlated well with the pulsation period; as also by the color indices \[H-K\] and \[K-12\](Whitelock et al. [@whit94]). While the trends in Fig 2 are certainly beyond the errors in EW, we find that the scatter is rather significant in the correlation plot between the CO(3-0) band EWs and \[H-K\] and other mass loss indicators (not shown) such as log P and \[K-12\], although it is much less in comparison to the plots for CO(2-0) band (only \[H-K\] is shown). Lancon & Wood [@lanc00] and Loidl, Lancon & Jorgensen [@loid01] argued that in the case of carbon-rich AGB stars, the CO lines arise deeper in side the atmospheres (close to the photospheres) and hence may not respond to dynamical effects such as pulsations as much as those lines that arise in the outer atmospheres. In our sample M stars, a large number of them are oxygen-rich and hence the pulsational phase effects may be significant and can possibly account for the scatter in our plots.
SPITZER Spectra on a few Post-AGB objects - PAH and 21 $\mu$m features:
-----------------------------------------------------------------------
We have searched the SPITZER archival spectroscopic data in the mid-infrared region of 6-30 $\mu$m and succeeded in obtaining 7 PAGB stars for which the data were available and unpublished and are not in the list of sources presented originally by Hrivnak, Volk & Kwok [@hriv00]. The raw spectra (post BCD Data) were analyzed using the Spitzer IRS Custom Extraction (SPICE) task under SPITZER data processing softwares. Interesting spectra in the wavelength range of 10-18 $\mu$m from these objects are shown in Fig 3 (where in addition to the 7 SPITZER stars, we have shown one more PAGB star, namely IRAS 22272+5435, re-analysed from the ISO spectral archives). Of these 8 stars, IRAS 19157-0247 and IRAS 22327-1731 are of early spectral types and showed nearly featureless continua (see Fig 3). For the spectra of those stars that showed the in-plane and out-of-plane bending modes of C-H bonds in PAH molecules (e.g., Kwok [@kwok04] and Tielens [@tiel05]), we have subtracted the continuum by (fifth order) polynomial fitting in segments, in order to bring out the comparative strengths of the features. Fig 4 shows the 6-10 $\mu$m spectra on two stars for which the data are available; these stars show the C-C stretching modes of 6.2 and 6.9 $\mu$m and the blended features around 8 $\mu$m due to C-C stretching and C-H in-plane bending modes. A comparison of the spectra in the region of 10-18 $\mu$m common to about 6 PAGB objects suggests that the intensity of the spectral features due to PAH molecules depends on the spectral type of the star - being weakest or shallowest for cooler stars than for relatively hotter stars. This can possibly be due to the UV radiation flux and the hardness that is available to excite the molecules (Tielens [@tiel05]).
We have computed the ratio of the EWs of the F$_{7.9}$/F$_{11.3}$ feature which is known to indicate whether the molecule is neutral or ionized (see Tielens [@tiel05]). We find that the PAGB star IRAS 04296+3429, having a ratio of 11.3, probably shows indications for ionized PAH molecules, while in IRAS 22272+5435, with a ratio of 0.8, neutrals may be predominant (see Figs 3 and 4). Further, the relative structure of PAHs may be inferred from the ratio of F$_{11.3}$/F$_{12.7}$: a large ratio indicates large compact PAHs, while a small ratio results when these molecules break up into smaller irregular structures (Hony et al. [@hony01]; Sloan et al. [@sloa05]; Peeters et al. [@peet02]; Tielens [@tiel05]). In our sample PAGB stars, we find (see Fig 3) that the objects IRAS 07134+1005 and 23304+5147 have ratios of 4.7 and 3.6 respectively and hence may have large compact PAH molecules compared to IRAS 04296+3429 that has a ratio of 0.8. In IRAS 05113+1347 and 22272+5435 the SPITZER spectra show wide features at 7-8 and 11-12 $\mu$m regions. Such broad features are usually attributed to more complex structures in PAH molecules (see Kwok [@kwok04]).
Since the PAH modes arise by absorption of UV photons, their presence indicates the onset of substantial UV flux from the central star and hence the onset of transition/proto-PN phase. Mt. Abu spectra taken on all these objects do not show the HI Brackett series in emission; they are seen in absorption. Thus it is possible that the UV radiation is soft ($ \leq $ 13.6 eV) as H ionization has not yet started in these objects.
Only three of our sample PAGB stars have SPITZER archival data beyond 18 $\mu$m, and all showed the 21 $\mu$m feature (Kwok, Volk & Hrivnak [@kwok89; @kwok99]; Volk, Kwok & Hrivnak [@volk99] and Decin et al. [@deci98]), which was attributed to TiC (von Helden et al. [@vonh00]) or SiC (Speck & Hofmeister [@spec04]). Since these objects are already known to have the 21 $\mu$m feature (Volk, Kwok & Hrivnak [@volk99]), in Table 4 we give only the EWs of the feature (in $\mu$m), along with the EW for IRAS 07134+1005 computed from ISO archival data. All the properties of the observed feature (the peak wavelength, width and red-side asymmetry) resemble the SiC feature as shown by Speck & Hofmeister [@spec04]).
Modeling of Infrared Spectral Energy Distribution:
--------------------------------------------------
Using the DUSTY code (Ivezic & Elitzur [@ivez97]), we have modeled the spectral energy distributions (SED) in the spectral region of 0.5 - 100 $\mu$m. This code incorporates full dynamical calculation for radiatively driven winds in AGB stars. The photometric data on all the programme objects were compiled mainly from 2MASS and IRAS archives. We have modelled the SEDs of all the programme stars except those that have upper-limits (quality = 1) in IRAS data in more than one band. For a majority of the programme stars IRAS LRS data are not available. Even in the ISO archives we could get only a few stars. Therefore, we have modelled only the photometric fluxes, assuming silicate dust for M, S and SR types; while for most of the PAGB stars, we required carbonaceous dust. The dust grain data were taken from in-built tables in DUSTY. In a few cases we obtained data from DENIS and MSX archives as well. In the model that assumes spherical geometry, we fix stellar parameters like the photometric temperature (based on the visible and the near-IR data), inner dust shell temperature, type of dust and size distribution (MRN assumed) and opacity at 0.5 $\mu$m (usually taken value between 0.1 and 1.0). We have used density distribution relevant for modelling the radiatively driven winds in AGB stars as incorporated in to DUSTY by its authors. There would be 10-20 % uncertainties in all the model output parameters. We have generated nearly 150 models using several combinations of the above parameters. We then take the fluxes computed by the code and normalize with the observed 2 $\mu$m flux from each object and construct the model SEDs. A $\chi^2$ test is performed to select the best fit model. In general the model shows that the M types have lower effective temperatures than the S and SR types. M types show higher opacities by a factor of 2, than the S and SR types; S type being lowest. However the inner dust shell temperature for all the types were found to be in the range 450-700 K. The SEDs of PAGB stars showed, in general, double-humped but flatter trend than their AGB counterparts, indicating detached circumstellar shells (see e.g., Hrivnak & Kwok [@hriv99]). Since some of these are transition objects their spectral classes are much earlier than M types. The inner dust shell temperatures for most of the PAGB stars, as listed in Table 4, are found to be in the range of $\sim$ 100-300 K, significantly cooler than the values for AGB stars. A warmer dust shell was required in addition to the cooler one in one of the PAGB stars (1200 K for IRAS 06556+1623). The dust parameters that we obtained are in good agreement with those in the published literature. A few examples of SEDs of M, S, SR and PAGB stars and their model fits are shown in Figs 5-7, respectively. The DUSTY AGB star model estimates mass loss rate also but with larger uncertainty of 30 %. The mass loss rates (in M$_{\odot}$/yr) thus obtained are listed in Tables 1-4 for all the types; the sign $\dagger$ shows stars for which the rates are independently available from CO rotational line observations by several authors, namely, Loup, Forveille & Omont [@loup93], Sahai & Liechti [@saha95], Groenewegen & de Jong [@groe98], Jorissen & Knapp [@jori98], Winters et al. [@wint00], Ramstedt et al. [@rams06], for M, S and SR stars; and Woodsworth, Kwok & Chan [@wood90], Likkel et al. [@likk91], Omont et al. [@omon93], Hrivnak & Kwok [@hriv99], Bujarrabal et al. [@buja01], Hoogzaad et al. [@hoog02] and Hrivnak & Bieging [@hriv05] for PAGB stars. Our model mass loss rates are in reasonably good agreement with those determined from CO rotational lines, for all cases. From Tables 1 and 4, one can see a clear trend of increase of mass loss rate with increase of the color \[K-12\] (e.g., Bieging, Rieke & Rieke [@beig02]). [**Basically most of our sample M type stars are in the ascending phase of their AGB stage and hence the mass loss rate increases with the \[K-12\] index (see Whitelock et al. [@whit94] and Habing [@habi96]); while the PAGB stars in our sample are already in the PPN stage, having just passed the superwind mass loss event. Following this phase the mass loss would decrease. In fact one can see (Table 4) that the mass loss rates for hot PAGB stars are distinctly less than those for cooler ones, possibly indicating the decrease of mass loss after a superwind event.**]{}
Spectral variations in some PAGB objects
========================================
Most of the PAGB stars in our sample are identified as Proto-PNe (see Ueta et al. [@ueta03] and references therein). Repeated spectroscopic observations were made on a few selected stars IRAS 17423-1755, IRAS 18237-0715, IRAS 19399+2312 and IRAS 22272+5435 (see Table 4 for their spectral types). Our results on several sources are compared with those taken earlier by Hrivnak, Kwok & Geballe [@hriv94], Oudmeijer et al. [@oudm95] and recently by Kelly & Hrivnak [@kell05]. Here we report spectral variations in the CO first overtone bands in IRAS 22272+5435; and in HI lines in stars of early spectral type (hot PAGB stars). In both cases the time scales of variability are short and may be associated with episodic mass loss.
Variability of CO first overtone bands in IRAS 22272+5435:
----------------------------------------------------------
The spectra of this object of spectral type G5Ia show signatures that are probably indicative of its advanced stage of evolution in the PAGB phase. It was shown by Hrivnak, Kwok & Geballe [@hriv94] that the object shows episodes of mass loss that was evident from month-scale variability in its spectra. We obtained its spectra in H and K bands on two occasions separated by about 47 days (4 Nov 2006 and 22 Dec 2006). EWs for 4 Nov 06 are given in Table 4. Fig 8 shows the spectra and it is clear that the CO first overtone bands in the K band seen on 4 Nov were absent on 22 Dec. But the Brackett HI absorption lines were seen in both H and K band spectra on both the observations. Interestingly the H band second overtone bands of CO were [*still present*]{} in the spectra on 22 Dec. This shows that the mass loss reached the outer regions of the atmosphere where the temperature is about a few hundreds of degrees K. The pulsations must be responsible for heating the matter in these regions resulting in the excitation of these lines and filling the absorption seen earlier.
We have found variability of spectra in a few other PAGB stars similar to IRAS 22272+5435. The H and K band spectra of IRAS 05113+1347, IRAS 19114+0002 and IRAS 22223+4327 were observed by Hrivnak, Kwok & Geballe [@hriv94]. Our observations, made about 12 years later, show clear indication of variability of CO in these stars which might have occurred during the intervening period. In all these cases, the variations were found only in CO first overtone bands while the HI lines did not change. [Oudmeijer et al. [@oudm95] found CO second overtone bands in emission in the object IRAS 19114+0002 while our observations show near absence of these bands which can be attributed to mass loss variations.]{}
Variability of HI lines in IRAS 19399+2312:
-------------------------------------------
On the basis of the IRAS colors, Gauba et al. [@gaub03] suggested that IRAS 19399+2312 (also known as V450 Vul) is a hot PAGB star possibly in transition to become a proto-PN. It was classified as a B1III star by Parthasarathy et al. [@part00] but as B1IIIe by Kohoutek & Wehmeyer [@koho99]. Recently, Greaves [@grea04] argued that being a possible member of the cluster NGC 6823, this star is unlikely to be a PAGB transition object but could be a spectroscopic type Be star. We made JHK band spectrometry at Mt Abu Observatory on this object on three occasions: first time on 1 Nov 2005 and then recently on 15 May 2007 and again on 5-6 Jun 2007 (the EWs obtained on 6 Jun 07 are given in Table 4). The 1 Nov 2005 spectra showed some what shallow absorption in HI Paschen $\beta$ and Brackett series lines. The later spectra however showed very prominent emissions in the HI lines, as well as a few permitted lines in HeI and even forbidden lines in \[FeII\] (1.60 $\mu$m) at relatively lower S/N (see Fig 9). We also notice at a rather low S/N but still discernable CO ($\Delta$v=3) lines in absorption. The appearance of the CO absorption lines is not usual in Be stars, as also the forbidden lines of metals (\[FeII\]). In view of the appearance of forbidden lines (which was not reported earlier), and faint but significant CO lines, we believe that this object may as well be a transition object. To our knowledge, our observations are the first to show spectral variability in this object. From the ratios of the Brackett series lines which deviate from the Case B assumption (see for example, Lynch et al. [@lync00]), we infer variations in electron density or opacity in the atmosphere of this object associated with the HI line ratio variations that occurred between 15 May 2007 and 5/6 June 2007.
We made repeated observations on two more PAGB stars of early spectral type: for IRAS 17423-1755 (Be) on 29 May 03 and 13 May 07 and for IRAS 18237-0715 (Be) on 28 May 04, 3 Nov 06 and 14 May 07. In Table 4, the EWs for IRAS 17423-1755 are for the date 29 May 03; and for IRAS 18237-0715 the EWs are for 3 Nov 06. We find variations in the intensities of HI Brackett series emission lines in both these objects. In the case of IRAS 18237-0715 the Brackett line ratios matched well with the Case B assumption on 28 May 04 and 3 Nov 06; while on 14 May 07 the ratios deviated from Case B. Earlier in a detailed study Miroshnichenko et al. [@miro05] found spectral line variations in this object. This star did not show CO lines (Table 4). In the case of IRAS 17423-1755 on both the days of our observations the ratios deviated from Case B. These observatons indicate the changes in electron density or opacity. This star showed CO first overtone lines in emission (Table 4). While Gauba et al. [@gaub03] found emission lines in the optical spectra of IRAS 17423-1755 and 18237-0715 and hence termed these as transition objects, the nature of the latter object is ambiguous and could be a luminous blue variable according to the study of Miroshnichenko et al. [@miro05]. Our IR spectra of this object closely resemble those of Miroshnichenko et al. [@miro05] and show P-Cygni type profiles with double peaks (see Fig 1 top panel). Since we do not find CO lines in the spectra of this object it is likely that this object can not possibly be a PAGB/transition object; or CO must have been dissociated.
In the case of the object IRAS 22327-1731 (type A0III) Oudmeijer et al. [@oudm95] found that the CO bands as well as the HI lines were absent; our results (Table 4) show that the object displays the CO bands and HI Paschen $\beta$ and Brackett $\gamma$ in absorption, thus showing variability.
Conclusions:
============
1. [**The equivalent widths of CO (3-0) show a trend of correlation with \[H-K\] in M type; while in S stars the CO (3-0) shows a positive correlation with \[K-12\].**]{}
2. [**The Mt Abu H and K band spectra also showed marked differences in the three types.**]{}
3. [**Archival SPITZER spectra of a few PAGB stars showing PAH emission features confirm their advanced stage of evolution into PPNe. There appears to be a dependence of the strength of the PAH features on the spectral type.**]{}
4. [**Modeling of SEDs of a number of the programme stars showed marked differences in various generic types. Model mass loss rates for M and PAGB stars show an increasing trend with the color \[K-12\] in our sample stars.**]{}
5. [**Our observations showed that several PAGB stars undergo short term spectral variability that is indicative of on-going episodic mass loss. Cooler PAGB stars showed variation in CO first overtone lines. In contrast, the hot PAGB stars showed variation in HI lines. The hot PAGB star IRAS 19399+2312 is identified as a transition object.**]{}
acknowledgements {#acknowledgements .unnumbered}
================
The authors sincerely thank the anonymous referee for critical comments that led to substantial improvement over the original version. This research was supported by the Department of Space, Government of India. We thank the Mt Abu Observatory staff for help in observations with NICMOS. We thank the SPITZER archival facilities. We acknowledge the DUSTY team for making available the code for the astronomy community.
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[^1]: vvenkat@prl.res.in
[^2]: anand@prl.res.in
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address: |
Mathematics Department\
CSULB\
Long Beach, CA 90840-1001
author:
- Scott Crass
title: |
New light on solving the sextic by iteration:\
An algorithm using reliable dynamics
---
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---
abstract: 'A continuous wave atom laser formed by the outcoupling of atoms from a trapped Bose-Einstein condensate (BEC) potentially has a range of metrological applications. However, in order for the device to be truly continuous, a mechanism to replenish the atoms in the BEC is required. Here we calculate the temporal coherence properties of a continuously pumped atom laser beam outcoupled from a trapped Bose-Einstein condensate which is replenished from a reservoir at finite temperature. We find that the thermal fluctuations of the condensate can significantly decrease the temporal coherence of the output beam due to atomic interactions between the trapped BEC and the beam, and this can impact the metrological usefulness of the device. We demonstrate that a Raman outcoupling scheme imparting a sufficient momentum kick to the atom laser beam can lead to a significantly reduced linewidth.'
author:
- 'Geoffrey M. Lee'
- 'Simon A. Haine'
- 'Ashton S. Bradley'
- 'Matthew J. Davis'
bibliography:
- 'atomlaser\_linewidth\_bib.bib'
title: Coherence and linewidth of a continuously pumped atom laser at finite temperature
---
Introduction
============
A rudimentary atom laser can be realised by outcoupling atoms from a trapped Bose-Einstein condensate. This has previously been achieved using either radio-frequency (RF) [@Mewes1997; @Bloch1999; @Coq2001; @Robins2004] or Raman transitions [@Hagley1999; @Robins2006] to transfer the BEC atoms to untrapped states which then propagate freely. The majority of experiments performed to date have outcoupled atoms from a BEC that is not replenished, leading to an overall decrease of beam flux as time proceeds. This places an upper limit on the integration time for any measurement [@Lahaye2005]. A truly continuous wave (CW) atom laser will not only extend the integration time of any measurement, but has other potential advantages such as a reduction in the linewidth of the beam due to gain narrowing [@Hope2000]. However, this will require a method of replenishing the number of atoms in the BEC so that the system achieves a steady state output. Continuous atom lasers have the potential for applications in metrology, interferometry and precision measurement [@Gustavson1997; @Shin2004], and the improved first-order coherence properties of BECs [@Bloch2000; @Esslinger2000; @Ritter2007] may provide a number of advantages over thermal atom sources [@szigeti2012] currently used in, for example, Sagnac interferometry for the detection of rotations [@Gustavson1997; @Durfee2006; @Eckert2006].
The first experiment that demonstrated a continuous source of Bose-condensed atoms was performed by Chikkatur *et al.* [@Chikkatur2002]. This was achieved by periodically replenishing a BEC held in an optical dipole trap with a new condensate transferred using optical tweezers. While the team was able to demonstrate a BEC of more than $10^6$ atoms at all times, this approach has the limitation that the coherence of the BEC is disturbed every time the condensates are combined due to their independent phases. An alternate approach was demonstrated by Lahaye *et al.* [@Lahaye2005] who performed evaporative cooling on a magnetically guided continuous atomic beam. They observed a significant increase in phase-space density, but did not reach Bose-Einstein condensation. More recently Robins *et al.* have demonstrated a pumped atom laser whereby atoms from one condensate acting as a source undergo a stimulated transition into a second condensate from which an atom laser beam was extracted using RF outcoupling [@Robins2008]. Recently a continuous strontium BEC formed by laser cooling has been demonstrated [@stellmer_pasquiou_13], which is potentially an excellent source for an atom laser. While these experiments have demonstrated the separate elements necessary for a continuous atom laser, they have yet to be combined in a single experiment.
Many theoretical proposals for continuously-pumped operation have been made, using either evaporative cooling [@Wiseman1996; @Holland1996a; @dennis2012] or spontaneous emission [@Bhongale2000; @Castin1998; @Wiseman1995], A number of studies have already considered the properties of a CW atom laser at zero temperature [@Edwards1999; @Schneider1999; @Naraschewski1997; @Hope2000a; @Haine2002; @Haine2003; @Haine2004; @Johnsson2005; @Johnsson2007; @Johnsson2007_1; @Szigeti2010]. However, one of the most likely experimental routes to a true CW atom laser will involve replenishing the BEC via cooling from a reservoir containing thermal atoms so that the condensate mode is maintained in a steady state via Bose-stimulated collisions, similar to the experiment of Stellmer *et al.* [@stellmer_pasquiou_13]. This will avoid the deleterious linewidth and heating effects associated with the merging of initially separated condensates with independent phases [@Chikkatur2002]. However, this will also mean that a CW atom laser will necessarily operate at finite temperature, and hence the effect of thermal fluctuations on the atom laser coherence should be accounted for. An initial study of such issues has been performed by Proukakis [@Proukakis2003] using a classical field model for the BEC replenished by an undepleted thermal reservoir. Here we build on this work by implementing a conceptually similar model but with specific improvements. In particular, we implement an appropriate energy cutoff in the harmonic oscillator basis for the classical field, and propagate the atom laser beam in a plane wave basis well beyond the influence of the trapped BEC, as illustrated in Fig. \[fig:temptrace\]. We focus on understanding the linewidth of the atom laser beam as a function of both the temperature and the momentum kick of the Raman outcoupling.
Model of a continuously pumped atom laser
=========================================
We model a trapped BEC at finite temperature using the simple growth theory of the Stochastic Projected Gross-Pitaevskii equation (SPGPE) formalism [@Blakie2008; @Gardiner2002; @Gardiner2003], in which the energy damping of the scattering terms are neglected. This has previously been successfully applied to model the formation of Bose-Einstein condensates from evaporative cooling [@weiler_neely_08]. Briefly, the second quantised Bose-field operator is divided at a cutoff energy $E_{\rm cut}$ into a high occupation ($N_k \gg 1$) coherent region, in which interaction effects and thermal fluctuations are significant, and a low occupation incoherent region that can be treated as a thermal gas. The dynamics of the entire coherent region can then be treated approximately as a classical field. For the problem to be tractable we treat the incoherent region as a undepleted reservoir at fixed temperature $T$ and chemical potential $\mu$. We expect this is a reasonable approximation in steady state if we imagine that we can replenish thermal atoms at a constant rate while undergoing continuous cooling, e.g. [@stellmer_pasquiou_13; @dennis2012]. The stationary incoherent region thus continuously replenishes the coherent region through Bose-stimulated collisions. The atom laser beam is formed by implementing Raman outcoupling from the coherent region, which depletes not only the condensate but also the non-condensed atoms, resulting in a thermally broadened linewidth for the atom laser beam.
We consider a magnetically trapped $^{87}$Rb condensate containing approximately $10^5$ atoms. We outcouple atoms into an atomic waveguide, as demonstrated by Guerin *et al.* [@Guerin2006], by transferring them to a magnetically insensitive state via a Raman transition which gives the outcoupled atoms a momentum kick in the positive $x$ direction. The waveguide provides confinement in the radial direction. We choose experimentally reasonable trapping frequencies of $(\omega_x,\omega_\perp)$ = $2\pi\times(10,200)$ Hz. For relatively tight transverse confinement, we neglect the dynamics in the radial direction and simulate the system in one dimension.
The equations of motion for the trapped field $\psi(x,t)$ and outcoupled atoms $\phi(x,t)$ are $$\begin{aligned}
\label{eqn:trapped}
d \psi(x) &=&-\frac{i}{\hbar}\mathcal{P} \left[(\hat{L} + g_{12} |\phi(x)|^2|)\psi(x)
-\hbar\Omega e^{-ik_0x}\phi(x)\right] dt \nonumber\\
&+& \mathcal{P}\left[\gamma(\mu-\hat{L}) \psi(x)dt+ dW_\gamma(x,t)\right],\\
d \phi(x) &=&-\frac{i}{\hbar}\bigg[ \left( \frac{-\hbar^2}{2m}\frac{\partial^2 }{\partial x^2}
+g_{12} |\psi(x)|^2 + g_{22}|\phi(x)|^2 - \hbar \delta \right) \phi(x) \nonumber \\
&-& \hbar\Omega e^{ik_0x}\psi(x) \bigg] dt \, ,\end{aligned}$$ with $$\label{eqn:gpoperator1d}
\hat{L}=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{1}{2}m\omega_x^2 x^2+g_{11} |\psi(x)|^2 \, ,$$ where $m$ is the atomic mass, and $\mu$ and $T$ are the chemical potential and temperature of the reservoir. The recoil momentum and Rabi frequency of the Raman transition are $\hbar k_0$ and $\Omega$ respectively, with a two-photon detuning of $\delta$. The operator ${\mathcal{P}}$ projects $\psi(x)$ onto the truncated basis of harmonic oscillator eigenstates with energy $\epsilon \le E_{\rm cut}$. The effective 1D interaction strengths are $g_{ij} = \frac{4\pi\hbar^2 a_{ij}}{2m\sigma_{\perp}}$, where $\sigma_\perp$ is determined by integrating out the transverse dimension using a Gaussian variational anstaz, and $a_{ij}$ is the $s$-wave scattering length for collisions between states $i$ and $j$. In this work we choose $a_{11} = a_{12} = a_{22} = 5.29 $ nm, which is approximately true for the $|F=1, m_F = -1\rangle$ and $|F=2, m_F=0\rangle$ states of $^{87}$Rb [@egorov2013]. The noise is complex Gaussian and has a non-vanishing correlator $
\langle dW^*_\gamma(x,t) dW_\gamma(x',t) \rangle = 2 \gamma \delta(x-x') dt$. The quantity $\gamma$ is the rate constant for the exchange of particles between the thermal reservoir and the classical field, for which we use the estimate $ \gamma= 12 m a_{11}^2 k_B T/\pi\hbar^3$ [@Bradley2008]. The steady state of the atom laser is independent of this quantity inside the weak outcoupling regime such that the condensate is not significantly depleted. We chose $\Omega = 1$ Hz for all our simulations in order to stay in the weak outcoupling regime [@Robins2005].
We simulate the classical field $\psi(x)$ using the harmonic oscillator basis as described in Ref. [@Blakie2005; @Blakie2008], and present results for a cutoff of $E_{\rm cut} = 150 \hbar \omega_x$. We have increased the cutoff to $E_{\rm cut} = 300 \hbar \omega_x$ in other simulations and found no significant difference in our results. The beam $\phi(x)$ is simulated on a rectangular grid, and we make use of numerical grid interpolation for the coupling terms. Classical field atoms are transferred to the beam with a momentum $\hbar k_0$, and we implement an imaginary absorbing potential at the edge of the atom laser beam grid.
We initialize our simulations with a $T=0$ condensate found from the solution of the time-independent Gross-Pitaevskii equation with the outcoupling turned off. The simulations are then run until we reach a steady state for the beam and condensate. We calculate observables by averaging 500 time samples of 64–95 independent trajectories, which gives converged results with relatively small statistical noise. Fig. \[fig:temptrace\] shows examples of the instantaneous density of the trapped Bose gas and the atom laser beam once the system has reached steady state at (a) $T=0$, and (b) $T= 144$ nK.
 Typical classical field and atom laser beam profiles in steady state. (a) T = 0, (b) $T=144$ nK. The instantaneous classical field density is shown in blue, whereas the instantaneous atom laser beam density is the black-dashed curve. The red dot-dash curve shows the time-averaged atom laser beam density for comparison. The atom laser beam density has been scaled by a factor of $2\times10^3$ so that is visible on the axes as the trapped classical field. Insets: Detailed view of the classical field region. In (a), the instantaneous atom laser beam density and time-average atom laser beam density lie on top of one another. For all plots: $\mu = 102$ $\hbar \omega_x$, $g_{11}/\hbar\omega_x x_0 = 0.0166$ where $x_0 = (\hbar/m\omega_x)^{1/2}$, $\Omega = 1$ Hz, $\hbar \gamma = 0.0028$, $k_0 = 2.3\times10^{7}$ m$^{-1}$. .](fig1.eps){width="0.85\columnwidth"}
Temporal coherence of the output beam
=====================================
The phase coherence of the atom laser beam can be quantified by the normalised equal position first-order temporal correlation function [@Walls1994] $$\label{eqn:g1}
g^{(1)}(x,\Delta t)=\frac{\left\langle \phi^*(x,t)\phi(x,t+\Delta t)\right\rangle}{\sqrt{\left\langle |\phi(x,t)|^2\right\rangle \left\langle |\phi(x,t+\Delta t)|^2\right\rangle }} \, ,$$ Figure \[fig:tempcomp\] shows $ |g^{(1)}(x,\Delta t)|$ for two different temperatures. Unsurprising, the coherence time decreases monotonically with increasing temperature for fixed $\mu$. In this parameter regime we find that the atom laser coherence directly reflects the first-order temporal coherence of the classical field $\psi(x)$ at trap centre, which is plotted for comparison in Fig. \[fig:tempcomp\]
To gain physical insight into the decrease of phase coherence in the condensate at finite temperature, we can approximate the evolution of the classical field by utilising the Bogoliubov approximation [@stringari_review]. Expressing the classical field as $$\begin{aligned}
{\psi}(x,t) &= e^{-i \mu t/\hbar}\left[\sqrt{N_0} \varphi_0(x) \nonumber \right.\\
& \left.+\sum_{j>0} \left(u_j(x)e^{-i\epsilon_j t/\hbar}{b}_j + v^*_j(x)e^{i\epsilon_j t/\hbar} {b}^*_j\right) \right]\, , \end{aligned}$$ where $N_0$ is the condensate number, $\varphi_0(x) $ is the condensate wave function, and $u_j(x)$ and $v_j(x)$ are the 1D Bogoliubov quasiparticle modes with energies $\epsilon_j$ measured relative to the chemical potential [@pitaevskii2003]. An approximate form for the Bogoliubov modes in the 1D Thomas-Fermi limit can be written as [@Petrov2000] $$f^{\pm}_j(\tilde{x}) = \sqrt{j+\frac{1}{2}} \left[ \frac{2\mu}{\epsilon_j}(1-\tilde{x}^2) \right]^{\pm\frac{1}{2}} P_j(\tilde{x}),
\label{eqn:bog_soln}$$ where the spatial dependence has been normalised to the Thomas-Fermi radius of the condensate $\tilde{x}=x/R_{\text{TF}}$ where $R_{\text{TF}} = (2 \mu/m \omega_x^2)^{1/2}$, and $P_j(\tilde{x})$ are Legendre polynomials. The functions $f^{\pm}_j(\tilde{x})$ are related to the Bogoliubov mode functions $u(\tilde{x})$ and $v(\tilde{x})$ via the relation $f^{\pm}_j(\tilde{x})=u(\tilde{x})\pm v(\tilde{x})$. The Bogoliubov modes have energies $\epsilon_j=\hbar\omega_x \sqrt{j(j+1)/2}$ for positive integers $j$ [@Petrov2000] in the one-dimensional Thomas-Fermi limit.
We can then construct an estimate of the first-order correlation function $g^{(1)}(x,\Delta t)\propto \langle{\psi}^*(x,t){\psi}(x,t +\Delta t)\rangle$ by substituting the Bogoliubov expansion and computing the sum numerically. We examine the temporal coherence at the centre of the condensate where outcoupling is resonant. We assume each Bogoliubov mode is thermally occupied with quasiparticles according to the equipartition distribution $$N_j \equiv \langle {b}^*_j {b}_j\rangle = \frac{k_B T}{\epsilon_j},$$ while the ground state has an occupation equal to the condensate number $N_0$ calculated from the one-body density matrix using the Penrose-Onsager criterion for condensation [@Blakie2008]. Assuming $N_j\gg 1$, we have
$$\langle\psi^*(0,t)\psi(0,t+\Delta t)\rangle = N_0 |\varphi_0(0)|^2
+ \sum_{j>0} \frac{k_BT}{\epsilon_j} \left(j+\frac{1}{2}\right)\frac{|P_j(0)|^2}{2}\left[\left(\frac{4\mu^2 + \epsilon_j^2}{2\mu\epsilon_j}\right)\cos\left(\epsilon_j \Delta t/\hbar\right)-2i\sin\left(\epsilon_j \Delta t/\hbar\right)\right],
\label{eqn:g1_est}$$
The estimate given by Eq. is also shown in Fig. \[fig:tempcomp\], where the correlation function has been normalised to compare with the SPGPE results. The estimate replicates the early decay of the numerical results, indicating that the mechanism for the loss of coherence observed in the condensate is the dephasing of thermal fluctuations. The revivals of coherence present in the Bogoliubov estimate are not present in the SPGPE result, as the noise term of the SPGPE causes a random drift in the phase of each of the modes and preventing the rephasing.
![\[fig:tempcomp\] (Color online) First-order temporal coherence of the atom laser beam and condensate quantified by $g^{(1)}(x,\Delta t)$ for temperatures (a) $T=48$ nK with $N_0 = 1.1\times 10^5$ and (b) $T=384$ nK with $N_0 =0.72\times 10^5$. For both plots — solid red line is the coherence of the condensate at $x=0$; dashed blue line: coherence of the atom laser beam for $x$ just outside the trapped gas; dash-dotted black line: estimate given by Bogoliubov approximation of Eq. . All other parameters are the same as for Fig. \[fig:temptrace\]. ](fig2.eps){width="0.85\columnwidth"}
Influence of the momentum kick on the linewidth
===============================================
We now investigate the effect of the magnitude of the momentum kick on the linewidth of the atom laser at finite temperature. At zero temperature the momentum transfer to the atom laser beam of $\hbar k_0$ would have no bearing on the temporal coherence and hence the linewidth of the atom laser. At finite temperature we find that the temporal coherence of the trapped BEC is also independent of the value of $k_0$. However, we find that this is not the case for the atom laser beam. Instead, below a critical value of $k_0$, we find that for a given temperature the temporal coherence of the beam decays significantly faster than that of the trapped BEC.
Defining the power spectrum of a field $\chi(x_0,t)$ as $$P_\chi(x_0,\omega) = \left|\int dt e^{-i \omega t} \chi(x_0,t)\right|^2,$$ we define the spectral linewidth of the field as $$\begin{aligned}
\Delta\omega_\chi(x_0)=\left[\int d\omega P_\chi(x_0,\omega) \omega^2 - \left(\int d\omega P_\chi(x_0,\omega) \omega\right)^2 \right]^{1/2} \, \nonumber \\\end{aligned}$$ This quantity is plotted for both the centre of the condensate (blue asterisks) and the middle of the atom laser beam (red diamonds) in Fig. \[fig:classicalcomp\] for a temperature of $T=384$ nK. It is clear that the atom laser beam exhibits a dramatic increase in linewidth for momentum kicks below $k_0 =6\times10^6$ m$^{-1}$.
We have found that this excess linewidth is due to the thermal density fluctuations of the classical field that the atom laser beam experiences as it propagates through the trapped BEC. If the outcoupling momentum $\hbar k$ is sufficiently large, the density of the trapped classical field is essentially frozen on the timescale required for particles to leave the region of the BEC, so the linewidth of the beam is determined by the spectral linewidth of the condensate at the spatial position where the outcoupling is resonant. However, for smaller momenta, and therefore longer exit times, the outcoupled atoms experience significant spatio-temporal fluctuations in the effective spatial potential due to the classical field, resulting in greater dispersion in velocity of the outcoupled atoms and a larger linewidth. A threshold momentum separating the different linewidth behaviour may be determined via a classical argument. Particles travelling significantly faster than the thermal density fluctuations can ‘outrun’ the fluctuations, and thus experience no velocity broadening. Taking the sound speed $c_s=(g_{11}N_0 |\varphi(0)|^2/m)^{1/2}$ as an estimate for the characteristic speed of density fluctuations, we note that the threshold momentum for the excess linewidth is approximately twice this value, consistent with our argument.
We have implemented a simple numerical model to confirm this explanation. We have simulated the dynamics of an ensemble of classical atom laser beam particles placed at the classical outcoupling position with initial momentum $\hbar k_0$. In a zero temperature system the effective potential experienced by the outcoupled atoms due to interactions with the BEC will be static. Therefore, outcoupling the atom laser beam from the centre of the trap will result in a momentum increase of the output as the atoms “roll down the hill” so their output momentum is $\hbar k=\sqrt{(\hbar k_0)^2+2m\mu}$ [@Johnsson2007_1], but there is no broadening of the initial distribution. However, at finite temperature, on exiting the trapped BEC, the momentum distribution of the beam will be broadened due to interactions with the fluctuating density of the classical field, and we quantify it by standard deviation of the output energy distribution. The resulting linewidth is plotted in Fig. \[fig:classicalcomp\] in comparison with the results from our SPGPE model, where we can see they are in broad agreement.
![\[fig:classicalcomp\] (Color online) Linewidths of the condensate and the atom laser as a function of momentum kick for $T=384$ nK. Blue asterisks: condensate linewidth at outcoupling position. Red diamonds: atom laser beam linewidth. Black dots: Classical model of the linewidth of Newtonian particles exiting the system experiencing the fluctuating classical field of the trapped BEC as described in the text. All other parameters are the same as for Fig. \[fig:temptrace\].](fig3.eps){width="0.85\columnwidth"}
As the excess linewidth of the atom laser beam is due to density fluctuations of the trapped BEC, the effect will disappear if there are no interactions between the trapped BEC and the atom laser beam. We have verified this by setting $g_{12}=0$ in our simulations (while appropriately adjusting the detuning $\delta= \hbar k_0^2/2m-\mu/\hbar$ in order to maintain resonant outcoupling in the centre of the condensate.) In this case, the linewidth of the atom laser beam reverts to the linewidth of the condensate for all values of the Raman kick momentum. It may be possible to exploit an inter-component Feshbach resonance to achieve this in an experiment, significantly reducing the linewidth. However, this may be technically challenging in practice.
A potential solution to reduce the atom laser linewidth for a fixed temperature and momentum kick is to alter the detuning such that the resonant position is closer to the edge of the condensate and the exit time of the beam is shorter. We do observe a monotonic decrease in the linewidth as the outcoupling position is moved towards $R_\text{TF}$, but it remains larger than that of the condensate. Beyond $R_\text{TF}$, the atom laser linewidth approaches that of the thermal reservoir, as expected — only thermal atoms are outcoupled. As a result, the linewidth of the atom laser beam is at least that of a thermal state for all values of the two-photon Raman detuning. We conclude that altering the detuning may decrease the overall atom laser linewidth, but cannot reduce it to that of the condensate. Furthermore, any reduction in linewidth is offset by the reduction in flux (for a fixed $\Omega$) associated with moving the outcoupling position away from the centre of the trap.
Another strategy to reduce the excess linewidth observed for a given temperature and momentum kick may be to remove the waveguide and allow outcoupled atoms to instead fall under gravity in the radial direction, potentially reducing the exit time of the atom laser beam. For the parameters used in this work, the gravitational acceleration results in an exit time equivalent to an initial momentum kick of $4.4\times10^6$ m$^{-1}$. This value is below the threshold momentum, and so the linewidth of the atom laser beam will still be excessive. Additionally, removing the waveguide will degrade the transverse profile of the atom laser beam [@Guerin2006; @Couvert2008], as was seen in [@Coq2001; @Busch2002; @Kohl2005; @Riou2006; @Jeppesen2008].
Conclusions
===========
We have modelled the temporal coherence and linewidth of a continuous atom laser at finite temperature. We find that when outcoupling with a large momentum kick, the first-order coherence of the atom laser beam mimics the first-order coherence of the trapped BEC which is determined by thermal fluctuations. However, below a threshold momentum kick related to the condensate speed of sound, there is a dramatic increase in the spectral linewidth of the atom laser beam. We have identified the origin of this as density fluctuations in the trapped BEC while the atom laser beam exits the condensate, further increasing the energy spread of the output beam. Our results will influence the design and optimisation of continuous atom lasers at finite temperature in the future.
This research was supported by Australian Research Council Discovery Projects DP1094025 and DE130100575. ASB acknowledges the support of The New Zealand Marsden Fund and the Royal Society of New Zealand. MJD acknowledges the support of the JILA Visiting Fellows program.
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abstract: 'We discuss variance reduced simulations for an individual-based model of chemotaxis of bacteria with internal dynamics. The variance reduction is achieved via a coupling of this model with a simpler process in which the internal dynamics has been replaced by a direct gradient sensing of the chemoattractants concentrations. In the companion paper [@limits], we have rigorously shown, using a pathwise probabilistic technique, that both processes converge towards the same advection-diffusion process in the diffusive asymptotics. In this work, a direct coupling is achieved between paths of individual bacteria simulated by both models, by using the same sets of random numbers in both simulations. This coupling is used to construct a hybrid scheme with reduced variance. We first compute a deterministic solution of the kinetic density description of the direct gradient sensing model; the deviations due to the presence of internal dynamics are then evaluated via the coupled individual-based simulations. We show that the resulting variance reduction is *asymptotic*, in the sense that, in the diffusive asymptotics, the difference between the two processes has a variance which vanishes according to the small parameter.'
author:
- 'Mathias Rousset[^1]'
- 'Giovanni Samaey[^2]'
bibliography:
- 'bib-papers.bib'
title: 'Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction'
---
[**MSC**]{}: 35Q80, 92B05, 65C35.\
[**Key words**]{}: asymptotic variance reduction, bacterial chemotaxis, velocity-jump process.
Introduction
============
The motion of flagellated bacteria can be modeled as a sequence of run phases, during which a bacterium moves in a straight line at constant speed. Between two run phases, the bacterium changes direction in a tumble phase, which takes much less time than the run phase and acts as a reorientation. To bias movement towards regions with high concentration of chemoattractant, the bacterium adjusts its turning rate by increasing, resp. decreasing, the probability of tumbling when moving in an unfavorable, resp. favorable, direction [@Alt:1980p8992; @Stock:1999p8984]. Since many species are unable to sense chemoattractant gradients reliably due to their small size, this adjustment is often done via an intracellular mechanism that allows the bacterium to retain information on the history of the chemoattractant concentrations along its path [@Bren:2000p7499]. The resulting model, which will be called the “internal state” or “fine-scale” model in this text, can be formulated as a velocity-jump process, combined with an ordinary differential equation (ODE) that describes the evolution of an internal state that incorporates this memory effect [@Erban:2005p4247].
The probability density distribution of the velocity-jump process evolves according to a kinetic equation, in which the internal variables appear as additional dimensions. A direct deterministic simulation of this equation is therefore prohibitively expensive, and one needs to resort to a stochastic particle method. Unfortunately, a direct fine-scale simulation using stochastic particles presents a large statistical variance, even in the diffusive asymptotic regime, where the behavior of the bacterial density is known explicitly to satisfy a Keller-Segel advection-diffusion equation. Consequently, it is difficult to assess accurately how the solutions of the fine-scale model differ from their advection-diffusion limit in intermediate regimes. We refer to [@Horst1; @Horst2; @KelSeg70] for numerous historical references on the Keller-Segel equation, to [@ErbOthm04; @Erban:2005p4247] for formal derivations (based on moment closures) of convergence of the the velocity-jump process with internal state to the Keller-Segel equation, and to the companion paper [@limits] for a proof of this convergence based on probabilistic arguments.
In this paper, we propose and analyze a numerical method to simulate individual-based models for chemotaxis of bacteria with internal dynamics with reduced variance. The variance reduction is based on a coupling technique (control variate): the main idea is to simultaneously simulate, using the same random numbers, a simpler, “coarse” process where the internal dynamics is replaced by a direct “gradient sensing” mechanism (see [@Alt:1980p7984; @Othmer:1988p7986; @Patlak:1953p7738] for references on such gradient sensing models). The probability density of the latter satisfies a kinetic equation without the additional dimensions of the internal state, and converges to a similar advection-diffusion limit, see e.g. [@Chalub:2004p7641; @Hillen:2000p4751; @Othmer:2002p4752; @limits].
More precisely, we consider the coupling of two velocity-jump processes with exactly the same advection-diffusion limit (for both processes, a proof of the latter convergence has been obtained in the companion paper [@limits]). The first one is the process with internal dynamics described by the system of equations , with jump rates satisfying and internal dynamics satisfying the two main assumptions and as well as other technical assumptions contained in Section \[sec:asympt\]. The second one is the process with direct gradient sensing , with jump rates satisfying with . The same random numbers are used for the two processes in the definition of both the jump times and the velocity directions. The main contributions of the present paper are then twofold :
- From a numerical point of view, we couple two systems of many particles consisting of different realizations of the fine-scale (internal state) and coarse (gradient sensing) processes, simulated with similar discretization schemes (see Section \[sec:coupl\_princ\]) and the same random numbers. Additionally, for the coarse model, also the continuum description is simulated on a spatial grid. Evolution of the fine-scale model is then evaluated on the grid at hand by adding the difference between the two coupled particle descriptions to the evolution of the coarse continuum description. This can be seen as using the coarse process as a control variate. As coupling is lost over time, a regular reinitialization of the two coupled particle systems is included in the algorithm. The asymptotic variance reduction method can be depicted using Table \[tab:table\] (see also Figure \[fig:pic\] in Section \[sec:method\]).
\[tab:table\]
Model Numerics
----------------------------------------------- ------------------------------
Internal state model–Individual description Monte-Carlo method
Gradient sensing model–Individual description Monte-Carlo method (coupled)
Gradient sensing model–Density description Grid method
: Models and numerical methods used in the variance reduction algorithm.
- From an analysis point of view, we rigorously prove, using probabilistic arguments, $L^p$-bounds on the position difference at diffusive times of both processes in terms of appropriate powers of the scaling parameter (Theorem \[thm:coupling\]). This analysis shows that the numerical variance reduction obtained by the method is *asymptotic* (the remaining statistical variance vanishes with the small scaling parameter), and provides upper bounds on the rate of the latter convergence. We argue in Section \[sec:sharp\] that this bound should be sharp in relevant regimes.
The idea of asymptotic variance reduction is a general and very recent idea in scientific computing that appears when using hybrid Monte-Carlo/PDE (partial differential equation) methods. While the use of control variates is fairly common in Monte Carlo simulation, the only explicit attempt to develop asymptotic variance reduction in hybrid methods, up to our knowledge, can be found in [@DimPar08] and related papers in the context of the Boltzmann equation, and is based on an importance sampling approach (see also [@10.1063/1.1899210] and related papers).
This paper is organized as follows. In Section \[sec:model\], we briefly review the mathematical models that we will consider, and summarize the results on their advection-diffusion limit that were obtained in the companion paper [@limits]. In Section \[sec:method\], we proceed to describe the coupled numerical method in detail. Section \[sec:variance\] is dedicated to a detailed analysis of the variance reduction of the coupled method. The main result is the following: up to a time shift perturbation, the variance of the difference between the two coupled process scales according to the small parameter of the diffusive asymptotics. We proceed with numerical illustrations in Sections \[sec:illustr\] and \[sec:appl\], and conclude in Section \[sec:concl\] with an outlook to future research.
Particle-based models for bacterial chemotaxis \[sec:model\]
============================================================
In this paper, we directly present the models of interest in a nondimensional form, incorporating the appropriate space and time scales. For a discussion on the chosen scaling, we refer to the companion paper [@limits].
Model with internal state\[sec:model-internal\]
-----------------------------------------------
We consider bacteria that are sensitive to the concentration of $m$ chemoattractants ${ \left(\rho_i(x)\right) }_{i=1}^m$, with $\rho_i(x) \geq 0$ for $x \in {\mathbb{R}}^d$. While we do not consider time dependence of chemoattractant via production or consumption by the bacteria, a generalization to this situation is straightforward, at least for the definition of the models and the numerical method. Bacteria move with a constant speed $v$ (run), and change direction at random instances in time (tumble), in an attempt to move towards regions with high chemoattractant concentrations. As in [@ErbOthm04], we describe this behavior by a velocity-jump process driven by some internal state $y \in \mathbb{Y}\subset{\mathbb{R}}^{n}$ of each individual bacterium. The internal state models the memory of the bacterium and is subject to an evolution mechanism attracted by a function $\psi: {\mathbb{R}}^m \to {\mathbb{R}}^n$ of the chemoattractants concentrations, $$S(x):= \psi(\rho_1(x),\ldots,\rho_m(x)) \in {\mathbb{R}}^n,$$ where $x$ is the present position of the bacterium. $S$ is assumed to be smooth with bounded derivatives up to order $2$. We refer to Section 2.2 in [@limits] for some typical choices; in the concrete example in this paper, we choose $m=n=1$, and $S(x)=\rho_1(x)$. By convention, the gradient of $S(x):{\mathbb{R}}^d \to {\mathbb{R}}^n$ is a matrix with dimension $$\nabla S(x) \in {\mathbb{R}}^{n\times d}.$$ We then denote the evolution of each individual bacterium position by $
t \mapsto X_t,
$ with normalized velocity $$\frac{{\mathrm{d}}X_t}{{\mathrm{d}}t} = \epsilon V_t, \qquad V_t\in \mathbb{V}={\mathbb{S}}^{d-1},$$ with ${\mathbb{S}}^{d-1}$ the unit sphere in ${\mathbb{R}}^d$. Hence, $V_t$ represents the direction and the parameter $\epsilon$ represents the size of the velocity. The evolution of the internal state is denoted by $t \mapsto Y_t$. The internal state adapts to the local chemoattractant concentration through an ordinary differential equation (ODE), $$\label{eq:internal}
\frac{{\mathrm{d}}Y_t}{{\mathrm{d}}t}= F_{\epsilon}(Y_t,S(X_t)),$$ which is required to have a unique fixed point $y^*=S(x^*)$ for every fixed value $x^* \in {\mathbb{R}}^d$. We also introduce the deviations from equilibrium $z=S(x)-y$. The evolution of these deviations is denoted as $$t \mapsto Z_t = S(X_t) -Y_t.$$ The velocity of each bacterium is switched at random jump times $(T_n)_{n \geq 1}$ that are generated via a Poisson process with a time dependent rate given by $\lambda(Z_t)$, where $z \mapsto \lambda(z)$ is a smooth function satisfying $$\label{eq:ratebound}
0 < \lambda_{\rm min} \leq \lambda { \left(z\right) } \leq \lambda_{\rm max},$$ as well as, $$\label{eq:lin_rate}
\lambda(z) = \lambda_0 - b^T z + c_\lambda{\mathcal{O}}{ \left( {\left | z\right |}^k \right) } ,$$ with $b \in {\mathbb{R}}^ n $, $k \geq 2$, and $c_\lambda >0$ is used to keep track of the non-linearity in the analysis. The new velocity at time $T_n$ is generated at random according to a centered probability distribution ${\mathcal M}(dv)$ with $\int v \, {\mathcal M}(dv) =0 $, typically $${\mathcal M}(dv) = \sigma_{{\mathbb{S}}^{d-1}}(dv),$$ where $\sigma_{{\mathbb{S}}^{d-1}}$ is the uniform distribution on the unit sphere.
The resulting fine-scale stochastic evolution of a bacterium is then described by a left continuous with right limits (lcrl) process $$t \mapsto { \left(X_t, V_t, Y_t\right) } ,$$ which satisfies the following differential velocity-jump equation : $$\label{eq:process_noscale}
\begin{cases}
{\displaystyle}\dfrac{{\mathrm{d}}X_t}{{\mathrm{d}}t} = \epsilon V_t, \\[8pt]
{\displaystyle}\dfrac{{\mathrm{d}}Y_t}{{\mathrm{d}}t} = F_{\epsilon}(Y_t,S(X_t)), \\[8pt]
{\displaystyle}\int_{T_n}^{T_{n+1}} \lambda(Z_t) {\mathrm{d}}t = \theta_{n+1}, \qquad \text {with }
Z_t := S(X_t)-Y_t ,\\[8pt]
{\displaystyle}V_{t} = {\mathcal{V}}_{n} \quad \text{ for $t \in [T_{n},T_{n+1}]$ } ,
\end{cases}$$ with initial condition $X_0, {\mathcal{V}}_0 \in {\mathbb{R}}^d$, $Y_0 \in {\mathbb{R}}^n$ and $T_0 = 0$. In , ${ \left(\theta _n \right) }_{n \geq 1}$ denote i.i.d. random variables with normalized exponential distribution, and ${ \left( {\mathcal{V}}_n \right) }_{n \geq 1}$ denote i.i.d. random variables with distribution $ \mathcal M (dv)$.
For concreteness, we provide a specific example, adapted from [@ErbOthm04], which will also be used later on to illustrate our results numerically. We consider $m=1$, i.e., there is only one chemoattractant $S(x)$ and the internal dynamics reduces to a scalar equation $$\label{e:scalar-y}
\frac{{\mathrm{d}}y(t)}{{\mathrm{d}}t} = \frac{S(x)-y}{\tau}=\frac{z}{\tau}.$$ For the turning rate $z\mapsto \lambda(z)$, we choose the following nonlinear strictly decreasing smooth function $$\label{eq:rate}
\lambda(z)=2\lambda_0 { \left(\frac{1}{2}-\frac{1}{\pi}\arctan { \left(\frac{\pi}{2\lambda_0}z\right) }\right) }.$$
Asymptotic regimes and linearizations\[sec:asympt\]
---------------------------------------------------
In the adimensional models -, a small parameter ${\epsilon}$ is introduced, which can be interpreted as the ratio of the typical time of random change of velocity direction to the typical time associated with chemoattractant variations as seen by the bacteria. In the present section, we present the main scaling assumptions on the internal dynamics that are necessary to consider the asymptotic regime ${\epsilon}\ll 1$, as well as the corresponding notation. Most of these assumptions were introduced in the companion paper [@limits], to which we refer for more details and motivation.
Throughout the text, the Landau symbol ${\mathcal{O}}$ denotes a *deterministic* and globally Lipschitz function satisfying ${\mathcal{O}}(0)=0$. Its precise value *may vary* from line to line, may depend on all parameters of the model, but its Lipschitz constant is uniform in ${\epsilon}$. In the same way, we will denote generically by $C > 0$ a deterministic constant that may depend on all the parameters of the model except ${\epsilon}$.
We assume that the ODE driving the internal state is well approximated by a near equilibrium evolution equation in following sense :
\[a:1\] We have $$\label{eq:tau_eps_def}
{{F_{{\epsilon}}}}(y,s) = - {{\tau_{{\epsilon}}}}^{-1} (y-s) + {\epsilon}^{1-\delta}c_F{\mathcal{O}}({\left | s-y\right |}^2),$$ where $\delta >0$ and ${{\tau_{{\epsilon}}}}\in{\mathbb{R}}^{n\times n}$ is an invertible constant matrix.
The constant $c_F$ is used to keep track of the dependance on the non-linearity of $F_{\epsilon}$ in the analysis. For technical reasons that are specific to the analysis of the time shift in the coupling (Lemma \[lem:DTestimbis\]), we need the following assumption on the scale of the non-linearity (this assumption was not necessary in the companion paper [@limits]):
\[a:1bis\] Assume that ${\left | y-s\right |} = {\mathcal{O}}({\epsilon}^\delta)$, then there is a $\gamma > 0$ such that: $$\label{eq:taueps_hypbis}
{\left | (y-s) - {{\tau_{{\epsilon}}}}{{F_{{\epsilon}}}}(y,s) \right |} = c_F{\mathcal{O}}( {\epsilon}^{1 + \gamma} ).$$
We now formulate the necessary assumptions on the scale of the matrix ${{\tau_{{\epsilon}}}}$ :
\[a:2\] There is a constant $C >0$ such that for any $t \geq 0$, one has $$\label{eq:tau_scales}
{\left\Vert{{\rm e}}^{- t {{\tau_{{\epsilon}}}}^{-1} }\right\Vert} \leq C {{\rm e}}^{- t {\epsilon}^{1-\delta}/C}.$$
\[a:3\] There is a constant $C >0$ such that $$\sup_{t \geq 0} {\left\Vertt{{\tau_{{\epsilon}}}}^{-1}{{\rm e}}^{- t {{\tau_{{\epsilon}}}}^{-1} } \right\Vert} \leq C {\epsilon}^{-1}.$$
Assumption \[a:2\] is used to ensure exponential convergence of the linear ODE with time scale at least of order ${\epsilon}^{1-\delta}$. Assumption \[a:3\] is necessary for technical reasons in the proof of Lemma \[lem:DTestim\] (which is given in [@limits]). Note that if ${{\tau_{{\epsilon}}}}^{-1}$ is symmetric and strictly positive with a lower bound of order ${\mathcal{O}}({\epsilon}^{1-\delta})$ on the spectrum, then Assumption \[a:2\] and Assumption \[a:3\] are satisfied. This implies that the slowest time scale of the ODE is at worse of order ${\epsilon}^{1-\delta}$. Indeed, integration over time in Assumption \[a:2\] implies ${\left\Vert{{\tau_{{\epsilon}}}}\right\Vert} \leq C {\epsilon}^{\delta-1}$. Moreover, we assume that the solution of the ODE driving the internal state in satisfies the following long time behavior :
\[ass:ODE\] Consider a path $t \mapsto S_t$ such that $ \sup_{t \in [0,+\infty] } {\left | \dfrac{ d S_t}{ dt} \right |}= {\mathcal{O}}( {\epsilon})$, and assume ${\left | Y_0 - S_0\right |} = {\mathcal{O}}({\epsilon}^\delta)$. Then the solution of $$\frac{ d Y_t}{ dt} = {{F_{{\epsilon}}}}(Y_t,s_t),$$ satisfies $$\sup_{t \in [0,+\infty] } {\left | Y_t - S_t \right |} = {\mathcal{O}}{ \left( {\epsilon}^\delta \right) }.$$
Assumption \[ass:ODE\] is motivated by the case of linear ODEs; indeed, in that case, Assumption \[ass:ODE\] is a consequence of Assumption \[a:2\] (see Section 3.3 in [@limits]). Recalling that $Z_t=S_t-Y_t$, we will assume in the remainder of the paper that the solution of satisfies, $${\left | Z_0\right |} = {\mathcal{O}}({\epsilon}^\delta),$$ as well as $${\left | {{\tau_{{\epsilon}}}}^{-1} Z_0\right |} = {\mathcal{O}}( 1 ).$$ Then, under Assumption \[a:1\], Assumption \[a:2\], Assumption \[a:3\], and Assumption \[ass:ODE\], we obtain the bounds (see Section 3.3 in [@limits]) $$\label{eq:techbound}
\sup_{t \in [0,+\infty] } {\left | {{\tau_{{\epsilon}}}}^{-1}{{\rm e}}^{- {{\tau_{{\epsilon}}}}^{-1} t } (S_t - Y_t) \right |} = {\mathcal{O}}{ \left( 1 \right) } ,$$ $$\sup_{t \in [0,+\infty]} { \left( {\left | Z_t\right |} \right) } = {\mathcal{O}}({\epsilon}^{\delta}),$$ as well as $$\sup_{t \in [0,+\infty]} { \left( {\left | {{\tau_{{\epsilon}}}}^{-1} {{\rm e}}^{- {{\tau_{{\epsilon}}}}^{-1} t} Z_t\right |} \right) } = {\mathcal{O}}(1) .$$
Finally, we will make use of the following notation. Denoting $c_S = {\left\Vert\nabla^{2} S\right\Vert}_{\infty},$ with $\nabla^2$ the Hessian, the error terms due to non-linearities of the problem will be handled through $$\label{eq:nl_err}
\begin{cases}
{ {\rm nl} }({\epsilon}) := c_{F} {\epsilon}^{\delta} + c_S {\epsilon}+ c_{\lambda} {\epsilon}^{ k \delta-1} = {o}(1),\\
{ {\rm nl_{2}} }({\epsilon}) := c_{F}( {\epsilon}^{\delta} + {\epsilon}^\gamma)+ c_S( {\epsilon}+ {\epsilon}^\delta) + c_{\lambda} {\epsilon}^{ k \delta-1} = {o}(1) .
\end{cases}$$ Note that $ { {\rm nl} }({\epsilon}) \leq { {\rm nl_{2}} }({\epsilon})$ and $ { {\rm nl} }({\epsilon}) \sim { {\rm nl_{2}} }({\epsilon})$ if $\delta \leq 1$ and $\delta \leq \gamma$. Recall that $k \geq 2$ is defined in .
Note that all the assumptions of the present section hold in the following case: (i) the ODE is linear ($F_{\epsilon}$ is linear), (ii) the associated matrix $\tau_{{\epsilon}}$ is symmetric positive and satisfies $$\tau_{{\epsilon}}^{-1} \geq C {\epsilon}^{1-\delta},$$ for some $\delta > 0$, (iii) the initial internal state is close to equilibrium in the sense that ${{\tau_{{\epsilon}}}}^{-1} Z_0={\mathcal{O}}(1)$. The assumptions of the present section can be seen as technical generalizations to non-linear ODEs with non-symmetric linear part.
Model with direct gradient sensing (control process)
----------------------------------------------------
We now turn to a simplified, “coarse” model, in which the internal process (\[eq:internal\]), and the corresponding state variables, are eliminated. Instead, the turning rate depends directly on the chemoattractant gradient. This process will be called the *control process* since it will be used in Section \[sec:method\] as a control variate to perform variance reduced simulations of (\[eq:process\_noscale\]).
The control process is a Markov process in position-velocity variables $$t \mapsto (X^c_t,V^c_t) ,$$ which evolves according to the following differential velocity-jump equations : $$\label{eq:cprocess}
\system{
& \frac{{\mathrm{d}}X_t^c}{{\mathrm{d}}t} = {\epsilon}V_t^c , &\\
& \int_{T_n^c}^{T_{n+1}^c} \lambda^c_{\epsilon}(X_t^c , V_t^c) {\mathrm{d}}t = \theta_{n+1} , & \\
& V_{t}^c = {\mathcal{V}}_n \quad \text{ for $t \in [T_{n}^c,T^c_{n+1}]$ } , &
}$$ with initial condition $X_0, {\mathcal{V}}_0 \in {\mathbb{R}}^d$. In (\[eq:cprocess\]), ${ \left(\theta _n \right) }_{n \geq 1}$ denote i.i.d. random variables with normalized exponential distribution, and ${ \left( {\mathcal{V}}_n \right) }_{n \geq 1}$ denote i.i.d. random variables with distribution $ {\mathcal M}(dv)$. The turning rate of the control process is assumed to satisfy $$\label{eq:rateboundc}
0 < \lambda_{\rm min} \leq \lambda^c_{{\epsilon}}(x,v) \leq \lambda_{\rm max},$$ as well as $$\label{eq:control_rate}
\lambda^c_{\epsilon}(x,v) := \lambda_0 - {\epsilon}\; A^T_{\epsilon}(x) v + {\mathcal{O}}({\epsilon}^2).$$
Typically, $\lambda^c_{{\epsilon}}(x,v)$ is a function of $\nabla S(x)$, so that the model may describe a large bacterium that is able to directly sense chemoattractant gradients. When $m=n=1$, and the turning rate (\[eq:control\_rate\]) is proportional to $ \nabla S(x) v \in {\mathbb{R}}$, it can be interpreted as follows: the rate at which a bacterium will change its velocity direction depends on the alignment of the velocity with the gradient of the chemoattractant concentration $\nabla S(x)$, resulting in a transport towards areas with higher chemoattractant concentrations. In the companion paper [@limits], it is shown that, when the vector field $A_{\epsilon}(x) \in {\mathbb{R}}^{d}$ is given by the formula $$\label{eq:Adef}
A_{\epsilon}(x) = b^T \frac{{{\tau_{{\epsilon}}}}}{\operatorname{{\rm Id}}+ \lambda_0 {{\tau_{{\epsilon}}}}} \nabla S(x),$$ asymptotic consistency with the internal state model is obtained in the limit when ${\epsilon}\to 0$ (see Section \[sec:model-hydro\]).
Kinetic formulation and advection-diffusion limits\[sec:model-hydro\]
---------------------------------------------------------------------
We now turn to the kinetic description of the probability distribution of the processes introduced above. The probability distribution density of the fine-scale process with internal state at time $t$ with respect to the measure ${\mathrm{d}}x \, {\mathcal M}( {\mathrm{d}}v ) \, {\mathrm{d}}y$ is denoted as $p(x,v,y,t)$, suppressing the dependence on ${\epsilon}$ for notational convenience, and evolves according to the Kolomogorov forward evolution equation (or master equation). In the present context, the latter is the following kinetic equation $$\label{e:kinetic}
\partial_t p + {\epsilon}v \cdot \nabla_x p + { {\rm div}}_y { \left( F_{\epsilon}(x,y) p \right) } = \lambda { \left(S(x)-y\right) } { \left( R(p) - p \right) },$$ where $$R(p) := \int_{v \in {\mathbb{S}}^{d-1}} p(\cdot,v,\cdot) \, {\mathcal M}(dv)$$ is the operator integrating velocities with respect to ${\mathcal M}$, and $F(x,y)$ and $\lambda(z)$ are defined as in Section \[sec:model-internal\]. Similarly, the distribution density of the control process is denoted as $p^c(x,v,t)$, and evolves according to the kinetic/master equation: $$\label{e:kinetic_control}
\partial_t p^c + {\epsilon}v \cdot \nabla_x p^c = { \left( R(\lambda^c_{\epsilon}p) - \lambda^c_{\epsilon}p \right) },$$ with $\lambda^c_{\epsilon}$ defined as in . The reader is referred to [@EthKur86] for the derivation of master equations associated to Markov jump processes.
In the companion paper [@limits], we show, using probabilistic arguments, that, in the limit ${\epsilon}\to 0$, both the equation for the control process and the equation for the process with internal state converge to an advection-diffusion limit on diffusive time scales. Convergence is to be understood *pathwise*, in the sense of convergence of probability distribution on paths endowed with the uniform topology.
Denote diffusive times by $${{\bar{t}}}:=t {\epsilon}^2,$$ and the processes considered on diffusive time scales as $$X^{{\epsilon}}_{{\bar{t}}}:=X_{{{\bar{t}}}/{\epsilon}^2}, \qquad X^{c,{\epsilon}}_{{\bar{t}}}:=X^c_{{{\bar{t}}}/{\epsilon}^2}.$$ For the control process, we then have the following theorem :
For ${\epsilon}\to 0$, the process ${{\bar{t}}}\mapsto X^{c,{\epsilon}}_{{\bar{t}}}$, solution of , converges towards an advection-diffusion process, satisfying the stochastic differential equation (SDE) $$\label{eq:cprocess_hydro}
{\mathrm{d}}X_{\bar{t}}^{c,0} ={ \left(\frac{D A_0(X_{\bar{t}}^{c,0})}{\lambda_0} {\mathrm{d}}{{\bar{t}}}+ { \left(\frac{2D}{\lambda_0}\right) }^{1/2} {\mathrm{d}}W_{\bar{t}}\right) },$$ where ${{\bar{t}}}\mapsto W_{{\bar{t}}}$ is a standard Brownian motion, the parameters $\lambda_0$ and $A_0:=\lim_{{\epsilon}\to 0} A_{\epsilon}$ originate from the turning rate , and the diffusion matrix is given by the covariance of the Maxwellian distribution : $$\label{eq:D}
D = \int_{{\mathbb{S}}^{d-1}} v \otimes v \, {\mathcal M}(dv) \in {\mathbb{R}}^{d \times d}.$$
In particular, this result implies that, at the level of the Kolomogorov/master evolution equation, the evolution of the position bacterial density, $$\label{eq:cdensity}
n^{c,{\epsilon}}(x,{{\bar{t}}}):= n^c(x,{{\bar{t}}}/{\epsilon}^2):=\int_{{\mathbb{V}}}p^c(x,v,{{\bar{t}}}/{\epsilon}^2) \, {\mathcal M}({\mathrm{d}}v)$$ converges to the advection-diffusion equation $$\label{eq:cdens_hydro}
\partial_{{\bar{t}}}n^{c,0} = \frac{1}{\lambda_0}{ {\rm div}}_x { \left( D \nabla_x n^{c,0} - D A_0(x) n^{c,0}\right) }$$ on diffusive time scales as $\epsilon \to 0$.
In the same way, a standard probabilistic diffusion approximation argument can be used to derive the pathwise diffusive limit of the process with internal state :
For ${\epsilon}\to 0$, the process ${{\bar{t}}}\mapsto X^{{\epsilon}}_{{\bar{t}}}$, solution of , converges towards an advection-diffusion process, satisfying the stochastic differential equation (SDE) , where $A_0$ originates from $$\label{eq:control_field}
A_0(x) = b^T \lim_{{\epsilon}\to 0} \frac{{{\tau_{{\epsilon}}}}}{\lambda_0 {{\tau_{{\epsilon}}}}+ \operatorname{{\rm Id}}} \nabla S(x),$$ in which, $b$, ${{\tau_{{\epsilon}}}}$, and $\lambda_0$ were introduced in - as parameters of the process with internal state, and $\operatorname{{\rm Id}}\in {\mathbb{R}}^{n\times n}$ is the identity matrix. Again, the diffusion matrix $D$ is given by the covariance of the Maxwellian distribution .
Also here, convergence needs to be understood in terms of probability distribution on paths endowed with the uniform topology.
Introducing the bacterial density of the process with internal state as $$n(x,t)= \int_{{\mathbb{Y}}}\int_{{\mathbb{V}}}p(x,v,y,t) {\mathcal M}({\mathrm{d}}v) {\mathrm{d}}y,$$ this implies that the evolution of $n$ converges to on diffusive time scales in the limit of $\epsilon \to 0$.
Numerical method\[sec:method\]
==============================
To simulate the process with internal state, solving the kinetic equation over diffusive time scales can be cumbersome, due to the additional dimensions associated with the internal state. The alternative is to use to stochastic particles. However, a particle-based simulation of equation is subject to a large statistical variance of the order $\mathcal{O}(N^{-1/2})$, where $N$ is the number of simulated particles. The asymptotic analysis shows that the position bacterial density approaches an advection-diffusion limit when ${\epsilon}\to 0$. Consequently, to accurately assess the deviations of the process with internal state as compared to its advection-diffusion limit (for small but non-vanishing (intermediate) values of ${\epsilon}$), the required number of particles needs to increase substantially with decreasing ${\epsilon}$, which may become prohibitive from a computational point of view.
In this section, we therefore propose a hybrid method, based on the principle of control variates, that couples the process with internal dynamics with the control process, which is simulated simultaneously using a grid-based method. We first describe the variance reduction technique (Section \[sec:coupl\_princ\]). The analysis in Section \[sec:variance\] will reveal that the variance reduction is *asymptotic*, in the sense that the variance vanishes in the diffusion limit. To ensure this asymptotic variance reduction during actual simulations, one needs to ensure that the time discretization preserves the diffusion limits of the time-continuous process. An appropriate time discretization is discussed in Section \[sec:discr\].
Coupling and asymptotic variance reduction\[sec:coupl\_princ\]
--------------------------------------------------------------
The proposed variance reduction technique is based on the introduction of a control variate that exploits a coupling between the process with internal state and the control process. While the idea of control variates for Monte Carlo simulation is already well known, see e.g. [@KloPla92] and references therein, the coupling that is proposed here is particular (we call it *asymptotic*), since the difference of the coupled processes, and hence the variance, vanishes with an estimable rate in the diffusion limit $\epsilon\to 0$, as will be shown in Section \[sec:variance\].
### The control variates
Let us first assume that we are able to compute the exact solution of the kinetic equation for the control process, , with infinite precision in space and time.
The algorithm of asymptotic variance reduction is based on a coupling between an ensemble of realizations evolving according to the process with internal state , denoted as $$\left\{X^i_t,V^i_t,Y^i_t\right\}_{i=1}^N,$$ and an ensemble of realization of the control process , denoted as $$\left\{X^{i,c}_t,V^{i,c}_{t}\right\}_{i=1}^N.$$ We denote the empirical measure of the particles with internal state in position-velocity space as $$\mu_{{\bar{t}}}^{N}(x,v) = \frac{1}{N} \sum_{i=1}^{N} \delta_{X^i_{{{\bar{t}}}/{\epsilon}^2},V^i_{{{\bar{t}}}/{\epsilon}^2}},$$ and, correspondingly, the empirical measure of the control particles as $$\mu_{{\bar{t}}}^{c,N} = \frac{1}{N} \sum_{i=1}^{N} \delta_{X^{c,i}_{{{\bar{t}}}/{\epsilon}^2},V^{c,i}_{{{\bar{t}}}/{\epsilon}^2}}.$$
A coupling between the two ensembles is obtained by ensuring that both simulations use *the same random numbers* $(\theta_n)_{n\ge 1}$ and $({\mathcal{V}}_n)_{n\ge 0}$, which results in a strong correlation between $(X^i_{t},V^{i}_t)$ and $(X_t^{i,c},V^{c,i}_t)$ for each realization. Simultaneously, the kinetic equation for the control process is also solved using a deterministic method (which, for now, is assumed to be exact). We formally denote the corresponding semi-group evolution as $${{\rm e}}^{{{\bar{t}}}L^c}, \qquad \text{ with } L^c(p^c)= - {\epsilon}v \cdot \nabla_x p^c + { \left( R(\lambda^c_{\epsilon}p^c) - \lambda^c_{\epsilon}p^c \right) }.$$ Besides the two particle measures $\mu^N_{\bar{t}}$ and $\mu^{c,N}_{\bar{t}}$, we denote by $\overline{\mu}^N_{\bar{t}}$ the variance reduced measure, which will be defined by the algorithm below. Since, with increasing diffusive time, the variance of the algorithm increases due to a loss of coupling between the particles with internal state and the control particles, the variance reduced algorithm will also make use of a reinitialization time step ${\overline{\delta t}_{ri}}$, which is defined on the diffusive time scale. The corresponding time instances are denoted as $\bar{t}_n=n{\overline{\delta t}_{ri}}$ on the diffusive time scale, or equivalently, on the original time scale as $t_n=n{\overline{\delta t}_{ri}}/{\epsilon}^2$.
Starting from an initial probability measure $\mu_0$ at time $t=0$, we sample $\mu_0$ to obtain the ensemble $\left\{X^i_t,V^i_t,Y^i_t\right\}_{i=1}^N$, corresponding to $\mu^N_0$, and then set $\mu^{c,N}_0:=\mu^N_0$, i.e., $X^{i,c}_{0}=X^{i}_{0}$ and $V^{i,c}_{0}=V^{i}_{0}$ for all $i=1,\ldots,N$. Furthermore, we set the variance reduced estimator as $\overline{\mu}_0^N:=\mu_0={ {\mathbb E} }(\mu_0^N)$. We then use the following algorithm to advance from $\bar{t}_n$ to $\bar{t}_{n+1}$, (see also Figure ) :
![\[fig:pic\] A schematic description of Algorithm \[algo\]. The dashed line represent the evolution of $N$ bacteria with internal state. The dotted line represent the coupled evolution of $N$ bacteria with gradient sensing, subject to regular reinitializations. The dashed-dotted line is computed according to a deterministic method simulating the density of the model with gradient sensing, and subject to regular reinitializations. The solid line is the variance reduced simulation of the internal state dynamics, and is computed by adding the difference between the particle computation with internal state, and the particle simulation with gradient sensing to the deterministic gradient sensing simulation. At each reinitialization step, the two simulations (deterministic and particles) of the gradient sensing dynamics are reinitialized to the values of their internal state simulation counterpart (as represented by the arrows).](figuren/fig-pic.pdf)
\[algo\] At time $t_n$, we have that the particle measure $\mu_{\bar{t}_n}^{c,N}= \mu_{\bar{t}_n}^{N}$, and the variance reduced measure is given by $\overline{\mu}^N_{\bar{t}_n}$. To advance from time $ \bar{t}_n$ to $\bar{t}_{n+1}$, we perform the following steps :
- Evolve the particles $\left\{X^i_t,V^i_t,Y^i_t\right\}_{i=1}^N$ from $t_n$ to $t_{n+1}$, according to ,
- Evolve the particles $\left\{X^{i,c}_t,V^{i,c}_{t}\right\}_{i=1}^N$ according to , using the same random numbers as for the process with internal state,
- Compute the variance reduced evolution $$\label{eq:var_reduced_estimation}
\overline{\mu}^N_{\bar{t}_{n+1}} = \overline{\mu}^N_{\bar{t}_n} {{\rm e}}^{{\overline{\delta t}_{ri}}/ {\epsilon}^2 L^c} + { \left(\mu_{\bar{t}_{n+1}}^{N} - \mu_{\bar{t}^{-}_{n+1}}^{c,N} \right) }.$$
- Reinitialize the control particles by setting $$X^{i,c}_{t_{n+1}}=X^{i}_{t_{n+1}}, \qquad V^{i,c}_{t_{n+1}}=V^{i}_{t_{n+1}}, \qquad i=1,\ldots,N,$$ i.e., we set the state of the control particles to be identical to the state of the particles with internal state.
In formula , we use the symbol $\bar{t}^{-}_{n+1}$ to emphasize that the involved particle positions and velocities are those obtained *before* the reinitialization. An easy computation shows that the algorithm is unbiased in the sense that for any $n \geq 0$, $${ {\mathbb E} }{ \left( \overline{\mu}^N_{\bar{t}_n}\right) } = { {\mathbb E} }{ \left(\mu^N_{\bar{t}_n}\right) },$$ since the particles with internal dynamics are unaffected by the reinitialization, and, additionally, $${ {\mathbb E} }\mu^{c,N}_{\bar{t}_{n+1}}= { {\mathbb E} }\overline{\mu}^N_{{{\bar{t}}}_n}{{\rm e}}^{{\overline{\delta t}_{ri}}/ {\epsilon}^2 L^c}.$$ Moreover, the variance is controlled by the coupling between the two processes. Indeed, using the independence of the random numbers between two steps of Algorithm \[algo\], and introducing ${\varphi}$ as a position and velocity dependent test function), we get (according to the main result of Theorem \[thm:coupling\]) $$\begin{aligned}
\operatorname{{\rm var}}( \overline{\mu}^N_{\bar{t}_n}({\varphi}) ) &= \sum_{k=1}^n { {\mathbb E} }{ \left(\mu_{}^{N}({\varphi}) - \mu_{\bar{t}_k^{-}}^{c,N}({\varphi}) \right) }^2 \nonumber \\
& \leq \sum_{k=1}^n \frac{{\left\Vert\nabla {\varphi}\right\Vert}_\infty}{N} { {\mathbb E} }{ \left( {\left | X_{\bar{t}_k/{\epsilon}^2} - X_{\bar{t}_k/{\epsilon}^2}^c \right |}^2 \right) } , \nonumber\end{aligned}$$ and thus $$\begin{aligned}
\label{eq:MC_bound}
\operatorname{{\rm var}}( \overline{\mu}^N_{\bar{t}_n}({\varphi}) )
& \leq C n\frac{ {\epsilon}+ {\epsilon}^\delta + { {\rm nl_{2}} }({\epsilon}) }{N},\end{aligned}$$ where in the last line, $C$ is independant of $n$, ${\epsilon}$, and $N$.
In some generic situations (see Section \[sec:sharp\]), we can argue that the statistical error in Algorithm \[algo\] coming from the coupling is “sharp” with respect to the order in ${\epsilon}$. This means that the difference between the probability distribution of the model with internal state and the probability distribution of the model with gradient sensing is of the same order. This would imply that, with the asymptotic variance reduction technique, one is able to reliably assess the true deviation of the process with internal variables from the control process using a number of particles $N$ that is independent of ${\epsilon}$.
### Time-space discretization of the kinetic equation
For both processes, the probability density distribution of the position and velocity can then be computed via binning in a histogram, or via standard kernel density estimation [@WScott:1992p8096; @WSilverman:1986p8151] using a kernel $K_h$, in which the chosen bandwidth $h$ can be based on the grid used for the deterministic simulation and on the data, for instance using the Silverman heuristic [@WSilverman:1986p8151]. This yields $$\label{eq:kde}
\hat{p}_N(x,v,t) = \frac{1}{N h} \sum_{i=1}^{N} K_h(x - X^i_{t},v-V^i_{t}),$$ and similarly $$\label{eq:kdec}
\hat{p}^c_N(x,v,t) = \frac{1}{N h} \sum_{i=1}^{N} K_h(x - X^{c,i}_{t},v-V^{c,i}_{t}).$$
The important point is that the solution $p^c(x,v,t)$ for the control process may be accurately approximated with a deterministic grid-based method, which ensures that $${\left | p^c(x,v,{{\bar{t}}}/ {\epsilon}^2)-p^c_{\rm grid}(x,v,{{\bar{t}}}/ {\epsilon}^2)\right |} = {\mathcal{O}}(\delta x^l) +{\mathcal{O}}(\delta t^k),$$ for some integers $k,l\ge 1$.
In this text, we will perform the simulations in one space dimension using a third-order upwind-biased scheme, and perform time integration using the standard fourth order Runge–Kutta method, i.e., $l=3$ and $k=4$.
Then, the unbiased nature of the variance-reduced estimator is conserved up to ${\mathcal{O}}(\delta x^l)+{\mathcal{O}}(\delta t^k)+{\mathcal{O}}(h)$ discretization errors.
Time discretization of velocity-jump processes \[sec:discr\]
------------------------------------------------------------
When simulating equation , a time discretization error originates from the fact that the equation for the evolution of the internal state, and hence the evolution of the fluctuations $Z_t$, is discretized in time. This results in an approximation of the jump times $(T_n)_{n\ge 1}$, and hence of $X_t$. To retain the diffusion limits of the time-continuous process, special care is needed. We now briefly recall the time discretization procedure that was proposed and analyzed in the companion paper [@limits]. For ease of exposition, we consider the scalar equation for the internal state; generalization to nonlinear systems of equations is briefly discussed in [@limits].
Consider first the linear turning rate . In that case, we define a numerical solution $({X^{\delta t}}_t,{V^{\delta t}}_t,{Y^{\delta t}}_t)$ as follows. Between jumps, we discretize the simulation in steps of size $\delta t$ and denote by $({X^{\delta t}}_{n,k},{Z^{\delta t}}_{n,k})$ the solution at $t_{n,k}={T^{\delta t}}_n+k\delta t$. The numerical solution for $t\in[t_{n,k},t_{n,k+1}]$ is given by $$\label{e:discr-nonlin}
\begin{cases}
{X^{\delta t}}_t = {X^{\delta t}}_{n,k}+{\epsilon}{\mathcal{V}}_{n}\;(t-t_{n,k}) \\
{Z^{\delta t}}_t = \exp(-(t-t_{n,k}) {{\tau_{{\epsilon}}}}^{-1}) {Z^{\delta t}}_{n,k} + {\epsilon}{{\tau_{{\epsilon}}}}{ \left(\operatorname{{\rm Id}}-\exp(-(t-t_{n,k}){{\tau_{{\epsilon}}}}^{-1})\right) } \nabla S({X^{\delta t}}_{n,k}) \, {\mathcal{V}}_n.
\end{cases}$$ We denote by $K\geq 0$ the integer such that the simulated jump time ${T^{\delta t}}_{n+1}\in [t_{n,K},t_{n,K+1}]$. To find ${T^{\delta t}}_{n+1}$, we first approximate the integral $ \int_{{T^{\delta t}}_n}^{{T^{\delta t}}_{n+1}}\lambda({Z^{\delta t}}_t){\mathrm{d}}t$ using $$\begin{aligned}
\label{eq:time-disc-lin}
\int_{{T^{\delta t}}_n}^{{T^{\delta t}}_{n+1}}\lambda({Z^{\delta t}}_t){\mathrm{d}}t &=\sum_{k=0}^{K-1}\int_{t_{n,k}}^{t_{n,k+1}}\lambda({Z^{\delta t}}_t){\mathrm{d}}t
+ \int_{t_{n,K}}^{{T^{\delta t}}_{n+1}}\lambda({Z^{\delta t}}_t){\mathrm{d}}t ,\end{aligned}$$ and then compute : $$\begin{gathered}
\int_{t_{n,k}}^{t_{n,k+1}}\lambda({Z^{\delta t}}_t){\mathrm{d}}t=
\lambda_0 \delta t - b^T { \left(\operatorname{{\rm Id}}- {{\rm e}}^{- {\delta t}{{{\tau_{{\epsilon}}}}^{-1}}} \right) }{{{\tau_{{\epsilon}}}}} {Z^{\delta t}}_{T_n} \\
- {\epsilon}b^T { \left(\delta t {{\tau_{{\epsilon}}}}-(\operatorname{{\rm Id}}-{{\rm e}}^{-\delta t {{\tau_{{\epsilon}}}}^{-1} } ){{\tau_{{\epsilon}}}}^2 \right) } \nabla S({X^{\delta t}}_{n,k}) {\mathcal{V}}_n .\end{gathered}$$ The jump time ${T^{\delta t}}_{n+1}$ can then be computed as the solution of $$\label{e:lin-Newton}
\int_{t_{n,K}}^{{T^{\delta t}}_{n+1}}\lambda({Z^{\delta t}}_t){\mathrm{d}}t=\theta_{n+1}- \sum_{k=0}^{K-1}\int_{t_{n,k}}^{t_{n,k+1}}\lambda({Z^{\delta t}}_t){\mathrm{d}}t,$$ using a Newton procedure. It is shown in [@limits] that the results on the diffusion limit (as outlined in Section \[sec:model-hydro\]) are not affected by the discretization.
We now consider a general nonlinear turning rate. We again discretize in time to obtain the time-discrete solution . The jump time ${T^{\delta t}}_{n+1}$ is now computed by linearizing in each time step, $$\label{eq:nonlinrate_disc}
{\lambda^{\delta t}}{ \left({Z^{\delta t}}_t,{Z^{\delta t}}_{n,k}\right) }=\lambda({Z^{\delta t}}_{n,k})+ \frac{{\mathrm{d}}\lambda({Z^{\delta t}}_{n,k})}{{\mathrm{d}}z}{ \left({Z^{\delta t}}_t-{Z^{\delta t}}_{n,k}\right) }.$$ One then replaces equation by $$\begin{aligned}
\int_{{T^{\delta t}}_n}^{{T^{\delta t}}_{n+1}}{\lambda^{\delta t}}({Z^{\delta t}}_t){\mathrm{d}}t &=\sum_{k=0}^{K-1}\int_{t_{n,k}}^{t_{n,k+1}}{\lambda^{\delta t}}({Z^{\delta t}}_t,{Z^{\delta t}}_{n,k}){\mathrm{d}}t
+ \int_{t_{n,K}}^{{T^{\delta t}}_{n+1}}{\lambda^{\delta t}}({Z^{\delta t}}_t,{Z^{\delta t}}_{n,K}){\mathrm{d}}t, \end{aligned}$$ and proceeds in the same way as for the linear case. Also in this case, the diffusive limit is recovered in an exact fashion for the time-discretized process.
For the control process, there is no internal state. Hence, the only time dependence of turning rate $\lambda^c$ is due to the spatial variation of $\nabla S(x)$, which can be treated by discretizing the integral of $\lambda$ in the same way as above.
Asymptotic variance reduction of the coupling\[sec:variance\]
=============================================================
In this section, we show how the difference between the two coupled processes on diffusive time scales $\bar{t}=t/{\epsilon}^2$ behaves in the limit of ${\epsilon}\to 0$. We first recall some notation from the companion paper [@limits] (Section \[sec:not\]), after which we state and prove the main theorem (Section \[sec:proof\]).
Notations and asymptotic estimates\[sec:not\]
---------------------------------------------
The main theorem in Section \[sec:proof\] relies on following auxiliary definitions and lemmas that were given in the companion paper [@limits]:
We denote by $m:{\mathbb{R}}\to {\mathbb{R}}^{n \times n}$ the function $$\label{eq:m(t)}
m(t) := t {{\tau_{{\epsilon}}}}- { \left(\operatorname{{\rm Id}}- {\rm e}^{-t {{\tau_{{\epsilon}}}}^{-1}}\right) } {{\tau_{{\epsilon}}}}^2,$$ whose derivative is given by $$m'(t) = {{\tau_{{\epsilon}}}}{ \left(\operatorname{{\rm Id}}- {\rm e}^{-t {{\tau_{{\epsilon}}}}^{-1}} \right) }.$$
This function satisfies the following lemmas :
Let $\theta$ be an exponential random variable of mean $1$. Then: $$\label{eq:av_m}
{ {\mathbb E} }{ \left( m\left( \frac{\theta}{\lambda_0} \right) \right) } = \frac{1}{\lambda_0} \frac{{{\tau_{{\epsilon}}}}}{\operatorname{{\rm Id}}+ \lambda_0 {{\tau_{{\epsilon}}}}}=A_{\epsilon}(x),$$ see equation .
For all $t \in {\mathbb{R}}$, we have ${\left\Vert m'(t) \right\Vert} \leq t$, as well as ${\left\Vert m(t) \right\Vert} \leq t^2/2$.
The difference between jump times is denoted as $$\Delta T_{n+1}^{c} := T_{n+1}^{c}- T_{n}^{c}, \qquad \Delta T_{n+1} := T_{n+1}- T_{n}.$$
The proofs of the asymptotic variance reduction make use of the following asymptotic expansions of the jump time differences of both processes :
\[lem:dt-control\] The difference between two jump times of the control process satisfies $$\label{eq:deltaTc}
\Delta T_{n+1}^{c} = \frac{\theta_{n+1}}{\lambda_0} + {\epsilon}\frac{\theta_{n+1}}{\lambda_0^2} A_{\epsilon}^T(X_{T^c_n}^c) {\mathcal{V}}_n + \theta_{n+1} {\mathcal{O}}({\epsilon}^2 ).$$
and
\[lem:DTestim\] The jump time variations of the process with internal state can be written in the following form (${ {\rm nl} }({\epsilon})$ being defined by ) : $$\label{eq:deltaT}
\Delta T_{n+1} = \Delta T_{n+1}^0 + {\epsilon}\Delta T_{n+1}^1 + (\theta_{n+1}^6+\theta_{n+1}){\mathcal{O}}{ \left({\epsilon}^2+ {\epsilon}\,{ {\rm nl} }({\epsilon})\right) } ,$$ where $$\begin{aligned}
\label{eq:estimDT0}
\Delta T_{n+1}^0 &= \frac{\theta_{n+1}}{\lambda_0} + \frac{b^T}{\lambda_0} m'(\Delta T_{n+1}^0 ) Z_{T_n} \nonumber \\
&= \frac{\theta_{n+1}}{\lambda_0} + \theta_{n+1} {\mathcal{O}}( {\epsilon}^\delta) \end{aligned}$$ and, correspondingly, $$\begin{aligned}
\label{eq:estimDT1}
\Delta T_{n+1}^1 &= \frac{1}{\lambda_0 - b^T {{\rm e}}^{- {\Delta T_{n+1}^0}{\tau^{-1}}} Z_{T_n} } b^T m(\Delta T_{n+1}^0) \nabla S(X_{T_n}) {\mathcal{V}}_{n}\nonumber \\
& = \frac{1}{\lambda_0} b^T m{ \left(\frac{\theta_{n+1}}{\lambda_0}\right) } \nabla S(X_{T_n}) {\mathcal{V}}_{n} + \theta_{n+1} {\mathcal{O}}({\epsilon}^\delta) .\end{aligned}$$
It is also useful to recall that according to -, the following hold : $${\left | \Delta T_{n+1}\right |} \leq C \theta_{n+1}, \qquad {\left | \Delta T_{n+1} ^c\right |} \leq C \theta_{n+1}.$$ Finally, we will also need a different expansion of the jump times of the process with internal state :
\[lem:DTestimbis\] When using ${ {\rm nl} }_2({\epsilon})$ as defined by , the jump time variations of the process with internal state can be written in the following form : $$\label{eq:deltaTbis}
\begin{split}
& \Delta T_{n+1} = \frac{\theta_{n+1}}{\lambda_0} - \frac{b^T {{\tau_{{\epsilon}}}}}{\lambda_0}(Z_{T_{n+1}}-Z_{T_{n}})+ {\epsilon}\Delta T _{0 } b^T {{\tau_{{\epsilon}}}}\nabla S (X_{T_n}) {\mathcal{V}}_n \\
& + { \left(\theta_{n+1}^3+\theta_{n+1}\right) }{\mathcal{O}}{ \left( {\epsilon}\,{ {\rm nl_{2}} }({\epsilon}) + {\epsilon}^{1+\delta} \right) }.
\end{split}$$
The proof is based on Duhamel integration of the ODE on $[T_n,T_{n+1}]$, as in the proof of Lemma 4.7 of the companion paper [@limits]. Following equation (4.24) in [@limits], we get for $t\in [T_n,T_{n+1}]$ : $$\begin{aligned}
\label{eq:key_Duhamel}
Z_t = {{\rm e}}^{- { \left(t-T_n\right) }{{{\tau_{{\epsilon}}}}^{-1}}} Z_{T_n} + {\epsilon}{ \left(\operatorname{{\rm Id}}- {{\rm e}}^{- { \left(t-T_n\right) }{{{\tau_{{\epsilon}}}}^{-1}}} \right) }{{{\tau_{{\epsilon}}}}} \nabla S(X_{T_n}) {\mathcal{V}}_{n} + (\theta_{n+1}^2+\theta_{n+1}){\mathcal{O}}{ \left(c_S {\epsilon}^2+ c_F {\epsilon}^{1+\delta} \right) }.\end{aligned}$$ Multiplying both sides by $\tau_{\epsilon}$, recalling that Assumption \[a:2\] implies $\|\tau_{\epsilon}\|={\mathcal{O}}({\epsilon}^{\delta-1})$, and using Assumption \[a:1bis\], we get for $t\in [T_n,T_{n+1}]$ : $$\begin{aligned}
\label{e:help}
{{\tau_{{\epsilon}}}}Z_t = {{\tau_{{\epsilon}}}}{{\rm e}}^{- { \left(t-T_n\right) }{{{\tau_{{\epsilon}}}}^{-1}}} Z_{T_n} + {\epsilon}{ \left(\operatorname{{\rm Id}}- {{\rm e}}^{- { \left(t-T_n\right) }{{{\tau_{{\epsilon}}}}^{-1}}} \right) }{{{\tau_{{\epsilon}}}}^2} \nabla S(X_{T_n}) {\mathcal{V}}_{n} + (\theta_{n+1}^2+\theta_{n+1}){\mathcal{O}}{ \left(c_S {\epsilon}^{1+\delta}+ c_F {\epsilon}^{1+\gamma} \right) }.\end{aligned}$$ Moreover, integrating on $t\in [T_n,T_{n+1}]$ yields $$\begin{aligned}
\label{eq:key_Duhamelint}
\begin{split}
\int_{T_n}^{T_{n+1}} Z_t {\mathrm{d}}t &= { \left(\operatorname{{\rm Id}}- {{\rm e}}^{- { \left(T_{n+1}-T_n\right) }{{{\tau_{{\epsilon}}}}^{-1}}} \right) }{{{\tau_{{\epsilon}}}}} { \left(Z_{T_n} -{\epsilon}{{\tau_{{\epsilon}}}}\nabla S(X_{T_n}) {\mathcal{V}}_{n} \right) } \\
& \quad +{\epsilon}(T_{n+1}-T_n){{\tau_{{\epsilon}}}}\nabla S(X_{T_n}) {\mathcal{V}}_{n} + (\theta_{n+1}^3+\theta_{n+1}^2){\mathcal{O}}{ \left(c_S {\epsilon}^2+ c_F {\epsilon}^{1+\delta} \right) }.
\end{split}\end{aligned}$$ Evaluating equation at $t=T_{n+1}$, add adding equation , we get $$\begin{aligned}
\int_{T_n}^{T_{n+1}} Z_t {\mathrm{d}}t + {{\tau_{{\epsilon}}}}Z_{T_{n+1}} = & {{\tau_{{\epsilon}}}}Z_{T_n} +{\epsilon}(T_{n+1}-T_n){{\tau_{{\epsilon}}}}\nabla S(X_{T_n}) {\mathcal{V}}_{n} \\
& \quad + { \left(\theta_{n+1}^3+\theta_{n+1}\right) }{\mathcal{O}}{ \left(c_S({\epsilon}^{1+\delta}+{\epsilon}^2 ) + c_F({\epsilon}^{1+\delta}+{\epsilon}^{1+\gamma}) \right) } . \end{aligned}$$ The estimates in Lemma $4.8$ from [@limits] (i.e., ${\left | \Delta T_{n+1}-\Delta T_{n+1}^0\right |} = (\theta_{n+1}^4+\theta_{n+1}){\mathcal{O}}({\epsilon}) $) yields $$\begin{aligned}
\int_{T_n}^{T_{n+1}} Z_t {\mathrm{d}}t + {{\tau_{{\epsilon}}}}Z_{T_{n+1}} = & {{\tau_{{\epsilon}}}}Z_{T_n} +{\epsilon}\Delta T_{n+1}^0 {{\tau_{{\epsilon}}}}\nabla S(X_{T_n}) {\mathcal{V}}_{n} \\
& \quad + { \left(\theta_{n+1}^3+\theta_{n+1}\right) }{\mathcal{O}}{ \left(c_S({\epsilon}^{1+\delta}+{\epsilon}^2 ) + (c_F+1){\epsilon}^{1+\delta}+c_F{\epsilon}^{1+\gamma}) \right) } . \end{aligned}$$
Finally, plugging the last equation in the estimate yields the result.
Analysis of variance of the coupling\[sec:proof\]
-------------------------------------------------
The present section is devoted to the proof of the following theorem:
\[thm:coupling\] Assume the assumptions of Section \[sec:asympt\] hold, and that $k \delta > 1$ where $k\geq 2$ is defined in . Then the difference between the process with internal state and the coupling process satisfies : $$\label{eq:coupl_estim}
{ {\mathbb E} }{ \left({ \left( X_{{{\bar{t}}}/{\epsilon}^2} - X^c_{{{\bar{t}}}/{\epsilon}^2} \right) } ^ p\right) }^{1/p} = {\mathcal{O}}( {\epsilon}+ {\epsilon}^\delta + { {\rm nl_{2}} }({\epsilon}) ), \qquad p\ge 1,$$ where ${ {\rm nl_{2}} }({\epsilon})$ is defined in .
The proof of Theorem \[thm:coupling\] relies on a number of steps that will be detailed by a series of lemmas. We will make use of the following random sequences, defined as the position of both processes after $n$ jumps, $$\label{eq:Xi}
\Xi_n := X_{T_n}= X_0 + {\epsilon}\sum_{m=0}^{n-1}\Delta T _{m+1} {\mathcal{V}}_m, \qquad n\geq 0,$$ as well as $$\label{eq:Xic}
\Xi^c_n := X_{T^c_n}^c= X_0 + {\epsilon}\sum_{m=0}^{n-1}\Delta T _{m+1}^c {\mathcal{V}}_m, \qquad n\geq 0.$$ We will also make use of the following random integers :
\[def:N\_Nc\] The random integers $N \geq 1$ and $N^c \geq 1$ are uniquely defined by: $$T_N \leq {{\bar{t}}}/ {\epsilon}^2 < T_{N+1}, \qquad T^c_{N^c} \leq {{\bar{t}}}/ {\epsilon}^2 < T^c_{N^c+1} .$$
The first lemma bounds the difference at time ${{\bar{t}}}/ {\epsilon}^2$ between the two processes $X_{{{\bar{t}}}/{\epsilon}^2}$ and $X^c_{{{\bar{t}}}/{\epsilon}^2}$ by expressing it in terms of differences of positions and jump times of both processes after the same random number of jumps, $\Xi_n$ and $\Xi_n^c$ :
The difference between the rescaled process with internal state $X^{\epsilon}_{{{\bar{t}}}}:= X_{{{\bar{t}}}/{\epsilon}^2}$ and the rescaled coupling process $X^{c,{\epsilon}}_{{{\bar{t}}}}:= X^c_{{{\bar{t}}}/{\epsilon}^2}$ satisfies: $$\begin{aligned}
{\left | X^{\epsilon}_{{{\bar{t}}}} - X^{{\epsilon},c}_{{{\bar{t}}}}\right |} & \leq {\left | \Xi_{N}-\Xi_{N}^c\right |} + {\epsilon}{\left | T_{N}-T_{N}^c\right |}+{ \left(\theta_{N+1}+\theta_{N_c+1}\right) } {\mathcal{O}}({\epsilon}) \label{eq:coupl_11} \\
& \leq {\left | \Xi_{N^c}-\Xi_{N^c}^c\right |} + {\epsilon}{\left | T_{N^c}-T_{N^c}^c\right |}+{ \left(\theta_{N+1}+\theta_{N_c+1}\right) } {\mathcal{O}}({\epsilon}) \label{eq:coupl_12} \end{aligned}$$
Only one of the two estimates - is necessary in the remainder of the proof, but we detail both to highlight the symmetry.
By definition, we have $$\begin{cases}
X^{\epsilon}_{{{\bar{t}}}}= \Xi_N+ {\epsilon}({{\bar{t}}}/{\epsilon}^2-T_N){\mathcal{V}}_{N}, \\
X^{c,{\epsilon}}_{{{\bar{t}}}}= \Xi_{N^c}^c+ {\epsilon}({{\bar{t}}}/{\epsilon}^2-T_{N^c}^c){\mathcal{V}}_{N},
\end{cases}$$ so that by Definition \[def:N\_Nc\] of $N$ and $N^c$, and by realizing that ${\left | {{\bar{t}}}/{\epsilon}^2-T_{N^c}^c\right |} \leq C \theta_{N_c+1}$ and ${\left | {{\bar{t}}}/{\epsilon}^2-T_{N}\right |} \leq C \theta_{N+1}$, we get $$\label{eq:diffX}
\begin{cases}
{\left | X^{\epsilon}_{{{\bar{t}}}}- X^{c,{\epsilon}}_{{{\bar{t}}}}\right |}\leq {\left | \Xi_N-\Xi^c_N\right |}+\underbrace{{\left | \Xi^c_{N^c}-\Xi^c_{N}\right |}}_{(a)}+{ \left(\theta_{N+1}+\theta_{N_c+1}\right) } {\mathcal{O}}({\epsilon}) , \\
{\left | X^{\epsilon}_{{{\bar{t}}}}- X^{c,{\epsilon}}_{{{\bar{t}}}}\right |} \leq {\left | \Xi_{N^c}-\Xi^c_{N^c}\right |}+\underbrace{{\left | \Xi_{N^c}-\Xi_{N}\right |}}_{(b)}+ { \left(\theta_{N+1}+\theta_{N_c+1}\right) } {\mathcal{O}}({\epsilon}).
\end{cases}$$ To analyze $(a)$, we consider three cases:
- Then $(a)=0$.
- Then $$\begin{aligned}
\frac{(a)}{{\epsilon}} & = {\left | \sum_{n = N_c}^{N-1} \Delta T _{n+1}^c {\mathcal{V}}_n \right |} \\
& \leq T _{N}^c-T_{N_c}^c=\Delta T^c_{N^c+1} + T^c_ N - T^c_{N^c+1} \\
&\leq T^c_ N - T_{N} + C \theta_{N^c+1},
\end{aligned}$$ where in the last line we have used that, $\Delta T^c_{N^c+1}\le C\theta_{N^c+1}$, and, by definition, $ T_{N} \leq {{\bar{t}}}/ {\epsilon}^2 < T^c_{N^c+1}$.
- Then $$\begin{aligned}
\frac{(a)}{{\epsilon}} & = {\left | \sum_{n = N}^{N_c-1} \Delta T _{n+1}^c {\mathcal{V}}_n \right |} \\
& \leq T _{N_c}^c-T_{N}^c \\
&\leq T_{N+1} - T_{N}^c = \Delta T_{N+1} + T_N-T_N^c \leq T_ N - T^c_{N} + C \theta_{N+1} ,
\end{aligned}$$ where in the last line we have used that by definition $ T^c_{N^c} \leq {{\bar{t}}}/ {\epsilon}^2 \leq T_{N+1}$.
This proves . By symmetry, we get: $$\frac{(b)}{{\epsilon}} \leq {\left | T_{N^c} - T^c_{N^c}\right |} + C{ \left(\theta_{N+1}+\theta_{N^c+1}\right) },$$ which yields .
In the next lemma, we estimate the supremum of the difference of the position of both processes after $n$ jumps for $n\in [0,n_{\epsilon}]$.
Let $n_{\epsilon}\geq 1$ be a deterministic integer verifying $n_{\epsilon}= {\mathcal{O}}({\epsilon}^{-2})$. Then $$\label{eq:estim_jumps}
{ {\mathbb E} }{ \left( \sup_{0 \leq n \leq n_{\epsilon}}{\left | \Xi_n-\Xi_n^c\right |}^p \right) } ^{1/p} = {\mathcal{O}}({\epsilon}+ {\epsilon}^\delta + { {\rm nl} }({\epsilon}) ).$$
First, using the estimates of jump times -, we can decompose the differences of positions of both processes as follows : $$\begin{aligned}
\Xi_{n}-\Xi_{n}^c &= \sum_{m=0}^{n-1}{\epsilon}(\Delta T _ {m+1}-\Delta T _ {m+1} ^c) {\mathcal{V}}_{m} \\
& = \sum_{m=0}^{n-1} \alpha_{m+1} + d_{m+1} + r_{m+1},
\end{aligned}$$ where, by definition, $$\begin{cases}
\alpha_{m+1} := {\epsilon}^2 \frac{{\mathcal{V}}_m}{\lambda_0^2} { \left(A_{\epsilon}(\Xi_{m})^T-A_{\epsilon}(\Xi_{m}^c)^T \right) } {\mathcal{V}}_m, \\
d_{m+1} :={\epsilon}^2\frac{{\mathcal{V}}_m}{\lambda_0^2} { \left(1-\theta_{m+1}\right) }A_{\epsilon}(\Xi_{m}^c)^T {\mathcal{V}}_m \\
\hspace{1.5cm} + {\epsilon}^2 \frac{{\mathcal{V}}_{m}}{\lambda_0^2} { \left( \lambda_0 b^T m{ \left(\frac{\theta_{m+1}}{\lambda_0}\right) } \nabla S(\Xi_m) - A_{\epsilon}(\Xi_{m})^T \right) } {\mathcal{V}}_{m} \\
\hspace{1.5cm} + {\epsilon}b^T m'(\Delta T_{m+1}^0 ) Z_{T_m} {\mathcal{V}}_m , \\
r_{m+1}:= { \left(\theta_{m+1}^6+\theta_{n+1}\right) } {\mathcal{O}}({\epsilon}^{3} + {\epsilon}^2 { {\rm nl} }({\epsilon}) ).
\end{cases}$$ Since $A_{\epsilon}$ is Lipschitz, we have that $\alpha_{m+1} \leq C {\epsilon}^2 {\left | \Xi_{m}-\Xi_{m}^c\right |}$, and we can write $${\left | \Xi_{n}-\Xi_{n}^c\right |} \leq C{\epsilon}^2 \sum_{m=0}^{n-1} {\left | \Xi_{m}-\Xi_{m}^c\right |} + \sup_{0 \leq l \leq n_{{\epsilon}}}{\left | \sum_{m=1}^{l} d_{m}\right |}+ \sum_{m=1}^{n_{\epsilon}}{\left | r_{m}\right |},$$ for $n \leq n_{{\epsilon}}$, which yields $$\begin{aligned}
{\left | \Xi_{n}-\Xi_{n}^c\right |} &\leq { \left(\sup_{0 \leq l \leq n_{{\epsilon}}}{\left | \sum_{m=1}^{l} d_{m}\right |}+ \sum_{m=1}^{n_{\epsilon}}{\left | r_{m}\right |}\right) } { \left(1+C {\epsilon}^2+ \dots + (C {\epsilon}^2)^{n_{\epsilon}}\right) }, \label{eq:key_xi}.\end{aligned}$$ Note that $${ \left(1+C {\epsilon}^2+ \dots + (C {\epsilon}^2)^{n_{\epsilon}}\right) } \leq C .$$ Now, we can remark that the discrete time process $
{ \left(M_l\right) }_{l \geq 0}:={ \left( \sum_{m=1}^{l} d_{m} \right) }_{l \geq 0}
$ is a martingale with respect to the filtration $\mathcal F _{l} = \sigma{ \left((\theta_{m},{\mathcal{V}}_{m-1}) \vert 1 \leq m \leq l \right) }$ with $M_0=0$, so that we can apply the Burkholder-Davies-Gundy upper bound (which is a simple consequence of Doob maximal inequality here, see [@Karatzasbook]), for some $p \geq 1$: $$\label{eq:BDG}
{ {\mathbb E} }{ \left(\sup_{0 \leq l \leq n_{\epsilon}} {\left | M_l\right |}^p\right) } \leq C_p n_{{\epsilon}}^{p/2 - 1} \sum_{l=1}^{n_{{\epsilon}}} { {\mathbb E} }{ \left({\left | M_l-M_{l-1}\right |}^p\right) }.$$ In the present case, since ${\left | Z_{T_n}\right |}={\mathcal{O}}({\epsilon}^\delta)$: $${ {\mathbb E} }{ \left({\left | d_l\right |}^p\right) } \leq C_p {\epsilon}^{p(1+\delta)}.$$ Moreover, $$\begin{aligned}
{ {\mathbb E} }{ \left({\left | \sum_{m=1}^{n_{\epsilon}}{\left | r_{m}\right |}\right |}^p\right) } &\leq n_{{\epsilon}}^{p - 1} \sum_{m=1}^{n_{\epsilon}}{ {\mathbb E} }{\left | r_{m}\right |}^p \\
& \leq C { \left( {\epsilon}+ { {\rm nl} }({\epsilon}) \right) }^p,\end{aligned}$$ and the result follows from .
In the same way, we can estimate the difference of the $n_{\epsilon}$-th jump time of both processes :
Let $n_{\epsilon}\geq 1$ a deterministic integer. Then $$\label{eq:estim_jumps_T}
{\epsilon}\, { {\mathbb E} }{ \left( \sup_{0 \leq n \leq n_{\epsilon}}{\left | T_n-T_n^c\right |}^p \right) }^{1/p} \leq C_p { \left( {\epsilon}^\delta + { {\rm nl_{2}} }({\epsilon}) \right) }$$
First, using the estimates of jump times -, we can decompose the differences of jump times of both processes as follows : $$\begin{aligned}
T_{n}-T_{n}^c &= \sum_{m=0}^{n-1} \Delta T _ {m+1}-\Delta T _ {m+1} ^c\\
& = \sum_{m=0}^{n-1} \delta_{m+1} + d_{m+1} + r_{m+1},
\end{aligned}$$ where, by definition, $$\begin{cases}
\delta_{m+1} := - \frac{b^T {{\tau_{{\epsilon}}}}}{\lambda_0}(Z_{T_{m+1}}-Z_{T_{m}}), \\
d_{m+1} :={\epsilon}\Delta T _{0 } b^T {{\tau_{{\epsilon}}}}\nabla S (X_{T_n}) {\mathcal{V}}_n \\
\hspace{1.5cm} - {\epsilon}\frac{\theta_{n+1}}{\lambda_0^2} A_{\epsilon}^T(X_{T^c_n}^c) {\mathcal{V}}_n \\
r_{m+1}:= { \left(\theta_{m+1}^3+ \theta_{n+1} \right) } {\epsilon}{\mathcal{O}}({\epsilon}^\delta + { {\rm nl_{2}} }({\epsilon})).
\end{cases}$$ We can write for $n \leq n_{{\epsilon}}$: $$\begin{aligned}
{\left | T_{n}-T_{n}^c\right |} \leq {\left | \frac{b^T {{\tau_{{\epsilon}}}}}{\lambda_0}(Z_{T_{n}}-Z_{T_{0}}) \right |} + \sup_{0 \leq l \leq n_{{\epsilon}}}{\left | \sum_{m=1}^{l} d_{m}\right |} + \sum_{m=1}^{n_{\epsilon}}{\left | r_{m}\right |}.\end{aligned}$$ Now, we can remark that the discrete time process $
{ \left(M_l\right) }_{l \geq 0}:={ \left( \sum_{m=1}^{l} d_{m} \right) }_{l \geq 0}
$ is a martingale with respect to the filtration $\mathcal F _{l} = \sigma{ \left((\theta_{m},{\mathcal{V}}_{m-1}) \vert 1 \leq m \leq l \right) }$ with $M_0=0$. Using the Burkholder-Davies-Gundy inequality , together with: $${ {\mathbb E} }{ \left({\left | m_{m}\right |}^p\right) } \leq C {\epsilon}^{p\delta},$$ yields $${ {\mathbb E} }{ \left( \sup_{0 \leq l \leq n_{{\epsilon}}}{\left | \sum_{m=1}^{l} d_{m}\right |}^p \right) }^{1/p} \leq C {\epsilon}^{\delta-1}.$$ Using: $${\left | \frac{b^T {{\tau_{{\epsilon}}}}}{\lambda_0}(Z_{T_{n}}-Z_{T_{0}}) \right |} = {\mathcal{O}}({\epsilon}^{2 \delta-1}) < {\mathcal{O}}({\epsilon}^{\delta-1}),$$ and plugging in the rest term, we finally get .
We can now conclude with the proof of Theorem \[thm:coupling\] :
We consider a given integer $n_{{\epsilon}} \geq 1$. We decompose the difference between both processes as $$\label{eq:proof1}
{\left | X^{\epsilon}_{{{\bar{t}}}}- X^{c,{\epsilon}}_{{{\bar{t}}}}\right |} = {\left | X^{\epsilon}_{{{\bar{t}}}}- X^{c,{\epsilon}}_{{{\bar{t}}}}\right |} {{\bf 1 }}_{N\leq n_{{\epsilon}}}+{\left | X^{\epsilon}_{{{\bar{t}}}}- X^{c,{\epsilon}}_{{{\bar{t}}}}\right |} {{\bf 1 }}_{N > n_{{\epsilon}}},$$ and use to write $$\label{eq:proof2}
{\left | X^{\epsilon}_{{{\bar{t}}}}- X^{c,{\epsilon}}_{{{\bar{t}}}}\right |} {{\bf 1 }}_{N\leq n_{{\epsilon}}} \leq \sup_{0 \leq n \leq n_{\epsilon}}{\left | \Xi_n-\Xi_n^c\right |} +{\epsilon}\sup_{0 \leq n \leq n_{\epsilon}}{\left | T_n-T_n^c\right |} + { \left(\theta_{N+1}+\theta_{N_c+1}\right) } {\mathcal{O}}({\epsilon}).$$ Using now the estimates and in , we get $$\begin{aligned}
\label{eq:estim_jumps_tot}
{ {\mathbb E} }{ \left( {\left | X^{\epsilon}_{{{\bar{t}}}}- X^{c,{\epsilon}}_{{{\bar{t}}}}\right |}^p {{\bf 1 }}_{N \leq n_{{\epsilon}}} \right) }^{1/p}& \leq C ( {\epsilon}+ {\epsilon}^\delta + { {\rm nl_{2}} }({\epsilon}) )\\
& \qquad + C{\epsilon}{ {\mathbb E} }(\theta_{N}^p+\theta_{N^c}^p)^{1/p} \nonumber \\
& = {\mathcal{O}}( {\epsilon}+ {\epsilon}^\delta + { {\rm nl_{2}} }({\epsilon}) ),\end{aligned}$$ where in the last line we have used a technical lemma (Lemma \[lem:tech1\]).
It remains to control the probability of the event $\{ N > n_{\epsilon}\}$. To this purpose, we consider $t \mapsto N_t \in \mathbb N$ the Poisson process uniquely defined by: $$\sum_{n=1}^{N_t} \theta_n \leq t < \sum_{n=1}^{N_t +1} \theta_n.$$ By definition of $N$ and $N_c$, there is a constant $C$ such that $N_{C{{\bar{t}}}/ {\epsilon}^2} \geq N$ and $N_{C{{\bar{t}}}/ {\epsilon}^2} \geq N^c$, so that we can write: $$\label{eq:proof3}
{\left | X^{\epsilon}_{{{\bar{t}}}}- X^{c,{\epsilon}}_{{{\bar{t}}}}\right |} \leq {\epsilon}C { \left(\sum_{n=1}^{N_{C{{\bar{t}}}/ {\epsilon}^2}+1}\theta_{n}\right) }{{\bf 1 }}_{N_{C{{\bar{t}}}/ {\epsilon}^2} > n_{{\epsilon}}}.$$ Let us choose $n_{{\epsilon}} \geq 4 {{\rm e}}C{{\bar{t}}}/ {\epsilon}^2 $, a choice that satisfies $n_{{\epsilon}}={\mathcal{O}}({\epsilon}^{-2})$. A second technical lemma (Lemma \[lem:tech2\]) implies that $$\label{eq:proof4}
{ {\mathbb E} }{ \left( {\left | X^{\epsilon}_{{{\bar{t}}}}- X^{c,{\epsilon}}_{{{\bar{t}}}}\right |}^p {{\bf 1 }}_{N > n_{{\epsilon}}} \right) } \leq {\epsilon}C_p 2^{-1/(C{\epsilon}^2)},$$ so that the contribution of the latter to the final estimate is negligible, since it is exponentially small with respect to ${\epsilon}^{-2}$. The proof is complete.
Discussion about the sharpness of the coupling {#sec:sharp}
----------------------------------------------
One may ask about the “sharpness” of the precise estimates of the coupling, given mainly by Theorem \[thm:coupling\]; (see also (position shift), and (time shift)). Let us discuss the case where $c_F=0$, $ \delta \in [1/k , 1/(k-1)]$. In this case the dominant error term in the jump times expansion of the internal state model in Lemma $4.8$ in [@limits] is due to the non-linearity of the turning rate: $c_\lambda {\epsilon}^{k \delta -1}$. Since this term is due to the internal state mechanism, it can be conjectured that the difference between the density of the internal state model and gradient sensing model will be at least of this order. On the other hand, the coupling estimate in Theorem \[thm:coupling\] is controlled by the same term: $c_\lambda {\epsilon}^{k \delta -1}$ , suggesting the sharpness of the latter.
Numerical illustration\[sec:illustr\]
=====================================
In this section, we demonstrate the validity of the analysis above. For the numerical experiments, we restrict ourselves to one space dimension. For the process with internal state, we use internal dynamics that are given by the scalar cartoon model . In all experiments, $\tau$ is chosen independently of ${\epsilon}$ so that $\delta=1$. The corresponding turning rate is given by with $\lambda_0=1$. For the control process, we choose the linear turning rate , with parameters such that the two processes have the same diffusion limit; in particular, $b=1$. The physical domain is $x\in [0,20]$, and we use reflecting boundary conditions, [*i.e*]{}. the bacterial velocity is reversed when $x=0$ or $x=20$. We fix a scalar bimodal chemoattractant concentration field $$\label{eq:chemoattractant}
S(x)=\alpha\left(\exp\left(-\beta\left(x-\xi\right)^2\right)+\exp\left(-\beta\left(x-\eta\right)^2\right)\right),$$ in which $\alpha=5$, $\beta=1$, $\xi=7.5$ and $\eta=12.5$.
#### Single bacterium as a function of time.
In a first experiment, we simulate a single bacterium evolving according to the fine-scale model (\[eq:process\_noscale\]), in which we set the parameters to ${\epsilon}=0.2$ and $\tau=1$. We compare this evolution to that of a bacterium that satisfies the corresponding control process . Both simulations are performed using the same random numbers starting from the initial condition $X_0=X_0^c=8$ and $V_0=V_0^c=+1$; the bacterium with internal state starts from $Y_0=S(X_0)$. The time step $\delta t=0.1$. The results are shown in figure \[fig:one-particle\].
![\[fig:one-particle\]Evolution of a single bacterium evolving according to the fine-scale process (\[eq:process\_noscale\]) (solid line) and the corresponding control process (\[eq:cprocess\]) (dashed) on short (left) and long (right) time-scales.](figuren/fig-one-particle)
We see a very good coupling initially, which degrades over time. Note the time shift in the short time picture. In the long time picture, coupling is completely lost at some point.
#### Expectation and variance as a function of $\epsilon$ and ${{\bar{t}}}$.
Next, we repeat the experiment using $N=10000$ particles and compute the empirical mean and variance of the coupling, i.e., $$E\left(\left|X_{{{\bar{t}}}}-X_{{{\bar{t}}}}^c\right|\right)=\frac{1}{N}\sum_i^N \left(\left|X^i_{{{\bar{t}}}}-X^{i,c}_{{{\bar{t}}}}\right|\right), \;\;\; \text{resp., } \;\;\; E\left((X_{{{\bar{t}}}}-X_{{{\bar{t}}}}^c)^2\right)=\frac{1}{N}\sum_i^N \left(X^i_{{{\bar{t}}}}-X^{i,c}_{{{\bar{t}}}}\right)^2.$$ As fine-scale parameters, we choose $\tau=1$, $\lambda_0=b=1$ and several values of ${\epsilon}$. The chemoattractant concentration is again given as , now with $\alpha=\beta=1$, $\xi=7.5$ and $\eta=12.5$. The time interval for the computation is $t\in [0,30/{\epsilon}^2]$.
![\[fig:eps-long\] Empirical mean (left) and variance (right) of the difference between the fine-scale process (\[eq:process\_noscale\]) and the corresponding control process (\[eq:cprocess\]) as a function of ${\epsilon}$ for different values of the reporting time $T$. The theoretical slope is indicated with a dashdotted line. The sample size is $N=10000$.](figuren/fig-eps-long)
Figure \[fig:eps-long\] shows the dependence in ${\epsilon}$ of the coupling, by plotting the empirical mean and variance defined above as a function of ${\epsilon}$ for different values of the reporting time $T$. The results shown in figure \[fig:eps-long\] are in clear accordance with the theoretical slope predicted by the asymptotic analysis.
![\[fig:T-long\]Evolution of the empirical mean (left) and variance (right) of the difference between the fine-scale process (\[eq:process\_noscale\]) and the corresponding control process (\[eq:cprocess\]) as a function of ${{\bar{t}}}=t/{\epsilon}^2$ for different values of ${\epsilon}$. The sample size is $N=10000$. ](figuren/fig-T-long)
Figure \[fig:T-long\] shows the mean and variance of the coupling difference as a function of time. The time dependence of the variance has not been analyzed mathematically in Section \[sec:variance\]; the specific behaviour viewed in figure \[fig:T-long\] and is probably due to: (i) sufficiently short diffusive times; (ii) the specific double-well form of the chemoattractant potential.
#### Expectation and variance as a function of $\tau$.
Finally, in a last experiment, we illustrate the dependence on $\tau$. To this end, we again simulate $N=10000$ particles choosing $\lambda_0=1$, ${\epsilon}=0.1$ and $X_0=7.5$, for different values of $\tau$. The results in figure \[fig:tau-long\] show that the variance quickly increases with $\tau$ until it reaches a plateau for $\tau>1$.
![\[fig:tau-long\] Empirical mean (left) and variance (right) of the difference between the fine-scale process (\[eq:process\_noscale\]) and the corresponding control process (\[eq:cprocess\]) as a function of $\tau$ for different values of the reporting time $T$. The sample size is $N=10000$.](figuren/fig-tau-long)
Simulation on diffusive time scales\[sec:appl\]
===============================================
In this section, we consider a simulation of the density of an ensemble of particles, with and without variance reduction and/or reinitialization, as described in Section \[sec:coupl\_princ\]. We again restrict ourselves to one space dimension, with domain $x\in [0,20]$ and periodic boundary conditions. In this case, the kinetic equation corresponding to the control process reduces to the system $$\label{eq:cprocess_hydro1d}
\system{
& \partial_t p^c_++\epsilon\partial_x p^c_+=-\frac{\lambda^c(x,+1)}{2}p_+^c+ \frac{\lambda^c(x,-1)}{2}p_-^c &\\
& \partial_t p^c_--\epsilon\partial_x p^c_-=\frac{\lambda^c(x,+1)}{2}p_+^c -\frac{\lambda^c(x,-1)}{2}p_-^c &
}.$$ of two PDEs, which is straightforward to simulate using finite differences.
We fix the chemoattractant concentration field as , with parameters $\alpha=2$, $\beta=1$, $\xi=7.5$ and $\eta=12.5$. For the internal dynamics, the same model (-, -) is used. The parameters are ${\epsilon}=0.5$, $\lambda_0=1$, $\tau=1$, $\delta t=0.1$.
All simulations are performed with $N=5000$ particles. The initial positions are uniformly distributed in the interval $x\in[13,15]$; the initial velocities are chosen uniformly, i.e., each particle has an equal probability of having an initial velocity of $\pm{\epsilon}$. The initial condition for the internal variable is chosen to be in local equilibrium, i.e., $Y_0^i=S(X^i_0)$. The initial positions and velocities of the control particles are chosen to be identical.
We discretize the continuum description on a mesh with $\Delta x = 0.1$ using a third-order upwind-biased scheme, and perform time integration using the standard fourth order Runge–Kutta method with time step $\delta t_{pde}= 10^{-1}$. The initial condition is given as $$p^+(x,0)=p^-(x,0)=\begin{cases}
0.25 , & {x\in [13,15],} \\
0, & \text{otherwise.}
\end{cases}$$
#### Simulation without variance reduction.
First, we simulate both stochastic processes up to time $\bar{t}=50$ ($t=50/{\epsilon}^2$) and estimate the density of each of these processes $\hat{n}_N(x,{{\bar{t}}})$, resp. $\hat{n}^c_N(x,{{\bar{t}}})$, without variance reduction. The density is obtained via binning in a histogram, in which the grid points of the deterministic simulation are the centers of the bins. Figure \[fig:no-variance-reduction\] (left)
![\[fig:no-variance-reduction\] Bacterial density as a function of space at $t=50/{\epsilon}^2$ without variance reduction. Left: one realization. Right: mean over $100$ realizations and $95\%$ confidence interval. The solid line is the estimated density from a particle simulation using the process with internal state; the dashed line is estimated from a particle simulation using the control process. Both used $N=5000$ particles. The dotted line is the solution of the deterministic PDE (\[eq:cdensity\]). ](figuren/fig-no-variance-reduction)
shows the results for a single realization. We see that, given the fluctuations on the obtained density, it is impossible to conclude on differences between the two models. This observation is confirmed by computing the average density of both processes over $100$ realizations. The mean densities are shown in figure \[fig:no-variance-reduction\] (right), which also reveals that the mean density of the control process is within the $95\%$ confidence interval of the process with internal state. Both figures also show the density that is computed using the continuum description, which coincides with the mean of the density of the control particles.
#### Simulation with variance reduction.
Next, we compare the variance reduced estimation with the density of the control PDE. We reinitialize the control particles after each coarse-scale step, i.e., each $k$ steps of the particle scheme, where $k \delta t = \delta t_{pde}$, (here $k=1$). The results are shown in figure \[fig:var-red-reinit\].
![\[fig:var-red-reinit\] Bacterial density as a function of space at $t=50/{\epsilon}^2$ with variance reduction and reinitialization. Left: variance reduced density estimation of one realization with $N=5000$ particles (solid) and deterministic solution for the control process (\[eq:cdensity\]) (dashed). Right: mean over $100$ realization and $95\%$ confidence interval (solid) and the deterministic solution for the control process (\[eq:cdensity\]) (dashed).](figuren/fig-variance-reduction)
We see that, using this reinitialization, the difference between the behaviour of the two processes is visually clear from one realization (left figure). Also, the resulting variance is such that the density of the control PDE is no longer within the $95\%$ confidence interval of the variance reduced density estimation (right figure). We see that there is a significant difference between both models: the density corresponding to the control process is more peaked, indicating that bacteria that follow the control process are more sensitive to sudden changes in chemoattractant gradient. This difference can be interpreted from the fact that the bacteria with internal state do not adjust themselves instantaneously to their environment, but instead with a time constant $\tau$.
#### Simulation with variance reduction but without reinitialization.
We also compare the variance reduced estimation with the density of the control PDE without performing any reinitialization of the control process to restore the coupling.
![\[fig:var-red-noreinit\] Bacterial density as a function of space at $t=50/{\epsilon}^2$ with variance reduction. Left: variance reduced density estimation of one realization with $N=5000$ particles (solid) and deterministic solution for the control process (\[eq:cdensity\]) (dashed). Right: mean over $200$ realization and $95\%$ confidence interval (solid) and the deterministic solution for the control process (\[eq:cdensity\]) (dashed).](figuren/fig-var-red-noreinit)
Figure \[fig:var-red-noreinit\] (right) shows the mean estimation of the density over $200$ realizations, as well as the density of the control process. As evidenced by the error bars, simulating a single realization of the process, even with variance reduction, is not able to reliably reveal this difference. This phenomenon is due to the degeneracy of the coupling on long diffusive times. This is also illustrated in figure \[fig:var-red-noreinit\] (left), which compares the variance reduced density estimation of a single realization with the density of the control PDE. Note that, as predicted by the analysis, the variance is much larger in regions where $\nabla S(x)$ is large.
Finally, we repeat the experiment with a larger chemoattractant gradient. We again choose the chemoattractant field as , with the same parameters as above, except that we now take $\alpha=5$. We also use the same discretization parameters as above. The variance reduced estimation , as well as the density of the control PDE, are shown in figure \[fig:var-red-reinit-alpha5\].
![\[fig:var-red-reinit-alpha5\] Bacterial density as a function of space at $t=50/{\epsilon}^2$ with variance reduction and reinitialization. Left: variance reduced density estimation of one realization with $N=5000$ particles (solid) and deterministic solution for the control process (\[eq:cdensity\]) (dashed). Chemoattractant given by with $\alpha=5$. Right: mean over $100$ realization and $95\%$ confidence interval (solid) and the deterministic solution for the control process (\[eq:cdensity\]) (dashed).](figuren/fig-variance-reduction-alpha5)
We see that, although the difference between the process with internal state and the control process becomes much larger, the variance is still very well controlled by the coupling.
Conclusions and discussion\[sec:concl\]
=======================================
In this paper, we studied the simulation of stochastic individual-based models for chemotaxis of bacteria with internal state. We have used a coupling with a simpler, direct gradient sensing model with the same diffusion limit to obtain an “asymptotic” variance reduction. Throughout this work, we have assumed that the computation of the bacterial density using the kinetic description of the direct gradient sensing model can be performed accurately with a grid-based method. Note that in higher spatial dimensions, this may become cumbersome, and the grid-based computation should be carried out at the level of the advection-diffusion description (or any desired moment system) only, using a second level of coupling between the velocity-jump process associated with the kinetic description, and the stochastic differential equation associated with the advection-diffusion limit. This is left for future work.
The current algorithm allows to explore the differences between the fine-scale process with internal state and the simpler, coarse process *using a fixed number of particles, independently of the small parameter ${\epsilon}$*. However, the computational cost to simulated an individual particle over diffusive time-scales becomes very large for ${\epsilon}\to 0$. In future work, we will therefore also study the development of truly “asymptotic preserving” schemes in the diffusion asymptotics, in the sense that the computational cost of the simulation of processes is independent of the small parameter ${\epsilon}$. This will require to deal with two kinds of difficulties: (i) We will need to use an asymptotic preserving method to solve the density evolution of the control model on the grid; and (ii) We will need to extrapolate forward in time the state of the fine-scale simulation.
The first difficulty implies that, instead of solving the full kinetic equation associated with the control process, we only solve its diffusion limit. This may imply the use of a second level of variance reduction, coupling the control velocity jump process and its limiting drift-diffusion process.
The second difficulty, extrapolation in time, is related to the equation-free [@KevrGearHymKevrRunTheo03; @Kevrekidis:2009p7484] and HMM [@EEng03; @E:2007p3747] types of methodologies; ideas of this type can be traced back to Erhenfest [@Ehrenfest:1990p9192]. One approach is to use a *coarse projective integration* method [@GearKevrTheo02; @KevrGearHymKevrRunTheo03]. One then extrapolates the bacterial density over a projective time step on diffusive time scales, after which the projected density needs to be *lifted* to an ensemble of individual bacteria. For such methods, besides the effect of extrapolation on the variance of the obtained results, also the effects of reconstructing the velocities and internal variables have to be systematically studied.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Radek Erban, Thierry Goudon, Yannis Kevrekidis and Tony Lelièvre for interesting discussions that eventually led to this work. This work was performed during a research stay of GS at SIMPAF (INRIA - Lille). GS warmly thanks the whole SIMPAF team for its hospitality. GS is a Postdoctoral Fellow of the Research Foundation – Flanders. This work was partially supported by the Research Foundation – Flanders through Research Project G.0130.03 and by the Interuniversity Attraction Poles Programme of the Belgian Science Policy Office through grant IUAP/V/22 (GS). The scientific responsibility rests with its authors.
Technical lemmas, used in proof of Theorem \[thm:coupling\]
===========================================================
\[lem:tech1\] Let $(\theta_n)_{n\geq 1}$ a sequence of independent exponential random variables with mean $1$, and $(T_{n})_{n \geq 1}$ a strictly increasing sequence of random times with $T_0=0$ such that:
- For any $m \geq 1$, the sequence $(\theta_n)_{n\geq m+1}$ is independent of the past $(\theta_n,T_n)_{n\leq m}$.
- There is a constant $C$ such that for all $n\geq 0 $, $$\label{eq:h1}
\frac{1}{C} \theta_{n+1} \leq T_{n+1} - T_n \leq C \theta_{n+1} .$$
Let $N_t$ a random integer such that: $$T_{N_t} \leq t \leq T_{N_t +1}.$$ Then for any integer $ k \geq 0$, there is a constant $C_k$ independent of $t$ such that: $${ {\mathbb E} }{ \left( \theta_{N_t+1} \right) }^k \leq C_k.$$
The result may seem rather intuitive, but the proof is slightly tricky. First, we consider sub-intervals of $[0,t]$ of the form: $$I_p = [\frac{p-1}{P} t, \frac{p}{P} t],$$ for some $P\geq 1$ and $p = 1, \dots , P$.
Step (i). The first step consists in proving the following estimate: $$\label{eq:app1}
{ {\mathbb E} }{ \left( \theta_{N_t+1} ^k \right) } \leq \sum_{p=0}^{P-1} { {\mathbb E} }{ \left(\theta^k {{\bf 1 }}{ \left( \theta \geq \frac{p}{C} \frac{t}{P} \right) } \right) } \sup_{p=1 \dots P} \sum_{n=0}^{+\infty} {\mathbb{P}}(T_n \in I_p)$$ Indeed, $$\begin{aligned}
{ {\mathbb E} }{ \left( \theta_{N_t+1} ^k \right) } & = \sum_{n=0}^{+ \infty} { {\mathbb E} }{ \left( \theta^k_{n+1} {{\bf 1 }}{ \left( T_{n} \leq t \leq T_{n +1} \right) } \right) } \\
& \leq \sum_{n=0}^{+ \infty} \sum_{p=1}^P { {\mathbb E} }{ \left( \theta^k_{n+1} {{\bf 1 }}{ \left( T_{n +1} -T_n \geq \frac{P-p}{P} t \vert T_n \in I_p \right) } \right) }{\mathbb{P}}{ \left(T_n \in I_p\right) }.\end{aligned}$$ Using the independence of $\theta_{n+1}$ with $T_n$, and the assumption , it yields, $${ {\mathbb E} }{ \left( \theta^k_{n+1} {{\bf 1 }}{ \left( T_{n +1} -T_n \geq \frac{P-p}{P} t \big \vert T_n \in I_p \right) } \right) }\leq { {\mathbb E} }{ \left( \theta^k {{\bf 1 }}_{ { \left( \theta \geq \frac{P-p}{C P} t \right) } } \right) }.$$ and finally changing the sum index $p$ to $P-p+1$ yields .
Step (ii). The second step consists in the following decomposition. Let $I=[a,b]$ be a finite interval of ${\mathbb{R}}^+$, $T_{-1}=-\infty$. Then, $$\label{eq:app2}
\sum_{n=0}^{+\infty} {\mathbb{P}}(T_n \in I) = \sum_{n=0}^{+\infty} { \left( {\mathbb{P}}( T_n \in I, T_{n-1} \notin I ) \sum_{m=0}^{+\infty} {\mathbb{P}}( T_{n+1},\dots,T_{n+m} \in I \vert T_n \in I, T_{n-1} \notin I ) \right) } .$$ The key is to write $$\begin{aligned}
\underbrace{\sum_{n=0}^{+\infty} {\mathbb{P}}(T_n \in I)}_{(a_0)} & = \sum_{n=0}^{+\infty} {\mathbb{P}}(T_{n-1} \notin I,T_n \in I) + \sum_{n=0}^{+\infty} {\mathbb{P}}(T_{n-1},T_n \in I) \\
& = \sum_{n=0}^{+\infty} {\mathbb{P}}(T_{n-1} \notin I,T_n \in I) + \underbrace{\sum_{n=0}^{+\infty} {\mathbb{P}}(T_{n} ,T_{n+1} \in I)}_{(a_1)} .\end{aligned}$$ We wish to iterate the decomposition of $(a_0)$ to $(a_1)$, $(a_2)$, etc... For this purpose, remark that $$\begin{aligned}
(a_m) &= \sum_{n=0}^{+\infty} {\mathbb{P}}(T_{n}, \dots ,T_{n+m} \in I) \\
& = \sum_{n=0}^{+\infty} {\mathbb{P}}(T_{n} \in I ) {\mathbb{P}}(T_{n+1}, \dots ,T_{n+m} \in I \vert T_{n} \in I ) \\
& \leq { \left( \sum_{n=0}^{+\infty} {\mathbb{P}}(T_{n} \in I ) \right) } {\mathbb{P}}( \theta_1+ \dots + \theta_m \leq C (b-a) ) \\
& \xrightarrow{m \to + \infty} 0 ,\end{aligned}$$ so that we get in the end $$(a_0) = \sum_{n=0}^{+\infty} \sum_{m=0}^{+\infty} {\mathbb{P}}(T_{n-1} \notin I,T_n, \dots T_{n+m} \in I).$$ Factorizing ${\mathbb{P}}( T_n \in I, T_{n-1} \notin I ) $ with Bayes’ formula yields .
Step (iii). The estimate yields $$\label{eq:app4}
{ {\mathbb E} }{ \left( \theta_{N_t+1} ^k \right) } \leq \sum_{p=0}^{+\infty} { {\mathbb E} }{ \left(\theta^k {{\bf 1 }}_{ { \left( \theta \geq \frac{p}{C} \frac{t}{P} \right) } } \right) } \sum_{m=0}^{+\infty} {\mathbb{P}}{ \left(\theta_1+\dots +\theta_{m} \leq \frac{t}{CP} \right) }.$$ Indeed for $I=[a,b]$, using the independence of $(\theta_{n+1}, \dots,\theta_{n+m})$ from $(T_n,T_{n-1})$: $$\sum_{m=0}^{+\infty} {\mathbb{P}}( T_{n+1},\dots,T_{n+m} \in I \vert T_n \in I, T_{n-1} \notin I ) \leq \sum_{m=0}^{+\infty} {\mathbb{P}}{ \left( \theta_{1} + \dots + \theta_{m} \leq C(b-a) \right) },$$ and $$\sum_{n=0}^{+\infty} {\mathbb{P}}( T_n \in I, T_{n-1} \notin I ) = {\mathbb{P}}(\exists n \geq 0 \vert T_n \in I ) \leq 1.$$ The two last estimates in yields: $$\sup_{p=1 \dots P} \sum_{n=0}^{+\infty} {\mathbb{P}}(T_n \in I_p) \leq \sum_{m=0}^{+\infty} {\mathbb{P}}{ \left(\theta_1+\dots +\theta_{m} \leq \frac{t}{CP} \right) },$$ and follows. Finally, we choose $P= \lfloor t +1\rfloor$ so that the right hand side of does not depend on $t$ any more. Finally we use the exponential convergence of the terms of the two series in the right hand side of (using for instnace large deviation estimates for the second) one, to conclude that the two series indeed converge, and conclude the proof.
\[lem:tech2\] Let ${ \left(\theta_n\right) }_{n \geq 0}$ be a sequence of exponential random variables of mean $1$, and $t \mapsto N_t \in \mathbb N$ the Poisson process uniquely defined by: $$\sum_{n=1}^{N_t} \theta_n \leq t < \sum_{n=1}^{N_t +1} \theta_n.$$ Let $n_0 \in \mathbb N$ such that $n_0 \geq 4 {{\rm e}}t$, $p_1 \geq 1$, and $p_2 \geq 1$. Then there is a constant $C_{p_1,p_2}$ independent of $(n_0,t)$ such that: $${ {\mathbb E} }{ \left( {\left | \sum_{n=1}^{N_t+1}\theta_n^{p_1} \right |}^{p_2} {{\bf 1 }}_{N_t > n_0}\right) } \leq C_{p_1,p_2} 2^{- n_0} .$$
Step (i).\
Let us condition on the different possible values of $N_t > n_0$: $$\begin{aligned}
{ {\mathbb E} }{ \left( {\left | \sum_{n=1}^{N_t+1}\theta_n^{p_1} \right |}^{p_2} {{\bf 1 }}_{N_t > n_0}\right) } & = \sum_{m=n_0+1}^{+\infty}{ {\mathbb E} }{ \left( {\left | \sum_{n=1}^{N_t+1}\theta_n^{p_1} \right |}^{p_2} \Big \vert N_t = m \right) } {\mathbb{P}}{ \left(N_t = m\right) } \\
& = \sum_{m=n_0+1}^{+\infty}{ {\mathbb E} }{ \left( {\left | \sum_{n=1}^{N_t+1}\theta_n^{p_1} \right |}^{p_2} \Big \vert N_t = m \right) } {{\rm e}}^{-t}\frac{t^{m}}{{m} ! } .
\end{aligned}$$
Step (ii).\
Let us denote $({\mathbb{P}}_{t,m},{ {\mathbb E} }_{t,m}) = ({\mathbb{P}}{ \left(\, . \, \vert N_t = m\right) },{ {\mathbb E} }{ \left(\, . \, \vert N_t = m\right) })$ the probability/expectation conditionally on the event ${\left\{N_t = m\right\}}$. Conditionally on ${\left\{N_t = m\right\}}$, the sequence $(\theta_1, \dots,\theta_{m+1})$ has the same distribution as $(t U_{(1)}, t U_{(2)}-t U_{(1)}, \dots, t U_{(m)}- t U_{(m-1)}, t- t U_{(m)}+\theta)$, where $(U_{(1)} < \dots < U_{(m)})$ is the order statistics of $m$ independent random variables $(U_1, \dots , U_m)$ uniformly distributed on the interval $[0,1]$, and $\theta$ is an independent exponential random variable of mean $1$ (see Section 2.4 of [@Nor98] for classical properties of Poisson processes). Thus denoting $U_{(0)}=0$ we can write: $$\begin{aligned}
\label{eq:app11}
& { {\mathbb E} }{ \left( {\left | \sum_{n=1}^{N_t+1}\theta_n^{p_1} \right |}^{p_2} {{\bf 1 }}_{N_t > n_0}\right) } = \nonumber\\
& \qquad \sum_{m=n_0+1}^{+\infty}{ {\mathbb E} }_{t,m} { \left( {\left | { \left(t- t U_{(m)}+\theta\right) }^{p_1} + t^{p_1}\sum_{n=1}^{m} { \left(U_{(n)}-U_{(n-1)}\right) }^{p_1} \right |}^{p_2} \right) } {{\rm e}}^{-t}\frac{t^{m}}{{m} ! } .
\end{aligned}$$ Using the inequality $(a+b)^p \leq C_p a^p + b^b$ for any $a,b > 0$, Jensen inequality, and denoting $U_{(m+1)}= 1$, there is a constant $C_{p_1,p_2}$ such that: $$\begin{aligned}
\label{eq:app12}
& {\left | { \left(t- t U_{(m)}+\theta\right) }^{p_1} + t^{p_1}\sum_{n=1}^{m} { \left(U_{(n)}-U_{(n-1)}\right) }^{p_1} \right |}^{p_2} \leq \nonumber \\
& \quad C_{p_1,p2}{ \left(\theta^{p_2p_1} + t^{p_1 p_2}(m+1)^{p_2-1} \sum_{n=1}^{m+1} { \left(U_{(n)}-U_{(n-1)}\right) }^{p_1 p_2} \right) }.\end{aligned}$$
Step (iii).\
Let us compute ${ {\mathbb E} }_{t,m} { \left( { \left(U_{(n)}-U_{(n-1)}\right) }^{p_1 p_2} \right) }$ for $n=1, \dots, m+1$. First, let us recall the probability density of a couple $(U_{(i)},U_{(j)})$ for $1 \leq i<j \leq m$ in an order statistics of size $m$: $$\begin{aligned}
\label{eq:orderstat}
& {\mathbb{P}}{ \left(U_{(i)}\in [u_i,u_i+d u_i], U_{(j)} \in [u_j,u_j + d u_j] \right) } \nonumber \\
& \qquad = m! {\mathbb{P}}{ \left(U_1 < \dots < U_{i-1} < u_{i}\right) } \times {\mathbb{P}}{ \left( U_{i} \in [u_{i},u_{i}+d u_{i}] \right) } \nonumber \\
& \qquad \quad \times {\mathbb{P}}{ \left( u_i < U_{i+1} < \dots < U_{j-1}<u_{j}\right) } \times {\mathbb{P}}{ \left( U_{j} \in [u_{j},u_{j}+d u_{j}]\right) } \times {\mathbb{P}}{ \left( u_j < U_{j+1} < \dots < U_m\right) } , \nonumber \\
& \qquad = m! \frac{u_{i}^{i-1}}{(i-1)!} \frac{(u_j-u_i)^{j-i-1}}{(j-i-1)!} \frac{(1-u_j)^{m-j}}{(m-j)!} d u_{i} \, d u_{j} .\end{aligned}$$ As a consequence, $$\begin{aligned}
{ {\mathbb E} }_{t,m} { \left( { \left(U_{(n)}-U_{(n-1)}\right) }^{p_1 p_2} \right) } & = \int_{0< u_{n-1} < u_n < 1} m! \frac{u_{n-1}^{n-2}}{(n-2)!} (u_n - u_{n-1} )^{p_1 p_2} \frac{(1-u_n)^{m-n}}{(m-n)!} d u_{n-1} \, d u_{n} \\
& = m! (p_1 p_2 ) ! \int_{0< u_{n-1} < u_n < 1} \frac{u_{n-1}^{n-2}}{(n-2)!} \frac{(u_n - u_{n-1} )^{p_1 p_2}}{(p_1 p_2 ) !} \frac{(1-u_n)^{m-n}}{(m-n)!} d u_{n-1} \, d u_{n} \\
& = \frac{m! (p_1 p_2 ) !}{( m+p_1p_2 )!},\end{aligned}$$ where in the last line we have used with appropriate $(i,j,m)$ to compute the integral. In the same way, the probability density of $U_{(i)}$ for $1 \leq i \leq m$ in an order statistics of size $m$ is given by: $${\mathbb{P}}{ \left(U_{(i)}\in [u_i,u_i+d u_i]\right) }=m! \frac{u_{i}^{i-1}}{(i-1)!} \frac{(1-u_i)^{m-i}}{(m-i)!} d u_i,$$ so that $$\begin{aligned}
{ {\mathbb E} }_{t,m} { \left( U_{(1)} ^{p_1 p_2} \right) } & = m! (p_1 p_2 ) ! \int_{0< u_{1} < 1} \frac{u_{1}^{p_1 p_2}}{(p_1 p_2)!} \frac{(1-u_1)^{m-1}}{(m-1)!} d u_1 \\
& = \frac{m! (p_1 p_2 ) !}{( m+p_1p_2 )!},\end{aligned}$$ and in the same way, $${ {\mathbb E} }_{t,m} { \left( (1-U_{(m)})^{p_1 p_2} \right) } = \frac{m! (p_1 p_2 ) !}{( m+p_1p_2 )!} .$$
Step (iv).\
Remarking that ${ {\mathbb E} }_m(\theta^{p_1p_2})=(p_1p_2)!$, and inserting the computations of Step (ii) in - yields: $$\begin{aligned}
& { {\mathbb E} }{ \left( {\left | \sum_{n=1}^{N_t+1}\theta_n^{p_1} \right |}^{p_2} {{\bf 1 }}_{N_t > n_0}\right) } \leq \nonumber\\
& \qquad C_{p_1p_2}\sum_{m=n_0+1}^{+\infty} { \left( (p_1p_2)! + t^{p_1 p_2}(m+1)^{p_2-1} \frac{m! (p_1 p_2 ) !}{( m+p_1p_2 )!} \right) }{{\rm e}}^{-t}\frac{t^{m}}{{m} ! } .\end{aligned}$$ Using the inequalities $ \frac{(m+1)^{p_2-1} m !}{( m+p_1p_2 )!} \leq 1$, ${{\rm e}}^{-t}\leq 1$ and $$\sum_{m=n_0+1}^{+\infty} \frac{t^{m}}{{m} ! } \leq \frac{t^{n_0}}{ n_0 ! },$$ it yields $${ {\mathbb E} }{ \left( {\left | \sum_{n=1}^{N_t+1}\theta_n^{p_1} \right |}^{p_2} {{\bf 1 }}_{N_t > n_0}\right) } \leq C_{p_1p_2} \frac{t^{n_0+p_1p_2}}{n_0 !},$$ so that in the end, using the assumption $n_0 \geq 4 {{\rm e}}t$ and the Stirling formula, $$\begin{aligned}
& { {\mathbb E} }{ \left( {\left | \sum_{n=1}^{N_t+1}\theta_n^{p_1} \right |}^{p_2} {{\bf 1 }}_{N_t > n_0}\right) } \leq C_{p_1p_2} (\frac{n_0}{4 {{\rm e}}})^{n_0+p_1p_2} \frac{1}{\sqrt{n_0}} (\frac{{{\rm e}}}{n_0})^{n_0} \\
& \qquad \leq C_{p_1p_2} \frac{n_0^{p_1 p_2}}{\sqrt{n_0} 2 ^{n_0}} 2^{- n _0} \leq C_{p_1p_2} 2^{- n _0} ,\end{aligned}$$ which yields the result.
[^1]: SIMPAF, INRIA Lille - Nord Europe, Lille, France, (mathias.rousset@inria.fr).
[^2]: Department of Computer Science, K.U. Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium, (giovanni.samaey@cs.kuleuven.be).
|
---
abstract: 'In this paper, we carry out a computational study using the spectral decomposition of the fluctuations of a two-pathogen epidemic model around its deterministic attractor, i.e., steady state or limit cycle, to examine the role of partial vaccination and between-host pathogen interaction on early pathogen replacement during seasonal epidemics of influenza and respiratory syncytial virus.'
author:
- 'Yury E. García, Marcos A. Capistrán [^1]'
bibliography:
- 'Bibliography.bib'
title: 'Early pathogen replacement in a model of Influenza and Respiratory Syncytial Virus with partial vaccination. A computational study'
---
[*Keywords:*]{} Power spectral density, Two-pathogen model, Influenza vaccination effects, Interaction between viruses .
Introduction {#sec:introduction}
============
In this paper, we study the impact of partial vaccination and between-host pathogen interaction on early pathogen replacement in a two-pathogen epidemic model of influenza and respiratory syncytial virus (RSV).\
It is known that influenza and RSV peak during the winter season in temperate regions, and have semi-annual activity near tropical areas [@bloom2013latitudinal]. Second, the outbreak interference between these viruses have been registered by years [@anestad1982interference; @aanestad1987surveillance; @anestad2009interference; @glezen1980influenza; @shinjoh2000vitro; @welliver2008respiratory]. Influenza is known to interact with other viruses, including RSV. Glenzen et al [@glezen1980influenza] studied the interaction between influenza and other respiratory viruses; one of their conclusions is that simultaneous viral infections are a competition for resources and the virus with the largest growth rate is the one that succeeds in the invasion. There are also in vitro experiments presented by Shinjoh et al [@shinjoh2000vitro] who have shown that the growth of RSV can be blocked by influenza A if they infect the host cells at the same time. In contrast, RSV can suppress the growth of influenza A if this occurs after RSV infection. Third, the effect of vaccination on epidemic synchrony patterns has been analyzed for several diseases. In particular, Rohani et al. [@rohani1999opposite] have shown that vaccination turned synchronous epidemics (measles) into irregular and spatially uncorrelated epidemics once vaccination was deployed, while whooping cough shifted from incoherence and spatial irregularity to regular dynamics as vaccination was introduced. Furthermore, in the context of vaccine-induced strain replacement, Martcheva et al [@martcheva2008vaccine] claim that [*“...the deployment of vaccination changes the proportion of hosts susceptible to either strain, ultimately shifting their relative and absolute abundances..."*]{}. On the other hand, Alonso et al [@alonso2007stochastic] explain transitions in epidemics between regular and irregular dynamics in terms of amplification of demographic noise.\
Based on the above results, in this paper, we have proposed a two-pathogen epidemic model with seasonality and partial vaccination as a continuous-time Markov jump process using typical kinetic parameter values of influenza and RSV. Furthermore, we have used standard theoretical and computational methods [@van2002reproduction; @kamo2002effect; @adams2007influence; @alonso2007stochastic; @black2010stochastic] to show that variations in coverage and efficacy of influenza vaccination may explain early pathogen replacement, e.g. either pathogen might invade first, and there is a second wave of infections where the second pathogen is dominant, see Anestad [@anestad1982interference; @aanestad1987surveillance].\
The joint probability distribution of the state variables in our model is governed by a *forward Kolmogorov* equation [@gardiner1986handbook]. The van Kampen asymptotic expansion [@van1992stochastic] applied to this equation separates state variables into a mean field equation that matches the thermodynamic limit of the stochastic process (and is amenable to stability analysis in the sense of van den Driessche and Watmough [@van2002reproduction]), and a Fokker-Plank equation governing the system fluctuations, which is equivalent to a Langevin equation in a neighborhood of the system attractor, e.g., steady state or limit cycle [@alonso2007stochastic; @black2010stochastic].\
First, we have used the mean field equation to carry out a standard analysis of the disease-free equilibrium in terms of the effective vaccination rate and cross-immunity parameter.\
Next, we have applied the Mckane approximation [@mckane2005predator] of the power spectral density (PSD) of the system fluctuations in a neighborhood of the system attractors [@alonso2007stochastic; @rozhnova2009fluctuations; @black2010stochastic; @rozhnova2010stochastic; @rozhnova2011stochastic]. Of note, this method of McKane to approximate the spectral decomposition of the system fluctuations is well suited to examine the role of between-host pathogen interaction and partial vaccination in seasonal patterns of respiratory diseases beyond the qualitative analysis of the mean field equation. Although, care must be taken since the PSD approximation does not hold near bifurcation points.\
Coexistence of stable attractors in one and two pathogen epidemic models with seasonality has been documented [@kuznetsov1994bifurcation; @kamo2002effect] at high contact rates. Likewise, it has been established that the intertwined basin of multiple attractors at low contact rate, or high vaccination the rate is not robust to spatial coupling [@alonso2007stochastic]. Since in this paper, we care about the role of partial vaccination in seasonal epidemics, we shall focus our analysis in the yearly regime in a two-pathogen epidemic model with low spatial coupling aiming at showing a route to pathogen switching in early epidemic season.\
The paper is organized as follows. Section \[sec:MatMet\] describes the mathematical model and the power spectral density for both seasonally forced and unforced models. Section \[sec:Results\] shows the results when some key epidemic parameters are varied. Finally, Section \[sec:discussion\] discusses our findings and offer some perspectives.
Theoretical background {#sec:MatMet}
======================
A Two-pathogen Epidemic Model with Partial Vaccination {#sec:model}
------------------------------------------------------
The nonlinear dynamics of infectious disease spread in communities is stochastic. Assuming spatial homogeneity, the populations of susceptible, infectious and recovered individuals follow a birth and death process in $\mathbb{N}^{n}$. Consequently, the epidemic model is posed as a continuous-time Markov jump process, whose forward Kolmogorov equation is known as the Chemical Master Equation (CME).
(SS) [$X_{SS}$]{}; (SI) \[right of=SS\] [$X_{SI}$]{}; (SR) \[right of=SI\] [$X_{SR}$]{}; (IS) \[below of=SS\] [$X_{IS}$]{}; (RS) \[below of=IS\] [$X_{RS}$]{}; (IR) \[below of=SR\] [$X_{IR}$]{}; (RR) \[below of=IR\] [$X_{RR}$]{}; (RI) \[right of=RS\] [$X_{RI}$]{}; (II) \[below of=SI\] [$X_{II}$]{}; (SS) [$X_{SS}$]{}; (end) \[right of=SR, node distance=2cm\];
(SS) – (1.0,-1.0) \[above right\] node [$\mu$]{}; (SI) – (2.5,1.0) \[above right\] node [$\eta$]{}; (SI) – (3.5,-1.0) \[above right\] node [$\mu$]{}; (SR) – (6.0,-1.0) \[above right\] node [$\mu$]{}; (IS) – (1.0,-3.5) \[above right\] node [$\mu$]{}; (IS) – (-1.0,-2.5) \[below left\] node [$\eta$]{}; (RS) – (1.0,-6.0) \[above right\] node [$\mu$]{}; (RI) – (3.5,-6.0) \[above right\] node [$\mu$]{}; (RR) – (6.0,-6.0) \[above right\] node [$\mu$]{}; (IR) – (6.0,-3.5) \[above right\] node [$\mu$]{}; (II) – (3.5,-3.5) \[above right\] node [$\mu$]{}; (0.5,1.8) –(4.5,1.8) node \[above,midway\] [RSV]{}; (-2.5,-0.5)–(-2.5,-5.0) node\[above, midway,rotate=90\] [Influenza]{};
(SS) edge node [$\beta_2 \lambda_2$]{} (SI); (SI) edge node [$\gamma$]{} (SR); (SR) edge node [$\sigma \beta_1 \lambda_1$]{} (IR); (SS) edge \[left\] node [$\beta_1 \lambda_1$]{} (IS); (IS) edge \[left\] node [$\gamma$]{} (RS); (RS) edge node [$\sigma \beta_2 \lambda_2$]{} (RI); (RI) edge node [$\gamma$]{} (RR); (IR) edge node [$\gamma$]{} (RR); (SS) edge \[loop left, looseness=1\] node [$\upsilon$]{} (RS); (IS) edge node [$\sigma \beta_2 \lambda_2$]{} (II); (II) edge node [$\gamma$]{} (IR); (SI) edge \[left\] node [$\sigma\beta_1 \lambda_1$]{} (II); (II) edge \[left\] node [$\gamma$]{} (RI);
To model the dynamics of two pathogens with partial vaccination, we extend a SIR model following [@alonso2007stochastic] and [@kamo2002effect]. Let $X_{kl}(t)$ denote the number of individuals at time $t$ in immunological status $k\in\{S , I, R\}$ for pathogen 1 (Influenza) and immunological status $l\in\{S, I, R\}$ for pathogen 2 (RSV). Reactions are illustrated in Figure \[fig:model1\].\
We use mass action with contact rates $\beta_1$ and $\beta_2$ to describe the flow of newly infected individuals from the susceptible group to the group of individuals infected with influenza or RSV respectively. The low spatial coupling is modeled with the immigration of infectious individuals with either disease at rate $\eta$. The average residence time for both diseases is $1/\gamma = 7$ days [@CDC2017]. The population is assumed constant $\Omega$. Therefore, we have set the birth rate equal to $\mu \Omega$, while life expectancy is set equal to $1/\mu=70$ years [@WHO2017]. Vaccination is not completely effective. Thus $\upsilon$ represents the effective vaccination rate. Vaccinated people either go to the recovered class $X_{RS}$ (concerning influenza) or remain in the susceptible class $X_{SS}$ depending on the vaccine efficacy. On the other hand, $\lambda_1$ and $\lambda_2$ represent the population infected with influenza and RSV respectively. Finally, to describe the relationship between RVS and influenza, we use a parameter $\sigma$ to describe either cross-immunity or cross-enhancement [@kamo2002effect; @adams2007influence]. There is pathogen cross-immunity when $0< \sigma<1$. This indicates that the presence of either pathogen inhibits the presence of the other one. $\sigma=0$ confers complete protection against a secondary infection and $\sigma=1$ confers no protection. While $\sigma>1$ represents increasing the degree of cross-enhancement, i.e., the presence of either pathogen enhances the presence of the other one [@adams2007influence].
### Chemical Master Equation
Let us consider a closed population of size $\Omega$ at a given time t, well mixed and homogeneously distributed, where individuals interact via $\mathcal{R}=25$ reactions depicted in Figure \[fig:model1\]. Transitions between states depend only on the time interval but not on absolute time, i.e., $X(\Delta t)$ and $X(t+\Delta t)-X(t)$ are identically distributed. Additionally, two or more transitions take place in the same time interval with zero probability. Finally, for small time increments $\Delta t$, the transition probabilities $a_j(y)$ are obtained by multiplying the rates shown in Figure \[fig:model1\] by $\Delta t$, see [@allen2008introduction; @gillespie2007stochastic]. These assumptions are encoded in the Kolmogorov forward equation (Chemical Master Equation, or CME). It represents the evolution of the probability distribution of finding the system in state $X=x$ at time t.
$$\label{eqs:ME}
\dfrac{dP_x(t)}{dt} = \sum_{j=1}^{\cal R}a_j(x-v_j)P_{x-v_j}(t)-\sum_{j=1}^{\cal R}a_j(x)P_x(t)$$
where $x(t)$ corresponds to the realizations of the random vector $X(t)=[X_{i}(t)]$ and $v_j(t)$ are the stoichiometric vectors e.g. vectors whose elements in $\{-1, 0 ,1\}$ describe the addition/subtraction of mass from a particular compartment. Let $S = S_{ij}, i=1,...,5$; $j=1,...,\mathcal{R}$ be the stoichiometric matrix that describes changes in the population size due to each of the $\mathcal{R}$ reactions and $S=[v_1,\ldots,v_{\mathcal{R}}]$. A list with the $\cal R$ reactions and the explicit form of these terms are defined in the supplementary material.
### Seasonal Forcing
Often, in order to analyze the full time-dependent master equation for the two pathogens model with seasonal forcing, authors describe the system dynamics using the same equations, e.g. equations (\[eqs:ME\]) and (\[eq:Langevin\]), except that $\beta_1$ and $\beta_2$ are functions of time, i.e.,
$$\label{eqs:beta}
\beta_{p}(t) = \beta_{i}(1+\delta \cos (2\pi t/T))$$
for $p,i=1,2$. Parameters $\beta_{i}$, are the baseline contact rate, $\delta$ is the magnitude of seasonal forcing and $T$ is the period of one year.
Theoretical and computational tools
-----------------------------------
### Van Kampen Expansion
For large populations, equation (\[eqs:ME\]) is computationally too expensive to be solved exactly. Hence, we assume that the linear noise approximation holds
$$\label{eqs:LNA}
X(t) = \Omega \phi(t) + \Omega^{1/2} \xi(t), \quad t\in[0, T],$$
namely, for large $\Omega$ the system states $X=x$ can be expressed as the sum of a macroscopic term $\phi(t)$ and a stochastic term $\xi(t)$, which describes the fluctuations and accounts for demographic stochasticity in the system. Combining equations (\[eqs:ME\]) and (\[eqs:LNA\]) gives rise to the van Kampen expansion [@van1992stochastic]. Assuming constant average concentration, the size of the stochastic component will increase as the square root of population size. The time-evolution of the terms of order $\Omega^{1/2}$ [@van1992stochastic] is governed by the ODE system
$$\label{eqs: detpartODE}
\begin{split}
\dfrac{d \phi_i (t)}{dt} & =\sum_{j=1}^{\cal R} S_{ij}f_j(\phi(t),t)\\
\phi_i (0) &= \phi_0
\end{split}$$
where $t\in [0,T], \; i=1,\ldots, \mbox{dim}\{X(t)\}$, $\phi_i(t)=\lim_{\Omega, X \longrightarrow \infty}X_{i}/\Omega$, and $f_j(\phi(t),t) = a_j(\phi(t))$. Collecting terms of order $\Omega^0$, we obtain a Fokker Plank equation for the joint distribution of the system fluctuations, see [@van1992stochastic]. Of note, there is a well known Langevin equation, which describes the temporal evolution of the normalized fluctuation of susceptible and infectious states [@alonso2007stochastic], and whose solution is the same as the Fokker-Planck equation for the system fluctuations in a neighborhood of the macroscopic steady state
$$\label{eq:Langevin}
\dot{\xi}(t) = A(t)\xi(t)+\zeta(t)$$
where $\zeta(t)$ is white noise with zero mean and correlation structure given by $\langle\zeta(t)
\zeta(t')^T\rangle = B(t)\delta(t-t')$. Here, $A(t) = \partial S f(\phi(t),t)/\partial \phi(t)$, $B(t) = EE^{T}$ and $E = S\,\mbox{diag}\{\sqrt{f(\phi(t), t)}\} $, see [@van1992stochastic; @gillespie2007stochastic; @komorowski2009bayesian].
### Power Spectral Density
We consider both, seasonally forced and unforced models. Our contributions rest on examining how the natural frequency of the epidemic outbreak varies when some key epidemic parameters are changed [@wang2012simple]. Consequently, in this Subsection, we describe the method first introduced by Newman and Mckane [@reuman2006power] to compute the analytical PSD in a neighborhood of the system attractor (steady state or limit cycle) to a two pathogen model.
### Unforced Model {#subsec:unforced}
We consider the power spectral density of the fluctuations obtained through Wiener-Khinchin theorem, [@champeney1987handbook] by Fourier transforming linear stochastic differential equation (\[eq:Langevin\])
$$P_k(\omega) = \langle|\tilde{\xi}_k(\omega)|^2\rangle$$
formally
$$\tilde{\xi}_k = \int_{-\infty}^{\infty}\xi_k(t)e^{-k\omega t}dt$$
for $k = 1,\ldots,5$. Rozhnova [@rozhnova2011stochastic], [@rozhnova2009fluctuations] provides a closed expression for the PSD in terms of matrices $A$ and $B$ obtained by the van Kampen expansion. To compute the PSD, matrices $A$ and $B$ are evaluated at the steady state of the system (\[eqs: detpartODE\]). The general solution is given by
$$\label{eq:psd}
P_{kl}(\omega) = \sum_{i,j} \Phi_{kj}(\omega)B_{ji} \Phi_{il}^\dagger(\omega),$$
$\Phi^\dagger(\omega) = (\Phi^H)^{-1} $ means the inverse of the conjuate transpose of $\Phi$ where $\Phi(\omega)=-i\omega I-A$. Equation (\[eq:psd\]) allows to compute the PSD for a wide range of frequencies and parameter ranges with moderate computational burden.
### Forced Model {#subsec:Floquet}
Matrices $A(t)=A(t+T)$ and $B(t)=B(t+T)$ are now periodic functions of time, instead of the method used in Subsection \[subsec:unforced\], we use Floquet’s theory to find the solution of Eqs. (\[eq:Langevin\]) and compute its power sprectrum density, see [@black2010stochastic].\
The solution of equation (\[eq:Langevin\]) can be written as a sum of the general solution of the homogeneous and a particular solution of the inhomogeneous system getting $$\frac{d\Phi(t)}{dt} = A(t)\Phi(t)$$
Where $\Phi(t)$ is the fundamental matrix [@grimshaw1991nonlinear], formed from the linearly independent solutions of homogeneous equation $\dot{\xi}(t) = A(t)\xi(t)$. Floquet’s theorem states that there exists a periodic non singlular matrix $M$ [@grimshaw1991nonlinear] such that $$\Phi(t+T)=\Phi(t)M$$
Matrix $M$ is sometimes referred as the monodromy matrix of the fundamental matrix $\Phi(t)$. This can be expressed in terms of the fundamental matrix by setting $t=0$ $$M = \Phi^{-1}(0)\Phi(T)$$
It is useful to choose $\Phi(t)$ to be the principal matrix, so that $\Phi(0)=I$, and then $M=\Phi(T)$. The eigenvalues of $M$, $\rho_1, \ldots, \rho_n$, are called the *caracteristic multipliers* and a related set of quantities are the *Floquet exponents* defined by $$\label{FloExp}
\vartheta_i = \dfrac{ln(\rho_i)}{T}$$
Of note, a limit cycle will be stable if $|\vartheta_i|<1$, see [@rozhnova2010stochastic], [@grimshaw1991nonlinear]. Using further Floquet’s theory and analytical expression, it is possible to obtain the auto-correlation function of the stochastic fluctuations [@black2010stochastic], [@boland2009limit], [@rozhnova2010stochastic], given by
$$C(\tau) = \dfrac{1}{T}\int_0^{T}\langle \xi(t+\tau)\xi^{'}(t)dt\rangle, \hspace{0.3cm} \xi \equiv \{\xi_1,\ldots,\xi_5\}$$
Taking the Fourier transform of this expression, we get an exact expression for power spectrum of the stochastic oscillations. The details are presented in references [@black2010stochastic; @boland2009limit] and the algorithm to compute the PSD is described by Black [@black2010thesis] (p. 111).
### Coherence
Based on Alonso et al [@alonso2007stochastic] definition of coherence as a measure of stochastic amplification, we consider the normalized cross-correlation as a measure of similarity of influenza and RSV spectral densities
$$\label{eq:coherence}
Q_{kl}(\omega)=\frac{P_{kl}(\omega)}{\sqrt{P_{k}(\omega)P_{l}(\omega)}},$$
where we denote $P_{kl}(\omega)=P_{k}(\omega)$ if $k=l$. In Section \[sec:Results\] we use equation to examine the out of phase relationship and correlation of influenza and RSV signals as a function of vaccination and cross-immunity rates at selected frequency ranges.
Results {#sec:Results}
=======
To carry out our analysis of the two pathogen model, we take a simplified Markov jump process whose elements are defined in terms of the Markov process defined in Figure \[fig:model1\] using the following identities
$$\begin{aligned}
Y_1(t) &= X_{SS}(t)\\
Y_2(t) &= X_{IS}(t)+X_{II}+X_{IR}(t)\\
Y_3(t) &= X_{SI}(t)+X_{II}+X_{RI}(t)\\
Y_4(t) &= X_{IS}(t)+X_{RS}(t)\\
Y_5(t) &= X_{SI}(t)+X_{SR}(t)\end{aligned}$$
where the $Y_{i}$, $i=1,...,5$ represent respectively, the number of those individuals who are susceptible to both pathogens, those who are infected with influenza only, those who are infected with RSV only, those who are susceptible to RSV and those who are susceptible to influenza respectively.
Role of seasonality on system fluctuations
------------------------------------------
According to Rozhnova and Nunes [@rozhnova2010stochastic], using Floquet’s theory described in (\[subsec:Floquet\]) we can show that the power spectral density has peaks at frequencies
$$\label{eqs:frecuency}
\dfrac{m}{T}\pm \dfrac{|Im(\vartheta_p)|}{2\pi}, \hspace{0.3cm} p=1,2$$
where $m$ is an integer and $\vartheta_p$ are the Floquet exponents. For the annual limit-cycle the dominant peak is at $Im(\vartheta_p)/2\pi$, with the others peaks being much smaller. Here, $|Im(\vartheta_p)|$ denotes the absolute value of the imaginary part of complex conjugate Floquet exponents.\
![[**PSD without and with seasonal forcing**]{}. Vertical helper lines mark the frequencies predicted by Eq. (\[eqs:frecuency\]). The parameters values are $\beta_1 = 93.88$ $year^{-1}$, $\beta_2 = 83.45$ $year^{-1}$, $\sigma=0.8$, $\delta=0.06$ and $\upsilon=0$. The Floquet exponents are $\vartheta_1 = -0.01341813\pm 0.74852756i $ and $\vartheta_2 = -0.01071943+0.58598082i $ and $\vartheta_3=-0.02753316+0.i$. []{data-label="fig:FigureSF1"}](both500.png "fig:"){width="100.00000%"}\
Figure \[fig:FigureSF1\] shows the analytic PSD with and without seasonal forcing. The parameters values are $\beta_1 = 93.88$ $year^{-1}$, $\beta_2 = 83.45$ $year^{-1}$, $\sigma=0.8$, $\delta=0.06$ and $\upsilon=0$. Floquet exponents are given by $\vartheta_1 = -0.01341813\pm 0.74852756i $ and $\vartheta_2 = -0.01071943+0.58598082i $ and $\vartheta_3=-0.02753316+0.i$. Thus, the dominant peak for Influenza is given by $Im(\vartheta_1)/2\pi= 0.119131$ $year^{-1}$ and the mean peak for RSV is given by $Im(\vartheta_2)/2\pi = 0.0932$ $year^{-1}$. We can see that the PSD for the non-seasonal case are qualitatively comparable with the PSD obtained for the seasonally forced system. The main difference is the addition of two annual peaks with seasonal transmission. The period for both peaks of influenza are $0.893549$ and $1.1352436$ $year$ and the period for the RSV peaks are $0.914694$ and $1.1028540$ $year$.\
Bifurcation diagram (not shown), indicates that there is a period doubling bifurcation as we increase either $\sigma$, or $\delta$. However, in the one hand we care about pathogen replacement during the early season epidemics. And on the other hand, the PSD analytic formulas do not hold near bifurcation points. Consequently, we limit our analysis to the yearly regime.
Epidemic criticality conditions
-------------------------------
Since seasonal and epidemic fluctuations are separated as indicated in Figure \[fig:FigureSF1\], in the remainder we focus on the analysis of the epidemics fluctuations. Let us analyze the steady states of the system without seasonality to understand the relationship between effective vaccination and cross-immunity. In terms of the deterministic model (\[eqs: detpartODE\]), the largest eigenvalue of the next generation matrix, or basic reproductive number is
$$R_0 = \max_{i=1,2} R_i$$
where, $R_{1}$, $R_{2}$ given by
$$\begin{split}
R_1 &= \dfrac{\mu\beta_1}{(\gamma+\mu)(\mu+\upsilon)}\\
R_2 &= \dfrac{\beta_2(\mu+\sigma\upsilon)}{(\gamma+\mu)(\mu+\upsilon)}
\end{split}$$
are the two only eigenvalues of the next generation matrix of van den Driessche and Watmough [@van2002reproduction] (see supplementary material for details). The two eigenvalues correspond to the reproduction numbers for each pathogen, $R_1$ for influenza and $R_2$ for RSV. If $\eta=0$ system (\[eqs: detpartODE\]) has a disease free equilibrium ($Y_{DFE}$)
$$\label{FixPoint1}
Y_{DFE} = [\mu/(\mu+\upsilon),0,0,\upsilon/(\mu+\upsilon),0].$$
The disease-free equilibrium is stable if $R_{0}<1$ and unstable if $R_{0}>1$. There is a steady state where only people infected with influenza is present, and similarly, there is a steady state where only people infected with RSV is present. Also, there is a steady state where both diseases coexist. We base this claim on Vasco et al [@vasco2007tracking] analysis of the case with no vaccination.
Analysis of cross-immunity
--------------------------
Let us study the disease-free equilibrium (\[FixPoint1\]), and conditions under which it is possible to eradicate both diseases assuming that there is no seasonal forcing. We will focus on the cross-immunity $\sigma$ and the effective vaccination rate $\upsilon$.\
The equilibrium point $Y_{DFE}$ is stable if $R_0<1$. Of note, $R_1$ and $R_2$ dependence on $\upsilon$ imply that by increasing the effective vaccination value we may reduce $R_0$ below 1, thus erradicating both diseases, even when vaccination is only against influenza. $R_0<1$ implies $R_1<1$ and $R_2<1$. We explore the scenarios that may take place given the conditions of stability for the free-disease equilibrium, i.e. $R_1<R_2<1$ and $R_2<R_1<1$. Consequently, we will plot the PSD for selected vaccination values in each case.\
If we consider the case when $R_1<R_2<1$, the conditions to effective vaccination rate and cross-immunity are
- $\upsilon>\dfrac{\mu(\beta_2-(\gamma+\mu))}{(\gamma+\mu-\sigma\beta_2)}$
- $\sigma<\dfrac{\gamma+\mu}{\beta_2}<1$
On the other hand, if we consider the case $R_2<R_1<1$ we have:
- $\upsilon>\dfrac{\mu(\beta_1-(\gamma+\mu))}{(\gamma+\mu)}$
- $\sigma<\dfrac{\gamma+\mu}{\beta_2}<1$
- $\beta_1>(\gamma+\mu)$
There are three cases to consider,
Case $\sigma<1$.
: See Fig. \[fig:vaccination\]. This condition assumes that getting sick from either virus confers some protection against the second one. Let us denote $\mathcal{R}_1$ and $\mathcal{R}_2$ respectively the values of $R_1$ and $R_2$ without the presence of effective vaccination, i.e. $\mathcal{R}_1 =\dfrac{\beta_1}{\gamma+\mu}$ and $\mathcal{R}_2 = \dfrac{\beta_2}{\gamma+\mu}$.\
Furthermore, let us consider the starting point in parameter space where $\mathcal{R}_1= 1.6$ and $\mathcal{R}_2=1.23$. This implies $\beta_1 = 83.45$ $year^{-1}$ and $\beta_2 = 64.15$ $year^{-1}$. Let us set $\sigma=0.8$. With these values, $\sigma$ and $\upsilon$ satisfy the condition above. Thus, the effective vaccination rate values for vanish both viruses are $\upsilon=0.008571$ $year^{-1}$ for influenza and $\upsilon = 0.2053999$ $year^{-1}$ for syncytial. This means that when there is some kind of cross-immunity between these viruses, it is possible to control both, even when vaccine acts only against influenza. On the other hand, it is possible to find vaccination values that, while decreasing the amplification of influenza, increase the amplification of RSV. It is noteworthy that there is a vaccination value ($\upsilon\approx0.0073$) where the PSD peak for both diseases trade places. Of note, this phenomenon has been observed in real data, e.g., [@noyola2005contribution].\
![Power Spectral Density for the case $\sigma<1$. This figure shows how the PSD for influenza and RSV vanish when crossing the respective thresholds $\upsilon=0.008571$ and $\upsilon = 0.2053999$ $year^{-1}$. PSD is shown at selected vaccination rates.[]{data-label="fig:vaccination"}](figure_s08_R1_16_R2_123){width="100.00000%"}
Case $\sigma=1$
: In this case the behavior of both pathogens is independent of each other. Note that $R_2 = \dfrac{\beta_2}{\gamma+\mu}$, i.e., it no longer depends on $ \upsilon $ or $\sigma $. Let us consider $\mathcal{R}_1=1.6$ and $\mathcal{R}_2=1.5$ implies $\beta_1 = 83.65$ and $\beta_2 = 78.42$\
In this case, the behavior of both diseases is independent, see Fig. \[case2\]. The frequency of the RSV peaks is the same in all cases and the frequency of influenza peaks increases when the effective vaccination rate is increased.
![Power Spectral Density for the case $\sigma=1$. Parameter values are $\sigma=1.0$, $\beta_1 = 83.65$ and $\beta_2 = 78.42$.[]{data-label="case2"}](figure_s1_R1_16_R2_15){width="100.00000%"}
Case $\sigma>1$.
: Let us consider the case when $\sigma=1.1$, that means that the presence of one pathogen enhances the presence of the second one. Thus, the vaccination rate not only changes the period of the influenza peak but also produces small changes in the RSV peak period.\
Figure \[fig:swap\] provides evidence that vaccination affects both influenza and RSV total amplification. Total amplification corresponds to the integral of the PSD over all frequencies. This phenomenon is observed when $\sigma$ is less and greater than one. After several experiments, we noticed that in some cases, by increasing the vaccination rate, we reduce the amplification of influenza but it can be amplified that of the RSV, see Fig. \[fig:vaccination\] when $\upsilon \approx 0.0083$ and Fig. \[fig:swap\] when $\upsilon \geq 0.0073$.
![Total Amplification for both pathogens. Parameter values: $\mathcal{R}_1=1.8$, $\beta_1=93.88$ $year^{-1}$, $\mathcal{R}_2=1.5$, $\beta_2=78.23$ $year^{-1}$, $\sigma=1.1$. $A_{Inf}$ and $A_{RSV}$ correspond to the total amplification of influenza and RSV respectively. They are calculated with different vaccination rate values ($\upsilon$)[]{data-label="fig:swap"}](swap2 "fig:"){width="100.00000%"}\
Discussion {#sec:discussion}
==========
There is evidence that before the introduction of influenza vaccination programs, seasonal patterns of RSV and influenza were regular, with an outbreak of RSV immediately followed by an influenza outbreak each year[@aanestad1987surveillance]. But, in the presence of vaccination against influenza either pathogen might invade first, see [@rozhnova2010stochastic; @noyola2005contribution]. Our analysis supports the claim that early season pathogen replacement depends on effective vaccination rate and relative virus fitness, e.g., $R_1$ and $R_2$. Vaccination changes the natural frequency and relative fitness of both virus, thus allowing either virus to peak first in a given season. If the strength of influenza infection ($R_1$) is greater than the strength of the RSV infection ($R_2$) and the vaccination rate is small, then the influenza peak can happen first. Otherwise, if the vaccination rate is large enough the RSV peak might happen first, even when $R_1>R_2$. However, the peak of RSV will appear first if $R_2>R_1$.\
![[**PSD Correlation as a measure of stochastic amplification among pathogens.**]{} Given fixed basic reproductive numbers for influenza ($\mathcal{R}_{1}=1.6$), and RSV ($\mathcal{R}_{2}=1.4$), we show from left to right, from top to bottom PSD Correlation at vaccination rates $\upsilon=(0,0.0036,0.0091,0.018,0.025,0.036)$ year$^{-1}$. This figure shows the correlation between the two pathogen amplifications when $\sigma$ is varied and how the frequency for both viruses change when $\upsilon$ increase. For $\sigma =1$ they are completely independent. []{data-label="fig:correlation"}](complex_coherence_sigma.png "fig:"){width="100.00000%"}\
Previous results [@anestad1982interference; @noyola2005contribution; @shinjoh2000vitro] support the existence of interference between outbreaks of RSV infection and influenza. We include the term $\sigma$ in the model to explore the immunity relationship between pathogens at the population level.\
When $\sigma\neq1$ there is a correlation in the fluctuations of both diseases as we can see in Figure \[fig:correlation\]. Consequently, partial vaccination not only affects the behavior of influenza but also the behavior of RSV. Of note, according to the model, it becomes possible to eradicate RSV when $\sigma<1$ by increasing the rate of vaccination, even if vaccination is directed only against influenza. On the other hand, $\sigma=1$ means that there is not immunity relation between the pathogens. In this case, we can vanish influenza by increasing the vaccination rate without having any effect on RSV. Moreover, when $\sigma\approx1$ and $R_1\approx R_2$, the periodicity of both diseases are similar and there is an overlap giving the shape of M that we can see in real time series [@anestad1982interference], [@noyola2005contribution]. But, when $R_1$ and $R_2$ have a considerable difference or $\sigma$ is far from one, the peaks have a totally different period.\
Another important factor is the seasonal forcing. Influenza and RSV are seasonally related [@mangtani2006association], [@bloom2013latitudinal]. Circulation of both often occur at similar times of the year in some temperate zones and peaks timing differ by less than one month[@bloom2013latitudinal], [@velasco2015superinfection]. Seasonality can induce epidemic cycles. When this is included in the model, the peaks periodicity are not affected but, non-seasonal peaks appear in frequencies $m/T \pm |Im(\vartheta)|/2\pi$. We predict the number and position of the dominant and non-seasonal peaks as a function of the epidemiological parameters.\
Finally, we consider that our analysis might serve as a basis to explore further the effect of partial vaccination on multi-pathogen epidemics. Of particular importance is to study the effect of vaccination aiming at reducing the morbidity caused by respiratory diseases.
[^1]: Centro de Investigación en Matemáticas A.C. Jalisco S/N Col. Valenciana, CP: 36240, Guanajuato, Gto. México. E-mail addresses: yury@cimat.mx (Y.E. García), marcos@cimat.mx (M.A. Capistrán)
|
---
abstract: 'Under the assumption that the density variation of the electrons can be approximated by an exponential function, the solar Mikheyev-Smirnov-Wolfenstein effect is treated for three generations of neutrinos. The generalized hypergeometric functions that result from the exact solution of this problem are studied in detail, and a method for their numerical evaluation is presented. This analysis plays a central role in the determination of neutrino masses, not only the differences of their squares, under the assumption of universal quark-lepton mixing.'
address:
- 'Department of Physics, University of Bergen, Allégaten 55, N-5007 Bergen, Norway'
- |
Gordon McKay Laboratory, Harvard University, Cambridge, Massachusetts 02138, and\
Theoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland
author:
- Per Osland
- and
- Tai Tsun Wu
title: |
Solar Mikheyev-Smirnov-Wolfenstein Effect\
with Three Generations of Neutrinos
---
[CERN-TH/99-286]{}\
[December 1999]{}
Introduction
============
Recently a program has been proposed to study neutrino masses [@OW-99-L]. This program is based on the following assumptions.
[**Assumption 1:**]{} There are three right-handed Dirac neutrinos in addition to the three known left-handed neutrinos.
[**Assumption 2:**]{} The lepton mass matrices have the same structure as the quark mass matrices, as described by Lehmann [*et al.*]{} [@LNW].
In carrying out this program, one of the technically most difficult aspects is to treat the Mikheyev-Smirnov-Wolfenstein (MSW) effect for the sun [@MSW]. It is the purpose of this paper to discuss this problem.
The difficulty arises in the following way. On the basis of the Super-Kamiokande data on atmospheric neutrinos [@Super-K-prl], the muon neutrino $\nu_\mu$ oscillates with another neutrino which is not the electron neutrino $\nu_e$. From Assumption 1, this neutrino must be the tau neutrino $\nu_\tau$. In other words, $\nu_\mu$ and $\nu_\tau$ are coupled significantly. Again from Assumption 1, the electron neutrinos $\nu_e$ from the sun must therefore oscillate with either $\nu_\mu$ or $\nu_\tau$. Since $\nu_\mu$ and $\nu_\tau$ are coupled significantly, the electron neutrinos $\nu_e$ from the sun must oscillate with [*both*]{} $\nu_\mu$ [*and*]{} $\nu_\tau$. It is accordingly necessary to understand the MSW effect with all three generations of neutrinos $\nu_e$, $\nu_\mu$ and $\nu_\tau$.
While there is an extensive literature on the MSW effect for two generations of neutrinos [@MSW-2-exp; @MSW-2-lin], relatively little is known for the case of three generations [@MSW-3]. One of the major differences between these two cases is the following. For two generations, the MSW effect occurs predominantly in the vicinity of a particular density in the sun. For this reason, the important parameter is the rate of change of the density at this particular density. In contrast, for three generations, the important contributions to the MSW effect can come from several regions. When the parameters, such as the neutrino masses, vary, these regions not only move but also merge and separate, features that are absent for the much simpler case of two generations.
Formulation of the Problem
==========================
There are many different ways to formulate the problem of the MSW [@MSW] effect in the sun for three generations of neutrinos. These different formulations are of course closely related to each other; it has been found that, at least for the present program [@OW-99-L], the following one is most appropriate.
In a current basis, the charged lepton currents are $$J_\mu\sim \bar\nu_{\rm L}\gamma_\mu \ell_{\rm L},$$ where $$\nu=
\left[\tighten\matrix{\nu_1 \cr \nu_2 \cr \nu_3\cr
}\right] \quad
{\rm and} \quad
\ell=\left[\tighten\matrix{
\ell_1 \cr \ell_2 \cr \ell_3\cr}\right]$$ denote the neutrino and charged lepton fields. The neutrino mass matrix $M$ [@Rosen], from the vacuum expectation value of the Higgs field in the neutrino-Higgs coupling, is $$\bar\nu_{\rm L} M \nu_{\rm R}+{\rm h.c.}$$
Let $p$ be the momentum of the neutrino, the value of which is taken to be much larger than the neutrino masses, which are the eigenvalues of $M$. Typically, $p$ is in the range from 0.4 MeV/$c$ to 10 MeV/$c$, while the neutrino masses are believed to be no more than about 1 eV; hence this assumption is well satisfied. Under this assumption, it is $M^2$ that enters in the MSW effect. Let $$M^2
=\left[\tighten\matrix{
M^2_{11} & M^2_{12} & M^2_{13}\cr
M^2_{21} & M^2_{22} & M^2_{23}\cr
M^2_{31} & M^2_{32} & M^2_{33}\cr}\right]$$ be real and symmetric. Note that $M_{ij}$ are not defined, only $M^2_{ij}\equiv(M^2)_{ij}$. Of course the eigenvalues of this matrix $M^2$ are the squares of the neutrino masses.
In terms of this $M^2$, the MSW effect for neutrino oscillations is described by the coupled ordinary differential equation [@MSW] $$\label{Eq:Schr-1}
i\,\frac{d}{dr}
\left[\tighten\matrix{
\phi_1(r)\cr \phi_2(r)\cr \phi_3(r)\cr}\right]
=\left(
\left[\tighten\matrix{
D(r) & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr}\right]
+\frac{1}{2p}
\left[\tighten\matrix{
M^2_{11} & M^2_{12} & M^2_{13}\cr
M^2_{21} & M^2_{22} & M^2_{23}\cr
M^2_{31} & M^2_{32} & M^2_{33}
\cr}\right]
\right)
\left[\tighten\matrix{
\phi_1(r)\cr \phi_2(r)\cr \phi_3(r)
\cr}\right],$$ where $$\label{Eq:define-D}
D(r)=\sqrt{2}\,G_{\rm F} N_e(r),$$ with $G_{\rm F}$ the Fermi weak-interaction constant and $N_e(r)$ the electron density at the point $r$.
So far this formulation is general. In view of the observation in Sec. I that, for three generations of neutrinos, important contributions to the MSW effect can come from several values of $r$, we propose to choose a suitable function $D(r)$. On the one hand, this $D(r)$ should describe approximately the electron density in the sun; on the other hand, this $D(r)$ should be sufficiently simple so that Eq. (\[Eq:Schr-1\]) can be solved explicitly. This choice is made in the following way.
First, since most neutrinos are produced by nuclear reactions [@sun] near the center of the sun, the simplifying assumption is made that the neutrinos are produced at the center of the sun. This has the consequence that the $r$ in Eqs. (\[Eq:Schr-1\]) and (\[Eq:define-D\]) is the radial distance from the center of the sun. This is the reason for the notation $r$.
Secondly, the density of the sun as a function of the radial distance is fairly accurately known [@BBP-98], as shown by the solid curve in Fig. 1, where the vertical axis is the density in g/cm$^3$ in a logarithmic scale, and the horizontal axis is $r/R_\odot$. The radius of the sun is $$R_\odot=6.96\times10^8\ {\rm m}.$$ From this solid curve in Fig. 1, it is seen that the solar density is approximately exponential, as shown by the dotted line of Fig. 1. This fit is reasonably good, to an accuracy of roughly 15%, for $$0.05 < r/R_\odot < 0.9.$$ In other words, $D(r)$ is taken to be given by $$\label{Eq:D-exponential}
D(r)=D(0)e^{-r/r_0}.$$
\[Fig:rho\]
(12,7.5) (2.0,1.0)
From the dotted line of Fig. 1, the following values are obtained $$D(0)=0.0458~{\rm km}^{-1}, \qquad r_0\simeq0.1\times R_\odot.$$ The problem is therefore to solve the MSW differential equation, for $r\ge0$, $$\label{Eq:Schr-2}
i\,\frac{d}{dr}
\left[\tighten\matrix{
\phi_1(r)\cr \phi_2(r)\cr \phi_3(r)
\cr}\right]
=\frac{1}{2p}
\left[\tighten\matrix{
2pD(0)e^{-r/r_0}+M^2_{11} & M^2_{12} & M^2_{13}\cr
M^2_{21} & M^2_{22} & M^2_{23}\cr
M^2_{31} & M^2_{32} & M^2_{33}
\cr}\right]
\left[\tighten\matrix{
\phi_1(r)\cr \phi_2(r)\cr \phi_3(r)
\cr}\right]$$ with the boundary conditions $$\label{Eq:boundary-cond}
\left[\tighten\matrix{
\phi_1(0)\cr \phi_2(0)\cr \phi_3(0)
\cr}\right]
=
\left[\tighten\matrix{
1\cr 0\cr 0\cr}\right].$$
Solution of the Differential Equation
=====================================
The differential equation (\[Eq:Schr-2\]) can be solved explicitly in terms of generalized hypergeometric functions. This is to be carried out in the present section. Indeed, this is the underlying reason for the choice of the exponential function (\[Eq:D-exponential\]).
Let $$u=r/r_0+u_0,$$ with $$\label{Eq:u0}
u_0=-\ln[D(0)r_0]\sim -8.07.$$ Then (\[Eq:Schr-2\]) is $$\label{Eq:Schr-3}
i\,\frac{d}{du}
\left[\tighten\matrix{
\phi_1\cr \phi_2\cr \phi_3
\cr}\right]
=
\left[\tighten\matrix{
e^{-u}+{\displaystyle \frac{r_0}{2p}}\,M^2_{11} &
{\displaystyle \frac{r_0}{2p}}\,M^2_{12} & {\displaystyle
\frac{r_0}{2p}}\,M^2_{13}\cr \noalign{\vskip4pt}
{\displaystyle
\frac{r_0}{2p}}\,M^2_{21} & {\displaystyle \frac{r_0}{2p}}\,M^2_{22} &
{\displaystyle
\frac{r_0}{2p}}\,M^2_{23}\cr \noalign{\vskip4pt}
{\displaystyle \frac{r_0}{2p}}\,M^2_{31} &
{\displaystyle \frac{r_0}{2p}}\,M^2_{32} & {\displaystyle
\frac{r_0}{2p}}\,M^2_{33}
\cr}\right]
\left[\tighten\matrix{
\phi_1\cr \phi_2\cr \phi_3
\cr}\right].$$ Note that $u=0$ corresponds to $$r=-r_0u_0\sim 0.81 R_\odot.$$
In view of the form of Eq. (\[Eq:Schr-3\]), it is convenient to diagonalize the $2\times2$ matrix $$\frac{r_0}{2p}
\left[\tighten\matrix{
M^2_{22} & M^2_{23}\cr
M^2_{32} & M^2_{33}
\cr}\right].$$ Let $\theta_0$ be the angle of rotation that diagonalizes this $2\times2$ matrix, i.e., $$\left[\tighten\matrix{
\cos\theta_0 & -\sin\theta_0\cr
\sin\theta_0 & \cos\theta_0
\cr}\right]
\frac{r_0}{2p}
\left[\tighten\matrix{
M^2_{22} & M^2_{23}\cr
M^2_{32} & M^2_{33}
\cr}\right]
\left[\tighten\matrix{
\cos\theta_0 & \sin\theta_0\cr
-\sin\theta_0 & \cos\theta_0
\cr}\right]
=
\left[\tighten\matrix{
\omega_2 & 0\cr
0 & \omega_3
\cr}\right]$$ with $$\label{Eq:omega-order}
\omega_2 < \omega_3.$$ The reason for this Eq. (\[Eq:omega-order\]) is that, if $\omega_2=\omega_3$, then the problem reduces to neutrino oscillations with only two neutrinos. Define also $$\omega_1=\frac{r_0}{2p}\,M^2_{11}, \qquad
\left[\tighten\matrix{
\chi_2\cr \chi_3
\cr}\right]
=\frac{r_0}{2p}
\left[\tighten\matrix{
\cos\theta_0 & -\sin\theta_0\cr
\sin\theta_0 & \cos\theta_0
\cr}\right]
\left[\tighten\matrix{
M^2_{12}\cr M^2_{13}
\cr}\right]$$ and $$\psi_1(u)=\phi_1(u), \qquad
\left[\tighten\matrix{
\psi_2(u)\cr \psi_3(u)
\cr}\right]
=\left[\tighten\matrix{
\cos\theta_0 & -\sin\theta_0\cr
\sin\theta_0 & \cos\theta_0
\cr}\right]
\left[\tighten\matrix{
\phi_2(u)\cr \phi_3(u)
\cr}\right];$$ then Eq. (\[Eq:Schr-3\]) becomes $$\label{Eq:Schr-4}
i\,\frac{d}{du}
\left[\tighten\matrix{
\psi_1(u)\cr \psi_2(u)\cr \psi_3(u)
\cr}\right]
=
\left[\tighten\matrix{
\omega_1+e^{-u} & \chi_2 & \chi_3\cr
\chi_2 & \omega_2 & 0\cr
\chi_3 & 0 & \omega_3
\cr}\right]
\left[\tighten\matrix{
\psi_1(u)\cr \psi_2(u)\cr \psi_3(u)
\cr}\right].$$
If $\chi_2$ or $\chi_3$ is zero, then once again this reduces to neutrino oscillations with two generations. By reversing the signs of $\psi_2(u)$ and/or $\psi_3(u)$ if necessary, it follows that, without loss of generality $$\label{Eq:chi-positive}
\chi_j>0, \qquad {\rm for \ \ }j=2,3.$$ Let $\mu_1$, $\mu_2$ and $\mu_3$ be the eigenvalues of the $3\times3$ matrix $$\left[\tighten\matrix{
\omega_1 & \chi_2 & \chi_3\cr
\chi_2 & \omega_2 & 0\cr
\chi_3 & 0 & \omega_3
\cr}\right];$$ then these three $\mu$’s are the squares of the neutrino masses multiplied by $r_0/(2p)$. Note that $$\sum_{j=1}^3 \omega_j
=\sum_{j=1}^3 \mu_j.$$ For cases of interest [@OW-99-L], the $\mu_j$ will typically be large, $\mu_j\gg 1$. This is of course not a question of units, but rather a reflection of the physical situation of having a large number of oscillations within the solar radius.
The interlacing property implies that, by renumbering the $\mu$’s if necessary, $$\label{Eq:interlace}
0\le \mu_1 < \omega_2 < \mu_2 < \omega_3 < \mu_3,$$ again assuming that there is no reduction to two-neutrino oscillation.
The following properties are easily verified: With fixed $\omega_2$, $\omega_3$, $\chi_2$ and $\chi_3$,
- $$\frac{\partial\mu_j}{\partial\omega_1}>0 \qquad {\rm for\ \ } j=1,2,3.$$
- As $\omega_1\to\infty$, $$\mu_1\to\omega_2, \qquad \mu_2\to\omega_3, \qquad {\rm and\ \ }
\mu_3\to\infty.$$
- As $\omega_1\to-\infty$, which is physically not possible, $$\mu_1\to-\infty, \qquad \mu_2\to\omega_2, \qquad {\rm and\ \ }
\mu_3\to\omega_3.$$
A third-order ordinary differential equation can be obtained from Eq. (\[Eq:Schr-4\]) as follows. Define $\psi_{11}$, $\psi_{22}$ and $\psi_{33}$ such that $$\begin{aligned}
\label{Eq:psi-i}
\psi_1&=&\left(i\,\frac{d}{du}-\omega_2\right)
\left(i\,\frac{d}{du}-\omega_3\right)\psi_{11}, \nonumber \\
\psi_2&=&\chi_2\left(i\,\frac{d}{du}-\omega_3\right)\psi_{22},
\nonumber
\\
\psi_3&=&\chi_3\left(i\,\frac{d}{du}-\omega_2\right)\psi_{33}.\end{aligned}$$ These $\psi_{11}$, $\psi_{22}$ and $\psi_{33}$ are of course not unique. It then follows from Eqs. (\[Eq:Schr-4\]) and (\[Eq:chi-positive\]) that $$\begin{aligned}
\left(i\,\frac{d}{du}-\omega_2\right)
\left(i\,\frac{d}{du}-\omega_3\right)(\psi_{11}-\psi_{22})&=&0,
\nonumber \\
\left(i\,\frac{d}{du}-\omega_2\right)
\left(i\,\frac{d}{du}-\omega_3\right)(\psi_{11}-\psi_{33})&=&0.\end{aligned}$$ Therefore, $$\begin{aligned}
\psi_{22}&=&\psi_{11}
+K_{22}e^{-i\omega_2u}+K_{23}e^{-i\omega_3u}, \nonumber \\
\psi_{33}&=&\psi_{11}
+K_{32}e^{-i\omega_2u}+K_{33}e^{-i\omega_3u}.\end{aligned}$$ In particular, $$\label{Eq:psi22-psi33}
\psi_{22}-(K_{23}-K_{33})e^{-i\omega_3u}
=\psi_{33}-(K_{32}-K_{22})e^{-i\omega_2u}.$$ Define this function of Eq. (\[Eq:psi22-psi33\]) to be $\psi$, then it follows that $$\begin{aligned}
\psi_{11}&=&\psi-K_{22}e^{-i\omega_2u}-K_{33}e^{-i\omega_3u}, \nonumber \\
\psi_{22}&=&\psi+(K_{23}-K_{33})e^{-i\omega_3u}, \nonumber \\
\psi_{33}&=&\psi+(K_{32}-K_{22})e^{-i\omega_2u}.\end{aligned}$$ Substitution into Eq. (\[Eq:psi-i\]) then expresses $\psi_1$, $\psi_2$ and $\psi_3$ in terms of the single function $\psi$: $$\begin{aligned}
\label{Eq:psi1-2-3}
\psi_1&=&\left(i\,\frac{d}{du}-\omega_2\right)
\left(i\,\frac{d}{du}-\omega_3\right)\psi, \nonumber \\
\psi_2&=&\chi_2\left(i\,\frac{d}{du}-\omega_3\right)\psi, \nonumber \\
\psi_3&=&\chi_3\left(i\,\frac{d}{du}-\omega_2\right)\psi.\end{aligned}$$ Furthermore, this $\psi$ is unique because of (\[Eq:omega-order\])
To obtain the third-order ordinary differential equation for $\psi$, it only remains to substitute (\[Eq:psi1-2-3\]) into the first equation of (\[Eq:Schr-4\]): $$\begin{aligned}
\lefteqn{\left(i\,\frac{d}{du}-\omega_1-e^{-u}\right)
\left(i\,\frac{d}{du}-\omega_2\right)
\left(i\,\frac{d}{du}-\omega_3\right)\psi}\qquad\nonumber \\
&&\mbox{}=\left[\chi_2^2\left(i\,\frac{d}{du}-\omega_3\right)
+\chi_3^2\left(i\,\frac{d}{du}-\omega_2\right)\right]\psi.\end{aligned}$$ From the definition of $\mu_1$, $\mu_2$ and $\mu_3$, this can be rewritten as $$\begin{aligned}
\label{Eq:Schr-5}
\lefteqn{\left(i\,\frac{d}{du}-\mu_1\right)
\left(i\,\frac{d}{du}-\mu_2\right)
\left(i\,\frac{d}{du}-\mu_3\right)\psi}\qquad\nonumber \\
&&\mbox{}=e^{-u}
\left(i\,\frac{d}{du}-\omega_2\right)
\left(i\,\frac{d}{du}-\omega_3\right)\psi.\end{aligned}$$ To put this equation into the form of the differential equation for the generalized hypergeometric function, let $$\label{Eq:z-u}
z=ie^{-u};$$ then the differential equation (\[Eq:Schr-5\]) for $\psi$ is $$\label{Eq:Schr-6}
\left[
\left(z\,\frac{d}{dz}-i\mu_1\right)
\left(z\,\frac{d}{dz}-i\mu_2\right)
\left(z\,\frac{d}{dz}-i\mu_3\right)
-z\left(z\,\frac{d}{dz}-i\omega_2\right)
\left(z\,\frac{d}{dz}-i\omega_3\right)
\right]\psi=0.$$ This is the differential equation for the generalized hypergeometric function $_2F_2$—see, for example, p. 184 of [@Bateman]. Three linearly independent solutions of this third-order differential equation (\[Eq:Schr-6\]) are $$\begin{aligned}
\psi^{(1)}
&=&K_1e^{-i\mu_1u}
{}_2F_2\left[\left.
\tighten\matrix{
-i(\omega_2-\mu_1), & -i(\omega_3-\mu_1)\cr
1-i(\mu_2-\mu_1), & 1-i(\mu_3-\mu_1)
\cr}
\right|ie^{-u} \right], \nonumber \\[5pt]
\psi^{(2)}
&=&K_2e^{-i\mu_2u}
{}_2F_2\left[\left.
\tighten\matrix{
-i(\omega_2-\mu_2), & -i(\omega_3-\mu_2)\cr
1-i(\mu_1-\mu_2), & 1-i(\mu_3-\mu_2)
\cr}
\right|ie^{-u} \right], \nonumber \\[5pt]
\psi^{(3)}
&=&K_3e^{-i\mu_3u}
{}_2F_2\left[\left.
\tighten\matrix{
-i(\omega_2-\mu_3), & -i(\omega_3-\mu_3)\cr
1-i(\mu_1-\mu_3), & 1-i(\mu_2-\mu_3)
\cr}
\right|ie^{-u} \right],\end{aligned}$$ where $K_1$, $K_2$ and $K_3$ are arbitrary non-zero constants. For the convenience of numerical calculations, we make the following choice: $$\begin{aligned}
K_1&=&\left[(\omega_2-\mu_1)(\omega_3-\mu_1)(\mu_2-\mu_1)(\mu_3-\mu_1)
\right]^{-1/2}, \nonumber \\
K_2&=&\left[(\omega_2-\mu_2)(\omega_3-\mu_2)(\mu_1-\mu_2)(\mu_3-\mu_2)
\right]^{-1/2}, \nonumber \\
K_3&=&\left[(\omega_2-\mu_3)(\omega_3-\mu_3)(\mu_1-\mu_3)(\mu_2-\mu_3)
\right]^{-1/2}.\end{aligned}$$
The function $_2F_2$ can be defined by the following series expansion[^1] [@Bateman] $$\label{Eq:F22-series}
{}_2F_2\left[\left.
\tighten\matrix{
\alpha_1, & \alpha_2\cr
\rho_1, & \rho_2
\cr}
\right|z \right]
=\sum_{k=0}^\infty
\frac{(\alpha_1)_k(\alpha_2)_k}{(\rho_1)_k(\rho_2)_k}\,
\frac{z^k}{k!},$$ where the Pochhammer symbol is defined as $$(\alpha)_k=\alpha(\alpha+1)\ldots(\alpha+k-1).$$
Since the Taylor series of $_2F_2$ about $z=0$, Eq. (\[Eq:F22-series\]), is always convergent, the differentiations specified in Eqs. (\[Eq:psi1-2-3\]) can be carried out term by term on these Taylor series. Thus these derivatives of $\psi^{(j)}$ can be easily expressed in terms of $_2F_2$ with parameters slightly modified. The result is $$\begin{aligned}
\label{Eq:psi1-2-3-b}
\psi_1&=&C_1\psi_1^{(1)}+C_2\psi_1^{(2)}+C_3\psi_1^{(3)}, \nonumber \\
\psi_2&=&C_1\psi_2^{(1)}+C_2\psi_2^{(2)}+C_3\psi_2^{(3)}, \nonumber \\
\psi_3&=&C_1\psi_3^{(1)}+C_2\psi_3^{(2)}+C_3\psi_3^{(3)},\end{aligned}$$ where $$\begin{aligned}
\label{Eq:psi1}
\psi_1^{(1)}&=&K_1(\mu_1-\omega_2)(\mu_1-\omega_3)e^{-i\mu_1u}
{}_2F_2\left[\left.
\tighten\matrix{
1-i(\omega_2-\mu_1), & 1-i(\omega_3-\mu_1)\cr
1-i(\mu_2-\mu_1), & 1-i(\mu_3-\mu_1)
\cr}
\right|ie^{-u} \right], \nonumber \\[5pt]
\psi_1^{(2)}&=&K_2(\mu_2-\omega_2)(\mu_2-\omega_3)e^{-i\mu_2u}
{}_2F_2\left[\left.
\tighten\matrix{
1-i(\omega_2-\mu_2), & 1-i(\omega_3-\mu_2)\cr
1-i(\mu_1-\mu_2), & 1-i(\mu_3-\mu_2)
\cr}
\right|ie^{-u} \right], \nonumber \\[5pt]
\psi_1^{(3)}&=&K_3(\mu_3-\omega_2)(\mu_3-\omega_3)e^{-i\mu_3u}
{}_2F_2\left[\left.
\tighten\matrix{
1-i(\omega_2-\mu_3), & 1-i(\omega_3-\mu_3)\cr
1-i(\mu_1-\mu_3), & 1-i(\mu_2-\mu_3)
\cr}
\right|ie^{-u} \right],\end{aligned}$$ $$\begin{aligned}
\label{Eq:psi2}
\psi_2^{(1)}&=&K_1\chi_2(\mu_1-\omega_3)e^{-i\mu_1u}
{}_2F_2\left[\left.
\tighten\matrix{
-i(\omega_2-\mu_1), & 1-i(\omega_3-\mu_1)\cr
1-i(\mu_2-\mu_1), & 1-i(\mu_3-\mu_1)
\cr}
\right|ie^{-u} \right], \nonumber \\[5pt]
\psi_2^{(2)}&=&K_2\chi_2(\mu_2-\omega_3)e^{-i\mu_2u}
{}_2F_2\left[\left.
\tighten\matrix{
-i(\omega_2-\mu_2), & 1-i(\omega_3-\mu_2)\cr
1-i(\mu_1-\mu_2), & 1-i(\mu_3-\mu_2)
\cr}
\right|ie^{-u} \right], \nonumber \\[5pt]
\psi_2^{(3)}&=&K_3\chi_2(\mu_3-\omega_3)e^{-i\mu_3u}
{}_2F_2\left[\left.
\tighten\matrix{
-i(\omega_2-\mu_3), & 1-i(\omega_3-\mu_3)\cr
1-i(\mu_1-\mu_3), & 1-i(\mu_2-\mu_3)
\cr}
\right|ie^{-u} \right],\end{aligned}$$ $$\begin{aligned}
\label{Eq:psi3}
\psi_3^{(1)}&=&K_1\chi_3(\mu_1-\omega_2)e^{-i\mu_1u}
{}_2F_2\left[\left.
\tighten\matrix{
1-i(\omega_2-\mu_1), & -i(\omega_3-\mu_1)\cr
1-i(\mu_2-\mu_1), & 1-i(\mu_3-\mu_1)
\cr}
\right|ie^{-u} \right], \nonumber \\[5pt]
\psi_3^{(2)}&=&K_2\chi_3(\mu_2-\omega_2)e^{-i\mu_2u}
{}_2F_2\left[\left.
\tighten\matrix{
1-i(\omega_2-\mu_2), & -i(\omega_3-\mu_2)\cr
1-i(\mu_1-\mu_2), & 1-i(\mu_3-\mu_2)
\cr}
\right|ie^{-u} \right], \nonumber \\[5pt]
\psi_3^{(3)}&=&K_3\chi_3(\mu_3-\omega_2)e^{-i\mu_3u}
{}_2F_2\left[\left.
\tighten\matrix{
1-i(\omega_2-\mu_3), & -i(\omega_3-\mu_3)\cr
1-i(\mu_1-\mu_3), & 1-i(\mu_2-\mu_3)
\cr}
\right|ie^{-u} \right].\end{aligned}$$ In Eqs. (\[Eq:psi1-2-3-b\]), the constants $C_1$, $C_2$ and $C_3$ are determined by the boundary conditions Eqs. (\[Eq:boundary-cond\]). For the exponential matter distribution (\[Eq:D-exponential\]), this gives completely the solution of Eq. (\[Eq:Schr-1\]), which describes the MSW effect in the sun.
We recall that by Eq. (\[Eq:u0\]), $e^{-u}=r_0D(0)e^{-r/r_0}$. Thus, the prefactors $e^{-i\mu_j u}$ are proportional to the exponentials describing propagation in vacuum, $e^{-i\mu_j r/r_0}$. In fact, as $r\to\infty$, all these functions approach a simple limit: $$\lim_{u\to\infty}{}_2F_2\left[\left.
\tighten\matrix{
\alpha_1, & \alpha_2\cr
\rho_1, & \rho_2
\cr}
\right|ie^{-u} \right]=1.$$
While this solution (\[Eq:psi1-2-3-b\]) is specifically for three generations of neutrinos, generalization, if desired, to $N$ generations is completely straightforward, leading to the generalized hypergeometric function $_{N-1}F_{N-1}$. Even for three generations, $_2F_2$ is a complicated function. From Eq. (\[Eq:u0\]), corresponding to the center of the sun, $e^{-u}$ is about 3000; for arguments of this order of magnitude, it is not practical to calculate the various $_2F_2$ of Eqs. (\[Eq:psi1\])–(\[Eq:psi3\]) using the Taylor series expansion.
Sections IV and V are devoted to the issue of how these $_2F_2$ can be calculated. Of the nine $_2F_2$ that appear in Eqs. (\[Eq:psi1\])–(\[Eq:psi3\]), three of them are evaluated approximately in Sec. IV. In Sec. V, the remaining six $_2F_2$ are then expressed in terms of another set of generalized hypergeometric functions $_3F_1$. These $_3F_1$ have convenient integral representations, useful for numerical evaluation. The approach described in the rest of this paper is the best we have found, but there may well be other methods that are superior.
$_2F_2$ that Appear in $\psi_{\lowercase{j}}^{(3)}$
===================================================
In this section, we study the three $_2F_2$ that appear in the expressions of $\psi_j^{(1)}$, $\psi_j^{(2)}$ and $\psi_j^{(3)}$ as given by Eqs. (\[Eq:psi1\])–(\[Eq:psi3\]). Define $$\begin{aligned}
\rho_1&=&\mu_3-\mu_1, \qquad \rho_2=\mu_3-\omega_2, \nonumber \\
\rho_3&=&\mu_3-\mu_2, \qquad \rho_4=\mu_3-\omega_3;\end{aligned}$$ then by (\[Eq:interlace\]) $$\rho_1 > \rho_2 > \rho_3 > \rho_4 > 0,$$ and the three $_2F_2$ of interest are $$\begin{aligned}
_2F_2(1)&=&
_2F_2\left[\left.
\tighten\matrix{
1+i\rho_2, & 1+i\rho_4\cr
1+i\rho_1, & 1+i\rho_3
\cr}\right|ie^{-u} \right], \nonumber \\[5pt]
_2F_2(2)&=&
_2F_2\left[\left.
\tighten\matrix{
i\rho_2, & 1+i\rho_4\cr
1+i\rho_1, & 1+i\rho_3
\cr}
\right|ie^{-u} \right], \nonumber \\[5pt]
_2F_2(3)&=&
_2F_2\left[\left.
\tighten\matrix{
1+i\rho_2, & i\rho_4\cr
1+i\rho_1, & 1+i\rho_3
\cr}
\right|ie^{-u} \right].\end{aligned}$$ Let $_2F_2(1)$ be considered in some detail; the treatments of $_2F_2(2)$ and $_2F_2(3)$ are similar.
From the definition of $_2F_2$ by its Taylor series, which is always convergent, it is straightforward to verify, by closing the contour of integration in the right half-plane, that $$\begin{aligned}
\label{Eq:F21-contour}
_2F_2(1)&=&\frac{1}{2\pi i}
\,\frac{\Gamma(1+i\rho_1)\Gamma(1+i\rho_3)}{\Gamma(1+i\rho_2)\Gamma(1+i\rho_
4)}
\nonumber \\[5pt]
&&\mbox{}\times\int_{\gamma-i\infty}^{\gamma+i\infty}
\frac{\Gamma(-s)\Gamma(1+i\rho_2+s)\Gamma(1+i\rho_4+s)}
{\Gamma(1+i\rho_1+s)\Gamma(1+i\rho_3+s)}
\left(e^{-i\pi/2}e^{-u}\right)^s ds\end{aligned}$$ where $$-1 < \gamma < 0.$$ This integral is to be evaluated approximately for $u$ large and negative. Except for an indentation near $s=0$, the contour of integration can be taken along the imaginary axis. It is therefore convenient to let $$s=i\tau.$$ The Stirling formula $$\label{Eq:Stirling-0}
\Gamma(z)\sim(2\pi)^{1/2} e^{-z} e^{(z-1/2)\ln z}$$ takes the form $$\label{Eq:Stirling}
\Gamma(ix)\sim\left(\frac{2\pi}{ix}\right)^{1/2}
e^{-\pi|x|/2} e^{ix(\ln |x|-1)}$$ when $z$ is purely imaginary. Using (\[Eq:Stirling\]), the integrand in (\[Eq:F21-contour\]) is $$\begin{aligned}
\lefteqn{\frac{\Gamma(-s)\Gamma(1+i\rho_2+s)\Gamma(1+i\rho_4+s)}
{\Gamma(1+i\rho_1+s)\Gamma(1+i\rho_3+s)}
\left(e^{-i\pi/2}e^{-u}\right)^s}\qquad\nonumber \\[5pt]
&=&\frac{\Gamma(-i\tau)\Gamma(1+i(\tau+\rho_2))\Gamma(1+i(\tau+\rho_4))}
{\Gamma(1+i(\tau+\rho_1))\Gamma(1+i(\tau+\rho_3))}\,
e^{\pi\tau/2}e^{i|u|\tau} \nonumber \\[5pt]
&\sim&\left(\frac{2\pi}{-i\tau}\right)^{1/2}
\frac{[i(\tau+\rho_2)]^{1/2}[i(\tau+\rho_4)]^{1/2}}
{[i(\tau+\rho_1)]^{1/2}[i(\tau+\rho_3)]^{1/2}}
\, e^{\pi\theta_1(\tau)/2}\,e^{i\theta_2(\tau)},\end{aligned}$$ with $$\theta_1(\tau)=|\tau+\rho_1|-|\tau+\rho_2|+|\tau+\rho_3|-|\tau+\rho_4|
-|\tau|+\tau,$$ and $$\begin{aligned}
\theta_2(\tau)
&=&-\tau(\ln|\tau|-1)-(\tau+\rho_1)(\ln|\tau+\rho_1|-1)
+(\tau+\rho_2)(\ln|\tau+\rho_2|-1) \nonumber \\
& &-(\tau+\rho_3)(\ln|\tau+\rho_3|-1)
+(\tau+\rho_4)(\ln|\tau+\rho_4|-1)+|u|\tau.\end{aligned}$$ The condition $$\frac{\partial\theta_2(\tau)}{\partial\tau}=0$$ for stationary phase gives $$\label{Eq:stat-phase}
\frac{|\tau||\tau+\rho_1||\tau+\rho_3|}{|\tau+\rho_2||\tau+\rho_4|}
=e^{|u|}.$$ Eq. (\[Eq:stat-phase\]) together with the fact that $\theta_1(\tau)$ is a non-decreasing function of $\tau$ implies that the relevant point of stationary phase occurs when $\tau$ is positive. Let $\tau_0$ be the unique positive root of the cubic equation $$\frac{\tau_0(\tau_0+\rho_1)(\tau_0+\rho_3)}{(\tau_0+\rho_2)(\tau_0+\rho_4)}
=e^{|u|},$$ then $\tau_0$ is the point of stationary phase that gives the leading contribution to $_2F_2(1)$. The result is, with all gamma functions replaced by the corresponding Stirling formula (\[Eq:Stirling-0\]), $$\begin{aligned}
\label{Eq:F22-Stirling-1}
_2F_2(1)
&\sim&\left(\frac{\rho_1\rho_3}{\rho_2\rho_4}\right)^{1/2}
\left[\frac{(\tau_0+\rho_2)(\tau_0+\rho_4)}
{(\tau_0+\rho_1)(\tau_0+\rho_3)}\right]^{1/2}\,
e^{i[\rho_1(\ln\rho_1-1)-\rho_2(\ln\rho_2-1)
+\rho_3(\ln\rho_3-1)-\rho_4(\ln\rho_4-1)]}
\nonumber \\
&&\mbox{}\times
\left[1+\frac{\tau_0}{\tau_0+\rho_1}-\frac{\tau_0}{\tau_0+\rho_2}
+\frac{\tau_0}{\tau_0+\rho_3}-\frac{\tau_0}{\tau_0+\rho_4}\right]^{-1/2}
e^{i\theta_2(\tau_0)},\end{aligned}$$ for $u$ negative and large. Similarly, $$\begin{aligned}
\label{Eq:F22-Stirling-2}
_2F_2(2)
&\sim&\frac{\rho_2}{\tau_0+\rho_2}\, _2F_2(1), \\[5pt]
\label{Eq:F22-Stirling-3}
_2F_2(3)
&\sim&\frac{\rho_4}{\tau_0+\rho_4}\, _2F_2(1).\end{aligned}$$ These are the desired results.
$_2F_2$ that Appear in $\psi_{\lowercase{j}}^{(1)}$ and $\psi_{\lowercase{j}}^{(2)}$
====================================================================================
The next step is to study the $_2F_2$ that appear in the expressions (\[Eq:psi1\])–(\[Eq:psi3\]) for $\psi_1^{(i)}$, $\psi_2^{(i)}$ and $\psi_3^{(i)}$ with $i=1,2$. It should be emphasized that, while the treatment in the preceding Sec. IV involves approximations, what is to be carried out in this section is exact. Since exact manipulations are usually more straightforward, the description will be relatively more brief in this section.
Consider the third-order ordinary differential equation $$\label{Eq:Schr-7}
\left[
\left(z\,\frac{d}{dz}+\beta_1\right)
\left(z\,\frac{d}{dz}+\beta_2\right)
\left(z\,\frac{d}{dz}+\beta_3\right)
-z\left(z\,\frac{d}{dz}+\alpha_1\right)
\left(z\,\frac{d}{dz}+\alpha_2\right)
\right]f=0.$$ A comparison with Eq. (\[Eq:Schr-6\]) shows that $$\label{Eq:beta-mu}
\beta_1=-i\mu_1, \qquad
\beta_2=-i\mu_2, \qquad
\beta_3=-i\mu_3.$$ The values of $\alpha_1$ and $\alpha_2$ are different for the various $\psi$’s: $$\begin{aligned}
\label{Eq:psi-1-alphas}
\text{for}\
\psi_1^{(j)}\text{:}\qquad\alpha_1=1-i\omega_2&\quad\text{and}\quad
\alpha_2=1-i\omega_3,\\
\text{for}\
\psi_2^{(j)}\text{:}\qquad\alpha_1=-i\omega_2\hspace{11pt}&\quad
\text{and}\quad \alpha_2=1-i\omega_3,\\
\text{for}\
\psi_3^{(j)}\text{:}\qquad \alpha_1=1-i\omega_2&\quad
\text{and}\quad \alpha_2=-i\omega_3,\hspace{11pt}\\
\text{for}\
\psi^{(j)}\text{:}\qquad\alpha_1=-i\omega_2\hspace{11pt}&\quad
\text{and}\quad \alpha_2=-i\omega_3.\hspace{11pt}\end{aligned}$$ The three linearly independent solutions of Eq. (\[Eq:Schr-7\]) are $$\begin{aligned}
\label{Eq:f1-f2-f3}
f^{(1)}(z)&=&z^{-\beta_1}
{}_2F_2\left[\left.
\tighten\matrix{
\alpha_1-\beta_1, & \alpha_2-\beta_1\cr
1+\beta_2-\beta_1, & 1+\beta_3-\beta_1
\cr}
\right|z \right], \nonumber \\[5pt]
f^{(2)}(z)&=&z^{-\beta_2}
{}_2F_2\left[\left.
\tighten\matrix{
\alpha_1-\beta_2, & \alpha_2-\beta_2\cr
1+\beta_1-\beta_2, & 1+\beta_3-\beta_2
\cr}
\right|z \right], \nonumber \\[5pt]
f^{(3)}(z)&=&z^{-\beta_3}
{}_2F_2\left[\left.
\tighten\matrix{
\alpha_1-\beta_3, & \alpha_2-\beta_3\cr
1+\beta_1-\beta_3, & 1+\beta_2-\beta_3
\cr}
\right|z \right],\end{aligned}$$ where $f^{(3)}(z)$ is the function treated in the preceding Sec. IV.
On the other hand, if in Eq. (\[Eq:Schr-7\]) the independent variable is changed to $$\hat z = z^{-1},$$ then the differential equation is $$\label{Eq:Schr-8}
\left[
\left(\hat z\,\frac{d}{d\hat z}-\alpha_1\right)
\left(\hat z\,\frac{d}{d\hat z}-\alpha_2\right)
+\hat z\left(\hat z\,\frac{d}{d\hat z}-\beta_1\right)
\left(\hat z\,\frac{d}{d\hat z}-\beta_2\right)
\left(\hat z\,\frac{d}{d\hat z}-\beta_3\right)
\right]f=0.$$ In this form, two of the three linearly independent solutions of Eq. (\[Eq:Schr-7\]) are (in terms of $z$): $$\begin{aligned}
\label{Eq:hat-f}
\hat f^{(1)}(z)
&=&z^{-\alpha_1}
{}_3F_1\left[\left.
\tighten\matrix{
-\beta_1+\alpha_1, & -\beta_2+\alpha_1, & -\beta_3+\alpha_1\cr
1-\alpha_2+\alpha_1 & &
\cr}
\right|-z^{-1} \right], \nonumber \\[5pt]
\hat f^{(2)}(z)
&=&z^{-\alpha_2}
{}_3F_1\left[\left.
\tighten\matrix{
-\beta_1+\alpha_2, & -\beta_2+\alpha_2, & -\beta_3+\alpha_2\cr
1-\alpha_1+\alpha_2 & &
\cr}
\right|-z^{-1} \right].\end{aligned}$$ For a discussion of the generalized hypergeometric functions $_pF_q$ with $p>q+1$, see for example Chapter V of [@Bateman].
It should be added parenthetically that, for the present purposes, $z^{-\beta_1}$, as an example, can sometimes be very small because by Eqs. (\[Eq:z-u\]) and (\[Eq:beta-mu\]) $$z^{-\beta_1}=e^{-\pi\mu_1/2}\,e^{-i\mu_1u}.$$ Thus, when $\mu_1$ is large, which is often the case, for numerical calculations it is convenient to replace $z^{-\beta_1}$ by $e^{-i\mu_1u}$, and similarly for the other powers of $z$ multiplying the $_2F_2$ and $_3F_1$.
Since Eq. (\[Eq:Schr-7\]) is a third-order linear ordinary differential equation, the two solutions as given by Eq. (\[Eq:hat-f\]) must be expressible as linear combinations of the three solutions of Eq. (\[Eq:f1-f2-f3\]). The result is $$\begin{aligned}
\label{Eq:hat-f1}
\hat f^{(1)}(z)
&=&\frac{\pi\Gamma(1-\alpha_2+\alpha_1)}
{\Gamma(-\beta_1+\alpha_1)\Gamma(-\beta_2+\alpha_1)\Gamma(-\beta_3+\alpha_1)
}
\nonumber \\[4pt]
&&\mbox{}\times\left[
\frac{\Gamma(-\beta_2+\beta_1)\Gamma(-\beta_3+\beta_1)}
{\Gamma(1-\alpha_1+\beta_1)\Gamma(1-\alpha_2+\beta_1)}
\,\frac{1}{\sin\pi(\alpha_1-\beta_1)}\, f^{(1)}(z)\right. \nonumber \\[4pt]
&&\mbox{}+\frac{\Gamma(-\beta_1+\beta_2)\Gamma(-\beta_3+\beta_2)}
{\Gamma(1-\alpha_1+\beta_2)\Gamma(1-\alpha_2+\beta_2)}
\,\frac{1}{\sin\pi(\alpha_1-\beta_2)}\, f^{(2)}(z) \nonumber \\[4pt]
&&\mbox{}+\left.\frac{\Gamma(-\beta_1+\beta_3)\Gamma(-\beta_2+\beta_3)}
{\Gamma(1-\alpha_1+\beta_3)\Gamma(1-\alpha_2+\beta_3)}
\,\frac{1}{\sin\pi(\alpha_1-\beta_3)}\, f^{(3)}(z) \right],\end{aligned}$$ $$\label{Eq:hat-f2}
\hat f^{(2)}(z)=\hat f^{(1)}(z)\Big|_{\alpha_1\leftrightarrow\alpha_2}.$$
Eqs. (\[Eq:hat-f1\]) and (\[Eq:hat-f2\]) can be considered to be the formulas that express $f^{(1)}(z)$ and $f^{(2)}(z)$ in terms of $\hat f^{(1)}(z)$, $\hat f^{(2)}(z)$ and $f^{(3)}(z)$. Since $f^{(3)}(z)$ is given by Eqs. (\[Eq:F22-Stirling-1\])–(\[Eq:F22-Stirling-3\]), it only remains to obtain $\hat f^{(1)}(z)$ and $\hat f^{(2)}(z)$, which are given by $_3F_1$ rather than $_2F_2$.
The advantage of $_3F_1$ over $_2F_2$ is due to the integral representation $$\label{Eq:F31-int-rep-1}
_3F_1\left[\left.
\tighten\matrix{
a_1, & a_2, & a_3 \cr
b & &
\cr}
\right|-x^{-1} \right]
=\frac{1}{\Gamma(a_1)}\,x^{a_1}
\int_0^\infty dt\, e^{-xt}\, t^{a_1-1}\,
{}_2F_1(a_2,a_3;b;-t),$$ where $_2F_1$ is the usual hypergeometric function. This Eq. (\[Eq:F31-int-rep-1\]) follows from Eq. (8) on p. 214 and Eq. (1) on p. 215 of [@Bateman], and has the great advantage of not containing in the integrand any gamma function that depends on the variable $t$, because such gamma functions require excessive computer time.
For the $_2F_1$ in the integrand of (\[Eq:F31-int-rep-1\]), the representation $$\begin{aligned}
\label{Eq:F21-Pochhammer}
_2F_1(a,b;c;-t)
&=&\frac{-\Gamma(c)e^{-i\pi c}}
{4\Gamma(b)\Gamma(c-b)\sin\pi b\sin\pi(c-b)} \nonumber \\[5pt]
&&\mbox{}\times\int_{\cal P} s^{b-1}(1-s)^{c-b-1}(1+ts)^{-a}ds\end{aligned}$$ is convenient, where ${\cal P}$ is the Pochhammer contour of Fig. 2. In the numerical evaluation through Eqs. (\[Eq:F31-int-rep-1\]) and (\[Eq:F21-Pochhammer\]), it is convenient to deform the contours of integration through the point of stationary phase. This point is discussed in more detail in the Appendix.
\[Fig:Pochhammer\]
(8,7.0) (-3.,2.0)
Summary and Discussion
======================
The Mikheyev-Smirnov-Wolfenstein effect in the sun has been studied and the differential equations solved for three types of neutrinos coupled through their mass matrix. Under the assumption, as expressed by Eq. (\[Eq:D-exponential\]), that the electron density can be approximated by an exponential function, the MSW differential equations are solved exactly in Sec. III, especially Eqs. (\[Eq:psi1-2-3-b\])–(\[Eq:psi3\]), in terms of the generalized hypergeometric function $_2F_2$. The method used there can be immediately generalized to $N$ types of neutrinos coupled in the same way, leading to $_{N-1}F_{N-1}$.
Since $_2F_2$ cannot be considered to be a familiar function, Secs. IV and V are devoted to the issue of how they can be calculated. Of the nine $_2F_2$ that appear in Eqs. (\[Eq:psi1\])–(\[Eq:psi3\]), three are evaluated approximately in Sec. IV. In Sec. V, the remaining six $_2F_2$ are then expressed in terms of another set of generalized hypergeometric functions, $_3F_1$. These $_3F_1$ have convenient integral representations given by Eqs. (\[Eq:F31-int-rep-1\]) and (\[Eq:F21-Pochhammer\]), useful for numerical evaluation. Eqs. (\[Eq:F31-int-rep-1\]) and (\[Eq:F21-Pochhammer\]) can also be used to get asymptotic expressions for $_3F_1$ and hence $_2F_2$, but this development is not discussed here because it is not needed for the study of neutrino oscillations [@OW-99-L].
Finally, it should be mentioned that, while the calculation of Sec. IV can be generalized immediately to $N$ types of neutrinos, there does not seem to be a similar straightforward generalization for Sec. V. More precisely, while $_{N-1}F_{N-1}$ can be related to $_{N}F_{N-2}$, the generalization of Eq. (\[Eq:F31-int-rep-1\]) involves $_{N-1}F_{N-2}$, which does not have an integral representation similar to Eq. (\[Eq:F21-Pochhammer\]) involving a single integral. We believe that, for $_{N}F_{N-2}$, it is necessary to use an $(N-1)$-fold integral to avoid having gamma functions that depend on the variables of integration.
{#section .unnumbered}
In this Appendix, Eqs. (\[Eq:F31-int-rep-1\]) and (\[Eq:F21-Pochhammer\]) are studied further. For definiteness, let $\alpha_1$ and $\alpha_2$ be defined by (\[Eq:psi-1-alphas\]), and consider the $_3F_1$ that appears in the first equation in (\[Eq:hat-f\]). Use the inequality (\[Eq:interlace\]) to define the four positive quantities $$\begin{aligned}
\label{Eq:xi-eta-zeta-zetap}
\xi &=&\mu_2-\omega_2, \nonumber \\
\eta &=&\omega_2-\mu_1, \nonumber \\
\zeta &=&\mu_3-\omega_2, \nonumber \\
\zeta'&=&\omega_3-\omega_2 < \zeta;\end{aligned}$$ then Eq. (\[Eq:F31-int-rep-1\]) gives, for this $_3F_1$ under consideration, $$\begin{aligned}
\label{Eq:F31-int-rep-2}
\lefteqn{_3F_1\left[\left.
\tighten\matrix{
1+i\xi, & 1-i\eta, & 1+i\zeta \cr
1+i\zeta', & &
\cr}
\right|\frac{i}{y} \right] }\qquad\nonumber \\[5pt]
&&\mbox{}=\frac{1}{\Gamma(1+i\xi)}\,\left(e^{i\pi/2}y\right)^{1+i\xi}
\int_0^\infty dt\, e^{-ity}\, t^{i\xi}\,
{}_2F_1(1-i\eta,1+i\zeta;1+i\zeta';-t),\end{aligned}$$ where, by Eq. (\[Eq:z-u\]), $$y=e^{-u}$$ is large for $u$ negative and large. The substitution of Eq. (\[Eq:F21-Pochhammer\]) into Eq. (\[Eq:F31-int-rep-2\]) then gives $$\begin{aligned}
\label{Eq:F31-int-rep-3}
\lefteqn{_3F_1\left[\left.
\tighten\matrix{
1+i\xi, & 1-i\eta, & 1+i\zeta \cr
1+i\zeta', & &
\cr}
\right|\frac{i}{y} \right]}\qquad \nonumber \\[5pt]
&&\mbox{}=\frac{1}{\Gamma(1+i\xi)}\,\left(e^{i\pi/2}y\right)^{1+i\xi}
\frac{\Gamma(1+i\zeta')e^{-i\pi(1+i\zeta')}}
{\Gamma(1-i\eta)\Gamma(i(\eta+\zeta'))4\sin\pi(1-i\eta)\sin\pi
i(\eta+\zeta')}
\, I,\end{aligned}$$ where $$\begin{aligned}
\label{Eq:I-t-s}
I&=&\int_0^\infty dt \int_{\cal P} ds\, e^{-ity}\, t^{i\xi}\,
s^{-i\eta}(1+s)^{i(\zeta-\zeta')}(1+s+st)^{-1-i\zeta} \nonumber \\[5pt]
&=&\int_0^\infty dt \int_{\cal P} ds\, (1+s+st)^{-1}\,
e^{-i\phi(t,s)},\end{aligned}$$ with $$\phi(t,s)=ty-\xi\ln t +\eta\ln s-(\zeta-\zeta')\ln(1+s)+\zeta\ln(1+s+st).$$ This $\phi$ is of course not related to the $\phi_j$ of Eq. (\[Eq:Schr-1\]).
For the double integral in Eq. (\[Eq:I-t-s\]), the point or points of stationary phase are given by $$\frac{\partial\phi(t,s)}{\partial t}
=\frac{\partial\phi(t,s)}{\partial s}=0,$$ or $$\label{Eq:stat-cond-1}
y-\frac{\xi}{t}+\frac{s\zeta}{1+s+st}=0,$$ and $$\label{Eq:stat-cond-2}
\frac{\eta}{s}-\frac{\zeta-\zeta'}{1+s}+\frac{(1+t)\zeta}{1+s+st}=0.$$ Elimination of $s$ between Eqs. (\[Eq:stat-cond-1\]) and (\[Eq:stat-cond-2\]) gives the cubic equation for $t$, $$\label{Eq:ell-cubic}
\ell(t)=0,$$ with $$\ell(t)=y^2t^3+y(y+\zeta-2\xi-\eta)t^2
+[(\zeta'-2\xi)y-(\zeta-\xi)(\xi+\eta)]t
-\xi(\zeta'-\xi).$$ When $\zeta'\to\zeta$, the three roots of Eq. (\[Eq:ell-cubic\]) are explicitly $$\begin{aligned}
t&=&t_+=(2y)^{-1}\bigl\{-(y-\xi-\eta)+[(y-\xi-\eta)^2+4\xi y]^{1/2}
\bigr\} > 0, \\
t&=&t_-=(2y)^{-1}\bigl\{-(y-\xi-\eta)-[(y-\xi-\eta)^2+4\xi y]^{1/2}
\bigr\} < 0, \\
t&=&-(\zeta-\xi)/y < 0.\end{aligned}$$ Since by (\[Eq:xi-eta-zeta-zetap\]) $$\ell(t_-)=-\frac{\eta t_-}{1+t_-}\,(\zeta-\zeta') > 0,$$ we have $$\label{Eq:t0-cases}
\ell(t)\tighten\cases{
< 0, & for $t$ sufficiently large and negative,\cr
> 0, & for $t=t_-<0$,\cr
< 0, & for $t=0$,\cr
> 0, & for $t$ sufficiently large and positive.\cr
}$$ It therefore follows that the cubic equation (\[Eq:ell-cubic\]) has one and only one positive root. Since the $t$ integral in Eq. (\[Eq:I-t-s\]) is along the positive real axis, this implies that there is exactly one relevant point of stationary phase, say ($t_0,s_0$), with $t_0$ given by this unique positive root of Eq. (\[Eq:ell-cubic\]). Furthermore, since $$\ell(t_+) < 0,$$ it follows from (\[Eq:t0-cases\]) that $$t_0 > t_+.$$
From Eq. (\[Eq:stat-cond-1\]), the corresponding $s_0$ is given by $$s_0 =-\left[1+t_0+\frac{\zeta t_0}{yt_0-\xi}\right]^{-1},$$ and hence $$-(1+t_0)^{-1} < s_0 < 0.$$ For the numerical evaluation of this $_3F_1$ of Eq. (\[Eq:F31-int-rep-3\]), using Eq. (\[Eq:I-t-s\]), it is convenient and efficient to choose the Pochhammer contour ${\cal P}$ to go through this value of $s_0$, and also to deform the contour for the $t$ integration so that at $t_0$ it is locally the path of steepest descent.
The other $_3F_1$’s can be treated in similar ways.
[99]{}
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[^1]: From this form, it is also rather obvious how the confluent hypergeometric function $_1F_1$ emerges as the solution when one of the $\alpha$’s equals one of the $\rho$’s, i.e., for the case of two generations.
|
---
abstract: 'For the linear damped wave equation (DW), the $L^p$-$L^q$ type estimates have been well studied. Recently, Watanabe [@Wat17] showed the Strichartz estimates for DW when $d=2,3$. In the present paper, we give Strichartz estimates for DW in higher dimensions. Moreover, by applying the estimates, we give the local well-posedness of the energy critical nonlinear damped wave equation (NLDW) $\partial_t^2 u - \Delta u +\partial_t u = |u|^{\frac{4}{d-2}}u$, $(t,x) \in [0,T) \times \mathbb{R}^d$, where $3 \leq d \leq 5$. Especially, we show the small data global existence for NLDW. In addition, we investigate the behavior of the solutions to NLDW. Namely, we give a decay result for solutions with finite Strichartz norm and a blow-up result for solutions with negative Nehari functional.'
address: 'Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan'
author:
- Takahisa Inui
title: The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation
---
Introduction
============
Backgroud
---------
We consider the damped wave equation. $$\begin{aligned}
\label{DW}
\left\{
\begin{array}{ll}
\partial_t^2 \phi - \Delta \phi +\partial_t \phi = 0, & (t,x) \in (0,\infty) \times \mathbb{R}^d,
\\
(\phi(0),\partial_t \phi(0)) = (\phi_0,\phi_1), & x\in \mathbb{R}^d,
\end{array}
\right.\end{aligned}$$ where $d \in \mathbb{N}$, $(\phi_0,\phi_1)$ is given, and $\phi$ is an unknown complex valued function.
Matsumura [@Mat76] applied the Fourier transform to and obtained the formula $$\begin{aligned}
\phi(t,x)={\mathcal{D}}(t) (\phi_0+\phi_1) +\partial_t {\mathcal{D}}(t) \phi_0,\end{aligned}$$ where ${\mathcal{D}}(t)$ is defined by $$\begin{aligned}
{\mathcal{D}}(t):=e^{-\frac{t}{2}}{\mathcal{F}}^{-1} L(t,\xi) {\mathcal{F}}\end{aligned}$$ with $$\begin{aligned}
L(t,\xi):=
{\left}\{
\begin{array}{ll}
\displaystyle
\frac{\sinh(t \sqrt{1/4-|\xi|^2})}{\sqrt{1/4-|\xi|^2}} & \text{if } |\xi|<1/2,
\\
\
\\
\displaystyle
\frac{\sin(t \sqrt{|\xi|^2-1/4})}{\sqrt{|\xi|^2-1/4}} & \text{if } |\xi|>1/2.
\end{array}
{\right}.\end{aligned}$$ By this formula, Matsumura [@Mat76] proved the $L^p$-$L^q$ type estimate: $$\begin{aligned}
\label{eq1.2}
\norm{\phi(t)}_{L^p}
{\lesssim}\jbra{t}^{-\frac{d}{2}{\left}( \frac{1}{q} - \frac{1}{p}{\right})} \norm{(\phi_0,\phi_1)}_{L^q\times L^q}
+e^{-\frac{t}{4}} {\left}( \norm{\phi_0}_{H^{{\left}[\frac{d}{2}{\right}]+1}} + \norm{\phi_1}_{H^{{\left}[\frac{d}{2}{\right}]}} {\right}),\end{aligned}$$ where $1\leq q \leq 2 \leq p \leq \infty$ and $[d/2]$ denotes the integer part of $d/2$. Such $L^p$-$L^q$ type estimates have been studied well. See [@Nis03; @HoOg04; @Nar05] and references therein. The $L^p$-$L^q$ type estimates for the heat equation and the wave equation are also well studied. We recall the $L^p$-$L^q$ type estimate for the heat equation $\partial_t v - \Delta v = 0$: $$\begin{aligned}
\norm{{\mathcal{G}}(t) g}_{L^p} {\lesssim}t^{-\frac{d}{2}{\left}( \frac{1}{q} - \frac{1}{p}{\right})} \norm{g}_{L^q},\end{aligned}$$ where $1\leq q \leq p \leq \infty$ and ${\mathcal{G}}(t):={\mathcal{F}}^{-1} e^{-t|\xi|^2} {\mathcal{F}}$. We also refer to the $L^p$-$L^q$ type estimate for the wave equation $\partial_t^2 w - \Delta w =0$: $$\begin{aligned}
\norm{{\mathcal{W}}(t)g}_{L^p} {\lesssim}|t|^{-2d{\left}(\frac{1}{2}-\frac{1}{p}{\right})} \norm{g}_{\dot{W}^{\gamma-1,p'}},\end{aligned}$$ for $2\leq p <\infty$ and $(d+1)(1/2-1/p) \leq \gamma < d$, where ${\mathcal{W}}(t):={\mathcal{F}}^{-1} \sin(t|\xi|) / |\xi|{\mathcal{F}}$. See [@Bre75]. Matsumura’s estimate shows that the solution of behaves like the solution of the heat equation and the wave equation in some sense. More precisely, the low frequency part of the solution to the damped wave equation behaves like the solution of the heat equation and the high frequency part behaves like the solution of the wave equation but decays exponentially (see [@IIOWp] for another $L^p$-$L^q$ estimate).
For the heat equation and the wave equation, by using the $L^p$-$L^q$ type estimates, we obtain the space-time estimates, what we call the Strichartz estimate. The Strichartz estimates for the heat equation are $$\begin{aligned}
\norm{v}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
{\lesssim}\norm{v_0}_{L^2} + \norm{F}_{L_{t}^{\tilde{q}'} (I: L_{x}^{\tilde{r}'}({{\mathbb R}}^d))},\end{aligned}$$ where $v$ satisfies $\partial_t v - \Delta v =F$ with $v(0)=v_0$ and $2/q+d/r=2/\tilde{q}+d/\tilde{r}=d/2$. See [@Wei81; @Gig86]. We also have the Strichartz estimates for the wave equation as follows. $$\begin{aligned}
\norm{w}_{L_{t}^{q} (I: L_{x}^{p}({{\mathbb R}}^d))}
{\lesssim}\norm{w_0}_{\dot{H}^1} +\norm{w_1}_{L^2} + \norm{F}_{L_{t}^{\tilde{q}'} (I: L_{x}^{\tilde{r}'}({{\mathbb R}}^d))},\end{aligned}$$ where $w$ satisfies $\partial_t^2 w - \Delta w=F$ with $(w(0),\partial_t w(0))=(w_0,w_1)$ and $1/q+d/r=d/2-1=1/\tilde{q}'+d/\tilde{r}'-2$. See [@GiVe95]. In the present paper, we give the Strichartz estimates for the damped wave equation. Recently, Watanabe [@Wat17] obtained the Strichartz estimates for the damped wave equation when $d=2,3$ by an energy method. In this paper, we give the Strichartz estimates by a duality argument for $d=2,3$ and higher dimensions. We also consider the energy critical nonlinear damped wave equation. $$\begin{aligned}
\label{NLDW}
\tag{NLDW}
\left\{
\begin{array}{ll}
\partial_t^2 u - \Delta u +\partial_t u = |u|^{\frac{4}{d-2}}u, & (t,x) \in [0,T) \times \mathbb{R}^d,
\\
(u(0),\partial_t u(0)) = (u_0,u_1), & x\in \mathbb{R}^d,
\end{array}
\right.\end{aligned}$$ where $d \geq 3$, $(u_0,u_1)$ is given, and $u$ is an unknown complex valued function. We will show the local well-posedness for when $3\leq d \leq 5$ by applying the Strichartz estimates. The existence of a local solution has been studied by [@Kap94; @IkIn16; @IkWa17p] (see also [@Kap87I; @Kap90II; @Kap90III]). However, the small data global existence has not been known. Using the Strichartz estimates which are proved in this paper, we can show not only the existence of a local solution but also the small data global existence for .
Moreover, we discuss the global behavior of the solutions to . For the energy critical nonlinear heat equation, the solution with a bounded global space-time norm decays to zero (see e.g. [@GuRo17p]). On the other hand, there exist finite time blow-up solutions by Levine [@Lev73]. For the energy critical nonlinear wave equation, the energy is conserved by the flow. There exist solutions which scatter to the solutions of the free wave equation and finite time blow-up solutions by Payne and Sattinger [@PaSa75]. See also [@KeMe08]. In the present paper, we show that the solution to with a finite space-time norm decays since the energy decays like the heat equation and there exist finite time blow-up solutions.
Main results
------------
We state main results. First, we obtain the Strichartz estimates for . The so-called admissible pairs can be taken as same as in the heat case since the $L^p$-$L^q$ type estimate of the low frequency part is similar to the heat estimate and the high frequency part decays exponentially in time. However, the derivative loss appears from the high frequency part which is wave-like part.
\[prop1.1\] Let $d \geq 2$, $2 \leq r < \infty$, and $2\leq q \leq \infty$. Set $\gamma:= \max\{ d(1/2 - 1/r)-1/q, \frac{d+1}{2}(1/2-1/r)\}$. Assume $$\begin{aligned}
\frac{d}{2}{\left}( \frac{1}{2} - \frac{1}{r}{\right}) \geq \frac{1}{q},\end{aligned}$$ Then, we have $$\begin{aligned}
\norm{{\mathcal{D}}(t)f}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
&{\lesssim}\norm{ \jbra{\nabla}^{\gamma-1} f}_{L^2},
\\
\norm{\partial_t {\mathcal{D}}(t)f}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
&{\lesssim}\norm{ \jbra{\nabla}^{\gamma} f}_{L^2},
\\
\norm{\partial_t^2 {\mathcal{D}}(t)f}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
&{\lesssim}\norm{ \jbra{\nabla}^{\gamma+1} f}_{L^2}.\end{aligned}$$
\[rmk1.1\] We note that the homogeneous Strichartz estimate holds in the heat end-point case *i.e.* $(q,r)=(2, 2d/(d-2))$ when $d\geq3$.
\[prop1.2\] Let $d\geq2$, $2\leq r,\tilde{r} < \infty$, and $2\leq q, \tilde{q} \leq \infty$. We set $\gamma:= \max \{ d(1/2-1/r)-1/q, \frac{d+1}{2}(1/2-1/r) \}$ and $\tilde{\gamma}$ in the same manner. Assume that $(q,r)$ and $(\tilde{q},\tilde{r})$ satisfies $$\begin{aligned}
\frac{d}{2}{\left}( \frac{1}{2} - \frac{1}{r}{\right}) +\frac{d}{2}{\left}( \frac{1}{2} - \frac{1}{\tilde{r}}{\right})
>\frac{1}{q} + \frac{1}{\tilde{q}},\end{aligned}$$ $$\begin{aligned}
\frac{d}{2} {\left}( \frac{1}{2} - \frac{1}{r}{\right}) + \frac{d}{2} {\left}( \frac{1}{2} - \frac{1}{\tilde{r}}{\right})
= \frac{1}{q} + \frac{1}{\tilde{q}}
\text{ and }
1< \tilde{q}' < q<\infty,\end{aligned}$$ or $$\begin{aligned}
(q,r)=(\tilde{q},\tilde{r})=(\infty,2).\end{aligned}$$ Moreover, we exclude the end-point case, that is, we assume $(q,r) \neq (2,2(d-1)/(d-3)))$ and $(\tilde{q},\tilde{r}) \neq (2,2(d-1)/(d-3)))$ when $d \geq 4$. Then, we have $$\begin{aligned}
\norm{\int_{0}^{t} {\mathcal{D}}(t-s) F(s) ds}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
&{\lesssim}\norm{ \jbra{\nabla}^{\gamma+\tilde{\gamma}+\delta-1} F}_{L_{t}^{\tilde{q}'} (I: L_{x}^{\tilde{r}'}({{\mathbb R}}^d))},
\\
\norm{\int_{0}^{t} \partial_t {\mathcal{D}}(t-s) F(s) ds}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
&{\lesssim}\norm{ \jbra{\nabla}^{\gamma+\tilde{\gamma}+\delta} F}_{L_{t}^{\tilde{q}'} (I: L_{x}^{\tilde{r}'}({{\mathbb R}}^d))},\end{aligned}$$ where $\delta = 0$ when $\frac{1}{\tilde{q}}(1/2-1/r)=\frac{1}{q}(1/2-1/\tilde{r})$ and in the other cases $\delta \geq 0$ is defined in the table 1 below.
-------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------
$\delta$ $\frac{1}{\tilde{q}} {\left}( \frac{1}{2}- \frac{1}{r}{\right}) < \frac{1}{q} {\left}( \frac{1}{2} - \frac{1}{\tilde{r}} {\right}) $ $\frac{1}{\tilde{q}} {\left}( \frac{1}{2}- \frac{1}{r}{\right}) > \frac{1}{q} {\left}( \frac{1}{2} - \frac{1}{\tilde{r}} {\right})$
\[4pt\] $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) \geq \frac{1}{q}$ $0$ $0$
$\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) \geq \frac{1}{\tilde{q}}$
\[5pt\] $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) \geq \frac{1}{q}$ $\times$ $\frac{\tilde{q}}{q} {\left}\{ \frac{1}{\tilde{q}}-\frac{d-1}{2} {\left}( \frac{1}{2} - \frac{1}{\tilde{r}}{\right}) {\right}\}$
$\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) < \frac{1}{\tilde{q}}$
\[5pt\] $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) < \frac{1}{q}$ $\frac{q}{\tilde{q}} {\left}\{ \frac{1}{q}-\frac{d-1}{2} {\left}( \frac{1}{2} - \frac{1}{r}{\right}) {\right}\}$ $\times$
$\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) \geq \frac{1}{\tilde{q}}$
\[5pt\] $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) < \frac{1}{q}$ $\frac{1}{\tilde{q}} \frac{d-1}{2} {\left}\{ \tilde{q}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) - q{\left}( \frac{1}{2} - \frac{1}{r}{\right}) {\right}\}$ $\frac{1}{q} \frac{d-1}{2} {\left}\{ q{\left}( \frac{1}{2}- \frac{1}{r}{\right}) - \tilde{q}{\left}( \frac{1}{2} - \frac{1}{\tilde{r}}{\right}) {\right}\}$
$\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right})< \frac{1}{\tilde{q}}$
-------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------
: The value of $\delta$. ($\times$ means that the case does not occur.)
If $(q,r)$ satisfies the wave admissible condition $(d-1)(1/2-1/r)/2 \geq 1/q$, then the derivative loss is same as that in the Strichartz estimates for the wave equation *i.e.* $\gamma=d(1/2-1/r)-1/q$. And thus, we need more derivative if $(q,r)$ is the pair between the wave case and the heat case, *i.e.* $d(1/2-1/r)/2 \geq 1/q > (d-1)(1/2-1/r)/2$.
Chen, Fang, and Zhang [@CFZ15] considered the damped fractional wave equation $\partial_t^2 v +(-\Delta)^{\alpha} v + 2\partial_t v=0$, where $\alpha>0$. They claimed that the better homogeneous Strichartz estimates can be obtained, where “better" means the derivative loss is less than that of the wave equation. However, at least when $\alpha=1$, their proof seems to be imcomplete.
Moreover, applying these Strichartz estimates, we can show the local well-posedness, especially small data global existence, for energy critical nonlinear damped wave equations .
Let $T \in (0,\infty]$. We say that $u$ is a solution to on $[0,T)$ if $u$ satisfies $(u,\partial_t u) \in C([0,T):H^1({{\mathbb R}}^d)\times L^2({{\mathbb R}}^d))$, $\jbra{\nabla}^{1/2} u \in L_{t,x}^{2(d+1)/(d-1)}(I)$ and $ u \in L_{t,x}^{2(d+1)/(d-2)}(I)$ for any compact interval $I \subset [0,T)$, $(u(0),\partial_t u(0)) = (u_0,u_1)$, and the Duhamel’s formula $$\begin{aligned}
u(t,x)={\mathcal{D}}(t) (u_0+u_1) +\partial_t {\mathcal{D}}(t) u_0 + \int_{0}^{t} {\mathcal{D}}(t-s) ( |u(s)|^{\frac{4}{d-2}}u(s) ) ds\end{aligned}$$ for all $t \in [0,T)$. We say that $u$ is global if $T=\infty$.
We have the following local well-posedness result when $3 \leq d \leq 5$.
\[thm1.3\] Let $d\in\{3,4,5\}$ and $T\in (0,\infty]$. Let $(u_0,u_1) \in H^1({{\mathbb R}}^d)\times L^2({{\mathbb R}}^d)$ satisfy $\| (u_0,u_1) \|_{H^1 \times L^2} \leq A$. Then, there exists $\delta=\delta(A)>0$ such that if $$\begin{aligned}
\norm{{\mathcal{D}}(t) (u_0+u_1) +\partial_t {\mathcal{D}}(t) u_0}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))} \leq \delta,\end{aligned}$$ then there exists a solution $u$ to with $\| u \|_{L_{t,x}^{2(d+1)/(d-2)}}([0,T)) \leq 2 \delta$. Moreover, we have the standard blow-up criterion, that is, if the maximal existence time $T_{+}=T_{+}(u_0,u_1)$ is finite, then the solution satisfies $\norm{u}_{L^{2(d+1)/(d-2)}([0,T_{+}))}=\infty$.
If $\| (u_0,u_1) \|_{H^1 \times L^2} \ll 1$, by the Strichartz estimates, we can take $T=\infty$. Namely, the small data global existence holds.
See the sequel paper [@InWap] for the local well-posedness and small data global existence of when $d \geq 6$. The difficulty of $d \geq 6$ comes from the loss of differentiability of the nonlinear term. We need to attention to the estimate of the difference.
The existence of local solution is well known (see [@IkIn16; @IkWa17p]). However, the small data global existence has not been known except for low dimension cases. (Watanabe [@Wat17] showed the small data global existence when $d=3$.)
As it is well known, we can obtain the local well-posedness of the nonlinear damped wave equation with the more general nonlinearity in the same way as Theorem \[thm1.3\]. Namely, we find the local well-posedness for the following equation. $$\begin{aligned}
\label{eq1.3}
{\left}\{
\begin{array}{ll}
\partial_t^2 u - \Delta u +\partial_t u = {\mathcal{N}}(u), & (t,x) \in (0,\infty) \times \mathbb{R}^d,
\\
(u(0),\partial_t u(0)) = (u_0,u_1), & x\in \mathbb{R}^d.
\end{array}
{\right}.\end{aligned}$$ Assume that the nonlinearity ${\mathcal{N}}: \mathbb{C} \to \mathbb{C}$ is continuously differentiable and obeys the power type estimates $$\begin{aligned}
{\mathcal{N}}(z)
&=O(|z|^{1+\frac{4}{d-2}}),
\\
{\mathcal{N}}_{z}(z), {\mathcal{N}}_{\bar{z}}(z)
&=O(|z|^{\frac{4}{d-2}}),
\\
{\mathcal{N}}_{z}(z)-{\mathcal{N}}_{z}(w), {\mathcal{N}}_{\bar{z}}(z)-{\mathcal{N}}_{\bar{z}}(w)
&=O(|z-w|^{\min\{1,\frac{4}{d-2}\}}(|z|+|w|)^{\max\{0,\frac{6-d}{d-2}\}} ),\end{aligned}$$ where ${\mathcal{N}}_{z}$ and ${\mathcal{N}}_{\bar{z}}$ are the usual derivatives $$\begin{aligned}
{\mathcal{N}}_{z}:= \frac{1}{2} {\left}( \frac{\partial {\mathcal{N}}}{\partial x} - i\frac{\partial {\mathcal{N}}}{\partial y} {\right}),
\quad
{\mathcal{N}}_{z}:= \frac{1}{2} {\left}( \frac{\partial {\mathcal{N}}}{\partial x} + i\frac{\partial {\mathcal{N}}}{\partial y} {\right})\end{aligned}$$ for $z=x+iy$. The typical examples are ${\mathcal{N}}(u)=\lambda |u|^{1+4/(d-2)}$ or $\lambda |u|^{4/(d-2)}u$ with $\lambda \in {{\mathbb C}}\setminus\{0\}$.
We have the energy $E$ of , which is defined by $$\begin{aligned}
E(u,\partial_t u) = \frac{1}{2} \norm{\nabla u}_{L^2}^2 + \frac{1}{2} \norm{\partial_t u}_{L^2}^2 -\frac{d-2}{2d} \norm{u}_{L^{\frac{2d}{d-2}}}^{\frac{2d}{d-2}}.\end{aligned}$$ If $u$ is a solution to , then the energy satisfies $$\begin{aligned}
\frac{d}{dt} E(u(t),\partial_t u(t)) = -\norm{\partial_t u(t)}_{L^2}^2\end{aligned}$$ for all $t \in (0,T_{\max})$. This means the energy decay. This observation shows us that some global solutions may decay. Indeed, we can prove that a global solution with a finite Strichartz norm decays to $0$ in the energy space as follows.
\[thm1.4\] If $u$ is a global solution of with $\| u \|_{L_{t,x}^{2(d+1)/(d-2)}([0,\infty))}<\infty$, then $u$ satisfies $$\begin{aligned}
\lim_{t \to \infty} {\left}(\norm{u(t)}_{H^1} + \norm{\partial_t u(t)}_{L^2}{\right}) =0.\end{aligned}$$
This is similar to the energy critical nonlinear heat equation. See Gustafson and Roxanas [@GuRo17p].
Theorem \[thm1.4\] holds for all dimensions $d \geq 3$ since we need to treat the estimate of the difference unlike the local well-posedness.
At last, we discuss the blow-up of the solutions to . We set $$\begin{aligned}
J(\varphi)
&:=\frac{1}{2} \norm{\nabla \varphi}_{L^2}^2 - \frac{d-2}{2d} \norm{\varphi}_{L^{\frac{2d}{d-2}}}^{\frac{2d}{d-2}},
\\
K(\varphi)
&:= \norm{ \nabla \varphi}_{L^2}^2-\norm{\varphi}_{L^{\frac{2d}{d-2}}}^{\frac{2d}{d-2}}. \end{aligned}$$ Then, it is well known that the minimizing problem $$\begin{aligned}
\mu := \inf {\left}\{ J(\varphi): \varphi \in \dot{H}^1 \setminus \{0\}, K(\varphi)=0 {\right}\}\end{aligned}$$ is attained by the Talenti function $$\begin{aligned}
W(x):= {\left}\{ 1+ \frac{|x|^2}{d(d-2)}{\right}\}^{-\frac{d-2}{2}}.\end{aligned}$$ See [@Tal76]. Here, the Talenti function satisfies the following nonlinear elliptic equation $$\begin{aligned}
-\Delta W = |W|^{\frac{4}{d-2}}W, \quad x \in {{\mathbb R}}^d. \end{aligned}$$ Therefore, $W$ is a static solution to . Then, we get the following blow-up result.
\[thm1.5\] Let $(u_0,u_1) \in H^1({{\mathbb R}}^d)\times L^2({{\mathbb R}}^d)$ belong to $$\begin{aligned}
{\mathscr{B}}:=\{(u_0,u_1) \in H^1({{\mathbb R}}^d)\times L^2({{\mathbb R}}^d) : E(u_0,u_1) < \mu, K(u_0)<0\}. \end{aligned}$$ Then the solution to blows up in finite time.
The theorem means that the static solution $W$ is strongly unstable for .
The proof of Theorem \[thm1.5\] is essentially given by Ohta [@Oht97]. He showed the blow-up result for abstract setting by the method of an ordinary differential inequality instead of by the so-called concavity argument, which is well applied to wave or Klein-Gordon equation.
We collect some notations. For the exponent $p$, we denote the Hölder conjugate of $p$ by $p'$. The bracket $\jbra{\cdot}$ is Japanese bracket *i.e.* $\jbra{a}:=(1+|a|^2)^{1/2}$.
We use $A \lesssim B$ to denote the estimate $A \leq CB$ with some constant $C>0$. The notation $A \sim B$ stands for $A \lesssim B$ and $A \lesssim B$.
Let $\chi_{\leq1} \in C_0^{\infty} (\mathbb{R})$ be a cut-off function satisfying $\chi_{\leq1} (r)=1$ for $|r| \le 1$ and $\chi_{\leq1}(r) =0$ for $|r| \ge 2$ and let $\chi_{>1}=1-\chi_{\leq 1}$.
For a function $f : \mathbb{R}^n \to \mathbb{C}$, we define the Fourier transform and the inverse Fourier transform by $$\begin{aligned}
\mathcal{F} [f] (\xi) = \hat{f} (\xi) = (2\pi)^{-n/2}
\int_{\mathbb{R}^n} e^{-i x \xi} f(x) \,dx,\quad
\mathcal{F}^{-1} [f] (x) = (2\pi)^{-n/2}
\int_{\mathbb{R}^n} e^{i x \xi} f(\xi) \,dx.\end{aligned}$$
For a measurable function $m = m(\xi)$, we denote the Fourier multiplier $m(\nabla)$ by $$\begin{aligned}
m(\nabla) f (x) = {\mathcal{F}}^{-1} \left[ m(\xi) \hat{f}(\xi) \right] (x).\end{aligned}$$
For $s \in \mathbb{R}$ and $1\leq p \leq \infty$, we denote the usual Sobolev space by $$\begin{aligned}
W^{s,p}(\mathbb{R}^d) := \left\{ f \in \mathcal{S}' (\mathbb{R}^d) :
\| f \|_{W^{s,p}} = \| \langle \nabla \rangle^s f \|_{L^p} < \infty \right\}.\end{aligned}$$ We write $H^s(\mathbb{R}^d) := W^{s,2}(\mathbb{R}^d)$ for simplicity. Let $\dot{W}^{s,p}(\mathbb{R}^d)$ and $\dot{H}^{s}({{\mathbb R}}^d)$ denote the corresponding homogeneous Sobolev spaces.
We define $P_{\leq1}:= {\mathcal{F}}^{-1} \chi_{\leq 1} {\mathcal{F}}$, $P_{>1}:={\mathcal{F}}^{-1} \chi_{> 1} {\mathcal{F}}$, and $$\begin{aligned}
P_{N} = {\mathcal{F}}^{-1} {\left}( \chi_{\leq1}{\left}( \frac{\xi}{N}{\right}) - \chi_{\leq1}{\left}( \frac{2\xi}{N}{\right}) {\right}){\mathcal{F}}\end{aligned}$$ for $N \in 2^{{{\mathbb Z}}}$. For a time interval $I$ and $F:I\times {{\mathbb R}}^d \to {{\mathbb C}}$, we set $$\begin{aligned}
\norm{F}_{L^{q}(I:L^r)({{\mathbb R}}^d)}:= {\left}(\int_{I} \norm{F(t,\cdot)}_{L^r({{\mathbb R}}^d)}^q dt {\right})^{1/q}\end{aligned}$$ and $L_{t,x}^{q}(I):=L^{q}(I:L^{q}({{\mathbb R}}^d))$. We sometimes use $L_s^p$ and $L_t^p$ to uncover time variables $s$ and $t$.
This paper is structured as follows. Section \[sec2\] is devoted to show the Strichartz estimates. In particular, we give the Strichartz estimates for low frequency part in Section \[sec2.1\] and that for high frequency part in Section \[sec2.2\]. In Setion \[sec3\], we prove the locall well-posedness of by the Strichartz estimates. In Section \[sec4\], we discuss the decay of the global solutions to with a bounded space-time norm. Section \[sec5\] contains the proof of the blow-up result.
The Strichartz estimates {#sec2}
========================
We split ${\mathcal{D}}$ to low frequency part ${\mathcal{D}}_{l}$ and high frequency part ${\mathcal{D}}_{h}$ as follows. $$\begin{aligned}
{\mathcal{D}}_{l}(t)&:={\mathcal{D}}(t)P_{\leq1},
\\
{\mathcal{D}}_{h}(t)&:={\mathcal{D}}(t)P_{>1}. \end{aligned}$$ In this section, we prove the Strichartz estimates for low and high frequency parts respectively.
The Strichartz estimates for low frequency part {#sec2.1}
-----------------------------------------------
We have the $L^p$-$L^q$ type estimates from [@CFZ14] and [@IIOWp]. These estimates are similar to those of the heat equation.
\[lem2.1\] Let $1 \leq \tilde{r} \leq r \leq \infty$ and $\sigma \geq 0$. Then, we have $$\begin{aligned}
\norm{|\nabla|^{\sigma} {\mathcal{D}}_{l}(t) f}_{L^r}
{\lesssim}\jbra{t}^{-\frac{d}{2}{\left}( \frac{1}{\tilde{r}} - \frac{1}{r}{\right})-\frac{\sigma}{2}} \norm{f}_{L^{\tilde{r}}}, \end{aligned}$$ for any $t > 0$ and $f \in L^{\tilde{r}}({{\mathbb R}}^d)$. We also have $$\begin{aligned}
\norm{|\nabla|^{\sigma} \partial_t {\mathcal{D}}_{l}(t) f}_{L^r}
&{\lesssim}\jbra{t}^{-\frac{d}{2}{\left}( \frac{1}{\tilde{r}} - \frac{1}{r}{\right})-\frac{\sigma}{2}-1} \norm{f}_{L^{\tilde{r}}},
\\
\norm{|\nabla|^{\sigma} \partial_t^2 {\mathcal{D}}_{l}(t) f}_{L^r}
&{\lesssim}\jbra{t}^{-\frac{d}{2}{\left}( \frac{1}{\tilde{r}} - \frac{1}{r}{\right})-\frac{\sigma}{2}-2} \norm{f}_{L^{\tilde{r}}}.\end{aligned}$$
By these $L^p$-$L^q$ type estimates, we obtain the following homogeneous Strichartz estimate.
\[lem2.2\] Let $\sigma\geq 0$. Let $1 \leq \tilde{r} \leq r \leq \infty$ and $1\leq q \leq \infty$. Assume that they satisfy $$\begin{aligned}
\frac{d}{2} {\left}( \frac{1}{\tilde{r}} - \frac{1}{r}{\right}) > \frac{1}{q},\end{aligned}$$ or $$\begin{aligned}
\frac{d}{2} {\left}( \frac{1}{\tilde{r}} - \frac{1}{r}{\right}) = \frac{1}{q}
\text{ and }
q >\tilde{r}>1.\end{aligned}$$ Then, for any $f \in L^{\tilde{r}}({{\mathbb R}}^d)$, $$\begin{aligned}
\norm{ \jbra{\nabla}^{\sigma} {\mathcal{D}}_{l}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
{\lesssim}\norm{f}_{L^{\tilde{r}}}, \end{aligned}$$ where $I \subset [0,\infty)$ is a time interval and the implicit constant is independent of $I$. Moreover, we also have $$\begin{aligned}
\norm{ \jbra{\nabla}^{\sigma} \partial_t {\mathcal{D}}_{l}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&{\lesssim}\norm{f}_{L^{\tilde{r}}},
\\
\norm{ \jbra{\nabla}^{\sigma} \partial_t^2 {\mathcal{D}}_{l}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&{\lesssim}\norm{f}_{L^{\tilde{r}}}.\end{aligned}$$
These Strichartz estimates are same as those of the heat equation. Thus, the same proof does work. However, we give the proof for reader’s convenience.
We first consider the case of $\frac{d}{2}(1/\tilde{r}-1/r)>1/q$. By the $L^r$-$L^{\tilde{r}}$ estimate (Lemma \[lem2.1\]), $$\begin{aligned}
\norm{ \jbra{\nabla}^{\sigma}{\mathcal{D}}_{l}(t) f}_{L^r}
&{\lesssim}\norm{{\mathcal{D}}_{l}(t) f}_{L^r}+\norm{|\nabla|^{\sigma}{\mathcal{D}}_{l}(t) f}_{L^r}
\\
&{\lesssim}\jbra{t}^{-\frac{d}{2}{\left}( \frac{1}{\tilde{r}} - \frac{1}{r}{\right})} \norm{f}_{L^{\tilde{r}}}
+ \jbra{t}^{-\frac{d}{2}{\left}( \frac{1}{\tilde{r}} - \frac{1}{r}{\right})-\frac{\sigma}{2}} \norm{f}_{L^{\tilde{r}}}
\\
&{\lesssim}\jbra{t}^{-\frac{d}{2}{\left}( \frac{1}{\tilde{r}} - \frac{1}{r}{\right})} \norm{f}_{L^{\tilde{r}}}.\end{aligned}$$ Then, we obtain $$\begin{aligned}
\norm{ \jbra{\nabla}^{\sigma}{\mathcal{D}}_{l}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
{\lesssim}\norm{\jbra{t}^{-\frac{d}{2} {\left}( \frac{1}{\tilde{r}} - \frac{1}{r}{\right}) } \norm{f}_{L^{\tilde{r}}}}_{L^q([0,\infty))}
{\lesssim}\norm{f}_{L^{\tilde{r}}}.\end{aligned}$$ Next, we consider the second case. We set $Tf:= \norm{ \jbra{\nabla}^{\sigma}{\mathcal{D}}_{l}(t) f}_{L^r({{\mathbb R}}^d)}$ and $(q_{1},r_{1}):=(\infty,r)$ and $(q_{2},r_{2})=(\rho,\gamma)$, where $(\rho,\gamma)$ satisfies $\frac{d}{2} {\left}( 1/\gamma - 1/r{\right}) = 1/\rho$ and $\rho,\gamma>1$. Then $T$ is sub-additive and we have $T:L^{r_{j}}({{\mathbb R}}^d) \to L^{q_{j},\infty}([0,\infty))$ for $j=1,2$. Indeed, we have $$\begin{aligned}
\norm{Tf(t)}_{L^\infty(I)}
&{\lesssim}\norm{\jbra{\nabla}^{\sigma} {\mathcal{D}}_{l}(t) f}_{L^{\infty}(I:L^r({{\mathbb R}}^d))}
{\lesssim}\norm{f}_{L^r},
\\
\norm{Tf(t)}_{L^{\rho,\infty}(I)}
&{\lesssim}\norm{\jbra{\nabla}^{\sigma} {\mathcal{D}}_{l}(t) f}_{L^{\rho,\infty}(I:L^r({{\mathbb R}}^d))}
{\lesssim}\norm{f}_{L^{\gamma}}.\end{aligned}$$ If $\rho \geq \gamma$, we can use the Marcinkiewicz interpolation theorem so that we have $$\begin{aligned}
\norm{{\mathcal{D}}_{l}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
{\lesssim}\norm{f}_{L^{\tilde{r}}}, \end{aligned}$$ for $(q,\tilde{r})$ satisfying $q>\tilde{r}>1$ and $$\begin{aligned}
\frac{1}{q} = \frac{1-\theta}{q_1} + \frac{\theta}{q_2}, \quad \frac{1}{\tilde{r}} = \frac{1-\theta}{r_1} + \frac{\theta}{r_2}, \quad 0< \theta <1. \end{aligned}$$ This means that the desired inequality holds for $(q,r)$ such that $\frac{d}{2}(1/\tilde{r}-1/r)=1/q$ and $q>\tilde{r}>1$. See also [@Wei81; @Gig86]. In the same way, we get the second and the third inequalities.
As stated in Remark \[rmk1.1\], the Strichartz estimate in the heat end-point case $(q,r)=(2,2d/(d-2))$ holds for $d \geq 3$. However, we exclude the end-point case in Lemma \[lem2.2\] since it is not clear whether the end-point Strichartz estimate holds or not for $q=\tilde{r}$ and $\tilde{r}\neq 2$. We give the proof of the Strichartz estimate in the end-point case $(q,r)=(2,2d/(d-2))$ for $d \geq 3$ in Section \[sec2.3\] (see Lemma \[lem2.11\]).
\[lem2.3\] Let $\sigma\geq 0$. Let $1 \leq \tilde{r}' \leq r \leq \infty$ and $1\leq q, \tilde{q} \leq \infty$. Assume that they satisfy $$\begin{aligned}
\frac{d}{2}{\left}( \frac{1}{2} - \frac{1}{r}{\right}) +\frac{d}{2}{\left}( \frac{1}{2} - \frac{1}{\tilde{r}}{\right})
>\frac{1}{q} + \frac{1}{\tilde{q}},\end{aligned}$$ $$\begin{aligned}
\frac{d}{2} {\left}( \frac{1}{2} - \frac{1}{r}{\right}) + \frac{d}{2} {\left}( \frac{1}{2} - \frac{1}{\tilde{r}}{\right})
= \frac{1}{q} + \frac{1}{\tilde{q}}
\text{ and }
1< \tilde{q}' < q<\infty,\end{aligned}$$ or $$\begin{aligned}
(q,r)=(\tilde{q},\tilde{r})=(\infty,2).\end{aligned}$$ Then it holds that $$\begin{aligned}
\norm{ \jbra{\nabla}^{\sigma} \int_{0}^{t} {\mathcal{D}}_{l}(t-s) F(s) ds}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&{\lesssim}\norm{F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))},
\\
\norm{\int_{0}^{t} \partial_t {\mathcal{D}}_{l}(t-s) F(s) ds}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&{\lesssim}\norm{F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))},\end{aligned}$$ where $I \subset [0,\infty)$ is a time interval such that $0 \in \overline{I}$ and the implicit constant is independent of $I$.
We only show the first estimate since the second can be proved similarly. Applying the $L^r$-$L^{\tilde{r}}$ estimate (Lemma \[lem2.1\]), we obtain $$\begin{aligned}
\norm{\jbra{\nabla}^{\sigma} \int_{0}^{t} {\mathcal{D}}_{l}(t-s) F(s) ds}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&{\lesssim}\norm{\int_{0}^{t} \norm{\jbra{\nabla}^{\sigma} {\mathcal{D}}_{l}(t-s) F(s)}_{L^r} ds}_{L^{q}(I)}
\\
& {\lesssim}\norm{\int_{0}^{t} \jbra{t-s}^{-\frac{d}{2}{\left}( \frac{1}{\tilde{r}'} - \frac{1}{r}{\right})} \norm{F(s)}_{L^{\tilde{r}'}} ds}_{L^{q}(I)}.\end{aligned}$$ When $\frac{d}{2} {\left}( \frac{1}{2} - \frac{1}{r}{\right}) + \frac{d}{2} {\left}( \frac{1}{2} - \frac{1}{\tilde{r}}{\right})> \frac{1}{q} + \frac{1}{\tilde{q}}$, by the Young inequality, we obtain $$\begin{aligned}
\norm{\int_{0}^{t} \jbra{t-s}^{-\frac{d}{2}{\left}( \frac{1}{\tilde{r}'} - \frac{1}{r}{\right})} \norm{F(s)}_{L^{\tilde{r}'}} ds}_{L^{q}(I)}
&{\lesssim}\norm{\jbra{t}^{-\frac{d}{2}{\left}( \frac{1}{\tilde{r}'} - \frac{1}{r}{\right})}}_{L^{\frac{q\tilde{q}}{\tilde{q}+q}}} \norm{F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))}
\\
&{\lesssim}\norm{F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))}.\end{aligned}$$ On the other hand, when $\frac{d}{2} {\left}( \frac{1}{2} - \frac{1}{r}{\right}) + \frac{d}{2} {\left}( \frac{1}{2} - \frac{1}{\tilde{r}}{\right})= \frac{1}{q} + \frac{1}{\tilde{q}}$ and $1<\tilde{q}'<q<\infty$, applying the Hardy–Littlewood–Sobolev inequality, we obtain $$\begin{aligned}
\norm{\int_{0}^{t} \jbra{t-s}^{-\frac{d}{2}{\left}( \frac{1}{\tilde{r}'} - \frac{1}{r}{\right})} \norm{F(s)}_{L^{\tilde{r}'}} ds}_{L^{q}(I)}
{\lesssim}\norm{F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))}.\end{aligned}$$ When $(q,r)=(\tilde{q},\tilde{r})=(\infty,2)$, the inequality is trivial. This completes the proof.
The Strichartz estimates for high frequency part {#sec2.2}
------------------------------------------------
Since we have $$\begin{aligned}
{\mathcal{D}}_{h}(t) = e^{-\frac{t}{2}} {\mathcal{F}}^{-1} \frac{e^{it \sqrt{|\xi|^2-1/4}} - e^{-it \sqrt{|\xi|^2-1/4}}}{2i \sqrt{|\xi|^2-1/4}} \chi_{>1} (\xi) {\mathcal{F}},\end{aligned}$$ it is enough to estimate $$\begin{aligned}
e^{-t/2} e^{\pm it\sqrt{-\Delta-1/4}} P_{>1}.\end{aligned}$$
\[lem2.4\] Let $d\geq 2$. Let $2\leq r < \infty$ and $2 \leq q \leq \infty$. We set $\gamma:= \max \{ d(1/2-1/r)-1/q, \frac{d+1}{2}(1/2-1/r) \}$. Then, we have $$\begin{aligned}
\norm{e^{-t/2} e^{\pm it\sqrt{-\Delta-1/4}} P_{>1} f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
{\lesssim}\norm{|\nabla|^{\gamma} f}_{L^2}\end{aligned}$$ where $I \subset [0,\infty)$ is a time interval and the implicit constant is independent of $I$. In particular, we have $$\begin{aligned}
\norm{{\mathcal{D}}_{h}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&{\lesssim}\norm{|\nabla|^{\gamma} \jbra{\nabla}^{-1} f}_{L^2},
\\
\norm{ \partial_t {\mathcal{D}}_{h}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&{\lesssim}\norm{|\nabla|^{\gamma} f}_{L^2},
\\
\norm{ \partial_t^2 {\mathcal{D}}_{h}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&{\lesssim}\norm{|\nabla|^{\gamma+1} f}_{L^2}.\end{aligned}$$
First, we consider $e^{it\sqrt{-\Delta-1/4}}$. We note that $$\begin{aligned}
e^{it \sqrt{-\Delta-1/4}} = e^{it|\nabla|} e^{it \left( \sqrt{-\Delta-1/4}-|\nabla| \right)}.\end{aligned}$$ Since we have $$\begin{aligned}
{\left}| \sqrt{|\xi|^2-1/4}-|\xi| {\right}|
= \frac{1}{4(\sqrt{|\xi|^2-1/4}+|\xi|)}
{\approx}|\xi|^{-1},\end{aligned}$$ a simple calculation shows $$\begin{aligned}
{\left}| \partial_{\xi}^{\alpha} e^{it {\left}( \sqrt{|\xi|^2-1/4}-|\xi| {\right})} {\right}|
{\lesssim}\jbra{t}^{|\alpha|} |\xi|^{-|\alpha|}\end{aligned}$$ for $\xi \neq 0$ and $\alpha \in \mathbb{Z}_{\ge 0}^n$. Thus, the Mihlin–Hörmander multiplier theorem (see [@Gra14 Theorem 6.2.7]) gives $$\begin{aligned}
\norm{ e^{ it\sqrt{-\Delta-1/4}} P_{>1} f}_{L^r}
{\lesssim}\jbra{t}^{\delta_r} \norm{ e^{ it |\nabla|} f }_{L^r}\end{aligned}$$ for some $\delta_r>0$. Therefore, we obtain $$\begin{aligned}
\norm{e^{-t/2} e^{ it\sqrt{-\Delta-1/4}} P_{>1} f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&= \norm{e^{-t/2} \norm{ e^{ it\sqrt{-\Delta-1/4}} P_{>1} f }_{L^r} }_{L^{q}(I)}
\\
&{\lesssim}\norm{e^{-t/2} \jbra{t}^{\delta_r} \norm{ e^{ it |\nabla|} f }_{L^r} }_{L^{q}(I)}
\\
&{\lesssim}\norm{ e^{ it |\nabla|} f }_{L^{\tilde{q}}(I:L^r({{\mathbb R}}^d))},\end{aligned}$$ where we have used the Hölder inequality in the last inequality and we take $\tilde{q}$ such that $$\begin{aligned}
\tilde{q}
=
{\left}\{
\begin{array}{ll}
q
& \text{ if } \frac{d-1}{2}{\left}( \frac{1}{2} - \frac{1}{r}{\right}) \geq \frac{1}{q},
\\
{\left}\{ \frac{d-1}{2} {\left}( \frac{1}{2} - \frac{1}{r} {\right}) {\right}\}^{-1}
& \text{ if } \frac{d-1}{2}{\left}( \frac{1}{2} - \frac{1}{r}{\right}) < \frac{1}{q}.
\end{array}
{\right}.\end{aligned}$$ Then, $(\tilde{q},r)$ is a wave admissible pair. Namely, it satisfies $$\begin{aligned}
\frac{1}{\tilde{q}} + \frac{d-1}{2r} \leq \frac{d-1}{4},
\ \tilde{q},r,d \geq 2,
\text{ and } (q,r,d) \neq (2,\infty,3)\end{aligned}$$ and $$\begin{aligned}
\frac{1}{\tilde{q}}+\frac{d}{r}=\frac{d}{2} - \gamma,\end{aligned}$$ where we note that $\gamma \geq 0$. Therefore, by the Strichartz estimate for the free wave equation (see [@GiVe95] or [@KTV14 Corollary 2.5 in p.233]), we get $$\begin{aligned}
\norm{e^{-t/2} e^{ it\sqrt{-\Delta-1/4}} P_{>1} f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&{\lesssim}\norm{ e^{ it |\nabla|} f }_{L^{\tilde{q}}(I:L^r({{\mathbb R}}^d))}
\\
& {\lesssim}\norm{|\nabla|^{\gamma} f}_{L^2}.\end{aligned}$$ Similarly, we also have $$\begin{aligned}
\norm{e^{-t/2} e^{- it\sqrt{-\Delta-1/4}} P_{>1} f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
{\lesssim}\norm{|\nabla|^{\gamma} f}_{L^2}. \end{aligned}$$ Combining them with the formula of ${\mathcal{D}}_{h}$, we obtain $$\begin{aligned}
\norm{{\mathcal{D}}_{h}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
{\lesssim}\norm{|\nabla|^{\gamma}\jbra{\nabla}^{-1} f}_{L^2},\end{aligned}$$ where we use $\sqrt{|\xi|^2-1/4} {\approx}\jbra{\xi}$ for $|\xi|\geq1$. Moreover, we also get the estimates related to $\partial_t {\mathcal{D}}_{h}(t)$ and $\partial_t^2 {\mathcal{D}}_{h}(t)$.
We can also obtain the homogeneous Strichartz estimates for high frequency part when $1\leq q < 2$. Indeed, taking $$\begin{aligned}
\tilde{q}
=
{\left}\{
\begin{array}{ll}
2
& \text{ if } \frac{d-1}{2}{\left}( \frac{1}{2} - \frac{1}{r}{\right}) \geq \frac{1}{2},
\\
&
\\
{\left}\{ \frac{d-1}{2} {\left}( \frac{1}{2} - \frac{1}{r} {\right}) {\right}\}^{-1}
& \text{ if } \frac{d-1}{2}{\left}( \frac{1}{2} - \frac{1}{r}{\right}) < \frac{1}{2},
\end{array}
{\right}.\end{aligned}$$ $(\tilde{q},r)$ is a wave admissible pair and thus the above argument does work. We note that, in this case, we need to redefine $\gamma$ such that $$\begin{aligned}
\gamma:= \max{\left}\{ \frac{d+1}{2}{\left}(\frac{1}{2}-\frac{1}{r}{\right}), d{\left}(\frac{1}{2}-\frac{1}{r}{\right})-\frac{1}{2}{\right}\} \geq 0.
\end{aligned}$$
To prove inhomogeneous Strichartz estimates for high frequency part, we show the $L^p$-$L^{q}$ type estimate.
\[lem2.5\] Let $d \geq 1$. Let $2 \leq r < \infty$. Then, it holds that $$\begin{aligned}
\norm{e^{\pm it \sqrt{-\Delta-1/4} }P_{>1} P_{N} f}_{L^{r}}
{\lesssim}\jbra{t}^{\delta_r} (1+|t|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } N^{d{\left}( 1 -\frac{2}{r} {\right})} \norm{P_{N} f}_{L^{r'}}\end{aligned}$$ for any $t>0$ and $N \in 2^{{{\mathbb Z}}}$, where $\delta_r$ is a positive constant.
Combining the $L^{p}$-$L^{q}$ type estimate for free wave equation (see [@Bre75] or [@KTV14 Lemma 2.1 in p.230]) and the Mihlin–Hörmander multiplier theorem, we get the statement.
\[lem2.6\] Let $d \geq 2$. Let $2 \leq r < \infty$ and $2 \leq q \leq \infty$. We set $\gamma:= \max \{ d(1/2-1/r)-1/q, \frac{d+1}{2}(1/2-1/r) \}$. We exclude the end-point case, that is, we assume that $(q,r) \neq (2, 2(d-1)/(d-3))$ when $d \geq 4$. Then, we have $$\begin{aligned}
\norm{\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) ds}_{L^{q}(I:L^r({{\mathbb R}}^d))}
{\lesssim}N^{2\gamma} \norm{P_{N}F}_{L^{q'}(I:L^{r'}({{\mathbb R}}^d))},\end{aligned}$$ where $I \subset [0,\infty)$ is a time interval such that $0 \in \overline{I}$ and the implicit constant is independent of $I$.
By the $L^{r}$-$L^{r'}$ estimate for high frequency part, Lemma \[lem2.5\], we get $$\begin{aligned}
\label{eq2.1}
&\norm{\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) ds}_{L^{q}(I:L^r({{\mathbb R}}^d))}
\\ \notag
&\leq \norm{\int_{0}^{t} e^{-\frac{t-s}{2}} \jbra{t-s}^{\delta_r} (1+|t-s|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } N^{d{\left}( 1 -\frac{2}{r} {\right})} \norm{P_{N} F(s)}_{L^{r'}} ds}_{L^{q}(I)}
\\ \notag
&{\lesssim}N^{d{\left}( 1 -\frac{2}{r} {\right})}
\norm{\int_{0}^{t} e^{-\frac{t-s}{4}} (1+|t-s|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } \norm{P_{N} F(s)}_{L^{r'}} ds}_{L^{q}(I)}.\end{aligned}$$ Here, by the Young inequality, we obtain $$\begin{aligned}
\label{eq2.2}
&N^{d{\left}( 1 -\frac{2}{r} {\right})}
\norm{\int_{0}^{t} e^{-\frac{t-s}{4}} (1+|t-s|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } \norm{P_{N} F(s)}_{L^{r'}} ds}_{L^{q}(I)}
\\ \notag
& {\lesssim}N^{d{\left}( 1 -\frac{2}{r} {\right})}
\norm{ e^{-\frac{\cdot}{4}} (1+|\cdot|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } }_{L^{q/2}([0,\infty))}
\norm{P_N F}_{L^{q'}(I: L^{r'}({{\mathbb R}}^d))}.\end{aligned}$$ In the case of $\frac{d-1}{2} (1-2/r)> 2/q$, since we have $$\begin{aligned}
\norm{ e^{-\frac{\cdot}{4}} (1+|\cdot|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } }_{L^{q/2}([0,\infty))}^{q/2}
\leq \int_{0}^{\infty} (1+|t|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) \frac{q}{2} } dt
{\lesssim}N^{-1},\end{aligned}$$ we obtain, from and , $$\begin{aligned}
\text{(L.H.S. of \eqref{eq2.1})}
& {\lesssim}N^{2 {\left}\{ d{\left}( \frac{1}{2} - \frac{1}{r} {\right}) - \frac{1}{q} {\right}\}} \norm{P_N F}_{L^{q'}(I: L^{r'}({{\mathbb R}}^d))}
\\
& = N^{2\gamma} \norm{P_N F}_{L^{q'}(I: L^{r'}({{\mathbb R}}^d))}. \end{aligned}$$ On the other hand, in the case of $\frac{d-1}{2} (1-2/r) < 2/q$, we have $$\begin{aligned}
&\norm{ e^{-\frac{\cdot}{4}} (1+|\cdot|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } }_{L^{q/2}([0,\infty))}^{q/2}
\\
&= \int_{0}^{\infty} e^{-\frac{q}{8}t} (1+|t|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) \frac{q}{2} } dt
\\
&\leq N^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) \frac{q}{2}} \int_{0}^{\infty} e^{-\frac{q}{8}t} t^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) \frac{q}{2} } dt
\\
&\leq N^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) \frac{q}{2}}
{\left}( \int_{0}^{1} t^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) \frac{q}{2} } dt
+ \int_{1}^{\infty} e^{-\frac{q}{8}t}dt {\right})
\\
& {\lesssim}N^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) \frac{q}{2}}.\end{aligned}$$ Therefore, we obtain, from and , $$\begin{aligned}
\text{(L.H.S. of \eqref{eq2.1})}
& {\lesssim}N^{2 {\left}\{ \frac{d+1}{2} {\left}( \frac{1}{2}-\frac{1}{r}{\right}) {\right}\}} \norm{P_N F}_{L^{q'}(I: L^{r'}({{\mathbb R}}^d))}
\\
& = N^{2\gamma} \norm{P_N F}_{L^{q'}(I: L^{r'}({{\mathbb R}}^d))}. \end{aligned}$$ At last, we consider the case of $\frac{d-1}{2} (1-2/r) = 2/q$. Then, we have $$\begin{aligned}
\label{eq2.3}
&N^{d{\left}( 1 -\frac{2}{r} {\right})}
\norm{\int_{0}^{t} e^{-\frac{t-s}{4}} (1+|t-s|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } \norm{P_{N} F(s)}_{L^{r'}} ds}_{L^{q}(I)}
\\ \notag
& {\lesssim}N^{d{\left}( 1 -\frac{2}{r} {\right})} N^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) }
\norm{\int_{0}^{t} |t-s|^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } \norm{P_{N} F(s)}_{L^{r'}} ds}_{L^{q}(I)}
\\ \notag
& = N^{2\gamma}
\norm{\int_{0}^{t} |t-s|^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } \norm{P_{N} F(s)}_{L^{r'}} ds}_{L^{q}(I)}\end{aligned}$$ and it follows from the Hardy–Littlewood–Sobolev inequality that $$\begin{aligned}
\label{eq2.4}
\norm{\int_{0}^{t} |t-s|^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } \norm{P_{N} F(s)}_{L^{r'}} ds}_{L^{q}(I)}
{\lesssim}\norm{P_{N} F(s)}_{L^{q'}(I:L^{r'}({{\mathbb R}}^d))},\end{aligned}$$ since $(q,r)$ is not the end-point. Combining , , and , we get the desired inequality.
In the previous lemma, we exclude the end-point case. However, we can easily obtain the Strichartz estimate in the end-point case if we permit additional derivative loss. Indeed, using and the following calculation: $$\begin{aligned}
&N^{d{\left}( 1 -\frac{2}{r} {\right})}
\norm{\int_{0}^{t} e^{-\frac{t-s}{4}} (1+|t-s|N)^{-\frac{d-1}{2} {\left}( 1-\frac{2}{r}{\right}) } \norm{P_{N} F(s)}_{L^{r'}} ds}_{L^{q}(I)}
\\
& {\lesssim}N^{d{\left}( 1 -\frac{2}{r} {\right})}
\norm{\int_{0}^{t} e^{-\frac{t-s}{4}} \norm{P_{N} F(s)}_{L^{r'}} ds}_{L^{q}(I)}
\\
& {\lesssim}N^{2d{\left}( \frac{1}{2} -\frac{1}{r} {\right})}
\norm{ e^{-\frac{\cdot}{4}} }_{L^{q/2}([0,\infty))}
\norm{P_N F}_{L^{q'}(I: L^{r'}({{\mathbb R}}^d))}
\\
& {\lesssim}N^{2d{\left}( \frac{1}{2} -\frac{1}{r} {\right})}
\norm{P_N F}_{L^{q'}(I: L^{r'}({{\mathbb R}}^d))},\end{aligned}$$ where we have used the Young inequality, we get the Strichartz estimate with the derivative loss $d(1/2-1/r)$ instead of $\gamma$. We note that the derivative loss $d(1/2 - 1/r)$ is larger than $\gamma$ in Lemma \[lem2.6\].
\[lem2.7\] Let $d \geq2$. Let $2 \leq r < \infty$ and $2 \leq q \leq \infty$. We set $\gamma:= \max \{ d(1/2-1/r)-1/q, \frac{d+1}{2}(1/2-1/r) \}$. We assume that $(q,r) \neq (2, 2(d-1)/(d-3))$ when $d \geq 4$. Then, we have $$\begin{aligned}
\norm{\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) ds}_{L^{\infty}(I:L^{2}({{\mathbb R}}^d))}
{\lesssim}N^{\gamma} \norm{P_{N}F}_{L^{q'}(I:L^{r'}({{\mathbb R}}^d))},\end{aligned}$$ where $I \subset [0,\infty)$ is a time interval such that $0 \in \overline{I}$ and the implicit constant is independent of $I$.
Now, we have $$\begin{aligned}
&\norm{\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) ds}_{L^{2}}^2
\\
&= \tbra{\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) ds}{\int_{0}^{t} e^{-\frac{t-\tau}{2}} e^{\pm i(t-\tau)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(\tau) d\tau}_{L^2}
\\
&=\int_{0}^{t} \int_{0}^{s} \tbra{e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) }{ e^{-\frac{t-\tau}{2}} e^{\pm i(t-\tau)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(\tau)}_{L^2} d\tau ds
\\
&\quad +\int_{0}^{t} \int_{0}^{\tau} \tbra{e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) }{ e^{-\frac{t-\tau}{2}} e^{\pm i(t-\tau)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(\tau)}_{L^2} ds d\tau
\\
&=I+I\!\!I.\end{aligned}$$ By the symmetry, it is enough to estimate $I$. By the Hölder inequality, $e^{-\frac{t-s}{2}}e^{-\frac{t-\tau}{2}}=e^{-(t-s)}e^{-\frac{s-\tau}{2}}$, and $e^{-(t-s)}\leq 1$ for $s \in [0,t]$ we obtain $$\begin{aligned}
I
&=\int_{0}^{t} \tbra{e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) }{ \int_{0}^{s} e^{-\frac{t-\tau}{2}} e^{\pm i(t-\tau)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(\tau) d\tau}_{L^2} ds
\\
&\leq \int_{0}^{t} e^{-(t-s)} \tbra{ {\left}| P_{N} F(s) {\right}|}{ {\left}| \int_{0}^{s} e^{-\frac{s-\tau}{2}} e^{\pm i(s-\tau)\sqrt{-\Delta-1/4}} P_{>1}^2 P_{N} F(\tau) d\tau {\right}|}_{L^2} ds
\\
&\leq \norm{P_{N} F}_{L^{q'}(I:L^{r'}({{\mathbb R}}^d))}
\norm{\int_{0}^{s} e^{-\frac{s-\tau}{2}} e^{\pm i(s-\tau)\sqrt{-\Delta-1/4}} P_{>1}^2 P_{N} F(\tau) d\tau}_{L_{s}^{q}((0,t):L^{r}({{\mathbb R}}^d))}.\end{aligned}$$ By Lemma \[lem2.6\], we obtain $$\begin{aligned}
I \leq N^{2\gamma} \norm{ P_{N} F}_{L^{q'}(I:L^{r'}({{\mathbb R}}^d))}^2.\end{aligned}$$ Thus, it follows that $$\begin{aligned}
\norm{\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) ds}_{L^{2}}^2
{\lesssim}N^{2\gamma} \norm{ P_{N} F}_{L^{q'}(I:L^{r'}({{\mathbb R}}^d))}^2.\end{aligned}$$ This finishes the proof.
\[rmk2.3\] Let $T>0$, $2 \leq r < \infty$ and $2\leq q \leq \infty$, $\gamma:= \max \{ d(1/2-1/r)-1/q, \frac{d+1}{2}(1/2-1/r) \}$, and $(q,r) \neq (2, 2(d-1)/(d-3))$ when $d \geq 4$. Then, we have the following inequality by the same argument as in Lemma \[lem2.6\]. $$\begin{aligned}
\label{eq2.5}
&\norm{\int_{s}^{t} e^{-(\tau-s)} e^{-\frac{t-\tau}{2}} e^{\mp i(t-\tau)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(\tau) d\tau}_{L_{t}^{q}((s,T):L^r({{\mathbb R}}^d))}
\\ \notag
&{\lesssim}N^{2\gamma} \norm{P_{N}F}_{L^{q'}(I:L^{r'}({{\mathbb R}}^d))},\end{aligned}$$ where $s<T$ is a parameter. Moreover, we also have the following estimate from and the similar argument to Lemma \[lem2.7\]. $$\begin{aligned}
\label{eq2.6}
&\norm{\int_{s}^{T} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(t) dt}_{L_{s}^{\infty}([0,T):L^{2}({{\mathbb R}}^d))}
\\ \notag
&{\lesssim}N^{\gamma} \norm{ P_{N} F}_{L^{q'}([0,T):L^{r'}({{\mathbb R}}^d))}.\end{aligned}$$
\[lem2.8\] Let $2 \leq r < \infty$ and $2 \leq q \leq \infty$. We set $\gamma:= \max \{ d(1/2-1/r)-1/q, \frac{d+1}{2}(1/2-1/r) \}$. We assume that $(q,r) \neq (2, 2(d-1)/(d-3))$ when $d \geq 4$. Then, we have $$\begin{aligned}
\norm{\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
{\lesssim}N^{\gamma} \norm{P_{N}F}_{L^{1}(I:L^{2}({{\mathbb R}}^d))},\end{aligned}$$ where $I \subset [0,\infty)$ is a time interval such that $0 \in \overline{I}$ and the implicit constant is independent of $I$.
We may write $I=[0,T)$. We use a standard duality argument. Let $G \in C_{0}^{\infty}(I \times {{\mathbb R}}^d)$ and $\tilde{P}_{N}:= P_{N/2}+P_{N} + P_{2N}$. Since we have $\tilde{P}_{N}P_{N}=P_{N}$, it follows from the Fubini theorem and Hölder inequality that $$\begin{aligned}
\label{eq2.7}
&\int_{0}^{T} \tbra{\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s) ds}{G(t)}dt
\\ \notag
&=\int_{0}^{T} \int_{0}^{t} e^{-\frac{t-s}{2}} \tbra{e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{N} F(s)}{G(t)} dsdt
\\ \notag
&=\int_{0}^{T} \int_{s}^{T} e^{-\frac{t-s}{2}} \tbra{P_{N} F(s)}{ e^{\mp i(t-s)\sqrt{-\Delta-1/4}} P_{>1} \tilde{P}_{N} G(t)} dt ds
\\ \notag
&=\int_{0}^{T} \tbra{P_{N} F(s)}{ \int_{s}^{T} e^{-\frac{t-s}{2}} e^{\mp i(t-s)\sqrt{-\Delta-1/4}} P_{>1} \tilde{P}_{N} G(t) dt} ds
\\ \notag
& \leq \norm{P_{N}F}_{L^{1}(I:L^{2}({{\mathbb R}}^d))} \norm{ \int_{s}^{T} e^{-\frac{t-s}{2}} e^{\mp i(t-s)\sqrt{-\Delta-1/4}} P_{>1} \tilde{P}_{N} G(t) dt}_{L^{\infty}(I:L^{2}({{\mathbb R}}^d))}\end{aligned}$$ By in Remark \[rmk2.3\], we get $$\begin{aligned}
\label{eq2.8}
&\norm{ \int_{s}^{T} e^{-\frac{t-s}{2}} e^{\mp i(t-s)\sqrt{-\Delta-1/4}} P_{>1} \tilde{P}_{N} G(t) dt}_{L^{\infty}(I:L^{2}({{\mathbb R}}^d))}
\\ \notag
&\leq \sum_{j=N/2,N,2N} \norm{ \int_{s}^{T} e^{-\frac{t-s}{2}} e^{\mp i(t-s)\sqrt{-\Delta-1/4}} P_{>1} P_{j} G(t) dt}_{L^{\infty}(I:L^{2}({{\mathbb R}}^d))}
\\ \notag
& {\lesssim}N^{\gamma} \sum_{j=N/2,N,2N} \norm{P_{j}G}_{L^{q'}(I:L^{r'}({{\mathbb R}}^d))}
\\ \notag
&{\lesssim}N^{\gamma} \norm{G}_{L^{q'}(I:L^{r'}({{\mathbb R}}^d))}.\end{aligned}$$ Since we have the duality $$\begin{aligned}
\norm{F}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
=\sup {\left}\{ \int_{I} \tbra{F(t)}{G(t)} dt: G\in C_{0}^{\infty}(I \times {{\mathbb R}}^d), \norm{G}_{L^{q'}(I:L^{r'}({{\mathbb R}}^d))}=1{\right}\},\end{aligned}$$ the desired estimate follows from and .
Combining these estimates, we obtain the following Strichartz estimates when $(1/q,1/r)$ and $(1/\tilde{q},1/\tilde{r})$ are on a same line.
\[lem2.9\] Let $2 \leq r, \tilde{r} < \infty$ and $2 \leq q, \tilde{q} \leq \infty$. We set $\gamma:= \max \{ d(1/2-1/r)-1/q, \frac{d+1}{2}(1/2-1/r) \}$ and $\tilde{\gamma}$ in the same manner. Assume that $$\begin{aligned}
\frac{1}{\tilde{q}} {\left}( \frac{1}{2}- \frac{1}{r}{\right}) = \frac{1}{q} {\left}( \frac{1}{2} - \frac{1}{\tilde{r}} {\right}).\end{aligned}$$ We also assume that $(q,r) \neq (2, 2(d-1)/(d-3))$ and $(\tilde{q},\tilde{r}) \neq (2, 2(d-1)/(d-3))$ when $d \geq 4$. Then, we have $$\begin{aligned}
\norm{\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
{\lesssim}\norm{|\nabla|^{\gamma+\tilde{\gamma}}F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))},\end{aligned}$$ where $I \subset [0,\infty)$ is a time interval such that $0 \in \overline{I}$ and the implicit constant is independent of $I$.
We set $$\begin{aligned}
\Psi[F](t,x):=\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} F(s) ds.\end{aligned}$$ First, we consider the case of $2 \leq r \leq \tilde{r}$. Then, $\tilde{q} \leq q$ and thus there exists $\theta \in [0,1]$ such that $$\begin{aligned}
\frac{1}{q} = \frac{\theta}{\tilde{q}} + \frac{1-\theta}{\infty},
\quad
\frac{1}{r} = \frac{\theta}{\tilde{r}} + \frac{1-\theta}{2}.\end{aligned}$$ By this formula, we have $\theta \tilde{\gamma}= \gamma$. Therefore, by the Hölder inequality, Lemmas \[lem2.6\] and \[lem2.7\], we obtain $$\begin{aligned}
&\norm{\Psi[F]}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
\\
&{\lesssim}\norm{\Psi[F]}_{L^{\tilde{q}}(I:L^{\tilde{r}}({{\mathbb R}}^d))}^{\theta} \norm{\Psi[F]}_{L^{\infty}(I:L^{2}({{\mathbb R}}^d))}^{1-\theta}
\\
&{\lesssim}{\left}(N^{2\tilde{\gamma}} \norm{P_N F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))} {\right})^{\theta}
{\left}(N^{\tilde{\gamma}} \norm{P_N F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))} {\right})^{1-\theta}
\\
&{\approx}N^{\gamma + \tilde{\gamma}} \norm{P_{N}F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))},\end{aligned}$$ where we use $\theta \tilde{\gamma}= \gamma$.
At second, we consider the case of $2 \leq \tilde{r} \leq r$. Then, we have $\tilde{q} \geq q$. Let $\eta \in [0,1]$ satisfy $$\begin{aligned}
\frac{1}{\tilde{q}'} = \frac{1-\eta}{1} + \frac{\eta}{q'},
\quad
\frac{1}{\tilde{r}'} = \frac{1-\eta}{2} + \frac{\eta}{r'}.\end{aligned}$$ Then, we have $\eta \gamma=\tilde{\gamma}$. By the interpolation, Lemmas \[lem2.6\], and \[lem2.8\], we get the desired inequality, where we note that $N^{(1-\eta) \gamma}N^{\eta 2\gamma} = N^{\gamma + \tilde{\gamma}}$. Taking summation for dyadic number $N$ gives the statement.
We can get Strichartz estimates even when $(1/q,1/r)$ and $(1/\tilde{q},1/\tilde{r})$ are not on a same line by permitting more derivative loss.
\[lem2.10\] Let $d \geq 2$. Let $2 \leq r, \tilde{r} < \infty$ and $2 \leq q, \tilde{q} \leq \infty$. We set $\gamma:= \max \{ d(1/2-1/r)-1/q, \frac{d+1}{2}(1/2-1/r) \}$ and $\tilde{\gamma}$ in the same manner. Assume that $$\begin{aligned}
\frac{1}{\tilde{q}} {\left}( \frac{1}{2}- \frac{1}{r}{\right}) \neq \frac{1}{q} {\left}( \frac{1}{2} - \frac{1}{\tilde{r}} {\right}).\end{aligned}$$ We also assume that $(q,r) \neq (2, 2(d-1)/(d-3))$ and $(\tilde{q},\tilde{r}) \neq (2, 2(d-1)/(d-3))$ when $d \geq 4$. Then, we have $$\begin{aligned}
\norm{\int_{0}^{t} e^{-\frac{t-s}{2}} e^{\pm i(t-s)\sqrt{-\Delta-1/4}} P_{>1} F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
{\lesssim}\norm{|\nabla|^{\gamma+\tilde{\gamma}+\delta}F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))},\end{aligned}$$ where $\delta \geq 0$ is defined in the table 1 (see Proposition \[prop1.2\]). Moreover, we have $$\begin{aligned}
\norm{\int_{0}^{t} {\mathcal{D}}_{h}(t-s) F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
&{\lesssim}\norm{|\nabla|^{\gamma+\tilde{\gamma}+\delta} \jbra{\nabla}^{-1}F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))},
\\
\norm{\int_{0}^{t} (\partial_t {\mathcal{D}}_{h} ) (t-s) F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
&{\lesssim}\norm{|\nabla|^{\gamma+\tilde{\gamma}+\delta}F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))}.\end{aligned}$$
We consider the following cases respectively.
1. $\frac{1}{\tilde{q}}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) < \frac{1}{q} {\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right})$
2. $\frac{1}{\tilde{q}}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) > \frac{1}{q} {\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right})$
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1. $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) \geq \frac{1}{q}$ and $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) \geq \frac{1}{\tilde{q}}$
2. $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) \geq \frac{1}{q}$ and $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) < \frac{1}{\tilde{q}}$
3. $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) < \frac{1}{q}$ and $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) \geq \frac{1}{\tilde{q}}$
4. $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) < \frac{1}{q}$ and $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) < \frac{1}{\tilde{q}}$
It is easy to show that Cases (1)-(b) and (2)-(c) do not occur.
[**Case(1).**]{} We treat the case of $\frac{1}{\tilde{q}}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) < \frac{1}{q} {\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right})$. Since $\frac{1}{\tilde{q}}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) <\frac{1}{q}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right})$, there exists $r_1 \in [2,\tilde{r})$ such that $$\begin{aligned}
\frac{1}{\tilde{q}}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) =\frac{1}{q}{\left}( \frac{1}{2}- \frac{1}{r_1}{\right}).\end{aligned}$$ Let $\gamma_1$ be the derivative loss for the pair $(\tilde{q},r_1)$. Then, by Lemma \[lem2.9\] and the Bernstein inequality, we get $$\begin{aligned}
\norm{\Psi[F]}_{L_t^qL_x^r}
&{\lesssim}N^{\gamma+\gamma_1} \norm{P_N F}_{L_t^{\tilde{q}'}L_x^{r_1'}}
\\
&{\lesssim}N^{\gamma+\gamma_1} N^{d{\left}(\frac{1}{\tilde{r}'} - \frac{1}{r_1'}{\right})} \norm{P_N F}_{L_t^{\tilde{q}'}L_x^{\tilde{r}'}}.\end{aligned}$$
[**Case(1)-(a).**]{} If $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) \geq \frac{1}{q}$, which also gives $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r_1}{\right}) \geq \frac{1}{\tilde{q}}$, we have $\gamma_1=d(1/2-1/{r_1})-1/{\tilde{q}}$. Thus, we obtain $$\begin{aligned}
\gamma+\gamma_1+d{\left}(\frac{1}{\tilde{r}'} - \frac{1}{r_1'}{\right})
&=\gamma + d{\left}( \frac{1}{2}- \frac{1}{r_1}{\right}) - \frac{1}{q_1}+d{\left}(\frac{1}{\tilde{r}'} - \frac{1}{r_1'}{\right})
\\
&=\gamma + d{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) - \frac{1}{\tilde{q}}
\\
&=\gamma + \tilde{\gamma}.\end{aligned}$$
[**Case(1)-(c).**]{} $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) < \frac{1}{q}$ gives $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r_1}{\right}) < \frac{1}{\tilde{q}}$. Then, we have $\gamma_1 = \frac{d+1}{2}(1/2-1/{r_1})$. Moreover, since $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) \geq \frac{1}{\tilde{q}}$, we have $\tilde{\gamma}=d(1/2-1/{\tilde{r}})-1/{\tilde{q}}$. Therefore, we obtain $$\begin{aligned}
\gamma+\gamma_1+d{\left}(\frac{1}{\tilde{r}'} - \frac{1}{r_1'}{\right})
&=\gamma + \tilde{\gamma} + \gamma_1 -\tilde{\gamma}+d{\left}(\frac{1}{\tilde{r}'} - \frac{1}{r_1'}{\right})
\\
&=\gamma + \tilde{\gamma}+\frac{q}{\tilde{q}} {\left}\{\frac{1}{q}- \frac{d-1}{2} {\left}(\frac{1}{2}-\frac{1}{r}{\right}) {\right}\},\end{aligned}$$ where we use $q{\left}( \frac{1}{2}- \frac{1}{r}{\right}) =\tilde{q}{\left}( \frac{1}{2}- \frac{1}{r_1}{\right})$ in the last equality.
[**Case(1)-(d).**]{} We have $\gamma_1 = \frac{d+1}{2}(1/2-1/{r_1})$ since $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) < \frac{1}{q}$. Since $\frac{d-1}{2}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) < \frac{1}{\tilde{q}}$, we have $\tilde{\gamma}=\frac{d+1}{2}(1/2-1/{\tilde{r}})$ and thus we obtain $$\begin{aligned}
\gamma+\gamma_1+d{\left}(\frac{1}{\tilde{r}'} - \frac{1}{r_1'}{\right})
&=\gamma + \tilde{\gamma} + \gamma_1 -\tilde{\gamma}+d{\left}(\frac{1}{\tilde{r}'} - \frac{1}{r_1'}{\right})
\\
&=\gamma + \tilde{\gamma}+\frac{1}{\tilde{q}} \frac{d-1}{2} {\left}\{ \tilde{q}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}) - q{\left}( \frac{1}{2} - \frac{1}{r}{\right}) {\right}\},\end{aligned}$$ where we use $\frac{1}{\tilde{q}}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) =\frac{1}{q} {\left}( \frac{1}{2}- \frac{1}{r_1}{\right})$ in the last equality.
[**Case(2).**]{} We treat the case of $\frac{1}{\tilde{q}}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) > \frac{1}{q}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right})$. Since $\frac{1}{\tilde{q}}{\left}( \frac{1}{2}- \frac{1}{r}{\right}) > \frac{1}{q}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right})$, there exists $r_2 \in [2,r)$ such that $$\begin{aligned}
\frac{1}{\tilde{q}}{\left}( \frac{1}{2}- \frac{1}{r_2}{\right}) =\frac{1}{q}{\left}( \frac{1}{2}- \frac{1}{\tilde{r}}{\right}).\end{aligned}$$ Let $\gamma_2$ be the derivative loss for the pair $(q,r_2)$. Then, by the Bernstein inequality and Lemma \[lem2.9\], we get $$\begin{aligned}
\norm{\Psi[F]}_{L_t^qL_x^r}
& {\lesssim}N^{d{\left}(\frac{1}{r_2} - \frac{1}{r}{\right})} \norm{\Psi[F]}_{L_t^{q}L_x^{r_2}}
\\
&{\lesssim}N^{d{\left}(\frac{1}{r_2} - \frac{1}{r}{\right})} N^{\gamma_2+\tilde{\gamma}} \norm{P_N F}_{L_t^{\tilde{q}'}L_x^{\tilde{r}'}}.\end{aligned}$$ By the symmetric argument, we get the desired statements.
Proof of the Strichartz estimates {#sec2.3}
---------------------------------
We only show the inequality for ${\mathcal{D}}$ since the similar argument works for $\partial_t {\mathcal{D}}$ and $\partial_t^2 {\mathcal{D}}$. By the integral inequality, we get $$\begin{aligned}
\norm{ {\mathcal{D}}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
\leq \norm{ {\mathcal{D}}_{l}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
+ \norm{ {\mathcal{D}}_{h}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}.\end{aligned}$$ By the assumption of $(q,r)$, we can apply Lemma \[lem2.2\] to the first term as $\tilde{r}=2$ and $\sigma=1$ and Lemma \[lem2.4\] to the second term. Then it follows that $$\begin{aligned}
\norm{ {\mathcal{D}}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
&\leq \norm{ {\mathcal{D}}_{l}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
+ \norm{ {\mathcal{D}}_{h}(t) f}_{L^{q}(I:L^r({{\mathbb R}}^d))}
\\
&{\lesssim}\norm{\jbra{\nabla}^{-1}f}_{L^2} + \norm{|\nabla|^{\gamma} \jbra{\nabla}^{-1}f}_{L^2}
\\
& {\approx}\norm{f}_{H^{\gamma-1}}\end{aligned}$$ This finishes the proof except for the heat end-point case. Next, we show the end-point estimate. First, we prove the following lemma, which is essentially obtained by Watanabe [@Wat17 Lemma 2.8]. However, we give a proof for reader’s convenience. Let $d \geq 3$ and $(q,r)=(2,2d/(d-2))$ from now on in this proof.
\[lem2.11\] Let $d \geq 3$ and $(q,r)=(2,2d/(d-2))$. Then, we have $$\begin{aligned}
\norm{{\mathcal{D}}(t)f}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
&{\lesssim}\norm{f}_{L^2},
\\
\norm{\partial_t {\mathcal{D}}(t)f}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
&{\lesssim}\norm{ \jbra{\nabla}f}_{L^2},
\\
\norm{\partial_{t}^2 {\mathcal{D}}(t)f}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d)) }
&{\lesssim}\norm{\jbra{\nabla}^2 f}_{L^{2}}.\end{aligned}$$
We use the energy method. Let $\phi$ be the solution of the linear equation with the initial data $(\phi(0),\partial_{t} \phi(0))=(\phi_0,\phi_1)$. Multiplying $\partial_{t} \phi$ by the linear equation and integrating it on $[0,t) \times {{\mathbb R}}^{d}$, we get $$\begin{aligned}
\frac{1}{2}{\left}( \norm{\partial_{t} \phi(t)}_{L^{2}}^{2} +\norm{\nabla \phi(t)}_{L^{2}}^{2} {\right})+ \int_{0}^{t} \norm{\partial_{t} \phi(s)}_{L^{2}}^{2} ds
\leq \norm{\phi_{0}}_{H^{1}}^{2} + \norm{\phi_{1}}_{L^{2}}^{2}.\end{aligned}$$ Moreover, multiplying $\phi$ by the linear equation and integrating it on $[0,t) \times {{\mathbb R}}^{d}$, we get $$\begin{aligned}
- \norm{\partial_{t} \phi(t)}_{L^{2}}^{2} + \frac{1}{4} \norm{\phi(t)}_{L^{2}}^{2}
- \int_{0}^{t} \norm{\partial_{t} \phi(s)}_{L^{2}}^{2} ds+ \int_{0}^{t} \norm{\nabla \phi(s)}_{L^{2}}^{2} ds
\leq \norm{\phi_{0}}_{H^{1}}^{2} + \norm{\phi_{1}}_{L^{2}}^{2}.\end{aligned}$$ Combining these estimates, we get $$\begin{aligned}
\norm{\phi(t)}_{H^{1}}^{2}
+\norm{\partial_{t} \phi(t)}_{L^{2}}^{2}
+ \int_{0}^{t} \norm{\partial_{t} \phi(s)}_{L^{2}}^{2} ds
+ \int_{0}^{t} \norm{\partial_{t} \phi(s)}_{L^{2}}^{2} ds
{\lesssim}\norm{\phi_{0}}_{H^{1}}^{2} + \norm{\phi_{1}}_{L^{2}}^{2}.\end{aligned}$$ Now, by the Sobolev embedding and this inequality, we have $$\begin{aligned}
\norm{\phi}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}^{2}
{\lesssim}\norm{\phi}_{L_{t}^{2}\dot{H}_{x}^{1}}^{2}
{\lesssim}\norm{\phi_{0}}_{H^{1}}^{2} + \norm{\phi_{1}}_{L^{2}}^{2}.\end{aligned}$$ Moreover, by differentiating the linear equation by $\partial_{x_i}$ for $i=1,2,\cdots,d$, multiplying $\partial_t \partial_{x_i} \phi$ and $\partial_{x_i} \phi$, and repeating the above argument, we get $$\begin{aligned}
\norm{\partial_{x_i} \phi(t)}_{H^{1}}^{2}
+\norm{\partial_{t} \partial_{x_i} \phi(t)}_{L^{2}}^{2}
&+ \int_{0}^{t} \norm{\partial_{t} \partial_{x_i} \phi(s)}_{L^{2}}^{2} ds
+ \int_{0}^{t} \norm{\partial_{t} \partial_{x_i} \phi(s)}_{L^{2}}^{2} ds
\\
&{\lesssim}\norm{\partial_{x_i} \phi_{0}}_{H^{1}}^{2} + \norm{\partial_{x_i} \phi_{1}}_{L^{2}}^{2}
\\
&{\lesssim}\norm{\phi_{0}}_{H^{2}}^{2} + \norm{\phi_{1}}_{H^{1}}^{2}.\end{aligned}$$ Thus, we can estimate $L_{t}^{2}L_{x}^{\frac{2d}{d-2}}$-norm of the time derivative of the solution as follows. $$\begin{aligned}
\norm{\partial_{t} \phi}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}}}^{2}
{\lesssim}\norm{\partial_{t} \phi}_{L_{t}^{2}\dot{H}_{x}^{1}}^{2}
{\lesssim}\norm{\phi_{0}}_{H^{2}}^{2} + \norm{\phi_{1}}_{H^{1}}^{2}.\end{aligned}$$ Here, ${\mathcal{D}}(t)f$ is the solution of the linear equation with initial data $(\phi_0,\phi_1)= (0,f)$. Therefore, we obtain $$\begin{aligned}
\norm{{\mathcal{D}}(t)f}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}} }
{\lesssim}\norm{f}_{L^{2}}.\end{aligned}$$ Since ${\mathcal{D}}(t)f+\partial_{t} {\mathcal{D}}(t)f$ is the solution of the linear equation with initial data $(\phi_0,\phi_1)= (f,0)$, it follows from the triangle inequality that $$\begin{aligned}
\norm{\partial_{t} {\mathcal{D}}(t)f}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}} }
&\leq \norm{{\mathcal{D}}(t)f+\partial_{t} {\mathcal{D}}(t)f}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}} } +\norm{{\mathcal{D}}(t)f}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}} }
\\
&{\lesssim}\norm{f}_{H^{1}}+\norm{f}_{L^{2}}
{\lesssim}\norm{f}_{H^{1}}.\end{aligned}$$ We can also estimate $\partial_t^2 {\mathcal{D}}(t)f$ as follows since $\partial_t({\mathcal{D}}(t)f+\partial_{t} {\mathcal{D}}(t)f)$ is the time derivative of the solution with the initial data $(\phi_0,\phi_1)= (f,0)$. $$\begin{aligned}
\norm{\partial_{t}^2 {\mathcal{D}}(t)f}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}} }
& \leq \norm{\partial_{t} ({\mathcal{D}}(t)f+\partial_{t} {\mathcal{D}}(t)f)}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}} } +\norm{\partial_{t} {\mathcal{D}}(t)f}_{L_{t}^{2}L_{x}^{\frac{2d}{d-2}} }
\\
&{\lesssim}\norm{f}_{H^{2}}+\norm{f}_{H^{1}}
{\lesssim}\norm{f}_{H^{2}}.\end{aligned}$$ This completes the proof of Lemma \[lem2.11\].
By the first estimate in Lemma \[lem2.11\], we have $$\begin{aligned}
\norm{\jbra{\nabla}^{\sigma} {\mathcal{D}}(t) P_{\leq1}f}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
{\lesssim}\norm{\jbra{\nabla}^{\sigma} P_{\leq1} f}_{L^2} {\lesssim}\norm{f}_{L^2},\end{aligned}$$ for $\sigma\geq 0$. Therefore, it follows from this inequality and Lemma \[lem2.4\] that $$\begin{aligned}
\norm{{\mathcal{D}}(t) f}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
&\leq \norm{{\mathcal{D}}_{l}(t) f}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))} + \norm{{\mathcal{D}}_{h}(t) f}_{L_{t}^{q} (I: L_{x}^{r}({{\mathbb R}}^d))}
\\
&{\lesssim}\norm{\jbra{\nabla}^{-1} f}_{L^{2}} + \norm{|\nabla|^{\gamma} \jbra{\nabla}^{-1}f}_{L^2}
\\
& {\approx}\norm{f}_{H^{\gamma-1}}.\end{aligned}$$ This completes the proof of the heat end-point homogeneous Strichartz estimate.
We only show the inequality for ${\mathcal{D}}$ since the similar argument works for $\partial_t {\mathcal{D}}$. By the integral inequality, we get $$\begin{aligned}
&\norm{\int_{0}^{t} {\mathcal{D}}(t-s) F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
\\
&\leq \norm{\int_{0}^{t} {\mathcal{D}}_{l}(t-s) F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
+\norm{\int_{0}^{t} {\mathcal{D}}_{h}(t-s) F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}\end{aligned}$$ By the assumption of $(q,r)$, we can apply Lemma \[lem2.3\] to the first term as $\tilde{r}=2$ and $\sigma=1$ and Lemmas \[lem2.9\], \[lem2.10\] to the second term. Then it follows that $$\begin{aligned}
&\norm{\int_{0}^{t} {\mathcal{D}}(t-s) F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
\\
&\leq \norm{\int_{0}^{t} {\mathcal{D}}_{l}(t-s) F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
+\norm{\int_{0}^{t} {\mathcal{D}}_{h}(t-s) F(s) ds}_{L^{q}(I:L^{r}({{\mathbb R}}^d))}
\\
&{\lesssim}\norm{\jbra{\nabla}^{-1} F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))}
+ \norm{|\nabla|^{\gamma+\tilde{\gamma}+\delta} \jbra{\nabla}^{-1}F}_{L^{\tilde{q}'}(I:L^{\tilde{r}'}({{\mathbb R}}^d))}
\\
&{\approx}\norm{F}_{L^{\tilde{q}'}(I:W^{\gamma+\tilde{\gamma}+\delta-1,\tilde{r}'}({{\mathbb R}}^d))}\end{aligned}$$ This is the desired estimate.
Well-posedness for the energy critical nonlinear damped wave equation {#sec3}
=====================================================================
In this section, we prove local well-posedness for , Theorem \[thm1.3\], by contraction mapping principle. We define the complete metric space $$\begin{aligned}
X(T,L,M):={\left}\{ v \text{ on } [0,T) \times {{\mathbb R}}^d : \norm{\jbra{\nabla}^{\frac{1}{2}} v }_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))} \leq L, \norm{v}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))} \leq M {\right}\}.\end{aligned}$$
\[rem3.1\] $(q,r)=(2(d+1)/(d-1),2(d+1)/(d-1))$ and $(2(d+1)/(d-2),2(d+1)/(d-2))$ satisfy the assumptions of the Strichartz estimates in Propositions \[prop1.1\] and \[prop1.2\]. Moreover, $\gamma=1/2$ when $(q,r)=(2(d+1)/(d-1),2(d+1)/(d-1))$ and $\gamma=1$ when $(q,r)=(2(d+1)/(d-2),2(d+1)/(d-2))$. We note that these exponents are same as in the local well-posedness for the critical nonlinear wave equation.
We define $$\begin{aligned}
\Phi[u](t)=\Phi_{u_0,u_1}[u](t):= {\mathcal{D}}(t) (u_0+u_1) +\partial_t {\mathcal{D}}(t) u_0 + \int_{0}^{t} {\mathcal{D}}(t-s) {\mathcal{N}}(u(s)) ds.\end{aligned}$$
As stated in Remark \[rem3.1\], the exponents are same as in the argument for the energy critical nonlinear wave equation. Thus, the proof if similar so that we only give sketch of the proof. See [@Pec84; @GSV92; @ShSt93; @KeMe08] for details. Since $(u_0,u_1) \in H^1({{\mathbb R}}^d) \times L^2({{\mathbb R}}^d)$, by the Strichartz estimates in Proposition \[prop1.1\], we obtain $$\begin{aligned}
\label{eq3.1}
&\norm{ {\mathcal{D}}(t) (u_0+u_1) +\partial_t {\mathcal{D}}(t) u_0}_{X(T)}
\\ \notag
&\leq \norm{\jbra{\nabla}^{\frac{1}{2}} {\mathcal{D}}(t) (u_0+u_1) }_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))}
+ \norm{\jbra{\nabla}^{\frac{1}{2}} \partial_t {\mathcal{D}}(t) u_0}_{L_{t,x}^{\frac{2(d+1)}{d-1}}}
\\ \notag
&\quad +\norm{ {\mathcal{D}}(t) (u_0+u_1) }_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}
+\norm{ \partial_t {\mathcal{D}}(t) u_0}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}
\\ \notag
&{\lesssim}\norm{u_0}_{H^1} + \norm{u_1}_{L^2} <A< \infty.\end{aligned}$$ We estimate the nonlinear term as follows. By the Strichartz estimates in Proposition \[prop1.2\] and the fractional Leibnitz rule (see [@KeMe08 Lemma 2.5] and references therein), we get $$\begin{aligned}
\label{eq3.3}
&\norm{\jbra{\nabla}^{\frac{1}{2}} \int_{0}^{t} {\mathcal{D}}(t-s) {\mathcal{N}}(u(s)) ds}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))}
\\ \notag
&\quad {\lesssim}\norm{\jbra{\nabla}^{\frac{1}{2}} {\mathcal{N}}(u)}_{L_{t,x}^{\frac{2(d+1)}{d+3}}([0,T))}
\\ \notag
&\quad {\lesssim}\norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}^{\frac{4}{d-2}}
\norm{\jbra{\nabla}^{\frac{1}{2}} u}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))}\end{aligned}$$ and $$\begin{aligned}
\label{eq3.4}
&\norm{ \int_{0}^{t} {\mathcal{D}}(t-s) {\mathcal{N}}(u(s)) ds}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}
\\ \notag
& \quad {\lesssim}\norm{\jbra{\nabla}^{\frac{1}{2}} {\mathcal{N}}(u)}_{L_{t,x}^{\frac{2(d+1)}{d+3}}([0,T))}
\\ \notag
&\quad {\lesssim}\norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}^{\frac{4}{d-2}}
\norm{\jbra{\nabla}^{\frac{1}{2}} u}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))}.\end{aligned}$$ Combining and , we obtain $$\begin{aligned}
\norm{\jbra{\nabla}^{\frac{1}{2}} \Phi[u]}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))}
&\leq \norm{\jbra{\nabla}^{\frac{1}{2}} {\mathcal{D}}(t) (u_0+u_1) +\partial_t {\mathcal{D}}(t) u_0}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))}
\\
&\quad+ \norm{\jbra{\nabla}^{\frac{1}{2}} \int_{0}^{t} {\mathcal{D}}(t-s) {\mathcal{N}}(u(s)) ds}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))}
\\
&\leq CA+ C L M^{\frac{4}{d-2}}
\\
&\leq L\end{aligned}$$ if we choose $L=2CA$ and $M$ such that $CM^{4/(d-2)} \leq 1/2$. By and , we get $$\begin{aligned}
\norm{\Phi[u]}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}
&\leq \norm{{\mathcal{D}}(t) (u_0+u_1) +\partial_t {\mathcal{D}}(t) u_0}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}
\\
&\quad+ \norm{ \int_{0}^{t} {\mathcal{D}}(t-s) {\mathcal{N}}(u(s)) ds}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}
\\
&\leq \delta + C L M^{\frac{4}{d-2}}
\\
&\leq M\end{aligned}$$ if we choose $\delta = M/2$ and $ L \leq(2C)^{-1}M^{(d-6)/(d-2)}$ (which is possible if $d \leq5$). Thus, $\Phi$ is a mapping on $X(T,L,M)$. $$\begin{aligned}
&\norm{\jbra{\nabla}^{\frac{1}{2}} (\Phi[u]-\Phi[v])}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))}
+\norm{\Phi[u]-\Phi[v]}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}
\\
& {\lesssim}\norm{\jbra{\nabla}^{\frac{1}{2}} ({\mathcal{N}}(u)-{\mathcal{N}}(v))}_{L_{t,x}^{\frac{2(d+1)}{d+3}}([0,T))}
\\
& {\lesssim}{\left}( \norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}^{\frac{4}{d-2}} + \norm{v}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}^{\frac{4}{d-2}} {\right})
\norm{\jbra{\nabla}^{\frac{1}{2}} ( u-v)}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))}
\\
&\quad +{\left}( \norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}^{\frac{6-d}{d-2}} + \norm{v}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}^{\frac{6-d}{d-2}} {\right})
\\
&\quad \quad \times {\left}( \norm{\jbra{\nabla}^{\frac{1}{2}} u}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))} +\norm{\jbra{\nabla}^{\frac{1}{2}} v}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))} {\right})
\\
&\quad \quad \times \norm{u-v}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}
\\
& \leq C M^{\frac{4}{d-2}} \norm{\jbra{\nabla}^{\frac{1}{2}} ( u-v)}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,T))}
+ CM^{\frac{6-d}{d-2}} L \norm{u-v}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([0,T))}. \end{aligned}$$ Taking $L$ and $M$ sufficiently small, $\Phi$ is a contraction mapping on $X(T,L,M)$. By the Banach fixed point theorem, we obtain the solution such that $u=\Phi[u]$. Then, $(u,\partial_t u)$ belongs to $C([0,T); H^1({{\mathbb R}}^d) \times L^2({{\mathbb R}}^d))$ because of the Strichartz estimates (Proposition \[prop1.1\] and \[prop1.2\]) and the nonlinear estimates (for example $\jbra{\nabla}^{\frac{1}{2}} {\mathcal{N}}(u) \in L_{t,x}^{\frac{2(d+1)}{d+3}}$). We give a proof of the standard blow-up criterion. We suppose that $T_{+}=T_{+}(u_0,u_1)<\infty$ and $\norm{u}_{L_{t,x}^{2(d+1)/d-2}([0,T_{+}))}<\infty$. Take $\tau$ and $T$ arbitrary such that $0<\tau<T<T_{+}$. By the Duhamel formula, we have $$\begin{aligned}
u(t) = {\mathcal{D}}(t-\tau) (u(\tau)+\partial_t u(\tau)) +\partial_t {\mathcal{D}}(t-\tau) u(\tau) + \int_{\tau}^{t} {\mathcal{D}}(t-s) {\mathcal{N}}(u(s)) ds,\end{aligned}$$ for $t>\tau$. By the Strichartz estimates, we obtain $$\begin{aligned}
& \norm{\jbra{\nabla}^{\frac{1}{2}} u}_{L_{t,x}^{\frac{2(d+1)}{d-1}}((\tau,T))}
\\
&{\lesssim}\norm{ (u(\tau),\partial_t u(\tau))}_{H^1 \times L^2}
+ \norm{ \int_{\tau}^{t} \jbra{\nabla}^{\frac{1}{2}} {\mathcal{D}}(t-s) {\mathcal{N}}(u(s)) ds}_{L_{t,x}^{\frac{2(d+1)}{d-1}}((\tau,T))}
\\
&{\lesssim}\norm{(u(\tau),\partial_t u(\tau))}_{H^1\times L^2}+ \norm{ \jbra{\nabla}^{\frac{1}{2}} {\mathcal{N}}(u(s)) }_{L_{t,x}^{\frac{2(d+1)}{d+3}}((\tau,T))}
\\
&{\lesssim}\norm{(u(\tau),\partial_t u(\tau))}_{H^1\times L^2}
+ \norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}((\tau,T))}^{\frac{4}{d-2}} \norm{ \jbra{\nabla}^{\frac{1}{2}} u }_{L_{t,x}^{\frac{2(d+1)}{d-1}}((\tau,T))}.\end{aligned}$$ Since $\norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}((\tau,T))} \ll 1$ for $\tau$ close to $T_{+}$, we obtain $$\begin{aligned}
\norm{ \jbra{\nabla}^{\frac{1}{2}} u }_{L_{t,x}^{\frac{2(d+1)}{d-1}}((\tau,T))}
{\lesssim}\norm{(u(\tau),\partial_t u(\tau))}_{H^1\times L^2}.\end{aligned}$$ Fix such $\tau$. Since $T$ is arbitrary, we get $$\begin{aligned}
\label{eq3.5}
\norm{ \jbra{\nabla}^{\frac{1}{2}} u }_{L_{t,x}^{\frac{2(d+1)}{d-1}}((\tau,T_{+}))}
{\lesssim}\norm{(u(\tau),\partial_t u(\tau))}_{H^1\times L^2}.\end{aligned}$$ Take a sequence $\{t_n\}$ such that $t_n \to T_{+}$ and $t_n >\tau$. Then, by the integral formula, the Strichartz estimates the assumption, and \[eq3.5\], we have $$\begin{aligned}
&\norm{ {\mathcal{D}}(t-t_n) (u(t_n)+\partial_t u(t_n)) +\partial_t {\mathcal{D}}(t-t_n) u(t_n)}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([t_n,T_{+}))}
\\
&{\lesssim}\norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([t_n,T_{+}))}
+ \norm{ \int_{t_n}^{t} {\mathcal{D}}(t-s) {\mathcal{N}}(u(s)) ds}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([t_n,T_{+}))}
\\
&{\lesssim}\norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([t_n,T_{+}))}
+ \norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}([t_n,T_{+}))}^{\frac{4}{d-2}}
\norm{\jbra{\nabla}^{\frac{1}{2}} u}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([t_n,T_{+}))}
\\
& \to 0 \text{ as } n \to \infty,\end{aligned}$$ Thus, $\norm{ {\mathcal{D}}(t-t_n) (u(t_n)+\partial_t u(t_n)) +\partial_t {\mathcal{D}}(t-t_n) u(t_n)}_{L_{t,x}^{2(d+1)/(d-2)}([t_n,T_{+}))}<\delta/2$ is true for large $n$. Then, for some ${\varepsilon}>0$, we get $$\begin{aligned}
\norm{ {\mathcal{D}}(t-t_n) (u(t_n)+\partial_t u(t_n)) +\partial_t {\mathcal{D}}(t-t_n) u(t_n)}_{L_{t,x}^{2(d+1)/(d-2)}([t_n,T_{+}+{\varepsilon}))}<\delta.\end{aligned}$$ The local well-posedness derives a contradiction.
Decay of global solution with finite Strichartz norm {#sec4}
====================================================
In this section, we give a proof of Theorem \[thm1.4\].
If $u$ is a global solution of with $\| u \|_{L_{t,x}^{2(d+1)/(d-2)}([0,\infty))}<\infty$, then $u$ satisfies $$\begin{aligned}
\norm{\jbra{\nabla}^{\frac{1}{2}} u}_{L_{t,x}^{\frac{2(d+1)}{d-1}}([0,\infty))}<\infty\end{aligned}$$
The proof is very similar to the proof of the standard blow-up criterion. Take $0 < \tau < T <\infty$ arbitrary. We know that the global solution belongs to $L_{t,x}^{\frac{2(d+1)}{d-1}}(K)$ for any compact interval $K \subset [0,\infty)$. It follows from the Duhamel’s formula and the Strichartz estimates that $$\begin{aligned}
\norm{\jbra{\nabla}^{\frac{1}{2}} u}_{L_{t,x}^{\frac{2(d+1)}{d-1}}((\tau,T))}
&{\lesssim}\norm{(u(\tau),\partial_t u(\tau))}_{H^1\times L^2}
\\
&\quad + \norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}((\tau,T))}^{\frac{4}{d-2}} \norm{ \jbra{\nabla}^{\frac{1}{2}} u }_{L_{t,x}^{\frac{2(d+1)}{d-1}}((\tau,T))}.\end{aligned}$$ Since $\norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}((\tau,T))} \ll 1$ for large $\tau$, we obtain $$\begin{aligned}
\norm{ \jbra{\nabla}^{\frac{1}{2}} u }_{L_{t,x}^{\frac{2(d+1)}{d-1}}((\tau,T))}
{\lesssim}\norm{(u(\tau),\partial_t u(\tau))}_{H^1\times L^2}\end{aligned}$$ for large $\tau>0$. Fix such $\tau$. Since $T$ is arbitrary, we get $\norm{ \jbra{\nabla}^{\frac{1}{2}} u }_{L_{t,x}^{2(d+1)/(d-1)}((\tau,\infty))}{\lesssim}\norm{(u(\tau),\partial_t u(\tau))}_{H^1\times L^2}$. Thus, we obtain $\norm{ \jbra{\nabla}^{\frac{1}{2}} u }_{L_{t,x}^{2(d+1)/(d-1)}([0,\infty))}<\infty$.
We have $$\begin{aligned}
\begin{pmatrix}
u
\\
\partial_t u
\end{pmatrix}
={\mathcal{A}}(t)
\begin{pmatrix}
u_0
\\
u_1
\end{pmatrix}
+\int_{0}^{t}
{\mathcal{A}}(t-s)
\begin{pmatrix}
0
\\
{\mathcal{N}}(u(s))
\end{pmatrix}
ds,\end{aligned}$$ where $$\begin{aligned}
{\mathcal{A}}(t)=
\begin{pmatrix}
{\mathcal{D}}(t)+\partial_t {\mathcal{D}}(t) & {\mathcal{D}}(t)
\\
\partial_t {\mathcal{D}}(t)+\partial_t^2 {\mathcal{D}}(t) &\partial_t {\mathcal{D}}(t)
\end{pmatrix}.\end{aligned}$$ We set $$\begin{aligned}
I
&:={\mathcal{A}}(t)
\begin{pmatrix}
u_0
\\
u_1
\end{pmatrix},
\\
I\!\!I
&:= \int_{0}^{\tau}
{\mathcal{A}}(t-s)
\begin{pmatrix}
0
\\
{\mathcal{N}}(u(s))
\end{pmatrix}
ds,
\\
I\!\!I\!\!I
&:= \int_{\tau}^{t}
{\mathcal{A}}(t-s)
\begin{pmatrix}
0
\\
{\mathcal{N}}(u(s))
\end{pmatrix}
ds.\end{aligned}$$ We begin with the estimate of $I$. Approximating $(u_0,u_1)$ by $(\psi_0,\psi_1) \in (C_{0}^{\infty}({{\mathbb R}}^d))^2$ in $H^1({{\mathbb R}}^d) \times L^2({{\mathbb R}}^d)$, we obtain $$\begin{aligned}
\norm{I}_{H^1 \times L^2}
&= \norm{{\mathcal{A}}(t) (u_0,u_1)^{T}}_{H^1 \times L^2}
\\
&\leq \norm{{\mathcal{A}}(t) \{(u_0,u_1)- (\psi_0,\psi_1)\}^{T}}_{H^1 \times L^2}
+ \norm{{\mathcal{A}}(t) (\psi_0,\psi_1)^{T}}_{H^1 \times L^2},\end{aligned}$$ where $^{T}$ denotes transposition. By [@IIOWp Theorem 1.1], we have the following $L^p$-$L^q$ type estimates: $$\begin{aligned}
\norm{{\mathcal{D}}(t)f}_{H^1}
&{\lesssim}\jbra{t}^{-\frac{d}{2} {\left}( \frac{1}{q}-\frac{1}{2}{\right})} \norm{f}_{L^q}+ e^{-\frac{t}{2}} \jbra{t}^{\delta} \norm{f}_{L^2},
\\
\norm{\partial_t {\mathcal{D}}(t)f}_{L^2}
&{\lesssim}\jbra{t}^{-\frac{d}{2} {\left}( \frac{1}{q}-\frac{1}{2}{\right})-1} \norm{f}_{L^q}+ e^{-\frac{t}{2}} \jbra{t}^{\delta} \norm{f}_{L^2},
\\
\norm{\partial_t {\mathcal{D}}(t)f}_{H^1}
&{\lesssim}\jbra{t}^{-\frac{d}{2} {\left}( \frac{1}{q}-\frac{1}{2}{\right})-1} \norm{f}_{W^{1,q}}+ e^{-\frac{t}{2}} \jbra{t}^{\delta} \norm{f}_{H^1},
\\
\norm{\partial_t^2 {\mathcal{D}}(t)f}_{L^2}
&{\lesssim}\jbra{t}^{-\frac{d}{2} {\left}( \frac{1}{q}-\frac{1}{2}{\right})-2} \norm{f}_{L^q}+ e^{-\frac{t}{2}} \jbra{t}^{\delta} \norm{\jbra{\nabla}f}_{L^2},\end{aligned}$$ for any $q \in [1,2]$ and some $\delta>0$. Therefore, applying these as $q=2$, we get $$\begin{aligned}
\norm{{\mathcal{A}}(t) \{(u_0,u_1)- (\psi_0,\psi_1)\}^{T}}_{H^1 \times L^2} {\lesssim}\norm{(u_0,u_1)- (\psi_0,\psi_1)}_{H^1\times L^2}.\end{aligned}$$ Thus, this can be made arbitrary small by the approximation. Applying the above $L^p$-$L^q$ type estimates as $q=1$, we obtain $$\begin{aligned}
\norm{{\mathcal{A}}(t) (\psi_0,\psi_1)^{T}}_{H^1 \times L^2}
&\leq \norm{{\mathcal{D}}(t) (\psi_0 +\psi_1)}_{H^1} + \norm{\partial_t {\mathcal{D}}(t) \psi_0 }_{H^1}
\\
&\quad + \norm{\partial_t {\mathcal{D}}(t) (\psi_0 +\psi_1)}_{L^2} + \norm{\partial_t^2 {\mathcal{D}}(t) \psi_0 }_{L^2}
\\
&{\lesssim}\jbra{t}^{-\frac{d}{4}} ( \norm{\psi_0}_{W^{1,1}}+\norm{\psi_1}_{L^1} )
+ e^{-\frac{t}{4}} (\norm{\psi_0}_{H^1}+ \norm{\psi_1}_{L^2} )
\\
&\to 0 \text{ as } t \to \infty. \end{aligned}$$ Next, we consider the estimate of $I\!\!I\!\!I$. By the Strichartz estimates, we have $$\begin{aligned}
\label{eq5.1}
\norm{I\!\!I\!\!I}_{H^1 \times L^2}
&= \norm{\jbra{\nabla} \int_{\tau}^{t} {\mathcal{D}}(t-s) {\mathcal{N}}(u(s)) ds}_{L^2}
\\ \notag
&\quad +\norm{ \int_{\tau}^{t} \partial_t {\mathcal{D}}(t-s) {\mathcal{N}}(u(s)) ds}_{L^2}
\\ \notag
&{\lesssim}\norm{\jbra{\nabla}^{\frac{1}{2}} {\mathcal{N}}(u)}_{L_{t,x}^{\frac{2(d+1)}{d+3}}((\tau,t))}
\\ \notag
&{\lesssim}\norm{u}_{L_{t,x}^{\frac{2(d+1)}{d-2}}((\tau,t))}^{\frac{4}{d-2}}
\norm{\jbra{\nabla}^{\frac{1}{2}} u}_{L_{t,x}^{\frac{2(d+1)}{d-1}}((\tau,t))}.\end{aligned}$$ Therefore, the term is arbitrary small taking $\tau$ sufficiently close to $t$. At Last, we calculate $I\!\!I$. We note that $$\begin{aligned}
I\!\!I
= \int_{0}^{\tau}
{\mathcal{A}}(t-s)
\begin{pmatrix}
0
\\
{\mathcal{N}}(u(s))
\end{pmatrix}
ds
=
{\mathcal{A}}(t-\tau)
\int_{0}^{\tau}
{\mathcal{A}}(\tau-s)
\begin{pmatrix}
0
\\
{\mathcal{N}}(u(s))
\end{pmatrix}
ds.\end{aligned}$$ Since by we know $$\begin{aligned}
\int_{0}^{\tau} {\mathcal{A}}(\tau-s) \begin{pmatrix} 0 \\ {\mathcal{N}}(u(s)) \end{pmatrix}\in H^1({{\mathbb R}}^d) \times L^2({{\mathbb R}}^d),\end{aligned}$$ approximating it by $\vec{\psi} \in (C_{0}^{\infty}({{\mathbb R}}^d)^2)$, we obtain $$\begin{aligned}
\norm{I\!\!I}_{H^1 \times L^2}
&\leq
\norm{
{\mathcal{A}}(t-\tau)
{\left}\{
\int_{0}^{\tau}
{\mathcal{A}}(\tau-s)
\begin{pmatrix}
0
\\
{\mathcal{N}}(u(s))
\end{pmatrix}
ds
- \vec{\psi}
{\right}\}
}_{H^1 \times L^2}
\\
& \quad +
\norm{{\mathcal{A}}(t-\tau) \vec{\psi}}_{H^1 \times L^2}.\end{aligned}$$ In the smae way as $I$, the first term is arbitrary small by the approximation and the second term tends to $0$ as $t \to \infty$. Combining the estimates of $I$, $I\!\!I$, and $I\!\!I\!\!I$, we get the decay.
Blow-up result {#sec5}
==============
This section is devoted to the proof of the blow-up result. The proof is essentially given by Ohta [@Oht97]. However, we give the full proof for the reader’s convenience. In [@Oht97], Ohta used argument of ordinary differential inequality instead of concavity argument. Indeed, the following lemma was used in [@Oht97]. See Li–Zhou [@LiZh95] and Souplet [@Sou95] for the proof.
\[lem5.1\] Let $h$ satisfy $$\begin{aligned}
{\left}\{
\begin{array}{ll}
h''(t) + h(t) \geq C h^\gamma(t), & t>0,
\\
h(0)>0, \quad h'(0)>0,
\end{array}
{\right}.\end{aligned}$$ for some constant $C>0$ and $\gamma>1$. Then $h$ can not exist for all $t>0$.
Setting $$\begin{aligned}
I(t):=\frac{1}{2} \norm{u(t)}_{L^2}^2, \end{aligned}$$ we will prove that $I$ satisfies the ordinary differential inequality for large $t$.
First, we show the positivity of $K$ near $0$.
Let $\{u_n\}_{n \in {{\mathbb N}}} \subset H^1({{\mathbb R}}^d)$ satisfy $u_n \to 0$ strongly in $H^1({{\mathbb R}}^d)$. Then, for large $n \in {{\mathbb N}}$, we have $$\begin{aligned}
K(u_n) \geq 0. \end{aligned}$$
By the Sobolev inequality, we get $$\begin{aligned}
K(u_n) &= \norm{\nabla u_n}_{L^2}^2 - \norm{u_n}_{L^{\frac{2d}{d-2}}}^{\frac{2d}{d-2}}
\\
&\geq \norm{\nabla u_n}_{L^2}^2 {\left}(1- C \norm{\nabla u_n}_{L^2}^{\frac{4}{d-2}}{\right}). \end{aligned}$$ Since $u_n \to 0$ strongly in $H^1({{\mathbb R}}^d)$, we have $K(u_n) \geq \frac{1}{2} \norm{\nabla u_n}_{L^2}^2 \geq 0$ for large $n$.
Secondly, we prove that the set ${\mathscr{B}}$ is invariant under the flow.
Let $(u_0,u_1)$ belong to ${\mathscr{B}}$. Then the solution $(u(t),\partial_t u(t))$ of belongs to ${\mathscr{B}}$ for all existence time $t \in [0,T_{\max})$.
Since the energy $E$ satisfies that $$\begin{aligned}
\label{eq6.1}
\frac{d}{dt} E(u(t),\partial_t u(t)) =- \norm{\partial_t u(t)}_{L^2}^2, \text{ for all } t \in (0,T_{\max}),\end{aligned}$$ we have $E(u(t), \partial_t u(t)) \leq E(u_0,u_1) \leq \mu$ for all $t \in [0,T_{\max})$. Thus, it is enough to prove $K(u(t)) < 0$ for all $t \in [0,T_{\max})$. We suppose that there exists a time $t_0 \in (0,T_{\max})$ such that $K(u(t_0))>0$. Then, by the continuity of the flow, there exists $t_* \in (0,T_{\max})$ such that $K(u(t_*))=0$ and $K(u(t))<0$ for $t \in (0,t_*)$. Assume that $u(t_*) \neq 0$. Then, by the definition of the minimizing problem $\mu$, we have $$\begin{aligned}
\mu \leq J(u(t_*)) \leq E(u(t_*),\partial_t u(t_*)) \leq E(u_0,u_1) < \mu.\end{aligned}$$ This is a contradiction. Therefore, we get $u(t_*)=0$. Since the flow is continuous, if we take $\{t_n\} \subset (0,t_*)$ such that $t_n \to t_*$, then $u(t_n) \to u(t_*)=0$ strongly in $H^1({{\mathbb R}}^d)$. By the positivity of $K$ near $0$, we have $K(u(t_n))\geq 0$ and $t_n \in (0,t_*)$ for large $n \in {{\mathbb N}}$. This contradicts $K(u(t))<0$ for all $t \in (0,t_*)$. Therefore, we get $K(u(t)) < 0$ for all $t \in [0,T_{\max})$.
We define $$\begin{aligned}
{\mathscr{G}}:= \{ (u_0,u_1) \in H^1({{\mathbb R}}^d) \times L^2({{\mathbb R}}^d): E(u_0,u_1) < \mu, K(u_0)\geq 0\}.\end{aligned}$$ If $(u_0,u_1)$ belongs to ${\mathscr{G}}$, then the solution $(u(t),\partial_t u(t))$ belongs to ${\mathscr{G}}$ for all existence time $t \in [0,T_{\max})$. This can be proved in the similar argument to the above.
We set $$\begin{aligned}
H(\varphi):= J(\varphi) - \frac{d-2}{2d} K(\varphi)
=\frac{1}{d} \norm{\nabla \varphi}_{L^2}^2. \end{aligned}$$
We have $$\begin{aligned}
\mu= \inf {\left}\{H(\varphi) : \varphi \in H^1({{\mathbb R}}^d) \setminus \{0\}, K(\varphi) \leq 0{\right}\}. \end{aligned}$$
We denote the right hand side by $\mu'$. It is trivial that $\mu \geq \mu'$. We prove $\mu \leq \mu'$. If $\varphi \in H^1({{\mathbb R}}^d)\setminus\{0\}$ satisfies $K(\varphi)=0$, it follows that $$\begin{aligned}
\mu \leq J(\varphi) = H(\varphi).\end{aligned}$$ If $\varphi \in H^1({{\mathbb R}}^d)\setminus\{0\}$ satisfies $K(\varphi)<0$, there exists $\lambda_0 \in (0,1)$ such that $K(\lambda_0 \varphi)=0$. Thus, we have $$\begin{aligned}
\mu \leq J(\lambda_0 \varphi) = H(\lambda_0 \varphi) < H(\varphi).\end{aligned}$$ Therefore, we obtain $\mu \leq H(\varphi)$ for any $\varphi \in H^1({{\mathbb R}}^d)\setminus\{0\}$. Take the infimum, we get $\mu \leq \mu'$.
\[lem5.5\] Let $u(t)$ be a solution to on $[0,T)$ with $(u_0,u_1) \in {\mathscr{B}}$. Then, we have $$\begin{aligned}
I''(t)+ I'(t) \geq {\left}(1+\frac{d}{d-2}{\right}) \norm{\partial_t u(t)}_{L^2}^2 + \frac{2d}{d-2} {\left}(\mu - E(u(t),\partial_t u(t)){\right})\end{aligned}$$ for all $t \in (0,T)$.
By direct calculations and the equation, we have $$\begin{aligned}
I'(t)&=\operatorname{Re}\tbra{u(t)}{\partial_t u(t)}_{L^2},
\\
I''(t)&=\norm{\partial_t u(t)}_{L^2}^2 +\operatorname{Re}\tbra{u(t)}{\partial_t^2 u(t)}_{L^2}
\\
&=\norm{\partial_t u(t)}_{L^2}^2 - \norm{\nabla u(t)}_{L^2}^2 - \operatorname{Re}\tbra{u(t)}{\partial_t u(t)}_{L^2} + \norm{u(t)}_{L^{\frac{2d}{d-2}}}^{\frac{2d}{d-2}},\end{aligned}$$ where $\tbra{u}{v}:= \int_{{{\mathbb R}}^d} u(x) \overline{v(x)} dx$ Therefore, we obtain $$\begin{aligned}
I''(t) + I'(t) = \norm{\partial_t u(t)}_{L^2}^2 - K(u(t)). \end{aligned}$$ Since $K(\varphi)=\frac{2d}{d-2} (J(\varphi) - H(\varphi))=\frac{2d}{d-2} (E(\varphi,\psi) - \frac{1}{2}\norm{\psi}_{L^2}^2 - H(\varphi)) $, it follows that $$\begin{aligned}
I''(t) + I'(t)
= {\left}( 1+ \frac{d}{d-2} {\right}) \norm{\partial_t u(t)}_{L^2}^2 + \frac{2d}{d-2} {\left}( H(u(t))- E(u(t), \partial_t u(t)) {\right})\end{aligned}$$ Since we have $u(t) \in {\mathscr{B}}$ for all $t \in [0,T_{\max})$ by the above lemma, we get $H(u(t)) \geq \mu$ by the above lemma. Thus, it follows that $$\begin{aligned}
I''(t) + I'(t)
\geq {\left}( 1+ \frac{d}{d-2} {\right}) \norm{\partial_t u(t)}_{L^2}^2 + \frac{2d}{d-2} {\left}( \mu - E(u(t), \partial_t u(t)){\right}). \end{aligned}$$
Assume that $u$ is a solution to on $[0,\infty)$ with $(u_0,u_1) \in {\mathscr{B}}$. Then, there exists $t_1>0$ such that $I(t)>0$, $I'(t)>0$, and $$\begin{aligned}
\frac{d}{dt} {\left}( \frac{E(u(t),\partial_t u(t)) - \mu }{I^{\frac{d-1}{d-2}}(t)}{\right}) \leq 0\end{aligned}$$ for all $t \in (t_1,\infty)$.
We set $E(t):=E(u(t),\partial_t u(t))$ for simplicity. We define $$\begin{aligned}
F(t):= I'(t) + {\left}(1+ \frac{d}{d-2}{\right}) {\left}( E(t) - \mu {\right}).\end{aligned}$$ It is follows from the energy and the lemma that $$\begin{aligned}
F'(t)
&=I''(t) + {\left}(1+ \frac{d}{d-2}{\right}) E'(t)
\\
&=I''(t) - {\left}(1+ \frac{d}{d-2}{\right}) \norm{\partial_t u(t)}_{L^2}^2
\\
&\geq -I'(t) + \frac{2d}{d-2} {\left}( \mu - E(t){\right})
\\
&= -F(t)+ \frac{2}{d-2} {\left}( \mu - E(t){\right})\end{aligned}$$ Therefore, we have $$\begin{aligned}
(e^{t} F(t))'
&= e^{t} F(t) + e^{t} F'(t)
\\
&\geq e^{t} \frac{2}{d-2} {\left}( \mu - E(t){\right})
\\
& \geq \frac{2}{d-2} {\left}( \mu - E(0){\right}) e^{t}\end{aligned}$$ Integrating this on $[0,t]$, we obtain $$\begin{aligned}
F(t) \geq F(0) e^{-t} + \frac{2}{d-2} {\left}( \mu - E(0){\right}) - \frac{2}{d-2} {\left}( \mu - E(0){\right}) e^{-t}.\end{aligned}$$ Thus there exists $t_0 >0$ such that $$\begin{aligned}
F(t) >0 \text{ for all } t \geq t_0,\end{aligned}$$ since $ \mu > E(u_0, u_1)$. This implies that, for all $t \geq t_0$, $$\begin{aligned}
\label{eq6.2}
I'(t)
&\geq - {\left}(1+ \frac{d}{d-2}{\right}) {\left}( E(t) - \mu {\right})
\\ \notag
&= {\left}(1+ \frac{d}{d-2}{\right}) {\left}( \mu - E(t){\right})
\\ \notag
&\geq {\left}(1+ \frac{d}{d-2}{\right}) {\left}( \mu - E(0){\right})
\\ \notag
&>0.\end{aligned}$$ Therefore, there exists $t_1 \geq t_0$ such that $$\begin{aligned}
I(t)>0 \text{ for all } t \geq t_1. \end{aligned}$$ For $t \geq t_1$, we have $$\begin{aligned}
\label{eq6.3}
\frac{d}{dt} {\left}( \frac{E(t) - \mu }{I^{\frac{d-1}{d-2}}(t)}{\right})
= \frac{E'(t) I(t) - \frac{d-1}{d-2} I'(t) (E(t) - \mu )}{I^{\frac{2d-3}{d-2}}(t)}.\end{aligned}$$ By and , we obtain $$\begin{aligned}
&E'(t) I(t) - \frac{d-1}{d-2} I'(t) (E(t) - \mu )
\\
&\leq - \norm{\partial_t u(t)}_{L^2}^2 I(t) - \frac{d-1}{d-2} I'(t) (E(t) - \mu )
\\
&\leq -2 \norm{\partial_t u(t)}_{L^2}^2 \norm{u(t)}_{L^2}^2 + \frac{d-1}{d-2} {\left}(1+ \frac{d}{d-2}{\right}) (I'(t))^{2}
\\
&= -2 \norm{\partial_t u(t)}_{L^2}^2 \norm{u(t)}_{L^2}^2 +2 (I'(t))^{2}
\\
&\leq 0, \end{aligned}$$ where we used the Cauchy–Schwarz inequality $I'(t) \leq \| u(t) \|_{L^2} \| \partial_t u(t) \|_{L^2}$ in the last inequality. This and implies the statement.
We suppose that the solution $u$ to with $(u_0,u_1) \in {\mathscr{B}}$ exists globally in time. Then, we have $$\begin{aligned}
I(t)>0,
\quad I'(t)>0,
\quad \mu- E(t) \geq C I^{\frac{d-1}{d-2}}(t)\end{aligned}$$ for large $t\geq t_1$, where we set $C=(\mu-E(t_1))/I^{\frac{d-1}{d-2}}(t_1)>0$ and we recall $E(t)=E(u(t),\partial_t u(t))$. By Lemma \[lem5.5\], we get $$\begin{aligned}
I''(t) + I'(t)
&\geq {\left}(1+\frac{d}{d-2}{\right}) \norm{\partial_t u(t)}_{L^2}^2 + \frac{2d}{d-2} {\left}(\mu - E(t){\right})
\\
&\geq \frac{2d}{d-2} {\left}(\mu - E(t){\right})
\\
&\geq C I^{\frac{d-1}{d-2}}(t),\end{aligned}$$ for $t \geq t_1$. We also have $I(t_1)>0$ and $I'(t_1)>0$. Thus, by Lemma \[lem5.1\], $I(t)$ can not exist globally. This contradicts the assumption that the solution $u$ is global. Therefore, we get the statement.
The author would like to express deep appreciation to Professor Masahito Ohta and Professor Yuta Wakasugi for many useful suggestions, valuable comments and warm-hearted encouragement. The author was partially supported by JSPS Grant-in-Aid for Early-Career Scientists JP18K13444.
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|
---
abstract: 'Gibbs sampling is a Markov chain Monte Carlo method that is often used for learning and inference on graphical models. Minibatching, in which a small random subset of the graph is used at each iteration, can help make Gibbs sampling scale to large graphical models by reducing its computational cost. In this paper, we propose a new auxiliary-variable minibatched Gibbs sampling method, [*Poisson-minibatching Gibbs*]{}, which both produces unbiased samples and has a theoretical guarantee on its convergence rate. In comparison to previous minibatched Gibbs algorithms, Poisson-minibatching Gibbs supports fast sampling from continuous state spaces and avoids the need for a Metropolis-Hastings correction on discrete state spaces. We demonstrate the effectiveness of our method on multiple applications and in comparison with both plain Gibbs and previous minibatched methods.'
author:
- |
Ruqi Zhang\
Cornell University\
`rz297@cornell.edu`\
Christopher De Sa\
Cornell University\
`cdesa@cs.cornell.edu`\
bibliography:
- 'neurips\_2019.bib'
title: 'Poisson-Minibatching for Gibbs Sampling with Convergence Rate Guarantees'
---
Introduction
============
Gibbs sampling is a Markov chain Monte Carlo (MCMC) method which is widely used for inference on graphical models [@koller2009probabilistic]. Gibbs sampling works by iteratively resampling a variable from its conditional distribution with the remaining variables fixed. Although Gibbs sampling is a powerful method, its utility can be limited by its computational cost when the model is large. One way to address this is to use *stochastic methods*, which use a subsample of the dataset or model—called a minibatch—to approximate the dataset or model used in an MCMC algorithm. Minibatched variants of many classical MCMC algorithms have been explored [@welling2011bayesian; @maclaurin2014firefly; @de2018minibatch; @li2017mini], including the MIN-Gibbs algorithm for Gibbs sampling [@de2018minibatch].
In this paper, we propose a new minibatched variant of Gibbs sampling on factor graphs called [*Poisson-minibatching Gibbs*]{} (Poisson-Gibbs). Like other minibatched MCMC methods, Poisson-minibatching Gibbs improves Gibbs sampling by reducing its computational cost. In comparison to prior work, our method improves upon MIN-Gibbs in two ways. First, it eliminates the need for a potentially expensive Metropolis-Hastings (M-H) acceptance step, giving it a better asymptotic per-iteration time complexity than MIN-Gibbs. Poisson-minibatching Gibbs is able to do this by choosing a minibatch in a way that depends on the current state of the variables, rather than choosing one that is independent of the current state as is usually done in stochastic algorithms. We show that such state-dependent minibatches can still be sampled quickly, and that an appropriately chosen state-dependent minibatch can result in a reversible Markov chain with the correct stationary distribution even without a Metropolis-Hastings correction step.
The second way that our method improves upon previous work is that it supports sampling over continuous state spaces, which are common in machine learning applications (in comparison, the previous work only supported sampling over discrete state spaces). The main difficulty here for Gibbs sampling is that resampling a continuous-valued variable from its conditional distribution requires sampling from a continuous distribution, and this is a nontrivial task (as compared with a discrete random variable, which can be sampled from by explicitly computing its probability mass function). Our approach is based on fast inverse transform sampling method, which works by approximating the probability density function (PDF) of a distribution with a polynomial [@olver2013fast].
In addition to these two new capabilities, we prove bounds on the convergence rate of Poisson-minibatching Gibbs in comparison to plain (i.e. not minibatched) Gibbs sampling. These bounds can provide a recipe for how to set the minibatch size in order to come close to the convergence rate of plain Gibbs sampling. If we set the minibatch size in this way, we can derive expressions for the per-iteration computational cost of our method compared with others; these bounds are summarized in Table \[tab:cost\]. In summary, the contributions of this paper are as follows:
- We introduce Poisson-minibatching Gibbs, a variant of Gibbs sampling which can reduce computational cost without adding bias or needing a Metropolis-Hastings correction step.
- We extend our method to sample from continuous-valued distributions.
- We prove bounds on the convergence rate of our algorithm, as measured by the spectral gap, on both discrete and continuous state spaces.
- We evaluate Poisson-minibatching Gibbs empirically, and show that its performance can match that of plain Gibbs sampling while using less computation at each iteration.
[**State Space**]{} [**Algorithm**]{} [**Computational Cost/Iter**]{}
--------------------- ----------------------------------------------- ---------------------------------
Discrete Gibbs sampling $O(D\Delta)$
MIN-Gibbs [@de2018minibatch] $O(D\Psi^2)$
MGPMH [@de2018minibatch] $O(DL^2 + \Delta)$
DoubleMIN-Gibbs [@de2018minibatch] $O(DL^2 + \Psi^2)$
Poisson-Gibbs $O(DL^2)$
Continuous Gibbs with rejection sampling $O(N\Delta)$
PGITS: Poisson-Gibbs with ITS $O(L^3)$
PGDA: Poisson-Gibbs with double approximation $O(L^2\log L)$
: Computational complexity cost for a single-iteration of Gibbs sampling. Here, $N$ is the required number of steps in rejection sampling to accept a sample, and the rest of the parameters are defined in Section \[sec:pre\].[]{data-label="tab:cost"}
Preliminaries and Definitions {#sec:pre}
-----------------------------
In this section, we present some background about Gibbs sampling and graphical models and give the definitions which will be used throughout the paper. In this paper, we consider Gibbs sampling on a factor graph [@koller2009probabilistic], a type of graphical model that defines a probability distribution in terms of its *factors*. Explicitly, a factor graph consists of a set of variables $\mathcal{V}$ (each of which can take on values in some set $\mathcal{X}$) and a set of factors $\Phi$, and it defines a probability distribution $\pi$ over a state space $\Omega = \mathcal{X}^{\mathcal{V}}$, where the probability of some $x \in \Omega$ is $$\textstyle
\pi(x) = \frac{1}{Z} \cdot \exp\left( \sum_{\phi\in\Phi} \phi(x) \right) = \frac{1}{Z} \cdot \prod_{\phi\in\Phi} \exp\left( \phi(x) \right).$$ Here, $Z$ denotes the scalar factor necessary for $\pi$ to be a distribution. Equivalently, we can think of this as the *Gibbs measure* with energy function $$\textstyle
U(x) = \sum_{\phi\in\Phi} \phi(x),
\hspace{2em}\text{where}\hspace{2em}
\pi(x) \propto \exp(U(x));$$ this formulation will prove to be useful in many of the derivations later in the paper. (Here, the $\propto$ notation denotes that the expression on the left is a distribution that is proportional to the expression on the right with the appropriate constant of proportionality to make it a distribution.) In a factor graph, the factors $\phi$ typically only depend on a subset of the variables; we can represent this as a bipartite graph where the nodesets are $\mathcal{V}$ and $\Phi$ and where we draw an edge between a variable $i \in \mathcal{V}$ and a factor $\phi \in \Phi$ if $\phi$ depends on $i$. For simplicity, in this paper we assume that the variables are indexed with natural numbers $\mathcal{V} = \{1, \ldots, n\}$. We denote the set of factors that depend on the $i$th variable, as $$A[i] = \{\phi | \phi \text{ depends on variable } i, \ \phi\in \Phi\}$$
[R]{}[0.45]{}
initial point $x$ **sample** variable $i \sim \operatorname{Unif}\{1, \ldots, n\}$ $x(i) \leftarrow v$ $U_v \leftarrow \sum_{\phi\in A[i]} \phi(x)$ **construct distribution** $\rho$ where $$\textstyle \rho(v) \propto \exp(U_v)$$ **sample** $v$ from $\rho$ **update** $x(i) \leftarrow v$ **output sample** $x$
An important property of a factor graph is that the conditional distribution of a variable can be computed using only the factors that depend on that variable. This lends to a particularly efficient implementation of Gibbs sampling, in which only these adjacent factors are used at each iteration (rather than needing to evaluate the whole energy function $U$): this is illustrated in Algorithm \[alg:gibbs\].
The performance of our algorithm will depend on several parameters of the graphical model, which we will now restate, from previous work on MIN-Gibbs [@de2018minibatch]. If the variables take on discrete values, we let $D = | \mathcal{X} |$ denote the number of values each can take on. We let $\Delta = \max_i |A[i]|$ denote the maximum degree of the graph. We assume that the magnitudes of the factor functions are all bounded, and for any $\phi$ we let $M_{\phi}$ denote this bound $$\textstyle
M_{\phi} = \left(\sup_{x \in \Omega} \phi(x) \right) - \left(\inf_{x \in \Omega} \phi(x) \right).$$ Without loss of generality (and as was done in previous works [@de2018minibatch]), we will assume that $0 \le \phi(x) \le M_{\phi}$ because we can always add a constant to any factor $\phi$ without changing the distribution $\pi$. We define the *local maximum energy* $L$ and *total maximum energy* $\Psi$ of the graph as bounds on the sum of $M_{\phi}$ over the set of the factors associated with a single variable $i$ and the whole graph, respectively, $$\textstyle
L = \max_{i\in\{1, 2, \dots, N\}}\sum_{\phi \in A[i]} M_{\phi}
\hspace{2em}\text{and}\hspace{2em}
\Psi = \sum_{\phi \in \Phi} M_{\phi}.$$ If the graph is very large and has many low-energy factors, the maximum energy of a graph can be much smaller than the maximum degree of the graph. All runtime analyses in this paper assume that evaluating a factor $\phi$ and sampling from a small discrete distribution can be done in constant time.
Poisson-Minibatching Gibbs Sampling
===================================
In this section, we will introduce the idea of Poisson-minibatching under the setting in which we assume we can sample from the conditional distribution of $x(i)$ exactly. One such example is when the state space of $x$ is discrete. We will consider how to sample from the conditional distribution when exact sampling is impossible in the next section.
In plain Gibbs sampling, we have to compute the sum over all the factors in $A[i]$ to get the energy in every step. When the graph is large, the computation of getting the energy can be expensive; for example, in the discrete case this cost is proportional to $D \Delta$. The main idea of Poisson-minibatching is to augment a desired distribution with extra Poisson random variables, which control how and whether a factor is used in the minibatch for a particular iteration. @maclaurin2014firefly used a similar idea to control whether a data point will be included in the minibatch or not with augmented Bernoulli variables. However, this method has been shown to be very inefficient when only updating a small fraction of Bernoulli variables in each iteration [@quiroz2016block]. Our method does not suffer from the same issue due to the usage of Poisson variables which we will explain further later in this section.
We define the conditional distribution of additional variable $s_{\phi}$ for each factor $\phi$ as $$\textstyle
s_{\phi}|x \sim \text{Poisson}\left(\frac{\lambda M_{\phi}}{L} + \phi(x)\right)$$ where $\lambda>0$ is a hyperparameter that controls the minibatch size. Then the joint distribution of variables $x$ and $s$, where $s$ is a variable vector including all $s_{\phi}$, is $\pi(x, s) = \pi(x) \cdot \mathbf{P}(s|x)$ and so $$\label{eq:pi}
\pi(x, s)
\propto
\exp\left(
\sum_{\phi\in\Phi} \left(s_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}}\phi(x) \right)
+ s_{\phi}\log\left(\frac{\lambda M_{\phi}}{L}\right) - \log\left( s_{\phi}! \right)\right)\right)$$ Using (\[eq:pi\]) allows us to compute conditional distributions (of the variables $x_i$) using only a subset of the factors. This is because the factor $\phi$ will not contribute to the energy unless $s_{\phi}$ is greater than zero. If many $s_{\phi}$ are zero, then we only need to compute the energy over a small set of factors. Since $$\textstyle
\mathbf{E}\left[ \left| \{ \phi \in A[i] \mid s_{\phi} > 0 \} \right| \right]
\le
\mathbf{E}\left[\sum_{\phi\in A[i]} s_{\phi} \right] = \sum_{\phi\in A[i]} \left(\frac{\lambda M_{\phi}}{L} + \phi(x)\right)\leq \lambda + L,$$ this implies that $\lambda + L$ is an upper bound of the expected number of non-zero $s_{\phi}$. When the graph is very large and has many low-energy factors, $\lambda + L$ can be much smaller than the factor set size, in which case only a small set of factors will contribute to the energy while most factor terms will disappear because $s_{\phi}$ is zero.
Using Poisson auxiliary variables has two benefits. First, compared with the Bernoulli auxiliary variables as described in FlyMC [@maclaurin2014firefly], there is a simple method for sampling $n$ Poisson random variables in total expected time proportional to the sum of their parameters, which can be much smaller than $n$ [@de2018minibatch]. This means that sampling $n$ Poisson variables can be much more efficient than sampling $n$ Bernoulli variables, which allows our method to avoid any inefficiencies caused by sampling Bernoulli variables as in FlyMC. Second, compared with a fixed-minibatch-size method such as the one used in [@welling2011bayesian], Poisson-minibatching has the important property that the variables $s_{\phi}$ are independent. Whether a factor will be contained in the minibatch is independent to each other. This property is necessary for proving convergence rate theorems in the paper.
In Poisson-Gibbs, we will sample from the joint distribution alternately. At each iteration we can (1) first re-sample all the $s_{\phi}$, then (2) choose a variable index $i$ and re-sample $x(i)$. Here, we can reduce the state back to only $x$, since the future distribution never depends on the current value of $s$. Essentially, we only bother to re-sample the $s_{\phi}$ on which our eventual re-sampling of $x(i)$ depends: statistically, this is equivalent to re-sampling all $s_{\phi}$. Doing this corresponds to Algorithm \[alg:poisson-gibbs\].
However, minibatching by itself does not mean that the method must be more effective than plain Gibbs sampling. It is possible that the convergence rate of the minibatched chain becomes much slower than the original rate, such that the total cost of the minibatch method is larger than that of the baseline method even if the cost of each step is smaller. To rule out this undesirable situation, we prove that the convergence speed of our chain is not slowed down, or at least not too much, after applying minibatching. To do this, we bound the convergence rate of our algorithm, as measured by the *spectral gap* [@levin2017markov], which is the gap between the largest and second-largest eigenvalues of the chain’s transition operator. This gap has been used previously to measure the convergence rate of minibatched MCMC [@de2018minibatch].
\[thm:discrete\] Poisson-Gibbs (Algorithm \[alg:poisson-gibbs\]) is reversible and has a stationary distribution $\pi$. Let $\bar{\gamma}$ denote its spectral gap, and let $\gamma$ denote the spectral gap of plain Gibbs sampling. If we use a minibatch size parameter $\lambda \ge 2 L$, then $$\bar{\gamma}
\ge
\exp\left( - \frac{4L^2}{\lambda} \right) \cdot\gamma.$$
This theorem guarantees that the convergence rate of Poisson-Gibbs will not be slowed down by more than a factor of $\exp( - 4L^2 / \lambda )$. If we set $\lambda = \Theta(L^2)$, then this factor becomes $O(1)$, which is independent of the size of the problem. We proved Theorem \[thm:discrete\] and the other theorems in this paper using the technique of Dirichlet forms, which is a standard way of comparing the spectral gaps of two chains by comparing their transition probabilities (more details are in the supplemental material).
Next, we derive expressions for the overall computational cost of Algorithm \[alg:poisson-gibbs\], supposing that we set $\lambda = \Theta(L^2)$ as suggested by Theorem \[thm:discrete\]. First, we need to evaluate the cost of sampling all the Poisson-distributed $s_{\phi}$. While a naïve approach to sample this would take $O(\Delta)$ time, we can do it substantially faster. For brevity, and because much of the technique is already described in the previous work [@de2018minibatch], we defer an explicit analysis to the supplementary material, and just state the following.
\[stmt:samplingphi\] Sampling all the auxiliary variables $s_{\phi}$ for $\phi \in A[i]$ can be done in average time $O(\lambda + L)$, resulting in a sparse vector $s_{\phi}$.
Now, to get an overall cost when assuming exact sampling from the conditional distribution, we consider discrete state spaces, in which we can sample from the conditional distribution of $x(i)$ exactly. In this case, the cost of a single iteration of Poisson-Gibbs will be dominated by the loop over $v$. This loop will run $D$ times, and each iteration will take $O(|S|)$ time to run. On average, this gives us an overall runtime $O((\lambda + L) \cdot D) = O(L^2 D)$ for Poisson-Gibbs. Note that due to the fast way we sample Poisson variables, the cost of sampling Poisson variables is negligible compared to other costs.
In comparison, the cost of the previous algorithms MIN-Gibbs, MGPMH and DoubleMIN-Gibbs [@de2018minibatch] are all larger in big-$O$ than that of Poisson-Gibbs, as showed in Table \[tab:cost\]. MGPMH and DoubleMIN-Gibbs need to conduct an M-H correction, which adds to the cost, and the cost of MIN-Gibbs and DoubleMIN-Gibbs depend on $\Psi$ which is a global statistic. By contrast, our method does not need additional M-H step and is not dependent on global statistics. Thus the total cost of Gibbs sampling can be reduced more by Poisson-minibatching compared to the previous methods.
#### Application of Poisson-Minibatching to Metropolis-Hastings.
Poisson-minibatching method can be applied to other MCMC methods, not just Gibbs sampling. To illustrate the general applicability of Poisson-minibatching method, we applied Poisson-minibatching to Metropolis-Hasting sampling and call it *Poisson-MH* (details of this algorithm and a demonstration on a mixture of Gaussians are given in the supplemental material). We get the following convergence rate bound.
\[thm:mh\] Poisson-MH is reversible and has a stationary distribution $\pi$. If we let $\bar{\gamma}$ denote its spectral gap, and let $\bar{\gamma}$ denote the spectral gap of plain M-H sampling with the same proposal and target distributions, then $$\textstyle
\bar{\gamma}
\ge
\frac{1}{2}\exp\left( - \frac{L^2}{\lambda + L}\right)\cdot\gamma.$$
Poisson-Gibbs on Continuous State Spaces
========================================
In this section, we consider how to sample from a continuous conditional distribution, i.e. when $\mathcal{X} = [a,b] \subset \mathbb{R}$, without sacrificing the benefits of Poisson-minibatching. The main difficulty is that sampling from an arbitrary continuous conditional distribution is not trivial in the same way as sampling from an arbitrary discrete conditional distribution is. Some additional sampling method is required. In principle, we can combine any sampling method with Poisson-minibatching, such as rejection sampling which is commonly used in Gibbs sampling. However, rejection sampling needs to evaluate the energy multiple times per sample, so even if we reduce the cost of evaluating the energy by minibatching, the total cost can still be large, besides which there is no good guarantee on the convergence rate of rejection sampling.
In order to sample from the conditional distribution efficiently, we propose a new sampling method based on inverse transform sampling (ITS) method. The main idea is to approximate the continuous distribution with a polynomial; this requires only a number of energy function evaluations proportional to the degree of the polynomial. We provide overall cost and theoretical analysis of convergence rate for our method.
\[alg:poisson-gibbs\] **given:** initial state $x \in \Omega$ **sample** variable $i \sim \operatorname{Unif}\{1, \ldots, n\}$. **sample** $s_{\phi} \sim \text{Poisson}\left( \frac{\lambda M_{\phi}}{L} + \phi(x) \right)$ $S \leftarrow \{ \phi | s_{\phi} > 0 \}$ $x(i) \leftarrow v$ $U_v \leftarrow \sum_{\phi \in S} s_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}}\phi(x) \right)$ **construct distribution** $\rho$ where $$\rho(v) \propto \exp(U_v)$$ **sample** $v$ from $\rho$ **update** $x(i) \leftarrow v$ **output sample** $x$
\[alg:PGDA\] **given:** state $x \in \Omega$, degree $m$ and $k$, domain $[a,b]$ **set** $i$, $s_{\phi}$, $S$, and $U$ as in Algorithm \[alg:poisson-gibbs\]. **construct** degree-$m$ Chebyshev polynomial approximation of energy $U_v$ on $[a,b]$: $\tilde{U}_v$ **construct** degree-$k$ Chebyshev polynomial approximation:$\tilde{f}(v) \approx \exp(\tilde U_v)$ **compute** the CDF polynomial $$\tilde{F}(v) = \left( \int_a^b \tilde f(y) \; dy \right)^{-1} \int_a^v \tilde f(y) \; dy$$ **sample** $u \sim \operatorname{Unif}[0,1]$. **solve** root-finding problem for $v$: $\tilde{F}(v) = u$ $\triangleright$ Metropolis-Hastings correction: $p \leftarrow \frac{\exp(U_{v})\tilde{f}(x(i))}{\exp(U_{x(i)})\tilde{f}(v)}$ **with probability** $\min(1,p)$, set $x(i) \leftarrow v$ **output sample** $x$
#### Poisson-Gibbs with Double Chebyshev Approximation.
Inverse transform sampling is a classical method that generates samples from a uniform distribution and then transforms them by the inverse of cumulative distribution function (CDF) of the desired distribution. Since the CDF is often intractable in practice, Fast Inverse Transform Sampling (FITS) [@olver2013fast] uses a Chebyshev polynomial approximation to estimate the PDF fast and then get the CDF by computing an integral of a polynomial. Inspired by FITS, we propose Poisson-Gibbs with double Chebyshev approximation (PGDA).
The main idea of double Chebyshev approximation is to approximate the energy function first and then the PDF by using Chebyshev approximation *twice*. Specifically, we first get a polynomial approximation to the energy function $U$ on $[a,b]$, denoted by $\tilde{U}$, the *Chebyshev interpolant* [@trefethen2013approximation] $$\begin{aligned}
\label{eq:energy}
\tilde{U}(x) = \sum_{k=0}^m \alpha_k T_k\left(\frac{2(x - a)}{b-a} - 1\right),\ \alpha_k\in \RR,\ x\in[a, b],\end{aligned}$$ where $T_k (x) = \cos(k \cos^{-1} x)$ is the degree-$k$ Chebyshev polynomial. Although the domain is continuous, we only need to evaluate $U$ on $m+1$ Chebyshev nodes to construct the interpolant, and the expansion coefficients $\alpha_k$ can be computed stably in $O(m \log m)$ time. The following theorem shows that the error of a Chebyshev approximation can be made arbitrarily small with large $m$. (Although stated for the case of $[a,b] = [-1,1]$, it easily generalizes to arbitrary $[a,b]$.)
\[thm:cheby\] Assume $U$ is analytic in the open Bernstein ellipse $B([-1, 1], \rho)$, where the Bernstein ellipse is a region in the complex plane bounded by an ellipse with foci at $\pm 1$ and semimajor-plus-semiminor axis length $\rho > 1$. If for all $x \in B([-1, 1], \rho)$, $|U(x)| \leq V$ for some constant $V > 0$, the error of the Chebyshev interpolant on $[-1,1]$ is bounded by $$\begin{aligned}
| \tilde{U}(x) - U(x) | \leq \delta_m
\hspace{2em}\text{where}\hspace{2em}
\delta_m = \frac{4V\rho^{-m}}{\rho - 1}.\end{aligned}$$
After getting the approximation of the energy, we can get the PDF by $\exp(\tilde{U})$. However, it is generally hard to get the CDF now since the integral of $\exp(\tilde{U})$ for polynomial $\tilde U$ is usually intractable. So, we use *another* Chebyshev approximation $\tilde{f}$ to estimate $\exp(\tilde{U})$. Constructing the second Chebyshev approximation requires no additional evaluations of energy functions; its total computational cost is $\tilde O(mk)$ because we need to evaluate a degree-$m$ polynomial $k$ times to compute the coefficients. After doing this, we are able to compute the CDF directly since it is the integral of a polynomial. With the CDF $\tilde{F}(x)$ in hand, inverse transform sampling is used to generate samples. First, a pseudo-random sample $u$ is generated from the uniform distribution on $[0,1]$, and then we solve the following root-finding problem for $x$: $\tilde{F}(x) = u$. Since $\tilde{F}(x)$ is a polynomial, this root-finding problem can be solved by many standard methods. We use bisection method to ensure the robustness of the algorithm [@olver2013fast].
Importantly, the sample we get here is actually from an *approximation* of the CDF. To correct the error introduced by the polynomial approximation, we add a M-H correction as the final step to make sure the samples come from the target distribution. Our algorithm is given in Algorithm \[alg:PGDA\]. As before, we prove a bound on PGDA in terms of the spectral gap, given the additional assumption that the factors $\phi$ are analytic.
\[thm:doupoly\] PGDA (Algorithm \[alg:PGDA\]) is reversible and has a stationary distribution $\pi$. Let $\bar{\gamma}$ denote its spectral gap, and let $\gamma$ denote the spectral gap of plain Gibbs sampling. Assume $\rho > 1$ is some constant such that every factor function $\phi$, treated as a function of any single variable $x_i$, must be analytically continuable to the Bernstein ellipse with radius parameter $\rho$ shifted-and-scaled so that its foci are at $a$ and $b$, such that it satisfies $| \phi(z) | \le M_{\phi}$ anywhere in that ellipse. Then, if $\lambda \log(2) \ge 4 L$, and if $m$ is set large enough that $4 \rho^{-m/2} \le \sqrt{\rho} - 1$, then it will hold that $$\begin{aligned}
\bar{\gamma}
&\geq
\left(1 - 4\sqrt{\digamma}\right)\exp\left(\frac{-4L^2}{\lambda}\right)\cdot \gamma,
\hspace{1em}\text{where}\hspace{1em}
\digamma
=
\frac{4\cdot \exp\left(8 L \right)\cdot\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}
+ \exp\left( \frac{16 L \cdot \rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1} \right) - 1.
\end{aligned}$$
Similar to Theorem \[thm:discrete\], this theorem implies that the convergence rate of PGDA can be slowed down by at most a constant factor relative to plain Gibbs. If we set $m = \Theta(\log L)$, $k = \Theta(L)$ and $\lambda = \Theta(L^2)$, then the ratio of the spectral gaps will also be $O(1)$, which is independent of the problem parameters. Note that it is possible to combine FITS with Poisson-Gibbs directly (i.e. use only one polynomial approximation to estimate the PDF directly), and we call this method [*Poisson-Gibbs with fast inverse transform sampling*]{} (PGITS). It turns out that PGDA is more efficient than PGITS since PGDA requires fewer evaluations of $U$ to achieve the same convergence rate. If we set the parameters as above, the total computational cost of PGDA is $O(m\cdot (\lambda + L) + m\cdot k) = O(\log L \cdot (L^2 + L)) = O(\log L\cdot L^2)$. On the other hand, the cost of PGITS to achieve the same constant-factor spectral gap ratio is $O(L^3)$. A derivation of this is given in the supplemental material.
Experiments
===========
We demonstrate our methods on three tasks including Potts models, continuous spin models and truncated Gaussian mixture in comparison with plain Gibbs sampling and previous minibatched Gibbs sampling. We release the code at <https://github.com/ruqizhang/poisson-gibbs>.
Potts Models {#sec:ising}
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We first test the performance of Poisson-minibatching Gibbs sampling on the Potts model [@potts1952some] as in @de2018minibatch. The Potts model is a generalization of the Ising model [@ising1925beitrag] with domain $\{1,\ldots, D\}$ over an $N\times N$ lattice. The energy of a configuration is the following: $$U(x) = \sum_{i=1}^n \sum_{j=1}^n\beta \cdot A_{ij} \cdot \delta\left(x(i), x(j)\right)$$ where the $\delta$ function equals one only when $x(i) = x(j)$ and zero otherwise. $A_{ij}$ is the interaction between two sites $i$ and $j$ and $\beta$ is the inverse temperature. As was done in previous work, we set the model to be fully connected and the interaction $A_{ij}$ is determined by the distance between site $i$ and site $j$ based on a Gaussian kernel [@de2018minibatch]. The graph has $n = N^2 = 400$ variables in total, $\beta = 4.6$ and $D=10$. On this model, $L = 5.09$.
We first compare our method with two other methods: plain Gibbs sampling and the most efficient MIN-Gibbs methods on this task, DoubleMIN-Gibbs. Note that, in comparison to our method, DoubleMIN-Gibbs needs an additional M-H correction step which requires a second minibatch to be sampled. We set $\lambda = 1\cdot L^2$ for all minibatch methods. We tried two values for the second minibatch size in DoubleMIN-Gibbs $\lambda_2 = 1\cdot L^2$ and $10^4\cdot L^2$. We compute run-average marginal distributions for each variable by collecting samples. By symmetry, the marginal for each variable in the stationary distribution is uniform, so the $\ell_2$-distance between the estimated marginals and the uniform distribution can be used to evaluate the convergence of Markov chain. We report this marginal error averaged over three runs.
Figure \[fig:ising\]a shows the $\ell_2$-distance marginal error as a function of iterations. We observe that Poisson-Gibbs performs comparably with plain Gibbs and it outperforms DoubleMIN-Gibbs significantly especially when $\lambda_2$ is not large enough. The performance of DoubleMIN-Gibbs is highly influenced by the size of the second minibatch. We have to increase the second minibatch to $10^4 \cdot L^2$ in order to make it converge. This is because the variance of M-H correction will be very large when the second minibatch is not large enough. On the other hand, Poisson-Gibbs does not require an additional M-H correction which not only reduces the computational cost but also improves stability. In Figure \[fig:ising\]b, we show the performance of our method with different values of $\lambda$. When we increase the minibatch size, the convergence speed of Poisson-Gibbs approaches plain Gibbs, which validates our theory. The number of factors being evaluated of Poisson-Gibbs varies each iteration, thus we report the average number which are 7, 28 and 132 respectively for $\lambda=0.1\cdot L^2$, $1\cdot L^2$ and $5\cdot L^2$.
The runtime comparisons with the same setup are reported in Figure \[fig:ising-time\]a and 2b to demonstrate the computational speed-up of Poisson-Gibbs empirically. We can see that the results align with our theoretical analysis: Poisson-Gibbs is significantly faster than plain Gibbs samping and faster than previous minibatched Gibbs sampling methods. Compared to plain Gibbs, Poisson-Gibbs speeds up the computation by evaluating only a subset of factors in each iteration. Compared to DoubleMIN-Gibbs, Poisson-Gibbs is faster because it removes the need of an additional M-H correction step.
Continuous Spin Models
----------------------
In this section, we study a more general setting of spin models where spins can take continuous values. Continuous spin models are of interest in both the statistics and physics communities [@michel2015event; @bruce1985universality; @dommers2017continuous]. This random graph model can also be used to describe complex networks such as social, information, and biological networks [@newman2003structure]. We consider the energy of a configuration as the following: $$U(x) = \sum_{i=1}^n \sum_{j=1}^n\beta \cdot A_{ij} \cdot \left(x(i) \cdot x(j) + 1\right)$$ where $x(i)\in [0, 1]$ and $\beta=1$. Notice that the existing minibatched Gibbs sampling methods [@de2018minibatch] are not applicable on this task since they can be used only on discrete state spaces. We compare PGITS, PGDA with: (1) Gibbs sampling with FITS (Gibbs-ITS); (2) Gibbs sampling with Double Chebyshev approximation (Gibbs-DA); (3) Gibbs with rejection sampling (Gibbs-rejection); and (4) Poisson-Gibbs with rejection sampling (PG-rejection). We use symmetric KL divergence to quantitatively evaluate the convergence. On this model, $L=13.71$ and we set $\lambda = L^2$. The degree of polynomial is $m=3$ for PGITS and the first approximation in PGDA. The degree of polynomial is $k=10$ for the second approximation in PGDA. In rejection sampling, we set the proposal distribution to be $w g$ where $g$ is the uniform distribution on $[0,1]$ and $w$ is a constant tuned for best performance. The ground truth stationary distribution is obtained by running Gibbs-ITS for $10^7$ iterations.
On this task, the average number of evaluated factors per iteration of Poisson-Gibbs is 190. Figure \[fig:ising\]c shows the symmetric KL divergence as a function of iterations, with results averaged over three runs. Observe that our methods achieve comparable performance to Gibbs sampling with only a fraction of factors. For rejection sampling, the average steps needed for a sample to be accepted is greater than 300 which means that the cost is much larger than that of PGITS and PGDA. Given the same time budget, it can only run for many fewer iterations (we run it for $10^4$ iterations). On the other hand, the two Chebyshebv approximation methods are much more efficient for both Poisson-Gibbs and plain Gibbs. The advantage of FITS over rejection sampling has also been discussed in previous work [@olver2013fast]. Also notice that PGDA converges faster than PGITS given the same degree of polynomial. This empirical result validates our theoretical results that suggest PGDA is more efficient than PGITS.
We also report the symmetric KL divergence as a function of runtime in Figure \[fig:ising-time\]c. Similar to the previous section, the two Poisson-Gibbs methods are faster than plain Gibbs sampling.
Truncated Gaussian Mixture {#sec:gmm}
--------------------------
We further demonstrate PGITS and PGDA on a truncated Gaussian mixture model. We consider the following Gaussian mixture with tied means as done in previous work [@welling2011bayesian; @li2017mini]: $$x_1 \sim \Ncal(0, \sigma_1^2), \; x_2 \sim \Ncal(0, \sigma_2^2), \;
y_i \sim \frac{1}{2} \Ncal( x_1, \sigma_y^2) + \frac{1}{2}\Ncal( x_1 + x_2, \sigma_y^2).$$ We used the same parameters as in @welling2011bayesian: $\sigma_1^2 = 10$, $\sigma_2^2 = 1$, $\sigma_y^2 = 2$, $ x_1 = 0$ and $ x_2 = 1$. This posterior has two modes at $(x_1, x_2) = (0,1)$ and $(x_1, x_2) = (1,-1)$. We truncate the posterior by bounding the variables $x_1$ and $x_2$ in $[-6, 6]$. The energy can be written as $$U(x) = \log p(x_1) + \log p(x_2) + \sum_{i=1}^N \log p(y_i|x_1, x_2)$$ which can be regarded as a factor graph with $N$ factors. We add a positive constant to the energy to ensure each factor is non-negative: this will not change the underlying distribution. As in @li2017mini, we set $N=10^6$. $L = 1581.14$ for this model and we set $\lambda = 500$, $m=20$ and $k=25$. We have also considered higher values of $\lambda$ and found that the results are very similar. We generate $10^6$ samples for all methods. A uniform distribution in $[-6, 6]$ is used as the proposal distribution in Gibbs with rejection sampling. We try varying values for $w$ but none of them results in reasonable density estimate which may be due to the inefficiency of rejection sampling [@olver2013fast]. We report the results when the average needed steps for a sample to be accepted is around 1000. The average number of factors being evaluated per iteration of Poisson-Gibbs is 1802. Our results are reported in Figure \[fig:mog\], where we observe visually that the density estimates of PGITS and PGDA are very accurate. In contrast, rejection sampling completely failed to estimate the density given the budget.
Conclusion
==========
We propose Poisson-minibatching Gibbs sampling to generate unbiased samples with theoretical guarantees on the convergence rate. Our method provably converges to the desired stationary distribution at a rate that is at most a constant factor slower than the full batch method, as measured by the spectral gap. We provide guidance about how to set the hyperparameters of our method to make the convergence speed arbitrarily close to the full batch method. On continuous state spaces, we propose two variants of Poisson-Gibbs based on fast inverse transform sampling and provide convergence analysis for both of them. We hope that our work will help inspire more exploration into unbiased and guaranteed-fast stochastic MCMC methods.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work was supported by a gift from Huawei. We thank Wing Wong for the helpful discussion.
**[Supplementary Material: Poisson-Minibatching for Gibbs Sampling with Convergence Rate Guarantees]{}**
Fast Sampling of the Auxiliary Variables
========================================
In this section, we describe in detail the method used to sample the auxiliary variables $s_{\phi}$ and prove Statement \[stmt:samplingphi\]. The method for doing so is described here in Algorithm \[alg:samplingphi\].
\[alg:samplingphi\] $\triangleright$ pre-computation step; happens once $\Lambda_i \leftarrow \sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{L} + M_{\phi}$ **compute distribution** $\rho_i$ over $A[i]$ where $$\rho_i(\phi) \propto \frac{\lambda M_{\phi}}{L} + M_{\phi}.$$ **process distribution** $\rho_i$ so that in future, it can be sampled from in constant time $\triangleright$ to actually re-sample the auxiliary variables **given:** current state $x \in \Omega$, variable $i$ to resample **initialize sparse vector** $s: A[i] \rightarrow \Z$ **sample** $B \sim \operatorname{Poisson}(\Lambda_i)$ **sample** $\phi \sim \rho_i$ **compute** $\phi(x)$ **with probability** $\frac{ \frac{\lambda M_{\phi}}{L} + \phi(x) }{ \frac{\lambda M_{\phi}}{L} + M_{\phi} }$ **update** sparse vector $s_{\phi} \leftarrow s_{\phi} + 1$
To see that this is valid, let $B = \sum_i^n s_i$ where $s_i$ are Poisson variables with parameters $\lambda_i$. We know that $B$ is also Poisson distributed with parameter $\Lambda = \sum_i^n \lambda_i$. Conditioned on the value of $B$, it is known that $s_i$ follows a multinomial distribution with event probabilities $\lambda_i/\Lambda$ and trial count $B$. Therefore, we can first sample $B \sim \text{Poisson}(\Lambda)$ and then sample $$(s_1, \dots s_n) \sim \operatorname{Multinomial}\left(B, \left(\frac{\lambda_1}{\Lambda}, \ldots, \frac{\lambda_n}{\Lambda} \right) \right).$$ Our Algorithm \[alg:samplingphi\] is only slightly more complicated than this process, in order to minimize the number of times that $\phi(x)$ is evaluated, but it can be seen to produce the valid distribution by the same reasoning.
The computational cost of Algorithm \[alg:samplingphi\] is clearly proportional to $B$, and since $$\mathbf{E}[B] = \Lambda_i = \sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{L} + M_{\phi} \le \lambda + L,$$ it follows that the overall average computational cost will also be $\lambda + L$. This proves Statement \[stmt:samplingphi\].
Poisson-Gibbs with Exact Sampling from the Conditional Distribution
===================================================================
Derivation of the joint distribution {#app:joint-dist}
------------------------------------
In this subsection, we derive the joint distribution (\[eq:pi\]) by substituting the distributions of $x$ and $s$ into the conditional distribution of $s$ given $x$. By the expression of Poisson distribution for $s_{\phi}$ and the independence of $s_{\phi}$, we have $$\begin{aligned}
\pi(x, s)
&=
\pi(x)\pi(s|x)\\
&\propto
\exp\left(\sum_{\phi\in\Phi} \phi(x)\right)\prod_{\phi\in\Phi}\pi(s_{\phi}|x)\\
&=
\exp\left(\sum_{\phi\in\Phi}\left( \phi(x)
+
\log\pi(s_{\phi}|x)\right)\right)\\
&=
\exp\left(\sum_{\phi\in\Phi}\left(\phi(x)
+
s_{\phi}\log\left(\frac{\lambda M_{\phi}}{L}+\phi(x)\right)
-\log (s_{\phi}!)
-\frac{\lambda M_{\phi}}{L}-\phi(x)\right)\right)\\
&=
\exp\left(\sum_{\phi\in\Phi}\left(
s_{\phi}\log\left(\frac{\lambda M_{\phi}}{L}
+
\phi(x)\right)
-\log (s_{\phi}!)-\frac{\lambda M_{\phi}}{L}\right)\right)\\
&\propto
\exp\left(\sum_{\phi\in\Phi}\left(
s_{\phi}\log\left(\frac{\lambda M_{\phi}}{L}
+
\phi(x)\right)
-\log (s_{\phi}!)\right)\right)\\
&=
\exp\left(
\sum_{\phi\in\Phi} \left(s_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}}\phi(x) \right) + s_{\phi}\log\left(\frac{\lambda M_{\phi}}{L}\right) - \log\left( s_{\phi}! \right)\right)\right).\end{aligned}$$
Proof of Theorem \[thm:discrete\]
---------------------------------
In this section, we prove that Poisson-Gibbs converges, and derive a bound on its convergence rate.
First, we will derive an expression for the transition operator of Poisson-Gibbs chain, and show it is reversible. Then we will bound the spectral gap.
If $x$ and $y$ are states which differ in only one variable $i$, the probability of transitioning from $x$ to $y$ will be the probability of choosing to sample variable $i$ times the expected value over the random choice of $s$ of the probability of sampling $y(i)$ from $\rho$. That is, $$\begin{aligned}
T(x, y)
&=
\frac{1}{n}
\cdot
{\mathbf{E}\left[
\rho(y(i))
\right]}\\
&=
\frac{1}{n}
\cdot
{\mathbf{E}\left[
\frac{
\exp(U_{y(i)})
}{
\int \exp(U_u) \; du
}
\right]}\\
&=
\frac{1}{n}
\cdot
\sum_s
\frac{
\exp(U_{y(i)})
}{
\int \exp(U_u) \; du
}
\cdot
\prod_{\phi \in A[i]}
\frac{1}{s_{\phi}!}
\left( \frac{\lambda M_{\phi}}{L} + \phi(x) \right)^{s_{\phi}}
\exp\left( - \left( \frac{\lambda M_{\phi}}{L} + \phi(x) \right) \right)\\
&=
\frac{1}{n}
\cdot
\sum_s
\frac{
\exp\left( \sum_{\phi \in A[i]} s_{\phi} \log\left( \frac{\lambda M_{\phi}}{L} + \phi(y) \right) \right)
}{
\int
\exp\left( \sum_{\phi \in A[i]} s_{\phi} \log\left( \frac{\lambda M_{\phi}}{L} + \phi(z_u) \right) \right) \; du
}
\\&\hspace{2em}\cdot
\prod_{\phi \in A[i]}
\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(x) \right)^{s_{\phi}}
\cdot\exp\left( - \phi(x) \right)
\\&\hspace{2em}\cdot
\prod_{\phi \in A[i]}
\frac{1}{s_{\phi}!}
\left( \frac{\lambda M_{\phi}}{L} \right)^{s_{\phi}}
\cdot\exp\left( - \frac{\lambda M_{\phi}}{L} \right)\end{aligned}$$ where $z_u$ denotes $x$ where $x(i)$ has been set equal to $u$. Note that $s_{\phi}$ here are non-negative integers that a Poisson variable can take, not variables. So if we let $r_{\phi} \sim \text{Poisson}\left( \frac{\lambda M_{\phi}}{L} \right)$ and $r_{\phi}$ to be all independent, we can write this as $$\begin{aligned}
T(x, y)
&=
\frac{1}{n}
\cdot
\textbf{E}_r \Bigg[
\frac{
\exp\left( \sum_{\phi \in A[i]} r_{\phi} \log\left( \frac{\lambda M_{\phi}}{L} + \phi(y) \right) \right)
}{
\int
\exp\left( \sum_{\phi \in A[i]} r_{\phi} \log\left( \frac{\lambda M_{\phi}}{L} + \phi(z_u) \right) \right) \; du
}
\\&\hspace{2em}\cdot
\prod_{\phi \in A[i]}
\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(x) \right)^{r_{\phi}}
\cdot\exp\left( - \phi(x) \right) \Bigg] \\
&=
\frac{1}{n}
\cdot
\textbf{E}_r \Bigg[
\frac{
\exp\left( \sum_{\phi \in A[i]} r_{\phi} \left(
\log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(y) \right)
+
\log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(x) \right)
\right) \right)
}{
\int
\exp\left( \sum_{\phi \in A[i]} r_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(z_u) \right) \right) \; du
}
\\&\hspace{2em}
\cdot\exp\left( - \sum_{\phi \in A[i]} \phi(x) \right) \Bigg] \\\end{aligned}$$ Therefore, since $$\pi(x) = \frac{1}{Z} \cdot \exp\left( \sum_{\phi\in\Phi} \phi(x) \right),$$ it follows that $$\begin{aligned}
&\hspace{-1em}\pi(x) T(x, y)\\
&=
\frac{1}{n Z}
\cdot
\textbf{E}_r \Bigg[
\frac{
\exp\left( \sum_{\phi \in A[i]} r_{\phi} \left(
\log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(y) \right)
+
\log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(x) \right)
\right) \right)
}{
\int
\exp\left( \sum_{\phi \in A[i]} r_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(z_u) \right) \right) \; du
}
\\&\hspace{2em}
\cdot\exp\left( \sum_{\phi \in \Phi} \phi(x) - \sum_{\phi \in A[i]} \phi(x) \right) \Bigg] \\
&=
\frac{\exp(U_{\neg i}(x))}{n Z}
\cdot
\textbf{E}_r\left[
\frac{
\exp\left( \sum_{\phi \in A[i]} r_{\phi} \left(
\log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(y) \right)
+
\log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(x) \right)
\right) \right)
}{
\int
\exp\left( \sum_{\phi \in A[i]} r_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(z_u) \right) \right) \; du
} \right].\end{aligned}$$ where we define $U_{\neg i}(x) = \sum_{\phi\notin A[i]}\phi(x)$. This expression is symmetric in $x$ and $y$ (note that $U_{\neg i}(x)$ does not depend on variable $i$), so it follows that the Markov chain is reversible, and its stationary distribution is indeed $\pi$.
We can proceed to try to bound its spectral gap, using the technique of Dirichlet forms. We start by simplifying our expression by defining $$\bar \phi(x) = \frac{ L \phi(x) }{ \lambda M_{\phi} }.$$ Using this, we get $$\begin{aligned}
\pi(x) T(x, y)
&=
\frac{\exp(U_{\neg i}(x))}{n Z}
\cdot
\textbf{E}_r\left[
\frac{
\exp\left( \sum_{\phi \in A[i]} r_{\phi} \left(
\log\left( 1 + \bar \phi(y) \right)
+
\log\left( 1 + \bar \phi(x) \right)
\right) \right)
}{
\int
\exp\left( \sum_{\phi \in A[i]} r_{\phi} \log\left( 1 + \bar \phi(z_u) \right) \right) \; du
} \right].\end{aligned}$$ We proceed by bringing the exponential on the top of this sum down to the bottom and inside the integral, which produces $$\begin{aligned}
\pi(x)T(x, y)
&=
\frac{\exp\left(U_{\neg i}(x)\right)}{n Z}
\cdot
\mathbf{E}_r\Bigg[
\Bigg(
\int
\exp\Bigg(
\sum_{\phi \in A[i]} r_{\phi} \Bigg(
\log\left( 1 + \bar \phi(z_u) \right)
\\&\hspace{2em} -
\log\left( 1 + \bar \phi(x) \right)
-
\log\left( 1 + \bar \phi(y) \right)
\Bigg)
\Bigg) \; du
\Bigg)^{-1}
\Bigg]\\
&\ge
\frac{\exp\left(U_{\neg i}(x)\right)}{n Z}
\cdot
\Bigg(\mathbf{E}_r\Bigg[
\int
\exp\Bigg(
\sum_{\phi \in A[i]} r_{\phi} \Bigg(
\log\left( 1 + \bar \phi(z_u) \right)
\\&\hspace{2em} -
\log\left( 1 + \bar \phi(x) \right)
-
\log\left( 1 + \bar \phi(y) \right)
\Bigg)
\Bigg) \; du
\Bigg]\Bigg)^{-1}\end{aligned}$$ where this inequality follows from Jensen’s inequality and the fact that $1/x$ is convex. By converting the exp-of-sum to a product-of-exp, and recalling that the $r_{\phi}$ are independent, we can further reduce this to $$\begin{aligned}
\pi(x) T(x, y)
&\ge
\frac{\exp\left(U_{\neg i}(x)\right)}{n Z}
\Bigg(
\int
\mathbf{E}_r\Bigg[
\prod_{\phi \in A[i]}
\exp\Bigg(
r_{\phi} \Bigg(
\log\left( 1 + \bar \phi(z_u) \right)
\\&\hspace{2em} -
\log\left( 1 + \bar \phi(x) \right)
-
\log\left( 1 + \bar \phi(y) \right)
\Bigg)
\Bigg)
\Bigg]du
\Bigg)^{-1}\\
&=
\frac{\exp\left(U_{\neg i}(x)\right)}{n Z}
\Bigg(
\int
\prod_{\phi \in A[i]}
\mathbf{E}_r\Bigg[
\exp\Bigg(
r_{\phi} \Bigg(
\log\left( 1 + \bar \phi(z_u) \right)
\\&\hspace{2em} -
\log\left( 1 + \bar \phi(x) \right)
-
\log\left( 1 + \bar \phi(y) \right)
\Bigg)
\Bigg)
\Bigg]du
\Bigg)^{-1}.\end{aligned}$$ This final expectation expression is just the moment generating function of the Poisson random variable $r_{\phi}$ evaluated at $$t
=
\log\left( 1 + \bar \phi(z_u) \right)
-
\log\left( 1 + \bar \phi(x) \right)
-
\log\left( 1 + \bar \phi(y) \right).$$ Here, from the standard formula for that MGF, we get $$\mathbf{E}_r [\exp(r_{\phi} t)]
=
\exp\left(
\frac{\lambda M_{\phi}}{L} \left(
\exp(t) - 1
\right)
\right)$$ So $$\begin{aligned}
&\exp(t) - 1\\
&=
\frac{
1 + \bar \phi(z_u)
}{
(1 + \bar \phi(x))
(1 + \bar \phi(y))
}
-
1\\
&=
\frac{
\bar \phi(z_u)
-
\bar \phi(x)
-
\bar \phi(y)
-
\bar \phi(x)
\bar \phi(y)
}{
(1 + \bar \phi(x))
(1 + \bar \phi(y))
}\\
&=
\bar \phi(z_u)
-
\bar \phi(x)
-
\bar \phi(y)
-
\frac{
\left(
\bar \phi(z_u)
-
\bar \phi(x)
-
\bar \phi(y)
\right)
\left(
\bar \phi(x)
+
\bar \phi(y)
+
\bar \phi(x)
\bar \phi(y)
\right)
+
\bar \phi(x)
\bar \phi(y)
}{
(1 + \bar \phi(x))
(1 + \bar \phi(y))
}.\end{aligned}$$ Since $$0 \le \bar \phi(x) = \frac{ L \phi(x) }{ \lambda M_{\phi} } \le \frac{L}{\lambda} \le \frac{1}{2}$$ (where here we’re using the condition in the theorem statement that $2 L \le \lambda$) we can bound this with $$\begin{aligned}
&\exp(t) - 1\\
&\le
\bar \phi(z_u)
-
\bar \phi(x)
-
\bar \phi(y)
-
\frac{
\left(
-
\bar \phi(x)
-
\bar \phi(y)
\right)
\left(
\bar \phi(x)
+
\bar \phi(y)
\right)
+
\left(
1
-
\bar \phi(x)
-
\bar \phi(y)
\right)
\bar \phi(x)
\bar \phi(y)
}{
(1 + \bar \phi(x))
(1 + \bar \phi(y))
}\\
&\le
\bar \phi(z_u)
-
\bar \phi(x)
-
\bar \phi(y)
+
\frac{
\left(
\bar \phi(x)
+
\bar \phi(y)
\right)
\left(
\bar \phi(x)
+
\bar \phi(y)
\right)
}{
(1 + \bar \phi(x))
(1 + \bar \phi(y))
}\\
&\le
\bar \phi(z_u)
-
\bar \phi(x)
-
\bar \phi(y)
+
\left(
\bar \phi(x)
+
\bar \phi(y)
\right)^2\\
&\le
\bar \phi(z_u)
-
\bar \phi(x)
-
\bar \phi(y)
+
\frac{4L^2}{\lambda^2}.\end{aligned}$$ So, $$\begin{aligned}
\mathbf{E}{\exp(r_{\phi} t)}
&=
\exp\left(
\frac{\lambda M_{\phi}}{L} \left(
\exp(t) - 1
\right)
\right)\\
&\le
\exp\left(
\frac{\lambda M_{\phi}}{L} \left(
\bar \phi(z_u)
-
\bar \phi(x)
-
\bar \phi(y)
+
\frac{4L^2}{\lambda^2}
\right)
\right)\\
&=
\exp\left(
\phi(z_u)
-
\phi(x)
-
\phi(y)
+
\frac{4 L M_{\phi}}{\lambda}
\right).\end{aligned}$$ Substituting this into the original expression produces $$\begin{aligned}
&\pi(x) T(x, y)\\
&\ge
\frac{\exp\left(U_{\neg i}(x)\right)}{n Z}
\left(
\int
\prod_{\phi \in A[i]}
\exp\left(
\phi(z_u)
-
\phi(x)
-
\phi(y)
+
\frac{4L M_{\phi}}{\lambda}
\right)du
\right)^{-1}\\
&=
\frac{\exp\left(U_{\neg i}(x)\right)}{n Z}
\left(
\int
\exp\left(
\sum_{\phi \in A[i]}
\phi(z_u)
-
\sum_{\phi \in A[i]}
\phi(x)
-
\sum_{\phi \in A[i]}
\phi(y)
+
\sum_{\phi \in A[i]}
\frac{4L M_{\phi}}{\lambda}
\right)du
\right)^{-1}\\
&\ge
\frac{\exp\left(U_{\neg i}(x)\right)}{n Z}
\left(
\int
\exp\left(
\sum_{\phi \in A[i]}
\phi(z_u)
-
\sum_{\phi \in A[i]}
\phi(x)
-
\sum_{\phi \in A[i]}
\phi(y)
+
\frac{4L^2}{\lambda}
\right)du
\right)^{-1}\\
&=
\exp\left( - \frac{4L^2}{\lambda} \right)
\frac{\exp\left(U_{\neg i}(x)\right)}{n Z}
\left(
\int
\exp\left(
\bar U_u
-
\bar U_{x(i)}
-
\bar U_{y(i)}
\right)du
\right)^{-1}\\
&=
\exp\left( - \frac{4L^2}{\lambda} \right)
\frac{\exp\left(U_{\neg i}(x)\right)}{n Z}
\frac{
\exp(\bar U_{x(i)}) \cdot \exp(\bar U_{y(i)})
}{
\int \exp( \bar U_u )du
}\\
&=
\exp\left( - \frac{4L^2}{\lambda} \right)
\frac{1}{n Z}
\frac{
\exp(U(x)) \cdot \exp(\bar U_{y(i)})
}{
\int \exp( \bar U_u )du
}\end{aligned}$$ where $\bar U_v$ denotes the assignment of $U_v$ in the plain Gibbs sampling algorithm (Algorithm \[alg:gibbs\]), $$\bar U_v = \sum_{\phi \in A[i]} \phi(z_v),$$ Finally, if we let $G$ denote the transition probability operator of plain Gibbs sampling, we notice right away that $$\begin{aligned}
\pi(x) T(x, y)
&\ge
\exp\left( - \frac{4L^2}{\lambda} \right)
\frac{1}{n Z}
\frac{
\exp(U(x)) \cdot \exp(\bar U_{y(i)})
}{
\int \exp( \bar U_u )du
}\\
&=
\exp\left( - \frac{4L^2}{\lambda} \right)
\pi(x) G(x, y).\end{aligned}$$
We will use the Dirichlet form argument to finish the proof. A real function $f$ is square integrable with respect to probability measure $\pi$, if it satisfies $$\int f(x)^2\pi(dx)<\infty.$$ Define $L^2(\pi)$ to be the Hilbert space of all such functions.
Let $L^2_0(\pi)\subset L^2(\pi)$ to be the Hilbert space that uses the same inner product but only contains functions such that $$\mathbf{E}_{\pi}[f] = \int f(x)\pi(dx) = 0.$$
We also define the notation $$\begin{aligned}
\langle f,g \rangle = \int f(x)g(x)\pi(dx).\end{aligned}$$ A special example is $\operatorname{Var}_{\pi}[f] = \langle f,f\rangle$.
From here, the Dirichlet form of a Markov chain associated with transition operator $T$ is given by [@fukushima2010dirichlet] $$\begin{aligned}
\mathbf{E}(f)
=
\frac{1}{2}\int\int\left(f(x)-f(y)\right)^2 T(x,y)\pi(x)dxdy.\end{aligned}$$
And the spectral gap can be written as [@aida1998uniform] $$\gamma = \inf_{f\in L^2_0(\pi): \operatorname{Var}_{\pi}[f] = 1} \mathbf{E}(f).$$ The spectral gap is related to other common measurement of the convergence of MCMC. For example, it has the following relationship with the mean squared error $e_{\pi}$ on a Markov chain $\{X_n\}_{n\in \mathbb{N}}$ [@rudolf2011explicit], $$e_{\pi}^2 \leq \frac{2}{n\gamma}\norm{f}_2^2.$$
With the expression of the spectral gap, it follows that $$\begin{aligned}
\bar{\gamma} &= \inf_{f\in L^2_0(\pi): Var_{\pi}[f] = 1} \left[\frac{1}{2}\int\int\left(f(x)-f(y)\right)^2T(x,y)\pi(x) \; dx \; dy\right]\\
&\geq \exp\left( - \frac{4L^2}{\lambda} \right)\cdot \inf_{f\in L^2_0(\pi): Var_{\pi}[f] = 1} \left[\frac{1}{2}\int\int\left(f(x)-f(y)\right)^2G(x,y)\pi(x) \; dx \; dy\right]\\
&= \exp\left( - \frac{4L^2}{\lambda} \right) \cdot \gamma.\end{aligned}$$
This proves the theorem.
Poisson-Gibbs on Continuous State Spaces
========================================
Poisson-Gibbs with Fast Inverse Transform Sampling (PGITS)
----------------------------------------------------------
In the main body of the paper, we mentioned the PGITS method, Poisson-Gibbs with Fast Inverse Transform Sampling. This method is to approximate the PDF by Chebyshev polynomials and then use inverse transform sampling. In this section, we will outline the algorithm and derive convergence rate results for it. These results will illustrate why PGITS can be expected to perform worse than PGDA.
PGITS operates by approximating the PDF with a Chebyshev polynomial approximation and then sampling from that polynomial approximation using inverse transform sampling. Specifically, if the PDF we want to sample from is $f(x)$, we can approximate $f$ by $\tilde{f}$ on $[a, b]$ using Chebyshev polynomials, $$\begin{aligned}
\label{eq:approx}
\tilde{f} = \sum_{k=0}^m\alpha_k T_k\left(\frac{2(x - a)}{b-a} - 1\right),\ \alpha_k\in \RR,\ x\in[a, b]\end{aligned}$$ where $T_k (x) = \cos(k \cos^{-1} x)$ is the degree $k$ Chebyshev polynomial, and $\alpha_k$ are the Chebyshev coefficients of the function $f$ [@trefethen2013approximation]. We do this by interpolating $f$ at its Chebyshev nodes, resulting in $\tilde f$ being the $m$th order *Chebyshev interpolant*. Once we have the polynomial approximation $\tilde{f}$ we can construct the corresponding CDF approximation $\tilde{F}$ by calculating the integral directly (since polynomials are straightforward to integrate). With the approximation $\tilde{F}$, we are able to use inverse transform sampling to generate samples. We call this whole algorithm [*PGITS*]{} and it is listed as Algorithm \[alg:PGITS\].
We show that PGITS is reversible and bound its spectral gap in the following theorem.
\[thm:poly\] PGITS (Algorithm \[alg:PGITS\]) is reversible and has a stationary distribution $\pi$. Let $\bar{\gamma}$ denote its spectral gap, and let $\gamma$ denote the spectral gap of plain Gibbs sampling. Assume $\rho > 1$ is some constant such that every factor function $\phi$, treated as a function of any single variable $x_i$, must be analytically continuable to the Bernstein ellipse with radius parameter $\rho$ shifted-and-scaled so that its foci are at $a$ and $b$, such that it satisfies $| \phi(z) | \le M_{\phi}$ anywhere in that ellipse. Then, if $\lambda \ge 2 L$ it will hold that $$\bar \gamma
\ge
\left(1 - \frac{8 \exp(L) \rho^{-m/2}}{\sqrt{\rho - 1}} \right)
\cdot
\exp\left(-\frac{4L^2}{\lambda}\right) \cdot \gamma.$$
We can set $m = \Theta(L)$ and $\lambda = \Theta(L^2)$ to make the ratio of the spectral gaps $O(1)$, which is independent of the size of the problem. If the parameters are set in this way, the total cost of PGITS is $O(m\cdot (\lambda + L)) = O(L\cdot L^2) = O(L^3)$.
\[alg:PGITS\] **given:** state $x \in \Omega$, degree $m$, domain $[a,b]$ **set** $i$, $s_{\phi}$, $S$, and $U$ as in Algorithm \[alg:poisson-gibbs\]. **construct** degree-$m$ Chebyshev polynomial approximation of polynomial PDF on $[a,b]$ $$\tilde{f}(v) \approx \exp(U_v)$$ **compute** the CDF polynomial $$\tilde{F}(v) = \left( \int_a^b \tilde f(y) \; dy \right)^{-1} \int_a^v \tilde f(y) \; dy$$ **sample** $u \sim \operatorname{Unif}[0,1]$. **solve** root-finding problem for $v$: $\tilde{F}(v) = u$ $\triangleright$ Metropolis-Hastings correction: $$p \leftarrow \frac{\exp(U_{v}) \cdot \tilde{f}(x(i))}{\exp(U_{x(i)}) \cdot \tilde{f}(v)}$$ **with probability** $\min(1,p)$, set $x(i) \leftarrow v$ **output sample** $x$
### Proof of Theorem \[thm:poly\] {#sec:proof-thm-poly}
Similar to the previous analysis of Poisson-Gibbs, we will show the PGITS is reversible by using the expression of the transition operator. Then we will bound the spectral gap.
Let $T_{i,s}(x, y)$ denote the probability of transitioning from state $x$ to $y$ given that we have already chosen to sample variable $i$ with minibatch coefficients $s$. Then, the overall transition operator will be $$T (x, y) = {\mathbf{E}\left[T_{i,s}(x, y)\right]}$$ where the expectation is taken over $i$ and $s$.
Let the polynomial interpolant for $\exp(U_v)$ be $\tilde{f}(v)$ which is given in (\[eq:approx\]). Note that this interpolant is a function of the index $i$ and the minibatch coefficients $s$. Then, $$\begin{aligned}
T_{i,s}(x, y)
&=
\rho(y(i))
\cdot
\min (1,a)\\
&=
\frac{
\tilde{f}(y(i))
}{
\int \tilde{f}(u)du
}
\cdot
\min\left(1, \frac{\exp(U_{y(i)})\tilde{f}(x(i))}{\exp(U_{x(i)})\tilde{f}(y(i))}\right)\end{aligned}$$ Therefore, $$\begin{aligned}
T(x, y)
&=
\frac{1}{n}\mathbf{E}{
\frac{
\tilde{f}(y(i))
}{
\int \tilde{f}(u)du
}
\cdot
\min\left(1, \frac{\exp(U_{y(i)})\tilde{f}(x(i))}{\exp(U_{x(i)})\tilde{f}(y(i))}\right)}\\
&=
\frac{1}{n}\mathbf{E}{
\frac{
1
}{
\int \tilde{f}(u)du
}
\cdot
\min\left(\tilde{f}(y(i)), \exp(U_{y(i)} - U_{x(i)})\tilde{f}(x(i))\right)}\\
&=
\frac{1}{n}\mathbf{E}{
\frac{
1
}{
\int \tilde{f}(u)du
}
\cdot
\min\left(\tilde{f}(y(i)), \exp\left(\sum_{\phi\in A[i]}s_{\phi}\log \frac{1+\frac{L}{\lambda M_{\phi}}\phi(y)}{1+\frac{L}{\lambda M_{\phi}}\phi(x)}\right)\tilde{f}(x(i))\right)}\\
&=
\frac{1}{n}\sum_s
\frac{
1
}{
\int \tilde{f}(u)du
}
\cdot
\min\left(\tilde{f}(y(i)), \exp\left(\sum_{\phi\in A[i]}s_{\phi}\log \frac{1+\frac{L}{\lambda M_{\phi}}\phi(y)}{1+\frac{L}{\lambda M_{\phi}}\phi(x)}\right)\tilde{f}(x(i))\right)
\\&\hspace{2em}\cdot
\exp\left(
\sum_{\phi\in A[i]}
s_{\phi} \log\left( \frac{\lambda M_{\phi}}{L} + \phi(x) \right)
-
\log\left( s_{\phi}! \right)
-
\left( \frac{\lambda M_{\phi}}{L} + \phi(x) \right)
\right)\\
&=
\frac{1}{n}\sum_s
\frac{
1
}{
\int \tilde{f}(u)du
}
\cdot
\min\Bigg(\tilde{f}(y(i))\exp\left(
\sum_{\phi\in A[i]}
s_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(x) \right)\right),
\\&\hspace{2em}\exp\left(\sum_{\phi\in A[i]}s_{\phi}\log\left( 1+\frac{L}{\lambda M_{\phi}}\phi(y)\right)\right)\tilde{f}(x(i))\Bigg)
\\&\hspace{2em}\cdot
\exp\left(
\sum_{\phi\in A[i]}\Bigg(
s_{\phi} \log\left( \frac{\lambda M_{\phi}}{L} \right)
-
\log\left( s_{\phi}! \right)
-
\left( \frac{\lambda M_{\phi}}{L} + \phi(x) \right)
\right)\Bigg)\\
&=
\frac{1}{n}\sum_s
\frac{
1
}{
\int \tilde{f}(u)du
}
\cdot
\min\Bigg(\tilde{f}(y(i))\exp\left(
\sum_{\phi\in A[i]}
s_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(x) \right)\right),
\\&\hspace{2em}\exp\left(\sum_{\phi\in A[i]}s_{\phi}\log\left( 1+\frac{L}{\lambda M_{\phi}}\phi(y)\right)\right)\tilde{f}(x(i))\Bigg)
\\&\hspace{2em}\cdot
\exp\left(
\sum_{\phi\in A[i]}\Bigg(
s_{\phi} \log\left( \frac{\lambda M_{\phi}}{L} \right)
-
\log\left( s_{\phi}! \right)
-
\left( \frac{\lambda M_{\phi}}{L} \right)\right)\Bigg)\exp(-U_{x(i)})\end{aligned}$$ Multiplying $\pi(x)$ on both sides, $$\begin{aligned}
&\hspace{1em}\pi(x)T(x, y)\\
&=
\frac{\exp(U_{\neg i}(x))}{nZ}\sum_s
\frac{1}{\int \tilde{f}(u)du}\cdot
\min\Bigg(\tilde{f}(y(i))\exp\left(
\sum_{\phi\in A[i]}
s_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(x) \right)\right),
\\&\hspace{2em}\tilde{f}(x(i)) \exp\left(\sum_{\phi\in A[i]}s_{\phi}\log\left( 1+\frac{L}{\lambda M_{\phi}}\phi(y)\right)\right)\Bigg)
\\&\hspace{2em}\cdot
\exp\left(
\sum_{\phi\in A[i]}\Bigg(
s_{\phi} \log\left( \frac{\lambda M_{\phi}}{L} \right)
-
\log\left( s_{\phi}! \right)
-
\left( \frac{\lambda M_{\phi}}{L} \right)\right)\Bigg)\end{aligned}$$ This expression is symmetric in $x$ and $y$, so it follows that $$\begin{aligned}
\pi(x) T(x, y)
=
\pi(y) T(y, x)\end{aligned}$$ Thus the Markov chain is reversible, and its stationary distribution is $\pi$.
We now bound its spectral gap, using the technique of Dirichlet forms. First, as before, we start by re-writing the chain in terms of an expectation of a new random variable $r_{\phi}$ where $r_{\phi} \sim \text{Poisson}\left( \frac{\lambda M_{\phi}}{L} \right)$ and the $r_{\phi}$ are all independent. We also define $\bar \phi(x) = \frac{L \phi(x)}{\lambda M_{\phi}}$ as before. This gives us $$\begin{aligned}
\pi(x)T(x, y)
&=
\frac{\exp(U_{\neg i}(x))}{nZ} \; \mathbf{E}_r\Bigg[
\frac{1}{\int \tilde{f}(u)du}\cdot
\min\Bigg(\tilde{f}(y(i))\exp\left(
\sum_{\phi\in A[i]}
r_{\phi} \log\left( 1 + \bar \phi(x) \right)\right),
\\&\hspace{4em}\tilde{f}(x(i)) \exp\left(\sum_{\phi\in A[i]} r_{\phi}\log\left( 1+ \bar \phi(y)\right)\right)\Bigg) \Bigg]
\\ &=
\frac{\exp(U_{\neg i}(x))}{nZ} \; \mathbf{E}_r\left[
\frac{1}{\int \tilde{f}(u)du}\cdot
\min\left(\tilde{f}(y(i))\exp\left( U_{x(i)} \right),
\tilde{f}(x(i)) \exp\left(U_{y(i)}\right)\right) \right]\end{aligned}$$ where now the $\tilde f$ are considered to be a function of $r_\phi$ rather than $s_{\phi}$ as before.
To proceed further we will need to use the fact that $\tilde f$ is a Chebyshev interpolant to bound its error compared with $U$. Recall that, here, $$U_v = \sum_{\phi \in A[i]} r_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(z_v) \right)
= \sum_{\phi \in A[i]} r_{\phi} \log\left( 1 + \bar \phi(z_v) \right),$$ and $\tilde f(v) \approx \exp(U_v)$ in the sense of being a degree-$m$ Chebyshev polynomial interpolant. Recall that we assumed that the each function $\phi$, treated as a function in any single variable, must be analytic on a (shifted) Bernstein ellipse on the interval $[a,b]$ with parameter $\rho$ (i.e. a standard Bernstein ellipse on $[-1,1]$ with parameter $\rho$ shifted and scaled to have its foci at $a$ and $b$), and that its magnitude must be bounded by $${\left|\phi(z)\right|} \le M_{\phi}$$ for any $z$ in this ellipse (keeping all the other parameters as usual within $[a,b]$. It follows that the magnitude of the function $U_v$ is bounded by $$\begin{aligned}
{\left| \exp(U_v) \right|}
&=
{\left| \exp\left( \sum_{\phi \in A[i]} r_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}} \phi(z_v) \right) \right) \right|} \\
&=
\prod_{\phi \in A[i]} {\left| 1 + \frac{L}{\lambda M_{\phi}} \phi(z_v) \right|}^{r_{\phi}} \\
&\le
\prod_{\phi \in A[i]} \left( 1 + \frac{L}{\lambda} \right)^{r_{\phi}}.\end{aligned}$$ Therefore, from Theorem \[thm:cheby\], we know that $${\left| \tilde f(v) - \exp(U_v) \right|}
\le
\frac{4 \rho^{-m}}{\rho - 1} \cdot \prod_{\phi \in A[i]} \left( 1 + \frac{L}{\lambda} \right)^{r_{\phi}}
=
\frac{4 \rho^{-m}}{\rho - 1} \cdot \left( 1 + \frac{L}{\lambda} \right)^{\sum_{\phi \in A[i]} r_{\phi}}.$$ Since we also assumed that $\phi(z)$ is always non-negative, $U_v$ must also be non-negative, and so in particular $\exp(-U_v) \le 1$, so $${\left| \frac{\tilde f(v)}{\exp(U_v)} - 1 \right|}
\le
\frac{4 \rho^{-m}}{\rho - 1} \cdot \left( 1 + \frac{L}{\lambda} \right)^{\sum_{\phi \in A[i]} r_{\phi}}
\le
\frac{4 \rho^{-m}}{\rho - 1} \cdot \exp\left( \frac{L}{\lambda} \sum_{\phi \in A[i]} r_{\phi} \right).$$ If we now define $$C = \frac{4 \rho^{-m}}{\rho - 1} \cdot \exp\left( \frac{L}{\lambda} \sum_{\phi \in A[i]} r_{\phi} \right),$$ then $$(1 - C) \cdot \exp(U_v) \le \tilde f(v) \le (1 + C) \cdot \exp(U_v).$$ In particular, this means that $$\min\left(\tilde{f}(y(i))\exp\left( U_{x(i)} \right),
\tilde{f}(x(i)) \exp\left(U_{y(i)}\right)\right)
\ge
(1 - C) \cdot \exp\left( U_{x(i)} + U_{y(i)} \right),$$ and $$\frac{1}{\int \tilde{f}(u) \; du}
\ge
\frac{1}{1 + C} \cdot \frac{1}{\int \exp(U_u) \; du}.$$ Substituting this into our bound above gives $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\frac{\exp(U_{\neg i}(x))}{nZ} \; \mathbf{E}_r\left[ \frac{1 - C}{1 + C} \cdot
\frac{\exp\left( U_{x(i)} + U_{y(i)} \right)}{\int \exp(U_u) \; du} \right].\end{aligned}$$ Now, recall that we set this up by sampling $r_{\phi}$ independently from a Poisson random variable $r_{\phi} \sim \text{Poisson}\left( \frac{\lambda M_{\phi}}{L} \right)$. This distribution is equivalent to assigning $$\Lambda = \sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{L},$$ sampling the random variable $B \sim \text{Poisson}\left( \Lambda \right)$, and then sampling $r_{\phi} \sim \operatorname{Multinomial}\left(B, \frac{\lambda M_{\phi}}{\Lambda L} \right)$. If we re-think our distribution as coming from this process, then by the Law of Total Expectation, $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\frac{\exp(U_{\neg i}(x))}{nZ} \; \mathbf{E}_B\left[ \frac{1 - C}{1 + C} \cdot \mathbf{E}_r\left[
\frac{\exp\left( U_{x(i)} + U_{y(i)} \right)}{\int \exp(U_u) \; du} \middle| B \right] \right],\end{aligned}$$ where we can pull out the terms in $C$ because we can write $C$ to depend only on $B$ as $$C
=
\frac{4 \rho^{-m}}{\rho - 1} \cdot \exp\left( \frac{L}{\lambda} \sum_{\phi \in A[i]} r_{\phi} \right)
=
\frac{4 \rho^{-m}}{\rho - 1} \cdot \exp\left( \frac{L B}{\lambda} \right).$$ Next, we can bound this inner expectation with $$\begin{aligned}
&\mathbf{E}_r\left[
\frac{\exp\left( U_{x(i)} + U_{y(i)} \right)}{\int \exp(U_u) \; du} \middle| B \right] \\
&=
\mathbf{E}_r\left[
\frac{1}{\int \exp(U_u - U_{x(i)} - U_{y(i)}) \; du} \middle| B \right] \\
&\ge
\mathbf{E}_r\left[
\int \exp(U_u - U_{x(i)} - U_{y(i)}) \; du \middle| B \right]^{-1} \\
&=
\mathbf{E}_r\left[
\int
\exp\left(\sum_{\phi \in A[i]} r_{\phi} \left(
\log\left( 1 + \bar \phi(z_u) \right)
-
\log\left( 1 + \bar \phi(x) \right)
-
\log\left( 1 + \bar \phi(y) \right)
\right) \right)
\; du \middle| B \right]^{-1} \\
&=
\mathbf{E}_r\left[
\int
\exp\left(\sum_{\phi \in A[i]} r_{\phi} t_{\phi} \right)
\; du \middle| B \right]^{-1} \\
&=
\left( \int \mathbf{E}_r\left[
\exp\left(\sum_{\phi \in A[i]} r_{\phi} t_{\phi} \right)
\middle| B \right] \; du \right)^{-1},\end{aligned}$$ where we define $$t_{\phi}
=
\log\left( 1 + \bar \phi(z_u) \right)
-
\log\left( 1 + \bar \phi(x) \right)
-
\log\left( 1 + \bar \phi(y) \right).$$ This inner expectation is now just the moment-generating function of the multinomial distribution. Applying the standard formula for that MGF gives us $$\mathbf{E}_r\left[
\exp\left(\sum_{\phi \in A[i]} r_{\phi} t_{\phi} \right)
\middle| B \right]
=
\left(
\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi})
\right)^B.$$ Substituting this back into our original expression gives $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\frac{\exp(U_{\neg i}(x))}{nZ} \; \mathbf{E}_B\left[ \frac{1 - C}{1 + C} \cdot \left( \int \left(
\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi})
\right)^B \; du \right)^{-1} \right].\end{aligned}$$ Next, let $\delta > 0$ be a small constant, to be assigned later. Recall that for any non-negative random variable $X$ and any event $A$, by the Law of Total Probability, $${\mathbf{E}\left[X\right]}
=
{\mathbf{E}\left[X | A\right]} \cdot \mathbf{P}(A) + {\mathbf{E}\left[X | \neg A\right]} \cdot \mathbf{P}(\neg A)
\ge
{\mathbf{E}\left[X | A\right]} \cdot \mathbf{P}(A).$$ So, since the interior of this expectation is a non-negative number, it follows that $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\frac{\exp(U_{\neg i}(x))}{nZ} \; \mathbf{E}_B\left[ \frac{1 - C}{1 + C} \cdot \left( \int \left(
\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi})
\right)^B \; du \right)^{-1} \middle| C \le \delta \right]
\\&\hspace{4em}\cdot \mathbf{P}_B(C \le \delta) \\
&\ge
\frac{\exp(U_{\neg i}(x))}{nZ} \cdot \frac{1 - \delta}{1 + \delta} \cdot \mathbf{E}_B\left[ \left( \int \left(
\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi})
\right)^B \; du \right)^{-1} \middle| C \le \delta \right]
\\&\hspace{4em}\cdot \mathbf{P}_B(C \le \delta).\end{aligned}$$ By Jensen’s inequality again, we get $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\frac{\exp(U_{\neg i}(x))}{nZ} \; \mathbf{E}_B\left[ \frac{1 - C}{1 + C} \cdot \left( \int \left(
\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi})
\right)^B \; du \right)^{-1} \middle| C \le \delta \right]
\\&\hspace{4em}\cdot \mathbf{P}_B(C \le \delta) \\
&\ge
\frac{\exp(U_{\neg i}(x))}{nZ} \cdot \frac{1 - \delta}{1 + \delta} \cdot \left( \int \mathbf{E}_B\left[\left(
\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi})
\right)^B \middle| C \le \delta \right] \; du \right)^{-1}
\\&\hspace{4em}\cdot \mathbf{P}_B(C \le \delta).\end{aligned}$$ Since this inner expectation is again non-negative, we can again apply our above inequality, but in the opposite direction, giving $${\mathbf{E}\left[X | A\right]} \le \frac{{\mathbf{E}\left[X\right]}}{\mathbf{P}(A)}.$$ This produces $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\frac{\exp(U_{\neg i}(x))}{nZ} \cdot \frac{1 - \delta}{1 + \delta} \cdot \left( \int \mathbf{E}_B\left[\left(
\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi})
\right)^B \right] \; du \right)^{-1}
\\&\hspace{4em}\cdot \mathbf{P}_B(C \le \delta)^2.\end{aligned}$$ Now, we are just left with the MGF of a Poisson-distributed random variable. This we already know to be $$\begin{aligned}
\mathbf{E}_B\left[\left(
\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi})
\right)^B \right]
&=
\mathbf{E}_B\left[ \exp\left( B \log\left(
\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi})
\right) \right) \right] \\
&=
\exp\left( \Lambda
\left( \left( \sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi}) \right) - 1 \right) \right) \\
&=
\exp\left( \sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{L} \cdot \left( \exp(t_{\phi}) - 1 \right) \right),\end{aligned}$$ where in the last line we can leverage the fact that $$\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} = 1$$ to justify pulling the $-1$ inside the sum. From the analysis of Poisson-Gibbs, we had that $$\exp(t_{\phi}) - 1
\le
\bar \phi(z_u) - \bar \phi(x) - \bar \phi(y) + \frac{4L^2}{\lambda^2}.$$ So, $$\begin{aligned}
\mathbf{E}_B\left[\left(
\sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{\Lambda L} \cdot \exp(t_{\phi})
\right)^B \right]
&\le
\exp\left( \sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{L} \cdot \left(
\bar \phi(z_u) - \bar \phi(x) - \bar \phi(y) + \frac{4L^2}{\lambda^2}
\right) \right) \\
&=
\exp\left( \sum_{\phi \in A[i]} \left(
\phi(z_u) - \phi(x) - \phi(y) + \frac{4L M_{\phi}}{\lambda}
\right) \right) \\
&\le
\exp\left( \bar U_u - \bar U_{x(i)} - \bar U_{y(i)} + \frac{4L^2}{\lambda} \right),\end{aligned}$$ where as in the analysis of Poisson-Gibbs, $\bar U_v$ denotes the assignment of $U_v$ in the plain Gibbs sampling algorithm (Algorithm \[alg:gibbs\]), $$\bar U_v = \sum_{\phi \in A[i]} \phi(z_v).$$ Substituting this expression in to our overall bound, we get $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\frac{\exp(U_{\neg i}(x))}{nZ} \cdot \frac{1 - \delta}{1 + \delta} \cdot \left( \int \exp\left( \bar U_u - \bar U_{x(i)} - \bar U_{y(i)} + \frac{4L^2}{\lambda} \right) \; du \right)^{-1}
\\&\hspace{4em}\cdot \mathbf{P}_B(C \le \delta)^2 \\
&=
\frac{\exp(U(x))}{nZ} \cdot \frac{1 - \delta}{1 + \delta} \cdot \frac{\exp(\bar U_{y(i)})}{\int \exp\left( \bar U_u \right) \; du}
\\&\hspace{4em}\cdot \exp\left(-\frac{4L^2}{\lambda}\right) \cdot \mathbf{P}_B(C \le \delta)^2.\end{aligned}$$ Finally, if we let $G$ denote the transition probability operator of plain Gibbs sampling, we notice right away that $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\frac{1 - \delta}{1 + \delta}
\cdot \exp\left(-\frac{4L^2}{\lambda}\right) \cdot \mathbf{P}_B(C \le \delta)^2 \cdot \pi(x) G(x,y) \\
&\ge
(1 - 2\delta) \cdot \exp\left(-\frac{4L^2}{\lambda}\right) \cdot \mathbf{P}_B(C \le \delta)^2 \cdot \pi(x) G(x,y).\end{aligned}$$ To get a final bound, all we need to do is bound $\mathbf{P}_B(C \le \delta)$. This is straightforward, since $$\begin{aligned}
\mathbf{P}_B(C \le \delta)
&=
\mathbf{P}_B\left(\frac{4 \rho^{-m}}{\rho - 1} \cdot \exp\left( \frac{L B}{\lambda} \right) \le \delta \right) \\
&=
\mathbf{P}_B\left( \exp\left( \frac{L B}{\lambda} \right) \le \frac{\rho - 1}{4 \rho^{-m}} \cdot \delta \right).\end{aligned}$$ Notice that by the MGF formula for $B$, $$\mathbf{E}_B\left[ \exp\left( \frac{L B}{\lambda} \right) \right]
\le
\exp\left(\Lambda \left(\exp\left( \frac{L}{\lambda} \right) - 1 \right) \right).$$ Since we chose a minibatch size parameter $\lambda \ge 2 L$, it follows that $L / \lambda \le 1/2$, and so $$\exp\left( \frac{L}{\lambda} \right) - 1 \le \frac{2L}{\lambda},$$ and so since also $$\Lambda = \sum_{\phi \in A[i]} \frac{\lambda M_{\phi}}{L} \le \lambda.$$ it follows that $$\mathbf{E}_B\left[ \exp\left( \frac{L B}{\lambda} \right) \right]
\le
\exp\left(\lambda \cdot \frac{2L}{\lambda} \right)
=
\exp(2 L).$$ Therefore, by Markov’s inequality, $$\begin{aligned}
\mathbf{P}_B(C \ge \delta)
&=
\mathbf{P}_B\left( \exp\left( \frac{L B}{\lambda} \right) \ge \frac{\rho - 1}{4 \rho^{-m}} \cdot \delta \right) \\
&\le
\frac{\exp(2L)}{\frac{\rho - 1}{4 \rho^{-m}} \cdot \delta} \\
&\le
\frac{4 \rho^{-m}}{\rho - 1} \cdot \frac{\exp(2L)}{\delta}.\end{aligned}$$ Thus, $$\begin{aligned}
\mathbf{P}_B(C \le \delta)
&=
1 - \mathbf{P}_B(C \ge \delta) \\
&\ge
1 - \frac{4 \rho^{-m}}{\rho - 1} \cdot \frac{\exp(2L)}{\delta},\end{aligned}$$ and in particular $$\begin{aligned}
\mathbf{P}_B(C \le \delta)^2
&=
\left( 1 - \mathbf{P}_B(C \ge \delta) \right)^2 \\
&\ge
1 - 2 \mathbf{P}_B(C \ge \delta) \\
&\ge
1 - \frac{8 \rho^{-m}}{\rho - 1} \cdot \frac{\exp(2L)}{\delta}.\end{aligned}$$ Substituting this back into our overall bound gives us $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\frac{1 - \delta}{1 + \delta}
\cdot \exp\left(-\frac{4L^2}{\lambda}\right) \cdot \mathbf{P}_B(C \le \delta)^2 \cdot \pi(x) G(x,y) \\
&\ge
(1 - 2 \delta)
\cdot
\left(1 - \frac{8 \rho^{-m}}{\rho - 1} \cdot \frac{\exp(2L)}{\delta} \right)
\cdot
\exp\left(-\frac{4L^2}{\lambda}\right) \cdot \pi(x) G(x,y) \\
&\ge
\left(1 - 2 \delta - \frac{8 \rho^{-m}}{\rho - 1} \cdot \frac{\exp(2L)}{\delta} \right)
\cdot
\exp\left(-\frac{4L^2}{\lambda}\right) \cdot \pi(x) G(x,y).\end{aligned}$$ Finally, choosing the value of $\delta$ as $$\delta = \frac{2 \exp(L)}{\rho^{m/2} \cdot \sqrt{\rho - 1}},$$ we get $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\left(1 - \frac{8 \exp(L) \rho^{-m/2}}{\sqrt{\rho - 1}} \right)
\cdot
\exp\left(-\frac{4L^2}{\lambda}\right) \cdot \pi(x) G(x,y).\end{aligned}$$ Now applying the standard Dirichlet form argument, we get $$\bar \gamma
\ge
\left(1 - \frac{8 \exp(L) \rho^{-m/2}}{\sqrt{\rho - 1}} \right)
\cdot
\exp\left(-\frac{4L^2}{\lambda}\right) \cdot \gamma,$$ which was the desired expression.
Proof of Theorem \[thm:doupoly\]
--------------------------------
The reversibility can be proved by the same procedure as in Section \[sec:proof-thm-poly\]. By applying that same analysis, which did not depend on the manner in which the approximation $\tilde f$ was constructed, we can arrive at the expression $$\begin{aligned}
\pi(x)T(x, y)
&=
\frac{\exp(U_{\neg i}(x))}{nZ} \; \mathbf{E}_r\left[
\frac{1}{\int \tilde{f}(u)du}\cdot
\min\left(\tilde{f}(y(i))\exp\left( U_{x(i)} \right),
\tilde{f}(x(i)) \exp\left(U_{y(i)}\right)\right) \right].\end{aligned}$$ By the assumption of $\phi(z)$, we have $$\begin{aligned}
{\left|U_v\right|} &={\left| \sum_{\phi \in A[i]} r_{\phi} \log\left( 1 + \bar \phi(z_v) \right)\right|}\\
&\le \sum_{\phi \in A[i]} r_{\phi} {\left|\log\left( 1 + \frac{L}{\lambda M_{\phi}}\phi(x) \right)\right|}\\
&\le \sum_{\phi \in A[i]} r_{\phi} {\left|\frac{2L}{\lambda M_{\phi}}\phi(x)\right|}\\
&\le \frac{2L}{\lambda} \sum_{\phi \in A[i]} r_{\phi}.\end{aligned}$$ where the second inequality holds because $${\left|z\right|}\le \frac{1}{2}
\hspace{1em}\Rightarrow\hspace{1em}
{\left|\log(1 + z)\right|}\le 2{\left|z\right|},$$ using the assumptions $\lambda\ge 2L$ and ${\left|\phi(x)\right|}\le M_{\phi}$. Now applying Lemma \[thm:chebyext\] in Section \[sec:chebyext\], assigning $\sigma=\sqrt{\rho}$ gives us, $$\begin{aligned}
{\left|\tilde{U}_v - U_v\right|}
&\le \frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot \frac{L}{\lambda} \sum_{\phi \in A[i]} r_{\phi},\end{aligned}$$ for any $v$ in the shifted-and-scaled Bernstein ellipse with parameter $\sqrt{\rho}$.
Next, since $\tilde{U}_v$ is a polynomial in $v$, $\exp(\tilde{U}_v)$ must be analytic everywhere in $\mathbb{C}$. In particular it must be analytic on the Bernstein ellipse on the interval $[a, b]$ with parameter $\sqrt{\rho}$. On that interval, it is bounded by $$\begin{aligned}
{\left|\exp(\tilde{U}_v)\right|}
&\le \exp\left({\left|\tilde{U}_v\right|}\right)\\
&\le \exp\left({\left|U_v\right|} + {\left|\tilde U_v - U_v\right|}\right)\\
&\le \exp\left(\frac{2L}{\lambda}\sum_{\phi \in A[i]} r_{\phi}\right) \cdot \exp\left(\frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot \frac{L}{\lambda} \sum_{\phi \in A[i]} r_{\phi}\right)\\
&\le \exp\left(\frac{4\rho^{-\frac{m}{2}}+ \sqrt{\rho} - 1}{\sqrt{\rho} - 1}\cdot \frac{2L}{\lambda} \sum_{\phi \in A[i]} r_{\phi}\right).\end{aligned}$$ Now applying Theorem \[thm:cheby\] using the Bernstein ellipse with parameter $\sqrt{\rho}$, we have, for any $v$ on the interval $[a,b]$, $${\left|\tilde{f}(v) - \exp(\tilde{U}_v)\right|}\le \frac{4\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}\cdot \exp\left(\frac{4\rho^{-\frac{m}{2}}+ \sqrt{\rho} - 1}{\sqrt{\rho} - 1}\cdot \frac{2L}{\lambda} \sum_{\phi \in A[i]} r_{\phi}\right)$$ Therefore, it follows that $$\begin{aligned}
{\left|\frac{\tilde{f}(v)}{\exp(U_v)} - 1\right|}
&\le
{\left|\frac{\tilde{f}(v) - \exp(\tilde U_v) + \exp(\tilde U_v)}{\exp(U_v)} - 1\right|} \\
&\le
\frac{{\left|\tilde{f}(v) - \exp(\tilde U_v)\right|}}{\exp(U_v)}
+
{\left|\exp(\tilde U_v - U_v) - 1\right|} \\
&\le
{\left|\tilde{f}(v) - \exp(\tilde U_v)\right|}
+
\exp\left({\left|\tilde U_v - U_v\right|}\right) - 1,\end{aligned}$$ where the last inequality is justified by the fact that $U_v$ is non-negative and for any $x$, ${\left|\exp(x) - 1\right|} \le \exp({\left|x\right|}) - 1$. Now substituting in our bounds from above gives us $$\begin{aligned}
&\hspace{-1em}{\left|\frac{\tilde{f}(v)}{\exp(U_v)} - 1\right|} \\
&\le \exp\left(\frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot \frac{L}{\lambda} \sum_{\phi \in A[i]} r_{\phi}\right) + \frac{4\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}\cdot \exp\left(\frac{4\rho^{-\frac{m}{2}}+ \sqrt{\rho} - 1}{\sqrt{\rho} - 1}\cdot \frac{2L}{\lambda} \sum_{\phi \in A[i]} r_{\phi}\right) - 1\end{aligned}$$ As before, we let $B = \sum_{\phi\in A[i]}r_{\phi}$ where $B\sim \text{Poisson}(\Lambda)$. Then $$\begin{aligned}
{\left|\frac{\tilde{f}(v)}{\exp(U_v)} - 1\right|}
&\le \exp\left(\frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot \frac{LB}{\lambda} \right) + \frac{4\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}\cdot \exp\left(\frac{4\rho^{-\frac{m}{2}}+ \sqrt{\rho} - 1}{\sqrt{\rho} - 1}\cdot \frac{2LB}{\lambda}\right) - 1\end{aligned}$$ We define $$E = \exp\left(\frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot \frac{LB}{\lambda} \right) + \frac{4\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}\cdot \exp\left(\frac{4\rho^{-\frac{m}{2}}+ \sqrt{\rho} - 1}{\sqrt{\rho} - 1}\cdot \frac{2LB}{\lambda}\right) - 1,$$ and by following the same steps as used in Section \[sec:proof-thm-poly\], with $E$ in place of the $C$ of that proof, we can get, for any constant $\delta > 0$, $$\begin{aligned}
\pi(x)T(x, y)
&\ge
(1 - 2\delta) \cdot \exp\left(-\frac{4L^2}{\lambda}\right) \cdot \mathbf{P}_B(E \le \delta)^2 \cdot \pi(x) G(x,y).\end{aligned}$$ All that remains is to bound $\mathbf{P}_B(E \le \delta)$. Using the MGF formula for $B$ twice, we get that $$\begin{aligned}
\mathbf{E}_B(E) &= \frac{4\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}\cdot \exp\left(\Lambda\left(\exp\left(\frac{4\rho^{-\frac{m}{2}}+ \sqrt{\rho} - 1}{\sqrt{\rho} - 1}\cdot \frac{2L}{\lambda}\right) - 1\right) \right)
\\&\hspace{2em} + \exp\left(\Lambda\left(\exp\left(\frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot\frac{L}{\lambda}\right) - 1\right) \right) - 1.\end{aligned}$$ If we require that $m$ is large enough that $$4\rho^{-\frac{m}{2}} \le \sqrt{\rho} - 1,$$ then $$\begin{aligned}
\mathbf{E}_B(E) &\le \frac{4\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}\cdot \exp\left(\Lambda\left(\exp\left( \frac{4L}{\lambda}\right) - 1\right) \right)
\\&\hspace{2em} + \exp\left(\Lambda\left(\exp\left(\frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot\frac{L}{\lambda}\right) - 1\right) \right) - 1.\end{aligned}$$ By Taylor’s theorem, for $x > 0$, $$\exp(x) - 1 = \exp(x) - \exp(0) \le x \cdot \exp(x).$$ So, since $\Lambda \le \lambda$, we can bound our expectation with $$\begin{aligned}
\mathbf{E}_B(E) &\le \frac{4\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}\cdot \exp\left(\Lambda \cdot \frac{4L}{\lambda} \cdot \exp\left( \frac{4L}{\lambda}\right) \right)
\\&\hspace{2em} + \exp\left(\Lambda \cdot \frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot\frac{L}{\lambda} \cdot \exp\left(\frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot\frac{L}{\lambda}\right) \right) - 1
\\&\le
\frac{4\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}\cdot \exp\left(4L \cdot \exp\left( \frac{4L}{\lambda}\right) \right)
\\&\hspace{2em} + \exp\left( \frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot L \cdot \exp\left(\frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot\frac{L}{\lambda}\right) \right) - 1
\\&\le
\frac{4\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}\cdot \exp\left(4L \cdot \exp\left( \frac{4L}{\lambda}\right) \right)
\\&\hspace{2em} + \exp\left( \frac{8\rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1}\cdot L \cdot \exp\left(\frac{4L}{\lambda}\right) \right) - 1.\end{aligned}$$ Since $\lambda \log(2) \ge 4 L$, we can bound $\exp(4L/\lambda) \le 2$, and so $$\begin{aligned}
\mathbf{E}_B(E) &\le
\frac{4\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}\cdot \exp\left(8 L \right)
+ \exp\left( \frac{16 L \rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1} \right) - 1.\end{aligned}$$ We now define $$F
=
\frac{4\cdot \exp\left(8 L \right)\cdot\rho^{-\frac{k}{2}}}{\sqrt{\rho} - 1}
+ \exp\left( \frac{16 L \rho^{-\frac{m}{2}}}{\sqrt{\rho} - 1} \right) - 1.$$ By Markov’s inequality, $$\begin{aligned}
&\hspace{-0em}\mathbf{P}_B(E \ge \delta)\ge \frac{\mathbf{E}_B(E)}{\delta}\ge F/\delta.\end{aligned}$$ It follows $$\begin{aligned}
\mathbf{P}_B(E \le \delta)^2 = \left(1 - \mathbf{P}_B(E \ge \delta)\right)^2\ge 1 - 2\mathbf{P}_B(E \ge \delta)\ge 1 - 2F/\delta.\end{aligned}$$ Substituting it back into the overall bound, $$\begin{aligned}
\pi(x)T(x, y)
&\ge
(1 - 2\delta) \cdot \exp\left(-\frac{4L^2}{\lambda}\right) \cdot \mathbf{P}_B(E \le \delta)^2 \cdot \pi(x) G(x,y)\\
&\ge
\left(1 - 2\delta - \frac{2F}{\delta}\right)\cdot\exp\left(-\frac{4L^2}{\lambda}\right)\cdot \pi(x) G(x,y)\end{aligned}$$ Let $$\delta = \sqrt{F},$$ it becomes $$\begin{aligned}
\pi(x)T(x, y)
&\ge
\left(1 - 4\sqrt{F}\right)\cdot\exp\left(-\frac{4L^2}{\lambda}\right)\cdot \pi(x) G(x,y)\end{aligned}$$ Again, using the Dirichlet form we bound the spectral gap, $$\begin{aligned}
\bar{\gamma}
&\geq
\left(1 - 4\sqrt{F}\right)\exp\left(\frac{-4L^2}{\lambda}\right)\cdot \gamma\end{aligned}$$
Poisson-MH {#sec:MH}
==========
We apply our Poisson-minibatching method to Metropolis-Hasting sampling. In Poisson-minibatching M-H (Poisson-MH), we first generate a candidate $x^*$ from the proposal distribution $q(x^*|x)$. Then the M-H ratio will be calculated as following $$\begin{aligned}
p = \frac{\exp\left(\sum_{\phi \in S} s_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}}\phi(x^*) \right)\right)q(x^*|x)}{\exp\left(\sum_{\phi \in S} s_{\phi} \log\left( 1 + \frac{L}{\lambda M_{\phi}}\phi(x) \right)\right)q(x|x^*)}\end{aligned}$$
We accept $x^*$ with the probability $\min(1, p)$. After applying Poisson-minibatching, the M-H ratio no longer needs to use the whole dataset which will reduce the computational cost significantly.
Theorem \[thm:mh\] is similar to the bounds of Poisson-Gibbs. As long as we set $\lambda = \Theta(L^2)$, the convergence is slowed down by at most a constant factor which is unrelated to the size of the problem.
Proof of Theorem \[thm:mh\]
---------------------------
We begin with the transition probability from $x$ to $x^*$ $$\begin{aligned}
&T( x^*, x)\\
&=
\mathbf{E}\left\{q( x^*| x)\min\left(1, \frac{q( x| x^*)\pi( x^*, s)}{q( x^*| x)\pi( x, s)}\right)\right\}\\
&=
\mathbf{E}\left\{q( x^*| x)\min\left(1, \frac{q( x| x^*)\exp\left( \sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x^*)\right) - \log s_{\phi}!\right]\right)}
{q( x^*| x)\exp\left( \sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x)\right) - \log s_{\phi}!\right]\right)}\right)\right\}\\
&=
\mathbf{E}\left\{q( x^*| x)\min\left(1, \frac{q( x| x^*)\exp\left( \sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x^*)\right)\right]\right)}
{q( x^*| x)\exp\left( \sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x)\right) \right]\right)}\right)\right\}\\
&=
\sum_s\left\{q( x^*| x)\min\left(1, \frac{q( x| x^*)\exp\left( \sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x^*)\right)\right]\right)}
{q( x^*| x)\exp\left( \sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x)\right) \right]\right)}\right)\right\}\prod_{\phi \in \Phi} p(s_{\phi}| x)\\
&=
\sum_s\left\{q( x^*| x)\min\left(\exp\left(\sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x)\right)
- \phi( x) - \frac{\lambda M_{\phi}}{L}- \log s_{\phi}!\right] \right),\right.\right.\\
&\hspace{2em}\left.\left.\frac{q( x| x^*)\exp\left( \sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x^*)\right)\right]\right)}
{q( x^*| x)\exp\left( \sum_{\phi\in \Phi}\phi( x) + \frac{\lambda M_{\phi}}{L}+ \log s_{\phi}! \right)}\right)\right\}\\
&=
\sum_s\left\{q( x^*| x)\min\left(\exp\left(\sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x)\right)
- \phi( x) - \frac{\lambda M_{\phi}}{L}- \log s_{\phi}!\right] \right),\right.\right.\\
&\hspace{2em}\left.\left.\frac{q( x| x^*) }
{q( x^*| x) }\exp\left(\sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x^*)\right)
- \phi( x) - \frac{\lambda M_{\phi}}{L}- \log s_{\phi}!\right] \right)\right)\right\}\end{aligned}$$
Multiplying $\pi(x)$ to both sides, $$\begin{aligned}
&\pi( x)T( x^*, x)\\
&=
\frac{1}{Z}\exp\left(\sum_{\phi\in \Phi}\phi( x) \right)T( x^*, x)\\
&=
\frac{1}{Z}\sum_s
\min\Bigg(q( x^*| x) \left(\exp\left(\sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x)\right) - \frac{\lambda M_{\phi}}{L}- \log s_{\phi}!\right] \right),\right.\\
&\hspace{2em}\left.q( x| x^*)
\exp\left(\sum_{\phi\in \Phi} \left[s_{\phi}\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x^*)\right) - \frac{\lambda M_{\phi}}{L}- \log s_{\phi}!\right] \right)\right)\Bigg) \end{aligned}$$
This implies the Markov chain is reversible.
We can continue to reduce this to $$\begin{aligned}
&\pi( x) T( x^*, x)\\
&=
\frac{1}{Z}\sum_s
\min\left(q( x^*| x) \exp\left(\sum_{\phi\in \Phi} s_{\phi}\left[\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x) \right)- \log \frac{\lambda M_{\phi}}{L}\right] \right),\right.\\
&\hspace{2em}\left.q( x| x^*)
\exp\left(\sum_{\phi\in \Phi} s_{\phi}\left[\log\left( \frac{\lambda M_{\phi}}{L}+ \phi( x^*) \right)- \log \frac{\lambda M_{\phi}}{L}\right] \right)\right)
\\&\hspace{2em}\cdot \prod_{\phi \in \Phi} \frac{1}{s_{\phi}!}\exp\left(- \frac{\lambda M_{\phi}}{L}\right)\left( \frac{\lambda M_{\phi}}{L}\right)^{s_{\phi}}\\
&=
\frac{1}{Z}\sum_s
\min\left(q( x^*| x) \exp\left(\sum_{\phi\in \Phi} s_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x) \right) \right),\right.\\
&\hspace{2em}\left.q( x| x^*)
\exp\left(\sum_{\phi\in \Phi} s_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x^*) \right) \right)\right)
\cdot \prod_{\phi \in \Phi} \frac{1}{s_{\phi}!}\exp\left(- \frac{\lambda M_{\phi}}{L}\right)\left( \frac{\lambda M_{\phi}}{L}\right)^{s_{\phi}} \end{aligned}$$
Similar to the previous proof, $s_{\phi}$ here are non-negative integers that a Poisson variable can take, not variables. So if we let $r_{\phi} \sim \text{Poisson}\left( \frac{\lambda M_{\phi}}{L} \right)$ and $r_{\phi}$ to be all independent, we can write this as $$\begin{aligned}
\pi( x) T( x^*, x)
&=
\frac{1}{Z}\mathbf{E}
\min\left(q( x^*| x) \exp\left(\sum_{\phi\in \Phi} r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x) \right) \right),\right.\\
&\left.q( x| x^*)
\exp\left(\sum_{\phi\in \Phi} r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x^*)\right)\right)\right)\end{aligned}$$
Assume $G( x^*, x)$ is the transition operator of a plain MCMC. Consider the ratio, $$\begin{aligned}
\frac{\pi( x) T( x^*, x)}
{\pi( x) G( x^*, x)}
&=
\frac{1}{Z}\mathbf{E}
\min\Bigg(q( x^*| x)\exp\left(\sum_{\phi\in \Phi} r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x) \right) \right),
\\&\hspace{2em}q( x| x^*)\exp\left(\sum_{\phi\in \Phi} r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x^*)\right) \right)\Bigg)
\\&\hspace{2em}\cdot\Bigg[1\bigg/\Bigg(\frac{1}{Z}
\min\left(q( x^*| x)\exp\left(\sum_{\phi \in \Phi}\phi( x)\right),
q( x| x^*)\exp\left(\sum_{\phi \in \Phi}\phi( x^*)\right) \right)\Bigg)\Bigg]\end{aligned}$$
We know that $\frac{\min(A, B)}{\min(C,D)} = \min\left(\frac{A}{\min(C,D)}, \frac{B}{\min(C,D)}\right) \geq \min\left(\frac{A}{C}, \frac{B}{D}\right)$. The last inequality is due to the fact that $\frac{1}{\min(C,D)}\geq \frac{1}{C}$ and $\frac{1}{\min(C,D)}\geq \frac{1}{D}$.
With this inequality, we can continue simplifying the ratio, $$\begin{aligned}
\frac{\pi( x) T( x^*, x)}
{\pi( x) G( x^*, x)}
&\geq
\mathbf{E}\Bigg[\min \Bigg(\frac{
\exp\left(\sum_{\phi\in \Phi} r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x) \right) \right)}
{
\exp\left(\sum_{\phi \in \Phi}\phi( x)\right)},
\\&\hspace{2em}\frac{
\exp\left(\sum_{\phi\in \Phi} r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x^*)\right) \right)}
{
\exp\left(\sum_{\phi \in \Phi}\phi( x^*)\right)}
\Bigg)\Bigg]\\
&=
\mathbf{E}\Bigg[\min \Bigg(
\exp\left(\sum_{\phi\in \Phi}\Bigg( r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x) \right) - \phi( x) \Bigg)\right),
\\&\hspace{2em}\exp\left(\sum_{\phi\in \Phi}\Bigg( r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x^*)\right) - \phi( x^*) \Bigg)\right)
\Bigg)\Bigg]\\
&=
\mathbf{E}\Bigg[\max \Bigg(
\exp\left(\sum_{\phi\in \Phi} \Bigg(\phi( x) - r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x) \right)\Bigg)\right),
\\&\hspace{2em}\exp\left(\sum_{\phi\in \Phi} \Bigg(\phi( x^*) - r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}}\phi( x^*)\right)\Bigg) \right)
\Bigg)^{-1}\Bigg]\end{aligned}$$
Because $f(x) = \frac{1}{x}$ is a convex function, by Jensen’s inequality it follows $$\begin{aligned}
\frac{\pi( x) T( x^*, x)}
{\pi( x) G( x^*, x)}
&\geq
\mathbf{E}\Bigg[\max \Bigg(
\exp\left(\sum_{\phi\in \Phi} \Bigg(\phi( x) - r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}} \phi( x) \right)\Bigg) \right),
\\&\hspace{2em}\exp\left(\sum_{\phi\in \Phi}\Bigg( \phi( x^*) - r_{\phi}\log\left(1+ \frac{L}{\lambda M_{\phi}}\phi( x^*)\right) \Bigg)\right)
\Bigg)\Bigg]^{-1}\end{aligned}$$ We have that the maximum of the product is less than the product of maximum, therefore $$\begin{aligned}
\frac{\pi( x) T( x^*, x)}
{\pi( x) G( x^*, x)}
&\geq
\prod_{\phi\in\Phi}\mathbf{E}\Bigg[\max \Bigg(
\exp \Bigg(\phi( x) - r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}} \phi( x) \right)\Bigg) ,
\\&\hspace{2em}\exp\Bigg( \phi( x^*) - r_{\phi}\log\left(1+ \frac{L}{\lambda M_{\phi}}\phi( x^*)\right) \Bigg)
\Bigg)\Bigg]^{-1}\end{aligned}$$ Since $\max(A,B)\leq A + B$ when $A$ and $B$ are positive, it follows $$\begin{aligned}
\frac{\pi( x) T( x^*, x)}
{\pi( x) G( x^*, x)}
&\geq
\prod_{\phi\in\Phi}\mathbf{E}\Bigg[
\exp \Bigg(\phi( x) - r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}} \phi( x) \right)\Bigg) +
\\&\hspace{2em}\exp\Bigg( \phi( x^*) - r_{\phi}\log\left(1+ \frac{L}{\lambda M_{\phi}}\phi( x^*)\right) \Bigg)\Bigg]^{-1}\end{aligned}$$ $\mathbf{E}\Bigg[
\exp \Bigg( - r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}} \phi( x) \right)\Bigg)\Bigg]$ is the moment generating function of the Poisson random variable $r_{\phi}$ evaluated at $$t = -\log\left( 1+ \frac{L}{\lambda M_{\phi}} \phi( x) \right)$$ We know that $$\begin{aligned}
\mathbf{E}\exp(r_{\phi}t)
&=
\exp\left( \frac{\lambda M_{\phi}}{L}\left(\exp(t) - 1\right)\right)\\\end{aligned}$$ Therefore, $$\begin{aligned}
\mathbf{E}\Bigg[
\exp \Bigg( - r_{\phi}\log\left( 1+ \frac{L}{\lambda M_{\phi}} \phi( x) \right)\Bigg)\Bigg]
&=
\exp\left(-\frac{\phi(x)}{1+\frac{L}{\lambda M_{\phi}}\phi(x)}\right)\end{aligned}$$ Substituting this into the original expression produces $$\begin{aligned}
\frac{\pi( x) T( x^*, x)}
{\pi( x) G( x^*, x)}
&\geq
\left[2\prod_{\phi\in\Phi}\exp \Bigg(-\frac{\phi(x)}{1+\frac{L}{\lambda M_{\phi}}\phi(x)} + \phi(x)\Bigg)\right]^{-1}\\
&\geq
\left[2\prod_{\phi\in\Phi}\exp(M_{\phi})\exp \Bigg(-\frac{1}{1+\frac{L}{\lambda }} + 1\Bigg)\right]^{-1}\\
&=
\left[2\prod_{\phi\in\Phi}\exp(M_{\phi})\exp \Bigg(\frac{L}{\lambda + L} \Bigg)\right]^{-1}\\
&=
\left[2\exp \Bigg(\frac{L^2}{\lambda + L} \Bigg)\right]^{-1}\\
&=
\frac{1}{2}\exp \Bigg(-\frac{L^2}{\lambda + L} \Bigg)\end{aligned}$$
From Dirichlet form argument, we get $$\bar{\gamma}
\ge
\frac{1}{2}\exp\left( - \frac{L^2}{\lambda + L}\right)\cdot\gamma.$$
Additional Experiment: Poisson-MH on Truncated Gaussian Mixture
---------------------------------------------------------------
We test Poisson-MH on the truncated Gaussian mixture as in Section \[sec:gmm\]. The proposal is $q(x^*|x)=\mathcal{N}(x, 0.45^2 I)$. We set $\lambda=500$. The estimated density is in Figure \[fig:mog-pmh\] which is very close to the true density. This demonstrates the effectiveness of Poisson-MH and the general applicability of Poisson-minibatching method.
{width="6.5cm"}
Extended Results about Chebyshev Interpolants {#sec:chebyext}
=============================================
In @trefethen2013approximation, Theorem 8.2 proves bounds on the error of a Chebyshev interpolant on the interval $[-1,1]$. However, in order to apply this theorem to a second Chebyshev interpolant that is a function of the first, we would need to bound the magnitude of that function *on a Bernstein ellipse*. To do this, we need the following extended version of Theorem 8.2, which bounds the error not only on the interval $[-1,1]$ but more generally on a Bernstein ellipse.
\[thm:chebyext\] Assume $U: \mathbb{C} \rightarrow \mathbb{C}$ is analytic in the open Bernstein ellipse $B([-1, 1], \rho)$, where the Bernstein ellipse is a region in the complex plane bounded by an ellipse with foci at $\pm 1$ and semimajor-plus-semiminor axis length $\rho > 1$. If for all $x \in B([-1, 1], \rho)$, $|U(x)| \leq V$ for some constant $V > 0$, then for any constant $1 < \sigma < \rho$, the error of the Chebyshev interpolant on the smaller Bernstein ellipse $B([-1,1], \sigma)$ is bounded by $$\begin{aligned}
| \tilde{U}(x) - U(x) | \leq \frac{4V}{\rho / \sigma - 1} \cdot \left( \frac{\rho}{\sigma} \right)^{-m}.\end{aligned}$$
This proof is essentially identical to that of Theorem 8.2 in @trefethen2013approximation, except that the error is bounded in a Bernstein ellipse rather than over only the real interval $[-1,1]$.
First, note that one parameterization of the boundary of the Bernstein ellipse with parameter $\rho$ is $$\left\{ \frac{z + z^{-1}}{2} \middle| z \in \mathbb{C}, \; {\left|z\right|} = \rho \right\},$$ and the open ellipse itself can be written as $$B([-1, 1], \rho) = \left\{ \frac{z + z^{-1}}{2} \middle| z \in \mathbb{C}, \; \rho^{-1} \le {\left|z\right|} \le \rho \right\}.$$ Now, Theorem 8.1 from @trefethen2013approximation states that the Chebyshev coefficients of a function that satisfies the conditions of this theorem (boundedness and analyticity in a Bernstein ellipse) are bounded by ${\left|a_0\right|} \le V$ and $${\left| a_k \right|} \le 2 V \rho^{-k}, \; k \ge 1.$$ That is, for $a_k$ bounded in this way, $$U(x) = \sum_{k=0}^{\infty} a_k T_k(x)$$ at least for all $x$ in the $\rho$-Bernstein ellipse on which $f$ is analytic. (While @trefethen2013approximation only states explicitly that this holds for $x \in [-1,1]$, the fact that it also holds on the rest of the Bernstein ellipse follows directly from the fact that both sides of the equation are analytic over that region, using the identity theory for holomorphic functions.) Formula (4.9) from @trefethen2013approximation states that $$U(x) - \tilde U_m(x)
=
\sum_{k=m+1}^{\infty} a_k \left( T_k(x) - T_{l(k,m)}(x) \right)$$ where $\tilde U_m$ denotes the degree-$m$ Chebyshev interpolant, and $$l(k,m) = {\left| \left( (k + m - 1) \bmod 2m \right) - (m - 1) \right|}.$$ Notice in particular that it always holds that $l(k,m) \le m+1$. Now, for $x$ inside the Bernstein ellipse $B([-1, 1], \sigma)$, there will always exist a $z \in \mathbb{C}$ such that $\sigma^{-1} \le {\left|z\right|} \le \sigma$ and $$x = \frac{z + z^{-1}}{2}.$$ For such an $x$, and for any $k$, $${\left|T_k(x)\right|}
=
{\left| T_k\left( \frac{z + z^{-1}}{2} \right) \right|}
=
{\left| \frac{z^k + z^{-k}}{2} \right|}
=
\frac{{\left|z\right|}^k + {\left|z\right|}^{-k}}{2}
\le
\sigma^k,$$ where the second equality is a well-known property of the Chebyshev polynomials. It follows that, for any $x$ in this Bernstein ellipse, $$\begin{aligned}
{\left| U(x) - \tilde U_m(x) \right|}
&=
{\left| \sum_{k=m+1}^{\infty} a_k \left( T_k(x) - T_{l(k,m)}(x) \right) \right|}
\\ &\le
\sum_{k=m+1}^{\infty} {\left|a_k\right|} \cdot {\left| T_k(x) - T_{l(k,m)}(x) \right|}
\\ &\le
\sum_{k=m+1}^{\infty} 2 V \rho^{-k} \cdot \left( \sigma^k + \sigma^{l(k,m)} \right)
\\ &\le
4 V \sum_{k=m+1}^{\infty} \rho^{-k} \sigma^k
\\ &\le
4 V \left(\frac{\sigma}{\rho} \right)^{m+1} \sum_{k=0}^{\infty} \left(\frac{\sigma}{\rho} \right)^k
\\ &\le
4 V \left(\frac{\sigma}{\rho} \right)^{m+1} \frac{1}{1 - \frac{\sigma}{\rho}}
\\ &\le
4 V \left(\frac{\sigma}{\rho} \right)^{m} \frac{1}{\rho / \sigma - 1}.\end{aligned}$$ This is the desired result.
|
---
abstract: 'Recent weak emission-line long-slit surveys and modelling studies of PNe have convincingly argued in favour of the existence of an unknown component in the planetary nebula plasma consisting of cold, hydrogen-deficient gas, as an explanation for the long-standing recombination-line versus forbidden-line temperature and abundance discrepancy problems. Here we describe the rationale and initial results from a detailed spectroscopic study of three Galactic PNe undertaken with the VLT FLAMES integral-field unit spectrograph, which advances our knowledge about the small-scale physical properties, chemical abundances and velocity structure of these objects across a two-dimensional field of view, and opens up for exploration an uncharted territory in the study and modelling of PNe and photoionized nebulae in general.'
date: '?? and in revised form ??'
---
Introduction
============
In recent years, extensive weak emission-line surveys of PNe have all focused on the optical recombination-line (ORL) spectra of carbon, nitrogen, oxygen and neon ions, and the comparison between abundances derived from ORLs versus those from the bright collisionally excited lines (CELs) of these objects (Tsamis et al. 2003b, 2004; Liu et al. 2004; Wesson et al. 2005). These studies show that for the majority of PNe abundances of the above elements, relative to hydrogen, from ORLs are a factor of 2–3 higher than the corresponding CEL values. For about 5–10% of PNe however, the abundance discrepancy factors (ADF) are in the range of 4–80. Important correlations have been demonstrated such as those between the ADFs for a range of ions and the difference between the \[O [iii]{}\] and Balmer jump temperatures, or between the ADF and the PN radius and intrinsic brightness, whereby larger/fainter (and arguably older) PNe display higher ADFs than more compact, young objects. All the above surveys, however, were based on data secured with spectrographs using long-slits (e.g. ESO 1.52-m Boller & Chivens, NTT 3.5-m EMMI) operated either at fixed nebular positions or in some cases scanned across the nebular surface, thus yielding average spectra of a given target with the unavoidable loss of spatial information in the direction perpendicular to the slit. A handful of studies charted the ORL/CEL abundances, the ADF behaviour and other nebular properties along the useful slit length for a few PNe (e.g. NGC6153, Liu et al. 2000; NGC6720, Garnett & Dinerstein 2001; NGC7009, Krabbe & Copetti 2006), as well as for the HII region 30 Doradus (Tsamis et al. 2003a). The spatial resolution of these however was only moderate (usually worse than 1$''$/pix) and the spectra did not sample the seeing. The FWHM spectral resolution, $\lambda$/$\Delta\lambda$ ($=R$), was also not better than $\sim$3000. Similar studies with échelle spectrographs, such as VLT UVES (e.g. Peimbert et al. 2004), fared better in spectral resolution ($\sim$8800), but the spatial coverage of those was even more restricted.
The advent of integral-field unit (IFU) spectrographs has opened up new options for PN research (see also Roth, this volume). IFUs provide high spatio-spectral resolution over a large field of view: with the use of arrays of small microlenses/fibres the seeing can be sampled and contiguous fibre coverage prevents flux losses, in contrast to long-slit techniques. Our instrument of choice, VLT FLAMES in Argus mode, provides coverage of a 12$''\times7''$ or 6.6$''\times4.2''$ field of view at spatial resolutions of 0.52$''$ or 0.30$''$, respectively, and at medium to high $R$. In this work we have used FLAMES/Argus to observe the Galactic disk PNe NGC 5882, 7009, and 6153. Our published long-slit studies show that these PNe have ADFs of $\sim$2, 5 and 10, respectively. It was argued for a variety of reasons, but primarily since the *ISO* fine-structure CELs yield consistent abundances with the optical CELs, that small-scale temperature fluctuations do not contribute significantly to the observed ORL/CEL abundance discrepancies, and the CEL abundances do provide a representative measure of the gas metallicity in these targets. The CNONe ORL emission, on the other hand, is best explained by the presence of a small amount of gas (1–2% of the total ionized nebular matter) which is hydrogen-deficient, and therefore cold ($\lesssim$1000K), and which does not emit high-excitation energy UV/optical CELs. Our Argus data allow us to map the two-dimensional distribution of heavy-element ORL vs. CEL emission across the PNe in unprecedented detail and are expected to yield new insights into the astrophysics of these objects.
FLAMES/Argus observations
=========================
![Single 0.52$''$ spaxel FLAMES/Argus spectrum of NGC6153 showing the optical recombination lines from CNONe ions. Note the full complement of O [ii]{} V1 multiplet ORLs near 4650Å.[]{data-label="fig:wave"}](tsamisf1.eps)
The spectra were obtained in queue mode between March–August 2005 in subarcsec seeing. NGC7009 was observed with both small and large IFUs, while NGC5882 & NGC6153 were observed with the large IFU only. The large IFU covered roughly one quadrant of the surface of NGC6153 & NGC7009, and was placed along the radial direction so as to sample both inner and outer nebular regions. The bright inner shell of NGC5882 was almost wholly covered with the 12$''\times7''$ IFU (Fig. 2). Spectra of NGC7009 (4188–4392, 4538–4759Å) secured with the small IFU have a $R$ of 32,500 ($=$9.2km/s) allowing us to measure gas radial velocities to an accuracy of a few km/s: we can thus probe whether heavy element ORLs arise from kinematically distinct regions than CELs in this nebula, by tracking subtle differences in the velocity profiles of CELs vs. ORLs as a function of position on the emission-line maps, and via the detection of asymmetrical profiles/faint line wings (Fig. 3). The 3964–5078Å spectra of all 3 PNe taken with the large IFU have a $R$ of $\sim$10,000–12,000 ($=$25–30km/s), comparable to the typical expansion velocity of a PN and thus optimal for the detection of faint ORLs. Fig. 1 shows a representative deep spectrum of NGC6153 extracted from a *single* 0.52$''$ spaxel, reaching weak O [ii]{} ORLs (of intensity less than 1% that of H$\beta$) at a S/N ratio greater than 10.
First results
=============
The data were reduced with the girBLDRS pipeline provided by the Geneva Observatory and flux calibrated within IRAF. Custom-made IRAF routines allowed us to create data cubes and to extract high S/N ratio emission-line maps by fitting Gaussians to the lines. The maps were then dereddened and converted to physical quantities, such as electron temperatures ($T_{\rm e}$), densities ($N_{\rm e}$) and abundances. In Fig. 2(*left*) the \[O [iii]{}\] $\lambda$4959 map of NGC5882 is shown. In Fig. 2(*centre*) the forbidden-line O$^{2+}$/H$^+$ abundance map of the SE quadrant of NGC6153 is shown: it was created using an \[O [iii]{}\] $\lambda$4959/H$\beta$ ratio map, an \[O [iii]{}\] $\lambda$4363/$\lambda$4959 $T_{\rm e}$ map and an \[Ar [iv]{}\] $\lambda$4711/$\lambda$4740 $N_{\rm e}$ map. The central star is at the bottom-right corner. The mean O$^{2+}$/H$^+$ abundance ratio from the map is 3.9$\times$10$^{-4}$, which compares well with the value of 4.3$\times$10$^{-4}$ from the scanning long-slit ESO 1.52-m study of the entire PN (27$''\times34''$) by Liu et al. (2000). In Fig. 2(*right*) the recombination-line O$^{2+}$/H$^+$ abundance map of the same region is shown: it was created from an O [ii]{} $\lambda$4649/H$\beta$ ratio map. The mean O$^{2+}$/H$^+$ value from that map is 4.0$\times$10$^{-3}$ compared to 2.8$\times$10$^{-3}$ from the scanning long-slit study. The derived ADF for O$^{2+}$ from the ratio of the two NGC6153 maps ranges from $\sim$5 to 20 with decreasing nebular radius and peaks around the central star; this directly implicates the central star itself, or the hard radiation field close to it, in the ORL/CEL discrepancy problem. Maps such as these will also allow us to track the pixel to pixel variation of crucial quantities such as, for example, the $T_{\rm e}$ from the O [ii]{} $\lambda4089$/$\lambda4649$ ORL ratio, or the $N_{\rm e}$ from the intramultiplet intensity ratios of O [ii]{} V1 ORLs and can throw new light on the nature of the hydrogen-poor nebular component.
Fig. 3 shows the high resolution spectrum of a 0.9$''\times0.9''$ region in NGC7009, near the projected edge of the bright inner nebular shell, covering the C [iii]{} 4647.42, 4650.25, 4651.47Å and O [ii]{} 4649.13, 4650.84Å ORLs. Each line exhibits 3 components with central wavelengths at $\sim-93, -60$ and $-28$km/s, probably corresponding to the edge of the inner PN shell (central peak) and the expanding fainter, outer shell (peaks on either side). In the upper panel observed (red) and synthetic (green) spectra are shown. The intrinsic synthetic spectra (before convolution with the instrumental profile) of C [iii]{} (orange) and O [ii]{} (blue) are overplotted. The synthetic O [ii]{} line profiles have been replaced by the profile of \[O [iii]{}\] $\lambda$4363. The bottom panel, which shows the difference between the observed and synthetic spectra, yields large residuals, especially for the central O [ii]{} peak, i.e. the O [ii]{} ORLs have significantly narrower widths than \[O [iii]{}\] $\lambda$4363. This indicates that, even though they are emitted from the same O$^{2+}$ ion, the O [ii]{} ORLs and \[O [iii]{}\] $\lambda$4363 cannot originate from material of identical physical properties.
2001, *ApJ* 558, 145
2006, *A&A* 450, 159
2000, *MNRAS* 312, 585
2004, *MNRAS* 353, 1251
2004, *ApJS* 150, 431
2003a, *MNRAS* 338, 687
2003b, *MNRAS* 345, 186
2004, *MNRAS* 353, 953
2005, *MNRAS* 362, 424
|
---
author:
- 'Daniel M. Kaplan[^1]'
- Klaus Kirch
- Derrick Mancini
- 'James D. Phillips'
- 'Thomas J. Phillips'
- 'Thomas J. Roberts'
- Jeff Terry
title: |
Measuring Antimatter Gravity with Muonium\
\
---
Introduction {#intro}
============
Indirect tests imply stringent limits on the gravitational acceleration of antimatter [@Alves]: $\bar{g} / g -1 < 10^{-7}$. Such limits are inferred based on the varying amounts of virtual antimatter presumed to constitute a portion of nuclear binding energies in various elements. (It is of course unclear to what extent these limits apply to muonium, as virtual antimuons surely play a negligible role in the nucleus.) Of the attempts at a direct test, none has yet achieved significance: the only published direct limit to date, on antihydrogen [@Amole], is $-65 < \bar{g} / g < 110$.
Besides antihydrogen, only one other experimental approach appears practical: measurements on muonium (M), a $\mu^+ e^-$ hydrogenic atom. We are developing a precision 3-grating muonium atom-beam interferometer to measure $\bar{g}$. Such a measurement will constitute a unique test of the gravitational interaction of leptonic and 2nd-generation matter with the gravitational field of the earth.
Experimental Approach
=====================
The key issue that must be addressed in a muonium experiment is the short lifetime of the muon, $\tau=2.2\,\mu s$. Thus in one lifetime, assuming the “null hypothesis” of standard gravity, an M atom has a gravitational deflection of only $$\frac{1}{2}g\tau^2=24\,{\rm pm}\,.$$ Measuring so small a deflection would be a substantial challenge even using state-of-the-art diffraction gratings and alignment and stabilization methods. This leads us to focus on measurements using “old” muons. It is straightforward to show that the statistical optimum is obtained with a measurement time of four lifetimes, for which the deflection is 16 times greater: $$\frac{1}{2}g(4\tau)^2=379\,{\rm pm}\,.$$ Whether statistics will dominate such a measurement is as yet unclear. A common-sense guesstimate is that, on the contrary, systematics will likely play a significant role. Thus, e.g., a measurement time of six lifetimes, statistically somewhat better than one lifetime, may prove to be optimal, giving a null-hypothesis deflection of 850pm.
Interferometer
--------------
We propose to measure the gravitational deflection by means of a three-grating Mach–Zehnder interferometer [@Phillips] (Fig. \[fig-MZ\]). In this, the de Broglie wave corresponding to an incident M atom interferes with itself, and the interference pattern created by the first two gratings is sampled by translating the third grating up and down by means of piezoelectric actuators. The result is a sinusoidally varying modulation of the counting rate, with period equal to the grating pitch. The gravitational deflection causes a phase offset of this interference pattern with respect to that of a straight-through path.
![Schematic diagram of measurement concept, with monoenergetic muonium (M) beam, incident from left, impinging on 3-grating Mach-Zehnder interferometer followed by muonium detector employing a coincidence measurement technique. Since the muonium beam undergoes a vertical gravitational acceleration, so does the interference pattern.[]{data-label="fig-MZ"}](MZ){width="13cm"}
Calibrating the straight-through phase to the required accuracy by direct measurement of the relative grating offsets would be challenging. We therefore propose to measure it concurrently with the muonium phase measurement using a soft X-ray source of wavelength comparable to that of the M beam. Another challenge is maintaining the desired alignment to a small fraction of 100pm. This can be done by means of Pound–Drever–Hall (PDH)-locked laser tracking frequency gauges [@Phillips-OL] mounted on at least one corner of each of two gratings (Fig. \[fig-PDH\]). The required dimensional stability can be achieved using Si${_3}$N${_4}$-film gratings mounted to a single-crystal silicon optical bench, operated cryogenically in order to take advantage of silicon’s nearly zero cryogenic temperature coefficient of expansion. (An alternate grating material, ultrananocrystalline diamond, is also under consideration.) Plausible state-of-the-art target grating parameters are 100nm pitch and 1cm$^2$ area.
Beam
----
It has been proposed to use a novel M beam produced by stopping decelerated surface muons in a thin layer of superfluid helium (SFHe) [@Kirch]. Such a beam is anticipated to be monoenergetic (a requirement in order not to wash out the M gravitational phase difference) due to the immiscibility of hydrogen in SFHe. Taking into account the mass-dependent chemical potential of hydrogenic atoms in SFHe, the M beam is expected to be expelled perpendicular to the SFHe surface at a speed of 6,300m/s, with energy spread $\Delta E/E=0.2$%. In order to obtain a horizontal M beam, a reflective SFHe surface at 45$^\circ$ is employed, as shown in Fig. \[fig-cryo\], taking advantage of the well-known tendency of SFHe to flow up walls and coat the inside of its container. Note that the exact horizontality of the beam is not crucial, thanks to the incident-angle independence of the Mach–Zehnder arrangement [@Chang-etal].
Sensitivity
-----------
A sensitivity estimate has been given by Kirch [@Kirch]: $$\Delta \bar{g}=0.3 g\, /\sqrt{\rm \#\,days} \,,$$ assuming an incident M rate of $10^5$/s, as estimated for the upgraded beam proposed at the Paul Scherrer Institute. At this rate the determination of the sign of $\bar{g}$ could be accomplished with minimal ($\sim$1day) running time (once the beam and apparatus are installed, shaken down, and calibrated), or in proportionately longer running time with a beam of lower intensity. A first measurement will suffice to determine whether we live in a so-called “Dirac–Milne” universe (one in which $\bar{g}=-g$) [@Dirac-Milne]. More challenging and time-consuming measurements could detect whether $\bar{g}/g=1\pm\epsilon$, for $\epsilon\sim10^{-2}$ or less.
![Schematic diagram of mounting concept, with PDH-locked laser gauge indicated at one grating corner[]{data-label="fig-PDH"}](PDH){width="10cm"}
![Experiment concept sketch[]{data-label="fig-cryo"}](cryo){width="10cm"}
Theoretical Considerations
==========================
We here briefly recap ideas developed more fully by Nieto and Goldman [@Nieto-Goldman]. As is well known, experimentally, gravity is consistent with being a pure tensor interaction. However, in the most general quantum field theories there should also be scalar and tensor terms. In the absence of a successful quantum field theory of gravity we are left in a somewhat speculative realm, but it seems not unreasonable to attribute the lack of experimental evidence for non-tensor behavior to a near-exact cancelation between the scalar and vector terms.
The vector terms are of opposite signs for matter–matter and matter–antimatter interactions. A good cancelation in the former case would then allow quite a large deviation between $\bar{g}$ and $g$. Whether this means $\epsilon\sim10^{-2}$, $10^{-10}$, something in between, or something very much smaller, is difficult to predict based on current knowledge. This would imply an experiment-driven regime for the present—as has been the case, e.g., for searches for $\mu\to e\gamma$ decays until recent years. More generally such a search has the potential to uncover an elusive fifth force, such as might be expected to couple to lepton- or lepton-generation number [@Lee-Yang]. To take the first experimental step into such an intriguing [*terra incognita*]{} would be a rare privilege indeed!
Cosmological Considerations {#sec-1}
===========================
Theories (see, e.g., [@Dirac-Milne]) in which antimatter repels matter (so-called “antigravity”) offer simple explanations of a number of key cosmological puzzles: (1) the cosmic baryon asymmetry; (2) galactic rotation curves and the binding of galaxy clusters; (3) the apparent cosmic acceleration; and (4) the “horizon” and “flatness” problems. The solutions of each of these problems in the antigravity scenario are briefly summarized in the following subsections.
Cosmic baryon asymmetry
-----------------------
Self-gravitating but mutually repulsive clusters of matter and antimatter form randomly interspersed matter and antimatter domains [@Dirac-Milne]. Thus there is no baryon asymmetry. The repulsion of matter and antimatter domains over billions of years ensures negligible production of annihilation radiation.
Anomalous gravitational clustering
----------------------------------
In an antigravity theory there must be virtual gravitational dipoles (matter–antimatter pairs). Unlike the electromagnetic case, these are repulsive, giving “anti-shielding,” and strengthening the force of gravity at large distances [@Hajdukovic-DM]. Thus there is no need for dark matter.
Cosmic acceleration
-------------------
Interspersed, mutually repulsive matter and antimatter clusters counteract the gravitational deceleration of the universe’s expansion, leading to a constant rate of recession [@Dirac-Milne]. Although differing from the usual, accelerating universe, interpretation, this is statistically consistent with the supernova data. Thus there is no need for dark energy.
Horizon and flatness problems
-----------------------------
Slower expansion of the early universe means that all parts are causally connected. Thus there is no need for inflation [@Dirac-Milne].
Occam’s razor
-------------
It is intriguing to note that the standard $\Lambda$CDM cosmology requires four postulates, for each of which there is little or no independent corroboration: (1) anomalous [*CP*]{} violation far in excess of the experimentally observed Standard Model level; (2) dark matter; (3) dark energy; and (4) inflation. What would Occam say?
Conclusion
==========
An experiment to measure the gravitational acceleration of muonium atoms is currently in an early R&D stage. If the effort comes to fruition and the measurement is made, the consequences of a “non-standard” result could be immense. Even the null result would be “one for the textbooks.” A thorough account of the current knowledge of gravity would include the disclaimer that it has never been directly tested with antimatter. If the experiment herein discussed succeeds, future textbooks will be able to say that experimental tests have demonstrated the applicability of the Principle of Equivalence even to antimatter—or not!
Acknowledgments {#acknowledgments .unnumbered}
===============
We have benefited from discussions with the late Andy Sessler, and from the support of the IPRO program [@IPRO] at Illinois Institute of Technology. The development of a suitable muon and muonium beam is supported by the Swiss National Science Foundation, grant \#200020\_146902.
[99]{}
D. S. M. Alves, M. Jankowiak, “Experimental constraints on the free fall acceleration of antimatter,” P. Saraswat, arXiv:0907.4110 \[hep-ph\] (2009). C. Amole [*et al.*]{} (ALPHA collaboration), “Description and first application of a new technique to measure the gravitational mass of antihydrogen,” Nature Commun. [**4**]{}, 1785 (2013). The ALPHA collaboration is pursuing plans for greater sensitivity, as discussed in P. Hamilton [*et al.*]{}, Phys. Rev. Lett. [**112**]{}, 121102 (2014). T. J. Phillips, “Antimatter gravity studies with interferometry,” Hyp. Int. [**109**]{}, 357 (1997). R. Thapa, J. D. Phillips, E. Rocco, R. D. Reasenberg, “Subpicometer length measurement using semiconductor laser tracking frequency gauge,” Optics Lett. [**36**]{}, 3759 (2011). K. Kirch, “Testing Gravity with Muonium,” arXiv:physics/0702143 (2007). B. J. Chang, R. Alferness and E. N. Leith, “Space-invariant achromatic grating interferometers: theory,” Appl. Optics [**14**]{}, 1592 (1975). A. Benoit-Lévy and G. Chardin, “Introducing the Dirac-Milne universe,” Astron. & Astrophys. [**537**]{}, A78 (2012). M. M. Nieto and T. Goldman, “The Arguments Against ‘Antigravity’ and the Gravitational Acceleration of Antimatter,” Phys. Rep. [**205**]{}, 221 (1991). T. D. Lee and C. N. Yang, “Conservation of heavy particles and generalized gauge transformations,” Phys. Rev. [**98**]{}, 1501 (1955). D. Hajdukovic, “Quantum vacuum and dark matter,” Astrophys. Space Sci. [**337**]{}, 9 (2012). IIT Interprofessional Projects website: .
[^1]:
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---
abstract: 'The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is $O(n^4)$ on an $n$-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we show that the expected absorption time is $\Omega(n\log n)$ and $O(n^2)$. We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.'
author:
- 'Josep Díaz[^1]'
- 'Leslie Ann Goldberg[^2]'
- David Richerby
- Maria Serna
bibliography:
- '\\jobname.bib'
title: 'Absorption Time of the Moran Process[^3]'
---
Introduction {#sec:intro}
============
The Moran process [@Moran58], as adapted by Lieberman, Hauert and Nowak [@LHN], is a stochastic model for the spread of genetic mutations through populations of organisms. Similar processes have been used to model the spread of epidemic diseases, the behaviour of voters, the spread of ideas in social networks, strategic interaction in evolutionary game theory, the emergence of monopolies, and cascading failures in power grids and transport networks [@ARLV2001:Influence; @Ber2001:Monopolies; @Gin2000:GT; @KKT2003:Influence; @Lig1999:IntSys].
In the Moran process, individuals are modelled as the vertices of a graph and, at each step of the discrete-time process, an individual is selected at random to reproduce. This vertex chooses one of its neighbours uniformly at random and replaces that neighbour with its offspring, a copy of itself. The probability that any given individual is chosen to reproduce is proportional to its *fitness*: individuals with the mutation have fitness $r>0$ and non-mutants have fitness $1$. The initial state has a single mutant placed uniformly at random in the graph, with every other vertex a non-mutant. On any finite, strongly connected graph, the process will terminate with probability $1$, either in the state where every vertex is a mutant (known as *fixation*) or in the state where no vertex is a mutant (known as *extinction*).
The principal quantities of interest are the *fixation probability* (the probability of reaching fixation) and the expected *absorption time* (the expected number of steps before fixation or extinction is reached). In general, these depend on both the graph topology and the mutant fitness. In principle, they can be computed by standard Markov chain techniques but doing so for an $n$-vertex graph involves solving a set of $2^n$ linear equations, which is computationally infeasible. Fixation probabilities have also been calculated by producing and approximately solving a set of differential equations that model the process [@HV2011:fixation]. These methods seem to work well in practice but there is no known bound on the error introduced by the conversion to differential equations and the approximations in their solution.
When the underlying graph is undirected and the mutant fitness $r$ is at least $1$, it is known how to approximate the fixation probability: The paper [@OurSODA] gave a fully polynomial randomised approximation scheme (FPRAS) for computing the fixation probability of the Moran process on undirected graphs. The approximation scheme uses the Markov chain Monte Carlo method. The fact that it provides a suitable approximation in polynomial time follows from the fact that the expected absorption time on an $n$-vertex graph is at most $\tfrac{r}{r-1} n^4$ for $r>1$.
Our contributions
-----------------
The main contribution of this paper is to determine the extent to which the polynomial bound on expected absorption time carries through to directed graphs. Throughout the paper, we assume that the mutant fitness $r$ exceeds $1$.
### Regular digraphs
We start by considering the absorption time on a strongly connected $\Delta$-regular digraph (where every vertex has in-degree $\Delta$ and out-degree $\Delta$). Regularity makes some calculations straightforward because the detailed topology of the graph is not relevant. We describe these first, and then discuss the more difficult questions (where topology does play a role) and state our results.
The following facts hold for $\Delta$-regular graphs, independent of the topology.
- It is well known [@LHN] that, on any regular graph on $n$ vertices, a single randomly placed mutant with fitness $r$ reaches fixation with probability $$\label{eq:isothermal}
\frac{1-r^{-1}}{1-r^{-n}}$$ To see this, note that if there are $k$ mutants, the total fitness of the population is $W_k = n+k(r-1)$. The probability that the next reproduction happens along the directed edge $(u,v)$ is $\tfrac{r}{W_k}\tfrac1\Delta$ if $u$ is a mutant and $\tfrac1{W_k}\tfrac1\Delta$, if it is not. Since the graph is $\Delta$-regular, there are exactly as many directed edges from mutants to non-mutants as there are from non-mutants to mutants. It follows that the probability that the number of mutants increases at the next step is exactly $r$ times the probability that it decreases, regardless of which vertices are currently mutants. Thus, the number of mutants in the population, observed every time it changes, forms a random walk on $\{0, \dots, n\}$ with initial state $1$, upward drift $r$ and absorbing barriers at $0$ and $n$. It is a standard result (e.g., [@GS2001:Probability Example 3.9.6]) that such a random walk reaches the barrier at $n$ with the probability given in .
- It is also well known (e.g., [@GS2001:Probability Example 3.9.6]) that the expected number of steps of this walk before absorption (which may be at either $0$ or $n$) is a function of $r$ and $n$ that tends to $n(1+\tfrac1r)$ in the limit as $n\to\infty$, independent of the graph structure beyond regularity. However, the number of steps taken by the random walk (often referred to as the “active steps” of the Moran process) is not the same as the original process’s absorption time, because the absorption time includes many steps at which the number of mutants does not change, either because a mutant reproduces to a mutant or because a non-mutant reproduces to a non-mutant.
In Section \[sec:regular\], we show that the expected absorption time of the Moran process is polynomial for regular digraphs. In contrast to the number of active steps, the absorption time does depend on the detailed structure of the graph. We prove the following upper and lower bounds, where $H_n$ denotes the $n$’th harmonic number, which is $\Theta(\log n)$.
\[thm:regbounds\][ The expected absorption time of the Moran process on a strongly connected $\Delta$-regular $n$-vertex digraph $G$ is at least $\left(\frac{r-1}{r^2}\right)n H_{n-1}$ and at most $n^2 \Delta$.]{}
In Section \[sec:family\], we prove the following theorem, which shows that the upper bound in Theorem \[thm:regbounds\] is tight up to a constant factor (which depends on $\Delta$ and $r$ but not on $n$).
\[thm:tight\][ Suppose that $r>1$ and $\Delta>2$. There is an infinite family $\mathcal{G}$ of $\Delta$-regular graphs such that, when the Moran process is run on an $n$-vertex graph $G\in \mathcal{G}$, the expected absorption time exceeds $\tfrac{1}{8r}(1-\tfrac{1}{r})\frac{n^2}{(\Delta-1)^2} $. ]{}
The digraphs in the family $\mathcal{G}$ are symmetric, so can be viewed as undirected graphs. The upper bound on the expected absorption time in Theorem \[thm:regbounds\] can be improved for certain classes of regular undirected graphs using the notion of the isoperimetric number $\iso(G)$ of a graph $G$, which is defined in Section \[sec:iso\]. There, we prove the following theorem.
\[thm:isoperimetric\] [ The expected absorption time of the Moran process on a connected $\Delta$-regular $n$-vertex undirected graph $G$ is at most $ 2\Delta n H_n /\iso(G)$. ]{}
Theorem \[thm:isoperimetric\] pinpoints the expected absorption time for $G=K_n$, up to a constant factor, since $\iso(K_n)=\lceil n/2\rceil$ [@Mohar] and Theorem \[thm:regbounds\] gives an $\Omega(n \log n)$ lower bound. Theorem \[thm:isoperimetric\] is worse than the upper bound of Theorem \[thm:regbounds\] by a factor of $O(\log n)$ for the cycle $C_n$ since $\iso(C_n)=2/\lfloor n/2\rfloor$ [@Mohar]. However, we often get an improvement by using the isoperimetric number. For example, the $\sqrt{n}$-by-$\sqrt{n}$ grid has $\iso(G) = \Theta(1/\sqrt{n})$ (see [@arrays]), giving an $O(n^{3/2} \log n)$ absorption time; the hypercube has $\iso(G)=1$ (see, e.g., [@Mohar]), giving an $O(n \log^2 n)$ absorption time. Bollob[á]{}s [@bollobas] showed that, for every $\Delta\geq 3$ there is a positive number $\eta<1$ such that, for almost all $\Delta$-regular $n$-vertex undirected graphs $G$ (as $n$ tends to infinity), $\iso(G)\geq (1-\eta) \Delta/2$, which gives an $O(n \log n)$ absorption time since these graphs are connected.
### Slow absorption
Theorem \[thm:regbounds\] shows that regular digraphs, like undirected graphs, reach absorption in expected polynomial time. In Section \[sec:directed\] we show that the same does not hold for general digraphs. In particular, we construct an infinite family $\{G_{r,N}\}$ of strongly connected digraphs indexed by a positive integer $N$. We then prove the following theorem.
\[thm:directed\] Fix $r>1$ and let $\epsilon_r = \min(r-1,\,1)$. For any positive integer $N$ that is sufficiently large with respect to $r$, the expected absorption time of the Moran process on $G_{r,N}$ is at least $$\frac{1}{16}{\left\lfloor {\left(\frac{\epsilon_r}{32}\right)(2^N-1)}\right\rfloor}
\frac{\epsilon_r}{4{\left\lceil {r}\right\rceil}+3}.$$
Theorem \[thm:directed\] shows that there is an infinite family of strongly connected digraphs such that the absorption time on $n$-vertex graphs in this family is exponentially large, as a function of $n$. It follows that the techniques from [@OurSODA] do not give a polynomial-time algorithm for approximating the fixation probability on digraphs.
The underlying structure of the graph $G_{r,N}$ is a large undirected clique on $N$ vertices and a long directed path. Each vertex of the clique sends an edge to the first vertex of the path, and each vertex of the clique receives an edge from the path’s last vertex. We refer to the first $N$ vertices of path as $P$ and the remainder as $Q$. Each vertex of $P$ has out-degree $1$ but receives $4{\left\lceil {r}\right\rceil}$ edges from $Q$. (See Figure \[fig:GrN\].)
Suppose that $N$ is sufficiently large with respect to $r$ and consider the Moran process on $G_{r,N}$. Given the relative sizes of the clique and the path, there is a reasonable probability (about $\tfrac{1}{4r+2}$) that the initial mutant is in the clique. The edges to and from the path have a negligible effect so it is reasonably likely (probability at least $1-\tfrac1r$) that we will then reach the state where half the clique vertices are mutants. To reach absorption from this state, one of two things must happen.
For the process to reach extinction, the mutants already in the clique must die out. Because the interaction between the clique and path is small, the number of mutants in the clique is very close to a random walk on $\{0, \dots, N\}$ with upward drift $r$, and the expected time before such a walk reaches zero from $N/2$ is exponential in $N$.
On the other hand, suppose the process reaches fixation. In particular, the vertices of $P$, the first part of the path, must become mutants. Note that no vertex of $Q$ can become a mutant before the last vertex of $P$ has done so. While all the vertices in $Q$ are non-mutants, the edges from that part of the path to $P$ ensure that each mutant in $P$ is more likely to be replaced by a non-mutant from $Q$ than it is to create a new mutant in $P$. As a result, the number of mutants in $P$ is bounded above by a random walk on $\{0,
\dots, N\}$ with a strictly greater probability of decreasing than increasing. Again, this walk is expected to take exponentially many steps before reaching $N$.
### Stochastic domination {#sec:introdom}
Our main technical tool is stochastic domination. Intuitively, one expects that the Moran process has a higher probability of reaching fixation when the set of mutants is $S$ than when it is some subset of $S$, and that it is likely to do so in fewer steps. It also seems obvious that modifying the process by continuing to allow all transitions that create new mutants but forbidding some transitions that remove mutants should make fixation faster and more probable. Such intuitions have been used in proofs in the literature; it turns out that they are essentially correct, but for rather subtle reasons.
The Moran process can be described as a Markov chain $(Y_t)_{t\geq 1}$ where $Y_t$ is the set $S\subseteq V(G)$ of mutants at the $t$’th step. The normal method to make the above intuitions formal would be to demonstrate a stochastic domination by coupling the Moran process $(Y_t)_{t\geq 1}$ with another copy $(Y'_t)_{t\geq 1}$ of the process where $Y_1 \subseteq Y'_1$. The coupling would be designed so that $Y_1\subseteq Y'_1$ would ensure that $Y_t\subseteq Y'_t$ for all $t>1$. However, a simple example shows that such a coupling does not always exist for the Moran process. Let $G$ be the undirected path with two edges: $V(G)=\{1,2,3\}$ and $E(G)=\{(1,2), (2,1), (2,3), (3,2)\}$. Let $(Y_t)_{t\geq 1}$ and $(Y'_t)_{t\geq 1}$ be Moran processes on $G$ with $Y_1=\{2\}$ and $Y'_1 = \{2,3\}$. With probability $\tfrac{r}{2(r+2)}$, we have $Y_2 = \{1,2\}$. The only possible value for $Y'_2$ that contains $Y_2$ is $\{1,2,3\}$ but this occurs with probability only $\tfrac{r}{2(2r+1)}$. Therefore, any coupling between the two processes fails because $$\Pr(Y_2\not\subseteq Y'_2) \geq \frac{r(r-1)}{2(r+2)(2r+1)}\,,$$ which is strictly positive for any $r>1$. The problem is that, when vertex $3$ becomes a mutant, it becomes more likely to be the next vertex to reproduce and, correspondingly, every other vertex becomes less likely. This can be seen as the new mutant “slowing down” all the other vertices in the graph.
To get around this problem, we consider a continuous-time version of the process, ${\widetilde{Y}}[t]$ ($t\geq 0$). Given the set of mutants ${\widetilde{Y}}[t]$ at time $t$, each vertex waits an amount of time before reproducing. For each vertex, this period of time is chosen according to the exponential distribution with parameter equal to the vertex’s fitness, independently of the other vertices. (Thus, the parameter is $r$ if the vertex is a mutant and $1$, otherwise.) If the first vertex to reproduce is $v$ at time $t+\tau$ then, as in the standard, discrete-time version of the process, one of its out-neighbours $w$ is chosen uniformly at random, the individual at $w$ is replaced by a copy of the one at $v$ and the time at which $w$ will next reproduce is exponentially distributed with parameter given by its new fitness. The discrete-time process is recovered by taking the sequence of configurations each time a vertex reproduces.
In continuous time, each member of the population reproduces at a rate given by its fitness, independently of the rest of the population whereas, in discrete time, the population has to co-ordinate to decide who will reproduce next. It is still true in continuous time that vertex $w$ becoming a mutant makes it less likely that each vertex $v\neq w$ will be the next to reproduce. However, the vertices are not slowed down as they are in discrete time: they continue to reproduce at rates determined by their fitnesses. This distinction allows us to establish the following coupling lemma, which formalises the intuitions discussed above.
\[lemma:couple\] [ Let $G=(V,E)$ be any digraph, let $Y\subseteq Y'\subseteq V(G)$ and $1\leq r\leq r'\!$. Let ${\widetilde{Y}}[t]$ and ${\widetilde{Y}}'[t]$ ($t\geq 0$) be continuous-time Moran processes on $G$ with mutant fitness $r$ and $r'\!$, respectively, and with ${\widetilde{Y}}[0]=Y$ and ${\widetilde{Y}}'[0]=Y'\!$. There is a coupling between the two processes such that ${\widetilde{Y}}[t]\subseteq {\widetilde{Y}}'[t]$ for all $t\geq 0$. ]{}
In the paper, we use the coupling lemma to establish stochastic dominations between *discrete* Moran processes. It also has consequences concerning fixation probabilities. Recall that “fixation” is the state of a (discrete) Moran process in which every vertex is a mutant. The fixation probability $f_{G,r}$ is the probability that this state is reached when the Moran process is run on a digraph $G=(V,E)$, starting from a state in which a single initial mutant is placed uniformly at random. For any set $S\subseteq
V$, let $f_{G,r}(S)$ be the probability of reaching fixation when the set of vertices initially occupied by mutants is $S$. Thus, $f_{G,r} = \tfrac{1}{|V|}\sum_{v\in V} f_{G,r}(\{v\})$. Using the coupling lemma, we can prove the following theorem.
\[thm:fix\] [ For any digraph $G$, if $0<r\leq r'$ and $S\subseteq
S'\subseteq V(G)$, then $f_{G,r}(S) \leq f_{G,r'}(S')$. ]{}
This theorem has two immediate corollaries. The first was conjectured by Shakarian, Roos and Johnson [@SRJ2012:review Conjecture 2.1]. It was known from [@SR2011:fastfix] that $f_{G,r}\geq f_{G,1}$ for any $r>1$.
\[cor:monotonicity\] If $0<r\leq r'$ then, for any digraph $G$, $f_{G,r}\leq f_{G,r'}$.
Corollary \[cor:monotonicity\] follows immediately from Theorem \[thm:fix\] since $f_{G,r}(\{v\})\leq f_{G,r'}(\{v\})$ for all $v\in V(G)$.
The second corollary can be stated informally as, “Adding more mutants can’t decrease the fixation probability,” and has been assumed in the literature, without proof. However, although it appears obvious at first, it is somewhat subtle: the example at the beginning of this section shows that adding more mutants can actually decrease the probability of a particular vertex becoming a mutant at the next step of the process.
\[cor:subsetdom\] For any digraph $G$ and any $r>0$, if $S\subseteq S'\subseteq
V(G)$, then $f_{G,r}(S)\leq f_{G,r}(S')$.
Note that, although we have introduced the continuous-time version of the process for technical reasons, to draw conclusions about the original, discrete-time Moran process, the continuous-time version may actually be a more realistic model than the discrete-time version.
Previous work
-------------
There is previous work on calculating the fixation probability of the Moran process. Fixation probabilities are known for regular graphs [@LHN] and for stars (complete bipartite graphs $K_{1,k}$) [@BR2008:fixprob]. Lieberman et al. [@LHN] have defined classes of directed graphs with a parameter $k$, for which they claim that the fixation probability tends to $1-r^{-k}$ for large graphs. While these graphs do seem to have very large fixation probability, we have shown this specific claim to be incorrect for $k=5$ [@DGMRSS2013:superstars]. Very recently, it has been claimed [@fixrevised] that for large $k$, the fixation probability is close to $1-\frac{1}{(k-2)r^4}$. Other work has investigated the possibility of so-called “suppressors”, graphs having fixation probability less than that given by for at least some range of values for $r$ [@MNRS2013:suppressors; @MS2013:strongbounds].
There is a more complicated version of the Moran process in which the fitness of a vertex is determined by its expected payoff when playing some two-player game against a randomly chosen neighbour [@TFSN2004:game; @SRJ2012:review]. In this version of the process, mutants play the game with one strategy and non-mutants play the game with another. The ordinary Moran process corresponds to the special case of this game in which the mutant and non-mutant strategies give payoffs $r$ and $1$, respectively, regardless of the strategy used by the opponent.
Most previous work on absorption times has been in the game-based version of the process, where the added complexity of the model limits analysis to very simple graphs, such as complete graphs, stars and cycles [@AS2006:fixtime; @BHR2010:speed].
Preliminaries
=============
When $k$ is a positive integer, $[k]$ denotes $\{1,\ldots,k\}$. We consider the evolution of the Moran process [@LHN] on a strongly connected directed graph (digraph). Consider such a digraph $G=(V,E)$ with $n=|V|$. When the process is run on $G$, each state is a set of vertices $S\subseteq V$. The vertices in $S$ are said to be “mutants”. If $|S|=k$ then the total fitness of the state is given by $W_k=n+(r-1)k$ — each of the $k$ mutants contributes fitness $r$ to the total fitness and each other vertex contributes fitness $1$. Except where stated otherwise, we assume that $r>1$. The starting state is chosen uniformly at random from the one-element subsets of $V(G)$. From a state $S$ with $|S|=k$, the process evolves as follows. First, a vertex $u$ is chosen to reproduce. The probability that vertex $u$ is chosen is $r/W_k$ if $u$ is a mutant and $1/W_k$ otherwise. Given that $u$ will reproduce, a directed edge $(u,v)$ is chosen uniformly at random from $\{(u,v') \mid (u,v')\in E\}$. The state of vertex $u$ in $S$ is copied to vertex $v$ to give the new state $$S|_{u\to v} = \begin{cases}
\ S\cup\{v\} & \text{if $u$ is a mutant,} \\
\ S\setminus\{v\} & \text{if $u$ is a non-mutant.}
\end{cases}$$
Let ${d}^+(u)$ denote the out-degree of vertex $u$ and let ${d}^-(u)$ denote its in-degree. A digraph $G=(V,E)$ is *$\Delta$-regular* if, for every vertex $u\in V$, ${d}^+(u)={d}^-(u)=\Delta$. $G$ is *regular* if it is $\Delta$-regular for some natural number $\Delta$. If the Moran process is run on a strongly connected digraph $G$, there are exactly two absorbing states — $\emptyset$ and $V(G)$. Once one of these states is reached, the process will stay in it forever. The *absorption time* is the number of reproduction steps until such a state is reached.
A digraph $G=(V,E)$ is weakly connected if the underlying undirected graph is connected. Given a subset $S\subseteq V$, let ${m^+_{S}}$ be the number of edges from vertices in $S$ to vertices in $V\setminus S$. Let ${m^-_{S}}$ be the number of edges from vertices in $V\setminus S$ to $S$. If $G$ is regular then, for every $S\subseteq V$, $$\begin{aligned}
{m^+_{S}} &= |\{(u,v)\in E\mid u\in S\}|
- |\{(u,v)\in E\mid u,v\in S\}| \\
&= |S|\Delta - |\{(u,v)\in E\mid u,v\in S\}| \\
&= |\{(u,v)\in E\mid v\in S\}|
- |\{(u,v)\in E\mid u,v\in S\}| \\
&= {m^-_{S}}\,.\end{aligned}$$ Thus, every regular digraph that is weakly connected is strongly connected.
We sometimes consider the Moran process on an undirected graph $G=(V,E)$. We view the undirected graph as a digraph in which the set $E$ of edges is symmetric (so $(u,v)\in E$ if and only if $(v,u)\in E$). If $G$ is undirected then, for every vertex $u$, ${d}^+(u)={d}^-(u)$ and in this case we just write ${d}(u)$ to denote this quantity.
Domination {#sec:domination}
==========
A useful proof technique is to show that the behaviour of the Moran process is stochastically dominated by that of a related process that is easier to analyse. Similarly, it is useful to compare the behaviour of the Moran process, evolving on a digraph $G$, with that of another Moran process on the same digraph, where the second process starts with more mutants. Recall that the Moran process can be described as a Markov chain $(Y_t)_{t\geq 1}$ where $Y_t$ is the set $S\subseteq V(G)$ of mutants at the $t$’th step. It would be natural to attempt to establish a coupling between Moran processes $(Y_t)_{t\geq 1}$ and $(Y'_t)_{t\geq 1}$ such that, if $Y_1\subseteq Y'_1$, then $Y_t\subseteq Y'_t$ for all $t\geq 1$ but, as we showed in Section \[sec:introdom\], this cannot be done.
To obtain useful dominations, we will consider a continuous-time version of the Moran process. The domination that we construct for the continuous-time process will allow us to draw conclusions about the original (discrete-time) Moran process. In a digraph $G=(V,E)$ where the set of mutants $Y$ have fitness $r$, let $r_{v,Y}=r$ if $v\in Y$ and $r_{v,Y}=1$, otherwise. We define the continuous-time version of the Moran process on a digraph $G=(V,E)$ as follows. Starting in configuration ${\widetilde{Y}}[t]$ at time $t$, each vertex $v$ waits for a period of time before reproducing. This period of time is chosen, independently of other vertices, according to an exponential distribution with parameter $r_{v,{\widetilde{Y}}[t]}$. Therefore, the probability that two vertices reproduce at once is zero. Suppose that the first vertex to reproduce after time $t$ is vertex $v$, at time $t+\tau$. As in the discrete-time version of the process, an out-neighbour $w$ of $v$ is chosen u.a.r. and the new configuration is given by ${\widetilde{Y}}[t+\tau]={\widetilde{Y}}[t]|_{v\to w}$.
From the definition of the exponential distribution, it is clear that the probability that a particular vertex $v$ is the next to reproduce, from configuration ${\widetilde{Y}}[t]$, is $r_{v,{\widetilde{Y}}[t]}/W_{|{\widetilde{Y}}[t]|}$. Thus, the Moran process (as generalised by Lieberman et al.) is recovered by taking the sequence of configurations each time a vertex reproduces.[^4] We can now give the proof of Lemma \[lemma:couple\] and Theorem \[thm:fix\].
[lemma:couple]{}[ Let $G=(V,E)$ be any digraph, let $Y\subseteq Y'\subseteq V(G)$ and $1\leq r\leq r'\!$. Let ${\widetilde{Y}}[t]$ and ${\widetilde{Y}}'[t]$ ($t\geq 0$) be continuous-time Moran processes on $G$ with mutant fitness $r$ and $r'\!$, respectively, and with ${\widetilde{Y}}[0]=Y$ and ${\widetilde{Y}}'[0]=Y'\!$. There is a coupling between the two processes such that ${\widetilde{Y}}[t]\subseteq {\widetilde{Y}}'[t]$ for all $t\geq 0$. ]{}
Suppose that ${\widetilde{Y}}[t]\subseteq {\widetilde{Y}}'[t]$ for some $t$. We couple the evolution of ${\widetilde{Y}}[t']$ and ${\widetilde{Y}}'[t']$ for $t'\geq t$ as follows. For ease of notation, we write $r_{v,t}$ and $r'_{\smash{v,t}}$ for $r_{\smash{v,{\widetilde{Y}}[t]}}$ and $r'_{\smash{v,{\widetilde{Y}}'[t]}}$, respectively. Let $$S = \{v\in V(G) \mid r_{v,t} < r'_{\smash{v,t}}\} \subseteq {\widetilde{Y}}'[t]$$ and note that, for $v\in V\setminus S$, $r'_{\smash{v,t}}=r_{v,t}$. For $v\in V\!$, let $t_v$ be a random variable drawn from ${\mathrm{Exp}}(r_{v,t})$ and, for $v\in S$, let $t'_v \sim
{\mathrm{Exp}}(r'_{\smash{v,t}} - r_{v,t})$. From the definition of the exponential distribution, it is easy to see that, for each $v\in
S$, $\min(t_v,t'_v) \sim {\mathrm{Exp}}(r'_{\smash{v,t}})$.
If some $t_v$ is minimal among $\{t_v\mid v\in V\} \cup \{t'_v\mid
v\in S\}$, then choose an out-neighbour $w$ of $v$ u.a.r. and set ${\widetilde{Y}}[t+t_v] = {\widetilde{Y}}[t]|_{v\to w}$ and ${\widetilde{Y}}'[t+t_v] = {\widetilde{Y}}'[t]|_{v\to w}$. It is clear that ${\widetilde{Y}}[t+t_v]\subseteq {\widetilde{Y}}'[t+t_v]$.
Otherwise, some $t'_v$ is minimal. In this case, set ${\widetilde{Y}}[t+t'_v]={\widetilde{Y}}[t]$; choose an out-neighbour $w$ of $v$ u.a.r. and set ${\widetilde{Y}}'[t+t'_v]={\widetilde{Y}}'[t]|_{v\to w}$. Since $v\in S\subseteq {\widetilde{Y}}'[t]$, we have $${\widetilde{Y}}[t+t'_v] = {\widetilde{Y}}[t]\subseteq {\widetilde{Y}}'[t] \subseteq {\widetilde{Y}}'[t+t'_v]\,.$$
In both cases, the continuous-time Moran process has been faithfully simulated up to time $t+\tau$, where $\tau=t_v$ in the first case and $\tau=t'_v$ in the second case, and the memorylessness of the exponential distribution allows the coupling to continue from ${\widetilde{Y}}[t+\tau]$ and ${\widetilde{Y}}'[t+\tau]$.
The coupling provided in Lemma \[lemma:couple\] could be translated to a coupling for the discrete-time Moran process, though the time steps in the two copies would not be the same since such a coupling was ruled out in Section \[sec:introdom\]. In fact, it will be easy for us to use Lemma \[lemma:couple\] directly.
[thm:fix]{}[ For any digraph $G$, if $0<r\leq r'$ and $S\subseteq
S'\subseteq V(G)$, then $f_{G,r}(S) \leq f_{G,r'}(S')$. ]{}
We split the proof into two parts: $1\leq r\leq r'$ and $r\leq
r'\leq 1$. The remaining case $r\leq 1\leq r'$ follows because $f_{G,r}(S)\leq f_{G,1}(S)\leq f_{G,r'}(S')$.
First, suppose that $1\leq r\leq r'\!$. Let ${\widetilde{Y}}[t]$ and ${\widetilde{Y}}'[t]$ be Moran processes on $G=(V,E)$ with mutant fitnesses $r$ and $r'$, respectively, with ${\widetilde{Y}}[0] = S$ and ${\widetilde{Y}}'[0] = S'\!$. By the coupling lemma, we can couple the processes such that ${\widetilde{Y}}[t]\subseteq {\widetilde{Y}}'[t]$ for all $t\geq 0$. In particular, if there is a $t$ such that ${\widetilde{Y}}[t]=V$, we must have ${\widetilde{Y}}'[t]=V$ also. Therefore, $f_{G,r'}(S')\geq f_{G,r}(S)$.
Now, suppose that $r\leq r'\leq 1$. Observe that the behaviour of the Moran process is independent of any consistent scaling of the mutant and non-mutant fitnesses, in the following sense. For any $\alpha>0$, the process where mutants have fitness $r$ and non-mutants have fitness $1$ is identical to the one where they have fitness $\alpha r$ and $\alpha$, respectively. Let ${\widetilde{Y}}[t]$ be the process where mutants have fitness $\tfrac{1}{r}\cdot r =
1$ and non-mutants have fitness $\tfrac{1}{r}\geq 1$, and let ${\widetilde{Y}}'[t]$ have mutant fitness $1$, non-mutant fitness $\tfrac{1}{r'}\geq 1$. Let ${\widetilde{Y}}[0]=S$ and ${\widetilde{Y}}'[0]=S'\!$. Now, $f_{G,r}(S)$ is the probability that the individuals with fitness $1$ take over the graph in ${\widetilde{Y}}[t]$, which is $1-f_{G,1/r}(V\setminus S)$; similarly, $f_{G,r'}(S') = 1 -
f_{G,1/r'}(V\setminus S')$. By the first part, $f_{G,1/r'}(V\setminus S')\leq f_{G,r}(V\setminus S)$ and the result follows.
Corollaries \[cor:monotonicity\] (monotonicity) and \[cor:subsetdom\] (subset domination) follow immediately from Theorem \[thm:fix\], as shown in Section \[sec:introdom\].
Regular digraphs {#sec:regular}
================
This section provides upper and lower bounds on the absorption time of the Moran process on regular digraphs.
An upper bound for undirected graphs
------------------------------------
It is clear that the absorption time bounds from [@OurSODA] do not apply to digraphs. For example, Theorem 7 of [@OurSODA] gives a polynomial upper bound on the expected absorption time for all connected undirected graphs, but Theorem \[thm:directed\] shows that process takes exponential time on some strongly connected digraphs.
Since we will be discussing both undirected graphs and digraphs in this section, we start by observing that Theorem 7 of [@OurSODA] can be improved to give an $O(n^3)$ bound in the special case in which the undirected graphs to which it applies are regular. This is certainly not tight (as we shall see below) but it is a natural place to begin.
\[prop:oldupper\] The expected absorption time of the Moran process on a connected $\Delta$-regular $n$-vertex undirected graph is at most $(r/(r-1))n^2 \Delta$.
Given an undirected graph $G$ and a set $S\subseteq V(G)$, let $\phi(S) = \sum_{v\in S} \tfrac{1}{{d}(v)}$. Let $\partial S$ be the set of (undirected) edges between vertices in $S$ and vertices in $V(G)\setminus S$. If $G$ is $\Delta$-regular and has $n$ vertices, then $\phi(V(G)) = n/\Delta$. The proof of Theorem 7 and Equation (1) of [@OurSODA] show that the absorption time is at most $$\phi(V(G)) \max \left\{
\left(
\frac{(n+(r-1)|S|)\, \Delta^2}{(r-1)\,|\partial S|}\right)
\mid \emptyset \subset S \subset V(G)
\right\}.$$ The bound follows using $|S|\leq n$ and $|\partial S|\geq 1$.
Definitions
-----------
We will use the following standard Markov chain definitions. For more detail, see, for example, [@Norris]. We use $(X_t)_{t\geq 0}$ to denote a discrete-time Markov chain $\calM$ with finite state space $\Omega$ and transition matrix $P$. $T_k = \inf\{ t \geq 1 \mid X_t=k\}$ is the first passage time for visiting state $k$ (not counting the initial state $X_0$). The time spent in state $i$ between visits to state $k$ is given by $$\gamma_i^k =
\sum_{t=0}^{T_k-1} 1_{X_t=i}, \mbox{ where $X_0=k$.}$$ The chain is said to be irreducible if, for every pair of states $(i,j)$ there is some $t\geq 0$ such that $\Pr(X_t=j \mid X_0=i)>0$. Since $\Omega$ is finite, this implies that the chain is recurrent, which means that, for every state $i\in \Omega$, $\Pr(\mbox{$X_t=i$ for infinitely many~$t$})=1$. We use the following proposition, which (up to minor notational differences) is the special case of [@Norris Theorem 1.7.6] corresponding to finite state spaces.
\[prop:Norristwo\] Let $\calM$ be an irreducible discrete-time Markov chain with finite state space $\Omega=\{0,\ldots,\omega-1\}$ and transition matrix $P$. For $k\in\Omega$, let $\lambda=(\lambda_0,\ldots,\lambda_{ \omega-1})$ be a vector of non-negative real numbers with $\lambda_k=1$ satisfying $\lambda P = \lambda$. Then, for every $j\in \Omega$, $E[\gamma^k_j]=\lambda_j$.
Active steps of the Moran process {#sec:active}
---------------------------------
We fix $r>1$ and study the Moran process on a strongly connected $\Delta$-regular $n$-vertex digraph $G=(V,E)$ with $n>1$. We refer to the steps of the process during which the number of mutants changes as “active steps”. As explained in the introduction, the evolution of the number of mutants, sampled after each active step, corresponds to a one-dimensional random walk on $\{0,\ldots,n\}$ which starts at state $1$, absorbs at states $0$ and $n$, and has upwards drift $p=r/(r+1)$. (To see this, note that the probability that the number of mutants increases from a size-$k$ state $S$ is $\sum_{e \in E \cap (S\times V(G)\setminus S)} \tfrac{r}{W_k \Delta}$ and the probability that it decreases is $\sum_{e \in E \cap (V(G)\setminus S \times S)} \tfrac{1}{W_k \Delta}$ but we showed earlier that the number of edges in each summation is equal when $G$ is $\Delta$-regular, so the ratio between these two probabilities is $r$ to $1$.)
To derive the properties that we need, we consider a Markov chain $\calM$ with state space $\Omega=\{0,\ldots,n+1\}$. The non-zero entries of the transition matrix $P$ of $\calM$ are as follows. $P_{0,n+1}=P_{n,n+1}=1$. Also, $P_{n+1,1}=1$. Finally, for $1\leq i \leq n-1$, $P_{i,i+1}=p$ and $P_{i,i-1}=1-p$. Starting from state $1$, the chain simulates the one-dimensional walk discussed above. State $n+1$ is a special state of the Markov chain that is visited after an absorbing state of the random walk is reached. From state $n+1$, the chain goes back to state $1$ and repeats the random walk. We use the following property of $\calM$.
\[lem:calc\] Let $f= (r^n-r^{n-1})/(r^n-1)$. Define the vector $\lambda = (\lambda_0,\ldots,\lambda_{n+1})$ as follows. $$\begin{aligned}
\lambda_0&=1- f,\\
\lambda_j &= (1+r)(1-f)(r^n-r^j)/(r^n-r),
\mbox{ for $1\leq j\leq n-1$,}\\
\lambda_n &= f,\\
\lambda_{n+1} &= 1.\end{aligned}$$ Then, for every $j\in \Omega$, $E[\gamma^{n+1}_j]=\lambda_j$.
Note that $P$ is irreducible. By Proposition \[prop:Norristwo\], it suffices to show that $\lambda P = \lambda$. First, consider the column vector $P_{*,0}$. This is all zero except the entry $P_{1,0}=1-p$ so $\lambda P_{*,0} = (1-p)\lambda_1 = \lambda_0$, as required. Then note that $1/(1-f)= r(r^n-1)/(r^n-r)$. So $$\begin{aligned}
\lambda P_{*,1} &= (1-p) \lambda_2+\lambda_{n+1}\\
&= (1-f)\left( \frac{r^n-r^2}{r^n-r}+ \frac{1}{1-f}\right) \\
& = (1-f)\left(\frac{r^n-r^2 + r(r^n-1)}{r^n-r}\right)\\
&= (1-f)(r+1) =\lambda_1,\end{aligned}$$ as required. Next, consider the column vector $P_{*,j}$ for $1<j<n-1$. In this case, $$\begin{aligned}
\lambda P_{*,j} &=
p \lambda_{j-1} + (1-p) \lambda_{j+1} \\
&=
(1-f)\left(\frac{r(r^n-r^{j-1}) + (r^n-r^{j+1})}
{r^n-r}\right)\\
&= (1-f)
\left(\frac{(1+r)(r^n-r^{j}) }
{r^n-r}\right) = \lambda_j,\\\end{aligned}$$ as required. Then $$\begin{aligned}
\lambda P_{*,n-1} &= p \lambda_{n-2} \\
&= \left(\frac{(1+r)(1-f)}{r^n-r}\right)\left(\frac{r(r^n-r^{n-2})}{r+1}\right) \\
&= \left(\frac{(1+r)(1-f)}{r^n-r}\right)\left( r^n-r^{n-1} \right) = \lambda_{n-1},
\end{aligned}$$ as required. Furthermore, $$\lambda P_{*,n} = p \lambda_{n-1} =
(1-f)\left(\frac{r^{n+1}-r^n}{r^n-r}\right).$$ Also, $$\frac{1-f}{f}= \frac{r^n-r}{r^{n+1}-r^n},$$ so $$\frac{\lambda P_{*,n}}{\lambda_n} =
\frac{p \lambda_{n-1}}{f} =
\frac{(1-f)}{f} \left(\frac{r^{n+1}-r^n}{r^n-r}\right) =
\left(\frac{r^n-r}{r^{n+1}-r^n}\right)\left(\frac{r^{n+1}-r^n}{r^n-r}\right)=1,$$ as required. Finally, $\lambda P_{*,n+1} = \lambda_0 + \lambda_n = 1 = \lambda_{n+1}$, as required. This completes the proof.
It is well known [@LHN] that $f$ is the fixation probability of the Moran process on a regular graph. This is an easy consequence of Lemma \[lem:calc\], but we don’t need it here. We will instead use the following corollary.
\[ourcor\] For all $j\in \{1,\ldots,n-1\}$, $ 1-\frac{1}{r^2}
\leq E[\gamma_j^{n+1}] \leq 1+\frac1r$.
From Lemma \[lem:calc\], $$E[\gamma_j^{n+1}] = \left(\frac{r+1}{r}\right)\left(\frac{r^n-r^j}{r^n-1}\right).$$ The upper bound follows from the fact that $r^n-r^j \leq r^n-1$ (a consequence of $r> 1$ and $j
\geq 0$). For the lower bound, note that $E[\gamma_j^{n+1}]$ is minimised at $j=n-1$ and $$E[\gamma_{n-1}^{n+1}] =
\left(\frac{r^n}{r^n-1}\right)\left(1-\frac{1}{r^2}\right)
.$$ The lower bound then follows from $r^n/(r^n-1)\geq 1$.
Absorption time
---------------
Now let the Moran process be $(Y_t)_{t\geq 1}$ where each state $Y_t$ is the set $S\subseteq V(G)$ of vertices of $G$ that are mutants at the $t$’th step. The state $Y_1$ is selected uniformly at random from the size-$1$ subsets of $V(G)$. For each state $S$, let $p(S) = \Pr(Y_{t+1}\neq S \mid Y_t=S)$ and let $\mu(S) = \inf\{t\geq 1 \mid Y_{t+1} \neq S, Y_1=S\}$. $\mu(S)$ is a random variable representing the number of times that the state $S$ appears when the process is run, starting from $S$, before another state is reached. It is geometrically distributed with parameter $p(S)$, so $E[\mu(S)]=1/p(S)$. The absorption time $\Tabs$ is the number of steps needed to get to state $0$ or state $n$, which is $\Tabs=\inf\{t\geq 1 \mid |Y_t|\in\{0,n\}\}-1$.
Let $\tau_1=1$. For $j>1$, let $\tau_j = \inf\{t>\tau_{j-1} \mid Y_t \neq Y_{t-1}\}$. The values $\tau_2,\tau_3,\ldots$ are the times at which the state changes. These are the active steps of the process. The sequence $Y_{\tau_1},Y_{\tau_2},\ldots$ is the same as the Moran process except that repeated states are omitted. Now recall the Markov chain $(X_t)_{t\geq 0}$ with start state $X_0=n+1$ and recall the definition of the first passage time $T_{n+1}$ which is the first time that the chain returns to state $n+1$. Note that the sequence $n+1, |Y_{\tau_1}|,|Y_{\tau_2}|,\ldots,|Y_{\tau_{(T_{n+1}-1)}}|$ is a faithful simulation of the Markov chain $X_0,X_1,\ldots,X_{T_{n+1}-1}$ starting from state $X_0=n+1$, up until it reaches state $0$ or state $n$. Also, the absorption time satisfies $$\Tabs = \tau_{(T_{n+1}-1)} -1 = \sum_{j=2}^{T_{n+1}-1} (\tau_j - \tau_{(j-1)})$$ and for $j\geq 2$, $\tau_j-\tau_{j-1}$ is distributed as $\mu(Y_{\tau_{j-1}})$, which is geometric with parameter $p(Y_{\tau_{j-1}})$.
To derive upper and lower bounds for $E[\Tabs]$, we break the sum into pieces. For $k\in\{1,\ldots,n-1\}$, let $${T_{\text{A},{k}}} = \sum_{j=2}^{T_{n+1}-1} \Psi_{k,j},$$ where $\Psi_{k,j}$ is geometrically distributed with parameter $p(Y_{\tau_{j-1}})$ if $|Y_{\tau_{j-1}}|=k$ and $\Psi_{k,j}=0$, otherwise. Then $\Tabs$ is distributed as $\sum_{k=1}^{n-1} {T_{\text{A},{k}}}$. In order to derive upper and lower bounds, let $p^+_k = \max\{p(S) \mid |S|=k\}$ and $p^-_k = \min\{p(S) \mid |S|=k\}$. Let ${T_{\text{A},{k}}}^+= \sum_{j=2}^{T_{n+1}-1} \Psi^+_{k,j}$ where $\Psi^+_{k,j}$ is geometrically distributed with parameter $p^-_k$ if $|Y_{\tau_{j-1}}|=k$ and $\Psi^+_{k,j}=0$, otherwise. Let ${T_{\text{A},{k}}}^-= \sum_{j=2}^{T_{n+1}-1} \Psi^-_{k,j}$ where $\Psi^-_{k,j}$ is geometrically distributed with parameter $p^+_k$ if $|Y_{\tau_{j-1}}|=k$ and $\Psi^-_{k,j}=0$, otherwise. Then by stochastic domination for the geometric distribution, $$\label{eq:forWald}
\sum_{k=1}^{n-1} E[{T_{\text{A},{k}}}^-] \leq E[\Tabs] \leq \sum_{k=1}^{n-1}
E[{T_{\text{A},{k}}}^+].$$
\[thm:abs\] $$\left(1-\frac{1}{r^2} \right)
\sum_{k=1}^{n-1} \frac{1}{p_k^+} \leq E[\Tabs] \leq
\left(1+\frac{1}{r}\right) \sum_{k=1}^{n-1} \frac{1}{p_k^-}.$$
By (\[eq:forWald\]), $E[\Tabs]$ is at most $\sum_{k=1}^{n-1}
E[{T_{\text{A},{k}}}^+]$. Now ${T_{\text{A},{k}}}^+$ is a sum of geometric random variables with parameter $p^-_k$. The number of random variables in the sum is $\gamma^{n+1}_k$ which is the number of times that state $k$ is visited between visits to state $n+1$ in the Markov chain $(X_i)$. Since $1/p^-_k$ and $E[\gamma^{n+1}_k]$ are both finite (see Corollary \[ourcor\]), Wald’s equality (see [@MU Theorem 12.3]) guarantees that $E[{T_{\text{A},{k}}}^+] = E[\gamma^{n+1}_k]/p^-_k$. The upper bound follows from Corollary \[ourcor\]. The lower bound is similar.
Upper and lower bounds
----------------------
We start with the following observation.
\[ObsmS\] If $|S|=k$ then $p(S) =
\frac{r {m^+_{S}}}{W_k \Delta} +
\frac{{m^-_{S}}}{W_k \Delta} $ so $\frac{1}{p(S)}= \frac{W_k \Delta }{ r {m^+_{S}} + {m^-_{S}}}
$.
Putting Theorem \[thm:abs\] together with Observation \[ObsmS\] we get the following.
\[cor:together\] The expected absorption time of the Moran process on a strongly connected $\Delta$-regular $n$-vertex digraph $G$ is at least $$\left(1-\frac{1}{r^2}\right)
W_1 \Delta
\sum_{k=1}^{n-1}
\frac{1}
{
\max \left\{
r {m^+_{S}}+{m^-_{S}} \mid |S|=k \right\}
}$$ and is at most $$\left(1+\frac1r\right)W_n\Delta \sum_{k=1}^{n-1}
\frac{1}
{
\min \left\{
r{m^+_{S}}+{m^-_{S}} \mid |S|=k \right\}} .$$
We can now prove Theorem \[thm:regbounds\].
[thm:regbounds]{}[ The expected absorption time of the Moran process on a strongly connected $\Delta$-regular $n$-vertex digraph $G$ is at least $\left(\frac{r-1}{r^2}\right)n H_{n-1}$ and at most $n^2 \Delta$.]{}
If $|S|=k$ then we have the trivial bound $r {m^+_{S}} + {m^-_{S}} \leq (r+1) k \Delta$, which, together with Corollary \[cor:together\], establishes the lower bound. If a digraph is strongly connected, then ${m^+_{S}}$ and ${m^-_{S}}$ are at least $1$ when $1\leq |S|\leq n-1$ so $r{m^+_{S}} + {m^-_{S}} \geq r+1$. This, together with Corollary \[cor:together\], establishes the upper bound.
Note that the upper bound in Theorem \[thm:regbounds\] generalises the one given in Proposition \[prop:oldupper\] to the directed case. The following observations follow from special cases of Corollary \[cor:together\].
Suppose that the graph $G$ is the undirected clique $K_n$ (which is $\Delta$-regular with $\Delta=n-1$). In this case, for $S$ of size $k$, ${m^+_{S}}= {m^-_{S}}=k(n-k)$, so Corollary \[cor:together\] shows that the absorption time is at most $$n \sum_{k=1}^{n-1} \frac{n-1}{k(n-k)}
\leq
n \sum_{k=1}^{n-1} \frac{n-k}{k(n-k)} +
n \sum_{k=1}^{n-1} \frac{k}{k(n-k)}
\leq 2 n H_{n-1},$$ matching the lower bound from Theorem \[thm:regbounds\] up to a constant factor (that depends only on $r$ but not on $n$).
\[obs:cycle\] Suppose that the graph $G$ is the undirected cycle $C_n$ (which is $\Delta$-regular with $\Delta=2$). Since the process starts with a single mutant, it is easy to see that the set of mutant vertices must be connected, if it is non-empty. Therefore, ${m^+_{S}}={m^-_{S}}=2$ for any non-trivial $S$ that is reachable from the initial configuration, so the absorption time is at least $ \left( 1-\frac{1}{r^2}\right) \frac{2n}{r+1} \sum_{k=1}^{n-1} \frac{1}{2} = \Omega(n^2)$, matching the upper bound from Theorem \[thm:regbounds\] up to a constant factor.
\[obs:dircycle\] Suppose that the graph $G$ is the directed $n$-vertex cycle (which is $\Delta$-regular with $\Delta=1$). Again, the mutants remain connected; in this case $ {m^+_{S}}={m^-_{S}}=1$ for any non-trivial $S$ so the absorption time is at least $ \left( 1-\frac{1}{r^2}\right) \frac{n}{r+1} \sum_{k=1}^{n-1} \frac{1}{2} = \Omega(n^2)$, matching the upper bound from Theorem \[thm:regbounds\] up to a constant factor.
Better upper bounds for undirected graphs via isoperimetric numbers {#sec:iso}
-------------------------------------------------------------------
Suppose that a graph $G$ is undirected. As in the proof of Proposition \[prop:oldupper\], let $\partial S$ be the set of (undirected) edges between vertices in $S$ and vertices in $V(G)\setminus S$. Then ${m^+_{S}}={m^-_{S}}=|\partial S|$. The isoperimetric number of the graph $G$ was defined by Buser [@Buser] as follows $$\iso(G) = \min\left\{
\frac{|\partial S|}{|S|} \mid S \subseteq V(G), 0 < |S| \leq |V(G)|/2
\right\}.$$ The quantity $\iso(G)$ is a discrete analogue of the Cheeger isoperimetric constant. For graphs with good expansion, Theorem \[thm:isoperimetric\] improves the upper bound in Theorem \[thm:regbounds\].
[thm:isoperimetric]{}[ The expected absorption time of the Moran process on a connected $\Delta$-regular $n$-vertex undirected graph $G$ is at most $ 2\Delta n H_n /\iso(G)$. ]{}
From Corollary \[cor:together\], the expected absorption time is at most
$$\frac{\Delta W_n}{r} \sum_{k=1}^{n-1}
\frac{1}
{ \min
\left\{ {|\partial_S|}
\mid S \subseteq V(G), |S|=k
\right\}}.$$ This is at most
$$\begin{aligned}
&\frac{\Delta W_n}{r}
\left(2 \sum_{k=1}^{ \lfloor n/2\rfloor }
\frac{1}
{ \min
\left\{ {|\partial_S|}
\mid S \subseteq V(G), |S|=k
\right\}}\right)
\\
& = \frac{\Delta W_n}{r} \left(
2 \sum_{k=1}^{\lfloor n/2 \rfloor}
\frac{1}
{ k \min
\left\{ \frac{|\partial_S|}{k}
\mid S \subseteq V(G), |S|=k
\right\}}\right)
\\
&\leq \frac{\Delta W_n}{i(G)r}
\left(
2 \sum_{k=1}^{\lfloor n/2 \rfloor }
\frac{1}
{ k }
\right)
= \frac{2 \Delta W_n H_{\lfloor n/2 \rfloor }}{i(G)r}. \qedhere
\end{aligned}$$
Families for which the upper bound is optimal {#sec:family}
---------------------------------------------
For every fixed $\Delta>2$, we construct an infinite family of connected, $\Delta$-regular undirected graphs for which the upper bound in Theorem \[thm:regbounds\] is optimal, up to a constant factor (which may depend upon $r$ and $\Delta$ but not on $n$).
To do this, we define the graph $H_\Delta$ to be $K_{\Delta-2,\Delta-1}$ with the addition of edges forming a cycle on the side with $\Delta-1$ vertices. Note that $\Delta-2$ vertices have degree $\Delta-1$ and the others have degree $\Delta$.
Now, let $G_{\ell,\Delta}$ be the $\Delta$-regular graph formed from a cycle $x_1\dots
x_\ell x_1$ and $\ell$ disjoint copies of $H_\Delta$ by adding an undirected edge between $x_i$ and each of the vertices of degree $\Delta-1$ in the $i$’th copy of $H_\Delta$, for each $i\in[\ell]$. Note that $|V(G_{\ell,\Delta})| = 2\ell(\Delta-1)$.
\[thm:new\] For $r>1$ and sufficiently large $\ell$ (with respect to $r$), the expected absorption time of the Moran process on $G_{\ell,\Delta}$ exceeds $\tfrac{1}{2r}(1-\tfrac{1}{r})\ell^2\!$.
Let $n=n(\ell,\Delta) = |V(G_{\ell,\Delta})|$ and let ${T_{\mathrm{d}}}=\ell^2/r$.
Let $(Y_t)_{t\geq 1}$ be the discrete-time Moran process on $G_{\ell,\Delta}$ and consider the following events. Let ${\ensuremath{\mathcal{F}}}^*$ be the event that $Y_{{T_{\mathrm{d}}}+1} = V(G_{\ell,\Delta})$, i.e., that the process reaches fixation in at most ${T_{\mathrm{d}}}$ steps. Let ${\ensuremath{\mathcal{E}}}^*$ be the event that $Y_{{T_{\mathrm{d}}}+1} = \emptyset$, i.e., that the process reaches extinction in at most ${T_{\mathrm{d}}}$ steps. $\Pr({\ensuremath{\mathcal{E}}}^*)$ is at most the extinction probability, which is less than $\tfrac{1}{r}$ since $G$ is regular, so the fixation probability is the quantity $f$ from Lemma \[lem:calc\] [@LHN] which exceeds $1-\tfrac1r$ for $r>1$.
To bound $\Pr({\ensuremath{\mathcal{F}}}^*)$, consider the continuous-time version of the process, ${\widetilde{Y}}[t]$. We will show that, by time ${T_{\mathrm{c}}}=
2{T_{\mathrm{d}}}/n = 2\ell^2/rn$, it is very likely that the continuous process will have had at least ${T_{\mathrm{d}}}$ reproductions. Let ${\ensuremath{\mathcal{S}}}$ (for “slow”) be the event that the continuous process has not had ${T_{\mathrm{d}}}$ reproductions by time ${T_{\mathrm{c}}}$. Let ${\widetilde{{\ensuremath{\mathcal{F}}}}}^*$ be the event that it has reached fixation by time ${T_{\mathrm{c}}}$. We have $$\Pr({\ensuremath{\mathcal{F}}}^*)
=
\Pr({\ensuremath{\mathcal{F}}}^* \wedge \neg {\ensuremath{\mathcal{S}}}) + \Pr({\ensuremath{\mathcal{F}}}^* \wedge {\ensuremath{\mathcal{S}}})
\leq \Pr({\widetilde{{\ensuremath{\mathcal{F}}}}}^*) + \Pr({\ensuremath{\mathcal{S}}}).$$
In the continuous process, each of the $n$ vertices reproduces at rate at least $1$ so the number $N$ of reproductions up to time ${T_{\mathrm{c}}}$ is stochastically bounded below by a Poisson random variable with parameter $n{T_{\mathrm{c}}}=2\ell^2/r$. By a Chernoff-type argument [@MU Theorem 5.4], we have $$\Pr({\ensuremath{\mathcal{S}}})
=
\Pr(N \leq {T_{\mathrm{d}}})
\leq e^{-n{T_{\mathrm{c}}}} \left(\frac{en{T_{\mathrm{c}}}}{{T_{\mathrm{d}}}}\right)^{{T_{\mathrm{d}}}}
= e^{-2\ell^2/r} (2e)^{\ell^2/r}
< \tfrac{1}{4}\left(1-\tfrac{1}{r}\right)\,,$$ for large enough $\ell$.
To bound $\Pr({\widetilde{{\ensuremath{\mathcal{F}}}}}^*)$, consider the process ${\widetilde{Z}}[t]$ on $G_{\ell,\Delta}$ that behaves like ${\widetilde{Y}}[t]$ except for the two following points.
- For some $i$, we have ${\widetilde{Y}}[0]=\{x_i\}$ or ${\widetilde{Y}}[0]$ is in the copy of $H_\Delta$ attached to $x_i$. Let ${\widetilde{Z}}[0] = {\widetilde{Y}}[0]\cup
\{x_i\}$.
- No mutant in the cycle $x_1\dots x_\ell x_1$ can ever be replaced by a non-mutant. That is, if, at time $t$, a non-mutant neighbour of some $x_i$ ($i\in [\ell]$) is selected to reproduce to $x_i$, then the state does not change.
We couple the processes ${\widetilde{Y}}[t]$ and ${\widetilde{Z}}[t]$ as follows. Let $t$ be such that ${\widetilde{Y}}[t]\subseteq {\widetilde{Z}}[t]$, noting that $t=0$ has this property. The coupling lemma (Lemma \[lemma:couple\]) allows us to maintain ${\widetilde{Y}}[t+\tau]\subseteq {\widetilde{Z}}[t+\tau]$ until the next time, $t'\!$, at which a mutant at one of the $x_i$ is replaced by a non-mutant in ${\widetilde{Y}}[t]$. This maintains the property that ${\widetilde{Y}}[t']\subseteq {\widetilde{Z}}[t']$, so the coupling can be restarted from this point.
Now $\Pr({\widetilde{{\ensuremath{\mathcal{F}}}}}^*) = \Pr({\widetilde{Y}}[{T_{\mathrm{c}}}]=V(G_{\ell,\Delta}))
\leq \Pr({\widetilde{Z}}[{T_{\mathrm{c}}}]=V(G_{\ell,\Delta}))$, so we will find an upper bound for $\Pr({\widetilde{Z}}[{T_{\mathrm{c}}}]=V(G_{\ell,\Delta}))$. Let $C = \{x_1, \dots, x_\ell\}$. The set ${\widetilde{Z}}[t]\cap C$ is non-empty, non-decreasing and connected in $G_{\ell,\Delta}$. If ${\widetilde{Z}}[t]\cap C$ is a proper subset of $C$ then it increases exactly when one of the two mutants in $C$ reproduces to its non-mutant neighbour in $C$ or, if there is only one mutant in $C$, when that mutant reproduces to either of its neighbours in the cycle. In both cases, this happens with rate $\tfrac{2r}{\Delta}$, so $|{\widetilde{Z}}[t]\cap C|$ is bounded from above by a Poisson random variable with parameter $\tfrac{2r}{\Delta}t
$. Let $\lambda^*=4\ell/9$. For $t={T_{\mathrm{c}}}$ the parameter is $\tfrac{2r}{\Delta}{T_{\mathrm{c}}}= \tfrac{4 \ell^2}{n \Delta }
<
\frac{4\ell}{\Delta^2}
\leq \lambda^*
$ so $|{\widetilde{Z}}[t]\cap C|$ is bounded above by a Poisson random variable $\Psi^*$ with parameter $\lambda^*$. Therefore, $ E[|{\widetilde{Z}}[{T_{\mathrm{c}}}] \cap C|] \leq \lambda^*$, and we have $$\begin{aligned}
\Pr\left(|{\widetilde{Z}}[{T_{\mathrm{c}}}]\cap C| \geq \tfrac{8}{9}\ell\right)
\leq \Pr(\Psi^* \geq 2 \lambda^*) \leq
\left(\tfrac{e}{4}\right)^{\lambda^*}
< \tfrac{1}{4}\left(1-\tfrac{1}{r}\right)\,,
\end{aligned}$$ for large enough $\ell$. Now, $$E[\Tabs] \geq E\left[\Tabs\mid
\overline{{\ensuremath{\mathcal{F}}}^* \cup {\ensuremath{\mathcal{E}}}^*}\,
\right]
\times \Pr\left(\overline{{\ensuremath{\mathcal{F}}}^* \cup
{\ensuremath{\mathcal{E}}}^*}\right)\,.$$ Clearly, the expected absorption time of the discrete process conditioned on absorption not occurring within $\ell^2/r$ steps is at least $\ell^2/r$. Meanwhile, $$\Pr\left(\overline{{\ensuremath{\mathcal{F}}}^* \cup {\ensuremath{\mathcal{E}}}^*}\right)
> 1 - \tfrac{1}{r} - 2\tfrac{1}{4}\left(1-\tfrac{1}{r}\right)
= \tfrac{1}{2}\left(1-\tfrac{1}{r}\right)$$ for large enough $\ell$ (with respect to $r$).
Thus, we have shown that the $O(n^2)$ upper bound of Theorem \[thm:regbounds\] is tight up to a constant factor (which may depend on $r$ and $\Delta$, but not on $n$).
[thm:tight]{}[ Suppose that $r>1$ and $\Delta>2$. There is an infinite family $\mathcal{G}$ of $\Delta$-regular graphs such that, when the Moran process is run on an $n$-vertex graph $G\in \mathcal{G}$, the expected absorption time exceeds $\tfrac{1}{8r}(1-\tfrac{1}{r})\frac{n^2}{(\Delta-1)^2} $. ]{}
For a given value of $r$, Let $\ell_r$ be the smallest value of $\ell$ for which that Theorem \[thm:new\] applies. Take $\mathcal{G}=\{ G_{\ell,\Delta} \mid \ell \geq \ell_r \}$ and the result is immediate from Theorem \[thm:new\] and the fact that $G_{\ell,\Delta}$ has $2\ell(\Delta-1)$ vertices.
General digraphs {#sec:directed}
================
Fix $r>1$ and let $\epsilon_r = \min(r-1,\,1)$. Theorem 7 of [@OurSODA] shows that the expected absorption time of the Moran process on a connected $n$-vertex undirected graph is at most $(r/(r-1)) n^4$. Theorem \[thm:regbounds\] shows that the expected absorption time on a strongly connected $\Delta$-regular digraph is at most $n^2 \Delta$. In contrast, we show that there is an infinite family of strongly connected digraphs such that the expected absorption time of the Moran process on an $n$-vertex graph from the family is $2^{\Omega(n)}\!$.
The family of graphs
--------------------
=\[fill=black, draw=black, circle, inner sep=1.5pt\]
in [0,...,11]{} () at (30\*:2) ;
in [0,...,10]{} in [,...,11]{} () – ();
(u1) at (4,-1) \[label=-90:[$u_1$]{}\] ; (u2) at (6,-1) \[label=-90:[$u_2$]{}\] ; at (7.5,-1) [$\cdots$]{}; (uN) at (9,-1) \[label=-90:[$u_N$]{}\] ;
(u1)–(u2); (u2)–(7,-1); (8,-1)–(uN);
(v0) at (2.75,1) \[label=[\[label distance=2.5pt\]90:[$\smash{v_0}$]{}]{}\] ;
(v1) at (3.5,1) \[label=[\[label distance=2.5pt\]90:[$\smash{v_1}$]{}]{}\] ; at (4,1) [$\cdots$]{}; (v2) at (4.5,1) \[label=[\[label distance=2.5pt\]90:[$\smash{v_{4{\left\lceil {r}\right\rceil}}}$]{}]{}\];
(v3) at (5.5,1) ; at (6,1) [$\cdots$]{}; (v4) at (6.5,1) \[label=[\[label distance=2.5pt\]90:[$\smash{v_{8{\left\lceil {r}\right\rceil}}}$]{}]{}\] ;
at (7.5,1) [$\cdots$]{};
(v5) at (8.5,1) ; at (9,1) [$\cdots$]{}; (v6) at (9.5,1) \[label=[\[label distance=2.5pt\]90:[$\smash{v_{4{\left\lceil {r}\right\rceil}N}}$]{}]{}\] ;
(v1)–(v0); (v3)–(v2); (7,1)–(v4); (v5)–(8,1);
in [0, ..., 11]{} ()–(u1);
in [0, ..., 11]{} (v0)–();
(v1)–(u1); (v2)–(u1);
(v3)–(u2); (v4)–(u2);
(v5)–(uN); (v6)–(uN);
(uN) .. controls (10,-0.5) and (11,0.5) .. (v6);
(4,0.75)–(u1); (6,0.75)–(u2); (9,0.75)–(uN);
at (0,0) [$K_N$]{};
Let $G_{r,N}$ be the disjoint union of the complete graph $K_N$ (with bidirectional edges), a directed path $P=u_1\dots u_N$ and a directed path $Q=v_{4{\left\lceil {r}\right\rceil}N}
\dots v_0$, along with the directed edge $(u_N,v_{4{\left\lceil {r}\right\rceil}N})$ and the following directed edges (see Figure \[fig:GrN\]):
- $(x,u_1)$ and $(v_0,x)$ for every $x\in K_N$;
- $(v_{4{\left\lceil {r}\right\rceil}(i-1)+j},u_i)$ for each $i\in[N]$, $j\in[4{\left\lceil {r}\right\rceil}]$.
The intuition is as follows. Consider the Moran process on $G_{r,N}$. With probability close to $\tfrac{1}{4{\left\lceil {r}\right\rceil}+2}$, the initial mutant is in the clique (Observation \[obs:StartClique\]). Conditioned on this, Lemmas \[lem:SometimePlentyClique\], \[lem:Extinction\] and \[lem:Fixation\] allow us to show that it is fairly likely that there is a time during the first $N^3$ steps when the clique is half full, but that absorption does not happen in the first $T^*(N)$ steps for a function $T^*$ which is exponential in $N$. Of course, the expected absorption time conditioned on this is at least $T^*(N)$, so we conclude (Theorem \[thm:directed\]) that the overall expected absorption time is at least $T^*(N)$.
The main challenge of the proof is the second step — showing that is it is fairly likely that there is a time during the first $N^3$ steps when the clique is half full, but that absorption does not happen in the first $T^*(N)$ steps. The fact that the clique becomes half full (Lemma \[lem:SometimePlentyClique\]) follows by dominating the number of mutants in the clique during the initial stages of the process by an appropriate one-dimensional random walk, and then showing that sufficiently many random-walk steps are actually taken during the first $N^3$ steps of the process. The fact that extinction is then unlikely in the first $T^*(N)$ steps (Lemma \[lem:Extinction\]) follows from the fact that the many mutants in the clique are unlikely to become extinct very quickly. On the other hand, the fact that fixation is unlikely in the first $T^*(N)$ steps (Lemma \[lem:Fixation\]) follows from the fact that mutants make slow progress along the path $P$ because because vertices in $Q$ tend to push the “mutant frontier” backwards towards $u_1$. However, the chain of mutants has to push all the way around this chain in order for fixation to occur.
The one-dimensional random walk
-------------------------------
The following Lemma is Example 3.9.6 from [@GS2001:Probability].
\[lemma:absorb\] Let $(Z_t)_{t\geq 0}$ be the random walk on $\{0, \dots, n\}$ with absorbing barriers at $0$ and $n$ and, for $0<Z_t<n$, let $Z_{t+1} =
Z_t+1$ with probability $p\neq\tfrac12$ and $Z_{t+1}=Z_{t}-1$ with probability $q=1-p$. Let $p_i$ be the probability of absorption at $0$, given that $Z_0=i$. Writing $\rho = q/p$, $$p_i = \frac{\rho^i - \rho^n}{1 - \rho^n}\,.$$
Bounding the absorption time
----------------------------
Consider the Moran process $(Y_t)_{t\geq 1}$ on $G_{r,N}$. We will assume that $N$ is sufficiently large with respect to $r$ — the exact inequalities that we need will be presented as they arise in the proof. Let $n = 1+(4{\left\lceil {r}\right\rceil}+2)N$ be the number of vertices of $G_{r,N}$. Let $W_t = n + (r-1)|Y_t|$ be the total fitness of $Y_t$. Let $\Tabs$ be the absorption time of the process. Our goal is to show that $E[\Tabs]$ is exponentially large, as a function of $N$. Let $$T^*(N) = {\left\lfloor {\left(\frac{\epsilon_r}{32}\right)(2^N-1)}\right\rfloor}$$ and let $N'$ denote ${\left\lfloor {N/2}\right\rfloor}$. We now identify various events which we will study in the lemmas that follow.
- Let ${\ensuremath{\mathcal{S_C}}}$ be the event that $Y_1 \subseteq K_N$ (mnemonic: ${\ensuremath{\mathcal{S_C}}}$ is the event that the initial mutant starts in the clique; $\mathcal S$ is for “Starts” and $\mathcal C$ is for “Clique”).
- Let ${\ensuremath{\mathcal H_{C,t}}}$ be the event that $|Y_t \cap K_N| \geq N'$ (mnemonic: ${\ensuremath{\mathcal H_{C,t}}}$ is the event that the clique is half full at time $t$. $\mathcal H$ is for “Half”).
- Let ${\ensuremath{\mathcal{H_C}}}= \bigcup_{t\in [N^3+1]} {\ensuremath{\mathcal H_{C,t}}}$.
- Let ${\ensuremath{\mathcal{F}}}^*$ be the event that $Y_{T^*(N)+1} = V(G_{r,N})$. (mnemonic: ${\ensuremath{\mathcal{F}}}^*$ is the event that fixation occurs after at most $T^*(N)$ steps; $\mathcal F$ is for “Fixation”).
- Let ${\ensuremath{\mathcal{E}}}^*$ be the event that $Y_{T^*(N)+1}=\emptyset$ (mnemonic: ${\ensuremath{\mathcal{E}}}^*$ is the event that extinction occurs after at most $T^*(N)$ steps; $\mathcal{E}$ is for “Extinction”).
\[obs:StartClique\] $$\Pr({\ensuremath{\mathcal{S_C}}}) =\frac{N}{n}= \frac{N}{N(4 {\left\lceil {r}\right\rceil}+2)+1}.$$
\[lem:SometimePlentyClique\] $\Pr({\ensuremath{\mathcal{H_C}}}\mid {\ensuremath{\mathcal{S_C}}}) \geq \epsilon_r/8.$
Let $Z'_t = |Y_t \cap K_N|$. We will condition on the fact that ${\ensuremath{\mathcal{S_C}}}$ occurs, so $Z'_1=1$. If $Z'_t \in \{1,\ldots,N'-1\}$ then for all $z_t$ and $w_t$, $$\Pr(Z'_{t+1}=z'_t+1\mid Z'_t=z_t, W_t=w_t)\geq \left(\frac{r z'_t}{w_t}\right)\left( \frac{N-z'_t}{N}\right).$$ (The probability is greater than this if $v_0$ is in $Y_t$.) Also, $$\Pr(Z'_{t+1}=z'_t-1\mid Z'_t=z'_t,W_t=w_t) \leq
\left( \frac{1}{w_t}\right)\left(\frac{z'_t}{N} \right)+ \left(\frac{N-z'_t}{w_t}\right)\left(\frac{z'_t}{N}
\right),$$ where the first term comes from reproduction from a non-mutant at $v_0$ and the second from reproduction within the clique. Also, $Z'_{t+1} \in \{Z'_t-1,Z'_t,Z'_t+1'\}$. Now let $$p' = \frac{r}{r+1+\tfrac{2}{N}}$$ and note that $$\frac{\Pr(Z'_{t+1}=z'_t+1 \mid Z'_t=z'_t)}
{\Pr(Z'_{t+1}=z'_t+1\mid Z'_t=z'_t) + \Pr(Z'_{t+1}=z'_t-1\mid Z'_t=z'_t)}
\geq
\frac{r(N-z'_t)}{r(N-z'_t)+1+(N-z'_t)}
\geq
p'.$$ The restriction of $(Z'_t)$ to steps where the state changes, stopping when $Z'_t$ reaches $0$ or $N'$ is a process which is dominated below by $Z_t$, a random walk on $\{0,\ldots,N'\}$ that starts at $1$ and absorbs at $0$ and $N'$ and has parameter $p'$.
Now let ${\ensuremath{\mathcal{E_C}}}$ be the event that there is a $t$ with $Y_t\cap K_N=\emptyset$ such that, for all $ t'<t$, we have $|Y_{t'}\cap K_N| < N'$. ${\ensuremath{\mathcal{E_C}}}$ is the event that the clique becomes empty before it becomes half full. Then applying Lemma \[lemma:absorb\] to the dominating random walk, $\Pr({\ensuremath{\mathcal{E_C}}}\mid {\ensuremath{\mathcal{S_C}}}) \leq
\frac{\rho - \rho^{N'}}{1 - \rho^{N'}}\leq \rho$, where $\rho=(1-p')/p' \leq (N+2)/(N+\epsilon_r N)$. Since we are taking $N$ to be sufficiently large with respect to $\epsilon_r$ (in particular, we will take $N\geq 4/\epsilon_r$) we have $\rho \leq (1+\epsilon_r/2)/(1+\epsilon_r)$ which is at most $1-\epsilon_r/4$ since $\epsilon_r \leq 1$.
Let ${\ensuremath{\mathcal{Q}}}$ be the event that there is a $t\in[N^3+1]$ with $|Y_t\cap K_n| \notin \{1,\ldots,N'-1\}$. (${\ensuremath{\mathcal{Q}}}$ is the event that the size of the clique changes quickly — it takes at most $N^3$ steps to either become empty or to become at least half full). We will show below that $\Pr({\ensuremath{\mathcal{Q}}}\mid {\ensuremath{\mathcal{S_C}}}) \geq 1- \epsilon_r/8$. Note that if ${\ensuremath{\mathcal{Q}}}$ occurs but ${\ensuremath{\mathcal{E_C}}}$ does not occur then ${\ensuremath{\mathcal{H_C}}}$ occurs. So $$\Pr({\ensuremath{\mathcal{H_C}}}\mid {\ensuremath{\mathcal{S_C}}}) \geq
\Pr({\ensuremath{\mathcal{Q}}}\setminus {\ensuremath{\mathcal{E_C}}}\mid {\ensuremath{\mathcal{S_C}}}) \geq
\Pr({\ensuremath{\mathcal{Q}}}\mid {\ensuremath{\mathcal{S_C}}}) - \Pr({\ensuremath{\mathcal{E_C}}}\mid {\ensuremath{\mathcal{S_C}}}) \geq
(1- \epsilon_r/8)-(1-\epsilon_r/4)\geq \epsilon_r/8,$$ which would complete the proof.
We conclude the proof, then, by showing $\Pr(\neg {\ensuremath{\mathcal{Q}}}\mid {\ensuremath{\mathcal{S_C}}}) \leq \epsilon_r/8$. So we will show that $\Pr(Z'_1,\ldots,Z'_{N^3+1} \in \{1,\ldots,N'-1\} \mid Z'_1=1) \leq \epsilon_r/8$. For this, let $\delta=\epsilon_r/20$ and let $\Psi = N'/\delta$. We require that $N$ is sufficiently large with respect to $r$, so $N^3 \geq 4 n \Psi$.
Let $\Upsilon_1,\ldots,\Upsilon_{1+\Psi}$ be $\Psi$ steps of a random walk on the integers that starts with $\Upsilon_1=1$ and has $\Upsilon_{t+1}=\Upsilon_t+1$ with probability $p'$ and $\Upsilon_{t+1}=\Upsilon_t-1$ with probability $1-p'$. It is likely that the state $\Upsilon_t$ increases at least $(1-\delta) p' \Psi$ times. By a Chernoff bound (e.g., [@MU Theorem 4.5]), the probability that this does not happen is at most $\exp(-p'\Psi \delta^2/2)$. Since $N$ is sufficiently large with respect to $r$ (and therefore $N'$ is sufficiently large with respect to $r$) and $p'\geq r/(r+2)$, this probability is at most $\epsilon_r/16$. (This calculation is not tight in any way — $\epsilon_r/16$ happens to be sufficient.) If there are at least $(1-\delta)p' \Psi$ increases then $$\Upsilon_{\Psi+1} - \Upsilon_1 \geq (2(1-\delta)p'-1)\Psi.$$ But since $$\delta = \frac{\left(\frac{\epsilon_r}{4}\right)}{5}< \frac{\frac{\epsilon_r}{2}-\frac{1}{N}}{2+\frac{3 \epsilon_r}{2}+\frac{1}{N}},$$ we have $$p' = \frac{1+\epsilon_r}{2(1+\frac{\epsilon_r}{2}+\frac{1}{N})} > \frac{1+\delta}{2(1-\delta)}.$$ so $$\label{eq:driftup}
\Upsilon_{\Psi+1} - \Upsilon_1 \geq (2(1-\delta)p'-1)\Psi \geq \delta \Psi = N'.$$
Now if $Z'_t \in \{1,\ldots,N'-1\}$ then $$\Pr(Z'_{t+1}\neq z'_t \mid Z'_t=z'_t,W_t=w_t)
\geq \frac{(r+1)z'_t (N-z'_t)}{N w_t} \geq \frac{1}{2n}.$$ (Again, this calculation is not tight, but $1/(2n)$ suffices.) If we select $N^3$ Bernoulli random variables, each with success probability $\frac{1}{2n}$, then, by another Chernoff bound, the probability that we fail to get at least $N^3/(4n)\geq \Psi$ successes is at most $\exp(- N^3/(16n)) \leq \epsilon_r/16$.
Now $\Pr(Z'_1,\ldots,Z'_{N^3+1} \in \{1,\ldots,N'-1\} \mid Z'_1=1)$ is at most the sum of two probabilities.
- The probability that $Z'_1,\ldots,Z'_{N^3+1}$ are all in $\{1,\ldots,N'-1\}$ and there fewer than $\Psi$ values $t$ with $Z'_{t+1}\neq Z'_t$. This is dominated by the the selection of $N^3$ Bernoulli random variables as above, and the probability is at most $\epsilon_r/16$.
- The probability that $Z'_1,\ldots,Z'_{N^3+1}$ are all in $\{1,\ldots,N'-1\}$ and there are at least $\Psi$ values $t$ with $Z'_{t+1}\neq Z'_t$. In order to bound this probability, imagine the evolution of the process proceeding in two sub-steps at each step. First, decide whether $Z'_{t+1}\neq Z'_t$ with the appropriate probability. If so, select $Z'_{t+1} \in \{Z'_t+1,Z'_t-1\}$ with the appropriate probability. The probability of the whole event is then dominated above by the probability that $\Upsilon_{\Psi+1} - \Upsilon_1 < N'$, which is at most $\epsilon_r/16$, as we showed above. (If this difference is at least $N'$ then either the $Z'_t$ process hits $0$ before it changes for the $\Psi$’th time, or $Z'_t$ reaches $N'$.)
\[lem:PlentyClique\] $\Pr(Y_{T^*(N)+t}=\emptyset \mid {\ensuremath{\mathcal H_{C,t}}}\wedge {\ensuremath{\mathcal{S_C}}}) \leq
\epsilon_r/(32 (N^3+1)).$
As in the proof of Lemma \[lem:SometimePlentyClique\], let $Z'_t = |Y_t \cap K_N|$. If ${\ensuremath{\mathcal H_{C,t}}}$ holds then $Z'_t \geq N'$. Let $\Gamma$ be the number of distinct values $t'$ with $t < t' < \inf\{t''>t \mid Z'_{t''}=0\}$ satisfying $Y_{t'}=N'$. We will show that $\Pr(\Gamma < T^*(N)) \leq \epsilon_r/(32 (N^3+1))$.
For any $T>t$, suppose that $Z'_T=N'-1$ and let $T' = \inf\{t''>t \mid Z'_{t''}\in\{0,N'\} \}$. Let $\pi = \Pr(Z'_{T'}=0)$. By the argument in the proof of Lemma \[lem:SometimePlentyClique\], $\pi$ is at most the probability that a random walk $Z_t$ on $\{0,\ldots,N'\}$ which starts at $N'-1$ and absorbs at $0$ and $N'$ and has parameter $p'$ absorbs at $0$. By Lemma \[lemma:absorb\], $$\pi \leq \frac{\rho^{N'-1}-\rho^N}{1-\rho^N} \leq \rho^{N'-1},$$ where $\rho=(N+2)/(N+\epsilon_r N)<1$. Then $\Pr(\Gamma < T^*(N)) \leq T^*(N) \rho^{N'-1}$. We will choose $N$ to be sufficiently large so that $$T^*(N)= {\left\lfloor {\left(\frac{\epsilon_r}{32}\right)(2^N-1)}\right\rfloor} \leq
\left(\frac{\epsilon_r}{32(N^3+1)}\right){\left(\frac{N+\epsilon_r N}{N+2}\right)}^{N'-1}.$$ Then $\Pr(\Gamma < T^*(N)) \leq T^*(N) \rho^{N'-1} \leq \epsilon_r/(32 (N^3+1))$, which completes the proof.
\[lem:Extinction\] $\Pr({\ensuremath{\mathcal{E}}}^* \mid {\ensuremath{\mathcal{H_C}}}\wedge {\ensuremath{\mathcal{S_C}}}) \leq \epsilon_r/32.$
This follows easily from Lemma \[lem:PlentyClique\] using the following summation. $$\begin{aligned}
\Pr({\ensuremath{\mathcal{E}}}^* \mid {\ensuremath{\mathcal{H_C}}}\wedge {\ensuremath{\mathcal{S_C}}})
&= \frac{\Pr({\ensuremath{\mathcal{E}}}^* \wedge {\ensuremath{\mathcal{H_C}}}\wedge {\ensuremath{\mathcal{S_C}}})}
{\Pr({\ensuremath{\mathcal{H_C}}}\wedge {\ensuremath{\mathcal{S_C}}})} \leq
\frac
{\sum_{t\in [N^3+1]} \Pr({\ensuremath{\mathcal{E}}}^* \wedge {\ensuremath{\mathcal H_{C,t}}} \wedge {\ensuremath{\mathcal{S_C}}})}
{\Pr({\ensuremath{\mathcal{H_C}}}\wedge {\ensuremath{\mathcal{S_C}}})} \\
&=
\sum_{t\in[N^3+1]} \frac
{\Pr({\ensuremath{\mathcal{E}}}^* \mid {\ensuremath{\mathcal H_{C,t}}} \wedge {\ensuremath{\mathcal{S_C}}})\Pr({\ensuremath{\mathcal H_{C,t}}} \mid {\ensuremath{\mathcal{S_C}}}) \Pr({\ensuremath{\mathcal{S_C}}})}
{\Pr({\ensuremath{\mathcal{H_C}}}\mid {\ensuremath{\mathcal{S_C}}}) \Pr({\ensuremath{\mathcal{S_C}}})}
\\
&\leq
\sum_{t\in[N^3+1]}
\Pr({\ensuremath{\mathcal{E}}}^* \mid {\ensuremath{\mathcal H_{C,t}}} \wedge {\ensuremath{\mathcal{S_C}}})
\\
& \leq
\sum_{t\in[N^3+1]}
\Pr( Y_{T^*(N)+t}=\emptyset \mid {\ensuremath{\mathcal H_{C,t}}} \wedge {\ensuremath{\mathcal{S_C}}}).\qedhere\end{aligned}$$
\[lem:Fixation\] $\Pr({\ensuremath{\mathcal{F}}}^* \mid {\ensuremath{\mathcal{S_C}}}) \leq \epsilon_r/32.$
Recall the paths $P=u_1\dots u_N$ and $Q=v_{4{\left\lceil {r}\right\rceil}N} \cdots v_0 $ in $G_{r,N}$. Define $U_t$ as follows.
- If $Y_t \cap Q$ is non-empty then $U_t=N$.
- If $Y_t \cap Q$ and $Y_t \cap P$ are both empty then $U_t=0$.
- If $Y_t \cap Q$ is empty and $Y_t \cap P$ is non-empty then $U_t = \max\{i \mid u_i \in Y_t\}$.
Let $\tau = \inf\{t \mid U_t=N\}$. We will show that $\Pr(\tau < T^*(N))\leq \epsilon_r/32$. If $U_t \in \{1,\ldots,N-1\}$ then $\Pr(U_{t+1}=U_t+1) = \frac{r}{W_t}$ and $\Pr(U_{t+1} = U_t-1) \geq \left(\frac{4{\left\lceil {r}\right\rceil}}{W_t}\right)
\left(\frac12\right)\geq \frac{2r}{W_t}$. Also, $U_{t+1} \in \{U_{t-1},U_t,U_t+1\}$.
Let $F_t = \inf\{t'>t \mid U_{t'}\in\{0,N\}\}$ and $\gamma = \Pr(U_{F_t}=N \mid U_t=1)$. $\gamma$ is at most the probability of absorbing at $N$ in a random walk on $\{0,\ldots,N\}$ that starts at $1$, absorbs at $0$ and $N$ and has parameter $1/3$ (twice as likely to go down as to go up). By Lemma \[lemma:absorb\] $$\gamma \leq 1-\left(\frac{ \rho-\rho^N}{1-\rho^N}\right),$$ where $\rho=2$, so $\gamma \leq1/(2^N-1)$. Now let $\Psi$ be the number of times $t<\tau$ with $U_t=0$. Then $\Psi \leq \tau$ so $$\Pr(\tau < T^*(N)) \leq \Pr (\Psi < T^*(N)) \leq T^*(N) \gamma \leq \epsilon_r/32.
\qedhere$$
Putting together the lemmas in this section, we prove **Theorem \[thm:directed\]**. Recall that $T^*(N) = {\left\lfloor {\left(\frac{\epsilon_r}{32}\right)(2^N-1)}\right\rfloor}$. For convenience, we restate the theorem using this notation.
\[thm:newdirected\] Fix $r>1$ and let $\epsilon_r = \min(r-1,\,1)$. Suppose that $N$ is sufficiently large with respect to $r$ and consider the Moran process $(Y_t)_{t\geq 1}$ on $G_{r,N}$. Let $\Tabs$ be the absorption time of the process. Then $E[\Tabs] > \tfrac{1}{16}T^*(N)\,\epsilon_r/(4{\left\lceil {r}\right\rceil}+3)$.
$$E[\Tabs] \geq E[\Tabs \mid {\ensuremath{\mathcal{S_C}}}\wedge {\ensuremath{\mathcal{H_C}}}\setminus
({\ensuremath{\mathcal{F}}}^* \cup {\ensuremath{\mathcal{E}}}^*)] \times \Pr({\ensuremath{\mathcal{H_C}}}\setminus ({\ensuremath{\mathcal{F}}}^* \cup {\ensuremath{\mathcal{E}}}^*)\mid {\ensuremath{\mathcal{S_C}}}) \times \Pr({\ensuremath{\mathcal{S_C}}}).$$
The first term on the right-hand side is greater than $T^*(N)$ by the definition of the excluded events ${\ensuremath{\mathcal{F}}}^*$ and ${\ensuremath{\mathcal{E}}}^*$. Lemmas \[lem:SometimePlentyClique\], \[lem:Extinction\] and \[lem:Fixation\] show that the second term on the right-hand side is at least $ \epsilon_r/8 - 2 \epsilon_r/32\leq \epsilon_r/16$. Finally, Observation \[obs:StartClique\] shows that the third term on the right-hand side is at least $ 1/(4{\left\lceil {r}\right\rceil}+3)$.
[^1]: Departament de Llenguatges i Sistemes Inform[à]{}tics, Universitat Polit[è]{}cnica de Catalunya, Spain.
[^2]: Department of Computer Science, University of Oxford, UK.
[^3]: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) ERC grant agreement no. 334828. The paper reflects only the authors’ views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.
[^4]: This is closely related to the jump chain, which is defined to be the discrete-time chain whose successive states are the states ${\widetilde{Y}}[t]$ for the successive times $t$ immediately after the state changes. Thus, the jump chain is the chain of “active” steps of the discrete-time Moran process (see Section \[sec:active\]).
|
---
abstract: 'Loop quantum cosmology is a symmetry-reduced application of loop quantum gravity that has led to the resolution of classical singularities such as the big bang, and those at the center of black holes. This can be seen through numerical simulations involving the quantum Hamiltonian constraint that is a partial [*difference*]{} equation. The equation allows one to study the evolution of sharply-peaked Gaussian wave packets that generically exhibit a quantum “bounce” or a non-singular passage through the classical singularity, thus offering complete singularity resolution. In addition, von-Neumann stability analysis of the difference equation – treated as a stencil for a numerical solution that steps through the triad variables – yields useful constraints on the model and the allowed space of states. In this paper, we develop a new method for the numerical solution of loop quantum cosmology models using a set of basis functions that offer a number of advantages over computing a solution by stepping through the triad variables. We use the Corichi and Singh model for the Schwarzschild interior as the main case study in this effort. The main advantage of this new method is computational efficiency and the ease of parallelization. In addition, we also discuss how the stability analysis appears in the context of this new approach.'
author:
- Alec Yonika
- Gaurav Khanna
title: |
Basis Function Method for Numerical Loop Quantum Cosmology:\
The Schwarzschild Black Hole Interior
---
Introduction
============
Classical general relativity is plagued with singularities such as those at the center of black holes and also the big bang in cosmological models. In loop quantum gravity, the spacetime continuum is replaced by a discrete quantum structure wherein geometric operators such as areas and volumes have discrete eigenvalues with a non-zero minimum [@lqg1; @lqg2; @lqg3]. The discrete structure of quantum spacetime only plays a significant role when the curvature approaches Planck scale; otherwise, strong agreement with general relativity is found. Nearly two decades of study has been performed in the context of singularity resolution in symmetry-reduced cosmological models with fairly robust results that replace the big bang with a big bounce [@lqc; @lqc1; @lqc2; @lqc3]. States sharply peaked on classical trajectories when the universe is large and expanding, can be evolved backward using the quantum Hamiltonian constraint; such states evolve in a stable and non-singular way, and bounce in the deep Planck regime into a contracting branch [@dgms14].
Several of the above mentioned results were obtained through the application of computational techniques in the field of loop quantum gravity (See Refs. [@nlqc; @nlqc1; @nlqc2] and references therein). Recently, similar numerical studies were performed in the context of the Corichi and Singh (CS) model of the Schwarzschild interior [@CS] resulting in a clear numerical demonstration of singularity resolution [@YKS]. The CS model improves upon previous models, offering a consistent and correct infra-red limit and independence from fiducial structures used in the quantization procedure. The quantum Hamiltonian constraint is a partial [*difference*]{} equation in two discrete triad variables making it somewhat more complicated in comparison with the previously studied cosmological models. Von-Neumann stability analysis of this equation results in a stability condition for black holes which have a very large mass compared to the Planck mass. In addition, further analysis for such large black holes leads to a constraint on the choice of the allowed states in numerical evolution. Evolution of a sharply peaked Gaussian wave packet yields a bounce in one of the triad variables, but for the other triad variable singularity resolution arises through a simple passage through the classical singularity. In addition, states are found to be peaked at the classical trajectory for a long time before and after the classical singularity [@YKS]. These results support a [*symmetric*]{} quantum black hole to white hole transition paradigm that has received tremendous attention in recent years [@bhwh; @bhwh1; @bhwh2].
Needless to say that the semi-classical behavior of the model is expected in the regime where the triad variables have large values in comparison to the Planck scale. This implies that the numerical triad grid required must span a very large domain, and is often limited by the finite computational resources (memory, compute time, numerical precision, etc.). In practice, our previous efforts succeeded in performing numerical simulations only on the scale of a few hundred Planck units in each triad dimension [@YKS]. The method used therein was a straightforward approach inspired by a finite-difference stencil computation – a recursive stepping through a 2D grid built using two discrete- valued triad variables. The method was difficult to parallelize owing to its intrinsically serial structure and was also constrained by finite floating-point numerical precision. In this paper, we develop a new numerical solution method inspired by the well-known spectral collocation approach for numerical solutions of partial differential equations that utilizes a set of basis functions in one triad variable, and performs an explicit stepping in the other variable. This allows for high computational efficiency, especially for the larger sized computations and reduced numerical precision requirements. Moreover, the basis function approach is readily parallelizable on multi- and many- core processors like modern CPUs and GPUs.
An alternative method of solution to quantum Hamiltonian constraints could also offer a different perspective on the role of the von-Neumann stability analysis in loop quantum cosmology models. After all, one may ask, are the discovered instabilities through that analysis a true feature of the model and possibly the underlying physics, or simply an artifact of some sort of how one solves the equations? The von-Neumann stability analysis is most commonly used to evaluate finite-difference stencil-based computations, i.e. a local stepping approach on a grid built using the triad variables’ discrete set of values. What if one took a more global approach towards solving the equation, that doesn’t involve any stepping? Does the same instability manifest itself in some other way? If so, then indeed, that would be a strong indication of the inherent nature of the instability and its relevance to a property of the model and possibly even something physical. In this paper, we study this question in some detail and show that the previously discovered instabilities are not simply artifacts of the manner in which the equations where solved, rather they are indicative of the issues within the CS model itself or something else of significance (see Ref. [@YKS] for different possibilities).
This paper is organized as follows: In Sec. \[Background\] we introduce the loop quantum CS model and briefly describe its key features. In Sec. \[BFM\] we present the new numerical method to solve such models, and in Sec. \[VS\] we apply this technique to a variable-separated representation of the CS model. In Sec. \[2DBFM\] the application is broadened to the full 2D form of the CS model. A discussion of some of the technical aspects of the implementation of the new numerical technique are also presented therein. A detailed discussion of the instability exhibited by the CS model is presented in Sec. \[INST\]. We conclude with a discussion of results in Sec. \[DnC\], and the novel benefit that the numerical technique offers is elaborated on in Sec. \[PARA\] and Sec. \[PREC\].
Background {#Background}
==========
The loop quantization of the Schwarzschild interior is performed using a Kantowski-Sachs vacuum spacetime with a phase space expressed in terms of holonomies of Ashtekar-Barbero connection components $b$ and $c$, and the two conjugate triad variables $p_b$ and $p_c$. The space-time metric in terms of these variables is given by $${\mathrm{d}} s^2 = - N^2 {\mathrm{d}} t^2 + \frac{p_b^2}{|p_c| L_o^2} {\mathrm{d}} x^2 + |p_c| ({\mathrm{d}} \theta^2 + \sin^2 \theta {\mathrm{d}} \phi^2) ~.$$ Here $L_o$ is a fiducial length scale in the $x$-direction of the spatial manifold. To relate this with the usual Schwarzschild metric variables, $p_b$ and $p_c$ satisfy $$\frac{p_b^2}{p_c} = \frac{2 m}{t} - 1 ~~~~ |p_c| = t^2 ~,$$ where $m = G M$, with $M$ as the ADM mass of the black hole space-time. In the classical theory, the horizon at $t=2m$ is identified with $p_b = 0$ and $p_c = 4 m^2$ while the central singularity is where both $p_b$ and $p_c$ vanish. In the quantum theory, the eigenvalues of triad operators are given by $$\label{pb_pc_ev}
\hat p_b \, |{\mu,\tau}\rangle = \frac{\gamma \lp^2}{2} \, \mu \, |{\mu,\tau}\rangle, ~~ \hat p_c \, |{\mu,\tau}\rangle = \gamma \lp^2 \, \tau \, |{\mu,\tau}\rangle$$ where $\gamma$ is the Immirzi parameter and $\lp$ is the Planck length. Loop quantization of the classical Hamiltonian constraint using the holonomies of the connection components $b$ and $c$ yields the following quantum difference equation [@CS] $$\begin{aligned}
\label{CS}
&& \nonumber \db(\sqrt{|\tau|} + \sqrt{|\tau + 2 \dc|}) \left(\Psi_{\mu + 2 \db, \tau + 2 \dc} -
\Psi_{\mu - 2 \db, \tau + 2 \dc} \right) \\
&& \nonumber \hskip-0.2cm + \tfrac{1}{2} \, (\sqrt{|\tau + \dc|} - \sqrt{|\tau - \dc|}) \bigg[(\mu+2\db)\Psi_{\mu + 4\db, \tau} \\
&& \nonumber \hskip0.4cm ~~~~~ + (\mu-2\db)
\Psi_{\mu - 4\db, \tau} - 2 \mu (1 + 2 \gamma^2 \db^2) \Psi_{\mu,\tau} \bigg]\\
&& \nonumber \hskip-0.2cm + \db(\sqrt{|\tau|} + \sqrt{|\tau - 2 \dc|}) \left(\Psi_{\mu - 2\db, \tau - 2 \dc} -
\Psi_{\mu+ 2\db, \tau - 2 \dc} \right) \\
&& = 0 ~.\end{aligned}$$ Here, for a Schwarzschild black hole interior corresponding to mass $m$, $$\label{dbdc}
\delta_b = \frac{\sqrt{\Delta}}{2 m}, ~~~ \mathrm{and} ~~~~ \delta_c = \frac{\sqrt{\Delta}}{{L_o}} ~~~$$ where $\Delta$ denotes the minimum area eigenvalue in loop quantum gravity, i.e. $\Delta = 4 \sqrt{3} \pi \gamma \lp^2$.
Basis Function Method {#BFM}
=====================
In this section, we demonstrate the use of our newly developed basis function method (BFM) to solve discrete quantum Hamiltonian constraints in loop quantum cosmology. As mentioned earlier, this approach is inspired by the well-known spectral collocation method use commonly for numerical solutions of partial differential equations. The main advantage of the basis function method is its high computational efficiency and ease of parallelization on modern computer hardware. We also compare the basis method based solutions with the previous approach of recursively stepping (RSM) through a grid of triad variable values.
We use the example of the CS model of the Schwarschild interior throughout. We set $\gamma\db \rightarrow 0$ (a requirement for stable solutions [@YKS]) and set $\delta_c = 2\delta_b=1$ without loss of generality.
Separable Solutions {#VS}
-------------------
Let us begin with performing a separation-of-variables solution of the CS model under consideration. To demonstrate the viability of the basis function method solution for this model, $\Psi$ is taken to be of the form $\Psi_{\mu,\tau} \rightarrow A(\mu)B(\tau)$, which reduces the CS model to $$\begin{aligned}
(\mu + 2\db)A(\mu+4\db) & + & (\mu-2\db)A(\mu-4\db)= 2\mu A(\mu)\nonumber \\
2\db\lambda(A(\mu-2\db)& - &A(\mu+2\db)) + 4\mu\gamma^2\db^2 A(\mu)
\label{Aeqn}\end{aligned}$$ and $$\begin{aligned}
&(\sqrt{|\tau|}+\sqrt{|\tau + 2\dc|})B(\tau+2\dc)& \nonumber \\ &-(\sqrt{|\tau|}+\sqrt{|\tau-2\dc|})B(\tau-2\dc)& \nonumber \\
&=-\lambda(\sqrt{|\tau+\dc|} - \sqrt{|\tau-\dc|})B(\tau)&
\label{Beqn}\end{aligned}$$ where $\lambda$ is the separation parameter.
The basic idea behind the basis function method is to start with a set of appropriate functions, and then solve the difference equation on a linear span of this set. The chosen basis should allow for highly variable behavior for small values of the triad variables $\mu$ and $\tau$, but only capture very smooth behavior for much larger values. This allows for very sharp quantum fluctuations in the deep Planck regime, and yet very smooth semi-classical behavior for large values of the triad variables. One approach to build such a basis is to seek inspiration from a Fourier basis as used in spectral collocation methods, but with a non-constant wavelength, i.e. $k(x)$. Note that our requirement of smoother behavior for larger values of $x$, then becomes the condition that $k(x)$ be a monotonically decreasing function of $x$. Specifically, we let $k$ drop rapidly (exponentially) with $x$. Thus our desired modified Fourier basis would take the form, $\exp(i n \theta(x))$ where $\theta'(x)=k(x)$ drops exponentially with $x$. Given such a form for a basis, we can then expand any solution as a linear combination of these basis elements and obtain the values of the coefficients by imposing the difference equation. Thus, ultimately we solve a linear system of equations for which a wide variety of efficient numerical algorithms and solvers are readily available.
More specifically, we choose the basis functions to take the form, $$\label{basis}
\Phi_n(\tau) = \exp(i n e^{-\frac{|\tau|}{N}\exp(\frac{2n}{N})})$$ which is a slightly modified version of a basis that was proposed by one of us in Ref. [@CK06] many years ago. Some sample basis elements are depicted in Fig. \[fig:basis\].
![Sample basis elements utilized throught this work. Note how they allow for highly oscillatory behavior at small values of the triad and then smoothen out as the values get larger. []{data-label="fig:basis"}](basis.pdf){width="\columnwidth"}
The basis function method ultimately involves solving a linear system of equations for a vector of weights $\omega_n$, where $n$ is an index spanning the number of basis elements. The system matrix is represented as an “interpolation matrix”, with each entry being the sequence in Eqn. \[Beqn\], and with each $B$ replaced with an appropriate representation of Eqn. \[basis\]. Each row in this matrix corresponds to an increasing value of $\tau$ in the domain of the computation. Thus, Eqn. \[Beqn\] would be represented as $$\begin{aligned}
&\sum^N_{n=0} \omega_n\big[(\sqrt{|\tau|}+\sqrt{|\tau + 2\dc|})\Phi_n(\tau+2\dc)& \nonumber \\
&-(\sqrt{|\tau|}+\sqrt{|\tau - 2\dc|})\Phi_n(\tau-2\dc)&\nonumber\\
&+\lambda(\sqrt{|\tau+\dc|} - \sqrt{|\tau-\dc|})\Phi_n(\tau)\big] = 0 & \label{Bseqinterp}\end{aligned}$$
To solve the above system for the function $B(\tau)$, other than the trivial $B(\tau)=0$, inspiration was taken from a standard technique for numerically solving partial differential equations. A row can be injected into the system matrix, as long as there is an accompanying column, to disrupt the homogeneity of the system without fundamentally changing the computation. This row can be interpreted as numerically incorporating initial or boundary conditions; and in our approach is treated as a restriction on the solution. For our results in this section, we used $\sum_n\omega_n\Phi_n(\tau=T)=1$ for a large value of $T$.
Using this approach, a reconstructed solution can be generated, which is in precise agreement with the recursively computed solution. In Fig. \[fig:fd\] we show a sample basis solution for the $B(\tau)$ equation as compared with a recursion based solution computed by stepping through the values of the $\tau$ variable. There, we set $\gamma\db \rightarrow 0$ (a requirement for stable solutions) and set $\delta_c = 2\delta_b=\lambda=1$ for simplicity.
![Solution of the $B(\tau)$ equation using the basis function method as compared with a solution computed by stepping through the $\tau$ values using a recursive approach. The upper panel shows both solutions plotted together. The lower panel depicts the absolute difference between the two solutions, which is on the scale of machine precision.[]{data-label="fig:fd"}](bseq.pdf "fig:"){width="\columnwidth"} ![Solution of the $B(\tau)$ equation using the basis function method as compared with a solution computed by stepping through the $\tau$ values using a recursive approach. The upper panel shows both solutions plotted together. The lower panel depicts the absolute difference between the two solutions, which is on the scale of machine precision.[]{data-label="fig:fd"}](berr.pdf "fig:"){width="\columnwidth"}
Similarly to solve Eqn. \[Aeqn\] we first make the substitution $C(\mu) \equiv A(\mu+2\db)-A(\mu-2\db)$ which results in $$\begin{aligned}
\mu\big(C(\mu+2\db)-C(\mu-2\db)\big)&+&\nonumber \\
2\db\big(C(\mu+2\db)+C(\mu-2\db)\big)&=&-2\lambda C(\mu) ~.\end{aligned}$$ This allows us to easily solve Eqn. \[Aeqn\] using both the basis function method and the recursive approach. The outcome is similar as that shown previously for the $B(\tau)$ function.
![Solution of the $C(\mu)$ equation using the basis function method as compared with a solution computed by stepping through the $\mu$ values using a recursive approach. The upper panel shows both solutions plotted together. The lower panel depicts the absolute difference between the two solutions.[]{data-label="fig:fd1"}](aseq.pdf "fig:"){width="\columnwidth"} ![Solution of the $C(\mu)$ equation using the basis function method as compared with a solution computed by stepping through the $\mu$ values using a recursive approach. The upper panel shows both solutions plotted together. The lower panel depicts the absolute difference between the two solutions.[]{data-label="fig:fd1"}](aerr.pdf "fig:"){width="\columnwidth"}
Full 2D Solution: Evolution with Basis Functions {#2DBFM}
------------------------------------------------
The numerical solution computation for the full 2D non-separable case, takes a similar approach. The solution is again taken to be a weighted-sum of the basis functions Eqn. \[basis\], but with $\tau$ dependent weights. This leads to Eqn. \[CS\] taking the form $$\begin{aligned}
\label{CS-mod}
&& \nonumber \sum^{N}_{n=0}\omega_n(\tau-2\dc)(\Phi_n(\mu+2\db)-\Phi_n(\mu-2\db)) = \\
&& \nonumber \hskip-0.2cm\tfrac{\sqrt{|\tau|} + \sqrt{|\tau + 2 \dc|}}{\sqrt{|\tau|} + \sqrt{|\tau - 2 \dc|}} \left(\Psi_{\mu + 2 \db, \tau + 2 \dc} -
\Psi_{\mu - 2 \db, \tau + 2 \dc} \right) \\
&& \nonumber \hskip-0.2cm + \tfrac{1}{2\db} \, \tfrac{\sqrt{|\tau+\dc|} - \sqrt{|\tau - \dc|}}{\sqrt{|\tau|} + \sqrt{|\tau - 2 \dc|}} \bigg[ (\mu+2\db)\Psi_{\mu + 4 \db, \tau} + \\
&& \hskip0.4cm (\mu-2\db)\Psi_{\mu - 4 \db, \tau} - 2 \mu (1 + 2 \gamma^2 \db^2) \Psi_{\mu,\tau} \bigg] ~.\end{aligned}$$ The solution can then be reconstructed on the full range of $\mu, \tau$ values via $$\label{interp}
\Psi_{\mu,\tau} = \sum^N_{n=0} \omega_n(\tau) \Phi_n(\mu) ~.$$ The $\tau$ dependence is captured by stepping through a $\tau$-valued grid of this reconstructed solution.
Similar to the 1D variable-separated case, the basis function method ultimately leads to a linear system of equations. The system involves finding the weights $\omega_n$, while the invertible system matrix represents the left-hand-side of Eqn. \[CS-mod\]. A boundary condition is implemented in the manner of row-column injection used previously, where $\lim_{\mu\rightarrow\infty}\Psi_{\mu,\tau}=0$ is represented as $\sum_n\omega_n\Phi_n(\mu=M)=0$. As typically done in such models, we impose a constraint that represents an initial “semi-classical” wave-packet, i.e. a Gaussian profile for the solution at large $\tau$, and then step backwards in $\tau$, ultimately evolving the system deep into the quantum regime and beyond.
![Solution of Eqn. \[CS\] using the basis function method (upper panel) as compared with a solution computed by stepping through the $\tau, \mu$ values using a recursive approach (lower panel).[]{data-label="fig:fd2"}](rbf.pdf "fig:"){width="\columnwidth"} ![Solution of Eqn. \[CS\] using the basis function method (upper panel) as compared with a solution computed by stepping through the $\tau, \mu$ values using a recursive approach (lower panel).[]{data-label="fig:fd2"}](fd.pdf "fig:"){width="\columnwidth"}
The error denoted by the $L_\infty$-norm as the maximum value of the residual between the recursive step method and basis method solutions at slices of constant $\tau$ is depicted in Fig. \[error\].
![$L_\infty$-norm of the error at each value of $\tau$ for different number of basis elements $N=20,23,25$.[]{data-label="error"}](err.pdf){width="\columnwidth"}
It is clear that the error stays low for the larger values of $\tau$, however it increases in the neighborhood of $\tau=0$. This is likely due to the use of a relatively small number of basis elements. In fact, as seen in Fig. \[error\], it is clear that overall error reduces dramatically with even a modest increase in the number of basis elements. Of course, one can also envision a minor tweak in the form of the basis to reduce the error in the small $\tau$ regime further. We do not attempt to do that in this work.
The agreement between the basis and recursive method can be even further inspected by examining the volume expectation value $\langle v\rangle$. This is shown in Fig. \[fig:expec\].
![$\langle v\rangle$ computed using both the basis function method and the recursive step method, and the associated relative difference.[]{data-label="fig:expec"}](volexpec.pdf "fig:"){width="\columnwidth"} ![$\langle v\rangle$ computed using both the basis function method and the recursive step method, and the associated relative difference.[]{data-label="fig:expec"}](voldiff.pdf "fig:"){width="\columnwidth"}
### Optimization {#OPT}
To reduce computational cost, we make use of a key property of the solutions, i.e. they [*must*]{} get smoother for the larger triad values. When the solutions are smooth, they can be interpolated very effectively, thus allowing for a significant reduction in the “sampling rate”. In order to take advantage of this, we only solve over a subset of triad values as computed through this expression $$\label{nodes}
\mu_{i+1} = \mu_i + \lfloor 1 + \big(\frac{2\mu_i}{25}\big)^2\rfloor$$ where $\lfloor \rfloor$ denotes the “integer part” or the [*gint*]{} function. Visually, this set can be represented as shown in Fig. \[fig:nodes\]. Through some experimentation, we discovered that this form yields significant computational benefit with relatively little loss of accuracy. This allows us to reduce the effective grid size by a factor of 10, offering us a tremendous speed-up!
![Subset of $\mu$ variable grid values compared with optimal $\mu$ variable grid values obtained through the empirical interpolation method.[]{data-label="fig:nodes"}](nodes.pdf){width="\columnwidth"}
In addition, while the choice of basis etc. was made largely based on physical considerations and numerical experimentation, we can show that this choice is reasonably optimal using a [*reduced basis*]{} approach [@BasReduc2010] with an [*empirical interpolation method*]{} framework [@EIM2008], calculated via the [*greedy algorithm*]{} [@AFHNT13]. This was done using the open source code [*rompy*]{} [@rompy1; @rompy2] which demonstrated that the optimal sampling nodes have a similar distribution to the nodes generated with Eqn. \[nodes\]. This is shown graphically in Fig. \[fig:nodes\]. Utilizing the chosen basis function in Eqn. \[basis\], one can generate an associated Vandermonde matrix, with the elements $$V_{i,j} = \Phi_{j}(\mu_i) ~.$$ If the basis elements $\Phi_j (\mu_i)$ are considered as a reduced basis, one can then perform a singular value decomposition (SVD) of $V_{i,j}$. This demonstrates that the chosen basis is approximately orthonormal and well-conditioned. This was further verified by performing Gram-Schmidt orthonormalization on the basis, which led to no perceived difference in the orthogonality of the basis, and only an insignificant reduction in the conditioning. Thus, we observe that while certain techniques may be used to improve our basis function method (e.g. Gram-Schmidt, Empirical Interpolation), they are not necessarily required.
Instability {#INST}
-----------
After performing von-Neumann stability analysis on Eqn. \[CS\], it was noted previously [@YKS] that the model is subject to an instability condition, $$\begin{aligned}
\mu & > & 4\tau ~.\end{aligned}$$ This condition $\mu > 4\tau$ is of particular interest as it only appears through use of von-Neumann stability analysis in the context of the full 2D system Eqn. \[CS\], and not in the variable-separable case. This is indicative of the fact that the $\mu > 4\tau$ condition is a consequence of the full 2D form of the quantum Hamiltonian constraint. In this section, we attempt to understand how exactly this instability appears in the 2D equation – both, in the context of the stencil-based, finite-difference-like stepping approach and, of course, the basis function method.
### The Large $\mu, \tau$ Limit {#CFLARG}
Beginning with the original Eqn. \[CS\] and taking the approximation $(\mu,\tau) >> (\db, \dc)$ and setting $\gamma\db \rightarrow 0$, we obtain $$\begin{aligned}
\label{pdecnvrt}
&& \nonumber (1 + \sqrt{|1+ 2 \frac{\dc}{\tau}|}) \left(\Psi_{\mu + 2 \db, \tau + 2 \dc} -
\Psi_{\mu - 2 \db, \tau + 2 \dc} \right) \nonumber \\
&& \nonumber \hskip-0.2cm + \tfrac{1}{2\db} \, (\sqrt{|1+ \frac{\dc}{\tau}|} - \sqrt{|1 - \frac{\dc}{\tau}|}) \bigg[(\mu)\Psi_{\mu + 4 \db, \tau} + \nonumber\\
&& \nonumber \hskip0.4cm(\mu)\Psi_{\mu - 4 \db, \tau} - 2 \mu \Psi_{\mu,\tau} \bigg]+ \nonumber\\
&& \nonumber \hskip-0.2cm (1 + \sqrt{|1 - 2 \frac{\dc|}{\tau}}) \left(\Psi_{\mu - 2 \db, \tau - 2 \dc} -
\Psi_{\mu + 2 \db, \tau - 2 \dc} \right) \\ && = 0 ~.\end{aligned}$$ Next, we insert the following Taylor approximation, making implicit use of the expectation that the solution tends towards smooth behavior for large values of $\mu, \tau$ $$\begin{aligned}
\Psi_{\mu\pm l\db,\tau \pm m\dc} \approx& \Psi(\mu\pm l\db,\tau\pm m\dc)\nonumber \\
\rightarrow& \Psi \pm l\db\tfrac{\partial}{\partial \mu}\Psi \pm m\dc\tfrac{\partial}{\partial \tau} \Psi \nonumber\\
&+\tfrac{(l\db)^2\partial^2}{2!\partial \mu^2} \Psi + \tfrac{(m\dc)^2\partial^2}{2!\partial \tau^2} \Psi \nonumber\\
& \pm 2lm\db\dc\tfrac{\partial^2}{\partial \mu\partial\tau}\Psi + \dots \nonumber\\
\sqrt{|1 \pm 2\frac{\dc}{\tau}|} \approx& 1 \pm \frac{\dc}{\tau} ~. \nonumber\end{aligned}$$ This substitution reduces Eqn. \[pdecnvrt\], to the expression $$\begin{aligned}
4\tau \frac{\partial^2}{\partial\mu\partial\tau}\Psi + \frac{\partial}{\partial \mu}\Psi + \mu \frac{\partial^2}{\partial \mu^2} \Psi = 0\end{aligned}$$ which can be further reduced into an equation of the form of the [*advection*]{} partial differential equation, $$\begin{aligned}
\label{CFL}
\dot\beta + \frac{\mu}{4\tau}\beta' = 0\nonumber\\
\beta \equiv \mu \frac{\partial}{\partial\mu}\Psi ~.\end{aligned}$$ Now, it is relatively easy to see why an instability appears for $\mu > 4\tau$ part of the computational domain. Making an analogy with the so-called, “CFL condition” [@cfl] that commonly appears in numerical partial differential equation methods, Eqn. \[CFL\] presents an advection equation with a locally defined speed of $v=\tfrac{\mu}{4\tau}$. This defines, at each point in ($\tau, \mu$), a local light cone extending into the past defined by the region traced out by the characteristic speed $v$. However, the finite-difference stencil as defined by the Hamiltonian constraint, has a triad stepping ratio of $\db / 2\dc = 1$. This results in an instability when $v>1$, since the local physical domain of dependence is no longer fully contained within the local computational domain as determined by the quantum Hamiltonian constraint.
![Evolution using the basis function method into the unstable regime of $\mu,\tau$.[]{data-label="fig:boost"}](rbfboost.pdf){width="\columnwidth"}
As mentioned before, the instability condition of $\mu>4\tau$ is a result of recursive stepping through both $\mu$ and $\tau$ as in Eqn. \[CS\]. With a global method like the basis method, it could be reasoned that such a local instability condition may be avoided. However, it does not; this can be seen by performing a “shift” of coordinates, $\mu \rightarrow \mu + 555, \tau \rightarrow \tau + 100$ and allowing part of the Gaussian wavepacket to evolve into the unstable regime. Fig. \[fig:boost\] depicts such a sample evolution.
We find that the instability condition as shown is still applicable even in the basis function method. It is suspected that this is due to an intrinsic issue with the CS model, or perhaps due to the lack of a complete Hilbert space of solutions or possibly even a reflection of some underlying physics. Thus, von-Neumann analysis continues to be beneficial towards providing insight on the viability of models in loop quantum cosmology. More specifically, it is able to provide various constraints on such models based on the requirement that evolutions must be stable [@nlqc; @YKS; @SS].
### Stability in the Basis Function Method {#BFMINSTAB}
To understand the instability in the context of the basis function approach, we take Eqns. \[CS\] and \[interp\] together, while noting that the only time-varying component in Eqn. \[interp\] is $\vec{\omega}$. We then rewrite Eqn. \[CS\] using vector and matrix notation $$\begin{aligned}
\Psi_{\mu\pm k\db,\tau\pm n\dc} \rightarrow& \sum^N_{i=0}\Phi_i(\mu\pm k\db)\omega_i(\tau\pm n\dc)\nonumber \\
=& \mathbf\Phi(\mu\pm k\db)\vec\omega_{\tau\pm n\dc}~.
\end{aligned}$$ where $\mathbf\Phi$ is a square matrix with the different basis elements arranged in columns and each row representing a different $\mu$ value. In this notation the CS model can be rewritten as
$$\begin{aligned}
&& \nonumber \big( \tfrac{\sqrt{|\tau|}+\sqrt{|\tau-2\dc|}}{\sqrt{|\tau|}+\sqrt{|\tau+2\dc|}}\big)\bigg[ \mathbf\Phi(\mu-2\db)-\mathbf\Phi(\mu+2\db)\bigg]\vec\omega_{\tau-2\dc}\\
&& \nonumber \hskip-.2cm+\frac{1}{2}\big( \tfrac{\sqrt{|\tau+\dc|}-\sqrt{|\tau-\dc|}}{\sqrt{|\tau|}+\sqrt{|\tau+2\dc|}}\big)\bigg[(\mu+2\db) \mathbf\Phi(\mu+4\db)+(\mu-2\db) \mathbf\Phi(\mu-4\db)-2\mu \mathbf\Phi(\mu) \bigg]\vec\omega_{\tau}\\
&& \hskip-.2cm - \bigg[\mathbf\Phi(\mu-2\db)- \mathbf\Phi(\mu+2\db)\bigg]\vec\omega_{\tau+2\dc}=\vec 0~.\end{aligned}$$
Given that $\mathbf\Phi$ is a matrix of full rank, we can compute the inverse of matrix $\mathbf A$ (defined below) and obtain $$\begin{aligned}
\label{rbfgrowth}
\nonumber \mathbf I\vec\omega_{\tau+2\dc} + \frac{1}{2}\big( \tfrac{\sqrt{|\tau-\dc|}-\sqrt{|\tau+\dc|}}{\sqrt{|\tau|}+\sqrt{|\tau+2\dc|}}\big) \mathbf{A}^{-1}\mathbf B\vec\omega_\tau-\big( \tfrac{\sqrt{|\tau|}+\sqrt{|\tau-2\dc|}}{\sqrt{|\tau|}+\sqrt{|\tau+2\dc|}}\big)\mathbf I\vec\omega_{\tau-2\dc} =&\; \vec0 \\
\nonumber \mathbf\Phi(\mu-2\db)-\mathbf\Phi(\mu+2\db)\equiv&\;\mathbf A\\
\big[(\mu+2\db)\mathbf\Phi(\mu+4\db) +(\mu-2\db)\mathbf\Phi(\mu-4\db) -2\mu\mathbf\Phi(\mu) \big]\equiv&\;\mathbf B ~.\end{aligned}$$
This can be further reduced down to $$\begin{aligned}
\label{rbfroot}
\nonumber \mathbf I\vec\omega_{\tau+2\dc} + \frac{1}{2}\alpha \mathbf D\vec\omega_{\tau} - \beta\mathbf I \vec\omega_{\tau - 2\dc} =& \vec 0 \\
\nonumber \mathbf D \equiv\; \mathbf A^{-1}\mathbf B& \\
\nonumber \alpha \equiv \big(\tfrac{\sqrt{|\tau-\dc|}-\sqrt{|\tau+\dc|}}{\sqrt{|\tau|}+\sqrt{|\tau+2\dc|}} \big)& \\
\beta \equiv \big(\tfrac{\sqrt{|\tau|}+\sqrt{|\tau-2\dc|}}{\sqrt{|\tau|}+\sqrt{|\tau+2\dc|}} \big)&~.\end{aligned}$$ Finally, if the following definitions are made, $$\begin{aligned}
\label{rbfreduc}
&\vec q_{\tau} \equiv
\begin{bmatrix}
\vec \omega_{\tau} \\
\vec \omega_{\tau-2\dc}
\end{bmatrix} &
\mathbf \Pi \equiv
\begin{bmatrix}
-\tfrac{1}{2}\alpha\mathbf{D} & \beta\mathbf I \\
\mathbf I & \mathbf 0
\end{bmatrix}&\end{aligned}$$ then one can use Eqn. \[rbfreduc\] to express Eqn. \[rbfroot\] in the following manner, $$\begin{aligned}
\label{rbfstab}
\vec q_{\tau + 2\dc} = \mathbf \Pi \vec q_{\tau}~.\end{aligned}$$ Now, we use the eigenvalues of matrix $\mathbf \Pi$ to test for stability; if any of the eigenvalue magnitudes exceed unity, that makes the solution of Eqn. \[rbfstab\] grow unboundedly. With some additional simplifications (in particular, $(\mu,\tau) >> (\db, \dc)$) accompanied with some numerical computations it is not difficult to see that one obtains the same unstable region ($\mu > 4\tau$) as derived before in Ref. [@YKS] using basic von Neumann stability analysis.
Discussion and Conclusions {#DnC}
==========================
In this paper, we have developed a new numerical method that is particularly suitable for computing solutions to the quantum Hamiltonian constraint of models in loop quantum cosmology. The method is inspired by the spectral collocation method that is commonly used for numerically solving partial differential equations. It uses a special set of basis functions that take full advantage of the expected behavior of physical solutions of the model. We also compared the new basis function method with the previously used approach of a stencil-style computation using a recursive computation over a grid of allowed values for the triad variables. In addition, we also discussed how the stability analysis appears in the context of the basis approach.
In this section we will document how the basis function method improves upon previous approaches. As pointed out earlier, the main advantage is computational efficiency and ease of parallelization.
Parallelizability {#PARA}
-----------------
The basis function method ultimately involves the computation of a solution of a linear system of equations of size proportional to the square of the number of basis elements in use. Since this is a very well-studied problem, there exist many highly optimized solvers available even for parallel hardware such as multi-core CPUs and many-core GPUs. On the other hand, the recursive step method is intrinsically serial and rather challenging to parallelize.
Cores BFM Run-Time (s)
------- ------------------
1 161.110
2 85.805
4 48.852
8 29.751
: Run-Time by number of cores with simple OpenMP multi-threading on $\mu,\tau\in [-1600,1600]$ grid[]{data-label="tab:performance"}
With a simple multi-threaded implementation for the BFM algorithm, the benefit of parallelization can be clearly seen. Increased benefit is expected when the number of basis elements gets larger; and we anticipate a more in-depth study of this feature for both multi-core CPUs and many-core GPUs in future work.
Precision {#PREC}
---------
As mentioned above, the basis function method involves solving a linear system of equations. Performing an analysis on the backward error of matrix inversion techniques yields $$\begin{aligned}
\mathbf{A}\vec{x} =& \vec{b} \nonumber \\
\epsilon_{BFM} = ||\mathbf{A}^{-1}\vec{b} - \vec{x}|| =& \mathcal{O}(\kappa_2\epsilon_{machine}) \nonumber \\
\kappa_2 \equiv& ||\mathbf{A}^{-1}||_2||\mathbf{A}||_2 \propto N ~.\end{aligned}$$ On the other hand it has been found that for a recursive step method with $\mu \in [\tfrac{-M}{2},\tfrac{M}{2}]$ and $\tau \in [\tfrac{-T}{2},\tfrac{T}{2}]$, the error $\epsilon_{RSM}\propto \mathcal{O}(T \times M\epsilon_{machine})$. Since standard backward error is proportional to the number of basis elements, we have shown that a small and constant number can compute solutions on a range of domains accurately; it stands to reason that as long as the non-zero regions of the computed solution are reasonably well sampled by the basis, the BFM approach can calculate a larger range of solutions with a set level of computational precision.
![Relative $L_2$ error for various size domains, and evolution with the basis function method over large domain, $\mu,\tau \in [-1600:1600]$[]{data-label="fig:rbflong"}](l2comp.pdf "fig:"){width=".45\columnwidth"} ![Relative $L_2$ error for various size domains, and evolution with the basis function method over large domain, $\mu,\tau \in [-1600:1600]$[]{data-label="fig:rbflong"}](rbflong.pdf "fig:"){width=".45\columnwidth"}
It can be seen in Fig. \[fig:rbflong\] that the recursive step method based solution experiences unbounded growth due to fixed, finite available precision; while the basis method computed solution maintains the correct overall behavior expected. For a closer analysis of the effects that finite precision can have on the recursive step method solutions a precision test can be performed. For such a test, we maintain a constant domain and solve [*identical*]{} simulations with varying precision. The error can then be analyzed by taking the highest precision solution as [*most accurate*]{}. Our standard error analysis follows as $$\begin{aligned}
\epsilon_P(\tau_i) &= \frac{\sum_n |\Psi_{Quad}(\mu_n,\tau_i)-\Psi_{P}(\mu_n,\tau_i)|^2}{\sum_n \Psi^2_{Quad}(\mu_n,\tau_i)} \end{aligned}$$ where $P$ is a precision other than quadruple-precision (e.g. double, single). The results are depicted in Fig. \[fig:prectest\]. They suggest that the recursive step method is far more prone to suffer from precision limitations over the basis method. This ultimately translates into a major computational advantage in favor of the basis method.
![Precision test for single (4-byte), and double (8-byte) precision. It is clear that error grows significantly as the evolution progresses (recall, the system evolves backward in $\tau$).[]{data-label="fig:prectest"}](prec.pdf){width="\columnwidth"}
In summary, we have developed a new numerical approach towards solving quantum Hamiltonian constraints in loop quantum cosmology. The approach makes use of a set of basis functions, specifically designed using key physical features of the solutions in mind. This basis method, borrows from the well-known spectral collocation method for solving partial differential equations. We demonstrate the efficacy of the method and compare it with the previously used recursive step method. We find that the basis method offers a number of benefits over the previously used approach, especially in the area of computational efficiency. Throughout this paper, we use the Corichi and Singh loop quantum model of the Schwarzschild interior for demonstration purposes.
[*Acknowledgements*]{}: The authors would like to thank Drs. Alfa Heryudono, Scott Field and Sigal Gottlieb for very helpful suggestions throughout this work. G. K. thanks research support from National Science Foundation (NSF) Grant No. PHY-1701284 and Office of Naval Research/Defense University Research Instrumentation Program (ONR/ DURIP) Grant No. N00014181255.
[10]{} T. Thiemann, [*Modern Canonical Quantum General Relativity*]{}, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2008). C. Rovelli, [*Quantum Gravity*]{}, Cambridge University Press (2007). A. Ashtekar, J. Pullin, [*Loop Quantum Gravity: The First 30 Years (100 Years of General Relativity)*]{}, World Scientific (2017). A. Ashtekar, and P. Singh, Class. Quant. Grav. 28, 213001 (2012). M. Bojowald, Living Rev. Rel., 11, 4 (2008). A. Ashtekar, T. Pawlowski, and P. Singh, Phys. Rev. Lett. 96, 141301 (2006). A. Ashtekar, T. Pawlowski, and P. Singh, Phys. Rev. D 74, 084003 (2006).
P. Singh, Computing in Science & Engineering 20(4), 26 (2018). D. Brizuela, D. Cartin, and G. Khanna, SIGMA 8, 001 (2012). P. Singh, Class. Quant. Grav. vol. 29, 244002 (2012). P. Diener, B. Gupt, M. Megevand, and P. Singh, Class.Quant. Grav., 31, 165006 (2014) A. Corichi, P. Singh, Class. Quantum Grav. 33, 055006 (2016). A. Yonika, G. Khanna, P. Singh, Class. Quantum Grav. 35, 045007 (2018). H. Haggard and C. Rovelli, Phys. Rev. D 92, 104020 (2015). E. Bianchi, M. Christodoulou, F. D’Ambrosio, H. Haggard and C. Rovelli, Class. Quantum Grav. 35, 225003 (2018). J. Olmedo, S. Saini and P. Singh, Class. Quantum Grav. 34, 225011 (2017). S. Connors, G. Khanna, Class. Quantum Grav. 23, 2919 (2006). S. Chaturantabut, D. Sorensen, SIAM J. of Sci. Comput. 32(5), (2010) Y. Maday, N. Nguyen, A. Patera, G. Pau, Communications on Pure and Applied Analysis 8(1), (2009) H. Antil, S. Field, F. Herrmann, R. Nochetto, M. Tiglio, Journal of Scientific Computing 57, (2013) S. Field, C. Galley, J. Hesthaven, J. Kaye, M. Tiglio, Phys. Rev. X 4, 031006 (2014) C. Galley, [*rompy*]{} package\
https://bitbucket.org/chadgalley/rompy/src/master R. Courant, K. O. Friedrichs, H. Lewy, IBM Journal 11, 215 (1967). P. Diener, B. Gupt and P. Singh, Class. Quant. Grav. 31, 025013 (2014) S. Saini and P. Singh, Class. Quant. Grav. 36, 105010 (2019)
|
---
abstract: 'Let $W$ denote a simply-laced Coxeter group with $n$ generators. We construct an $n$-dimensional representation $\phi$ of $W$ over the finite field $F_2$ of two elements. The action of $\phi(W)$ on $F_2^n$ by left multiplication is corresponding to a combinatorial structure extracted and generalized from Vogan diagrams. In each case W of types A, D and E, we determine the orbits of $F_2^n$ under the action of $\phi(W)$, and find that the kernel of $\phi$ is the center $Z(W)$ of $W.$'
author:
- 'Hau-wen Huang[^1]'
- 'Chih-wen Weng[^2]'
date: 'April 14, 2008'
title: 'Combinatorial representations of Coxeter groups over a field of two elements[^3]'
---
[ Coxeter groups; Dynkin diagrams; Group representations; Vogan diagrams.]{}
Introduction {#s1}
============
A [*simply-laced Coxeter group*]{} is a group $W_S(m)$ with a finite set of generators $S\subseteq W_S(m)$ subject only to relations $$(ss')^{m(s, s')}=1,$$ where $m(s, s)=1$ and $m(s, s')=m(s', s)\in\{2, 3\}$ for $s\not=s'$ in $S$. When $m$ is specified, we write $W_S$ for $W_S(m)$, and if both $S$ and $m$ are specified, we write $W$ for $W_S(m)$. A [*Coxeter graph*]{} $S$ represents a simply-laced Coxeter group $W_S(m)$, and vice versa. The vertex set of this graph is $S$, and there is an edge joining two vertices $s$ and $s'$ whenever $m(s, s')=3.$ There are Coxter groups which are not simple-laced. In this article we always assume simple-laced property in a Coxter group to make the corresponding Coxeter graph $S$ a simple graph.
We shall investigate a kind of [*flipping puzzle*]{}, which is also studied in [@xw:07; @wu:06], associated with a given Coxeter graph $S$. The [*configuration*]{} of the flipping puzzle is $S$, together with an [*assignment*]{} of a unique state, white or black, on each vertex of $S$. A [*move*]{} in the puzzle is to select a vertex $s$ which has black state, and then flip the state of each neighbor of $s$. When $S$ is one of the Dynkin diagrams described in Figure 1, the configuration above is essentially a [*Vogan diagram with identity involution*]{}, which was first defined in [@k:96], in a more general way, as a combinatorial object representing the real form of the corresponding complex simple Lie algebra and a system of choices. See also [@pb:00; @pb:02; @mkc:04].
(100, 90) (0,89)(30,90) (38,90) (46,90) (50,90) (54,90) (58,90) (62,90) (70,90) (78,90) (30.75,90)[( 1,0)[6.5]{}]{} (38.75,90)[( 1, 0)[6.5]{}]{} (62.75,90)[( 1,0)[6.5]{}]{} (70.75,90)[( 1, 0)[6.5]{}]{} (29,87) (36,87) (45,87) (61,87) (69,87) (77,87)
(0,74)(30,70) (30,80) (38,75) (46,75) (50,75) (54,75) (58,75) (62,75) (70,75) (78,75) (30.6,70.2)[( 5, 3)[6.9]{}]{} (30.6,79.8)[(5,-3)[6.9]{}]{} (38.75,75)[( 1, 0)[6.5]{}]{} (62.75,75)[(1, 0)[6.5]{}]{} (70.75,75)[( 1, 0)[6.5]{}]{} (29,67) (28,77) (37,72) (45,72) (61,72) (69,72) (77,72)
(0,55)(30,55) (38,55) (46,55) (54,55) (62,55) (46,63) (30.75,55)[( 1, 0)[6.5]{}]{} (38.75,55)[( 1, 0)[6.5]{}]{} (46.75,55)[( 1, 0)[6.5]{}]{} (54.75,55)[( 1, 0)[6.5]{}]{} (45.75,55.75)[( 0, 1)[6.5]{}]{} (30,52) (38,52) (46,52) (54,52) (62,52) (46,65)
(0,35)(30,35) (38,35) (46,35) (54,35) (62,35) (46,43) (70,35) (30.75,35)[( 1,0)[6.5]{}]{} (38.75,35)[( 1, 0)[6.5]{}]{} (46.75,35)[( 1,0)[6.5]{}]{} (54.75,35)[( 1, 0)[6.5]{}]{} (46,35.75)[( 0,1)[6.5]{}]{} (62.75,35)[( 1, 0)[6.5]{}]{} (30,32) (38,32) (46,32) (54,32) (62,32) (46,45) (70,32)
(0,15)(30,15) (38,15) (46,15) (54,15) (62,15) (46,23) (70,15) (78,15) (30.75,15)[( 1, 0)[6.5]{}]{} (38.75,15)[( 1, 0)[6.5]{}]{} (46.75,15)[( 1, 0)[6.5]{}]{} (54.75,15)[( 1, 0)[6.5]{}]{} (46,15.75)[( 0, 1)[6.5]{}]{} (62.75,15)[( 1, 0)[6.5]{}]{} (70.75,15)[( 1, 0)[6.5]{}]{} (30,12) (38,12) (46,12) (54,12) (62,12) (46,25) (70,12) (78,12)
(0,3)[[**Figure 1:**]{} Simply-laced Dynkin diagrams.]{}
We fix a simply-laced Coxeter group $W$ and its Coxeter graph $S,$ where $|S|=n.$ Let $F_2$ denote the finite field of two elements $0$ and $1$. In Section \[s2\], we use the column vector set $F_2^S=F_2^n$ to describe the set of configurations in the flipping puzzle associated with $S$ by setting that $\ell_s=1$ iff the configuration $\ell\in F_2^n$ has black state in the vertex $s.$ For each vertex $s\in S$, we find a way to associate the move with selecting vertex $s$ as an $n\times n$ invertible matrix $\mathbf{s}$ over $F_2.$ This $\mathbf{s}$ acts on a configuration $\ell\in F_2^n$ by left multiplication to become a new configuration $\mathbf{s}\ell$ which has the desired property as stated in the definition of the flipping puzzle when $\ell_s=1.$ Unlike in the definition, our move $\mathbf{s}$ does not select configuration $\ell$, but if a configuration has white state in $s$, it makes no effect; i.e. if $\ell_s=0$ then $\mathbf{s}\ell=\ell.$
Let ${\rm GL}_n(F_2)$ denote the set of $n\times n$ invertible matrices over $F_2$ and let $\mathbf{W}$ denote the subgroup of ${\rm GL}_n(F_2)$ generated by the moves $\mathbf{s}$ for $s\in
S.$ We refer $\mathbf{W}$ to a [*flipping group*]{} of $S.$ In Section \[s3\], we find that the canonical map $\phi:W\rightarrow GL_n(F_2),$ lifted from $\phi(s)=\mathbf{s}$ for $s\in S$, is a homomorphism with $\phi(W)=\mathbf{W}.$ Due to its origination, we refer such a map to the [*Vogan representation*]{} of $W.$ Then we find that the flipping group $\mathbf{W}$ has trivial center in Section \[s4\]. In Sections \[s5\], \[s6\] and \[s7\], we assume $W$ to be $A_n$, $D_n$ and $E_n$ respectively. By using the finiteness of $W$, we can determine the size of the corresponding flipping group $\mathbf{W}$. We find that the kernel of the Vogan representation of $W$ is the center $Z(W)$ of $W$ when $W$ is finite.
In the flipping puzzle on a Coxeter graph $S$, two configurations are said to be [*equivalent*]{} if one can be obtained from the other by a sequence of moves. Let $\mathcal{P}$ denote the partition of configurations (i.e. $F_2^n$) according to the above equivalent relation. As a byproduct of our work, we solve the flipping puzzle associated with $S$ when $S$ is each of $A_n$, $D_n$ and $E_n$ by determining $\mathcal{P}$. Note that when $S$ is a [*tree*]{}, a generalization of Dynkin diagrams, some partial results on $\mathcal{P}$ are obtained in [@wu:06] and [@xw:07].
Flipping groups {#s2}
===============
Throughout this article, $W$ will be a simply-laced Coxeter group with corresponding Coxter graph $S$ of $n$ elements and edge set $R=\{ss'~|~m(s, s')=3\}$. We shall construct a matrix group associated with the flipping puzzle on the Coxeter graph $S$. Let ${\rm Mat}_n(F_2)$ denote the set of $n\times n$ matrices over $F_2$ with rows and columns indexed by $S.$ Let $F_2^n$ denote the set of $n$-dimensional column vectors over $F_2$ indexed by $S$. For $s\in S$, let $\widetilde{s}$ denote the characteristic vector of $s$ in $F_2^n$; that is $\widetilde{s}=(0, 0, \ldots, 0, 1, 0,
\ldots, 0)^t,$ where $1$ is in the position corresponding to $s.$
\[d2.1\] For $s\in S,$ we associate a matrix $\mathbf{s}\in {\rm
Mat}_n(F_2)$, denoted by the bold type of $s$, as $$\mathbf{s}_{uv}=\left\{
\begin{array}{ll}
1, & \hbox{if $u=v,$ or $v=s$ and $uv\in R$;} \\
0, & \hbox{else,} \\\end{array} \right.$$ where $u,v\in S$.
The following is a reformulating of Definition \[d2.1\].
\[l2.2\] For $s, v\in S,$ $$\mathbf{s}\widetilde{v}=\left\{\begin{array}{ll}
\widetilde{v} , & \hbox{if $v\not= s$;} \\ \widetilde{v}+\sum\limits_{uv\in R}\widetilde{u} & \hbox{if $v=s$.} \\
\end{array}\right.$$$\Box$
The flipping puzzle associated with $S$, which is described in the introduction, is now restated as follows. A configuration is simply a vector $\ell\in F_2^n,$ where $\ell_s=1$ (resp. $\ell_s=0$) means that the vertex $s\in S$ has black state (resp. white state). In this setting, if $\ell_s=1$ then $\mathbf{s}\ell$ is the new configuration after the move to select the vertex $s$. Note that if $\ell_s=0$, we have $\mathbf{s}\ell=\ell$ from Lemma \[l2.2\], so we can view the action of $\mathbf{s}$ on $\ell$ as a [*feigning move*]{} on $\ell$ which is not originally defined as a move in the flipping puzzle. The following lemma is immediate from this combinatorial realization.
\[l2.3\] For $s \in S,$ $\mathbf{s}$ is an involution; that is ${\mathbf{s}}^2=I$, the identity matrix. $\Box$
From Lemma \[l2.3\], $\mathbf{s}$ is invertible, so we can give the following definition.
\[d2.4\] Let $\mathbf{W}$ denote the subgroup of ${\rm GL}_n(F_2)$ generated by the set $\{\mathbf{s}~|~s\in S\}$. $\mathbf{W}$ is referring to the [*flipping group*]{} of $S$.
Coxeter groups and their combinatorial representations {#s3}
======================================================
Let $W$ denote a simply-laced Coxeter group. Recall that an [*$n$-dimensional representation*]{} of $W$ over $F_2$ is a homomorphism of $W$ into ${\rm GL}_n(F_2).$ It is notorious difficult in the study of groups only defined by generators and relations. Hence the representation theory of Coxeter groups plays an important role in the study. In [@h:90 Section 5.3], Humphreys gives “geometric representations” of Coxeter groups and use these representations to show that the finite Coxeter groups are essentially those associated with Dynkin diagrams. In this section we shall show that the flipping groups defined in the last section give “combinatorial representations” of simply-laced Coxeter groups. First we need a lemma.
\[l3.2\] Let $W$ denote a simply-laced Coxeter group with Coxeter graph $S.$ For $s\in S$, set $E_s\in {\rm Mat}_n(F_2)$ by $$\label{e3.1}E_s\widetilde{v}=\left\{\begin{array}{ll}
0, & \hbox{if $v\not=s$;} \\
\sum\limits_{uv\in R}\widetilde{u}, & \hbox{if $v=s$} \\
\end{array}\right. \qquad {\rm for}~v\in S.$$ Then with referring to the notation in Definition \[d2.1\], the following (i)-(iii) hold.
1. $\mathbf{s}=I+E_s$ for $s\in S$
2. $E_{s'}E_s=0$, if $s's\notin R$.
3. If $s_is_{i-1}\in R$ for $i=1,2,\ldots,t$, then $$E_{s_t}E_{s_{t-1}}\cdots E_{s_0}=\left\{\begin{array}{ll}
E_{s_0}, & \hbox{if $s_t=s_0$;} \\
E_{s_t}E_{s_0}, & \hbox{if $s_ts_0\in R$.}\\
\end{array}\right.$$
\(i) is immediate from Lemma \[l2.2\]. Note that $E_{s'}E_s\widetilde{v}=0$ by (\[e3.1\]) for any $v, s, s'\in S$ with $s's\not\in R,$ and hence we have (ii). (iii) follows from the same reason as in (ii) by applying the product of matrices in either side of the equation to $\widetilde{v}$ and obtaining the desired equality in each case.
\[t3.2\] Let $W$ denote a simply-laced Coxeter group with Coxeter graph $S$. Let $\mathbf{W}$ denote the flipping group of $S$. Then there exists a surjective homomorphism $\phi:W\rightarrow \mathbf{W}$ such that $\phi(s)=\mathbf{s}$ for $s\in S.$ In particular, $\phi$ is a representation of $W$ over $F_2.$
We have seen $\mathbf{s}^2=I$ for $s\in S.$ It remains to show $(\mathbf{s}\mathbf{s}')^2=I$ if $s\neq s'$ and $ss'\not\in R,$ and to show $(\mathbf{s}\mathbf{s}')^3=I$ if $ss'\in R.$ For $s, s'\in
S$, $$\begin{aligned}
\mathbf{s}\mathbf{s}'&=&(I+E_s)(I+E_{s'}) \\
&=&I+E_s+E_{s'}+E_sE_{s'}\end{aligned}$$ by Lemma \[l3.2\](i). In the case $s\neq s'$ and $ss'\not\in R$, $$\begin{aligned}
(\mathbf{s}\mathbf{s}')^2&=&(I+E_s+E_{s'})(I+E_s+E_{s'}) \\
&=&I+2E_s+2E_{s'}\\
&=&I\end{aligned}$$ by Lemma \[l3.2\](ii). In the case $ss'\in R$, $$\begin{aligned}
(\mathbf{s}\mathbf{s}')^2&=&(I+E_s+E_{s'}+E_sE_{s'})(I+E_s+E_{s'}+E_sE_{s'}) \\
&=&I+3E_s+3E_{s'}+4E_sE_{s'}+E_{s'}E_s \\
&=&I+E_s+E_{s'}+E_{s'}E_s\end{aligned}$$ and $$\begin{aligned}
(\mathbf{s}\mathbf{s}')^3&=&(\mathbf{s}\mathbf{s}')^2(\mathbf{s}\mathbf{s}')\\
&=&(I+E_s+E_{s'}+E_{s'}E_s)(I+E_s+E_{s'}+E_sE_{s'})\\
&=&I+2E_s+4E_{s'}+2E_sE_{s'}+2E_{s'}E_s\\
&=&I\end{aligned}$$ by Lemma \[l3.2\](iii).
\[d3.4\] The representation $\phi$ defined in Theorem \[t3.2\] is called the [*Vogan representation*]{} of $W$.
Suppose $J\subseteq S$. Let $\mathbf{W}_J$ denote the subgroup of $\mathbf{W}$ generated by the set $\{\mathbf{s}~|~s\in J\}$ and $W_J$ denote simply-laced Coxeter group with the set $J$ of generators with the function $m\upharpoonright J\times J$, the restriction of $m$ to $J\times J.$ Note that $W_J$ is isomorphic to the subgroup of $W$ generated by the set $\{s~|~s\in J\}$ [@h:90 Section 5.5]. Hence we use the same symbol $W_J$ to express these two isomorphic groups. It makes no confused if the place that $W_J$ appears is also considered. For example, the first $W_J$ in (iii) of the following lemma is in the first meaning and the remaining two $W_J$ are in the second meaning. Note that $\mathbf{W_J},$ which is different to $\mathbf{W}_J$, is the flipping group on $J.$ Let $G[J]$ denote the submatrix of $G\in {\rm Mat}_n(F_2)$ with rows and columns indexed by $J$, and $\mathbf{W}_J[J]:=\{G[J]~|~G\in
\mathbf{W}_J\}$.
\[l3.5\] Suppose $J\subseteq S.$ The following (i)-(iii) hold.
1. $\mathbf{W}_J[J]=\mathbf{W_J}$.
2. The map $\psi:\mathbf{W}_J\rightarrow \mathbf{W_J}$, defined by $\psi(G)=G[J]$ for $G\in \mathbf{W}_J$, is a surjective homomorphism.
3. Let $\phi$ and $\phi'$ denote the Vogan representations of $W_S$ and $W_J$ respectively. Then $\phi'=\psi\circ \phi\upharpoonright W_J$. In particular, ${\rm Ker}~\phi\upharpoonright W_J\subseteq{\rm
Ker~}\phi'.$
By Definition \[d2.1\], $\mathbf{s}_{uv}=0$ for $s,u\in J$ and $v\in
S-J$. By this, each matrix $G\in \mathbf{W}_J$ has the form $$G=\left(
\begin{array}{clr}
A & 0\\
B & C
\end{array}
\right)$$ if indices in $J$ are placed in the beginning of rows and columns, where $A$ is a $|J|\times |J|$ matrix, $B$ is an $(n-|J|)\times
|J|$ matrix, $C$ is an $(n-|J|)\times (n-|J|)$ matrix, and $0$ is a $|J|\times (n-|J|)$ zero matrix. Then (i) and (ii) follow from the following matrix product rule in block form: $$\left(
\begin{array}{clr}
A & 0\\
B & C
\end{array}
\right) \left(
\begin{array}{clr}
A' & 0\\
B' & C'
\end{array}
\right) = \left(
\begin{array}{clr}
AA' & 0\\
BA'+CB' & CC'
\end{array}
\right).$$ Since $\psi\circ\phi(s)=\mathbf{s}[J]=\phi'(s)$ by (i) for all $s\in J$, we see $\phi'=\psi\circ \phi\upharpoonright W_J$, and this implies ${\rm Ker}~\phi\upharpoonright W_J\subseteq{\rm
Ker~}\phi'.$
The center of a flipping group {#s4}
==============================
As we shall see in Proposition \[p5.12\] that the Coxeter group of type $D_n$ has nontrivial center when $n$ is even. In this section, we show that the center $Z(\mathbf{W})$ of any flipping group $\mathbf{W}$ of a Coxeter graph $S$ is trivial. Therefore, the center $Z(W)$ of any Coxeter group $W$ is contained in the kernel of the Vogan representation of $W.$ Recall that a Coxeter graph $S$ is [*disconnected*]{} if there is a partition of $S=S'\cup S''$ with $S', S''\not=\emptyset$ and there is no edge $uv\in R$ with $u\in S'$ and $v\in S''.$ In this case the Coxeter group $W$ is isomorphic to the direct product $W'\times W''$ of the Coxeter groups $W'=W_{S'}$ and $W''=W_{S''}.$ $S$ is [*connected*]{} if $S$ is not disconnected.
\[p4.1\] Let $W$ denote a simple-laced Coxeter group with Coxeter graph $S.$ Then the center $Z(\mathbf{W})$ of the flipping group $\mathbf{W}$ of $S$ is trivial.
It suffices to assume that $S$ is connected with at least two vertices. Let $Z$ be an element in the center of $\mathbf{W}$ and let $u,v$ be two distinct elements in $S$. We show that $Z_{vu}=0$ to conclude $Z=I.$ Suppose $Z_{vu}=1.$ On the one hand $\mathbf{v}Z\widetilde{u}\not=Z\widetilde{u}$ since $Z\widetilde{u}$ has $1$ in the $v$th position. On the other hand, $\mathbf{v}Z\widetilde{u} =
Z\mathbf{v}\widetilde{u}=Z\widetilde{u}$ since $\mathbf{v}\widetilde{u}=\widetilde{u}.$ Hence we have a contradiction.
From the above Proposition \[p4.1\] we immediately have the following corollary.
\[c4.2\] Let $W$ denote a simply-laced Coxeter group. Then the center $Z(W)$ is contained in the kernel of the Vogan representation of $W.$ $\Box$
Coxeter groups of type $A_n$ {#s5}
============================
Recall that the Vogan representation $\phi$ of $W$ is [*faithful*]{} whenever $\phi$ is injective. Also $\phi$ is [*irreducible*]{} if there is no subspace $V\subseteq F_2^n$, $V\not=0,
F_2^n,$ such that $\phi(W)V\subseteq V.$ For $a\in F_2^n$, the subset of $F_2^n$ consisting of all elements $Ga$ with $G\in
\phi(W)$ is called the [*orbit*]{} of $F_2^n$ containing $a$ under the action of $\phi(W).$
In this section we assume that $W$ is of type $A_n$ with the Coxeter graph $S$ as shown in Fig. 1, and determine the orbits of $F_2^n$ under the action of $\phi(W)$. We also show that the kernel of the Vogan representation $\phi$ of $W$ is the center $Z(W)$ of $W$ and determine the reducibility of $\phi.$ The trivial case is given in the following.
Let $W$ be a Coxeter group of type $A_1$ with the Vogan representation $\phi$. Then the orbits of $F_2$ are $\{0\}$, $\{1\}$ under the action of $\phi(W),$ ${\rm Ker}~\phi=\{1,s_1\}=W=Z(W)$, and $\phi$ is irreducible.
This follows from that $W=\{1,s_1\}$ and $\phi(W)$ is a trivial group.
In the remaining of this section, we always assume $n\geq 2$. Set $$\label{e4.1}
\overline{1}=\widetilde{s}_1,~\overline{i+1}=\mathbf{s_i}\mathbf{s_{i-1}}\cdots
\mathbf{s_1}\overline{1}\quad{\rm for}~ 1\leq i\leq n.$$ Note that $$\label{e4.0}
\overline{i}=\widetilde{s}_{i-1}+ \widetilde{s}_i \quad {\rm for}~
2\leq i\leq n,$$ and $$\label{e4.2}
\overline{n+1}=\widetilde{s}_n=\overline{1}+\overline{2}+\cdots+\overline{n}.$$ Set $\Delta=\Delta(A_n):=\{\overline{1},\overline{2},\ldots,\overline{n}\}.$ Note that $\Delta$ is a basis of $F_2^n.$ We refer $\Delta$ to a [*simple basis*]{} of $F_2^n.$ For $a\in F_2^n$, let $\Delta(a)$ denote the subset of $\Delta$ consisting of all the elements appeared in the expression of $a$ as a linear combination of elements in $\Delta.$ The [*weight*]{} of an element $a\in F_2^n$ is $wt(a):=|\Delta(a)|.$ For example, $\Delta(\overline{n+1})=\Delta$ and $wt(\overline{n+1})=n.$
\[l4.2\] $\mathbf{s_i}\overline{i}=\overline{i+1},$ $\mathbf{s_i}\overline{i+1}=\overline{i}$ and $\mathbf{s_i}$ fixes other vectors in $ \{\overline{1},$ $\overline{2},$ $\ldots,$ $\overline{n+1}\}-\{\overline{i},\overline{i+1}\}$ for $1\leq i\leq
n.$
This is immediate by applying Lemma \[l2.2\], (\[e4.1\]) and (\[e4.0\]).
Let $S_{n+1}$ denote the group of permutations on $\{\overline{1},\overline{2},\ldots ,\overline{n+1}\}.$ By Lemma \[l4.2\], we can give the following definition.
\[d4.3\] Let $\alpha~:~\mathbf{W}\rightarrow S_{n+1}$ be the homomorphism defined by $$\alpha(G)\overline{j}=G\overline{j}$$ for each $1\leq j\leq n+1$ and $G\in \mathbf{W}$.
Note that $\alpha(\mathbf{s_i})$ is the transposition $(\overline{i},\overline{i+1})$ in $S_{n+1}$ for each $1\leq i\leq
n$.
\[p4.4\] $\alpha$ is an isomorphism from $\mathbf{W}$ onto $S_{n+1}.$
$\alpha$ is surjective since the transpositions $\alpha(\mathbf{s_1})$, $\alpha(\mathbf{s_2})$,$\ldots$, $\alpha(\mathbf{s_{n}})$ generate $S_{n+1}$. Since $\Delta\cup
\{\overline{n+1}\}$ spans $F_2^n$, $\alpha$ is injective.
The next proposition determines the orbits of $F_2^n$ under the action of $\mathbf{W}$.
\[l4.4\] For $0 \leq i\leq \lfloor\frac{n+1}{2}\rfloor,$ $$O_i=\{a\in F_2^n~|~wt(a)=i~{\rm or}~n+1-i\}$$ is an orbit of $F_2^n$ under the action of $\mathbf{W},$ where $\lfloor t\rfloor$ is the largest integer less than or equal to $t.$
Suppose $a\in F_2^n$ with $wt(a)=i.$ Observe that from Lemma \[p4.4\] and (\[e4.2\]), $$\Delta(Ga)=\left\{
\begin{array}{ll}
\alpha(G)\Delta(a), & \hbox{if $\overline{n+1}\not\in \alpha(G)\Delta(a)$ ;} \\
\Delta-\alpha(G)\Delta(a), & \hbox{if $\overline{n+1}\in \alpha(G)\Delta(a)$}\\
\end{array}\right.$$ for $G\in \mathbf{W}.$ The proposition follows from this observation because the subgroup of $\alpha(\mathbf{W})=S_{n+1}$ generated by the transpositions $\alpha(\mathbf{s_1})$, $\alpha(\mathbf{s_2})$,$\ldots$, $\alpha(\mathbf{s_{n-1}})$ acts transitively on the fixed size subsets of $\Delta$, and $\mathbf{s_n}\overline{n}=\overline{1}+\overline{2}+\cdots+\overline{n}$ by Lemma \[l4.2\] and (\[e4.2\]).
In the following propositions, we study the reducibility of $\phi$ and ${\rm Ker}~\phi.$
\[p4.6\] The Vogan representation $\phi$ of $W$ is irreducible if and only if $n$ is even.
Let $V$ denote a nontrivial proper subspace of $F_2^n$ such that $\phi(W)V\subseteq V$. Referring to Proposition \[l4.4\], note that $$\label{e5.4} V=\bigcup\limits_{i\in J}O_i$$ for some proper subset $J\subseteq \{0, 1, \ldots,
\lfloor\frac{n+1}{2}\rfloor\}$ with $J\not=\{0\}.$ Note that the set in the right side of (\[e5.4\]) to be closed under addition is when it is the set of even weight vectors, and this occurs if and only if $n$ is odd.
\[p4.5\] The Vogan representation $\phi$ of $W$ is faithful. In particular, ${\rm Ker}~\phi=Z(W)$ is the trivial group.
The first statement follows from that Proposition \[p4.4\] and $W$ is isomorphic to $S_{n+1}$ [@h:90 p41]. The second follows from the first and Corollary \[c4.2\].
Coxeter groups of type $D_n$ {#s6}
============================
Fix an integer $n\geq 4.$ Let $W$ denote the Coxeter group of type $D_n$ with the Coxeter graph in Fig. 1. Let $\phi$ denote the Vogan representation of $W$, and $\mathbf{W}=\phi(W)$ be the flipping group of $S.$ Set $$\begin{aligned}
\label{e5.0}
\begin{split}
&\overline{1}=\widetilde{s}_1,~\overline{i+1}=\mathbf{s_i}\mathbf{s_{i-1}}\cdots
\mathbf{s_1}\overline{1}\quad {\rm for}~ 1\leq i\leq n-1,~{\rm
and~} \overline{n+1}=\widetilde{s}_n.
\end{split}\end{aligned}$$ Note that $$\begin{aligned}
\label{e5.1}
\begin{split}
&\overline{i}=\widetilde{s}_{i-1}+\widetilde{s}_{i}~~~~{\rm~for}~~2\leq
i\leq n-2,\\
&\overline{n-1}=\widetilde{s}_{n-2}+\widetilde{s}_{n-1}+\widetilde{s}_n,
\end{split}\end{aligned}$$ and $$\label{e5.2}
\overline{n}=\widetilde{s}_{n-1}+\widetilde{s}_n=\overline{1}+\overline{2}+\cdots+\overline{n-1}.$$ Set $\Delta=\Delta(D_{n}):=\{\overline{1},
\overline{2},\ldots,\overline{n-1}, \overline{n+1}\}$ to be the simple basis of $F_2^n$ in the case of type $D_n$. Set $\Delta(a)$ and $wt(a)$ as before for $a\in F_2^n.$ For example, $\Delta(\overline{n})=\Delta-\{\overline{n+1}\}$ by (\[e5.2\]), and $wt(\overline{n})=n-1.$
\[l5.1\] The following (i),(ii) hold.
1. For each $1\leq i\leq n-1$, $\mathbf{s_i}\overline{i}=\overline{i+1},$ $\mathbf{s_i}\overline{i+1}=\overline{i},$ and $$\mathbf{s_i}\overline{j}=\overline{j}\qquad {\rm for~~} j\in
\{\overline{1},\overline{2},\ldots,\overline{n+1}\}-\{\overline{i},\overline{i+1}\}.$$
2. $\mathbf{s_n}\overline{n-1}=\overline{n},$ $\mathbf{s_n}\overline{n}=\overline{n-1},$ $\mathbf{s_n}\overline{n+1}=\overline{n-1}+\overline{n}+\overline{n+1},$ and $$\mathbf{s_n}\overline{j}=\overline{j}\qquad {\rm for~~} j\in\{\overline{1}, \overline{2}, \ldots, \overline{n-2}\}.$$
In particular, $\overline{n+1}\in \Delta(G\overline{n+1})$ and $G(\{\overline{1}, \overline{2}, \ldots, \overline{n}\})\subseteq
\{\overline{1}, \overline{2}, \ldots, \overline{n}\}$ for all $G\in
\mathbf{W}.$
This follows immediately from Lemma \[l2.2\], (\[e5.0\]) and (\[e5.1\]).
Let $S_n$ denote the group of permutations on $\{\overline{1},\overline{2},\ldots,\overline{n}\}$. By Lemma \[l5.1\], we can give the following definition.
\[d5.2\] Let $\beta~:~\mathbf{W}\rightarrow S_n$ denote the homomorphism defined by $$\beta(G)(\overline{j})=G\overline{j}$$ for $1\leq j\leq n$ and $G\in\mathbf{W}.$
In fact, $\beta$ is surjective since the $n-1$ transpositions $\beta(\mathbf{s_1}),$ $\beta(\mathbf{s_2}),$ $\ldots,$ $\beta(\mathbf{s_{n-1}})$ generate $S_n.$ Let $Z$ denote the additive group of $(n-1)$-dimensional subspace of $F_2^n$ spanned by the set $\{\overline{1},\overline{ 2}, \ldots, \overline{n-1}\}.$ Note that $a\in Z$ iff $\overline{n+1}\not\in \Delta(a)$ for $a\in
F_2^n.$ By Lemma \[l5.1\] and (\[e5.2\]), $Z$ is closed under the left multiplication of matrices in $\mathbf{W}.$
The Vogan representation $\phi$ of $W$ is not irreducible. In particular $\phi(W) Z\subseteq Z.$ $\Box$
Hence $Z$ is a disjoint union of orbits of $F_2^n$ under the action of $\mathbf{W}.$ Note that $F_2^n-Z$ is also a disjoint union of orbits of $F_2^n$ under the action of $\mathbf{W}$. The following proposition determines all the orbits of $F_2^n$ under the action of $\mathbf{W}.$
\[p5.3\] The following are orbits of $F_2^n$ under the action of $\mathbf{W}.$ $$\begin{aligned}
O_i&=&\{a\in Z~|~wt(a)=i~{\rm or}~n-i\},\\
\Omega_o&=&\{a\in F_2^n-Z~|~wt(a)\equiv 1~{\rm
or}~n-1~\pmod{2}\},\\
\Omega_e&=&\{a\in F_2^n-Z~|~wt(a)\equiv 0~{\rm or}~n~\pmod{2}\},\end{aligned}$$ where $0\leq i\leq \lfloor\frac{n}{2}\rfloor.$ In particular $\Omega_o=\Omega_e=F_2^n-Z$ is an orbit when $n$ is odd.
The proof is similar to the proof of Proposition \[l4.4\]. The reason that $O_i$ is an orbit follows from two facts: (i) $\beta(\mathbf{s_1})$, $\beta(\mathbf{s_2}),$ $\ldots,$ $\beta(\mathbf{s_{n-2}})$ generate the subgroup $S_{n-1}$ of $S_n$ consisting of permutations on $\Delta-\{\overline{n+1}\}$ and $S_{n-1}$ acts transitively on fixed size subsets of $\Delta-\{\overline{n+1}\}$, and (ii) $$\mathbf{s_{n-1}}\overline{n-1}=\mathbf{s_{n}}\overline{n-1}=\overline{n}=
\overline{1}+\overline{2}+\cdots+\overline{n-1}$$ by Lemma \[l5.1\](i),(ii) and (\[e5.2\]). The reason that $\Omega_o$ and $\Omega_e$ are orbits follows from an additional fact that $$wt(\mathbf{s_n}\overline{n+1})=wt(\overline{n-1}+\overline{n}+\overline{n+1})
=wt(\overline{1}+\overline{2}+\cdots+\overline{n-2}+\overline{n+1})=n-1.$$
We study the structure of $\mathbf{W}.$
\[d5.6\] Let $\gamma~:~\mathbf{W}\rightarrow {\rm Aut}(Z)$ denote the homomorphism from $\mathbf{W}$ into the group ${\rm Aut}(Z)$ of automorphisms of $Z$ such that $$\gamma(G)(u)=Gu$$ for $G\in \mathbf{W}$ and $u\in Z.$
\[l5.7\] There exists a unique homomorphism $\theta: S_n\rightarrow {\rm Aut}(Z)$ such that $\gamma=\theta \circ \beta.$
Since $\beta$ is surjective, it suffices to show that the kernel of $\beta$ is contained in the kernel of $\gamma.$ Suppose $G\in {\rm Ker}~\beta$. Then $G\overline{i}=\overline{i}$ for $1\leq i\leq n;$ in particular $G$ fixes each element in the basis $\Delta-\{\overline{n+1}\}$ of $Z$. Thus $G\in{\rm Ker}~\gamma.$
Let $Z\rtimes_{\theta} S_n$ denote the group of [*external semidirect product*]{} of $Z$ and $S_n$ with respect to $\theta$[@dj:02 p.155]; i.e. $Z\rtimes_{\theta} S_n$ is the set $Z\times S_n$ with the following product rule: $$(u,\sigma)(v,\tau)=(u+\theta(\sigma)(v),\sigma\tau),$$ where $u, v\in Z$ and $\sigma, \tau\in S_n.$ Note that $\overline{n+1}+G\overline{n+1}\in Z$ for any $G\in \mathbf{W}$ by Lemma \[l5.1\].
\[d5.8\] Let $\delta~:~\mathbf{W}\rightarrow Z\rtimes_{\theta} S_n$ denote the map defined by $$\delta(G)=(\overline{n+1}+G\overline{n+1},\beta(G))$$ for any $G\in \mathbf{W}.$
\[l5.9\] $\delta$ is an injective homomorphism of $\mathbf{W}$ into $Z\rtimes_{\theta} S_n$.
For $G, H\in \mathbf{W},$ $$\begin{aligned}
\delta(G)\delta(H)&=&(\overline{n+1}+G\overline{n+1},
\beta(G))(\overline{n+1}+H\overline{n+1}, \beta(H))\\
&=& (\overline{n+1}+G\overline{n+1}+\theta(\beta(G))(\overline{n+1}+H\overline{n+1}),~\beta(G)\beta(H))\\
&=& (\overline{n+1}+G\overline{n+1}+G(\overline{n+1}+H\overline{n+1}),~\beta(G)\beta(H))\\
&=& (\overline{n+1}+GH\overline{n+1},~\beta(GH))\\
&=&\delta(GH).\end{aligned}$$ This shows that $\delta$ is a homomorphism. $\delta$ is injective since if $\overline{n+1}+G\overline{n+1}=0$ and $G\in {\rm
Ker}~\beta$ , then $G$ fixes all vectors in $\Delta$, so $G$ is the identity matrix.
Note that $Z=\overline{n+1}+\Omega_o$ if $n$ is odd, and $Z=(\overline{n+1}+\Omega_o)\cup (\overline{n+1}+\Omega_e)$ if $n$ is even.
\[p5.10\] $\delta(\mathbf{W})=(\overline{n+1}+\Omega_o)\rtimes_{\theta} S_n.$ In particular $\delta(\mathbf{W})=Z\rtimes_{\theta} S_n$ if $n$ is odd; $\delta(\mathbf{W})$ has index $2$ in $Z\rtimes_{\theta} S_n$ if $n$ is even.
Note that $\delta(\mathbf{s_1}), \delta(\mathbf{s_2}), \ldots,
\delta(\mathbf{s_{n-2}}), \delta(\mathbf{s_{n-1}})$ generate $0\rtimes_{\theta} S_n.$ Since $\Omega_o$ is an orbit containing $\overline{n+1},$ we have $\delta(\mathbf{W})=(\overline{n+1}+\Omega_o)\rtimes_{\theta} S_n.$ The second part follows from Proposition \[p5.3\].
\[p5.12\] The Vogan representation $\phi$ of $W$ is faithful when $n$ is odd; ${\rm Ker}~\phi$ has order 2 when $n$ is even. Moreover, ${\rm
Ker}~\phi$ is the center $Z(W)$ of $W.$
Note that $W$ is isomorphic to the semidirect product $Z\rtimes
S_n$ of $Z$ and $S_n$ [@h:90 p.42]. By Lemma \[p5.10\], $\phi$ is faithful when $n$ is odd, and ${\rm Ker}~\phi$ has order 2 when $n$ is even. From Corollary \[c4.2\], $Z(W)\subseteq {\rm
Ker}~\phi,$ and from the fact that a normal subgroup of order 2 is contained in the center, we have ${\rm Ker}~\phi\subseteq Z(W).$
Coxeter groups of type $E_n$ {#s7}
============================
Fix an integer $n\geq 6.$ Let $W$ denote the Coxeter group of type $E_n$ with the Coxeter graph $S$ in Fig. 2. In this section we shall determine the orbits of $F_2^n$ under the action of the flipping group $\mathbf{W}$ of $S.$ Restricting the attention to the case $n=6, 7$ or $8$ in which $W$ is finite, we show that the kernel of the Vogan representation $\phi$ of $W$ is the center $Z(W)$ of $W$.
(50, 50) (0,25)(30,25) (38,25) (46,25) (54,25) (62,25) (46,33) (66,25) (70,25) (74,25) (78,25) (86,25)(94,25) (30.75,25)[( 1, 0)[6.5]{}]{} (38.75,25)[( 1, 0)[6.5]{}]{} (46.75,25)[( 1, 0)[6.5]{}]{} (54.75,25)[( 1, 0)[6.5]{}]{} (46,25.75)[( 0, 1)[6.5]{}]{} (78.75,25)[( 1, 0)[6.5]{}]{} (86.75,25)[( 1, 0)[6.5]{}]{} (30,22) (38,22) (46,22) (54,22) (62,22) (46,35) (78,22) (86,22) (94,22)
(0,5)[[**Figure 2:**]{} The Coxeter graph of type $E_n.$]{}
Set $\overline{1}=\widetilde{s}_1$, $\overline{i+1}=\mathbf{s_i}\mathbf{s_{i-1}}\cdots
\mathbf{s_1}\overline{1}$ for $1\leq i\leq n-1$ and $\overline{n+1}=\widetilde{s}_n.$ Note that $$\begin{aligned}
\label{e6.1}
\overline{i}&=&\widetilde{s}_i+\widetilde{s}_{i-1}\quad{\rm for}~
2\leq i\leq n-3,\nonumber\\
\overline{n-2}&=&\widetilde{s}_{n-3}+\widetilde{s}_{n-2}+\widetilde{s}_n,\\
\overline{n-1}&=&\widetilde{s}_{n-2}+\widetilde{s}_{n-1}+\widetilde{s}_n,\nonumber\\
\overline{n}&=&\widetilde{s}_{n-1}+\widetilde{s}_n.\nonumber\end{aligned}$$ Set $\Delta=\Delta(E_n):=\{\overline{1},
\overline{2},\ldots,\overline{n}\}$ to be the simple basis of $F_2^n$ in this case. Observe that $$\label{e6.2}
\overline{n+1}=\overline{1}+\overline{2}+\cdots+\overline{n}.$$ Set $\Delta(a)$ and $wt(a)$ as before for $a\in F_2^n.$ For example, $\Delta(\overline{n+1})=\Delta$ and $wt(\overline{n+1})=n.$
\[p6.1\] The following (i),(ii) hold.
1. For each $1\leq i\leq n-1$, $\mathbf{s_i}\overline{i}=\overline{i+1}$, $\mathbf{s_i}\overline{i+1}=\overline{i}$, and $$\mathbf{s_i}\overline{j}=\overline{j} \qquad {\rm for~~}
\overline{j}\in\{\overline{1},\overline{2},\ldots,\overline{n+1}\}-\{\overline{i},\overline{i+1}\}.$$
2. $\mathbf{s_n}\overline{n+1}=\overline{n-2}+\overline{n-1}+\overline{n}$, $\mathbf{s_n}\overline{n}=\overline{n-2}+\overline{n-1}+\overline{n+1}$, $\mathbf{s_n}\overline{n-1}=\overline{n-2}+\overline{n}+\overline{n+1}$, $\mathbf{s_n}\overline{n-2}=\overline{n-1}+\overline{n}+\overline{n+1}$ and $$\mathbf{s_n}\overline{j}=\overline{j} \qquad {\rm for~~} 1\leq
j\leq n-3.$$
This is immediate by applying Lemma \[l2.2\] and (\[e6.1\]).
Let $S_n$ denote the group of permutations on $\Delta=\{\overline{1},\overline{2},\ldots,\overline{n}\}$. Set $T:=\{s_1,s_2,\ldots,s_{n-1}\}$. Recall that $\mathbf{W}_T$ is the subgroup of $\mathbf{W}$ generated by $\{\mathbf{s}~|~s\in T\}$. By Lemma \[p6.1\], we find that the set $\Delta$ is closed under the left multiplication of elements in $\mathbf{W}_T$.
\[d7.2\] Let $\epsilon:\mathbf{W}_T\rightarrow S_n$ denote the homomorphism satisfying $$\epsilon(G)(\overline{j})=G\overline{j}$$ for $1\leq j\leq n$ and $G\in \mathbf{W}_T.$
In fact, $\epsilon$ is an isomorphism since $\Delta$ is a spanning set and the $n-1$ transpositions $\epsilon(\mathbf{s_1}),\epsilon(\mathbf{s_2}),\ldots,\epsilon(\mathbf{s_{n-1}})$ generate $S_n$.
\[p6.2\] The following are orbits of $F_2^n$ under the action of $\mathbf{W}.$ $$\begin{aligned}
\label{ee7.3}
O_0&=&\{0\},\nonumber\\
O_1&=&\{a\in F_2^n~|~a\not=0, wt(a)\equiv 1~{\rm or~}
n-2~\pmod{4}\},\\
O_2&=&\{a\in F_2^n~|~a\not=0, wt(a)\equiv 2~{\rm or~}
n-3~\pmod{4}\},\nonumber\\
O_3&=&\{a\in F_2^n~|~a\not=0, wt(a)\equiv 3~{\rm or~}
n~\pmod{4}\},\nonumber\\
O_4&=&\{a\in F_2^n~|~a\not=0, wt(a)\equiv 0~{\rm or~}
n-1~\pmod{4}\}.\nonumber\end{aligned}$$ In particular $O_1=O_3$ when $n\equiv 1{\pmod 4}$, $O_1=O_4$ and $O_2=O_3$ when $n\equiv 2{\pmod 4}$, $O_2=O_4$ when $n\equiv
3{\pmod 4}$, and $O_1=O_2$ and $O_3=O_4$ when $n\equiv 0{\pmod
4}$.
It is clear that $O_0$ is an orbit. There are four cases to put nonzero vectors $a,$ $b$ in an orbit. (a)$wt(a)=wt(b):$ This is because $\epsilon(\mathbf{W}_T)=S_n$ acts transitively on the fixed size subsets of $\Delta$; (b) $wt(b)=n+3-wt(a),$ or $n-1-wt(a):$ This is from (a) and the observation that $$\label{e6.3}
wt(\mathbf{s_n}a)=\left\{
\begin{array}{llr}
n+3-wt(a), & \hbox{if $|\Delta(a)\cap \{\overline{n},\overline{n-1},\overline{n-2}\}|=3$;} \\
n-1-wt(a), & \hbox{if $|\Delta(a)\cap \{\overline{n},\overline{n-1},\overline{n-2}\}|=1$;} \\
w(a), & \hbox{else} \\
\end{array}
\right.$$ by Lemma \[p6.1\](ii) and (\[e6.2\]); (c) $wt(a)=wt(b)-4:$ This is by applying the first case of (\[e6.3\]) and then applying the second case of (\[e6.3\]); and (d) $wt(a)=wt(b)+4:$ This is by applying the second case of (\[e6.3\]) and then the first case of (\[e6.3\]). The proposition follows from the above cases (a)-(d).
With reference to Proposition \[p6.2\], for each orbit $O$ of $F_2^n$ with $O\not=O_0$ there is $1\leq i\leq n$ such that $\widetilde{s}_{i}\in O.$ For example $\widetilde{s}_i\in O_i$ for $i=1,2,3$ and $\widetilde{s}_{n-1}\in O_4.$
Similar to case of $A_n$, we determine the reducibility of $\phi$ from Proposition \[p6.2\] immediately.
The Vogan representation $\phi$ is irreducible if and only if $n$ is even.$\Box$
Recall that for $a\in F_2^n,$ the [*isotropy group*]{} of $a$ in $\mathbf{W}$ is $\{G\in \mathbf{W}~|~Ga=a\}$, and the cardinality of the orbit of $a$ is equal to the index of the isotropy group of $a.$
\[c6.1\] For $J:=\{s_2,s_3,\ldots,s_n\}.$ the number $|\mathbf{W}_J||O_1|$ divides $|\mathbf{W}|$, where $$\begin{aligned}
\label{e6.4}
|O_1|=\left\{
\begin{array}{ll}
2^{n-1}-(-1)^{\frac{n}{4}}2^{\frac{n-2}{2}}, & \hbox{if $n\equiv 0\pmod{4},$}\\
2^{n-1}, & \hbox{if $n\equiv 1\pmod{4},$ } \\
2^{n-1}+(-1)^{\frac{n-2}{4}} 2^{\frac{n-2}{2}}-1, & \hbox{if $n\equiv 2\pmod{4},$} \\
2^{n-2}+(-1)^{\frac{n-3}{4}} 2^{\frac{n-3}{2}}, & \hbox{if $n\equiv 3\pmod{4}.$} \\
\end{array}
\right.\end{aligned}$$
Since $ \mathbf{W}_J$ is a subgroup of the isotropy group of $\overline{1}$, the number $|\mathbf{W}_J||O_1|$ divides $|\mathbf{W}|$. Note that by (\[ee7.3\]) $$|O_1|=\left\{
\begin{array}{ll}
\sum\limits_{k\equiv 1,2(\bmod{4})\atop 1\leq k \leq n}{n \choose k}, & \hbox{if $n\equiv 0\pmod{4},$}\\
\sum\limits_{k\equiv 1(\bmod{2})\atop 1\leq k \leq n}{n \choose k}, & \hbox{if $n\equiv 1\pmod{4},$}\\
\sum\limits_{k\equiv 0,1(\bmod{4})\atop 1\leq k \leq n}{n \choose k}, & \hbox{if $n\equiv 2\pmod{4},$}\\
\sum\limits_{k\equiv 1(\bmod{4})\atop 1\leq k \leq n}{n \choose k}, & \hbox{if $n\equiv 3\pmod{4},$}\\
\end{array}
\right.$$ where $n\choose k$ is the binomial coefficient. From this, we routinely prove (\[e6.4\]) by induction on $n$.
We need to quote a lemma.
\[l6.3\]([@bcn:89 Lemma 10.2.11]) If $W$ is of type $E_7$ or $E_8$ then $Z(W)=\{1,w_0\}$, where $w_0$ is the longest element of $W$. $\Box$
Recall that $T=\{s_1,s_2,\ldots,s_{n-1}\}$ and $J=\{s_2,s_3,\ldots,s_n\}$ when we indicate that the Coxeter group $W$ is of type $E_n.$
\[c6.2\] The Vogan representation $\phi$ of $W$ is faithful if $W$ is of type $E_6$, and $|{\rm Ker}~\phi|=2$ if $W$ is of type $E_7$. Moreover, ${\rm Ker}~\phi=Z(W)$ if $W$ is of type $E_6$ or $E_7$.
Suppose $W$ is of type $E_6$. With referring to Corollary \[c6.1\], we have $|O_1|=27$. By Lemma \[l3.5\](iii) and Proposition \[p5.12\] (the case $D_5$), we know $|\mathbf{W}_J|=2^45!$, where $J$ is of type $D_5$. Since $|\mathbf{W}_J||O_1|$ divides $|\mathbf{W}|$, we have $|\mathbf{W}|\geq 2^45!\cdot27=2^73^45$. Since $|W|=2^73^45$ [@h:90 p.44], $W$ is isomorphic to $\mathbf{W}$ and ${\rm
Ker}~\phi$ is trivial. By this and Corollary \[c4.2\], $Z(W)$ is trivial.
Suppose $W$ is of type $E_7$. From Corollary \[c4.2\] and Lemma \[l6.3\], $|{\rm Ker}~\phi|\geq 2$. Since $|W|=2^{10}3^45\cdot7$ [@h:90 p.44], we see that $|\mathbf{W}|\leq2^93^45\cdot7$. On the other hand, according to a similar counting argument as above, we have $|O_1|=28$, $|\mathbf{W}_J|=2^73^45,$ where $J$ is of type $E_6,$ and hence $|\mathbf{W}|\geq2^93^45\cdot7$. Thus, $|\mathbf{W}|=2^93^45\cdot7$ and $|Z(W)|=|{\rm Ker}~\phi|=2$.
We now go to the last case $W$ of type $E_8.$ Note that $J=\{s_2, s_3, \ldots, s_8\}$ is of type $E_7$ and $T\cap J=\{s_2, s_3, \ldots, s_7\}$ is of type $A_6.$ We need more information of the nontrivial element $w_0$ in the center $Z(W_J)$ of $W_J.$ It is quite complicate to describe $w_0$ directly as a product of elements in $J.$ We borrow two notations to describe $w_0.$ Let $\phi$ denote the Vogan representation of $W$. Note that $\phi\upharpoonright W_{T\cap J}$ is an isomorphism of $W_{T\cap J}$ onto $\mathbf{W}_{T\cap J}$ by Lemma \[l3.5\](ii) and Proposition \[p4.5\]. Also $\epsilon\upharpoonright\mathbf{W}_{T\cap J}:\mathbf{W}_{T\cap
J}\rightarrow S_7$ is an isomorphism , where $\epsilon$ is as in Definition \[d7.2\] and $S_7$ is the group of permutations on $\{\overline{2},\overline{3},\ldots,\overline{8}\}.$ The expression of $w_0$ is as follows. $$\label{e7.6}
\left.
\begin{array}{ll}
w_0=&\phi^{-1}(\epsilon^{-1}( (\overline{2},
\overline{8},\overline{3}, \overline{7}, \overline{4}, \overline{6},
\overline{5})))s_8\phi^{-1}(\epsilon^{-1}( (\overline{5},
\overline{8})(\overline{4}, \overline{7})(\overline{3},
\overline{6})))s_8\\
&\phi^{-1}(\epsilon^{-1}( (\overline{4}, \overline{8})(\overline{3},
\overline{7})(\overline{2},
\overline{6})))s_8\phi^{-1}(\epsilon^{-1}( (\overline{5},
\overline{8})(\overline{4},
\overline{7})))s_8\\&\phi^{-1}(\epsilon^{-1}( (\overline{3},
\overline{7})(\overline{2}, \overline{6})))s_8.
\end{array}
\right.$$ It is routine to check that the above $w_0$ maps to $-I$ by the faithful representation defined in [@ab:05 Proposition 8] with $c=0$ or in [@he:94 p. 291] to conclude $w_0$ is in the center of $W_J$ and indeed is the longest element of $W_J$ by [@ab:05 Proposition 21]. Thus, we have the following lemma.
\[l7.8\] Let $W$ be of type $E_8$ with the Vogan representation $\phi$ and $w_0\in Z(W_J)$ be not identity. Then $\phi(w_0)$ is $$\begin{aligned}
&& \epsilon^{-1}((\overline{2}, \overline{8},\overline{3},
\overline{7}, \overline{4}, \overline{6},
\overline{5}))\mathbf{s_8}\epsilon^{-1}((\overline{5},
\overline{8})(\overline{4}, \overline{7})(\overline{3},
\overline{6}))\mathbf{s_8} \epsilon^{-1}((\overline{4},
\overline{8})(\overline{3}, \overline{7})(\overline{2},
\overline{6}))\mathbf{s_8}\\&\times& \epsilon^{-1}((\overline{5},
\overline{8})(\overline{4},
\overline{7}))\mathbf{s_8}\epsilon^{-1}((\overline{3},
\overline{7})(\overline{2}, \overline{6}))\mathbf{s_8}.\end{aligned}$$ $\Box$
Note that $W_J$ is not isomorphic to its flipping group $\mathbf{W_J}$ by Proposition \[c6.2\]. The following lemma claims that $W_J$ is isomorphic to the subgroup $\mathbf{W}_J$ of $\mathbf{W}$.
\[l6.4\] Let $W$ be of type $E_8$ with the Vogan representation $\phi.$ Then the restriction $\phi\upharpoonright W_J$ of $\phi$ to $J$ is injective.
Let $\phi':W_J\rightarrow \mathbf{W_J}$ denote the Vogan representation of $W_J$. From Lemma \[l3.5\](iii) and Proposition \[c6.2\], we see that ${\rm Ker}~\phi\upharpoonright
W_J\subseteq {\rm Ker}~\phi'=\{1,w_0\}$, where $w_0$ is given in (\[e7.6\]). To prove that ${\rm Ker}~\phi\upharpoonright W_J$ is trivial, it suffices to show that $\phi(w_0)\not=I.$ This follows from the computation $$\phi(w_0)\overline{8}=\overline{1}+\overline{8}$$ by applying the expression $\phi(w_0)$ in Lemma \[l7.8\] to $\overline{8}$ and using Lemma \[p6.1\] and (\[e6.2\]) for $n=8$ to simplify.
There is a similar result about $W$ of type $E_8.$
If $W$ is of type $E_8$ then ${\rm Ker}~\phi$ has order 2. Moreover, ${\rm Ker}~\phi=Z(W)$.
We have $|O_1|=2^3\cdot 3\cdot 5$ from (\[e6.4\]), $|\mathbf{W}_J|=|W_J|=2^{10}3^45\cdot7$ from Lemma \[l6.4\] and $|W|=2^{14}3^55^27$[@h:90 p.44]. Therefore, as the proof of Proposition \[c6.2\], ${\rm Ker}~\phi$ has order 2 and ${\rm
Ker}~\phi=Z(W)$.
Concluding remarks
==================
We list the main results of this article as follows.
[ccc]{} Dynkin diagram &reducibility of $\phi$ &$|{\rm Ker}~\phi|$\
\
$A_n$ &[$\phi$ is irr. iff $n=1$ or $n$ is even.]{} &[$\left\{
\begin{array}{ll}
2,& \hbox{if $n=1,$}\\
1,& \hbox{else.} \\
\end{array}
\right. $ ]{}\
\
$\left.
\begin{array}{c}
D_n\\
(n\geq 4)\\
\end{array}
\right.$ &[$\phi$ is not irr.]{} &[$\left\{
\begin{array}{ll}
2,& \hbox{if $n$ is even,}\\
1,& \hbox{else.} \\
\end{array}
\right.$]{}\
\
$E_6$ &[$\phi$ is irr.]{} &$1$\
\
$E_7$ &[$\phi$ is not irr.]{} &$2$\
\
$E_8$ &[$\phi$ is irr.]{} &$2$\
\
[**Table 1:**]{} The reducibility and the kernel of a Vogan representation $\phi.$
[cc]{} Coxeter graph &orbits\
\
$A_n$ &[$
\left.
\begin{array}{l}
O_i=\{a\in F_2^n~|~wt(a)=i~{\rm or}~n+1-i\}
\hbox{$(0 \leq i\leq \lfloor\frac{n+1}{2}\rfloor$).}\\
\end{array}
\right. $]{}\
\
$\left.
\begin{array}{c}
D_n\\
(n\geq 4)\\
\end{array}
\right.$ &[$ \left.
\begin{array}{l}
O_i=\{a\in Z~|~wt(a)=i~{\rm or}~n-i\}~~~(0\leq i\leq \lfloor\frac{n}{2}\rfloor),\\
\Omega_o=\{a\in F_2^n-Z~|~wt(a)\equiv 1~{\rm
or}~n-1~\pmod{2}\},\\
\Omega_e=\{a\in F_2^n-Z~|~wt(a)\equiv 0~{\rm or}~n~\pmod{2}\},\\
\hbox {$\Omega_o=\Omega_e=F_2^n-Z$ when $n$ is odd.}\\
\end{array}
\right. $]{}\
\
$\left.
\begin{array}{c}
E_n\\
(n\geq 6)\\
\end{array}
\right.$ &[$ \left.
\begin{array}{l}
O_0=\{0\},\\
O_1=\{a\in F_2^n~|~a\not=0, wt(a)\equiv 1~{\rm or~}
n-2~\pmod{4}\},\\
O_2=\{a\in F_2^n~|~a\not=0, wt(a)\equiv 2~{\rm or~}
n-3~\pmod{4}\},\\
O_3=\{a\in F_2^n~|~a\not=0, wt(a)\equiv 3~{\rm or~}
n~\pmod{4}\},\\
O_4=\{a\in F_2^n~|~a\not=0, wt(a)\equiv 0~{\rm or~}
n-1~\pmod{4}\}.\\
\hbox{$O_1=O_3$ when $n\equiv 1{\pmod
4}$,}\\
\hbox{$O_1=O_4$ and $O_2=O_3$ when $n\equiv 2{\pmod 4}$,}\\
\hbox{$O_2=O_4$ when $n\equiv 3{\pmod 4}$,}\\
\hbox{$O_1=O_2$ and $O_3=O_4$ when $n\equiv 0{\pmod 4}$}.\\
\end{array}
\right.$]{}\
\
[**Table 2:**]{} The orbits of $F_2^n$ under the action of the flipping group of a Coxeter graph $S.$
[10]{}
P. Batra, , 223 (2000) [208–236]{}.
P. Batra, , 251 (2002) [80–97]{}.
A. Björner and F. Brenti, , , 2005.
A.E. Brouwer, A.M. Cohen, and A. Neumaier, , Springer-Verlag, Berlin, 1989.
Meng-Kiat Chuah and Chu-Chin Hu, , 279 (2004) [22–37]{}.
H. Eriksson, , , KTH, Stockholm, Sweden, 1994.
J. E. Humphreys, , , Cambridge, 1990.
D. Joyner, , , Baltimore and London, 2002.
A. W. Knapp, , , 1996.
Xinmao Wang and Yaokun Wu, , 381 (2007) [292–300]{}.
Hsin-Jung Wu and Gerard J. Chang, , , NTU, Taiwan, 2006.
Hau-wen Huang Department of Applied Mathematics National Chiao Tung University 1001 Ta Hsueh Road Hsinchu, Taiwan 30050, R.O.C.Email: [poker80@msn.com]{} Fax: +886-3-5724679
[^1]: Email address: poker80@msn.com (Hau-wen Huang).
[^2]: $~^\dag$Department of Applied Mathematics National Chiao Tung University 1001 Ta Hsueh Road Hsinchu, Taiwan 300, R.O.C..
[^3]: Research partially supported by the NSC grant 96-2628-M-009-015 of Taiwan R.O.C..
|
---
abstract: |
We prove that the sequence of averaged quantities $\int_{\R^m}u_n(\mx,\msnop)$ $\rho(\msnop)d\msnop$, is strongly precompact in $\Ldl\Rd$, where $\rho\in \Ldc{\R^m}$, and $u_n\in
\Ld{\R^m; \pL s\Rd}$, $s\geq 2$, are weak solutions to differential operator equations with variable coefficients. In particular, this includes differential operators of hyperbolic, parabolic or ultraparabolic type, but also fractional differential operators. If $s>2$ then the coefficients can be discontinuous with respect to the space variable $\mx\in \R^d$, otherwise, the coefficients are continuous functions. In order to obtain the result we prove a representation theorem for an extension of the H-measures.
address:
- ' Martin Lazar, University of Dubrovnik, Department of Electrical Engineering, 20000 Dubrovnik, Croatia'
- ' Darko Mitrović, University of Montenegro, Faculty of Mathematics, Cetinjski put bb, 81000 Podgorica, Montenegro'
author:
- Martin Lazar
- Darko Mitrović
date:
-
-
title: 'Velocity averaging – a general framework'
---
Introduction
============
The main subject of the paper is the following sequence of equations:
$$\label{main-sys}
\begin{split}
{\cal P}u_n(\mx,\msnop)&=\sum\limits_{k=1}^d\pa^{\alpha_k}_{x_k}
\left(a_k(\mx,\msnop) u_n(\mx,\msnop)\right)=\pa^\mkappa_\msnop G_n(\mx,\msnop),
\end{split}$$
where $u_n$ are weak solutions to such that $u_n\dscon 0$ in $\Ld{\Rm; \pL s{\Rd}}$, $s\geq 2$, while:
- $\alpha_k> 0$ are real numbers and $\pa^{\alpha_k}_{x_k}$ are (the Fourier) multiplier operators with the symbols $(2 \pi i\xi_k)^{\alpha_k}$, $i^{\alpha_k}:=e^{\frac{i \alpha_k \pi}{2}}$, $k=1,\dots,d$;
- $$a_k\in \begin{cases} \Ld{\Rm;\CB{\R^d}}, & s=2\\
\Ld{\Rm; \pL r{\Rd}}, & 2/s+1/r=1, \quad s>2,
\end{cases}$$ where $\CB\Rd$ stands for a space of continuous and bounded functions;
- $\pa_\msnop^\mkappa=\pa^{\kappa_1}_{p_1}\dots \pa^{\kappa_m}_{p_m}$ for a multi-index $\mkappa=(\kappa_1,\dots,\kappa_m)\in \N^m$, and $$G_n\to 0 \ \ {\rm in} \ \
\Ld{\R^m;\W{-\malpha}{s'}\Rd}, \ \ \malpha=(\alpha_1,\dots,\alpha_d),$$ where $\W{-\malpha}{s'}\Rd$ is a dual of $\W{\malpha}{s}\Rd=\{u\in \pL s\Rd: \pa_k^{\alpha_k}u\in \pL s\Rd, k=1,\dots,d \}$ (for details on anisotropic Sobolev spaces see e.g. [@VP]).
Equations involve the space variable $\mx\in \R^d$, with respect to which we have derivatives of solutions $(u_n)$, and the variable $\msnop\in \R^m$, which is usually called the velocity variable.
Notice that if $\alpha_k\in \N$ then equation is a standard partial differential equation. In particular, for $\alpha_1=\dots=\alpha_d=1$ one gets a transport equation (considered in e.g. [@Ger; @Per]; see more detailed discussion below). In general, we have a linear fractional differential equation.
First, we introduce a definition of a weak solution to . Assume for the moment that the sub-index $n$ is removed in .
\[weaksol\] We say that a function $u\in \Ld{\Rm; \pL s{\Rd}}$ is a weak solution to if for every $g\in
\Wc{|\mkappa|}2{\R^m;\W{\malpha}{s}\Rd}$ it holds $$\begin{split}
\label{defws}
&\int\limits_{\R^{m+d}}\!\sum\limits_{k=1}^d
a_k(\mx,\msnop) u(\mx,\msnop)\overline{(-{\pa}_{x_k})^{\alpha_k}(g(\mx,\msnop))}d\mx
d\msnop
= (-1)^{|\mkappa|}\!\!\int\limits_{\R^{m}}\!\Dupp{ G(\cdot,\msnop)}
{\overline{\pa^{\mkappa}_\msnop g(\cdot,\msnop)}} d\msnop\,,
\end{split}$$ where duality on $\W{\malpha}{s}\Rd$ is considered.
In this paper, we are concerned with compactness properties of sequence $(u_n)$. It is not difficult to find examples of equations of type such that the sequence $(u_n)$ does not converge strongly in $\Ll s{\R^m\times\R^d}$ for any $s\geq 1$. Indeed, a trivial example $u_n=\sin n\msnop$ solving with coefficients being independent of $\mx\in \R^d$ and $\alpha_k\in \N$, $k=1,\dots,d$, does not converge strongly in ${\rm L}^s_{loc}$ for any $s\geq 1$.
Still, from the viewpoint of applications, it is almost always enough to analyse the sequence $(u_n)$ averaged with respect to the velocity variable $(\int_{\R^m}\rho(\msnop)u_n(\mx,\msnop)d\msnop)$, $\rho\in
\Cc{\R^m}$ (see e.g. famous papers [@8; @LPT]) which, as firstly noticed by Agoshkov [@Ago] in the homogeneous hyperbolic case, can be strongly precompact in $\Ll s\Rd$ for an appropriate $s\geq 1$ even when the sequence $(u_n(\mx,\msnop))$ is not. Such results are usually called velocity averaging lemmas.
After Agoshkov’s paper, the investigations in this directions continued rather intensively. Still, in most of the previous works on the subject, the symbol $P(i\mxi, \mx, \msnop)$ of the differential operator ${\cal P}$ was of the first order and independent of $\mx\in \R^d$. Thus the corresponding equation describes a transport process occurring in a homogeneous medium. On the other hand, most of natural phenomena take place in heterogeneous media (flow in heterogeneous porous media, sedimentation processes, blood flow, gas flow in a variable duct, etc). However, it appears that it is much more complicated to work on heterogeneous transport equations than on homogeneous ones.
This fact could be explained by the following simple observation. Assume that the coefficients in do not depend on $\mx\in \R^d$. If we apply the Fourier transform in $\mx\in \R^{d}$ on equation , at least informally, we can separate solutions $(u_n)$ and the known coefficients. To be more precise, let us consider the sequence of homogeneous transport equations from [@Per]:
$$\label{perth} \pa_t u_n + a(\msnop)\cdot \nabla_\mx
u_n=\sum\limits_{j=1}^d\pa_{x_j} \pa_\msnop^\mkappa g^n_j, \ \ (t, \mx, \msnop)\in
\R^+\times\R^d\times\R^d,$$
where, for some $s>1$, $u_n\rightharpoonup 0$ weakly in $\pL s{\R^{d+1}}$, while $g^n_j\to 0$ strongly in $\Ll s{\R^+\times \R^d\times \R^d}$, $j=1,\dots,d$. The function $a:\R^d\to \R^d$ is continuous.
By finding the Fourier transform of with respect to $(t,\mx)\in \R^+\times \R^d$ (denoted by $\hat{}$ below), we conclude from the above $$(\tau+a(\msnop)\cdot \mxi)\hat{u}=\sum\limits_{j=1}^d \xi_j \pa_\msnop^\mkappa
\hat{g}_j,$$ and from here, for any $\beta>0$, $$\hat{u}=\frac{\beta^2|\mxi|^2 \hat{u}+\sum\limits_{j=1}^d
(\tau+a(\msnop)\cdot \mxi) \xi_j \pa_\msnop^\mkappa \hat{g}_j}{(\tau+a(\msnop)\cdot
\mxi)^2+\beta^2 |\mxi|^2}.$$ As the term containing $\hat{u}$ on the right-hand side can be controlled by constant $\beta$, it was proved in [@Per] that the sequence of averaged quantities $(\int_{\R^m} \rho(\msnop)u_n(t,\mx,\msnop)d\msnop)$, $\rho\in \pL{s'}{\R^m}$, $1/s+1/s'=1$, converges to zero strongly in $\pL s{\R^{d+1}}$.
Actually, such framework is probably the main approach used on the subject [@15; @20; @Gol; @Tao]. Other approaches include the use of wavelet decomposition [@16], “real-space methods" in time [@5; @52] and “real-space methods" in space using the Radon transform [@12; @54], $X$-transform [@31], duality based dispersion estimates [@26], etc.
In the heterogeneous case, the method applied on is not at our disposal (since $\hat u$ can not be separated). Probably the only possible way to tackle the heterogeneous velocity averaging problem is through a variant of defect measures [@Ant2; @Ger; @MI; @Pan_IHP; @Tar]. In [@Ger Theorem 2.5] the concrete application of defect measures on the averaging lemmas can be found. The result from [@Ger] claims that the sequence of solutions $(u_n)$ of equations satisfying conditions a)–c) with $\alpha_1=\alpha_2=\dots=\alpha_d\in \N$ and $s=2$, is such that the sequence of averaged quantities $(\int
u_n(\mx,\msnop)\rho(\msnop)d\msnop)$ strongly converges to zero in $\Ld\Rd$.
In this paper, we shall generalise Gerard’s result on a wider class of equations, and we shall allow the coefficients to be discontinuous if the solutions $u_n$ are from $\Ld{\R^m; \pL s\Rd}$ for $s>2$ (which is the situation in a numerous applications; e.g. [@BKT; @Mitr; @pan_jms]). We remark again that the result from [@Per] can be applied only in the case of homogeneous transport equations, but it is optimal in the sense that a sequence of solutions can belong to $\pL s{\R^{d+1}\times\R^d}$ for any $s>1$ (in the current contribution, we must have $s\geq 2$).
Let us now describe defect measures that we are going to use. A defect measure is an object describing loss of compactness of a family of functions. Originally, the notion of the defect measure was systematically studied for sequences satisfying elliptic estimates by P.L.Lions [@Liocc]. Since elliptic estimates automatically eliminate oscillations, the defect measures used in [@Liocc] were not appropriate enough for studying loss of compactness caused by oscillations, which typically appear in the case of e.g. hyperbolic problems.
In order to control oscillations, a natural idea was to introduce an object which distinguishes oscillations of different frequencies. The idea was formalised by P. Gerard [@Ger] and independently by L. Tartar [@Tar]. P. Gerard named the appropriate defect measure as the microlocal defect measure (mdm in the sequel), while L. Tartar used the term H-measure. Let us recall Tartar’s theorem introducing the H-measures.
[@Tar] \[H-meas\] If $(\vu_n)=((u_n^1,\dots, u_n^r))$ is a sequence in $\Ld{\R^d;\R^r}$ such that $\vu_n\rightharpoonup 0$ in $\Ld{\R^d;\R^r}$, then there exists its subsequence $(\vu_{n'})$ and a positive definite matrix of complex Radon measures $\mmu=\{\mu^{ij}\}_{i,j=1,\dots,r}$ on $\R^d\times S^{d-1}$ such that for all $\varphi_1,\varphi_2\in
C_0(\R^d)$ and $\psi\in C(S^{d-1})$
$$\label{basic1}
\begin{split}
\lim\limits_{n'\to \infty}\int_{\R^d}&
{\cal A}_\psi(\varphi_1 u^i_{n'})(\mx)\overline{(\varphi_2
u^j_{n'})(\mx)}d\mx
=\langle\mu^{ij},\varphi_1\overline{\varphi_2}\psi
\rangle\\&= \int_{\R^d\times
S^{d-1}}\varphi_1(\mx)\overline{\varphi_2(\mx)}\psi(\mxi)d\mu^{ij}(\mx,\mxi),
\ \ i,j=1,\dots,r,
\end{split}$$
where ${\cal A}_\psi$ is a multiplier operator with symbol $\psi\in C(S^{d-1})$ (see Definition \[multiplier\]).
G' erard’s approach generalises the above results to $\LLd$-sequences taking values in an infinite-dimensional, separable Hilbert space $H$. In the case when $H=\Ld{\R^m}$, G' erard’s mdm is an object belonging to ${\cal M}_+(S^\star \Omega, {\cal L}^1(H))$, i.e. to the space of non-negative Radon measures on the cospherical bundle $S^\star
\Omega$ (the set $\Omega\times S^{d-1}$ endowed with the natural structure of manifold) with values in the space of trace class operators on $H$. It is important to mention an extension of H-measures in the case of sequences which (basically) have the form $({\rm sgn}(\lambda-u_n(\mx)))$, $u_n\in \Lb\Rd$, given by Panov [@pan_ms]. There, it was proved that for almost every $\lambda_1,\lambda_2\in \R$ there exists a measure $\mu^{\lambda_1\lambda_2}$ defined by for $u^{\lambda_i}(\mx)={\rm sgn}(\lambda_i-u_n(\mx))$, $i=1,2$. This notion appeared to be very useful, and it was successfully applied in many recent papers [@AM; @AMP; @HKM; @MI; @Pan_IHP; @pan_arma; @sazh]. Here, we extend Panov’s results to sequences belonging to $\Ld{\R^m; \pL s\Rd}$, $s\geq 2$.
Moreover, our result represents a generalisation of the original H-measures from two aspects. First, test functions (in applications these are given by coefficients entering equations of interest) in our case can be more general, even discontinuous with respect to the space variable. Second, our generalisation of the H-measures is constructed for use on a large class of equations (unlike original H-measures [@Ger; @Tar] which were adapted only for hyperbolic type problems).
In a view of the last observation, remark that parabolic [@Ant1; @Ant2], and ultra-parabolic [@Pan_IHP] variants of the H-measures, and finally the H-measures adapted to large class of manifolds [@MI] were introduced. The last one is the main tool used in this paper. Its description, as well as the introduction to the main result (Theorem \[main-result\]) is given in the next section.
In Section 3 we shall further develop the H-measure concept, which will be used in Section 4 for proving the precompactness property of a sequence of solutions to . The proof is based on a special (trivial) form of the variant H-measure corresponding to the sequence $(u_n)$.
In Section 5 we shall apply our result on ultra-parabolic equations with discontinuous flux under different assumptions on coefficients than the ones from [@pan_jms] (which is the most up-to-date result and which comprises the results from [@pan_arma]).
Statement of the main result
============================
To formulate the main result of the paper, we need to introduce the variant of H-measures that we are going to use. First, we need some auxiliary notions.
\[multiplier\] A multiplier operator ${\cal A}_\psi:\Ld\Rd\to
\Ld\Rd$ associated to a bounded function $\psi\in \CB\Rd$ (see e.g. [@ste]), is a mapping by $${\cal A}_\psi(u)=\bar{\F}(\psi \hat{u}),$$where $\hat{u}(\mxi)=\F(u)(\mxi)=\int_{\R^d}e^{-2\pi i \mx\cdot
\mxi}u(x)dx$ is the Fourier transform while $\bar{\F}$ (or $^\vee$) is the inverse Fourier transform.
If the multiplier operator ${\cal
A}_\psi$ satisfies $$\nor{{\cal A}_\psi (u)}{\LLp p} \leq C \nor{ u}{\LLp p}, \qquad u\in \pL p\Rd\cap\Ld\Rd,$$ where $C$ is a positive constant, then the function $\psi$ is called the $\LLp p$-multiplier.
Let $l$ be a minimal number such that $l \alpha_k > d$ for each $k$. We shall introduce the following manifolds, denoted by $\Pd$ and determined by the order of the derivatives from : $$\label{kin2} \Pd=\{\mxi\in \R^d: \; \sum\limits_{k=1}^d
|\xi_k|^{l\alpha_k}=1 \}.$$ On such manifolds, which are smooth according to the choice of $l$, we shall define the necessary H-measures. Remark that it can seem more natural to take $\Pd=\{\mxi\in
\R^d: \; \sum\limits_{k=1}^d |\xi_k|^{\alpha_k}=1 \}$ but the latter manifold is not smooth enough. Namely, we shall need the following corollary of the Marzinkiewicz multiplier theorem [@ste Theorem IV.6.6’]:
\[m1\] Suppose that $\psi\in \pC{d}{\R^d\backslash \{\vnul\}}$ is such that for some constant $C>0$ it holds $$\label{c-mar} |\mxi^\mbeta \partial^\mbeta \psi(\mxi)|\leq C, \ \ \mxi\in
\R^d\backslash \{\vnul\}$$ for every multi-index $\mbeta=(\beta_1,\dots,\beta_d)\in\Z_+^d$ such that $|\mbeta|=\beta_1+\beta_2+\dots+\beta_d \leq d$. Then, the function $\psi$ is an $\LLp p$-multiplier for $p\in \oi 1\infty$, and the operator norm of ${\cal A}_\psi$ depends only on $C, p$ and $d$.
The next lemma is an easy corollary of Lemma \[m1\]. First, denote by $$\pi_{\Pd}(\mxi)=\left(\frac{\xi_1}{\left(\xi_1^{l\alpha_1}+\dots+\xi_d^{l\alpha_d}
\right)^{1/l
\alpha_1}},\dots,\frac{\xi_d}{\left(\xi_1^{l\alpha_1}+\dots+\xi_d^{l\alpha_d}
\right)^{1/l \alpha_d}}\right), \ \ \mxi\in \R^d\backslash\{0\},$$ a projection of $\R^d\backslash \{\vnul\}$ on $\Pd$. The following result holds.
\[marz\] For any $\psi\in \pC{d}\Pd$, the composition $\psi\circ
\pi_{\Pd}$ is an $\LLp p$-multiplier, $p\in \oi 1\infty$, and the norm of the corresponding multiplier operator depends on $\nor \psi {\pC{d}\Pd},\, p$ and $d$.
Due to the Fa' a di Bruno formula, it is enough to prove that the conditions of Lemma \[m1\] are satisfied for $\pi_k(\mxi)=\frac{\xi_k}{\left(\xi_1^{l\alpha_1}+\dots+\xi_d^{l\alpha_d}
\right)^{1/l \alpha_k}}$, $k=1,\dots,d$.
The statement will be proved by the induction argument.
- $n=1$
In this case, we compute $$\pa_{j}\pi_k(\mxi)=\begin{cases}
-\frac{\alpha_j}{\alpha_k}\frac{1}{\xi_j}\pi_k(\mxi)\pi_j^{l \alpha_j}(\mxi), & j\neq k\\
-\frac{1}{\xi_k} \pi_k(\mxi) \left(1-\pi_k^{l \alpha_k}(\mxi)
\right), & j=k.
\end{cases}$$ and it obviously holds $|\xi_j \pa_{j} \pi_k(\mxi)| \leq C$.
- $n=m$
Our inductive hypothesis is $$\label{ih} \partial^\mbeta
\pi_k(\mxi)=\frac{1}{\mxi^\mbeta}P_\mbeta(\pi_1(\mxi),\dots,\pi_d(\mxi)),
\ \ |\mbeta|=m,$$ for a polynomial $P_\mbeta$.
- $n=m+1$
To prove that holds for $|\mbeta|=m+1$ it is enough to notice that $\mbeta=\ve_j+\mbeta'$, where $|\mbeta'|=m$, and to notice $$\partial^\mbeta\pi_k(\mxi)=\partial_j \partial^{\mbeta'}
\pi_k(\mxi)=\partial_j\left(\frac{1}{\mxi^{\mbeta'}}P_{\mbeta'}(\pi_1(\mxi),\dots,\pi_d(\mxi))\right)$$ and from here, repeating the procedure from the case $n=1$, we conclude that holds for $n=m+1$.
From here, immediately follows for $\pi_k$ and consequently for $\psi\circ\pi_\Pd$.
To proceed, we introduce a family of curves $$\label{kin3} \eta_k=\xi_k t^{1/l \alpha_k }, \ t\in\Rpl, $$ by points $\mxi=(\xi_1,\dots,\xi_d)\in \Pd$. They are disjoint and fibrate entire space $\R^d$. They play the same role as the rays $\mxi/|\mxi|$ in the definition of the H-measures. Moreover, we see that the curves respect the scaling given by the differential operator from . Indeed, if we have the classical situation $\alpha_k=1$, $k=1,\dots,d$, then curves are rays and we can use the classical H-measures [@Ger; @Tar].
The following theorem is essentially proved in [@MI], but here we provide its more elegant proof based on the ideas of L. Tartar.
\[tbasic1\] For fixed $\alpha_k>0$, $k=1,\dots,d$, denote by $\Pd$ the manifold given by , and by $\pi_{\Pd}:\R^d\to
\Pd$ projection on the manifold $\Pd$ along the fibres . If $(\vu_n)=((u_n^1,\dots, u_n^r))$ is a sequence in $\Ld{\R^d;\R^r}$ such that $\vu_n {\buildrel {\rm L}^{2}\over\dscon} \;0$ (weakly), then there exists its subsequence $(u_{n'})$ and a positive definite matrix of complex Radon measures $\mmu=\{\mu^{ij}\}_{i,j=1,\dots,d}$ from ${\cal M}_{b}(\Rd\times \Pd)$ such that for all $\varphi_1,\varphi_2\in \Cnl\Rd$ and ${\psi}\in \Cp\Pd$ $$\label{basic1_new}
\begin{split}
\lim\limits_{n'\to \infty}\int_{\R^d}
&{\cal A}_{\psi_\Pd}(\varphi_1
u^i_{n'})(\mx)\overline{(\varphi_2
u^j_{n'})(\mx)}dx=\langle\mu^{ij},\varphi_1\overline{\varphi_2}\psi
\rangle\\&= \int_{\R^d\times
\Pd}\varphi_1(\mx)\overline{\varphi_2(\mx)}\psi(\mxi)d\mu^{ij}(\mx,\mxi), \
\ (\mx,\mxi)\in \R^d\times \Pd,
\end{split}$$ where ${\cal A}_{{\psi_\Pd}}$ is a multiplier operator with the symbol ${\psi_\Pd}:= \psi\circ \pi_\Pd$.
The measure $\mmu$ we call the ${\rm H}_\Pd$-measure corresponding to the sequence $(\vu_n)$.
First, we shall prove that the fibration satisfies conditions of the variant of the first commutation lemma [@tar_book Lemma 28.2]. More precisely, we shall prove that any symbol $\psi$ on the manifold $\Pd$ satisfies
$$\label{fcl-o}
\begin{split}
&\Svaki{ r, {\varepsilon}\in \Rpl} \ \ \Postoji{M\in \Rpl} \ \ \\
&|\meta_1-\meta_2|\leq r, \; |\meta_1|, |\meta_2| > M \ \ \implies \ \
|\psi(\pi_{\Pd}(\meta_1))-\psi(\pi_{\Pd}(\meta_2))|\leq {\varepsilon},
\end{split}$$
where $\pi_{\Pd}$ is the projection on the manifold $\Pd$ along the fibres .
As $\psi$ is an uniformly continuous on $\Pd$, it is enough to show that for fixed $r$ and ${\varepsilon}$, the difference $|\pi_{\Pd}(\meta_1)-\pi_{\Pd}(\meta_2)|$ is arbitrary small for $M$ large enough. According to the mean value theorem $$|\pi_{\Pd}(\meta_1)-\pi_{\Pd}(\meta_2)|\leq |\nabla\pi_{\Pd}(\mzeta)| |\meta_1-\meta_2|,$$ where $\mzeta = \vartheta \meta_1 + (1- \vartheta) \meta_2$ for some $\vartheta\in \oi 01$, and the statement follows as $\nabla \pi_\Pd (\meta)$ tends to zero when $|\meta|$ approaches infinity.
Now, we can use [@tar_book Lemma 28.2] to conclude that the mappings $$(\varphi_1\overline{\varphi_2},\psi)\mapsto \lim\limits_{n'\to
\infty}\int_{\R^d}{\cal
A}_{\psi_\Pd}(\varphi_1 u^i_{n'})(\mx)\overline{(\varphi_2 u^j_{n'})(\mx)}d\mx, \ \ i,j=1,\dots,d,$$ form a positive definite matrix of bilinear functionals on $\Cnl\Rd\times \Cp\Pd$. According to the Schwartz kernel theorem, the functionals can be extended to a continuous linear functionals on ${\cal D}(\R^d\times \Pd)$. Due to its non-negative definiteness, the Schwartz theorem on non-negative distributions [@Sw Theorem I.V] provides its extension on the Radon measures.
Notice that, using the Plancherel theorem, can be conveniently rewritten via the Fourier transform as follows: $$\begin{split}
\lim\limits_{n'\to \infty}\int_{\R^d}&\F(\varphi_1
u^i_{n'})(\mxi)\overline{\F(\varphi_2 u^j_{n'})(\mxi)}\, {\psi\circ\pi_\Pd}(\mxi)
d\mxi=\langle\mu^{ij},\varphi_1\overline{\varphi_2}\psi \rangle\\&=
\int_{\R^d\times
\Pd}\varphi_1(\mx)\overline{\varphi_2(\mx)}\psi(\mxi)d\mu^{ij}(\mx,\mxi).
\end{split}$$
Now, we can formulate the main theorem of the paper.
\[main-result\] Assume that $u_n\dscon 0$ weakly in $\Ld{\R^m;\pL s\Rd}\cap \Ld{\R^{m+d}}$, $s\geq 2$, where $u_n$ represent weak solutions to in the sense of Definition \[weaksol\].
Furthermore, for $s=2$ we assume that for every $(\mx,\mxi)\in
\R^d\times\Pd$ $$\label{kingnl} A(\mx,\mxi,\msnop):=\sum\limits_{k=1}^d
a_k(\mx,\msnop)(2 \pi i\xi_k)^{\alpha_k}\not= 0 \quad\ae{ \msnop \in \Rm}\,.$$ If $s>2$, the last assumption is reduced to almost every $\mx\in \R^d$ and every $\mxi\in \Pd$.
Then, for any $\rho\in \pLc{2}{\R^m}$, $$\int_{\R^m}u_n(\mx,\msnop)\rho(\msnop)d\msnop
\str 0 \ \ \text{ strongly in $\Ldl\Rd$}.$$
Before we continue, remark that the conditions of the theorem can be relaxed by assuming that $(u_n)$ is merely bounded in $\Ld{\R^m;\pL s\Rd}$, while $(G_n)$ strongly precompact in $\Ld{\R^m;\W{-\malpha}{s'}\Rd}$. In that case there exists a subsequence $(u_{n'})$ such that for any $\rho\in
\pLc{2}{\R^m}$ the sequence $(\int_{\R^m}\rho(\msnop)u_{n'}(\mx,\msnop)d\msnop)$ converges toward $\int_{\R^m}\rho(\msnop)u(\mx,\msnop)d\msnop$, where $u$ denotes the weak limit of $(u_n')$.
Auxiliary results
=================
In this section, we shall extend Theorem \[tbasic1\] on sequences with uncountable indexing. A similar procedure we used in the case of the parabolic variant H-measures [@osijek], and for the sake of completeness we reproduce some results here. These will be substantially extended by Proposition \[prop\_repr\] and Theorem \[lfeb518-r\] containing representation results of ${\rm H}_\Pd$-measures associated to sequences of functions $u_n\in \Ld{\R^m;\pL s\Rd}$, $s>2$, which turn to be crucial for the proof of the main theorem.
Let us take an arbitrary sequence of functions $(u_n)$ in variables $\mx\in \R^d$ and $\msnop\in\R^m$, weakly converging to zero in $\Ld{\R^m
\times \R^d}$. Introduce a regularising kernel $\omega\in
\Cbc{\R^m}$, where $\omega$ is a non-negative smooth function with total mass one. For $k\in\N$ denote $ \omega_k(\msnop)=k^m\omega(k \msnop)$ and convolute it with $(u_n(\mx,\msnop))$ in $\msnop$: $$u_n^k(\mx,\msnop):=\Bigl(u_n(\mx,\cdot)\ast\omega_k\Bigr)(\msnop)=\int_{\R^m}
u_n(\mx,\my)\omega_k(\msnop-\my)d\my.$$ By the Young inequality functions $u_n^k$ are bounded in $\Ld{\R^{m+d}}$ uniformly with respect to both $k$ and $n$. Meanwhile, for every fixed $k$, sequence of functions $u_n^k(\cdot, \msnop)$ is bounded in $\Ld\Rd$, uniformly in $\msnop$, and converges weakly to zero. Furthermore, $u^k_n$ are Lipschitz continuous as functions from $\R^m$ to $\Ld\Rd$, with an $n$-independent Lipschitz constant. Having all this in mind, we can prove the following lemma.
There exists a subsequence $ (u_{n'})$ of the sequence $(u_n)$, and a family $\{\mu^{\msnop{\mq}}_{k}: \msnop,{\mq}\in\R^m\}$ of ${\rm H}_\Pd$-measures on $\R^d\times \Pd$ such that for every $k\in \N$, $\varphi_i\in
C_0(\R^d)$, $i=1,2$, and $\psi\in C(\Pd)$: $$\label{jan2718}
\begin{split}
\lim\limits_{n'}\int\limits_{\Rd}\Bigl({\cal A}_{\psi_\Pd} \,\ph_1
u^{k}_{n'}(\cdot, \msnop)\Bigr)(\mx)
&\,\overline{\ph_2(\mx)u^{k}_{n'}(\mx,\mq )}d\mx\\&=
\!\!\!\!\int\limits_{\Rd\times \Pd}\!\!\!\ph_1(\mx)\overline{\ph_2(\mx)}\psi(\mxi)d\mu^{\msnop\mq}_k(\mx,\mxi).\\
\end{split}$$
According to Theorem \[tbasic1\], for fixed $\msnop,\mq\in \R^m$ and $k\in {\bf N}$, there exist a subsequence of $(u_n)$ and corresponding complex Radon measure $\mu_{k}^{\msnop{\mq}}$ over $\Rd\times \Pd$ such that holds. Using the diagonalisation procedure, we conclude that for a countable dense subset $D\times {D} \subset \R^m\times \R^{m}$ there exists a subsequence $(u_{n'})\subset (u_n)$ such that holds for every $(\msnop,{\mq})\in D\times {D}$ and every $k\in \N$.
Let us take an arbitrary $k\in {\bf N}$ and $(\msnop,{\mq})\in
\R^m\times \R^{m}$. Let $(\msnop_m,{\mq}_m)$ be a sequence in $D\times {D}$ converging to $(\msnop,{\mq})$. The sequence $(\msnop_m,{\mq}_m)$ defines sequence of Radon measures $(\mu_k^{\msnop_m {\mq}_m})$, which is bounded in ${\cal M}_b(\Rd\times \Pd)$, due to the bounds of $(u_n^{k})$ in $\Lb{\R^m; \Ld\Rd}$. Therefore, there exists a complex Radon measure ${\mu}_k^{\msnop\mq}$ such that, along a subsequence, $\mu_k^{\msnop_m {\mq}_m}\rightharpoonup {\mu}_k^{\msnop\mq}$. Thus for arbitrary test functions $\ph=\ph_1 \bar\ph_2$ and $\psi$ we have: $$\label{lema1}
\begin{split}
\int \ph(\mx)\psi(\mxi)\, d\mu_k^{\msnop {\mq}} (\mx, \mxi)
&=\lim_m \int \ph(\mx)\psi(\mxi) \,d\mu_k^{\msnop_m {\mq}_m}(\mx,\mxi)\\
&=\lim_m \lim\limits_{n'} V_{n'}^k (\msnop_m, \mq_m) ,\,
\end{split}$$ where $V_n^k$ denotes the function by $$\label{Vnk} V_n^k (\msnop, \mq):= \int\limits_{\Rd}\Bigl({\cal
A}_{\psi_\Pd} \,\ph_1 u^{k}_{n}(\cdot, \msnop)\Bigr)(\mx)
\,\overline{\ph_2(\mx)u^{k}_{n}(\mx,\mq )} d\mx\,.$$ On the other hand $$\begin{split}
V_{n'}^k (\msnop_m, \mq_m) - V_{n'}^k (\msnop, \mq)
&=V_{n'}^k (\msnop_m, \mq_m) - V_{n'}^k (\msnop, \mq_m) + V_{n'}^k (\msnop, \mq_m) -V_{n'}^k (\msnop, \mq)\\
&\leq C(k)\Bigl(|\msnop_{m}-\msnop|_{\R^{m}}+
|{\mq}_{m}-{\mq}|_{\R^{m}}\Bigr),\,
\end{split}$$ where on the last step we combined the Cauchy-Schwartz inequality, boundedness of the multiplier ${\cal A}_{\psi_\Pd}$ on $\Ld\Rd$, and the Lipschitz continuity of the functions $u_n^k$. The constant $C(k)$ appearing above is independent of $n'$, and we can exchange limits in . This actually means that the functional $\mu^{\msnop \mq}_k$ does not depend on the defining subsequence (i.e. it is well for every $\msnop,\mq\in \R^m$), which completes the proof.
Using the previous assertion, we prove the existence of ${\rm H}_\Pd$-measures associated to functions taking values in $\Ld{\R^m}$. First, we need to recall a few basic notions of $\LLd$ functions taking values in an arbitrary Banach space $E$.
We say that $f: \R^m \to E'$ is weakly $\ast$ measurable if it is measurable with respect to weak $\ast$ $\sigma(E', E)$ topology. The dual of $\Ld{\R^m, E}$ corresponds to the Banach space $\Ldws{\R^{m}; E'}$ of weakly $\ast$ measurable functions $f:
\R^{m}\! \to \!E'$ such that $\int_{\R^{m}} \nor{f(\mx)}{E'}^2 d\mx
<\infty$ (for details see [@Ed p. 606]).
By taking $E=\Cnl{\Rd\times \Pd}$, the topological dual of $\Ld{\R^{2m}; \Cnl{\Rd\times \Pd}}$ corresponds to the Banach space $\Ldws{\R^{2m}; {\cal M}_{b}(\Rd\times \Pd)}$ of weakly $\ast$ measurable functions $\mu: \R^{2m}\! \to \!{\cal
M}_{b}(\Rd\times \Pd)$ such that $\int_{\R^{2m}}
\Nor{\mu(\msnop, \mq)}^2 d\msnop d\mq <\infty$.
\[lfeb518\] For the subsequence $(u_{n'})\subseteq (u_{n})$ extracted in Lemma 8, there exists a measure $\mu \in \Ldws{\R^{2m};
{\cal M}_{b}(\Rd\times \Pd)}$ such that for all $v\in
\Ldc{\R^{2m}}$, $\varphi_i\in C_0(\Rd)$, $i=1,2$, and $\psi\in
C(\Pd)$: $$\label{jan0218}
\begin{split}
\lim\limits_{n'}\int\limits_{\R^{2m}}\int\limits_{\Rd}v(\msnop,{\mq})
&\Bigl({\cal A}_{\psi_\Pd} \,\ph_1 u_{n'}(\cdot, \msnop)\Bigr)(\mx)
\,\overline{\ph_2(\mx) u_{n'}(\mx,\mq )} d\mx d\msnop d\mq\\
&=\int\limits_{\R^{2m}} v(\msnop,{\mq})\,\langle \mu(\msnop,\mq,\cdot,\cdot),\varphi_1\bar{\varphi}_2 \otimes\psi\rangle d\msnop d\mq.\\
\end{split}$$
\[rem1P\] Notice that the new object has inherited the hermitian character of H-measures. Indeed, with the help of Plancherel’s theorem, we can rewrite as $$\begin{split}
\lim\limits_{n'}\int\limits_{\R^{2m}}\int\limits_{\Rd}v(\msnop,{\mq})
&\psi(\mxi) \F(\,\ph_1 u_{n'}(\cdot, \msnop))(\mxi)
\,\overline{\F(\ph_2(\cdot) u_{n'}(\cdot,\mq ))(\mxi)} d\mxi d\msnop d\mq\\
&=\int\limits_{\R^{2m}} v(\msnop,{\mq})\,\langle \mu(\msnop,\mq,\cdot,\cdot),\varphi_1\bar{\varphi}_2 \otimes\psi\rangle d\msnop d\mq\,,\\
\end{split}$$ from which it easily follows that $$\mu(\msnop,\mq,\cdot,\cdot)=\overline{\mu(\mq,\msnop,\cdot,\cdot)}\,.$$ Also, notice that we can take $\varphi_1\in \CB\Rd$ (since $\varphi_2\in C_0(\R^d)$).
For fixed test functions $\varphi_{1,2}$ and $\psi$, similarly to , we denote $$F_k (\msnop, \mq):=\lim\limits_{n'} V_{n'}^k (\msnop,
\mq)=\langle\mu_k^{\msnop\mq},\varphi_1\bar{\varphi}_2\psi \rangle .$$
Due to the uniform bound of $u_n^{k}$ in $\Ld{\R^{m+d}}$, the functions $V_n^k$ belong to the space $\Ld{\R^{2m}}$, with norm depending on $\nor{\varphi_{1,2}}{\LLb}$ and $\nor{\psi}{\LLb}$, but not on $n$ and $k$. Thus the Fatou lemma asserts the sequence $(F_k)$ is bounded in $\Ld{\R^{2m}}$, as well.
Furthermore, for a fixed $k$, the sequence $(V_n^k)$ is bounded in $\Lb{\R^{2m}}$. By taking an arbitrary $v\in \Ldc{\R^{2m}}$, we have $$\label{mar1918}
\begin{split}
\lim\limits_{k}\int\limits_{\R^{2m}} v(\msnop, \mq) F_k(\msnop, \mq)
d\msnop d\mq
&= \lim\limits_{k}\int\limits_{\R^{2m}} v(\msnop, \mq)\lim\limits_{n'} V_{n'}^k (\msnop, \mq) d\msnop d\mq\\
&= \lim\limits_{k} \lim\limits_{n'}\int\limits_{\R^{2m}}v(\msnop,
\mq) V_{n'}^k (\msnop, \mq) d\msnop d\mq\,
\end{split}$$ where on the last step we have used the Lebesgue dominated convergence theorem.
As the functions $u_n^k$ are uniformly bounded in $\Ld{\R^{m+d}}$, the sequence of averaged quantities $\int\limits_{\R^{2m}}v(\msnop, \mq)
V_{n}^k (\msnop, \mq) d\msnop d\mq$ converges to $\int\limits_{\R^{2m}} v(\msnop, \mq) V_n (\msnop, \mq)
d \msnop d\mq$ uniformly with respect to $n$, where $V_n$ is similarly to $V_n^k$, with $u_n^{k}$ replaced by $u_n$ in .
Thus we can exchange the limits in providing $$\label{jan2618} \lim\limits_{k}\int\limits_{\R^{2m}} v(\msnop, \mq)
F_k(\msnop, \mq) d\msnop d\mq
=\lim\limits_{n'}\int\limits_{\R^{2m}}v(\msnop, \mq) V_{n'} (\msnop,
\mq) d\msnop d\mq\,.$$
On the other hand, the boundedness of $(F_k)$ in $\Ld{\R^{2m}}$ enables us to define a bounded sequence of operators $\mu_k\in
\Ldws{\R^{2m}; {\cal M}_{b}(\Rd\times \Pd)}$: $$\mu_k (\msnop, \mq)(\phi): =\Dup{\mu_k^{\msnop\mq}}{\phi}, \quad \phi \in \Cnl{\Rd\times \Pd}.$$
Therefore, there exists a subsequence $(\mu_{k'})\subseteq(\mu_k)$ such that $\mu_{k'}\def\Dscon{\relbar\joinrel\dscon} \povrhsk\ast \mu$ in $\Ldws{\R^{2m}; {\cal
M}_{b}(\Rd\times \Pd)}$. By passing to the limit on the left side of , we get the relation .
\[remark\_3\]
Notice that the last theorem remains valid in the case when the test functions $\varphi_{1,2}$ depend on the velocity variable ($\msnop$ or $\mq$) as well, i.e. when $\varphi_{1,2}$ are taken from the space $\Ldc{\R^{m};\Cnl\Rd}$ (with function $v$ removed from ). As it is enough to prove the statement for test functions from a dense set, we take arbitrary $\varphi_{1,2}\in \Ldc{\R^{m};\Cnl\Rd}$ compactly supported in $\mx$ and approximate them by sums $\sum\limits_{l=1}^{N} v^l_1(\msnop)\varphi_1^l(\mx)$ and $\sum\limits_{j=1}^{N}v_2^j(\mq) \varphi_2^j(\mx)$ such that $$\begin{aligned}
&\|\sum\limits_{l=1}^{N} v^l_1\otimes\varphi_1^l-\varphi_1\|_{\Ld{\R^{m};\Cnl\Rd}}\leq 1/N,\\
&\|\sum\limits_{j=1}^{N}v_2^j\otimes \varphi_2^j -\varphi_2\|_{\Ld{\R^{m};\Cnl\Rd}}\leq 1/N.\end{aligned}$$
Then it holds for any $\psi\in \Cp\Pd$ $$\begin{aligned}
&\Bigg|\int\limits_{\R^{2m}}\langle \mu(\msnop,\mq,\cdot,\cdot),\varphi_1(\cdot,\msnop)\bar{\varphi}_2(\cdot,\mq) \otimes\psi\rangle d\msnop d\mq\\&\qquad
-\lim\limits_{n'}\int\limits_{\R^{2m}}\int\limits_{\R^d}
\Bigl({\cal A}_{\psi_\Pd} \,(\ph_1 u_{n'})(\cdot, \msnop)\Bigr)(\mx)
\,\overline{(\ph_2 u_{n'})}(\mx,\mq ) d\mx d\msnop d\mq\;\Bigg|\\
&\leq \Bigg|\int\limits_{\R^{2m}}\Dupp{ \mu(\msnop,\mq,\cdot,\cdot)}{\varphi_1(\cdot,\msnop)\bar{\varphi}_2(\cdot,\mq) \otimes\psi
-\Bigl(\sum\limits_{l=1}^{N} v^l_1(\msnop)\otimes\varphi_1^l\Bigr) \Bigl(\sum\limits_{j=1}^{N} \overline{v_2^j(\mq)\otimes\varphi_2^j}\Bigr) \otimes\psi } d\msnop d\mq \Big|\\
&+\Bigg|\lim\limits_{n'}\int\limits_{\R^{2m}}\int\limits_{\R^d}
\Biggl(\sum\limits_{l,j}^{N} v^l_1(\msnop) \bar v_2^j(\mq)
\Bigl({\cal A}_{\psi_\Pd} \,\ph_1^l u_{n'}(\cdot, \msnop)\Bigr)(\mx)
\,\overline{(\ph_2^j u_{n'})}(\mx,\mq )
\\&\qquad\qquad\qquad- \Bigl({\cal A}_{\psi_\Pd} \,(\ph_1 u_{n'})(\cdot, \msnop)\Bigr)(\mx)
\,\overline{(\ph_2 u_{n'})}(\mx,\mq )\Biggr) d\mx d\msnop d\mq\Bigg|\\&=\lim\limits_{n'}\Bigg|\int\limits_{\R^{2m}}\int\limits_{\R^d}
\sum\limits_{l,j}^{N}
\Bigl({\cal A}_{\psi_\Pd} \left(( v^l_1\varphi_1^l-\varphi_1) u_{n'}\right)(\cdot, \msnop)\Bigr)\!(\mx)
\overline{\left(( v_2^j\varphi_2^j-\varphi_2)u_{n'}\right)}(\mx,\mq )d\mx d\msnop d\mq\Bigg|\\
&+
\lim\limits_{n'}\Bigg|\int\limits_{\R^{2m}}\int\limits_{\R^d}
\sum\limits_{l}^{N}
\Bigl({\cal A}_{\psi_\Pd} \left(( v^l_1\varphi_1^l-\varphi_1) u_{n'}\right)(\cdot, \msnop)\Bigr)(\mx)
\overline{(\ph_2 u_{n'})}(\mx,\mq )
d\mx d\msnop d\mq \Bigg|\\
&+
\lim\limits_{n'}\Bigg|\int\limits_{\R^{2m}}\int\limits_{\R^d}
\sum\limits_{j}^{N}
\Bigl({\cal A}_{\psi_\Pd} \,\ph_1 u_{n'}(\cdot, \msnop)\Bigr)(\mx)
\overline{\left(( v_2^j\varphi_2^j-\varphi_2)u_{n'}\right)}(\mx,\mq )
d\mx d\msnop d\mq\Bigg|
+{\cal O}(1/N)\\&={\cal O}(1/N),\end{aligned}$$ which proves the remark.
Now, we shall describe the object $\mu$ in Theorem \[lfeb518\] more precisely by showing that it can be represented as $\mu(\msnop, \mq,\cdot)=f(\msnop, \mq,\cdot)\nu$, where $\nu\in {\cal
M}_{b}(\R^d\times \Pd)$ is a positive Radon measure, and $f\in
\Ld{\R^{2m}; \Lj{\Rd\times \Pd:\nu}}$. If we could conclude that for every $\phi\in \Cnl{\R^d\times \Pd}$, the function $\Dup{\mu(\msnop, \mq, \cdot)}\phi$ represents a kernel of a trace class operator, then we could rely on [@Ger Proposition A.1.] to state the latter representation. The most famous sufficient condition for a function to be a kernel of a trace class operator is given by the Mercer theorem. It demands the kernel to be continuous, symmetric and positive definite. The function $\Dup{\mu(\msnop, \mq, \cdot)}\phi)$ has the last two properties, but it is not necessarily continuous. Therefore, we need the following proposition.
\[prop\_repr\] The operator $\mu\in \Ldws{\R^{2m}; {\cal
M}_{b}(\Rd\times \Pd)}$ in Theorem \[lfeb518\] has the form $$\label{repr_1} \mu(\msnop, \mq,\mx,\mxi)=f(\msnop, \mq,\mx,\mxi)\nu(\mx,\mxi),$$ where $\nu\in {\cal M}_b(\R^d\times \Pd)$ is a non-negative scalar Radon measure, while $f$ is a function from $\Ld{\R^{2m}; \Lj{\Rd\times \Pd:\nu}}$ satisfying $$\int_{\R^{2m}}\int_{\R^d\times \Pd}\rho(\msnop)\bar \rho(\mq)\phi(\mx,\mxi)f(\msnop, \mq,\mx,\mxi)d\nu(\mx,\mxi) d\msnop d\mq\geq 0$$ for any $\rho\in \Ldc{\R^m}$, $\phi\in \Cnl{\R^d\times \Pd}$, $\phi\geq 0$.
The proof is based on rewriting the measure $\mu(\msnop,
\mq,\mx,\mxi)$ via the basis in the (Hilbert) space $\Ld{\R^{2m}}$.
Accordingly, let $\{e_i\}_{i\in \N}$ be an orthonormal basis in $\Ld{\R^m}$. Denote by $\mu_{ij}\in {\cal M}_b(\R^d\times \Pd)$ an ${\rm H}_\Pd$-measure generated by the sequences $\int_{\R^m}e_i(\msnop)u_n(\mx,\msnop)d\msnop$ and $\int_{\R^m}e_j(\mq)u_n(\mx,\mq)d\mq$. We claim: $$\label{hs-repr} \mu(\msnop, \mq,\mx,\mxi)=\sum\limits_{i,j=1}^\infty
\mu_{ij}(\mx,\mxi)\bar e_i(\msnop) e_j(\mq).$$ Indeed, take arbitrary $\rho\in \Ld{\R^{2m}}$, $\varphi_{1,2}\in \Cnl{\R^d}$, $\psi\in \Cp\Pd$, and notice that $$\rho(\msnop, \mq)=\sum\limits_{i,j=1}^\infty c_{ij}
e_i(\msnop)\bar e_j(\mq),$$ where $(c_{ij})_{i,j\in \N}$ is a square sumable sequence.
According to the definition of the functional $\mu$, we have $$\begin{aligned}
&\int_{\R^{2m}}\rho(\msnop, \mq)\langle \mu(\msnop, \mq,\cdot),\varphi_1\bar\varphi_2\psi \rangle d\msnop d\mq\\
&=\lim\limits_{n\to \infty}\int_{\R^{2m}}\int_{\R^d} \rho(\msnop, \mq)
\Bigl({\cal A}_{\psi}\,\varphi_1 u_n(\cdot,\msnop)\Bigr)(\mx)
\,\bar\varphi_2 \bar u_n(\mx,\mq) d\mx d\msnop d\mq\\
&=\sum\limits_{i,j=1}^\infty\lim\limits_{n\to \infty}\int_{\R^d} \!c_{ij}
\Biggl({\cal A}_{\psi}\,\varphi_1 \int_{\R^m} u_n(\cdot,\msnop)e_i(\msnop)d\msnop \Biggr)(\mx)\,
\bar \varphi_2 (\mx)\int_{\R^m} \!\bar u_n(\mx,\mq)\bar e_j(\mq)d\mq d\mx\\
&=\sum\limits_{i,j=1}^\infty c_{ij} \langle
\mu_{ij}(\mx,\mxi),\varphi_1\bar\varphi_2\psi\rangle
=\sum\limits_{i,j=1}^\infty \langle
\mu_{ij}(\mx,\mxi),\varphi_1\bar\varphi_2\psi\rangle
\int_{\R^{2m}}\rho(\msnop, \mq)\bar e_i(\msnop) e_j(\mq)d\msnop d\mq,\end{aligned}$$ which completes the proof of . Remark that in the last derivation we have used the square integrability of the sequence $(c_{ij})$ and the Lebesgue dominated convergence theorem.
We introduce a positive bounded measure $$\nu(\mx,\mxi)=\sum\limits_{i=1}^\infty \frac{1}{2^i} \mu_{ii}(\mx,\mxi),$$ as the weighted trace of the measure matrix $(\mu_{ij})_{i,j=1,\infty}$ and claim that $$\mu(\msnop, \mq,\cdot) << \nu,$$for almost every $\msnop, \mq\in \R^m$. Indeed, if $\nu(E)=0$ for some Borel set $E\subset \R^d\times\Pd$, then $\mu_{ii}(E)=0$ for every $i\in \N$. On the other hand, due to the hermitian character of matrix ${\rm H}_\Pd$ measures, we have $$|\mu_{ij}(E)|\leq \mu_{ii}(E)^{1/2}\mu_{jj}(E)^{1/2}.$$ From here, it follows that $\mu_{ij}(E)=0$ for every $i,j\in \N$, and thus, according to , $\mu(\msnop, \mq,\mx,\mxi)(E)=0$ for almost every $\msnop, \mq\in \R^m$.
Now, the conclusion follows from the Radon-Nikodym theorem.
Next, we shall make an extension of Theorem \[lfeb518\].
Notice that if in Theorem \[H-meas\] we assume $u_n \in \pL
s{\Rd}$ for some $s>2$, then the $\R^d$–projection of a corresponding H-measure is absolutely continuous with respect to the Lebesgue measure (see [@Tar Corollary 1.5] and [@pan_arma Remark 2, a)]). Furthermore, in that case we can assume that the test function $\varphi_{1}$ is merely in $\pL
{r}{\R^d}$.
The result generalises to sequences of functions taking values in a function space. More precisely, the following theorem holds.
\[lfeb518-r\] Assume that the sequence $(u_n)=(u_n(\mx,\msnop))$, converges weakly to zero in $\Ld{\R^{m+d}}\cap \Ld{\Rm; \pL s{\Rd}}$, $s>2$. Then the $\R^d$ projection $\int_{\Pd}d\nu(\mx,\mxi)$ of the measure $\nu$ from the last proposition can be extended to a bounded functional on $\pL{r}\Rd$, where $r$ is the dual index of $s/2$. Furthermore, for all $\varphi_1\in \Ld{\Rm; \pL {r}{\Rd}}
$, $\varphi_2\in \Ldc{\R^{m};\Cnl\Rd}$, and $\psi\in \pC d{\Pd}$, it holds: $$\label{jan0218-r}
\begin{split}
\lim\limits_{n'}\int\limits_{\R^{2m}}\int\limits_{\Rd}
&(\ph_1 u_{n'})(\mx,
\msnop)
\,\Bigl(\overline{{\cal A}_{\psi_\Pd} \,\ph_2 u_{n'}(\cdot,
\mq)}\Bigr)(\mx) d\mx d\msnop d\mq\\
&=\int\limits_{\R^{2m}} \langle
\mu(\msnop,\mq,\cdot,\cdot),{\varphi}_1(\cdot,\msnop)\bar{\varphi}_2
(\cdot,\mq)\otimes\bar\psi\rangle d\msnop d\mq.
\end{split}$$
Let $\varphi_1^{\varepsilon}\in \Cc{\R^{m+d}}$, ${\varepsilon}>0$, be a family of continuous functions such that $\|\varphi_1 -\varphi_1^{\varepsilon}\|_{\Ld{\Rm; \pL {r}{\Rd}}}\to 0$ as ${\varepsilon}\to 0$. By means of Remarks \[rem1P\] and \[remark\_3\] we define
$$\begin{aligned}
\label{cshy}
\begin{split}
&\int\limits_{\R^{2m}} \langle
\mu(\msnop,\mq,\cdot,\cdot),\varphi_1\bar{\varphi}_2
\otimes\bar\psi\rangle d\msnop d\mq: =\lim\limits_{{\varepsilon}\to
0}\int\limits_{\R^{2m}} \langle
\mu(\msnop,\mq,\cdot,\cdot),\varphi^{\varepsilon}_1\bar{\varphi}_2
\otimes\bar\psi\rangle d\msnop d\mq\\
&=\lim\limits_{{\varepsilon}\to 0}\lim\limits_{n'\to \infty}\int\limits_{\R^{2m}}\int\limits_{\Rd}
(\ph_1^{\varepsilon}u_{n'})(\mx,
\msnop)
\,\Bigl(\overline{{\cal A}_{\psi_\Pd} \,\ph_2 u_{n'}(\cdot,
\mq)}\Bigr)(\mx) d\mx d\msnop d\mq.
\end{split}\end{aligned}$$
The latter limit exists since for ${\varepsilon}_1,{\varepsilon}_2>0$ it holds
$$\begin{aligned}
\nonumber
&\Bigg|\,\int\limits_{\R^{2m}} \langle
\mu(\msnop,\mq,\cdot,\cdot),(\varphi_1^{{\varepsilon}_1}-\varphi_1^{{\varepsilon}_2})(\cdot,\msnop)\bar{\varphi}_2(\cdot,\mq)
\otimes\bar\psi\rangle d\msnop d\mq\Bigg|\\
&\leq\limsup\limits_{n'} \int\limits_{\R^{2m}}\int\limits_{\Rd}
\Big| (\varphi_1^{{\varepsilon}_1}-\varphi_1^{{\varepsilon}_2} ) u_{n'}(\mx,\msnop)
\,\Bigl(\overline{{\cal A}_{\psi_\Pd} \,\ph_2 u_{n'}(\cdot,
\mq)}\Bigr)(\mx) \Big| d\mx d\msnop d\mq \nonumber\\& \leq
\limsup\limits_{n'} C \int_{\R^{2m}} \|
(\varphi_1^{{\varepsilon}_1}-\varphi_1^{{\varepsilon}_2} ) u_{n'}(\cdot,\msnop)
\|_{\pL {s'}{\Rd}} \|(\varphi_2 u_{n'})(\cdot,\mq)\|_{\pL s{\Rd}}
d\msnop d\mq
\nonumber\\
& \leq \limsup\limits_{n'} C \int_{\R^{m}}
\|(\varphi_1^{{\varepsilon}_1}-\varphi_1^{{\varepsilon}_2})(\cdot,\msnop)\|_{\pL
r{\Rd}} \|( u_{n'}(\cdot,\msnop)\|_{\pL s{\Rd}} d\msnop
\nonumber\\&\hskip26mm \cdot
\int_{\R^{m}}\|\varphi_2 (\cdot,\mq)\|_{\Lb{\Rd}}\| u_{n'}(\cdot,\mq)\|_{\pL s{\Rd}} d\mq
\nonumber\\
& \leq \limsup\limits_{n'} C \,\,
\|(\varphi_1^{{\varepsilon}_1}-\varphi_1^{{\varepsilon}_2})\|_{\Ld{\Rm; \pL r{\Rd}}}
\|\varphi_2 \|_{\Ld{\Rm; \Lb{\Rd}}} \,\| u_{n'}\|_{\Ld{\Rm; \pL
s{\Rd}}}^2 \,, \nonumber\end{aligned}$$
where $C$ depends on $s$, $d$, and $\|\psi\|_{\pC d{\Pd}}$. Since $\|\varphi_1-\varphi_1^{\varepsilon}\|_{\Ld{\Rm; \pL {r}{\Rd}}}\to 0$, the limit in exists.
The same analysis from the above implies $$\lim\limits_{{\varepsilon}\to 0 }\int\limits_{\R^{2m}}\int\limits_{\Rd}
\Big| (\varphi_1^{{\varepsilon}}-\varphi_1 ) u_{n'}(\mx,\msnop)
\,\Bigl(\overline{{\cal A}_{\psi_\Pd} \,\ph_2 u_{n'}(\cdot,
\mq)}\Bigr)(\mx) \Big|
d\mx d\msnop d\mq = 0$$ and the convergence is uniform with respect to $n'$. Thus we can exchange limits in the second line of , which proves .
In order to prove that the $\R^d$ projection $\int_{\Pd}d\nu(\mx,\mxi)$ of the measure $\nu$ belongs to $\pL{r'}\Rd$ take an arbitrary $\varphi\in \Cc\Rd$ and consider $$\begin{aligned}
&\Big|\int_{\R^d} \varphi(\mx)\int_{\Pd}d\nu(\mx,\mxi) \bigg|
=\sum\limits_{i=1}^\infty \Big| \langle \frac{1}{2^i}\mu^{ii},\varphi \otimes 1\rangle \Big|\\
&\leq\sum\limits_{i=1}^\infty\frac{1}{2^i}\lim\limits_{n'\to
\infty}\int_{\R^{2m}}\int_{\Rd}\Big|\varphi(\mx)u_{n'}(\mx,\msnop)
e_i(\msnop)\bar u_{n'}(\mx,\mq)\bar e_i(\mq) \Big|d\mx d\msnop d\mq\\
&\leq
\sum\limits_{i=1}^\infty\frac{1}{2^i}\|\varphi\|_{\pL{r}\Rd}
\limsup\limits_{n'\to \infty}\|u_{n'}\|^2_{\Ld{\R^{2m}; \pL s{\R^d}}}\leq C\|\varphi\|_{\pL{r}\Rd}.\end{aligned}$$ Thus $\int_{\Pd}d\nu(\mx,\mxi)$ can be extended to a bounded functional on $\pL{r}\Rd$, i.e. there exists an $h\in \pL{r'}\Rd$ such that $\int_{\Pd}d\nu(\mx,\mxi) =h(\mx) d\mx$.
The following statement on the measure $\nu$ now follows from results on slicing measures [@Evans Theorem 1.5.1].
\[lemma-slicing\] Under assumptions of the last theorem, for $\hbox{\rm a.e. } \mx\in \Rd$ there exists a Radon probability measure $\nu_\mx$ such that $d\nu(\mx, \mxi)= d \nu_\mx (\mxi) h(\mx) d\mx$, where $h$ is a $\LLp {r'}$ function introduced above. More precisely, for each $\phi\in\Cnl{\Rd\times \Pd}$ $$\int_{\Rd\times \Pd} \phi(\mx, \mxi)d\nu(\mx, \mxi) = \int_\Rd \left( \int_\Pd \phi(\mx, \mxi) d \nu_\mx (\mxi) \right)h(\mx) d\mx\,.$$ The above result is also valid if we take a test function $\phi \in \pL{r}{\Rd; {\Cp \Pd}}$.
Proof of the main theorem
=========================
In this section, we shall prove Theorem \[main-result\]. The proof is based on the special choice of the test function to be applied in , and the ${\rm H}_\Pd$-measures techniques developed in the previous section.
We introduce the multiplier operator ${\cal I}$ with the symbol $\frac{1-\theta(\mxi)}{\left(|\xi_1|^{l\alpha_1}+\dots+|\xi_d|^{l\alpha_d}\right)^{1/l}}$, where $\theta\in\Cbc\Rd$ is a cut-off function, such that $\theta\equiv1$ on a neighbourhood of the origin.
According to Lemma \[marz\], for any $\psi\in \pC d\Pd$, the multiplier operator ${\cal I} \circ {\cal A}_{\psi\circ\pi_{\Pd}}:
\Ld\Rd\cap \pL{s}\Rd \to \W\malpha{s}{\R^d}$ is bounded (with $\LLp {s}$ norm considered on the domain). Indeed, it is enough to notice that the symbol of $\pa^{\alpha}_{x_k}\left({\cal I}\circ {\cal A}_{\psi\circ\pi_{\Pd}}\right)$: $$(\psi\circ\pi)(\mxi)\frac{(1-\theta(\mxi))(2\pi i\xi_k)^{\alpha_k}}{\left(|\xi_1|^{l\alpha_1}+\dots+|\xi_d|^{l\alpha_d}\right)^{1/l}}$$ is a smooth, bounded function that satisfies conditions of Lemma \[m1\].
Insert in (with reintroduced sub-index $n$) the test function $g_n$ given by (a similar procedure was firstly applied in [@sazh]): $$\label{jul0218} g_n(\mx,\msnop)=\rho_1(\msnop)\int_{\Rm}
({\cal I}\circ {\cal A}_{\psi\circ\pi_{\Pd}})\bigl(\ph u_n(\cdot, \mq)\bigr)(\mx)\rho_2(\mq)d\mq,$$ where $\psi\in \pC d\Pd$, $\ph\in \Cbc\Rd$, $\rho_1, \rho_2 \in {\rm
C}_c^{|\mkappa|}(\R^m)$, and $\mkappa$ is the multi-index appearing in . Due to the boundedness properties of the operator ${\cal I}\circ {\cal A}_{\psi\circ\pi_{\Pd}}$ discussed above, the sequence $(g_n)$ is bounded in ${\rm
C}_c^{|\mkappa|}(\R^m) \times \W\malpha s\Rd$.
Letting $n\to \infty$ in , we get after taking into account Theorem \[lfeb518-r\] and the strong convergence of $(G_n)$ $$\begin{split}
\int_{\R^{2m}}\int_{\R^d\times \Pd}
A(\mx,\mxi,\msnop)\overline{\rho_1(\msnop)\rho_2(\mq){\ph}(x)\psi(\mxi)}d\mu(\msnop,\mq,\mx,\mxi)d\msnop d\mq=0,
\end{split}$$ where, let it be repeated, $A(\mx,\mxi,\msnop)=\sum_{k=1}^d (2\pi i\xi_k)^{\alpha_k}
a_k$. As the test functions $\rho_i$, $\ph$, and $\psi$ are taken from dense subsets in appropriate spaces, we conclude $$\label{jul0328} A(\mx,\mxi,\msnop)
d\mu(\msnop,\mq,\mx,\mxi)=0, \quad \ae{\msnop, \mq \in\R^{2m}}\,.$$
For $s=2$ the non-degeneracy condition directly implies that $\mu=0$. In order to show the same result for $s>2$ fix an arbitrary $\delta>0$, and for a $\rho\in
\Ldc{\R^m}$ and $\phi \in \Cnl{\R^d\times \Pd}$ consider the test function $$\frac{\rho(\msnop)\bar\rho(\mq)\phi(\mx,\mxi)\overline{A(\mx,\mxi,\msnop)}}{|A(\mx,\mxi,\msnop)|^2
+\delta}.$$ From , we obtain
$$\int_{\R^{2m}}\int_{\R^d\times
\Pd}\frac{\rho(\msnop)\bar\rho(\mq)\phi(\mx,\mxi)|A(\mx,\mxi,\msnop)|^2}{
|A(\mx,\mxi,\msnop)|^2+\delta} d\mu(\msnop,\mq,\mx,\mxi) d\msnop d\mq=0,$$
which by means of representation and Fubini’s theorem takes the form $$\label{fin_4} \int_{\R^d\times \Pd}\int_{\R^{2m}}
\frac{\rho(\msnop)\bar\rho(\mq)\phi(\mx,\mxi)|A(\mx,\mxi,\msnop)|^2}{ |A(\mx,\mxi,\msnop)|^2
+\delta}f(\msnop,\mq,\mx,\mxi) d\msnop d\mq d\nu(\mx,\mxi)=0.$$ Let us denote $$I_\delta(\mx, \mxi)= \int_{\R^{2m}}
\rho(\msnop)\bar\rho(\mq)\frac{|A(\mx,\mxi,\msnop)|^2}{ |A(\mx,\mxi,\msnop)|^2
+\delta}f(\msnop,\mq,\mx,\mxi)d\msnop d\mq\,.$$ According to the non-degeneracy condition and the representation of the measure $\nu$ given in Lemma \[lemma-slicing\], for $s>2$ we have $$I_\delta(\mx, \mxi)\to \int_{\R^{2m}}
\rho(\msnop)\bar\rho(\mq)f(\msnop,\mq,\mx,\mxi)d\msnop d\mq,$$ as $\delta\to 0$ for $\nu- {\rm a.e. }\, (\mx, \mxi) \in \R^d\times \Pd$. By using the Lebesgue dominated convergence theorem, it follows from after letting $\delta\to 0$: $$\begin{aligned}
&\int_{\R^d\times
\Pd}\int_{\R^{2m}}\rho(\msnop)\bar\rho(\mq)\phi(\mx,\mxi)f(\msnop,\mq,\mx,\mxi) d\msnop d\mq d\nu(\mx,\mxi)
\\
&=\int_{\R^{2m}}\rho(\msnop)\bar\rho(\mq)\,\langle\mu(\msnop,\mq,\cdot,\cdot), \phi\rangle d\msnop d\mq=0. \nonumber\end{aligned}$$ Having in mind the definition of the measure $\mu$ from Theorem \[lfeb518\], by putting here $\phi(\mx,\mxi)=|\varphi(\mx)|^2$ for $\varphi\in \Cnl{\R^d}$, we immediately obtain $$\lim\limits_{n'\to
\infty}\int_{\R^d}\left|\int_{\Rm}\rho(\msnop)u_{n'}(\mx,\msnop)d\msnop \right|^2
|\varphi(\mx)|^2d\mx =0.$$ Due to arbitrariness of $\varphi$, this concludes the proof. $\Box$
We conclude the section by remarking that our results easily extend to equations containing mixed derivatives with respect to the space variables (see also [@Ger Theorem 2.1]):
$$\label{general}
\begin{split}
{\cal P}u_n(\mx,\msnop)&=\sum\limits_{s\in I}\partial^{\malpha_s}_\mx
\left(a_s(\mx,\msnop) u_n(\mx,\msnop)\right)=\pa^\mkappa_\msnop
G_n(\mx,\msnop),
\end{split}$$
where $I$ is a finite set of indices, and $\partial_\mx^{\malpha_s}=\partial_{x_1}^{\alpha_{1s}}\dots \partial_{x_d}^{\alpha_{ds}}$, for a multi-index $\malpha_s=(\alpha_{s1},\dots,\alpha_{sd})\in
\R^d$.
Denote by $A$ the principal symbol of the (pseudo-)differential operator ${\cal P}$, which is of the form $$A(\mxi,\mx,\msnop)=\sum\limits_{s\in I'} (2\pi i\mxi)^{\malpha_s}a_s(\mx,\msnop),$$ where the upper sum goes above all terms from whose order of derivative $\malpha_s$ is not dominated by any other multiindex from $I$.
For $A$ we must additionally assume that there exist $\alpha_1, \cdots, \alpha_d\in \Rpl$ such that for any positive $\lambda\in \R$, it holds $$A(\lambda^{1/\alpha_1}\xi_1,\dots,\lambda^{1/\alpha_d}\xi_d,\mx,\msnop)=
\lambda A(\mxi,\mx,\msnop),$$ and that it satisfies genuine non-degeneracy condition: for almost every $x\in \R^d$, every $\xi\in \Pd$, it holds $$A(\mx,\mxi,\msnop)\not= 0 \quad\ae{ \msnop \in \Rm}\,;$$
The proof of Theorem \[main-result\] for equation of form goes along the same lines as for the equation .
Let us finally remark that in the case when derivative orders $\alpha_k$, $k=1,\dots,d$, are non-negative integers, we can assume that the sequence $u_n$ is only locally bounded in $\Ldl{\R^m; \Ll s\Rd}$.
In that case we simply take
$$g_n(\mx,\msnop)=\rho_1(\msnop)\ph(\mx) \int_{\Rm} ({\cal I}\circ
{\cal A}_{\psi\circ\pi_{\Pd}})\bigl(\ph u_n(\cdot,
\mq)\bigr)(\mx)\rho_2(\mq)d\mq,$$
instead of $g_n$ from . By repeating the rest of the procedure from this section, we conclude that the measure $\mu$ from Theorem \[lfeb518\] corresponding to $(\ph u_n)$ equals zero. Due to arbitrariness of $\ph$, we conclude that for any $\rho\in \Ldc{\R^m}$, the sequence $(\int u_n(\mx,\msnop) \rho(\msnop)d\msnop)$ is strongly precompact in $\Ldl\Rd$.
Ultra-parabolic equation with discontinuous coefficients
========================================================
In this section, we consider an ultra-parabolic equation with discontinuous coefficients in a domain $\Omega$ (an open subset of $\Rd$). Ultraparabolic equations (with regular coefficients) were first considered by Graetz [@Gr] and Nusselt [@Nus] in their investigations concerning the heat transfer. A specific situation modelled by such equations is the one when diffusion can be neglected in the directions $x_{l+1}, \dots, x_d$, $l\geq 0$. Recently, such equations were investigated in [@pan_jms] and we aim to extend results from there.
More precisely, the equation that we are going to consider here has the form $$\label{scldf} \dv \vf(\mx,u)- \dv \dv \mB(\mx,u) +\psi(\mx,u)=0,$$ where $\mB(x,u)=(b_{jk})_{j,k=1,\dots,d}$ is a symmetric matrix such that for some $l<d$ it holds $(b_{jk})\equiv 0$ for $\min(j, k)\leq l$, while $\tilde{\mB}=(b_{jk})_{j,k=l+1,\dots,d}$ satisfies an ellipticity condition on $\R^{d-l}$ in the following sense: for every $\tilde{\mxi}\in \R^{d-l}$, $\lambda_1,\lambda_2\in \R$ and $\mx\in \Omega$, $$\left(\lambda_1>\lambda_2\right) \povlaci
(\tilde{\mB}(\mx,\lambda_1)-\tilde{\mB}(\mx,\lambda_2))\tilde{\mxi} \cdot \tilde{\mxi}\geq c |\tilde{\mxi}|^2, \ \ c>0.$$
Accordingly, we shall use anisotropic spaces like $\W{({\sf 1,
2})}q\Omega$, where $({\sf 1, 2})\in \Rd$ is a multiindex with first $l$ components equal to 1.
Furthermore, we assume that $\psi\in \Lj{\Omega;\Lb\R}$, while $\vf=(f_1,\dots,f_d)$ and $\mB$ are such that for every $j, k=1,\dots,d$ $$\partial_\lambda f_k, \partial_\lambda b_{jk} \in \Ldl{\R;
{\Ll r\Omega}}, \ \ r>1.$$
We also need to assume a kind of uniform continuity of $\vf$ and $\mB$ in the sense that there exists an increasing function $w$ on $\R^+$, vanishing and continuous at $0$ (i.e. $w$ is a modulus of continuity type function), and $\sigma\in \Ll {1+{\varepsilon}}\Omega, \,{\varepsilon}>0$ such that $$\label{uc} |\vf(x,\lambda_1)-\vf(x,\lambda_2)|,
|\mB(x,\lambda_1)-\mB(x,\lambda_2)| \leq
w\left(\big|\lambda_1-\lambda_2)\big|\right)\big|\sigma(x)\big|.$$
Concerning regularity with respect to $\mx\in \R^d$ of the functions $\vf$ and $\mB$, we assume that for every $\lambda\in \R$ $$ \dv \vf(\mx,\lambda)-\dv \dv \mB(\mx,\lambda)=\gamma(\mx,\lambda)\in
{\cal M}(\R^d).$$
To proceed, denote $\gamma(\mx,\lambda)=\omega(\mx,\lambda)d\mx+\gamma^s(\mx,\lambda)$ where $\omega(\mx,\lambda)d\mx$ denotes the regular, and $\gamma^s(\mx,\lambda)$ denotes the singular part of the measure $\gamma$ with respect to the Lebesgue measure. The following definition is used in [@pan_jms].
\[eac\] We say that a function $u\in \Lb\Omega$ represents an entropy admissible weak solution to if for every $\lambda \in \R$ it holds $$\begin{aligned}
\label{p5}
&\dv \Bigl({\rm sgn}(u(\mx) - \lambda)\bigl(\vf(\mx, u(\mx)) - \vf(\mx, \lambda)\bigr)\Bigr) \\&-
\dv \dv \Bigl({\rm sgn}(u(\mx) - \lambda)\bigl(\mB(\mx, u(\mx)) - \mB(\mx, \lambda)\bigr)\Bigr) \nonumber\\
&+
{\rm sgn}(u(\mx)- \lambda)\bigl(\omega(\mx,\lambda) + \psi(\mx, u(\mx))\bigr) - |\gamma(\mx,\lambda)| \leq 0
\nonumber\end{aligned}$$ in the sense of distributions on $\R^d$.
We shall prove a result similar to those from [@pan_jms; @pan_arma], stating the assumptions under which a sequence of entropy solutions is strongly precompact in $\Ldl\Omega$. There it is assumed that $\max\limits_{\lambda\in
\oi {-M}M}|\vf (\cdot,\lambda)|, \max\limits_{\lambda\in
\oi {-M}M}|\mB (\cdot,\lambda)|\in \Ldl\Omega$, for $M=\limsup_n \nor{u_n}{\Lb\Omega}$, while we demand $ \partial_\lambda \vf, \partial_\lambda \mB \in \Ldl{\R;
{\Ll r\Omega}}$ for an $r>1$. Remark that we have increased regularity with respect to $\lambda\in \R$ (there the continuity is merely assumed), but we have decreased it with respect to $\mx\in \Omega$. However, in the case $r\geq 2$ the statement of the next theorem also follows from the more general results of [@pan_jms].
Assume that the coefficients of equation satisfy the genuine nonlinearity conditions analogical to :
- for every $\mxi=(\hat{\mxi},\tilde{\mxi})\in
\Pd=\{(\hat{\mxi},\tilde{\mxi})\in \R^{l}\times \R^{d-l}:\, |\hat{\mxi}|^2+|\tilde{\mxi}|^4=1 \}$ and almost every $x\in \R^d$ $$\label{pgnl} 2\pi i\sum\limits_{k=1}^l \xi_k \pa_\lambda
f_k(\mx,\lambda)+4 \pi^2 \langle \pa_\lambda\mB(\mx,\lambda)\mxi,\mxi
\rangle
\not=0 \quad\ae{\lambda\in \R}\,.$$
Then, a sequence of entropy solutions $(u_n)$ to such that $\nor{u_n}{\Lb\Omega}<M$ for every $n\in \N$ is strongly precompact in $\Ldl\Omega$.
To prove the theorem, remark first that, according to the Schwartz theorem [@Sw Theorem I.V], for every $\lambda\in \R$ we can rewrite as
$$\begin{aligned}
\label{p6}
&\dv \Bigl({\rm sgn}(u_n(\mx) - \lambda)\big(\vf(\mx, u_n(\mx)) - \vf(\mx, \lambda)\big)\Bigr) \\&-
\dv \dv \Bigl({\rm sgn}(u_n(\mx) - \lambda)\big(\mB(\mx, u_n(\mx)) - \mB(\mx, \lambda)\big)\Bigr) \nonumber\\
&= G_n(\mx,\lambda)
\nonumber\end{aligned}$$
where $G_n(\cdot,\lambda)\in {\cal M}(\Omega
)$ are Radon measure on $\Omega$, locally uniformly bounded with respect to $n$. According to [@Evans Theorem 1.6] (see also [@pan_jms Proposition 7]), the sequence of measures $(G_n(\cdot,\lambda))$ is strongly precompact in $\Wl{({\sf -1, -2})}q\Omega$ for each $q\in \oi 1{\frac{d}{d-1}}$. Furthermore, for every $\varphi\in
C_c^{{\sf 1,2}}(\Omega)$, we have according to $$\begin{aligned}
\label{1/2}
&|\langle
G_n(\cdot,\lambda_1)-G_n(\cdot,\lambda_2), \varphi \rangle|
\\&
=\int_{\Omega}\Big|\Big({\rm sgn}(u_n\!-\!\lambda_1)\big(\mB(\cdot,u_n)\!-\! \mB(\cdot,\lambda_1)\big)\!-\!{\rm sgn}(u_n-\lambda_2)\big(\mB(\cdot,u_n)\!-\! \mB(\cdot,\lambda_2)\bigr) \Bigr)\cdot(\nabla\otimes\nabla) \varphi \Big| d\mx
\nonumber\\&
\qquad +\int_{\Omega}\Big|\Big({\rm sgn}(u_n\!-\!\lambda_1)\big(\vf(\cdot,u_n)\!-\! \vf(\cdot,\lambda_1)\big)\!-\!{\rm sgn}(u_n\!-\!\lambda_2)\big(\vf(\cdot,u_n)\!-\!\vf(\cdot,\lambda_2)\big)\Bigr)\cdot\nabla \varphi \Big|d\mx
\nonumber
\\& \leq
C w\left(\big|\lambda_1-\lambda_2)\big|\right) \nor\varphi{\WW{({\sf 1, 2})}{q'}},
\nonumber\end{aligned}$$ for a constant $C$ independent of $n$ (it depends only on $f$, $\mB$, and $\sigma$). Indeed, according to it holds $$\begin{aligned}
&\big|{\rm sgn}(u-\lambda_1)(\vf(\mx, u)- \vf(\mx, \lambda_1)-{\rm sgn}(u-\lambda_2)(\vf(\mx, u)- \vf(\mx, \lambda_2) \big| \quad \\
&\leq \begin{cases}
\aps{ \vf(\mx, \lambda_1)-\vf(\mx, \lambda_2)}, &(u-\lambda_1)(u-\lambda_2)\geq 0\\
\aps{ \vf(\mx, u)-\vf(\mx, \lambda_1)} + \Apslr{\vf(\mx, u)-\vf(\mx, \lambda_2) }, &(u-\lambda_1)(u-\lambda_2)\leq 0
\end{cases} \\
&\leq\;2 w\left(\big|\lambda_1-\lambda_2)\big|\right) \big|\sigma(\mx)\big|,\end{aligned}$$ and similarly for $\vf$ replaced by $\mB$, from where immediately follows.
Take now a countable dense subset $D$ of $\R$ and for every $\lambda_m\in D$ denote by $G(\cdot,\lambda_m)\in {\cal M}(\Omega)$ such that $G_n(\cdot,\lambda_m)\dstr G(\cdot,\lambda_m)$ strongly in $\Wl{({\sf -1, -2})}q\Omega$ along a subsequence. Since $D$ is countable, we can choose the same subsequence (which we denote the same as the original one) for every $\lambda_m\in D$. Now, we extend $G(\cdot,\lambda)$, $\lambda\in D$, by continuity on entire $\R$: for every $\lambda\in \R$, we choose a sequence $(\lambda_m)$ from $D$ converging to $\lambda$ and define for every $\varphi\in C_c^{{\sf 1,2}}(\Omega)$: $$\label{wd}
\langle G(\cdot,\lambda), \varphi \rangle:=\lim\limits_{m\to \infty} \langle G(\cdot,\lambda_m), \varphi \rangle.$$ The latter is well defined since for any $\lambda_{1},\lambda_{2}\in D$ and any ${\varepsilon}>0$ one can find an $n>0$ such that $$\begin{aligned}
&\Big| \langle G(\cdot,\lambda_{1})-G(\cdot,\lambda_{2}), \varphi \rangle \Big| \leq \\&
\Big| \langle G_n(\cdot,\lambda_{1})-G(\cdot,\lambda_{1}), \varphi\rangle \Big|+
\Big| \langle G_n(\cdot, \lambda_{1})-G_n(\cdot,\lambda_{2}), \varphi \rangle \Big|
+\\& | \langle G_n(\cdot,\lambda_{2})-G(\cdot,\lambda_{2}), \varphi \rangle \Big|\leq
\Big({\varepsilon}+C w\left(\big|\lambda_1-\lambda_2)\big|\right)+{\varepsilon}\Big)\nor\varphi{\WW{({\sf 1, 2})}{q'}} .\end{aligned}$$ From here, the Cauchy criterion will provide properness of . Furthermore, since $G(\cdot,\lambda_m)$ are Radon measures, the functional $G(\cdot, \lambda)$ is also a Radon measure.
Using the same arguments, it is not difficult to prove that for every $\lambda \in \R$, $$G_n(\cdot,\lambda)\to G(\cdot,\lambda) \ \ {\rm in} \ \
\Wl{({\sf -1, -2})}q\Omega.$$
According to the Lebesgue dominated convergence theorem, we conclude from the latter that $$\label{uvjet51} G_n\to G \ \ {\rm in} \ \
\Ldl{\R;\Wl{({\sf -1, -2})}q\Omega}.$$
By finding derivative of with respect to $\lambda$, we reach to (the kinetic formulation of ; see [@PC])
$$\begin{aligned}
&\dv \Bigl(h_n(\mx,\lambda)\partial_\lambda\vf(\mx,\lambda)\Bigr) -
\dv \dv (h_n(\mx,\lambda) \partial_\lambda\mB(\mx, \lambda)) = -\pa_\lambda
G_n(\mx,\lambda)\end{aligned}$$
where $h_n(\mx,\lambda)={\rm sgn}(u_n(\mx)-\lambda)$, and this is the special case of equation . From here, we see that, due to Remark 16, the convergence , and the genuine nonlinearity conditions , the sequence $(\varphi h_n)$ satisfies conditions of Theorem \[main-result\] (see also Remark 15). Thus it follows that $(\int_{-M}^{M} h_n(\mx,\lambda)d\lambda)$ is strongly precompact in $\Ldl\Omega$. Since $$\int_{-M}^{M} h_n(\mx,\lambda)d\lambda=2
u_n(\mx),$$ we conclude that $(u_n)$ is strongly $\Ldl\Omega$ precompact itself.
[**Acknowledgement**]{} Darko Mitrović is engaged as a part time researcher at the University of Bergen in the frame of the project “Mathematical and Numerical Modeling over Multiple Scales” of the Research Council of Norway whose support we gratefully acknowledge.
The work presented in this paper was also supported in part by the Ministry of Science, Education and Sports of the Republic of Croatia (project 037-0372787-2795), as well as by the DAAD project [*Center of Excellence for Applications of Mathematics*]{}.
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|
---
abstract: 'The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $\Lambda$, which equals the bias voltage $V$, if $V$ is the largest energy scale compared to temperature $T$ and inverse wire length $L^{-1}$, and interpolates between these quantities in the crossover regimes.'
author:
- 'D.N. Aristov'
- 'P. Wölfle'
title: 'Transport properties of a two-lead Luttinger liquid junction out of equilibrium: fermionic representation'
---
Introduction \[sec:Intro\]
==========================
In the past few years, exactly one-dimensional quantum wires have become available for experimental investigation in the form of carbon nanotubes, chains of metal atoms or weakly side-coupled molecular chains in solids. The new data emerging from these experiments [@Prokudina2014; @Mebrahtu2013; @Jezouin2013], in particular in non-equilibrium situations, require a more detailed and more general theoretical description than presently available. Electron transport in nanowires has been studied theoretically for more than two decades. In the first papers it was found that electron-electron interaction affects even the conductance of a clean wire [@Apel1982; @Kane1992]. In the case of realistic boundary conditions, namely adiabatically attaching ideal leads to the interacting quantum wire, the two-point conductance of a clean wire is that of the leads, equal to one conductance quantum per channel, irrespective of the (forward scattering) interaction [@Maslov1995]. The work of Kane and Fisher [@Kane1992] and Furusaki and Nagaosa [@Furusaki1993] showed, that interaction has a dramatic effect on the conductance in the presence of a potential barrier. Namely, for repulsive interaction these authors found that the conductance tends to zero as the temperature $T$, or more generally, the excitation energy of the electrons approaches zero, while for attractive interaction the conductance scales to its maximum value. This behavior has been shown to carry over to the dependence on bias voltage, at sufficiently low temperatures (and for long wires). There exists a large body of theoretical work addressing different aspects of transport through Luttinger liquids without or with impurities, such as the effect of the leads on a finite length wire, [@Safi1995; @Furusaki1996] the response to an a.c. electric field [@Sassetti1996], the appearance of oscillatory behavior in the nonlinear conductance [@Dolcini2003; @Dolcini2005] and the emergence of bistability for the very strong interaction and bias voltage [@Egger2000]. The transport through Luttinger liquid junctions at not too strong interaction has also been calculated using the Functional Renormalization Group method as reviewed in [@Metzner2012]. The results mentioned above have been mostly obtained within the bosonization method, which needs to be amended by a correction taking care of the physical situation of a wire of finite length attached to reservoirs (see above). Experimentally, the predictions of theory have been found to be observed, at least qualitatively [@Bockrath1997; @Milliken1996; @Yacoby1996; @Tarucha1995; @Tans1997].
A proper treatment of the two-point conductance in the limit of weak interaction, taking into account the gradual build-up of the Friedel oscillations around the barrier as the infrared cutoff is lowered has been given by Yue, Glazman and Matveev [@Yue1994]. These authors used the perturbative RG for fermions to derive the conductance for an arbitrary (but short) potential barrier (“fermionic representation”). In this paper we extend the approach of Yue et al. to transport under stationary non-equilibrium conditions. Following our extensive work on transport in the linear regime through junctions of Luttinger liquids at arbitrary strength of interaction [@Aristov2008; @Aristov2009; @Aristov2010; @Aristov2011; @Aristov2011a; @Aristov2012; @Aristov2012a; @Aristov2013] we derive in the following RG-equations for the conductance at finite bias voltage and for any interaction strength. We use the fact that the $\beta$- function of the RG-equation for the conductance can be obtained in very good approximation by summing a class of contributions in perturbation theory in all orders of the interaction [@Aristov2009]. A comparison of our previous results on the linear response conductances of two- [@Aristov2009] and three-lead junctions [@Aristov2010; @Aristov2011a; @Aristov2013] with or without additional symmetries, or an applied magnetic flux [Aristov2012a,Aristov2013]{}, with the results of the bosonization method, of conformal field theory methods, of Bethe ansatz, where available, are in full agreement provided those results were well-founded. In a few cases where the conformal field theory result was based on an additional assumption we found disagreement, which we interpret as saying that the assumption was not justified.
In this paper we consider the transport of spinless fermions, which begs the question of how our results may be applied to experiment. The spinless Luttinger liquid model has actually been used to describe transport through spin-polarized quantum wires, as considered in Ref. [@Mebrahtu2013] A generalization of our theory to spinful fermions is in progress.
The model \[sec:Model\]
========================
We consider a system of spin-less fermions in one dimension, interacting in the region $a<|x|<L$ (the “wire”), adiabatically connected to reservoirs at $%
|x|>L$. There is a barrier in the narrow regime $|x|<a$ , which scatters the fermions as described by the $S$-matrix (up to overall phase factors in the individual wires) $$S=%
\begin{pmatrix}
r & t \\
\widetilde{t} & r%
\end{pmatrix}%
=%
\begin{pmatrix}
\sin \theta & i\cos \theta e^{-i\varphi } \\
i\cos \theta e^{i\varphi } & \sin \theta
\end{pmatrix}%$$We choose this parametrization in terms of the transmission and reflection amplitudes $t,r$ , since it is readily generalizable to the case of multi-wire junctions ($n$ wires, $n>2$ ). The above form of the $S$-matrix is completely general.
In the continuum limit, linearizing the spectrum at the Fermi energy and adding forward scattering interaction of strength $g_{j}$ in wire $j$ , we may write the Hamiltonian $\mathcal{H}$ in the representation of incoming and outgoing waves as $$\begin{aligned}
\mathcal{H} &=\int_{0}^{\infty
}dx\sum_{j=1}^{2}[H_{j}^{0}+H_{j}^{int}\Theta (a<x<L)]\,, \\
H_{j}^{0} &=v_{j}\psi _{j,in}^{\dagger }i\nabla \psi _{j,in}-v_{j}\psi
_{j,out}^{\dagger }i\nabla \psi _{j,out}\,, \\
H_{j}^{int} &=2\pi v_{j}g_{j}\psi _{j,in}^{\dagger }\psi _{j,in}\psi
_{j,out}^{\dagger }\psi _{j,out}\,.
\end{aligned}$$We are using the chiral representation, labeling electrons in lead $j$ by $%
(j,\eta )\equiv j_{\eta }$ where $\eta =+1$ for outgoing and $\eta =-1$for incoming electrons and all position arguments $x$ are on the positive semi-axis. The range of the interaction lies within the interval $(a,L)$, where $a>0$ serves as an ultraviolet cutoff (at energy scale $v_{j}/a$) and separates the domains of interaction and potential scattering on the junction; non-interacting leads are attached adiabatically at large $x$ beyond $L$. In terms of the doublet of incoming fermions $\Psi _{-}=(\psi
_{1,-},\psi _{2,-})$ the outgoing fermion operators may be expressed with the aid of the $S$-matrix as $\Psi _{+}(x)=S\cdot \Psi _{-}(x)$ . For later use we define density operators $\widehat{\rho }_{j,\eta =-1}=\psi
_{j,-}^{\dagger }\psi _{j,-}=\Psi _{-}^{+}\rho _{j}\Psi _{-}$, and $\widehat{%
\rho }_{j,\eta =1}=\psi _{j,+}^{\dagger }\psi _{j,+}=\Psi _{-}^{+}\widetilde{%
\rho }_{j}\Psi _{-}$, where $\widetilde{\rho }_{j}=S^{+}\cdot
\rho _{j}\cdot S$. The $2\times 2$ matrices are defined by $%
(\rho _{j})_{\alpha \beta }=\delta _{\alpha \beta }\delta _{\alpha j}$ and $(%
\widetilde{\rho }_{j})_{\alpha \beta }=S_{\alpha j}^{+}S_{j\beta }$.
Charge current of free fermions \[sec:FreeCurrent\]
===================================================
The net current flowing outward through the point $z$ in wire $j$ is composed out of two chiral components, moving towards ($\eta =-1$) and from ($\eta =1$) the junction, $$J_{j}(z)=v_{j}\Big(\langle \widehat{\rho }_{j,+}(z)\rangle -\langle \widehat{%
\rho }_{j,-}(z)\rangle \Big)$$where $v_{j}$ is the group velocity of the fermions. We use units where electrical charge $e=1$, also $ \hbar =1$ and Boltzmann’s constant $k_{B}=1.
$
We work with the Green’s functions in this chiral basis and in Keldysh formulation (we denote matrices in Keldysh space by an underbar), $$\underline{G}=%
\begin{pmatrix}
G^{R} & G^{K} \\
0 & G^{A}%
\end{pmatrix}%$$
Here retarded, advanced and Keldysh components of the Green’s functions, in matrix form in the chiral basis are given by ($2\times 2$ matrices in the chiral basis are denoted by a hat, $\widehat{G}_{\eta _{l}\eta
_{j}}(l,y|j,x)=G(l,\eta _{l},y|j,\eta _{j},x)$) $$\begin{aligned}
\widehat{G}_{\omega }^{R}(l,y|j,x) &=-\frac{i}{\sqrt{v_{l}v_{j}}}\theta
(\tau )e^{i\omega \tau }%
\begin{bmatrix}
\delta _{lj} & 0 \\
S_{lj} & \delta _{lj}%
\end{bmatrix}
\label{eq:Green} \\
\widehat{G}_{\omega }^{A}(l,y|j,x) &=\frac{i}{\sqrt{v_{l}v_{j}}}\theta
(-\tau )e^{i\omega \tau }%
\begin{bmatrix}
\delta _{lj} & S_{jl}^{\ast } \\
0 & \delta _{lj}%
\end{bmatrix}
\\
\widehat{G}_{\omega }^{K}(l,y|j,x) &=-\frac{i}{\sqrt{v_{l}v_{j}}}e^{i\omega
\tau }%
\begin{bmatrix}
\delta _{lj}h_{l} & S_{jl}^{\ast }h_{l} \\
S_{lj}h_{j} & S_{jm}^{\ast }S_{lm}h_{m}%
\end{bmatrix}
\\
\tau &= \eta_{l} y/v_{l}- \eta_{j} x/v_{j}
\end{aligned}$$where $h_{j}(\omega )=\tanh [(\omega -\mu _{j})/2T]$ is the equilibrium distribution function in the reservoir of lead $j$ , characterized by the chemical potential $\mu _{j}$ . We shall assume the temperatures $T$ in the leads to be equal.
The average density of the chiral current at point $z$ , $\langle \rho
_{j,\eta }(z)\rangle $ , is represented by the diagram in Fig. \[fig:tadpole\].
![ The diagram showing the zeroth order contribution to the current. \[fig:tadpole\]](tadpole){width=".5\columnwidth"}
In terms of the Green’s function matrix and defining the external vertex by the Keldysh matrix $\underline{\gamma }_{ext}$
$$\underline{\gamma }_{ext}=\frac{i}{2}%
\begin{pmatrix}
1 & 1 \\
-1 & -1%
\end{pmatrix}%$$
we have
$$\langle \rho _{j,\eta }(z)\rangle =\int \frac{d\Omega }{2\pi } \mbox{Tr}_{K}\left[
\underline{\gamma }_{ext}\cdot \underline{G}_{\Omega }(j,\eta ,z|j,\eta ,z)%
\right]$$
with the trace $\mbox{Tr}_{K}$ taken over the Keldysh indices.
Using the expressions (\[eq:Green\]), we obtain $$\begin{aligned}
v_{j}\langle \rho _{j,-}(z)\rangle &=\tfrac{1}{2}\int \frac{d\Omega }{2\pi
}\,(1-h_{j}(\Omega )), \\
v_{j}\langle \rho _{j,+}(z)\rangle &=\tfrac{1}{2}\int \frac{d\Omega }{2\pi
}\,(1-\sum_{m}|S_{jm}|^{2}h_{m}(\Omega ))
\end{aligned}$$Notice here that the incoming current in the $j$th wire is characterized by the distribution function referring to the same wire. The outgoing current in the $j$th wire is characterized by the distribution functions referring to the remaining wires. The dependence on $z$ vanishes in the d.c. limit considered here.
Using the unitarity property (i.e. charge conservation), $%
\sum_{m}|S_{jm}|^{2}=1$, we may represent the net current in the form $$J_{j}^{(0)}(z)=\tfrac{1}{2}\int \frac{d\Omega }{2\pi }\,%
\sum_{m}|S_{jm}|^{2}(h_{j}(\Omega )-h_{m}(\Omega ))$$which is a well-known expression. For the above choice of the weight function $h_{j}(\Omega )$ , the remaining integration can be easily done with the result $$J_{j}^{(0)}(z)=\frac{1}{2\pi }\sum_{m}|S_{jm}|^{2}(\mu _{m}-\mu _{j})$$The conductance (in units of the conductance quantum $e^{2}/2\pi \hbar $ ) of a two-lead junction is in lowest order given by $$G_{0}=J/V=|S_{12}|^{2}=t^{2}$$where $V=\mu _{1}-\mu _{2}$ is the applied bias voltage. In the following we will find it convenient to introduce the quantity $Y=1-2G_{0}$ characterizing the conductance.
Current to first order in the interaction \[sec:1stOrder\]
==========================================================
The first order correction to the current in the non-equilibrium case is represented as the diagram depicted in Fig. \[fig:triangle-diag\]. Here the wavy line stands for the electronic interaction, taking place at the point $x$ in the wire $l$. The contribution to the current of chirality $\eta _{n}$ in the $n$-th wire can be expressed as
$$\begin{aligned}
J_{j_{\eta }}^{(1)}(z) &=v_{j}\int \frac{d\Omega \,d\omega }{(2\pi )^{2}}%
\,\int dx\,dy\sum_{\mu =1,2}\sum_{l,\eta _{l}}\,\mbox{Tr}_{K}
[\underline{\gamma }_{ext} \\
& \times \underline{G}_{\Omega }(j_{\eta },z|l_{\eta },y)%
\underline{\bar{\gamma}}^{\mu }\underline{G}_{\Omega +\omega }(l_{\eta
},y|l_{\eta },x) \underline{\gamma }^{\mu } \\
& \times \underline{G}_{\Omega }(l_{\eta },x|j_{\eta
},z)](2\pi ig_{l}v_{l})\delta (x-y)
\end{aligned}$$
The trace $\mbox{Tr}_{K}$ is over the lower (fermionic) Keldysh indices; the fermion-boson vertices, $\gamma _{ij}^{\mu }\rightarrow \underline{\gamma }%
^{\mu }$, $\bar{\gamma}_{ij}^{\mu }\rightarrow \underline{\bar{\gamma}}^{\mu
}$, tensors of rank $3$ defined in Keldysh space, are given by
$$\gamma^{1}_{ij} =\bar\gamma^{2}_{ij} = \tfrac1{\sqrt{2}} \delta_{ij}, \quad
\gamma^{2}_{ij} =\bar\gamma^{1}_{ij} = \tfrac1{\sqrt{2}} \tau^{1}_{ij},
\label{def:gamma}$$
with $\tau^{1}$ the first Pauli matrix.
Notice that, similarly to the case of zeroth order in the interaction, the factor $v_{j}$ at the external point $z$ is compensated by the prefactor coming from the Green’s function, Eq. (\[eq:Green\]). If the point of the observation $z$ lies outside the interacting region, $z>L$, then the dependence on $z$ disappears in the outgoing current, $%
J_{j,+}^{(1)}(z>L)=J_{j}^{(1)}$, whereas the corrections to the incoming current are alltogether absent, $J_{j,-}^{(1)}(z>L)=0$. In what follows we discuss the corrections to the outgoing current. In view of the later generalization involving an infinite summation of higher order terms it is useful to represent the above first order expression as
$$\begin{aligned}
J_{j}^{(1)} & = i \int \frac{d\omega }{2\pi }\,\int dx\,dy\,\sum_{l_{\eta
},m_{\eta }}
\\ & \times
\mbox{Tr}_{K}[\ \underline{T}_{\omega }(m_{\eta },y|l_{\eta },x;j,+,z)%
\underline{L}_{0,\omega }(l_{\eta },x|m_{\eta },y)] \label{1stL0}
\end{aligned}$$
where we recall the definitions $$l_{\eta }=(l,\eta _{l})\,$$etc. Here we defined a “boson propagator” representing the interaction line
$$\underline{L}_{0,\omega }(l,\eta _{l},x|m,\eta _{m},y)=(2\pi
g_{l}v_{l})\delta (x-y)\delta _{lm}\tau _{\eta _{l},\eta _{m}}^{1}%
\begin{pmatrix}
1 & 0 \\
0 & 1%
\end{pmatrix}%$$
and the quantity $T$ representing the triangle of Green’s functions in Fig. \[fig:triangle-diag\]
$$\begin{aligned}
\label{def:T}
T_{\omega }^{ \nu\mu }&(
m_{\eta },y|l_{\eta },x ;j_{\eta},z)
\\
&=v_{j}\int
\frac{d\Omega }{2\pi }\mbox{Tr}_{K}\left[ \underline{\gamma }_{ext}\underline{%
\widehat{G}}_{\Omega }(j_{\eta},z|m_{\eta },y ) \underline{\bar{%
\gamma}}^{\nu } \right. \\
&\times
\left. \underline{\widehat{G}}_{\Omega +\omega }(m_{\eta },y|l_{\eta },x)%
\underline{\gamma }^{\mu }\underline{\widehat{G}}_{\Omega }(l_{\eta },x |j_{\eta},z) \right]\,,
\end{aligned}$$
this diagram should be combined with the one, where the arrows on the fermonic lines are reverted.
The triangle diagram is characterized by two Keldysh indices and thus is subdivided into four blocks. Symbolically, we write $$\mbox{Tr}_{K}[\ TL]=T^{11}L^{R}+T^{22}L^{A}$$anticipating that $T^{21}=0,L^{21}=0$ (to be shown later) .
When integrating over $\Omega $ in (\[def:T\]) we find two generic integrals. One of them is $$\int d\Omega \,(h_{j}(\Omega +\omega )-h_{j}(\Omega ))=2\omega$$and the other is $$\int d\Omega \,[1-h_{j}(\Omega +\omega )h_{m}(\Omega )]=2F(\omega +\mu
_{m}-\mu _{j}). \label{def:F}$$For the above form of $h_{j}(\Omega )$, we have $F(x)=x\coth (x/2T)$.
As mentioned above there are no corrections to the incoming currents. In addition to this observation we should recall Kirchhoff’s law, stating the conservation of charge. Given that the total incoming current is equal to the total outgoing current, we should have $J_{1}+J_{2}=0$, which is indeed confirmed by direct calculation.
Taking these facts into account, only the difference of the currents, $J^{(1)} = \frac12 (J_{2}^{(1)}-J_{1}^{(1)})$, is of interest. This involves the difference of the components of $T$ belonging to different leads. Accordingly, for the case of two leads, we define the weighted difference (denoted by the same symbol, $T$, but dependent on fewer variables),
$$\begin{aligned}
\label{Tdiff}
T_{\omega }^{\mu \nu }(m_{\eta },y|l_{\eta },x) % & \\
& = \frac{1}{2}[T_{\omega }^{\mu \nu }(m_{\eta },y|l_{\eta },x;1,+,z
>L) \\ & -T_{\omega }^{\mu \nu }(m_{\eta },y|l_{\eta },x;2,+,z>L)]
\end{aligned}$$
The $4\times 4$ matrices appearing here are now direct products of $2\times 2$ matrices in chiral space (outer block structure) and $2\times 2$ matrices in lead space (inner block structure, see Table \[tab:Indices\] ) $$\begin{aligned}
\label{oddT}
T^{11} _{}&=
\frac{ r^{2}t^{2}} {8\pi }
[F(\omega +V)-F(\omega -V)]%
\\ &\times
\Phi_{\omega} (y) \begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & -1 & 0 & 0 \\
-1 & 1 & 0 & 0%
\end{bmatrix}%
\Phi_{\omega}^{\ast}(x)
\,, \\
T^{22} _{}&=-(T^{11}_{})^{\dagger } \vert_{x\leftrightarrow y}
\,,\quad T^{21}_{}=0\,.
\\
\Phi_{\omega} (x) &=
diag\left [\frac{e^{-i\omega x/v_1}}{v_1} , \frac{e^{-i\omega x/v_2}}{v_2} ,
\frac{ e^{i\omega x/v_1}}{v_1} , \frac{e^{i\omega x/v_2}}{v_2} \right]
\end{aligned}$$The vanishing of $T^{21}_{}$ implies that the Keldysh component of the renormalized interaction, $L^{K}$, does not enter. Inserting the components of $T^{\mu \nu }_{}$ and $L_{0}$ into the expression (\[1stL0\]) for the current for two equal wires, $g_{j}=g$, $v_{j}=v$, we find
$$J^{(1)}=-\frac{ gt^{2}r^{2}}{\pi}\int_{0}^{\omega _{c}}\frac{d\omega }{\omega }%
[F(\omega +V)-F(\omega -V)]\sin ^{2}\frac{\omega L}{v}$$
Here we apply an upper cut-off $\omega _{c}$ given in the microscopic model either as $\omega _{c}=v/a$ as mentioned above or $\omega _{c}=W$, the band width. The conductance as a function of voltage $V$, temperature $T$, wire length $L$, is found from there as
$$G^{(1)}=-2g\, G_{0}(1-G_{0})\;\Lambda (V,T,L) \,.$$
Here we introduced the scaling variable $\Lambda $
$$\Lambda (V,T,L)=\int_{0}^{\omega _{c}}\frac{d\omega }{\omega }%
\frac{F(\omega +V)-F(\omega -V)}{V}\sin ^{2}\frac{\omega L}{v} \,.
\label{Lambda-gen}$$
The factor $\sin ^{2}(\omega L/v)]$ guarantees convergence of the integral at $\omega <1/t_{0}=\pi v/L$. At $\omega >1/t_{0}$ we may average this rapidly oscillating function and approximate $\sin ^{2}(\omega L/v)]\simeq
1/2$. Employing this and analogous procedures for the cases of small $V,L^{-1}$ or small $V,T$ we may approximate $\Lambda $ as
$$\Lambda (V,T,L)\simeq \ln \left( \frac{\omega _{c}}{\max \{V,T,v/L\}}\right) \,.$$
$\alpha$ = 1 2 3 4
-------------- --- --- --- ---
before/after
wire \# 1 2 1 2
: Convention for the indices []{data-label="tab:Indices"}
Scale-dependent part of the current: Summation to infinite order in the interaction \[sec:summation\]
=====================================================================================================
As shown in our previous work, the perturbative treatment may be extended into the strong coupling regime by summing up an infinite series of relevant scale-dependent contributions to the conductance in all orders (“ladder approximation”). These represent a self-energy renormalization of the “boson propagator” $L_{0}$ introduced above. They thus technically constitute a renormalized one-loop contribution to the RG equation. This series can still be represented by the generic diagram of Fig. \[fig:triangle-diag\], but the wavy line of electronic interaction should be dressed by screening effects, as discussed below.
As a result of this summation the interaction line, $g_{l}$, acquires non-locality and retardation effects. Moreover, if we have initially only diagonal components in Keldysh space, after the summation we generate a Keldysh component and in general a rather complicated structure. Schematically, we replace $L_{0}$ by $L$ in Eq. (\[1stL0\]):
$$\underline{L}_{0}\rightarrow \underline{L}=%
\begin{pmatrix}
L_{\omega }^{R}(l_{\eta },x|m_{\eta },y) & L_{\omega }^{K}(l_{\eta },x|m_{\eta },y) \\
0 & L_{\omega }^{A}(l_{\eta },x|m_{\eta },y)%
\end{pmatrix}%$$
We now embark on the calculation of $\underline{L}$ . Introducing compact notation, we express the lowest order result $\underline{L}_{0}$ in the form $L_{0}^{\mu\nu}(l_{\eta },x|m_{\eta },y)=
\delta _{\mu\nu}\tau _{\eta _{l},\eta _{m}}^{1}\delta
_{lm}\, g\delta (x-y)=\mathbf{1}_{K}\otimes {\tau }_{C}^{1}\otimes \mathbf{1}%
_{w}\, g\delta (x-y)$.
The integral equation describing the summation of the relevant infinite class of diagrams (see Fig. \[fig:ladder\]) takes the form
$$\begin{aligned}
\underline{L} &=\underline{L}_{0}+\underline{L}_{0}\ast \underline{\Pi }%
\ast \underline{L}_{0}+\underline{L}_{0}\ast \underline{\Pi }\ast \underline{%
L}_{0}\ast \underline{\Pi }\ast \underline{L}_{0}+\ldots \\
&=\underline{L}_{0}+\underline{L}_{0}\ast \underline{\Pi }\ast \underline{L}
\end{aligned}$$
where $\underline{\Pi }$ represents a fermion bubble
$$\begin{aligned}
\Pi _{\omega }^{ \mu \nu}(l_{\eta },x | j_{\eta },y) &= i \int \frac{d\Omega }{2\pi
}\mbox{Tr}_{K}[\underline{\bar{\gamma}}^{\mu }\underline{G}_{\Omega +\omega
}(l_{\eta },x|j_{\eta },y)
\\ & \times
\underline{\gamma }^{\nu }\underline{G}_{\Omega
}(j_{\eta },y|l_{\eta },x) ]
\end{aligned}$$
The multiplication $\ast $ is defined as
$$\begin{aligned}
(\Pi \ast L)_{\omega }^{\mu \nu }(j_{\eta },y|n_{\eta },x)&=\int
dz\sum_{l_{\eta }}\sum_{\lambda =1,2}\Pi _{\omega }^{\mu \lambda }(j_{\eta
},y|l_{\eta },z)
\\ & \times
L_{\omega }^{\lambda \nu }(l_{\eta },z|n_{\eta },x)
\end{aligned}$$
At the level of Keldysh structure we have
$$\begin{aligned}
\begin{pmatrix}
L^{R} & L^{K} \\
0 & L^{A}%
\end{pmatrix}
&=%
\begin{pmatrix}
L_{0} & 0 \\
0 & L_{0}%
\end{pmatrix}%
+ \\
&+%
\begin{pmatrix}
L_{0} & 0 \\
0 & L_{0}%
\end{pmatrix}%
\ast
\begin{pmatrix}
\Pi ^{R} & \Pi ^{K} \\
0 & \Pi ^{A}%
\end{pmatrix}%
\ast
\begin{pmatrix}
L^{R} & L^{K} \\
0 & L^{A}%
\end{pmatrix}%
\end{aligned}$$
which means that we can solve the integral equation in three steps.
First, we solve the coupled integral equations in the retarded sector $$L^{R}=L_{0}+L_{0}\ast \Pi ^{R}\ast L^{R} \label{eq:LR}$$Second, considering that $$L^{A}=L_{0}+L_{0}\ast \Pi ^{A}\ast L^{A}$$ if we are using the relation between $\Pi ^{A}$ and $\Pi ^{R}$, we need not solve this equation separately. Third, we notice for completeness that $$L^{K}=L_{0}\ast \Pi ^{R}\ast L^{K}+L_{0}\ast \Pi ^{K}\ast L^{A}$$and hence, $$\begin{aligned}
L^{K} &=(1-L_{0}\ast \Pi ^{R})^{-1}\ast L_{0}\ast \Pi ^{K}\ast L^{A} \\
&=L^{R}\ast \Pi ^{K}\ast L^{A}
\end{aligned}$$where we used (\[eq:LR\]) to obtain the second equality. This means that once we have $L^{R}$, we can easily find the two remaining components, $L^{A}
$ and $L^{K}$. We recall, however, that as pointed out above the component $%
L^{K}$ does not enter the calculation of the current.
The solution for $L^{R}$ in the linear response case was presented previously in our work [@Aristov2012]. We follow that derivation but reformulate it somewhat for the present purposes. First we define the integral (scalar) kernel $$\begin{aligned}
P _{\omega }(j,x|l,z) &=(2\pi v_{j}v_{l})^{-1}(\delta (\tau )+i\omega \theta
(\tau )e^{i\omega \tau }) \\
\tau &=x/v_{j}-z/v_{l}
\end{aligned}$$and the matrix quantity $$\begin{aligned}
\label{PiR}
\widehat{\mathbf{\Pi} }^{R} &=
\begin{pmatrix}
\mathbf{\Pi } (-x|-z) , & \mathbf{Y}^{T} \Pi (-x|z) \\
\mathbf{Y} \Pi (x|-z) , & \mathbf{\Pi } (x|z) %
\end{pmatrix}%
\end{aligned}$$ where $ \Pi _{jl}(x|z) = \delta_{jl}P _{\omega }(j,x|l,z) $, $\mathbf{Y} \Pi (x|z) = Y_{jl} P _{\omega }(j,x|l,z)$ and $\mathbf{Y}^{T} \Pi (x|z) = Y_{lj} P _{\omega }(j,x|l,z)$ with $Y_{jl}=|S_{jl}|^{2}$. In the case of two leads we have $\mathbf{Y}=\mathbf{Y}^{T}$. Notice that $\mathbf{Y}^{T} \Pi (-x|z) = 0 $ for $x,z>0$, and we use the full form (\[PiR\]) for future reference.
Then we express the integral equation for $\mathbf{L}^{R}$ as a $2\times 2$ matrix equation in the chiral space $$\begin{aligned}
&\widehat{\mathbf{L}}^{R}(x|y)=2\pi \delta (x-y)%
\begin{pmatrix}
0 & \mathbf{g} \\
\mathbf{g} & 0%
\end{pmatrix}
\\
&-2\pi \int_{a}^{L}dz%
\begin{pmatrix}
\mathbf{g}\mathbf{Y} \Pi (x|-z) & \mathbf{g} \mathbf{\Pi} (x|z) \\
\mathbf{g} \mathbf{\Pi} (-x|-z) & 0%
\end{pmatrix}%
\widehat{\mathbf{L}}^{R}(z|y)\,,
\end{aligned}$$Here $\widehat{\mathbf{L}}^{R}$ is a $4\times 4$ (in the general case of $n$ leads $2n\times 2n$ ) matrix. The elements of the $2\times 2$ matrices in chiral space (denoted by a hat) are $2\times 2$ matrices in the space of the $2$ leads (denoted by bold letters). The matrix of interaction constants is then given by $\mathbf{g}=diag[ g_{1}v_{1} , g_{2}v_{2} ] $. The scattering properties of the junction are encoded in the $2\times 2$ matrix $\mathbf{Y}$. The equation for $\mathbf{L}^{A}$ is similar to the above, but $$\widehat{\mathbf{\Pi} }^{A} =
\big( \widehat{\mathbf{\Pi} }^{R} \big) ^{\dagger}
\Big|_{x \leftrightarrow z } =
\begin{pmatrix}
0 & \mathbf{1} \\
\mathbf{1} & 0%
\end{pmatrix}%
\widehat{\mathbf{\Pi} }^{R}%
\begin{pmatrix}
0 & \mathbf{1} \\
\mathbf{1} & 0%
\end{pmatrix}%
\Big|_{\omega \rightarrow -\omega ,\ \mathbf{Y}\rightarrow \mathbf{Y}^{T}}$$Because $L_{0}$ does not contain $\omega $, $Y$, it follows that $$\widehat{\mathbf{L}}^{A}=
\big( \widehat{\mathbf{L} }^{R} \big) ^{\dagger}
\Big|_{x \leftrightarrow y }
% \begin{pmatrix} 0 & \mathbf{1} \\ \mathbf{1} & 0 \end{pmatrix} \widehat{\mathbf{L}}^{R}
%\begin{pmatrix} 0 & \mathbf{1} \\ \mathbf{1} & 0 \end{pmatrix}%
%\Big|_{\omega \rightarrow -\omega ,\ \mathbf{Y}\rightarrow \mathbf{Y}^{T}}
\label{LAviaLR}$$
The Keldysh part of the kernel takes the form presented in the appendix [KeldyshPi]{}. We show there, that $\Pi^{K}$ is an even function of $V$, and therefore $L^{K}$ does not contribute to the current.
Following the method of solution of the integral equation described in [Aristov2012]{} we first solve the equation for the case $Y_{jl}=0$ , to give a partial summation resulting in a auxiliary quantity $\mathbf{C}$
$$\begin{aligned}
&\widehat{\mathbf{C}}(x|y)=2\pi \delta (x-y)%
\begin{pmatrix}
0 & \mathbf{g} \\
\mathbf{g} & 0%
\end{pmatrix}
\\
&-2\pi \int_{a}^{L}dz%
\begin{pmatrix}
0 & \mathbf{g}\mathbf{\Pi} (x|z) \\
\mathbf{g}\mathbf{\Pi} (-x|-z) & 0%
\end{pmatrix}%
\widehat{\mathbf{C}}(z|y)\,,
\label{eq:defC}
\end{aligned}$$
In terms of $\widehat{\mathbf{C}}$ the integral equation for $\widehat{%
\mathbf{L}}^{R}$ may be expressed as $$\begin{aligned}
\label{eq:LviaC}
\widehat{\mathbf{L}}^{R}(x|y) & =\widehat{\mathbf{C}}(x|y)
-2\pi \int_{a}^{L}dz_{1}dz_{2}\widehat{\mathbf{C}}(x|z_{1})%
\\ & \times
\begin{pmatrix}
0 & 0 \\
\mathbf{Y} {\Pi} (z_{1}|-z_{2}) & 0%
\end{pmatrix}%
\widehat{\mathbf{L}}^{R}(z_{2}|y)\,,
\end{aligned}$$The solution of the integral equation for $\widehat{\mathbf{C}}$, which is of the Wiener-Hopf type, may be found by an appropriate ansatz described in [@Aristov2012]. By construction, $\widehat{\mathbf{C}}(x|y)$ is diagonal in wire space, $\widehat{\mathbf{C}}_{jl}(x|y) = \delta_{jl}\widehat{C}_{j}(x|y)$. The explicit expressions for $\widehat{C}_{j}(x|y)$ are presented in the Appendix \[sec:FullCL\].
Returning to the integral equation (\[eq:LviaC\]) for $\widehat{\mathbf{L}}^{R}$ in terms of $\widehat{\mathbf{C}}$ we observe that its kernel is separable and thus the solution may be readily obtained. The explicit expressions and some details of the derivation of this result are given in Appendix [sec:FullCL]{}.
An inspection of Eqs. (\[1stL0\]), (\[oddT\]) shows that the $x$, $y$ dependence of $T^{\mu\nu}_{}$ comes only from the matrices $\Phi_{\omega}^{\ast}(x)$, $\Phi_{\omega}(y)$. We combine these matrices with $\widehat{\mathbf{L}}^{R}$ and integrate over the position $$\widehat{\mathbf{L}}^{R}_{simple}=
\int_{a}^{L} dx\,dy\,
\Phi_{\omega}^{\ast} (x) \,
\widehat{\mathbf{L}}^{R}(x|y)\, \Phi_{\omega} (y) \,.$$ Making now use of relations (\[oddT\]), (\[LAviaLR\]), we arrive at a much simpler algebraic expression for the current. Instead of (\[1stL0\]) we have $$J^{(L)} = -2\, \mbox{Im} \int \frac{d\omega }{2\pi }\, \mbox{Tr } [\widehat{T}_{core} ^{11}
\widehat{\mathbf{L}}^{R}_{simple} ]
\label{TLmatrix}$$with $T^{\mu\nu}_{core}$ obtained by putting $\Phi_{\omega} (x) =\Phi_{\omega} (y) = \mathbf{1}$ in Eq. (\[oddT\]). We show the algebraic relation (\[TLmatrix\]) diagrammatically in Fig. \[fig:triangle-diag2\].
Introducing the quantities $d_{j}$ and $q_{j}$ $$\begin{aligned}
\label{def:dq}
d_{j}^{2} &=1-g_{j}^{2} \,, \\
q_{j}^{-1} &=\frac{g_{j}}{1+id_{j}\cot (\frac{\omega L}{v_{j}d_{j}})} \,, %
%\rightarrow \frac{g_{j}}{1+d_{j}}
\end{aligned}$$we present the simpler expressions of $\widehat{C}$, $\widehat{L}$ integrated over position.
$$\begin{aligned}
\label{integL}
\widehat{ {L}}^{R}_{jk,simple} & = \frac{2\pi i }{\omega} \left(
\delta_{jk} \widehat{ {C}} _{j,simple}
\right . \\ & \left .
+\Upsilon _{jk} \begin{bmatrix}
V_{1,j} V_{2,k} , & V_{1,j}V_{1,k} \\
V_{2,j}V_{2,k} , & V_{2,j}V_{1,k}%
\end{bmatrix}
\right)
\end{aligned}$$
where $$\begin{aligned}
\label{integCV}
\widehat{ {C}} _{j,simple} &=
\left( \begin{bmatrix} -1, & 0 \\ 0,& -1 \end{bmatrix}
\right . \\ & \left.
+ q_{j}^{-1}
\begin{bmatrix} \frac{i d_{j}e^{-iL\omega/v}}{g_{j} \sin(\omega L/v_{j}d_{j})}, & 1 \\ e^{-2iL\omega/v},&
\frac{i d_{j}e^{-iL\omega/v}}{g_{j} \sin(\omega L/v_{j}d_{j})} \end{bmatrix}
\right ) \,, \\
V_{1,j} & = ( \widehat{ {C}} _{j,simple} )_{12} \,, \quad
V_{2,j} = ( \widehat{ {C}} _{j,simple} )_{11} \,, \\
\Upsilon _{jk} &=Y_{jl}(1-Q^{-1}\cdot Y)_{lk}^{-1},\quad Q_{jk}^{-1}=\delta
_{jk}q_{j}^{-1},
\end{aligned}$$
Combining the above results, (\[oddT\]), (\[TLmatrix\]), (\[integL\]), (\[integCV\]), we find the current for two equal wires, $g_{j}=g$, $v_{j}=v$, as
$$\begin{aligned}
\label{ladderCurrent}
J^{(L)}=&\frac{g}{8\pi} \int_{0}^{\omega _{c}}d\omega \frac{F(\omega +V)-F(\omega -V)}{%
\omega }
\\ & \times
\mbox{Re}\left\{ \frac{2(1-Y^{2})}{1-gY+id\cot \left( \frac{L\omega
}{dv}\right) }\right\}
\end{aligned}$$
Again the $\frac{1}{\omega }-$singularity of the integrand leads to a logarithmically divergent contribution, which we identify as a scaling contribution. The singularity is controlled by the largest of the three energy scales, (i) energy scale $\omega _{L}=v/L$ controlled by the length $L
$, (ii) temperature $\omega _{T}=T$ , (iii) bias voltage $\omega _{V}=V$. In the limit $V\rightarrow 0,T\rightarrow 0$ we have $F(\omega +V)-F(\omega
-V)=2Vsign(\omega )$. The $\frac{1}{\omega }-$singularity is in this case cut off at the scale $\omega _{L}=dv/L$ by the $\cot \left( \frac{L\omega
}{dv}\right) -$term in the denominator. Above this scale we may average the rapidly oscillating function in the curly brackets in (\[ladderCurrent\]) over one oscillation period, $\omega _{0}<\omega <$ $\omega _{0}+(\pi /t_{0})$ , with $%
t_{0}=L/dv$ :
$$\frac{t_{0}}{\pi }\int_{\omega _{1}}^{\omega _{1}+\pi /t_{0}}\frac{d\omega }{%
1-gY\pm id\cot \omega t_{0}}=(1-gY+d)^{-1} \,,
\label{average}$$
such that the correction to the conductance is obtained as $$G^{(L)}=- g\frac{(1-Y^{2})}{1-gY+d}\ln \frac{L}{a}$$in agreement with [@Aristov2009]. In the general case we find accordingly
$$G^{(L)}=- g\frac{(1-Y^{2})}{1-gY+d}\Lambda
\label{Gladder}$$
where $\Lambda =\ln (\omega _{c}/\max \{V,T,v/L\}$ . In the limit of long wires, $L\to \infty$, a closed expression is found in Appendix \[sec:Lambda\] in the form $$\label{def:Lambda}
\Lambda = \ln\left( \frac{\omega _{c} }{2\pi T} \right)
- \mbox{Re}
\left[ \psi \left( 1 + \frac{iV}{2\pi T} \right)
\right] \,,$$ with $\psi(x)$ digamma function. This function shows a smooth interpolation between the regimes with $\ln\left( \frac{\omega _{c} }{2\pi T} \right) + 0.577$ at $ V\ll T$ and $\ln\left( {\omega _{c} }/{V} \right) $ at $ V\gg T$.
Further corrections not considered here are generated by the Hartree diagrams of the self energy: in the nonequilibrium situation the local chemical potential is renormalized by a molecular field term involving the bare interaction and the local particle density. In Ref. [@Egger2000] this effect is included by applying a corresponding boundary condition to the thermodynamic Bethe ansatz fields. As a consequence a bistability of the current has been found at very strong interaction. We have not included this correction term into our analysis, as it would require a separate calculation of the single particle Green?s function, especially of the local chemical potential shift, which is beyond the scope of the present paper. We are therefore confining our considerations from weak up to moderately strong interaction, such that $K > 0.2$.
Renormalization group equation for the conductance \[sec:deriveRG\]
===================================================================
The above calculation of the leading scale-dependent contribution to the current allows us to derive a renormalization group (RG) equation for the conductance $G=I/V$ as a function of the scaling variable $\Lambda =\ln
(\omega _{c}/\max [V,T,v/L])$ , $G=G(\Lambda )$. We thereby use the scaling property of $G$ , $G(V,T,v/L,G_{0};g)=G(\Lambda ;g)$ . In our previous works [@Aristov2009; @Aristov2011] we explicitly checked this property in the equilibrium situation. We directly calculated all the contributions to the conductance up to third order in the interaction, which involves about $%
10^{4}$ diagrams. It was shown that the principal contribution near the fixed points (FPs) of this equation is obtained in one-loop order, with the interaction being dressed as described above, $g\rightarrow L$. The scaling exponents obtained this way are identical to those found earlier by the method of bosonization.
Away from the FPs one finds in general additional non-universal contributions, appearing first in the third order. These determine the prefactor in the scaling law near the FP and also fine details of the conductance in the intermediate regime. In the present study focused on the transport far out of equilibrium it would be too costly to perform a similar direct computation of all contributions up to third order. Instead we assume that even out of equilibrium we have the scaling property and the scaling exponents are fully determined by the contribution provided by the approximation of fully dressing the interaction line of the one-loop calculation.
This assumption may be justified by at least two facts. First, in the renormalized one-loop calculation presented here the bias voltage $V$ appears as an infrared cutoff in the scale dependent terms, replacing the cutoff energy scales temperature $T$ or level splitting $v/L$ present in equilibrium. It may therefore be expected that the structure of the scale-dependent terms generated by the cutoff $V$ is analogous to that of the terms generated by the cutoff $T$ or $v/L$. No additional scale dependent terms are found in non-equilibrium and none of the scale-dependent terms present in equilibrium disappears in non-equilibrium. This suggests that the structure of the scale-dependent terms is preserved and therefore the scaling property is preserved even out of equilibrium. Secondly, as will be shown below, the results of our theory are in agreement with exact results obtained by other methods.
We now briefly review the logic by which the RG-equation is derived from the perturbative result. We start from the result for the renormalized conductance $G$ as a power series expansion in the interaction, and dependent on the scattering properties of the junction (encapsulated in the conductance $G_{0}$ in the absence of interaction) obtained above, which takes the general form ,
$$G=G_{0}-gf(g,G_{0})\Lambda +\mathcal{O(}g^{2}\Lambda ^{2})
\label{generalpert}$$
In the approximation of summing up the leading terms in each order, considered above, a very good approximation $f_{app}$ of the function $f(g,G)
$ has been obtained, see Eq.(\[Gladder\]). We do not calculate the terms of order $g^{2}\Lambda ^{2}$ and higher in this paper. The relation Eq. (\[generalpert\]) is valid in the asymptotic regime $g\Lambda \rightarrow 0$ . With the aid of the scaling property we may find the analytic continuation to finite values of $g\Lambda $. To this end we first invert the relation (making use of $G=G_{0}+\mathcal{O}(g\Lambda )$ ) and write $$G_{0}(g,G;\Lambda )=G+gf(g,G)\Lambda +\mathcal{O}(g^{2}\Lambda ^{2})$$Formally $G_{0}$ here is a function of $G,g$ and $\Lambda $ . We now employ the crucial property that the value of the bare conductance, $G_{0}$, should not depend on the scaling variable $\Lambda $ , which means $$0=\frac{\partial G_{0}}{\partial \Lambda }+\frac{\partial G_{0}}{\partial G}%
\frac{dG}{d\Lambda }$$and hence $$\frac{dG}{d\Lambda } =-\frac{gf(g,G)+\mathcal{O}(g^{2}\Lambda )}{1+g \Lambda [\partial
f(g,G)/\partial G] + \mathcal{O}(g^{2}\Lambda^{2} ) }
\label{betaLambda}$$ The scaling property of $G$ implies that the explicit $\Lambda$-dependence in (\[betaLambda\]) cancels. This leads to the definition of the RG $\beta $-function $$\frac{dG}{d\Lambda } =\beta (g,G)=-gf(g,G) \,.$$ Our earlier direct third order calculation in [@Aristov2009; @Aristov2011] showed that the above ratio was indeed independent of $\Lambda $ to the considered accuracy $g^{3}$. The function $gf(g,G)$ has been calculated beyond the ladder approximation in [@Aristov2009] for the present case of a two-lead junction with the result $$\tfrac{d}{d\Lambda }G=-gf_{app}(g,G)+c_{3}g^{3}G^{2}(1-G)^{2}+\mathcal{O}%
(g^{4})\, \label{RGeq-generic}$$ The second term here of order $g^{3}$originates from terms not contained in the perturbation series for $L$ considered above. This term is subleading in the sense that it vanishes more rapidly on approach to the fixed points at $%
G=0,1$ than the first term and does therefore not influence the critical properties. There are indications that this is also the case with the higher order contributions not captured by the ladder summation.
A similar conclusion regarding the relative unimportance of corrections beyond the ladder summation $g\rightarrow L$ was reached in [Aristov2011]{} for the more general case of the three-lead Y-junction. In the symmetrical setup the Y-junction was characterized by two conductances, and after extensive computer analysis of perturbative corrections we arrived at a set of two coupled RG equations. We found that the three-loop corrections, not contained in the ladder series of diagrams, did not contribute to the scaling exponents.
We expect that non-universal contributions to the $\beta -$function will also exist in the case of non-equilibrium, but those terms will again be unimportant when it comes to determine the critical behavior at the fixed points. We will therefore approximate the exact function $f(g,G)$ by the one determined in the ladder approximation and given through eq.([generalpert]{}), which gives rise to the $\beta -$function
$$\frac{d}{d\Lambda }G= -4 g\frac{G(1-G)}{1-g(1-2G)+d}\,
\label{eq:beta1}$$
Introducing the Luttinger parameter $K = \sqrt{(1-g)/(1+g)}$, Eq. (\[eq:beta1\]) may be re-expressed as $$\frac{d}{d\Lambda }G =-2 (1-K)\frac{G(1-G)}{K +(1- K) G}\,,
\label{eq:beta2}$$ which is explicitly solved in the next section.
solution of RG equation \[sec:RGsolution\]
==========================================
Inverting Eq. (\[eq:beta2\]) we write $$2 (1-K) d\Lambda =- d G \left ( \frac{K}{G } +\frac{1}{1-G}
\right )
\, ,$$ which is integrated with the result $$\frac {1-G}{G^{K}} = \frac{1-G_{0}} {G_{0}^{K}} e^{2 (1-K) \Lambda}
\label{explicitGLambda}
\, .$$
It is more instructive to exclude here the bare conductance, $G_{0}$, and to represent our result as (cf. [@Jezouin2013]) $$\frac {(1-G)/G^{K} |_{V_{1},T_{1}}}{(1-G)/G^{K} |_{V_{2},T_{2}}}
% = \frac {\exp2 (1-K) \Lambda (V_{1},T_{1}) } { \exp 2 (1-K) \Lambda (V_{2},T_{2}) }
= \frac { e^{2 (1-K) \Lambda (V_{1},T_{1}) }} { e^{ 2 (1-K) \Lambda (V_{2},T_{2}) } }
\label{ratio-conduc}
\, .$$
The latter exponential can be written as $$e^{2 (1-K) \Lambda} = \left(\frac{\omega _{c}}{\max \{V,T,v/L\}} \right) ^{2 (1-K)} \,.$$ We see that near the two fixed points of the RG equation, $G=0$, $G=1$, we have the well-established scaling behavior [@Kane1992] $$\begin{aligned}
G & \simeq \left(V/ {\omega _{0}} \right) ^{2 (K^{-1}-1)} \,, \quad G \to0 \,, \\
1- G & \simeq c_{\ast}\, \left( {\omega _{0}}/{V } \right) ^{2 (1- K )} \,, \quad G \to1 \,.
\label{GviaV}
\end{aligned}$$ with an appropriate $\omega_{0}$ and where $V$ should be replaced by $\exp(\Lambda(V,T,v_{F}/l))$ in the more general situation. At the same time, (\[explicitGLambda\]) provides a smooth crossover between the fixed points, i.e. for those values of $G$ which, strictly speaking, are inaccessible by the bosonization approach.
We notice further, that if the overall energy scale $\omega_{0}$ is fixed near one fixed point, then the constant $c_{\ast}$ is entirely defined by the three-loop and higher-loop terms in the RG equation. In the approximation of neglecting the three-loop terms, as in Eqs. (\[eq:beta2\]), (\[def:Lambda\]), the coefficient $c_{\ast} = 1$. Keeping the three-loop terms, Eq. (\[explicitGLambda\]) is approximately given by [@Aristov2009] $$\frac{G^{K}} {1-G} \left(1+G\tfrac{1-K}{K}\right)^{4c_{3}(1-K)} =
\left[\frac{\max \{V,T,v/L\}} {\omega _{0}}\right] ^{2 (1-K)}
\label{explicitG-c3}
\, ,$$ which implies $c_{\ast} = K ^{-4c_{3}(1-K)}$.
Comparison with the exact solution at $K=1/2$
---------------------------------------------
To understand better the limitations of our formula (\[explicitGLambda\]), we compare it with the exact result at $K=1/2$. Explicitly, our expression in this case reads as $$G= 1- \frac{\sqrt{1+4x^{2}}-1}{2x^{2}}, \quad
x = \frac{T}{T^{\ast}} \exp{\mbox{Re }
\psi \left[ 1 + \tfrac{iV}{2\pi T} \right]
} \,. \label{ourG12}$$
The exact formula, obtained with the aid of the Bethe ansatz [@Weiss1995] is
$$G_{1/2} = 1- \frac{4\pi T^{\ast} }{V} \mbox{Im } \psi \left[ \frac12 + \frac{T^{\ast}}{T}+ i \frac {V}{4\pi T}\right]
\,,$$
with $T^{\ast}$ depending on the impurity backscattering amplitude and the ultraviolet cutoff. In two important limiting cases we have for the linear conductance $$\begin{aligned}
G_{1/2}(T) & = G_{1/2}(V\to0,T) \,, \\
&= 1- \tfrac{ T^{\ast} }{T} \psi' \left( \tfrac12 + \tfrac{T^{\ast}}{T} \right) \,,
\label{G12V0T} \\
& \simeq \frac{1}{12} \left(\frac T{ T^{\ast} }\right)^{2}, \quad T \ll T^{\ast} \,,\\
& \simeq 1- \frac{\pi^{2}}2 \frac{ T^{\ast} }{T} , \quad T \gg T^{\ast} \,.
\end{aligned}$$ And for the nonlinear conductance $$\begin{aligned}
G_{1/2}(V) & = G_{1/2}(V,T\to0) \,, \\
&= 1- \tfrac{ 4\pi T^{\ast} }{V} \arctan \tfrac{V}{4\pi T^{\ast}} \,,
\label{G12VT0} \\
& \simeq \frac{1}{12} \left(\frac V{ 2\pi T^{\ast} }\right)^{2}, \quad V \ll T^{\ast} \,, \\
& \simeq 1- \pi \frac{ 2\pi T^{\ast} }{V} , \quad V \gg T^{\ast} \,.
\end{aligned}$$ These expressions indicate the existence of non-universal three-loop terms in the RG $\beta$-function. Indeed, fixing the overall scale at small $T$ by $G_{1/2}(T) = (T/T^{\ast}_{1})^{2}$ with $T_{1}^{\ast} = \sqrt{12} T^{\ast}$ gives the above constant $c_{\ast}=\pi^{2}/(4\sqrt{3}) \simeq 1.424 $. At the same time, fixing the scale at small $V$ by $G_{1/2}(V) = (V/T^{\ast}_{2})^{2}$ with $T_{2}^{\ast} = 2\pi \sqrt{12} T^{\ast}$ produces $c_{\ast}=\pi/(2\sqrt{3}) \simeq 0.91$.
This means, firstly, that the three-loop term $\sim c_{3} g^{3} G^{2}(1-G)^{2}$ in the RG equation (\[RGeq-generic\]) has a *different* prefactor $c_{3}$, depending on whether the choice of low-energy cutoff is $T$ or $V$. This fact was noted in [@Aristov2009] on the basis of direct computation of perturbative corrections. From the above estimate $c_{\ast} = K ^{-4c_{3}(1-K)}$ we retrieve $c_{3} \simeq 0.255$ and $c_{3}\simeq -0.070$ for $G_{1/2}(T)$ and $G_{1/2}(V)$, respectively.
Secondly, in the absence of three-loop RG terms ($c_{\ast}=1$) the ratio $G^{K}/(1-G)$, appearing in (\[ratio-conduc\]), should be a linear function of $V$, $T$ at $K=1/2$. Plotting this ratio for the functions (\[G12V0T\]), (\[G12VT0\]), we compare it with the straight line corresponding to Eq. (\[ourG12\]). We confirm much better agreement with the straight line in the case of the non-linear conductance $G_{1/2}(V,T\to0)$, see Fig. \[fig:plotK12\].
In practical terms these observations mean the following. When fitting experimental data with one universal curve for the whole range of conductances, one should use slightly different expressions for $G(V=0,T)$ and $G(V,T=0)$. The generic formula is (\[explicitG-c3\]), where the value $c_{3}=1/4$ is appropriate for $G(V=0,T)$, while $c_{3}\simeq -0.07$ is better suited for $G(V,T=0)$.
Oscillatory nonlinear conductance
----------------------------------
Let us also discuss the case $T=0$ and $VL$ finite. The expression (\[Lambda-gen\]) is reduced in this case to $$\label{oscLambda}
\begin{aligned}
\Lambda &= \ln\left( \frac{\omega _{c} e }{V} \right) + f (2VL/v) \,, \\
f(x) &= \mbox{Ci }(x) - \sin x/x \,,\\
& \simeq - \cos x/x^{2} \,,\quad x\gg1\,, \\
&\simeq \gamma_{E} -1 +\ln x \,, \quad x \ll 1 \,,
\end{aligned}$$ with $\gamma_{E} \simeq 0.577\ldots$. We see the appearance of oscillations in $2VL$, discussed in [@Dolcini2003] for the case of weak impurity. In our treatment it corresponds to $G\simeq 1$, and from (\[explicitGLambda\]) may may represent the conductance as follows $$G = 1 - c_{\ast}\, \left( {\omega _{0}}/{V } \right) ^{2 (1- K )} \exp\left (2(1-K) f ({2VL}/{v}) \right)$$ cf. (\[GviaV\]). In the limit of large $2VL/v$ we have $|f ({2VL}/{v})|\ll 1$ and $$\exp\left (2(1-K) f ({2VL}/{v}) \right) \simeq 1 + 2(1-K) f ({2VL}/{v})$$ This is in agreement with [@Dolcini2003] where the corresponding expression in this limit and in our notation reads as $$1+f^{\mbox{\small osc}}_{\mbox{\small BS}} \simeq 1 - 2(1-K) \frac{ \cos(2VL/v)}{(2VL/v)^{2}} \,.$$
Conclusion \[sec:conclusion\]
=============================
Electron transport through one-dimensional quantum wires of various types has been studied experimentally in several recent works. In a typical set-up stationary charge transport is measured in a two-point geometry of a system of one or several wires connected by a junction. The quantum wires are adiabatically connected to reservoirs kept at a fixed chemical potential and temperature. These systems are described by modeling the quantum wires as Luttinger liquids (spinless, or spinful) of fermions with linear dispersion subject to point-like interaction and treating the reservoirs as non-interacting. A useful picture of the transport process is to think of individual electrons entering the interaction region (quantum wires plus junction) from an initial reservoir and leaving as individual electrons into the final reservoir. If we model the reservoirs as non-interacting systems there is no room for collective excitations such as fractional quasiparticles or multiple quasiparticles in the final state.
Conventionally this problem has been addressed by the bosonization method, which takes advantage of the fact that the exact excitations of a clean Luttinger wire are bosons, at least in the infinite wire. The problem of including the transformation of incoming electrons into bosons has been addressed for the clean wire, and is believed to be solved. For the case of semi-wires connected by a junction there is no convincing calculation of the above transformation available. In order to avoid this difficulty we are using a fermionic representation.
Our approach starts with determining the leading scale-dependent contributions to the conductances in all orders of perturbation theory. We have demonstrated in the linear response case that by summing up these terms one arrives at a description of the critical properties near the fixed points (i.e. the location of the FPs and the critical exponents describing the power laws followed by the conductances). For this it is necessary to establish the scaling property of the conductances (or else to assume its validity, which is usually done), allowing to derive a set of renormalization group equations out of the perturbative result.
In the present paper we followed this approach for the case of stationary non-equilibrium transport. We first derived a general result for the scale-dependent terms in the conductances of an $n$-lead junction in first order of the interaction. Then we presented the infinite order summation for the dressed interaction. At this point we specialized our considerations to the case of two symmetric semi-wires. We derived the corresponding RG-equation for the conductance. In general the scaling is dependent on three energy scales, bias voltage $V$, temperature $T$, and infrared cut-off provided by the wire length $v/L$. Whenever one of these energy scales dominates, the scaling variable is varying logarithmically, $\Lambda =\ln (%
\frac{v/a}{\max \{V,T,v/L\}})$. In the case $V,T \gg v/L$ we were able to determine the form of the scaling variable describing the crossover from the regime characterized by $V\gg T$ to $V\ll T$ , as well.
The intermediate results presented for the general case of an $n$-lead junction should be a good starting point for analyzing the behavior of non-equilibrium transport through $Y$-junctions or even four-lead junctions. Work in this direction is in progress.
We are grateful to D. A. Bagrets, I. V. Gornyi and D. G. Polyakov for useful discussions. This work was partly supported by the German-Israeli Foundation (GIF) and by a BMBF grant. The research of D.A. was supported by the Russian Scientific Foundation grant (project 14-22-00281).
Normalization of wave functions \[sec:app:wavefunctions\]
=========================================================
The usual summation over the quantum states in the infinite medium is done as an integration over the momentum $\int dk/(2\pi)$ or the summation, $\sum
_{n}$ over the quasi-momentum $k = 2\pi n/L$ in a ring geometry with a finite length $L$. In our situation with a broken translational symmetry we should resort to the integration over the energy, then the correct normalization factor is given by the density of states, which is the inverse Fermi velocity in the simplest situation, $\sum _{n} \to \int dE/v_{F}$. In case with several Fermi velocities, $v_{j}$, in different wires we shall keep the integration over the energy, and the normalization factor enters the definition of the wave functions.
Thus in the formula for the retarded Green’s function, $$\widehat{G}_{E}^{R}(l,y|j,x)=\int dE \frac{\phi_{E,l}(y) \phi^{\ast}_{E,j}(x)}{\omega-E+i0} ,
\label{app:defGreen}$$ we adopt the wave functions in the $j$th wire in the form $$\phi_{E,j}(y) = e^{iE\,y/v_{j}} /\sqrt{2\pi v_{j}}$$ and come to the formula (\[eq:Green\]). Notice also that the integration in (\[app:defGreen\]) should be restricted by the electronic bandwidth $%
|E| < W =E_{F}$, which can be modeled by introducing the density of states function, $N_{j} (E)$, with the property $N_{j} (0) = v_{j}^{-1}$. So strictly speaking the formulas (\[eq:Green\]) are defined at $|\omega| \ll
W$, which justifies the upper cutoff in energy in the calculation of logarithmic corrections and the RG procedure.
Keldysh structure of the triangle $T$
=====================================
The straightforward calculation shows that only a few terms in the complicated expression for $T$ contribute to the final result. Let us sketch here the derivation and present arguments showing the selection of the relevant terms.
To condense our writing, we use the position dependent notation, $%
G^{R}_{\omega} \to R$, and position in the product denotes the position in the initial expression, (\[def:T\]). So that $G^{R}_{\Omega} (z|x)
G^{L}_{\omega+\Omega}(x|y) G^{A}_{\Omega} (y|z) \leftrightarrow R L A$ etc. Up to a numerical factor we have
$$\begin{aligned}
T_{11} &= RAA + (K+A)(RK - RR + KA) , \\
T_{22} & = RRA+ (RK + KA + AA )(K-R) , \\
T_{21} &= RKA + KAA + RRK - RRR + AAA . \label{eq:TviaRKA}
\end{aligned}$$
The combinations $RRR$ and $AAA$ are necessarily zero for the point $z$ outside the interacting region. We may suggest (and it is confirmed by the direct calculation), that the contributing terms in (\[eq:TviaRKA\]) are those which contain two Keldysh components, $K$. In this sense, we may keep only the terms $$\begin{aligned}
T_{11} &\simeq KRK + KKA , \\
T_{22} & \simeq RKK + KAK , \quad T_{21} \simeq 0 \label{eq:TviaRKA2}
\end{aligned}$$ Note that the notation “$\simeq$” here also means that the combination $KK$ should be *regularized* at $\Omega \to \pm \infty$ by subtracting 1 from the product of distribution factors $h_{j}(\omega+\Omega) h_{l}(\Omega)$. This regularization is suggested by inspection of the corresponding expressions in the direct calculation. A closer inspection shows that the combinations $%
h_{j}(\Omega) h_{l}(\Omega)$, not containing $\omega$ do not contribute to the corrections, when multiplied by $L(\omega)$.
Thus the expressions for $T_{ij}$ can be simplified even further : $$\begin{aligned}
T_{11} &\simeq KKA , \quad T_{22} \simeq RKK , \quad T_{21} \simeq 0
\end{aligned}$$ The last expression means that the corrections to the *incoming* current are absent, because $G^{A}_{\Omega} (y|z)=0$ and $G^{R}_{\Omega}
(z|x) =0$ in this case, due to the step functions in (\[eq:Green\]).
Keldysh kernel of integral equation \[KeldyshPi\]
=================================================
$$\Pi ^{K}=\frac{i}{2\pi }
\Phi_{\omega}^{\ast} (x)
\begin{pmatrix}
\mathbf{1} F(\omega ) & \mathbf{Y} F(\omega) \\
\mathbf{Y} F(\omega ) & \mathbf{K} (\omega)
\end{pmatrix} \Phi_{\omega} (y)$$
with $$K_{jl}(\omega )=\int_{-\infty }^{\infty }\frac{d\Omega }{2}%
\sum_{m,n}S_{jm}^{\ast }S_{lm}S_{ln}^{\ast }S_{jn}(1-h_{m}(\Omega
)h_{n}(\Omega +\omega ))$$The latter quantity may be cast in the form $$\begin{aligned}
\mathbf{K} (\omega ) &=F(\omega )%
\begin{pmatrix}
1 & 0 \\
0 & 1%
\end{pmatrix}
+r^{2}t^{2} F_{2}(\omega,V)
\begin{pmatrix}
1 & -1 \\
-1 & 1%
\end{pmatrix} \,,
\\
F_{2} (\omega,V) &= F (\omega +V)+F (\omega -V)-2F (\omega ) \,.
\end{aligned}$$ with $F(\omega)$ defined in (\[def:F\]). Importantly, $\mathbf{K}(\omega )$ is an even function of $\omega$.
Full form of the solution for $\protect\widehat{L}_{jk}^{R}(x|y) $ \[sec:FullCL\]
=================================================================================
The solution of (\[eq:defC\]) can be found as follows. We iterate the right-hand side of the equation once, to arrive at the diagonal kernel with components of the form $$\frac {g^{2}}v \left[ \delta \left(\frac{x-z}v\right) + i \frac\omega2 (e^{i\omega(x+z)/v } + e^{i\omega |x-z|/v })
\right]$$ and another component obtained from here by changing $x \to L - x$, $y \to L-y$. We pick first the easier part of this iterated kernel, $\propto \delta(x-z)$, and arrive at the equation for $\widehat{C}_{j}(x|y)$ with more complicated inhomogeneity instead of $L_{0}$ and non-singular kernel. This latter kernel shows a jump in its derivative at $x=z$, which we use by twice differentiating $\widehat{C}_{j}(x|y)$ with respect to $x$. We thus arrive at a second-order differential equation, similarly to what was done in [@Aristov2012]. The difference now is that we deal with a $2\times2$ matrix for $\widehat{C}_{j}(x|y)$ for each wire $j$. We determine the solution to this differential equation dependent on the $x$ variable up to terms proportional to $e^{\pm i\omega x/(v_{j}d_{j})}$ which are multiplied by as yet unknown matrices $\hat{A}(y)$, $\hat{B}(y)$, respectively. Considering the initial Eq. (\[eq:defC\]) for $\widehat{C}_{j}(x|y)$ in the simpler cases $x=0$, $x=L$, we form a set of two coupled (matrix) equations for $\hat{A}(y)$, $\hat{B}(y)$, which is eventually solved. As a result we obtain the quantity $\widehat{\mathbf{C}}$ diagonal in wire space, with its diagonal elements $\widehat{C}_{j}$ of the form
$$\begin{aligned}
&\widehat{C}_{j}(x|y)=\frac{2\pi v_{j}g_{j}}{d_{j}^{2}}\delta (x-y)%
\begin{bmatrix}
-g_{j}, & 1 \\
1, & -g_{j}%
\end{bmatrix}%
+\frac{i\pi \omega g_{j}^{2}}{d_{j}^{3}}
\\
&\times e^{{i\omega |x-y|}/{v_{j}d_{j}}}
% \exp (\frac{i\omega |x-y|}{v_{j}d_{j}})%
\begin{bmatrix}
d_{j}sgn(y-x)-1, & g_{j} \\
g_{j}, & d_{j}sgn(x-y)-1%
\end{bmatrix}
\\
&+\frac{i\pi \omega g_{j}^{2}}{d_{j}^{4}q_{j}}\left[
% \exp (\frac{i\omega x}{v_{j}d_{j}})
e^{i\omega x/v_{j}d_{j}}
\hat{A}_{j}(y)+
\frac{d_{j} e^{{i\omega (2L-x)}/{v_{j}d_{j}}}}
{\sin (\frac{\omega L}{v_{j}d_{j}})}
%\exp (\frac{i\omega (2L-x)}{v_{j}d_{j}})
\hat{B}_{j}(y)\right]
\end{aligned}$$
with $d_{j}$, $q_{j}$ defined in (\[def:dq\]) and
$$\begin{aligned}
\hat{A}_{j}(y) &=%
\begin{bmatrix}
g_{j}(q_{j}^{-1}-g_{j}), & g_{j}(1-g_{j}q_{j}^{-1}) \\
(d_{j}-1)(q_{j}^{-1}-g_{j}) , & (d_{j}-1)(1-g_{j}q_{j}^{-1})%
\end{bmatrix}
\\
& \times \cos (\frac{\omega y}{v_{j}d_{j}})
+id_{j}\sin (\frac{\omega y}{v_{j}d_{j}})%
\begin{bmatrix}
-q_{j}^{-1} , & 1 \\
-q_{j}^{-1} , & 1%
\end{bmatrix}
\\
\hat{B}_{j}(y) &=i\cos (\frac{\omega y}{v_{j}d_{j}})%
\begin{bmatrix}
\frac{(d_{j}-1)}{g_{j}} , & (1-d_{j}) \\
1, & -g_{j}%
\end{bmatrix}
\\
&+d_{j}\sin (\frac{\omega y}{v_{j}d_{j}})%
\begin{bmatrix}
\frac{(d_{j}-1)}{g_{j}} , & 0 \\
1 , & 0%
\end{bmatrix}%
\end{aligned}$$
We next use these expressions in Eq. (\[eq:LviaC\]), which can be schematically represented as $$\begin{aligned}
L & = C + C \ast Y \ast L % \\ &
%= C + C \ast Y \ast C + C \ast Y \ast C \ast Y \ast C + \ldots \\ &
= C + C \ast \Upsilon \ast C , \\ \Upsilon &= Y + Y \ast C \ast Y + \ldots = Y \ast (1- C \ast Y)^{-1}
\end{aligned}$$and obtain finally $$\begin{aligned}
\label{eq:completeL}
\widehat{L}_{jk}^{R}(x|y) &=\delta _{jk}\widehat{C}_{j}(x|y)-i\omega \frac{%
2\pi g_{j}g_{k}}{d_{j}^{2}d_{k}^{2}}\Upsilon _{jk}
\\ & \times
\begin{pmatrix}
V_{1,j}(x)V_{2,k}(y) & V_{1,j}(x)V_{1,k}(y) \\
V_{2,j}(x)V_{2,k}(y) & V_{2,j}(x)V_{1,k}(y)%
\end{pmatrix}%
\\
V_{1,j}(x) &=(1-g_{j}q_{j}^{-1})\cos (\frac{\omega y}{v_{j}d_{j}}%
)+id_{j}\sin (\frac{\omega y}{v_{j}d_{j}}) \\
V_{2,j}(x) &=(q_{j}^{-1}-g_{j})\cos (\frac{\omega y}{v_{j}d_{j}}%
)-id_{j}q_{j}^{-1}\sin (\frac{\omega y}{v_{j}d_{j}})
\end{aligned}$$ with $ \Upsilon$ given in Eq. (\[integCV\])
Analytic properties of $L(\omega)$. \[sec:analyticL\]
=====================================================
In contrast to our previous studies [@Aristov2009; @Aristov2012], we see now the appearance of poles in the $\omega$-plane of the quantities $q_{j}$, Eq. (\[def:dq\]), which we further integrate over $\omega$. Given the arbitrariness of the $S$-matrix, reflected in $Y_{ik} = |S_{ik}|^{2}$, we check here the absence of singularities in $L^{R}(\omega)$ in the upper semiplane of complex $\omega$.
The poles of $q_{j}$ correspond to the solution of $$\tan \bar \omega = - i d_{j} ,$$ where we introduced $\bar \omega = \omega L/{v_{j}d_{j}}$. The last equation means that we have an infinite sequence of roots $$\bar \omega = - i \mbox{ arctanh } d_{j} + \pi n ,\quad n = 0, \pm 1, \pm 2
, \ldots$$ hence the poles of $q_{j}^{-1}$ are always in the lower semiplane of complex $\omega$, as it should be for a retarded function.
Less trivial is the question about the position of the poles of the above expression $%
(1-\mathbf{Q}^{-1} \cdot \mathbf{Y} )^{-1}$. We consider it for a simpler situation with identical wires, $g_{j}= g$, $d_{j} =d= \sqrt{1-g^{2}}$, $v_{j} = v$.
The poles are defined by $$det (1-\mathbf{Q}^{-1} \cdot \mathbf{Y} )=
det \left [\mathbf{1}- \frac{g \mathbf{Y} }{ 1 + i d \cot
\bar\omega} \right] = 0 .$$ Since the denominator in the last expression cannot modify the location of poles, and $\sin \bar\omega = 0 $ is not a solution, we can rewrite $$det \left [i d \cot \bar\omega + 1- g \mathbf{Y} \right] = 0 .$$ Defining the eigenvalues of $\mathbf{Y} $ as $y_{j}$, with $j = 1,\ldots N$ (for a junction connecting $N$ wires), the poles are are defined by conditions (cf.above) $
\tan \bar \omega = - i \frac{d } {1- gy_{j} } ,
$ or $$\bar \omega = - i \mbox{ arctanh } \frac{d } {1- gy_{j} } + \pi n ,\quad j =
1,\ldots N .$$
Generally we have $|y_{j}| \le 1$ for all $j$. This is evident for $N=2$, and can be easily extended to any $N$. The proof of this statement is as follows. [@Aristov2011] Consider a set of diagonal $N\times N$ matrices $\lambda_{j}$ with values $1$ in the $j$th row and $0$ otherwise. This is a set of $N$ generators of a Cartan subalgebra of the algebra $U(N)$, normalized according to $\mbox{Tr}(\lambda_{j}%
\lambda_{k})=\delta_{jk}$. A rotation of these generators is defined as $\tilde\lambda_{j} = S^{\dagger} \lambda_{j} S$ where $S$ is the unitary matrix. Obviously the new set $\tilde\lambda_{j}$ remains orthonormal $\mbox{Tr}(\tilde\lambda_{j}%
\tilde\lambda_{k})=\delta_{jk}$. The operator $P$, defined by $PA = \sum_{j}
\tilde\lambda_{j} \mbox{Tr}(\tilde\lambda_{j} A)$, is a projection operator, $%
P^{2}=P$. We have $P \lambda_{k} = \sum_{j}\tilde \lambda_{j} Y_{jk}$ because $\mathbf{Y}$ can be written as $Y_{jk} = |S_{jk}|^{2} =
\mbox{Tr}(\tilde\lambda_{j} \lambda_{k})$. Let $\{ c_{j}\}$ be an eigenvector of $\mathbf{Y}$, i.e. $\sum_{k} Y_{jk}
c_{k} = y c_{j} $. Introducing the diagonal matrix $\lambda_{\ast} = \sum_{j} c_{j} \lambda_{j}$, we obtain $P\lambda_{\ast} = y \tilde \lambda_{\ast}$, with $\tilde
\lambda_{\ast}$ is the rotated vector $\lambda_{\ast}$. Since $\|
P\lambda_{\ast} \| \leq \| \lambda_{\ast} \| $ and $\| \tilde \lambda_{\ast}
\| = \| \lambda_{\ast} \| $, we conclude that $|y| \le 1$.
It follows that the above ratio $\frac{d } {1- gy_{j} }$ is always positive. As a result, the poles of $(1-\mathbf{Q}^{-1} \cdot \mathbf{Y} )^{-1}$ lie in the lower semiplane of $\omega$.
Scaling variable $\Lambda$ in the crossover regime. \[sec:Lambda\]
==================================================================
In the limit $L\to \infty $ we have to evaluate the integral $$P(T,V) = \int _{-W}^{W}\frac{d\omega}{\omega} [F(\omega+V)-F(\omega-V) ] \,,$$ with the upper cutoff being $W = v_{F}/a$ or the bandwidth.
After the rescaling $\omega = 2T x$, $V=2T y$, $W=2T w$ we have $$P(T,V) = 2T % \int _{-w}^{w}\frac{dx}{x} [ (x+y)\coth(x+y) - (x-y)\coth(x-y) ] \,.
\int _{-w}^{w}\frac{dx}{x} \left[ \frac{x+y}{\tanh(x+y)} - \frac{x-y}{\tanh(x-y)} \right] \,.$$ We can make a shift $x \to x\pm y$ in two terms here; the contributions from the limits $\pm w$ do not cancel upon this shift, but add a constant in the limit $ w \to \infty$. After simple calculation we get in this limit $$P(T,V) = 4Ty \left( 2 +
\int _{-w}^{w}\frac{dx \, x\coth x}{x^{2}-y^{2}} \right) \,,$$ where the principal value of the integral should be taken. Next we close the contour of integration in the upper semiplane of complex $x$, by adding a semicircle of radius $w$. The contribution from this semicircle vanishes as ${\cal O}(w^{-2})$ and we reduce the remaining integral to the sum over the residues of $\coth x$ at $x=i\pi n$ as follows : $$\int _{-w}^{w}\frac{dx \, x\coth x}{x^{2}-y^{2}}
\to \sum_{n=1}^{w/\pi} \frac{2 n}{n^{2}+(y/\pi)^{2}}
\,,$$ The last sum is easily evaluated with the final result of the form $P(T,V) = 4 V \Lambda$ with $\Lambda$ given by $$%\label{def:Lambda}
\Lambda = \ln\left( \tfrac{W e}{2\pi T} \right)
- \tfrac12
\left[ \psi \left( 1 + \tfrac{iV}{2\pi T} \right) +\psi \left( 1 - \tfrac{iV}{2\pi T} \right)
\right] \,.$$ with $e= 2.718\ldots$ and $\psi(x)$ digamma function.
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---
abstract: 'Random skew plane partitions of large size distributed according to an appropriately scaled Schur process develop limit shapes. In the present work we consider the limit of large random skew plane partitions where the inner boundary approaches a piecewise linear curve with non-lattice slopes, describing the limit shape and the local fluctuations in various regions. This analysis is fairly similar to that in [@OR2], but we do find some new behavior. For instance, the boundary of the limit shape is now a single smooth (not algebraic) curve, whereas the boundary in [@OR2] is singular. We also observe the bead process introduced in [@beads] appearing in the asymptotics at the top of the limit shape.'
address:
- 'Cedric BoutillierUPMC Univ Paris 06UMR 7599, LPMAF-75005, Paris, France &Département de Mathématiques et Applications (DMA), École Normale Supérieure 75230 Paris, FRANCE '
- 'Sevak Mkrtchyan Rice University, Math Department - MS1366100 S. Main St,Houston, TX, 77005'
- 'Nicolai Reshetikhin UC Berkeley, Department of Mathematics970 Evans Hall 3840,Berkeley, CA 94720'
- 'Peter Tingley MIT dept. of math77 Massachusetts AveCambridge, MA, 02139'
author:
- Cedric Boutillier
- Sevak Mkrtchyan
- Nicolai Reshetikhin
- Peter Tingley
title: Random skew plane partitions with a piecewise periodic back wall
---
[^1]
Introduction {#sec:intro}
============
A *skew plane partition* with inner boundary being given by a partition $\lambda$ is any array of positive integers $\pi= \{\pi_{i,j} \}$ defined for pairs $(i,j)$ such that $i\geq 1$ and $j\geq \lambda_i$, which is non-increasing in both $i$ and $j$. A skew plane partition is called [*bounded*]{} by $N$ and $M$ if $\pi_{ij}=0$ when $i>N$ or $j>M$. An example of a skew plane partition with inner shape $\lambda=(3,2,2)$ is given on Figure \[fig:skew\_partition\]. Looking at this picture we can also see that a bounded skew plane partition can be regarded as monotonic piles of cubes in a certain “semi-infinite room” with no roof (see also Figure \[fig:empty\]).
A natural question to ask is: what is the shape of a typical pile containing a large number of cubes? One must first make this precise by fixing a probability distribution. One natural distribution to choose would be the uniform distribution on all skew plane partitions with a fixed number of cubes. This is actually quite difficult to deal with, and we instead consider the following probability measure (*grand-canonical ensemble*) on bounded skew plane partitions: $$\label{eq:distr}
P(\pi)=\frac{1}{Z} q^{|\pi|},$$ where $|\pi|=\sum_{i,j}\pi_{i,j}$ is the total number of boxes in the piles corresponding to the plane partition $\pi$, $0<q<1$ and $$Z=\sum_{\pi} q^{|\pi|}$$ is the normalization factor known as the *partition function* in statistical mechanics. The larger $q$ is, the more likely it is to have many cubes. We will study certain limits of this system as $q$ approaches $1$, the bounds $M$ and $N$ approach infinity, and the partition $\lambda$ defining the back wall of the room grows in a specified way.
There are many ways to take such a limit. For instance, one can specify that, after an appropriate rescaling, $\lambda$ approaches a fixed curve. The limiting behavior of the system will certainly depend on this curve. Some such cases have previously been studied:
- The case when $\lambda$ is empty was studied in the unbounded case in [@CerfKenyon] and in the bounded case in [@OR1].
- The case when $\lambda$ approaches a piecewise linear curve with lattice slopes was studied in [@OR2].
In this article we consider a limit where $\lambda$ approaches a piecewise linear function with a (non-lattice) slope staying *strictly* between $-1$ and $1$. The large size limit is studied via *correlation functions* using the formalism and notations developed in [@OR1; @OR2].
Before we start, let us mention some other related work. Using results of this article, the limit where $\lambda$ approaches a piecewise linear function with arbitrary slopes has been studied in [@M]. Local fluctuations and the free energy for the partition function on a torus were first computed in [@NHB].
Random skew plane partitions and Correlation functions
------------------------------------------------------
Our piles of unit cubes are in ${\mathbb R}^3$ with coordinates $(x,y,z)$ centered at points $({\mathbb Z}+\frac{1}{2},{\mathbb Z}+\frac{1}{2},{\mathbb Z}+\frac{1}{2})$ and bounded in $(x,y)$ directions as it was described above. That is, the $(x,y)$ coordinates of centers of cubes satisfy conditions: $$\begin{cases}
\frac{1}{2} \leq x \leq N-\frac{1}{2},\\
\lambda_{x+1/2} \leq y \leq M.
\end{cases}$$ The partition $\lambda $ is given and describes the configuration of the back wall. Denote this region of ${\mathbb R}^3$ by $D_{\lambda, M,N}$.
![ \[fig:empty\] An empty skew plane partition. Here the inner shape is given by the partition $(6,3,3,3,1)$.](back-wall.pdf){width="8cm"}
The mapping $$(x,y,z)\mapsto \bigl(t=y-x,h=z-\frac{x+y}{2}\bigr)$$ projects ${\mathbb R}^3$ to ${\mathbb R}^2$. This projection maps piles of cubes to tilings of the region $$E_{M, N}=\bigl\{(t,h)\ |\ -M\leq t\leq N;
h\geq \max\{-M-t/2, N+t/2\}\bigr\}$$ which differ from the one corresponding to the empty pile in finitely many places. This is evident from Fig. \[fig:skew\_partition\].
![ A skew plane partition represented as a stepped surface. Notice that the same picture can be viewed as a tiling of a portion of the plane by three types of rhombi. []{data-label="fig:skew_partition"}](3d_skew_part2.pdf){width="5cm"}
A convenient characterization of a system of random skew plane partitions is the collection of local correlation functions $\rho_{(t_1,h_1),\dots(t_k,h_k)}$ giving the probability to see horizontal rhombi at $(t_1,h_1),\dots,(t_k,h_k)$: $$\rho_{(t_1,h_1),\dots, (t_k, h_k)}=\sum_{\pi} P(\pi)\prod_{a=1}^k\sigma_{i_a,j_a}(\pi)$$ where $\sigma_{t,h}(\pi)=1$ when $(t,h)$ is the position of the center of a horizontal rhombus and $\sigma_{t,h}(\pi)=0$ if not.
An explicit formula was found in [@OR1; @OR2] for such correlation functions. To describe this formula, we first encode the boundary of the partition $\lambda$ as a function $h=-b(t)$, where $$\label{eq:bt}
b(t)=\frac{1}{2}\sum_{i=1}^n|t-v_i|-\frac{1}{2}\sum_{i=1}^{n-1}|t-u_i|.$$ The points $u_1,\dots,u_{n-1}$ (resp. $v_1,\dots,v_n$) represent the inner (resp. the outer) corners of the back wall. See Figure \[fig:empty\]. It is convenient to set $u_0=-M$ and $u_n=N$. The lines $t=u_0$ and $t=u_n$ are limiting on the left and on the right the region tiled by rhombi.
Then we introduce the partition of the horizontal interval ${\mathbb Z}+ \frac{1}{2} \cap (u_0,u_n)$ into $D_-\cup D_+$ where $$\begin{aligned}
D^+ &=\bigl\{m \in {{\mathbb Z}}+ \frac{1}{2} \cap (u_0,u_n)\ |\ \text{where the boundary $\lambda$ in $(t,h)$ has slope $-\frac{1}{2}$ at $m$} \}, \\
D^- &=\bigl\{m \in {{\mathbb Z}}+ 1/2 \cap (u_0,u_n)\ |\ \text{where the boundary $\lambda$ in $(t,h)$ has slope $+\frac{1}{2}$ at } m \bigr\}.\end{aligned}$$
Define the functions $\Phi_\pm(z,t)$ by: $$\Phi_+(z,t)=\prod_{\substack{m\in D^+\\ m > t }}(1-zq^m), \quad
\Phi_-(z,t)=\prod_{\substack{ m\in D^-\\ m < t }}(1-z^{-1}q^{-m}).$$
The following holds:
[@OR2] \[thm:OR2\_main\] Correlation functions are determinants of the correlation kernel: $$\label{eq:det}
\rho_{(t_1,h_1),\dots, (t_k, h_k)}=\det \bigl[K\bigl((t_i,h_i),(t_j,h_j)\bigr)\bigr]_{1\leq i, j \leq k}$$ The correlation kernel $K$ is given by $$\begin{gathered}
\label{eq:main-corr2}
K((t_1,h_1),(t_2,h_2))= \frac{1}{(2\pi i)^2}
\oint_{z\in C_1}\oint_{w\in C_2}
\frac{\Phi_-(z,t_1)\Phi_+(w,t_2)}{\Phi_+(z,t_1)\Phi_-(w,t_2)}\\
\times\frac{\sqrt{zw}}{z-w}z^{-h_1-b(t_1)-1/2}w^{h_2+b(t_2)-1/2}\frac{dzdw}{zw},\end{gathered}$$ where $C_1$ is a simple positively oriented contour around $0$ such that its interior contains none of the zeroes of $\Phi_+(z,t_1)$ . Similarly $C_2$ is a simple positively oriented contour around $0$ such that its exterior does not contain zeroes of $\Phi_-(w,t_2)$. Moreover, if $t_1< t_2$, then $C_1$ is contained in the interior of $C_2$, and otherwise, $C_2$ is contained in the interior of $C_1$.
Main results {#subsec:main_intro}
------------
We are interested in the behavior of random skew plane partitions sampled according to the probability distribution for *large domains*, meaning that a limit of the following form is taken:
- Fix a sequence $(r_k)$ of positive real numbers such that $\lim_{k\to \infty} r_k=0$ and for all $k$, let $q_k=\exp(-r_k)$.
- Choose sequences $(M_k)$, $(N_k)$ and $(\lambda_k)$ such that as $k$ goes to infinity, the 3-dimensional domain $r_k D_{\lambda_k, M_k, N_k}$ converges to some region $D$ of ${\mathbb R}^3$.
- Find characteristic scales of fluctuations for large $k$ and describe random plane partitions at appropriate scales in the limit $k\to \infty$.
The increasing sequence of domains where $r_k =\frac{1}{k}$ and the corners $u^{(k)}_i$ and $v^{(k)}_i$ describing the partition $\lambda_k$ are given by: $$u^{(k)}_i=kU_i, \quad v^{(k)}_i=kV_i,$$ where $n$ and $U_0 \leq V_1 \leq \dots \leq V_n \leq U_n$ are fixed integers was studied in [@OR2]. As $k\to \infty$, the function $(x,y)\mapsto h_k(x,y)=\pi_{kx,ky}/k$ converges in probability to a fixed limit shape. This function is infinitely differentiable everywhere except along a curve where the surface it defines merges with the walls. Along the curve, its second derivative is discontinuous (the first derivative has a square root singularity). The projection of this curve to the $(t,h)$-plane is called the *arctic circle* or the boundary of the frozen region. Indeed outside this curve the tiling by rhombi is “frozen", which is to say that with probability tending to 1, the tiles outside of the curve are of only one type.
The results in [@OR2] are obtained by using the steepest descent method to evaluate the integrals in Equation (\[eq:main-corr2\]). In the limit one obtains exact formulas for the limit shape, as well as descriptions of the behavior of local correlation functions.
In this paper, we focus on a different type of increasing sequences of domains. We keep $r_k$ equal to $\frac{1}{k}$, but we assume that the partitions $\lambda_k$ describing the back wall consist of $n$ segments growing in length, and that each of the segments is not straight anymore, but is a “staircase” with fixed slope. To be precise, the $j^{th}$ segment of $\lambda_k$ consists of $k L_j$ copies of the corner with sides $(a_j, b_j)$ for some $L_j,a_j, b_j \in {\mathbb Z}_{>0}$ fixed, see Figure \[fig:NotationGraph\].
In the framework described above, we compute the limit shape of the random pile of boxes as $k\to \infty$ and correlation functions. We show that:
- The system rescaled by a factor $r_k$ converges to a deterministic limit shape. This is a smooth transcendental function in the bulk. The frozen boundary consists of a single smooth curve.
- As in [@OR2], correlation functions in the bulk are given by the incomplete Beta kernel.
- For most values of $t$, if one fixes $t$ and allows $h$ to approach infinity, the slope of the limit shape turns vertical, and the determinantal process with the incomplete Beta kernel describing the statistics of horizontal tiles degenerates into the bead process introduced in [@beads].
Relation to Dimer models
------------------------
The limit shapes for rhombi tilings of bounded domains with piecewise linear boundary parallel to axes $h=const, h\pm t/2=const$ were studied in [@KO] where it was shown that uniformly distributed tilings develop a limit shape given by an algebraic curve. This generalizes the arctic circle theorem for large hexagons [@CohnLarsenPropp; @J]. Local correlation functions for large hexagonal domains with the uniform distribution of rhombi tilings were studied in [@J; @G], and in [@Ke:GFF2].
The rhombi used in these tilings can be thought of as *dimers*, since they are the union of two adjacent faces of the triangular lattice (*monomers*). These random tilings are thus examples of *dimer models*. Dimer models have a long history in statistical mechanics, see for example [@Ka; @TempFish; @MCW]. Corresponding tiling models with height functions were first studied in the physics literature, where many ideas about the limit shape and the structure of correlation functions in the thermodynamical limit can be traced, see for example [@NHB].
Acknowledgments
---------------
We are grateful to A. Borodin, A. Okounkov, and P. van Moerbeke for interesting discussions.
Setup: a piecewise periodic back wall
=====================================
We are studying a system of random skew plane partitions, as described in Section \[sec:intro\]. The precise system is described in Section \[subsec:main\_intro\]. Let us start by introducing some more notation.
Let $\tau = rt$ and $\chi = rh$ be the rescaled coordinates. The $\tau$ coordinates where the slope of the back wall changes are denoted by $A_0, A_1,\dots ,A_n$, where $A_0$ and $A_n$ are the bounds of the shape. The $t$ coordinates that correspond to these corners will be $A_1/r,\ldots,A_n/r$. Between $A_{k-1}/r$ and $A_k/r$ we have a succession of patterns made of $a_k$ ascending steps and $b_k$ descending steps, repeated $L_k/r$ times. The slope of the piece of wall between $A_{k-1}$ and $A_k$ is $\alpha_k = \frac{a_k-b_k}{2(a_k+b_k)}$. It will be convenient to define $\alpha_0=-\frac{1}{2}$ and $\alpha_{n+1}=\frac{1}{2}$.
![The setup []{data-label="fig:NotationGraph"}](NotationGraph.pdf){width="14.6cm"}
Let $j$ be the integer in $\{1,\dots,n\}$ such that $\tau \in (A_{j-1},A_j]$. Define the function $B(\tau)$ by $$\label{eq:B}
-B(\tau) = -\frac{1}{2}A_0+ \sum_{k=1}^{j-1}\alpha_k (A_{k}-A_{k-1}) + \alpha_j (\tau-A_{j-1}) = -\sum_{k=0}^{j-1} A_k (\alpha_{k+1} -\alpha_k) + \alpha_j \tau,$$ if $ A_{j-1} <\tau < A_j$. $B(\tau)$ is the limiting rescaled boundary function $b(t)$ of Equation expressed in the rescaled variables $\tau$ and $\chi$. We can decide to define $B(\tau)$ from the rightmost corner $A_n$ instead of the leftmost one $A_0$, yielding an alternative expression: $$\label{eq:B2}
-B(\tau) = \frac{1}{2}A_n -\sum_{k=j+1}^{n}\alpha_k(A_{k}-A_{k-1}) -\alpha_j(A_{j}-\tau)
= \alpha_j \tau + \sum_{k=j}^{n}(\alpha_{k+1}-\alpha_k) A_k.$$
We are interested in the correlation functions in the scaling limit when $\lim_{r\rightarrow +0} r t_1=\lim_{r\rightarrow +0}r t_2 \text{ and } t_1-t_2=\Delta(t) \text{ is a constant}.$
The function S(z)
-----------------
We want to use the steepest descent method in order to find the asymptotics of local correlation functions as the size of the system increases and $r\to 0$. The first step in doing this is to rewrite the integral defining the correlation kernel in Theorem \[thm:OR2\_main\] as $$K\bigl( (t_1,h_1),(t_2,h_2) \bigr) = \frac{1}{(2i\pi)^2} \oint \oint e^{\frac{S^{(r)}_{t_1,h_1}(z)-S^{(r)}_{t_2,h_2}(w)}{r}} \frac{1}{z-w} \frac{dz}{2i\pi z} \frac{dw}{2i\pi w},$$ show that $S^{(r)}_{t,h}(z)$ converges to some function $S_{\tau,\chi}(z)$ as $r\to 0$, $r t\to \tau$, $r h\to \chi$ and find the critical points of $z\mapsto S_{\tau,\chi}(z)$.
The function $S^{(r)}_{t,h}(z)$ is given by: $$\begin{gathered}
S^{(r)}_{t,h}(z)= r\log \left(z^{-h-b(t)}\frac{\Phi_-(z,t)}{\Phi_+(z,t)}\right) \\
=-r(h+b(t))\log z + \sum_{\stackrel{m<t, m\in D^-,}{ m\in {\mathbb Z}+{\tfrac12}}}r\log(1-z^{-1}e^{rm}) -\sum_{\stackrel{m>t, m\in D^+,}{ m\in {\mathbb Z}+{\tfrac12}}}r\log(1-ze^{-rm}).\end{gathered}$$ Here and in the sequel, $\log$ denotes the branch of the logarithm with an imaginary part in $(-\pi,\pi)$, with a cut along $\mathbb{R}_{-}$.
In order to find the limit $S_{\tau,\chi}(z)$ of this quantity, we first state a lemma:
\[lem:CD\] If $[C,D]$ corresponds to a piece of wall with parameters $a,b$ then $$\label{eq:oon} \lim_{r\rightarrow 0}
\sum_{\stackrel{\tfrac Cr<m'<\tfrac Dr, m'\in D^-,}
{m'\in{\mathbb Z}+{\tfrac12}}}
r\log(1-z^{-1}e^{rm'})=\frac{a}{a+b}
\int_{C}^{D} \log (1-z^{-1}e^v)dv$$ and $$\begin{aligned}
\lim_{r\rightarrow 0}
\sum_{\stackrel{\tfrac Cr<m'<\tfrac Dr, m'\in D^+,}
{m'\in{\mathbb Z}+{\tfrac12}}}
r\log(1-z e^{-rm'})=
\frac{b}{a+b}
\int_{C}^{D} \log (1-z e^{-v})dv.\end{aligned}$$
The proof is elementary. Here is how it goes for : $$\begin{aligned}
\quad&
\lim_{r\rightarrow 0}
\sum_{\stackrel{\tfrac Cr<m'<\tfrac Dr, m'\in D^-,}
{m'\in{\mathbb Z}+{\tfrac12}}}
r\log(1-z^{-1}e^{rm'})=
\lim_{r\rightarrow 0}
\sum_{\stackrel{C<m<D, m\in D^-,}{ m\in r({\mathbb Z}+{\tfrac12})}}
r\log(1-z^{-1}e^{m})
\\=\quad&
\lim_{r\rightarrow 0}
\sum_{\stackrel{C<m<D,}{ m\in r(a+b)({\mathbb Z}+{\tfrac12})}}
\sum_{k=0}^{a-1} r\log(1-z^{-1}e^{m+kr})
\\=\quad&
\frac{1}{a+b}
\sum_{k=0}^{a-1}
\lim_{r\rightarrow 0}
\sum_{\stackrel{C<m<D,}{ m\in r(a+b)({\mathbb Z}+{\tfrac12})}}
r(a+b)\log(1-z^{-1}e^{m+kr})
\\\stackrel{\text{defn. of } \int}{=}&
\frac{1}{a+b}
\sum_{k=0}^{a-1}
\int_{C}^{D} \log (1-z^{-1}e^v)dv =
\frac{a}{a+b}
\int_{C}^{D} \log (1-z^{-1}e^v)dv.\end{aligned}$$ The convergence is uniform in $z$ on compact sets of $\mathbb{C}\setminus [e^C,e^D]$.
Using Lemma \[lem:CD\] for each interval $[A_k,A_{k+1}], [A_{j-1},\tau],[\tau,A_j]$, and the fact that $$\lim_{rt \to \tau} r b(t) = B(\tau),$$ we get $$\begin{aligned}
S_{\tau,\chi}(z):=&\lim_{r\rightarrow 0} S^{(r)}_{t,h}(z)
= \nonumber
-(\chi + B(\tau))\log z
\\&
+\sum_{k=1}^{j-1}
\frac{a_k}{a_k+b_k}
\int_{A_{k-1}}^{A_k}
\log \bigl(1-z^{-1}e^v\bigr) dv
+\frac{a_j}{a_j+b_j}
\int_{A_{j-1}}^\tau
\log\bigl(1-z^{-1}e^v) dv
\\ \nonumber &
-\frac{b_j}{a_j+b_j}
\int_{\tau}^{A_j}
\log\big(1-ze^{-v}\bigr) dv
-\sum_{k=j+1}^{n}
\frac{b_k}{a_k+b_k}
\int_{A_{k-1}}^{A_k}
\log \bigl(1-z e^{-v}\bigr)dv.\end{aligned}$$
Critical points of S(z)
=======================
The localization of the critical points of the function $S_{\tau,\chi}$ is crucial in order to apply the steepest descent method. The following proposition gives a complete understanding of their nature.
\[prop:regions\] Fix $A_0 < \tau < A_n$. Then,
- there is a unique real number $\chi_0(\tau)$ such that
1. For any $\chi> \chi_0(\tau)$, $S_{\tau,\chi}(z)$ has exactly two non-real critical points, which are complex conjugates.
2. For any $\chi<\chi_0(\tau)$, $S_{\tau,\chi}(z)$ has only real critical points.
- If $\tau\in (A_{j-1}, A_j)$ ,as $\chi\to +\infty$ the two non-real critical points $z_c,\bar{z}_c$ have the following asymptotic: $$\label{eq:crit_point_asymp}
z_c,\bar{z}_c= e^{\tau} \exp\left( -e^{-\chi} \rho(\tau) e^{\pm i\pi\left(\frac{1}{2}+\alpha_j\right)}\right) (1+O(e^{-\chi})),$$ where $$\rho(\tau)=\prod_{k=0}^n \left| 2 \sinh \left(\frac{\tau-A_k}{2}\right)\right|^{\alpha_{k+1}-\alpha_k},$$ and $z_c$ is chosen to be the critical point with positive imaginary part.
- If $\tau=A_j+\delta$, $\chi\to +\infty $ and $\delta\to 0$ such that $$p=e^{\chi-\chi^{(j)}}|\delta|^{1+\alpha_j-\alpha_{j+1}}$$ is fixed, with $$e^{\chi^{(j)}} = \rho^{(j)} = \prod_{\substack{k=0 \\ k\neq j}}^{n} \Bigl| 2\sinh\bigl(\frac{A_j-A_k}{2}\bigr)\Bigr|^{\alpha_{k+1}-\alpha_k},$$ then the pair of complex conjugate solutions behave as $z=e^{\tau-s|\delta|}$ where $s$ is a solution to the equation $$\label{eq:t-scaling}
p=e^{\pm i\pi (\frac{1}{2}+\alpha_j)}\frac{(s-\mathrm{sign}(\delta))^{(\alpha_{j+1}-\alpha_j)}}{s}.$$
The proof is rather technical, and requires a number of intermediate statements. It will be completed at the end of this section.
An expression for S’(z)
-----------------------
Using the identities $$\frac{a_k}{a_k+b_k}=\frac{1}{2}+\alpha_k,\quad \frac{b_k}{a_k+b_k}=\frac{1}{2}-\alpha_k,
\quad \log \bigl(\frac{1-z e^{-v}}{1-z e^{-u}}\bigr)=u-v + \log\bigl(\frac{z-e^{v}}{z-e^{u}}\bigr),$$ and the formula for $B(\tau)$ , one gets the following expression for the derivative of $S$ with respect to $z$:
$$\begin{aligned}
z\frac{d}{dz}S(z) \nonumber =&
-(\chi+B(\tau))
+\sum_{k=1}^{j-1}
\frac{a_k}{a_k+b_k}
\int_{A_{k-1}}^{A_k}
\frac{z^{-1}e^v}{1-z^{-1}e^v}dv
+\frac{a_j}{a_j+b_j}
\int_{A_{j-1}}^{\tau}
\frac{z^{-1}e^v}{1-z^{-1}e^v}dv
\\ \nonumber &\quad
+\frac{b_j}{a_j+b_j}
\int_{\tau}^{A_j}
\frac{z e^{-v}}{1-z e^{-v}}dv
+\sum_{k=j+1}^n
\frac{b_k}{a_k+b_k}
\int_{A_{k-1}}^{A_k}
\frac{z e^{-v}}{1-z e^{-v}}dv
\\ \nonumber =&
-(\chi+B(\tau))
-\sum_{k=1}^{j-1}
\frac{a_k}{a_k+b_k}
\log \left( \frac{z-e^{A_k}}{z-e^{A_{k-1}}} \right)
-\frac{a_j}{a_j+b_j}
\log \left( \frac{z-e^{\tau}}{z-e^{A_{j-1}}} \right)
\\ \nonumber &
+\frac{b_j}{a_j+b_j}
\log \left( \frac{1-ze^{-A_j}}{1-ze^{-\tau}} \right)
+\sum_{k=j+1}^{n}
\frac{b_k}{a_k+b_k}
\log \left( \frac{1-ze^{-A_k}}{1-z e^{-A_{k-1}}} \right)
\\ \nonumber =&
-(\chi+B(\tau))
-\sum_{k=1}^{j-1}
\frac{a_k}{a_k+b_k}
\log\left( \frac{z-e^{A_k}}{z-e^{A_{k-1}}} \right)
-\frac{a_j}{a_j+b_j}
\log\left(\frac{z-e^{\tau}}{z-e^{A_{j-1}}}\right)
\\ \nonumber &
+\frac{b_j}{a_j+b_j}
\log\left( \frac{z-e^{A_j}}{z-e^{\tau}} \right)
+\sum_{k=j+1}^{n}
\frac{b_k}{a_k+b_k}
\log\left( \frac{z-e^{A_k}}{z- e^{A_{k-1}}} \right)
\\ \nonumber &
\underbrace{
-\frac{b_j}{a_j+b_j}(A_{j}-\tau)
-\sum_{k=j+1}^{n}
\frac{b_k}{a_k+b_k}(A_{k}-A_{k-1})}
_{B(\tau)+\frac{\tau}{2}}
\\=&\label{eq:SPrime}
-(\chi-\frac{\tau}{2})
-\sum_{k=1}^{j-1}
(\frac{1}{2}+\alpha_k)
\log\left(\frac{z-e^{A_k}}{z-e^{A_{k-1}}}\right)
-(\frac{1}{2}+\alpha_j)
\log\left(\frac{z-e^{\tau}}{z-e^{A_{j-1}}}\right)
\\ \nonumber &
+(\frac{1}{2}-\alpha_j)
\log\left(\frac{z-e^{A_{j}}}{z-e^{\tau}}\right)
+\sum_{k=j+1}^{n}
(\frac{1}{2}-\alpha_k)
\log\left(\frac{z-e^{A_k}}{z-e^{A_{k-1}}}\right).\end{aligned}$$
Note that each of the terms $\log\left(\frac{z-e^{A_k}}{z-e^{A_{k-1}}}\right)$ is a holomorphic function of $z$ on $\mathbb{C}\setminus[e^{A_{k-1}},e^{A_k}]$. Its imaginary part is in $(-\pi,\pi)$ and represents the angle between the vectors $z-e^{A_k}$ and $z-e^{A_k-1}$. This angle is 0 when $z\in\mathbb{R}\setminus[e^{A_{k-1}},e^{A_k}]$, and approaches $\pi$ (resp. $-\pi$) when $z$ approaches $(e^{A_{k-1}},e^{A_k})$ from above (resp. from below). See Figure \[fig:angles\].
Asymptotic of critical points when chi goes to infinity {#sec:crit_points_large}
-------------------------------------------------------
Here we will prove the last two statements of Proposition \[prop:regions\] concerning the asymptotics of the non real critical points when $\chi$ is large.
\[lem:largeXcrit\] Fix $A_0 < \tau < A_n$. Then, for sufficiently large $\chi$, $S_{\tau, \chi}(z)$ has exactly two critical points, which are complex conjugates and have non-zero imaginary parts. Their asymptotic as $\chi\to +\infty$ is described by the last part of proposition \[prop:regions\].
$A_{j-1} < \tau < A_j$.
Critical points are solutions to the equation $$\label{eq:crit}
z\frac{d}{dz}S(z) = 0.$$
The real part of this equation can be written as $$\begin{aligned}
0=
\Re \left( z\frac{d}{dz}S(z) \right)
&=
-(\chi-\frac{\tau}{2})
-\sum_{k=1}^{j-1}
(\frac{1}{2}+\alpha_k)
\log\left|\frac{z-e^{A_k}}{z-e^{A_{k-1}}}\right|
-(\frac{1}{2}+\alpha_j)
\log\left|\frac{z-e^{\tau}}{z-e^{A_{j-1}}}\right|
\\&
+(\frac{1}{2}-\alpha_j)
\log\left|\frac{z-e^{A_{j}}}{z-e^{\tau}}\right|
+\sum_{k=j+1}^{n}
(\frac{1}{2}-\alpha_k)
\log\left|\frac{z-e^{A_k}}{z-e^{A_{k-1}}}\right|
\\&=
-(\chi-\frac{\tau}{2})
-\log\bigl|1-ze^{-\tau}\bigr|
+\sum_{k=0}^n
(\alpha_{k+1}-\alpha_k)
\log\bigl|1-z e^{-A_k}\bigr|.\end{aligned}$$
From here we can see that a solution $z$ as $\chi\to +\infty$ should approach to $e^\tau$, or to $e^{A_l}$, for some $l\in\{1,\dots,n\}$ with $\alpha_{l+1}< \alpha_l$ (inner corner).
The imaginary part of equation (\[eq:crit\]) for the critical points is: $$\begin{gathered}
\label{eq:im-part}
0 =
-\sum_{k=0}^{j-1}
(\frac{1}{2}+\alpha_k)
\mathrm{angle}(z-e^{A_{k-1}},z-e^{A_k})
-(\frac{1}{2}+\alpha_j)\mathrm{angle}(z-e^{A_{j-1}},z-e^{\tau})
\\
+(\frac{1}{2}-\alpha_j)\mathrm{angle}(z-e^\tau,z-e^{A_j})
+\sum_{k=j+1}^{n}(\frac{1}{2}-\alpha_k)
\mathrm{angle}(z-e^{A_{k-1}},z-e^{A_k}).\end{gathered}$$
![Geometric representation of the imaginary part of $\log\left( \frac{z-e^{A_l}}{z-e^{A_{l-1}}} \right)$ as the angle between $z-e^{A_l}$ and $z-e^{A_{l-1}}$.[]{data-label="fig:angles"}](angles.pdf){width="7cm"}
Assume that $z$ is a critical point with positive imaginary part, and $z\to e^{A_l}$ for some $l$ (since the equation has real coefficients, the non real solutions will come as pairs of conjugate complex numbers). Since $z$ approaches to the real axis, the angle between $z-e^{B}$ and $z-e^C$ approaches to zero, unless $B$ or $C$ is equal to $A_l$. Therefore as $\chi\to +\infty$ , two terms in Equation will dominate. These are $$\begin{cases}
(\frac{1}{2}+\alpha_l)\mathrm{angle}(z-e^{A_{l-1}},z-e^{A_l}) + (\frac{1}{2}+\alpha_{l+1}) \mathrm{angle}(z-e^{A_l},z-e^{A_{l+1}}) & \text{if $\tau<A_l$},\\
(\frac{1}{2}-\alpha_l)\mathrm{angle}(z-e^{A_{l-1}},z-e^{A_l}) +(\frac{1}{2}-\alpha_{l+1})
\mathrm{angle}(z-e^{A_l},z-e^{A_{l+1}}) & \text{if $\tau > A_l$}.
\end{cases}$$ Thus, these expressions should vanish as $\chi\to +\infty$.
But this is impossible, because the sum $$\mathrm{angle}(z-e^{A_{l-1}},z-e^{A_l})+
\mathrm{angle}(z-e^{A_l},z-e^{A_{l+1}})=
\mathrm{angle}(z-e^{A_{l-1}},z-e^{A_{l+1}})$$ is positive and close to $\pi$ (see Figure \[fig:angles\]). Therefore this option for a critical point is impossible.
The only possibility left is that $z\to e^{\tau}$ as $\chi\to +\infty$. Let $z=e^{\tau-{\varepsilon}}$, where ${\varepsilon}\to 0$ as $\chi\to \infty$ and $\arg{\varepsilon}\in(-\pi,\pi)$.
As ${\varepsilon}\to 0$, we have for $k\neq j$: $$\begin{aligned}
\log\left(\frac{z-e^{A_k}}{z-e^{A_{k-1}}}\right) &=\log\left(\frac{e^{\tau}-e^{A_{k}}}{e^{\tau}-e^{A_{k-1}}}\right)+O({\varepsilon}) \\
&= \frac{A_k-A_{k-1}}{2}+\log 2\sinh\left| \frac{\tau-A_k}{2} \right|-\log 2\sinh\left| \frac{\tau-A_{k-1}}{2} \right|+O({\varepsilon}), \\
\log\left(\frac{z-e^{\tau}}{z-e^{A_{j-1}}} \right) &= \frac{\tau-A_{j-1}}{2}+\log(-{\varepsilon})-\log 2\sinh\left|\frac{\tau-A_{j-1}}{2}\right| +O({\varepsilon}),\\
\log\left( \frac{z-e^{A_j}}{z-e^{\tau}} \right)&= \frac{A_j-\tau}{2}-\log({\varepsilon})+\log 2\sinh \left|\frac{\tau-A_j}{2}\right| +O({\varepsilon}).\end{aligned}$$ Substitute these expressions in the equation for critical points: $$\begin{aligned}
0 \nonumber &=
-(\chi-\frac{\tau}{2})
-\sum_{k=1}^{j-1}
(\alpha_k+\frac{1}{2})
\left(\frac{A_k-A_{k-1}}{2}
+\log 2\sinh\Bigl| \frac{\tau-A_k}{2} \Bigr|
-\log 2\sinh\Bigl| \frac{\tau-A_{k-1}}{2}\Bigr|\right)
\\ \nonumber &
-(\alpha_j+\frac{1}{2})
\left( \frac{\tau-A_{j-1}}{2}+\log(-{\varepsilon})
-\log 2\sinh\Bigl|\frac{\tau-A_{j-1}}{2}\Bigr|
\right)
\\ \nonumber &
+(\frac{1}{2}-\alpha_j)
\left(\frac{A_j-\tau}{2}+
\log 2\sinh \Bigl|\frac{\tau-A_j}{2}\Bigr|
-\log({\varepsilon})\right)
\\ \nonumber &
+\sum_{k=j+1}^{n}
(\frac{1}{2}-\alpha_k)
\left( \frac{A_k-A_{k-1}}{2}
+\log 2\sinh\Bigl| \frac{\tau-A_k}{2} \Bigr|
-\log 2\sinh\Bigl| \frac{\tau-A_{k-1}}{2} \Bigr|\right)
+O({\varepsilon})
\\&=
-\chi
-\log({\varepsilon})
\mp (\alpha_j+\frac{1}{2})i\pi
+ \sum_{k=0}^n
(\alpha_{k+1}-\alpha_k)
\log 2\sinh\Bigl|\frac{\tau-A_k}{2}\Bigr|
+O({\varepsilon}).
\label{eq:critpoints_multislope}\end{aligned}$$ Here we used the identity $$\log(-\varepsilon)=
\begin{cases}
\log \varepsilon -i\pi&\text{if $\arg\varepsilon >0$}, \\
\log \varepsilon + i\pi&\text{if $\arg\varepsilon <0$},
\end{cases}$$ which holds for our choice of branch for $\log$. The cancellation of the linear combination of the $A_i$ comes from the two possible expressions for $-B(\tau)$ given by Equations and .
Solving this equation for ${\varepsilon}$, we arrive to the asymptotical formula: $$\begin{aligned}
{\varepsilon}= e^{-\chi} e^{\pm i\pi(\frac{1}{2}+\alpha_j)} \underbrace{\prod_{k=0}^n \left| 2 \sinh \left(\frac{\tau-A_k}{2}\right)\right|^{\alpha_{k+1}-\alpha_k}}_{\rho(\tau)}\left(1+O(e^{-\chi})\right),\end{aligned}$$ which implies that the asymptotic of the pair of complex conjugate solutions is given by the formulae in the last part of proposition \[prop:regions\].
$\tau$ becomes close to $A_j$ for $1\leq j\leq N-1$ as $\chi$ goes to $+\infty$.
When $\tau=A_j+\delta$, $\delta\to 0$ and $\chi\to +\infty$, the same arguments as above show that complex critical points still accumulate near $z=e^\tau$ but instead of the Equation for ${\varepsilon}$, where $z=e^{\tau-\varepsilon}$ we will have
$$\label{eq:CritPtsNearCorn}
\chi-\chi^{(j)}=-(\alpha_j+\frac{1}{2})\log(-\varepsilon)
+(\frac{1}{2}-\alpha_j)\log\left(\frac{\varepsilon-\delta}{\varepsilon}\right) - (\frac{1}{2}-\alpha_{j+1})\log(\varepsilon-\delta) +O({\varepsilon}).$$
From here we obtain equation (\[eq:t-scaling\]) for $s=\varepsilon/|\delta|$.
This completes the proof of Lemma \[lem:largeXcrit\].
Whereas the previous lemma describes what happens when $\chi$ goes to $+\infty$, the situation when $\chi$ goes to $-\infty$ is given in the following:
\[lem:smallXcrit\] For each $A_0 < \tau < A_n$, and sufficiently negative $\chi$, every critical point of $S(z)$ is real.
This is proven in the same way as Lemma \[lem:largeXcrit\]. To have $\Re(z\frac{d}{dz}S(z))=0$, $z$ needs to be very close to $e^{A_l}$, for some $l\in\{0,1,\ldots,n\}$, such that $\alpha_{l+1}>\alpha_l$. Exactly as in the case $\chi\rightarrow\infty$, one can show that if $l\neq 0$ or $n$, then $\Im(z\frac{d}{dz}S(z))$ cannot be $0$. If $z$ is near $e^{A_0}$ or $e^{A_n}$, then $z$ is actually real and respectively $z<e^{A_0}$ or $z>e^{A_n}$. This shows that when $\chi$ is sufficiently negative, $z\frac{d}{dz}S(z)=0$ has no complex solutions.
Proof of Proposition \[prop:regions\]
-------------------------------------
Note that the last two parts of the proposition were proven in the previous section. To prove the first part we first show that for fixed $\tau\in (A_0, A_n)$ there is a unique value of $\chi=\chi_0(\tau)$ such that the number of critical points of $S(z)$ with non-zero imaginary part changes as $\chi$ passes through $\chi_0$. That is:
\[prop:one\_double\] For each $\tau\in (A_0, A_n)$ there exists $\chi_0$ such that the number of critical points of $S(z)$ with $\Im(z)\neq 0$ is the same for all $\chi>\chi_0$, and it decreases by $2$ when $\chi$ passes the point $\chi_0$ into the region $\chi<\chi_0$.
We prove Proposition \[prop:one\_double\] at the end of this section, after setting up some technical tools.
Define $f(z)=z\frac{d}{dz} S(z)$. It is clear that $z=0$ is not a critical point of $S(z)$ and therefore critical points are zeroes of $f(z)$ in $\mathbb{C}\setminus\{0\}$. It is also clear that $z$ is a double critical point if and only if $$f(z)=f'(z)=0.$$
The following two lemmas give some information about the location of the double critical points of $S(z)$.
\[lem:noin\] There are no solutions to $f(z)=0$ in $[e^{A_0},e^{A_n}]$.
Consider $z=x+i\epsilon$, where $x\in(e^{A_{i-1}},e^{A_i})$ and $|\epsilon|<<1$. Looking at the terms in $f(z)$ in formula (\[eq:SPrime\]), it is easy to see that for such $z$ we have $$\Im\left(f(z)-c\log\left(\frac{z-e^{A_i}}{z-e^{A_{i-1}}}\right)\right) = O(\epsilon),$$ where $c\neq 0$ is the coefficient of $\log\left(\frac{z-e^{A_i}}{z-e^{A_{i-1}}}\right)$ in $f(z)$. Since $\log$ is the branch of the logarithm with an imaginary part in $(-\pi,\pi)$ with a cut along $\mathbb{R}_{-}$, we have that $\Im\left(c\log\left(\frac{z-e^{A_i}}{z-e^{A_{i-1}}}\right)\right) = \pm c \pi + O(\epsilon)$, which in turn implies $\Im(f(z)) = \pm c \pi + O(\epsilon)$. This shows that there are no solutions to $f(z)=0$ in $(e^{A_0},e^{A_n})$.
The only remaining points are $e^{A_j}$, but the function $f(z)$ is singular at these points, and therefore they cannot be solutions either.
\[lem:noout\] $f'(z)=0$ has exactly one real solution outside the interval $[e^{A_0},e^{A_n}]$ (possibly at $\infty$). \[prop:T(z)=0OneSoln\]
Recall that, by definition, the equation $f'(z)=0$ means $$\begin{aligned}
f'(z) = \begin{cases}
\frac{d}{dz} f(z)=0 & z \neq \infty \\
\lim_{z \rightarrow \infty} z^2 \frac{d}{dz} f(z)=0 & z=\infty.
\end{cases}\end{aligned}$$ Thus solving $f'(z)=0$ in ${{\mathbb R}}\cup \{ \infty \} \backslash [e^{A_0},e^{A_n}]$ is equivalent to solving $G(z)=0$ in this region, where $$G(z) := (z-e^{A_j})(z-e^\tau)f'(z).$$ When $\tau\in (A_{j-1}, A_j)$ we have $$\begin{aligned}
&G(z)=\label{eq:G}
-\sum_{k=1}^{j-1}
\left(\frac{1}{2}+\alpha_k\right) (e^{A_k}-e^{A_{k-1}})
\frac{(z-e^{A_j})(z-e^\tau)}{(z-e^{A_k})(z-e^{A_{k-1}})} \\ \nonumber -&
\left(\frac{1}{2}+\alpha_j\right) (e^{\tau}-e^{A_{j-1}})
\frac{(z-e^{A_j})(z-e^\tau)}{(z-e^{\tau})(z-e^{A_{j-1}})}
+\left(\frac{1}{2}-\alpha_j\right) (e^{A_j}-e^{\tau})
\frac{(z-e^{A_j})(z-e^\tau)}{(z-e^{A_j})(z-e^{\tau})}
\\ \nonumber +&
\sum_{k=j+1}^{n}
\left(\frac{1}{2}-\alpha_k\right) (e^{A_k}-e^{A_{k-1}})
\frac{(z-e^{A_j})(z-e^\tau)}{(z-e^{A_k})(z-e^{A_{k-1}})}.\end{aligned}$$ Notice that each term has the form $$\nonumber
c\frac{(z-u_1)(z-u_2)}{(z-v_1)(z-v_2)}$$ where $$c<0, u_1,u_2>v_1,v_2,\text{ OR }c>0, u_1,u_2<v_1,v_2,$$ and $$z>u_1,u_2,v_1,v_2,\text{ OR }z<u_1,u_2,v_1,v_2.$$
Under these conditions $$\frac{d}{dz}\left(c\log
\left(\frac{(z-u_1)(z-u_2)}{(z-v_1)(z-v_2)}\right)\right)= c\left(\frac{1}{z-u_1}+\frac{1}{z-u_2}-
\frac{1}{z-v_1}-\frac{1}{z-v_2}\right)<0,$$ for $z \in {{\mathbb R}}\backslash [e^{A_0},e^{A_n}]$, so $G(z)$ is decreasing in $z$ for all $z \in {{\mathbb R}}\backslash [e^{A_0},e^{A_n}]$. Furthermore, it is clear from formula that $G(z)$ is regular at $\infty$. Thus $G(z)$ is decreasing as one moves along $(e^{A_n}, \infty]$, and then “wraps around infinity” and moves from $-\infty$ to $e^{A_0}$.
The function $G(z)$ can also be written as $$G(z)= (z-e^{A_j})(z-e^\tau) \left(
\frac{\frac{1}{2}+\alpha_1}{z-e^{A_0}}
-\sum_{k=1}^{n-1} \frac{\alpha_k-\alpha_{k+1}}{z-e^{A_k}}
+\frac{\frac{1}{2}-\alpha_n}{z-e^{A_n}}
-\frac{1}{z-e^\tau} \right).$$
Using this formula and the fact that $\frac{1}{2}+\alpha_1>0$ and $\frac{1}{2}-\alpha_n>0$, one can see that $$\lim_{z\to e^{A_0}-} G(z)<0, \lim_{z\to e^{A_n}+}G(z)>0.$$ this completes the proof of the Lemma.
As $\chi$ varies, the number of non-real critical points can change only at the point $\chi_0$ when there is a double solution to $f(z)=0$ on the real line, or at infinity. Lemma \[lem:noin\] shows that this never happens with $z \in [e^{A_0},e^{A_n}]$.
Lemma \[lem:noout\] shows that $f'(z)=0$ has exactly one solution $z_0 \in {{\mathbb R}}\backslash [e^{A_0},e^{A_n}]$ (or possibly at $\infty$). Since $f(z)=g(z)+\chi$, where $g(z)$ does not depend on $\chi$, it implies that there is exactly one point $\chi_0=-g(z_0)$ where $f(z)=f'(z)=0$ and that this happens when $z=z_0$.
By Lemma \[lem:largeXcrit\], for sufficiently large $\chi$ there are exactly two critical points of $S_{\chi,\tau}(z)$ with $\Im(z)\neq 0$. By Lemma \[lem:smallXcrit\], for sufficiently negative $\chi$ there are no such critical points. Taking into account Proposition \[prop:one\_double\], this implies there is a unique value $\chi_0$ such that all critical points of $S(z)$ are real for $\chi<\chi_0$ and there is a pair of complex conjugate critical points $z_c, \bar{z_c}$ with $\Im(z_c)>0$ for $\chi>\chi_0$. This completes the proof of Proposition \[prop:regions\].
The asymptotic of critical points near corners
----------------------------------------------
The following proposition describes what happens in situations interpolating the two regimes described in Proposition \[prop:regions\].
\[prop:near\_corner\] Complex conjugate solutions to (\[eq:t-scaling\]) have the following asymptotic when $p\to 0$:
$$s=p^{-\frac{1}{1+\alpha_j-\alpha_{j+1}}}\exp\left(\frac{\pm i\pi(\frac{1}{2}+\alpha_j)}{1+\alpha_j-\alpha_{j+1}}\right)(1+O(p)).$$
If $p\to \infty$ the asymptotic is different for different signs of $\delta$:
1. When $\delta\to +0$ $$s=p^{-1}\exp\left(\pm i\pi(\frac{1}{2}+\alpha_{j+1})\right)\left(1+O\Bigl(\frac{1}{p}\Bigr)\right).$$
2. When $\delta\to -0$ $$s=p^{-1}\exp\left(\pm i\pi(\frac{1}{2}+\alpha_j)\right)\left(1+O\Bigl(\frac{1}{p}\Bigr)\right).$$
We will do the case when $p\rightarrow \infty, \delta>0$. The other cases are similar. From (\[eq:t-scaling\]) we see that if $p\rightarrow\infty$ then either $s\rightarrow 0$ or, if $(\alpha_{j+1}-\alpha_j)<0$, $s\rightarrow 1$. If $s\rightarrow 0$, then $s$ can be easily calculated to have the asymptotics given in the statement of the proposition. We need to show that the case $s\rightarrow 1$ is not possible.
We have $z=e^{\tau-\varepsilon}, \varepsilon\rightarrow 0$, and we can assume that $Im(\varepsilon)\in(0,\pi)$.
Write $s$ as $s=1+te^{i\theta}$. Since $Im(\varepsilon)\in(0,\pi)$, we can assume $\theta\in(0,\pi)$ as well.
Let’s rewrite (\[eq:CritPtsNearCorn\]) as $$\begin{aligned}
&\chi - \chi^{(j)}+(1+\alpha_j-\alpha_{j+1})\log(\delta)=
\\&=-(\alpha_j+\frac 12)\log(-s) + (\frac 12 - \alpha_j)\log(\frac{s-1}{s}) - (\frac 12 - \alpha_{j+1})\log(s-1) + O(\delta)
\\&=-(\alpha_j+\frac 12)\log(-1-te^{i\theta}) + (\frac 12 - \alpha_j)\log(\frac{te^{i\theta}}{1+te^{i\theta}}) - (\frac 12 - \alpha_{j+1})\log(te^{i\theta}) + O(\delta).\end{aligned}$$
Looking at the real part of this equation we get
$$\chi - \chi^{(j)}+(1+\alpha_j-\alpha_{j+1})\log(\delta)
= (\frac 12 - \alpha_j)\log(t) - (\frac 12 - \alpha_{j+1})\log(t) + O(1).$$
From here we get
$$t=e^\frac{\chi-\chi^{(j)}+ (1+\alpha_j-\alpha_{j+1}) \log(\delta)}{\alpha_{j+1}-\alpha_j}\rightarrow 0.$$
Let’s look at the imaginary part. We have $$0 = -(\alpha_j+\frac 12)(-\pi+O(t))
+ (\frac 12 - \alpha_j)(\theta - O(t)) - (\frac 12 - \alpha_{j+1})\theta + O(\delta).$$
From here
$$\theta=\pi\frac{\alpha_j+\frac 12}{\alpha_j-\alpha_{j+1}} + O(t) + O(\delta).$$
But this is impossible because $$\theta\in(0,\pi)$$ and $$0<\alpha_j-\alpha_{j+1}<\alpha_j+\frac 12.$$
These asymptotics for $s$ as a function of $p$ imply the following asymptotical formulae for complex conjugate critical points $z=e^{\tau-{\varepsilon}}$of $S(z)$.
- When $\chi\to +\infty$ and $\delta\to 0$ such that $e^\chi |\delta|^{1+\alpha_j-\alpha_{j+1}}\to 0$, we have $${\varepsilon}=e^{\pm i\pi \frac{\alpha_j+\frac{1}{2}}{1+\alpha_j-\alpha_{j+1}}}
e^{-\frac{\chi}{1+\alpha_j-\alpha_{j+1}}}(1+o(1)).$$
- When $\chi\to +\infty$ and $\delta\to +0$, such that $p\to\infty $, $${\varepsilon}=e^{\pm i\pi (\alpha_j+\frac{1}{2})}e^{-\chi}|\delta|^{\alpha_{j+1}-\alpha_j}
(1+o(1)).$$
- When $\chi\to +\infty$ and $\delta\to -0$, such that $p\to \infty $, $${\varepsilon}=e^{\pm i\pi (\alpha_{j+1}+\frac{1}{2})}e^{-\chi}|\delta|^{\alpha_{j+1}-\alpha_j}
(1+o(1)).$$
Notice that the limit $\chi\to \infty$ of complex conjugate critical points for fixed $\tau$ given by (\[eq:crit\_point\_asymp\]) agrees with the last two asymptotics when $\tau\to A_j \pm 0$.
Correlation functions, the frozen boundary, and the limit shape {#sec:corr}
===============================================================
In this section we will study correlation functions in the limit of the infinitely large system. In particular, the one-point correlation function gives the macroscopic density of horizontal rhombi.
In our analysis we will follow [@OR1] and [@OR2]. The basic idea is to use the steepest descent method for computing the asymptotical behavior of the double integral defining the correlation kernel.
It has been shown in [@OR1] that if $(\tau,\chi)$ is such that all critical points of $S(z)$ are real, the region in the vicinity of this point is *frozen*, i.e. in terms of tiling by rhombi it is tiled with probability 1 by rhombi of one type (tilted to the left, tilted to the right, or horizontal).
The region where two simple real critical points collapse into one degenerate critical point is a curve in the $(\tau,\chi)$-plane which separates the *frozen region* from the *disordered region*, where the function $S(z)$ has a pair of complex conjugate simple critical points. This curve is the analogue of the arctic circle for tilings of large regular hexagons [@CohnLarsenPropp].
As it follows from Proposition \[prop:regions\], and in opposition to what occurs in [@OR1], the boundary of the frozen region in our case projects bijectively to the interval $(A_0,A_n)$ on the $\tau$ axis. As we will see below, the region above this curve is disordered, the region below is frozen.
![The frozen boundary when the parameters are $A = \{-8, -7, 0, 4, 6, 7\}$, $\alpha = \{-0.1, 0.4, -0.45, 0.4, -0.25\}$ []{data-label="fig:FrozenBoundary2"}](FrozenBoundary.pdf){width="10cm"}
Local correlation functions in the bulk of the disordered region {#subsec:corr-disord}
----------------------------------------------------------------
In this section we will prove the following theorem.
\[thm:cor\_beta\] In the limit when $r$ goes to $0$, the correlation functions of the system near a point $(\chi, \tau)$ in the bulk are given by determinants of the incomplete beta kernel $$\label{eq:lim_kernel}
K_{\tau,\chi}^b(\Delta t, \Delta h) = \int_\gamma (1- e^{-\tau} z)^{\Delta t} z^{-\Delta h +\frac{\Delta t}{2}} \frac{dz}{2i\pi z},$$ where the integration contour connects the two non-real critical points of $S_{\tau,\chi}(z)$, passing through the real line in the interval $(0,1)$ if $\Delta(t)\geq 0$ and through $(-\infty, 0)$ otherwise.
The proof is completely parallel to the similar statements from [@OR1; @OR2]. Here is the outline.
The correlation functions are given by Formula (\[eq:main-corr2\]). When $r\rightarrow 0$, the leading asymptotics of the integrand is given by $e^\frac{S_{\tau,\chi}(z)-S_{\tau,\chi}(w)}{r}\frac{1}{z-w}$. In the region where $S_{\tau,\chi}(z)$ has exactly two complex conjugate critical points we calculate the asymptotic of the integral by deforming the contours of integration as follows.
The poles of $\frac{\Phi_-(z,t_1)}{\Phi_+(z,t_1)}$ are real and lie in the interval $(e^{rt_1},\infty)$. The poles of $\frac{\Phi_+(w,t_2)}{\Phi_-(w,t_2)}$ are also real and lie in the interval $(0,e^{rt_2})$. When $r\rightarrow 0$, they accumulate along the intervals $(e^\tau,\infty)$ and $(0,e^\tau)$ respectively.
It is clear that any deformation of the contours of integration away from the real line doesn’t change the integral, as long as the point where the $z$-contour crosses the positive real line does not move to the right, the point where the $w$-contour intersect the real line does not move to the left and the contours do not cross one another.
Note that whether the $z$-contour is inside the $w$-contour or vice-versa, depends on the sign of $\Delta t=t_2-t_1$. Now look at the level curves of $\Re(S(z))$ passing through $z_{c}$ and $\bar{z}_c$. In the case when the $z$-contour is outside, we can deform the original contours $C_z,C_w$, to $C'_z,C'_w$ as in Figure \[fig:ContourDeform\]. The only poles we cross are the poles at $z=w$, so we need to take into account the contribution from the residues at $z=w$. We obtain $$\begin{gathered}
\label{eq:kernel_deformed_contours}
K((t_1,h_1),(t_2,h_2))=
\int_{C'_z}\int_{C'_w}\text{same as in }(\ref{eq:main-corr2}) +\\
\int_{\mathcal{C}}\frac{\Phi_-(z,t_1)\Phi_+(z,t_2)}{\Phi_+(z,t_1)\Phi_-(z,t_2)} z^{h_2-h_1+B(t_2)-B(t_1)} \frac{dz}{2i\pi z},\end{gathered}$$ where $\mathcal{C}$ is an arc connecting points $\bar{z}_{c}$ and $z_{c}$ which crosses the real line on the positive side. In the case when the $w$-contour is inside of the $z$-contour, $\mathcal{C}$ crosses the real line on the negative side.
![ Deformation of contours when the $z$ contour is outside the $w$ contour. []{data-label="fig:ContourDeform"}](contour_deformation.pdf){width="9cm"}
Now, let’s show that in the limit $r\rightarrow 0$ the first integral is $0$. Since $\lim_{z\rightarrow 0} \Re S(z)=+\infty$, the gradient of the real part of $S(z)$ looks as in Picture \[fig:ContourDeform\], which implies that everywhere along the contours $C'_z,C'_w$ except when $z=w=$ critical points, we have $\Re(S(z))<\Re(S(w))$. Because the asymptotically leading term of the integrand is $e^\frac{S_{\tau,\chi}(z)-S_{\tau,\chi}(w)}{r}\frac{1}{z-w}$, we conclude that the integral is $0$ as $r\to 0$ exponentially fast. Since $z-$ and $w-$ contours intersect transversally, there are no problems with the integrability of $\frac{1}{z-w}$ in the neighborhood of the critical points.
We conclude that the correlation kernel is equal to the second integral in .
The second integral in , i.e. $$\int_{\mathcal{C}}\frac{\Phi_-(z,t_1)\Phi_+(z,t_2)}{\Phi_+(z,t_1)\Phi_-(z,t_2)} z^{h_2-h_1+B(t_2)-B(t_1)} \frac{dz}{2i\pi z},$$ is asymptotic, as $r\rightarrow 0$, $t_i r \rightarrow \tau$, $t_1-t_2=\Delta t$, $h_1-h_2=\Delta h$ to:
$$\label{eq:lim_kernel0}
\left(-e^\tau\right)^{N_-(t_1)-N_-(t_2)}\int_{\mathcal{C}}\left(1-e^{-\tau} z\right)^{\Delta t}z ^{-\Delta h + \frac{\Delta t }{2}}\frac{d z }{2i\pi z}$$
where $N_-(t_i)$ is the number of descending unit pieces of back wall between the leftmost corner $A_0$ and $t_i$. The prefactor in front of the integral is of the form $$\frac{g(t_1,h_1)}{g(t_2,h_2)},$$ which disappears by multi-linearity when we compute correlation functions, as determinants of the form $\det(K( (t_i,h_i),(t_j,h_j) ) )$.
The kernel $K_{\chi,\tau}^b$ has a simple expression on the diagonal: when $\Delta t=\Delta h =0$, we have $$K_{\chi, \tau}^b(0,0)=\int_{\mathcal{C}}
\frac{dz}{2i\pi z}= \frac{\theta}{\pi},$$ where $\theta$ is the argument of the critical point $z_c$ in the upper-half plane. This quantity is exactly the density of horizontal tiles. It is therefore the vertical gradient of the limit shape height function.
The correlation kernel depends only on one complex parameter $z_c$. It is determined uniquely by coordinates $(\tau,\chi)$ or by the slope of the limit shape at this point.
The distribution of the tiles in the neighborhood of a point $(\chi, \tau)$ converges to the translation invariant ergodic Gibbs measure on tilings of the plane with rhombi, with activities $1$, $e^{-\tau}|z_c|$ and $|z_c|$ for the three types of tiles in absence of a magnetic field.
Indeed, from [@Ke:LocStat; @KOS] the correlation functions for the translation invariant probability measure with activities $a$, $b$, $c$ on rhombus configurations (with no external field) are given by determinants of the correlation kernel $$K_{a:b:c}((t_1,h_1), (t_2,h_2)) = \iint_{{\mathbb T}^2} \frac{z^{-\Delta h+ \frac{\Delta t}{2}}w^{\Delta t}}{a+\frac{1}{w}(b+\frac{c}{z})} \frac{dz}{2i\pi z} \frac{dw}{2i\pi w}.$$ Taking the proposed choice for the activities, and computing the integral over $w$ by residues leads to a simple integral along a sector of the unit circle. After a change of variable in the integral by multiplying by $|z_c|$, one gets the same expression as in , see [@OR1]. The convergence of finite dimensional distributions follows from Theorem \[thm:cor\_beta\]. Since the space of tilings is compact for the product topology, the convergence of finite dimensional distributions implies the convergence of the whole distribution.
This result gives a precise description of the local behavior of the system in the neighborhood of a point inside the liquid region of the limit shape. However, the expressions of the critical points $z_c$ and $\overline{z_c}$ are not explicit for a finite $\chi$. In the next subsection, we investigate the asymptotic when $\chi \to \infty$ .
Similar arguments can be applied to the case when all critical points are real. In this case the integral (\[eq:main-corr2\]) tends either to zero or to one depending on the sign of the critical points. The steepest descent method results in the following asymptotic: $$\rho_{(t_1,h_1),\dots, (t_k,h_k)}=\rho+ O(e^{-\frac{\alpha}{r}}),$$ where $\alpha>0$, $\rho=1$ when $(\tau,\chi)$ is below the segment of the boundary of the limit shape between two points where it touches the boundary, and $\rho=0$ when $(\tau,\chi)$ is below the boundary of the limit shape but above the turning points.
The correlation functions for large chi and the bead model
----------------------------------------------------------
In this section, we investigate the behavior of the high piles of cubes of the random skew plane partition, that are close to the back wall. The asymptotic of local correlation functions as $\chi\to \infty$ depends on whether $\tau$ is in a vicinity of $A_j$ or not.
A\_{j-1}<tau<A\_{j}
-------------------------
In this subsection we consider the case when $A_{j-1}<\tau <A_j$.
Let $$t_i = \frac{\tau}{r}+\eta_i, \quad h_i = \frac{\chi}{r}+e^{\chi}\xi_i, \quad i=1,2,$$ so that $\Delta \eta = \Delta t$ and $\Delta \xi = e^{-\chi} \Delta h$.
In the limit where $\chi\to \infty$, such that $\Delta \eta$ and $\Delta\xi$ are fixed, the kernel $K_{\tau,\chi}^b( (t_1,h_1),(t_2,h_2))$ of Equation divided by $e^{\chi}$ and a factor which does not affect the determinant (\[eq:det\]), converges to $$K^{(\gamma)}((\eta_1,\xi_1),(\eta_2,\xi_2))=
\begin{cases}
{\displaystyle d\int_{-1}^{1} \left(\gamma +i \varphi \sqrt{1-\gamma^2}\right)^{\eta_1-\eta_2} e^{-i\varphi d (\xi_1-\xi_2)} \frac{d \varphi}{2\pi}}, & \eta_1 \geq \eta_2,\\
\displaystyle -d \int_{\mathbb{R}\setminus[-1,1]} \left(\gamma +i \varphi \sqrt{1-\gamma^2}\right)^{\eta_1-\eta_2} e^{-i\varphi d (\xi_1-\xi_2)} \frac{d \varphi}{2\pi}, & \eta_1 < \eta_2,
\end{cases}$$ where $\alpha_j$ is the slope of the piece of the back wall between $A_{j-1}$ and $A_j$, $\gamma=\sin \pi \alpha_j$, and $$d=\rho(\tau)\cos(\pi\alpha_j)=\cos(\pi\alpha_j)\prod_{k=0}^n \left| 2 \sinh \left(\frac{\tau-A_k}{2}\right)\right|^{\alpha_{k+1}-\alpha_k}.$$
Suppose $t_1 \geq t_2$. Recall that the two complex conjugate critical points $z_c$ and $\bar{z}_c$ for this value of $\tau$ have the asymptotics when $\chi\to \infty$. The curve $\mathcal{C}$ joining $z_c$ and $\bar{z}_c$ can be chosen as the positively oriented arc of the circle centered at 0, of radius $|z_c|$. A possible parametrization of this arc is: $$z= z(\varphi)= e^ \tau \exp\left(e^{-\chi} \rho(\tau) \left(\sin(\pi\alpha_j)+ i\varphi \cos(\pi\alpha_j)\right)\right)(1+o(1)),$$ where $\varphi$ runs from $-1$ to $+1$.
The expression of the kernel becomes $$\begin{aligned}
K_{\chi,\tau}^b(\Delta t, \Delta h) &= e^{-\chi}\rho(\tau) \cos(\pi \alpha_j)
\int_{-1}^{1} \left ( 1-e^{-\tau} z(\varphi)\right )^{\Delta t} \bigl( z(\varphi)\bigr)^{-\Delta h + \Delta t/2}\frac{d\varphi}{2\pi}.\end{aligned}$$
When $\chi$ goes to infinity, one has $$\begin{aligned}
\left( 1-e^{-\tau} z(\varphi)\right)^{t_1-t_2} &= (-e^{-\chi}\rho(\tau))^{\eta_1-\eta_2} \left(\sin(\pi \alpha_j) + i\varphi \cos(\pi \alpha_j)\right)^{\eta_1-\eta_2} (1+o(1)),\\
\bigl( z(\varphi)\bigr)^{-(h_1-h_2)} &= \left(e^{\tau e^\chi}e^{\rho(\tau)\sin(\pi\alpha_j)}\right)^{-(\xi_1-\xi_2)} e^{-i\rho(\tau)\varphi \cos(\pi\alpha_j)(\xi_1-\xi_2)} (1+o(1)).\end{aligned}$$
The factors $$(e^{-\chi}\rho(\tau))^{\eta_1-\eta_2}\left(e^{\tau e^\chi}e^{\rho(\tau)\sin(\pi\alpha_j)}\right)^{-(\xi_1-\xi_2)}$$ are of the form $\frac{g(\xi_1,\eta_1)}{g(\xi_2,\eta_2)}$. These factors have no effect on the computations of the probabilities, since they cancel out when computing, in the limit, the determinant $\det(K_{\chi,\tau}( t_i-t_j,h_i-h_j))$. We can thus drop them out, and obtain a new kernel defining the same determinantal process.
The integral converges as $\chi \rightarrow +\infty$ to $$\rho(\tau)\cos(\pi \alpha_j)\int_{-1}^{1} \left(\sin(\pi \alpha_j) +i \varphi \cos(\pi\alpha_j)\right)^{\eta_1-\eta_2} e^{-i\varphi \rho(\tau) \cos(\pi\alpha_j)(\xi_1-\xi_2)} \frac{d \varphi}{2\pi}.$$
Similarly, when $t_1 < t_2$, we use the same change of variable $z=z(\varphi)$. But now, $\varphi\in ( -\frac{\pi e^\chi}{\rho(\tau)\cos(\pi\alpha_j)} , -1)\cup ( 1, \frac{\pi e^\chi}{\rho(\tau)\cos(\pi \alpha_j)})$. The same computations as above show that $K_{\chi,\tau}(\Delta t, \Delta h)$ divided by $\frac{g(\xi_1,\eta_1)}{g(\xi_2,\eta_2)}$ converges to $$-\rho(\tau)\cos(\pi \alpha_j)\int_{\mathbb{R}\setminus[-1,1]} \left(\sin(\pi \alpha_j) +i \varphi \cos (\pi \alpha_j)\right)^{\eta_1-\eta_2} e^{-i\varphi \rho(\tau) \cos(\pi\alpha_j )(\xi_1-\xi_2)} \frac{d \varphi}{2\pi}.$$
The kernel divided by $e^{-\chi}$ and some factors $\frac{g(\eta_1,\xi_1)}{g(\eta_2,\xi_2)}$ which cancel in the determinant, converges to $$\begin{cases}
\displaystyle \rho(\tau) \cos(\pi\alpha_j) \int_{-1}^{1} \left(\sin(\pi\alpha_j)+i\varphi\cos(\pi\alpha_j)\right)^{\eta_1-\eta_2} e^{-i\cos(\pi\alpha_j)\rho(\tau)(\xi_1-\xi_2)\varphi} \frac{d\varphi}{2\pi}, & t_1\geq t_2 \\
\displaystyle -\rho(\tau) \cos(\pi\alpha_j) \int_{\mathbb{R}\setminus[-1,1]}\!\!\! \left(\sin(\pi\alpha_j)+i\varphi\cos(\pi\alpha_j)\right)^{\eta_1-\eta_2} e^{-i\cos(\pi\alpha_j)\rho(\tau)(\xi_1-\xi_2)\varphi} \frac{d\varphi}{2\pi}, & t_1 < t_2
\end{cases}$$
This ends the proof.
As a consequence, using the same argument as in [@beads], we have the following theorem
The point process describing the positions of horizontal rhombi in the neighborhood of a non frozen point $(\tau,\chi)$, with $A_{j-1}<\tau <A_j$ in the limit shape $r\rightarrow 0$ converges to the bead process on $\mathbb{Z}\times \mathbb{R}$ with parameter $\gamma=\sin\pi\alpha_j$ and density $$\frac{1}{\pi}\rho(\tau)\cos(\pi\alpha_j) = \frac{1}{\pi} \cos(\pi\alpha_j)\prod_{k=0}^n\left| 2 \sinh\left(\frac{\tau-A_{k}}{2} \right)\right|^{\alpha_{k+1}-\alpha_k}.$$
Note that the density is singular when at the corners: it tends to $0$ at inner corners (where $\alpha_{k+1} > \alpha_k$), and goes to $\infty$ at outer corners. This is a remnant of the singularities of the limit shape observed in [@OR1]. This suggests that the scaling has to be modified to observe a non trivial phenomenon in the vicinity of the corners $A_k$.
### $\tau$ is in the vicinity of $A_j$ for $0<j<n$
Now assume that $\tau=A_j+\delta$, and $\delta\to 0$ as $\chi\to \infty$ in such a way that $p=e^{\chi-\chi^{(j)}}|\delta|^{1+\alpha_j-\alpha_{j+1}}$ is fixed. Recall that in this case, the critical point of $S_{\tau,\chi}(z)$ are given by Equation . In this limit the curve $\mathcal{C}$ joining two complex conjugate critical points can be parameterized as $$z(\phi)=e^\tau \exp(-t^{\beta_j}e^{-\beta_j(\chi-\chi^{(j)})}(s'+is''\phi)),$$ where $s'$ and $s''$ are real and imaginary parts of $s$ respectively, and $\beta_j=\frac{1}{1+\alpha_j-\alpha_{j+1}}$.
\[thm:scaling\]Assume that $\chi\to \infty$ and $\delta\to 0$ as above, and that $\Delta\eta=\Delta t$ and $\Delta\zeta=e^{-\beta_j\chi}\Delta h$ are fixed. Then the correlation kernel , divided by $e^{-\beta_j\chi}$ and some factors not affecting the determinant, converges to the same expression as in the previous theorem $$K^{(\gamma')}((\eta_1,\zeta_1),(\eta_2,\zeta_2))=
\begin{cases}
d' \int_{-1}^{1} \left(\gamma' +i \varphi \sqrt{1-\gamma'^2}\right)^{\eta_1-\eta_2} e^{-i\varphi d' (\zeta_1-\zeta_2)} \frac{-d \varphi}{2\pi} & \text{if $\eta_1 \geq \eta_2$}\\
- d' \int_{\mathbb{R}\setminus[-1,1]} \left(\gamma' +i \varphi \sqrt{1-\gamma'^2}\right)^{\eta_1-\eta_2} e^{-i\varphi d' (\zeta_1-\zeta_2)} \frac{d \varphi}{2\pi} & \text{if $\eta_1 < \eta_2$},
\end{cases}$$ with $$d'=e^{\beta_j\chi^{(j)}}s'', \quad \gamma'=-\frac{s''}{|s|}.$$
The proof is completely parallel to the proof of the previous statement, so we will skip it.
It is easy to see that these formulae when $\delta\to\pm 0$ and $p\to \infty$ agree with the previous result when $\tau\to A_j \pm 0$. let us verify it for $\delta\to -0$, the calculations being almost identical for $\delta\to +0$. We have in the limit $\delta\rightarrow 0-$: $$s\sim p^{-1}e^{i\pi (\alpha_j+\frac{1}{2})}.$$ From the definition of $p$ and a simple algebra we have $$( p e^{-\chi})^{\beta_j-1}=|\delta|^\frac{\beta_j-1}{\beta_j}
e^{-(\beta_j-1)\chi^{(j)}}.$$
Taking into account the definition of $\Delta\xi$ and $\Delta\zeta$ we obtain: $$\Delta\zeta e^{\beta_j\chi^{(j)}}s'' \sim |\delta|^{\alpha_{j+1}-\alpha_j}e^{\beta_j\chi^{(j)}}\cos(\pi\alpha_j)
\Delta\xi.$$ Moreover, when $\tau\to A_j-0$ $$\rho(\tau) \sim \rho^{(j)} |\delta|^{\alpha_{j+1}-\alpha_j} = |\delta|^{\alpha_{j+1}-\alpha_j}e^{\chi^{(j)}},$$ where $\delta=\tau-A_j$. Comparing the two kernels in these regimes yields:
$$e^{-\beta_j\chi} K^{(\gamma')}((\eta_1,\zeta_1),(\eta_2,\zeta_2))\sim
e^{-\chi} K^{(\gamma)}((\eta_1,\xi_1),(\eta_2,\xi_2)).$$
It is also easy to find the asymptotic when $t\to 0$. In this case: $$\gamma'\to \sin\left(\frac{\pi}{2}\frac{\alpha_{j+1}+\alpha_j}{1+\alpha_j-\alpha_{j+1}}\right),$$ and for the density we have: $$e^{\beta_j\chi^{(j)}}t^{\beta_j}s''\to e^{\beta_j\chi_j}\cos(\frac{\pi}{2}\frac{\alpha_{j+1}+\alpha_j}{1+\alpha_j-\alpha_{j+1}}).$$
Again, as a corollary of Theorem \[thm:scaling\] the statistics of the horizontal tiles, rescaled vertically by a factor $e^{-\chi\beta_j}$, converge as $\chi\to\infty$ to the bead process with parameter $\gamma=\sin\left(\frac{\pi(\alpha_{j+1}+\alpha_j)}{2(1+\alpha_j-\alpha_{j+1})}\right)$ and density $\frac{1}{\pi} \left(\rho^{(j)}\right)^{\beta_j}\cos\left(\frac{\pi(\alpha_{j+1}+\alpha_j)}{2(1+\alpha_j-\alpha_{j+1})}\right)$.
Notice that in the vicinity of $\tau=A_j$ one can get a bead process with any parameter in the interval $[\sin(\pi\alpha_j), \sin(\pi \alpha_{j+1})]$ by appropriately tuning $p$.
[E-G-S]{}
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[^1]: The third and fourth authors were partially supported by the NSF grant DMS-0354321.
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---
address: 'Tonantzintla, Puebla, Mexico'
author:
- Ricardo Chávez
bibliography:
- 'bib/thesis.bib'
title: Constraining the Parameter Space of the Dark Energy Equation of State Using Alternative Cosmic Tracers
---
To my parents, for their tireless support.
|
---
abstract: 'It is shown that the strong Coulomb coupling in intrinsic suspended semiconducting transition metal dichalcogenides can exceed the critical value needed for an excitonic ground state. The dipole-allowed optical excitations then correspond to intra-excitonic transitions such that the optically bright excitonic transitions near the Dirac points have a $p$-like symmetry whereas the $s$-like states are dipole forbidden. The large intrinsic coupling strength seems to be a generic property of the semiconducting transition metal dichalcogenides and strong Coulomb-coupling signatures in the form of the optical selection rules can be observed even in samples grown on typical substrates like SiO$_2$. For the examples of WS$_2$ and WSe$_2$, excellent agreement of the computed excitonic resonance energies with recent experiments is demonstrated.'
author:
- 'T. Stroucken'
- 'S.W. Koch'
title: 'Optically bright $p$-excitons indicating strong Coulomb coupling in transition-metal dichalcogenides'
---
Introduction
============
Monolayers of transition metal dichalcogenides (TMDs) are novel two-dimensional (2D) semiconductors. Similar to graphene, these materials display exciting new physical properties, distinct from their bulk counterparts[@Mak2010; @Terrones2013; @Splendiani2010; @Cappelluti2013; @Lambrecht2012; @Liu2013; @Xiao2012; @Molina-Sanchez2013; @Cao2012; @Zeng2012; @Mak2012; @Zeng2013; @Chernikov2014; @Ye2014]. The linear optical spectra are dominated by strong resonances with peak absorption values of more than 10% of the incoming light. Furthermore, the absence of inversion symmetry allows for efficient second harmonic generation (SHG). Though initially assigned to free-particle transitions, meanwhile the excitonic nature of the strong optical response is widely accepted. On the basis of recent linear and nonlinear optical experiments on WS$_2$ \[1-3\], the respective authors report large binding energies for the energetically lowest, optically bright excitons ranging from 0.32-0.71 eV. At the same time, strong deviations from the usual hydrogenic Rydberg series are observed. Due to the difficulty to directly measure the single-particle bandgap energy, the binding energies have been deduced either from the energetic separation between different resonances or by the comparison with theoretical predictions.
In general, the proper analysis of the measured spectra depends critically on the correct identification of the optically active states. For this purpose, one derives optical selection rules that result from the conservation of the total angular momentum in the excitation and emission processes. The $\Gamma$-point transitions in conventional direct-gap GaAs-type semiconductors occur between the $sp^3$-hybridized near-parabolic $s$-like conduction and $p$-like valence bands. The near-bandgap optical properties are well described by the so-called Elliot formula[@HaugKoch], expressing the energetic position and oscillator strength of the excitonic transitions in terms of the excitonic wave functions. As a result of the simple dipole selection rules, only $s$-like excitonic states couple to the light field. This leads to the well-known excitonic Rydberg series where the lowest optical active state is the 1s exciton resonance.
The derivation of the Elliot formula[@HaugKoch] is based on the implicit assumption that the system is excited from the noninteracting ground state. In contrast, if one considers an excitonic insulator[@Keldysh1965; @Keldysh1968; @Jerome1967; @Halperin1968; @Littlewood1996], the ground state itself is excitonic and light absorption or emission involves intra-excitonic transitions that are governed by fundamentally different optical selection rules. The energetically lowest transition from an $1s$-type excitonic ground state will lead to optically excited $p$-type excitons.
In this paper, we present strong evidence that not only for suspended TMDs, but also for TMDs on a SiO$_2$ substrate, the optically active states correspond to the $p$-type excited states of a two-dimensional (2D) hydrogenic Rydberg series, while the lowest lying $1s$-exciton is merged with the ground-state level. Our predictions are based on the analysis of the Hamiltonian for 2D massive Dirac Fermions with Coulomb interaction. To support our theory, we compare its predictions with recent experiments on WS$_2$ [@Chernikov2014; @Zhu2014; @Ye2014] and WSe$_2$[@Wang2014], demonstrating that the assumption of bright $p$-states allows for a simple interpretation of a wide range of different experimental results.
Microscopic Theory
==================
Starting point of our analysis is the low-energy Hamiltonian for the bandstructure in the vincinity of the Dirac-points. Dictated by the symmetry properties of the hexagonal lattice, the lowest order ${{\bf{k}}}\cdot{{\bf{p}}}$ Hamiltonian has the form[@Xiao2012] $$H_0 = \sum_{\tau,{{\bf{k}}}}{\hat{\Psi}}^\dagger_{\tau{{\bf{k}}}} \left( at{{{\bf{k}}}} \cdot {\hat{\bf
\sigma}}_\tau+\frac{\Delta}{2}\hat\sigma_z-\tau\lambda\frac{\hat\sigma_z-1}{2}
\hat s_z \right){\hat{\Psi}}_{\tau{{\bf{k}}}},
\label{Hamiltonian}$$ where $\tau=\pm 1$ is the so-called valley index identifying the two non-equivalent Dirac points, and ${\hat{\Psi}}^\dagger_{\tau{{\bf{k}}}}$ is the tensor product of the electron spin state and a two-component quasi-spinor. The Pauli matrices ${\hat{\bf
\sigma}}_\tau=(\tau\hat\sigma_x,\hat\sigma_y)$ and $\hat\sigma_z$ act in the pseudo-spin space and $\hat s_z$ in the real spin space, respectively. The basis functions for the pseudo-spinors are a linear combination of the relevant atomic orbitals that contribute to the valence and conduction bands and depend on the specific material system under consideration. Within a tight-binding model, the parameters $\Delta$, $t$ and $a$ correspond to the bias in on-site energies, the effective hopping matrix element, and the lattice constant, respectively. Furthermore, $2\lambda$ is the spin-splitting of the valence band due to the intra-atomic spin-orbit coupling (SOC). The Hamiltonian (\[Hamiltonian\]) is valid for the entire class of monolayer hexagonal structures including graphene with a zero gap and negligible SOC, h-BN with a large gap and also negligible SOC, and the variety of TMDs with a gap in the optical range and strong SOC.
Independent of the explicit expressions for the pseudo-spinor basis functions, the Hamiltonian describing the light-matter (LM) interaction can be found by the minimal substitution $\hbar {{\bf{k}}}\rightarrow\hbar{{\bf{k}}}-e{{\bf{A}}}/c$. Defining the Fermi velocity by $ v_F=at/\hbar$, one obtains $$\label{HIspinor}
H_I=-e\frac{v_F}{c}\sum_{\tau,{{\bf{k}}}}{\hat{\Psi}}^\dagger_{\tau{{\bf{k}}}} {{{\bf{A}}}} \cdot {\hat{\bf
\sigma}}_\tau{\hat{\Psi}}_{\tau{{\bf{k}}}}.$$
Using the eigenstates of $H_0$, the LM Hamiltonian can be written as
$$\begin{aligned}
H_I&=&-e\frac{v_F}{c}\sum_{s\tau{{\bf{k}}}}\frac{\hbar v_F k}{2\epsilon_{s\tau{{\bf{k}}}}}
\left({\rm e}^{-i\tau\theta_{{{\bf{k}}}}} A^\tau+{\rm e}^{i\tau\theta_{{{\bf{k}}}}}A^{-\tau}\right)
\left(c^\dagger_{s\tau{{\bf{k}}}}c_{s\tau{{\bf{k}}}}-\nu^\dagger_{s\tau{{\bf{k}}}}\nu_{s,\tau,{{\bf{k}}}}\right)\nonumber\\
&+&e\frac{v_F}{c}\sum_{s,\tau,{{\bf{k}}}}\frac{1}{2\epsilon_{s\tau{{\bf{k}}}}}\left\{
\left((\epsilon_{s\tau{{\bf{k}}}}-\frac{\Delta_{s\tau}}{2}){\rm e}^{-i\tau\theta_{{{\bf{k}}}}}A^\tau
-(\epsilon_{s\tau{{\bf{k}}}}+\frac{\Delta_{s\tau}}{2}){\rm e}^{i\tau\theta_{{{\bf{k}}}}}A^{-\tau}\right)
c^\dagger_{s\tau{{\bf{k}}}}\nu_{s\tau{{\bf{k}}}}
+h.c.\right\} .
\label{HIbands}\end{aligned}$$
Here, $c^\dagger_{s\tau{{\bf{k}}}}$ ($\nu^\dagger_{s\tau{{\bf{k}}}}$) creates an electron with spin $s$ and valley index $\tau$ in the conduction (valence) band, $\Delta_{s\tau}=\Delta-s\tau\lambda$ is the spin and valley dependent gap, $\epsilon_{s\tau{{\bf{k}}}}=\sqrt{\left(\frac{\Delta_{s\tau}}{2}\right)^2+(\hbar v_Fk)^2}$ is the relativistic dispersion of a quasi-particle with rest energy $m_{s\tau}v_F^2=\Delta_{s\tau}/2$, $A^\tau=A_x+i\tau A_y$, and $\theta_{{\bf{k}}}$ is the angle in $k$-space defined by ${\rm tan}\theta_{{{\bf{k}}}}=\frac{k_y}{k_x}$.
In Eq. (\[HIbands\]), the first line describes the intraband and the second line the interband transitions, respectively. Contributions proportional to $ A^\tau$ correspond to the absorption of a photon with circular polarization $\sigma^\tau$. In zero-gap materials like graphene or silicene, the transition matrix elements for both intra- and interband absorption are equal in magnitude. Differences in the transition probabilities exclusively result from the different occupation numbers in the initial state. In contrast, in a wide gap system, the leading orders of the transition matrix elements are given by $(\epsilon_{s\tau{{\bf{k}}}}-\frac{\Delta_{s\tau}}{2})/2\epsilon_{s\tau{{\bf{k}}}}\approx (\hbar v_F k)^2/\Delta_{s\tau}^2$, $\hbar v_F k/2\epsilon_{s\tau{{\bf{k}}}}\approx \hbar v_Fk/\Delta_{s\tau}$, and $(\epsilon_{s\tau{{\bf{k}}}}+\frac{\Delta_{s\tau}}{2})/2\epsilon_{s\tau{{\bf{k}}}}\approx 1$. Keeping only contributions up to first order in $\hbar v_F k/\Delta$ gives
$$\begin{aligned}
H_I&\approx&
-\frac{1}{c}\sum_{s\tau{{\bf{k}}}}{\bf A}\cdot{\bf j}_{s\tau{{\bf{k}}}}\left(c^\dagger_{s\tau{{\bf{k}}}}c_{s\tau{{\bf{k}}}}-\nu^\dagger_{s\tau{{\bf{k}}}}\nu_{s\tau{{\bf{k}}}}
\right)
\nonumber\\
&-&
e\frac{v_F}{c}\sum_{s\tau{{\bf{k}}}}\left\{ {\rm e}^{i\tau\theta_{{{\bf{k}}}}} A^{-\tau}
c^\dagger_{s\tau{{\bf{k}}}}\nu_{s\tau{{\bf{k}}}}
+h.c.\right\},
\label{HIwidegap}\end{aligned}$$
where the intraband contributions have been rewritten with the aid of the intraband current matrix element ${\bf j}_{s\tau{{\bf{k}}}}=-\frac{e}{\hbar}\nabla_{{{\bf{k}}}}\epsilon_{s\tau{{\bf{k}}}}$. From this simplified LM Hamiltonian, Eq. (\[HIwidegap\]), one recognizes clearly that the valence-to-conduction-band excitations at the $K^\pm$ valley require the absorption of a $\sigma^\mp$ polarized photon, as has been shown by Xiao et al.[@Xiao2012].
The optically induced current can be obtained from the LM Hamiltonian via ${\bf j}=-c\langle \frac{\delta H_I}{\delta {\bf A}}\rangle$ and contains both intra- and interband transitions. From Eq. (\[HIwidegap\]) it is clear that a finite macroscopic interband current requires an even part of the microscopic transition amplitudes $P_{s\tau{{\bf{k}}}}={\rm e}^{-i\tau\theta_{{{\bf{k}}}}}\langle\nu^\dagger_{s\tau{{\bf{k}}}}c_{s\tau{{\bf{k}}}}\rangle$, while intraband transitions contribute to the macroscopic current only if $g_{s\tau{{\bf{k}}}}=\langle\nu^\dagger_{s\tau{{\bf{k}}}}\nu_{s\tau{{\bf{k}}}}\rangle-\langle c^\dagger_{s\tau{{\bf{k}}}}c_{s\tau{{\bf{k}}}}\rangle$ contains an odd part.
The microscopic transition amplitudes $P_{s\tau{{\bf{k}}}}$ and the Pauli blocking factor $g_{s\tau{{\bf{k}}}}$ can be computed from the Heisenberg equations of motion yielding the semiconductor Bloch equations (SBE)[@HaugKoch]. Using Eq. (\[HIwidegap\]) and the standard Coulomb-interaction Hamiltonian, one obtains $$\begin{aligned}
\label{SBEP}
i\hbar \partial_t P_{s\tau{{\bf{k}}}}&=&2\left(\Sigma_{s\tau{{\bf{k}}}}-\frac{1}{c}{\bf A}\cdot{\bf j}_{s\tau{{\bf{k}}}}\right)
P_{s\tau{{\bf{k}}}}
\nonumber\\
&-&g_{s\tau{{\bf{k}}}}\left(\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}P_{s\tau{{\bf{k}}}'}
-e\frac{v_F}{c}
A^{-\tau}\right),\\
i\hbar \partial_t g_{s\tau{{\bf{k}}}}
&=&
2P^*_{s\tau{{\bf{k}}}}\left(\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}P_{s\tau{{\bf{k}}}'}
-e\frac{v_F}{c}
A^{-\tau}\right)
-c.c.
\label{SBEf}\end{aligned}$$ where $\Sigma_{s\tau{{\bf{k}}}}
=\epsilon_{s\tau{{\bf{k}}}}+\frac{1}{2}\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}g_{s\tau{{\bf{k}}}'}$ denotes the renormalized single particle energy that includes the Coulomb renormalization of the band-gap.
Regime of weak Coulomb coupling
-------------------------------
Assuming the noninteracting ground state as the initial state before optical excitation, the linear response SBE for the polarization is given by $$\begin{aligned}
\label{SBENI}
i\hbar \partial_t P_{s\tau{{\bf{k}}}}&=&2\Sigma_{s\tau{{\bf{k}}}} P_{s\tau{{\bf{k}}}}-\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}P_{s\tau{{\bf{k}}}'}
-e\frac{v_F}{c} A^{-\tau}.\end{aligned}$$ Since the populations are at least second order in the exciting field, we use $g_{s\tau{{\bf{k}}}}=1$. The expansion into the eigenstates of the Wannier equation with the relativistic single-particle dispersion $$\begin{aligned}
2\Sigma_{s\tau{{\bf{k}}}}\phi_{\mu}^{s\tau}({{\bf{k}}})-\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}
\phi_{\mu}^{s\tau}({{\bf{k}}}')
=
E_{\mu}^{s\tau}\phi_{\mu}^{s\tau}({{\bf{k}}})
\label{wannier}\end{aligned}$$ yields $$\begin{aligned}
\chi^\sigma(\omega)&=&-\frac{e^2 v_F^2}{\omega^2}\sum_{s,\mu}\frac{ {\cal{F}}^{s,-\sigma}_\mu}
{\hbar(\omega+i\gamma)-E_{\mu}^{s,-\sigma}}\nonumber\\
&+&\frac{e^2 v_F^2}{\omega^2}\sum_{s,\mu}\frac{ {\cal{F}}^{s,\sigma}_\mu}
{\hbar(\omega+i\gamma)+E_{\mu}^{s,\sigma}} .
\label{Elliot}\end{aligned}$$ for the linear susceptibility $\chi$, describing the optical response via $$\begin{aligned}
j^\sigma=\frac{\omega^2}{c}\chi^\sigma(\omega)A^ \sigma.\end{aligned}$$ Here, the oscillator strength is given as ${\cal{F}}^{s,\tau}_\mu=\left|\sum_{{{\bf{k}}}}\phi_{\mu}^{s,\tau}({{\bf{k}}})\right|^2\label{oscillatorstrength}$.
The fully relativistic Wannier equation (\[wannier\]) corresponds to the $k$-space representation of the Dirac Coulomb problem. In real space, the wave functions are two-component spinors[@Pereira2008; @Rodin2013] $${\hat{\Psi}}^{s\tau}_{jn}({{\bf{r}}})=\left(
\begin{array}{c}{\rm e}^{i( j-\tau/2)\varphi}A^{s\tau}_{j n}(r)\\ {\rm e}^{i( j+\tau/2)\varphi}B^{s\tau}_{j n}(r)\end{array}
\right)$$ obeying the wave equation $$\left(2 v_F\hat{\bf \sigma}\cdot{{\bf{p}}}+ \Delta_{s\tau}
\hat\sigma_z-\frac{e^2}{r}\right)\Psi_{jn}^{s\tau}({{\bf{r}}})=E_{jn}^{s\tau}\Psi_{jn}^{s\tau}({{\bf{r}}}).
\label{wannierrel}$$ Here $j$ is the eigenvalue of the total angular momentum $\hat L_z+\frac{\tau}{2}\hat\sigma_z$, and $n$ is the principle quantum number. We can identify the set of quantum numbers $\{\mu\}=(j,n)$ such that, e.g., an $s$ -type wave function has $j=\pm 1/2$. The eigenfunctions $\phi_{jn}^{s\tau}$ of Eq. (\[wannier\]) correspond to the solutions of Eq. (\[wannierrel\]) with positive energy and are given by $$\begin{aligned}
\phi_{jn}^{s\tau}({{\bf{k}}})&=&\sqrt{\frac{\epsilon_{s\tau{{\bf{k}}}}+\frac{\Delta_{s\tau}}{2}}{2\epsilon_{s\tau{{\bf{k}}}}}}{\rm e}^{i( j-\tau/2)\theta_{{{\bf{k}}}}}A^{s\tau}_{j n}(k)\nonumber\\
&+&\sqrt{\frac{\epsilon_{s\tau{{\bf{k}}}}-\frac{\Delta_{s\tau}}{2}}{2\epsilon_{s\tau{{\bf{k}}}}}}{\rm e}^{i( j+\tau/2)\theta_{{{\bf{k}}}}}B^{s\tau}_{j n}(k)\nonumber\\
&\approx&
{\rm e}^{i( j-\tau/2)\theta_{{{\bf{k}}}}}A^{s\tau}_{j n}(k),\end{aligned}$$ while the solutions corresponding to negative energies are of no physical relevance here.
For the 2D Coulomb interaction potential, the eigenvalues of the relativistic hydrogen problem are given by[@Pereira2008; @Rodin2013] $$E^{s\tau}_{nj}= \Delta_{s\tau}
\left\{
\frac{n+\sqrt{j^2-\left(\frac{\alpha}{2}\right)^2}}
{\sqrt{\left(\frac{\alpha}{2}\right)^2+\left(n+\sqrt{j^2-\left(\frac{\alpha}{2}\right)^2}\right)^2}}\right\}$$ with $n=0,1,\dots$ and $j=\pm1/2,\pm3/2,..$. Here, $\alpha=e^2/\kappa\hbar v_F$ is the coupling constant and $\kappa$ is the effective dielectric constant of the environment. The eigenvalues are real for those $j$-values where $j^2-\left(\frac{\alpha}{2}\right)^2\ge 0$. For small values of $\alpha$, the eigenvalue spectrum reduces to that of the nonrelativistic Rydberg series $$E_{nj}^{s\tau} = \Delta_{s\tau}-\alpha^2\Delta_{s\tau}/8(n+|j|)^2
= \Delta_{s\tau}-\alpha^2\Delta_{s\tau}/8(N-1/2)^2$$ with $N=n+|j|+1/2=1,2,\dots$.
From the expression (\[Elliot\]) for the linear susceptibility, one can derive the exciton spectrum and optical selection rules for dipole allowed optical transitions. At the poles of Eq. (\[Elliot\]), the reflection and absorption spectra display resonances. The oscillator strength is a measure for the transition probability from the noninteracting groundstate to the excitonic state with quantum number $\mu$.
Thus, we see that only excitonic transitions with $s$-type orbital angular momentum are dipole allowed. Furthermore, the resonant contributions for a given circular polarization component $\sigma$ stem from the $K^{-\sigma}$ valley, while the nonresonant contributions stem from the $K^{\sigma}$ valley. This opens the possibility of a valley selective excitation with circularly polarized light. Moreover, the dependence of the resonance energies $E_\mu^{s,\sigma}$ on the product of the spin and valley index couples the valley dynamics to the spin dynamics. Both effects have been predicted in and demonstrated experimentally for various monolayers of TMDs.
Regime of Strong Coulomb Coupling
---------------------------------
Interestingly, in the regime of strong Coulomb coupling characterized by $\alpha>1$, the optical spectrum contains no $s$-type wave functions with $j=\pm 1/2$. In this case, total angular momentum conservation cannot be fulfilled for optical transitions from the noninteracting groundstate such that no one-photon transitions should be possible.
To check if any of the TMD systems can be in the regime of strong Coulomb coupling, we use the material parameters given in Ref. . Thus, we obtain the nominal values $\alpha= 3.29 /\kappa$ for WS$_2$, $\alpha= 3.66 /\kappa$ for WSe$_2$ and $\alpha= 4.11 /\kappa$ for MoS$_2$ respectively. In free standing monolayers of TMDs, these values are very large and even on top of a typical substrate like SiO$_2$ with $\kappa=(3.9+1)/2$, the effective coupling exceeds the critical value. Thus, the assumption of a noninteracting ground state predicts [*the absence of the one-photon optical transitions*]{} for all of these systems, clearly contradicting the experimental findings.
However, in the regime of strong Coulomb coupling, the real part of the lowest exciton energy $E_{0\pm\frac{1}{2}}$ vanishes, i.e., it is degenerate with the noninteracting groundstate and the system transitions into an excitonic insulator state. This excitonic insulator state is a BEC-like condensate of excitons, i.e. a coherent superposition of the noninteracting groundstate and all exciton states with $\Re\left[E_\mu\right]= 0$[@Keldysh1965; @Keldysh1968; @Jerome1967; @Halperin1968; @Littlewood1996]. This state exhibits a static interband polarization $P_{s\tau{{\bf{k}}}}^{\left[0\right]}=\Omega_{s\tau{{\bf{k}}}}^{\left[0\right]}/2{\cal E}_{s\tau{{\bf{k}}}}$ and static blocking factor $g_{s\tau{{\bf{k}}}}^{\left[0\right]}=\Sigma_{s\tau{{\bf{k}}}}^{\left[0\right]}/{\cal E}_{s\tau{{\bf{k}}}}$ that can be obtained as nontrivial solutions of the gap equations[@stroucken2011; @stroucken2013] $$\begin{aligned}
\Omega_{s\tau{{\bf{k}}}}^{\left[0\right]}&=&\frac{1}{2}\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}\frac{\Omega_{s\tau{{\bf{k}}}'}^{\left[0\right]}}{{\cal E}_{s\tau{{\bf{k}}}'}},\\
\Sigma_{s\tau{{\bf{k}}}}^{\left[0\right]}&=&\epsilon_{s\tau{{\bf{k}}}}+\frac{1}{2}\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}
\frac{\Sigma_{s\tau{{\bf{k}}}'}^{\left[0\right]}}{{\cal E}_{s\tau{{\bf{k}}}'}} \, .
\end{aligned}$$ Here, ${\cal E}_{s\tau{{\bf{k}}}}=\sqrt{ \left.\Omega_{s\tau{{\bf{k}}}}^{\left[0\right]}\right.^2+
\left.\Sigma_{s\tau{{\bf{k}}}}^{\left[0\right]}\right.^2}$ is the dispersion of the Bogoliubov bands.
The static groundstate polarizations and populations induce a coupling between the linearized equations for the optical polarization and populations and add an additional source term to the equation of motion for the optical interband transition amplitude. Explicitely, we have
$$\begin{aligned}
i\hbar\partial_t P_{s\tau{{\bf{k}}}}^{\left[1\right]}&=&2\Sigma_{s\tau{{\bf{k}}}}^{\left[0\right]} P^{\left[1\right]}_{s\tau{{\bf{k}}}}-
g_{s\tau{{\bf{k}}}}^{\left[0\right]}\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}P^{\left[1\right]}_{s\tau{{\bf{k}}}'}\nonumber\\
&+&2 P_{s\tau{{\bf{k}}}}^{\left[0\right]}\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}g^{\left[1\right]}_{s\tau{{\bf{k}}}'}
-
\Omega^{\left[0\right]}_{s\tau{{\bf{k}}}'}g_{s\tau{{\bf{k}}}}^{\left[1\right]}
\nonumber\\
&-&e\frac{v_F}{c}g_{s\tau{{\bf{k}}}}^{\left[0\right]} A^{-\tau}
-\frac{2}{c}{\bf A}\cdot{\bf j}_{s\tau{{\bf{k}}}} P_{s\tau{{\bf{k}}}}^{\left[0\right]},
\label{SBEPEx}
\\
i\hbar \partial_t g_{s\tau{{\bf{k}}}}^{\left[1\right]}
&=&
2 \Omega^{\left[0\right]}_{s\tau{{\bf{k}}}'} P_{s\tau{{\bf{k}}}}^{\left[1\right]*}
+2 P_{s\tau{{\bf{k}}}}^{\left[0\right]*}\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}P^{\left[1\right]}_{s\tau{{\bf{k}}}'}-c.c.
\nonumber\\
\label{SBEfEx}\end{aligned}$$
As the additional source term is proportional to the intraband current matrixelement with odd parity, it mixes states with different parity. Though the odd terms of the linear interband polarization do not couple to the optical field directly, they serve as source terms for the optically induced intraband current. Eqs. (\[SBEPEx\]) and (\[SBEfEx\]) can be decoupled by the transformation $$\begin{aligned}
\label{trafo}
\Pi_{s\tau{{\bf{k}}}}^{\left[1\right]}&=&\frac{{\cal E}_{s\tau{{\bf{k}}}}+\Sigma_{s\tau{{\bf{k}}}}^{\left[0\right]}}{{\cal E}_{s\tau{{\bf{k}}}}}P_{s\tau{{\bf{k}}}}^{\left[1\right]}
-\frac{{\cal E}_{s\tau{{\bf{k}}}}-\Sigma_{s\tau{{\bf{k}}}}^{\left[0\right]}}{{\cal E}_{s\tau{{\bf{k}}}}}P_{s\tau{{\bf{k}}}}^{\left[1\right]*}
-\frac{\Omega_{s\tau{{\bf{k}}}}^{\left[0\right]}}{{\cal E}_{s\tau{{\bf{k}}}}}g_{s\tau{{\bf{k}}}}^{\left[1\right]},\\
\Gamma_{s\tau{{\bf{k}}}}^{\left[1\right]}&=&\frac{\Omega_{s\tau{{\bf{k}}}}^{\left[0\right]}}{{\cal E}_{s\tau{{\bf{k}}}}}\left(
P_{s\tau{{\bf{k}}}}^{\left[1\right]}+P_{s\tau{{\bf{k}}}}^{\left[1\right]*}\right)+
\frac{\Sigma_{s\tau{{\bf{k}}}}^{\left[0\right]}}{{\cal E}_{s\tau{{\bf{k}}}}}
g_{s\tau{{\bf{k}}}}^{\left[1\right]},
\end{aligned}$$ yielding $$\begin{aligned}
i\hbar\partial_t \Pi_{s\tau{{\bf{k}}}}^{\left[1\right]}&=&2{\cal E}_{s\tau{{\bf{k}}}}^{\left[0\right]}
-\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}\Pi^{\left[1\right]}_{s\tau{{\bf{k}}}'}\nonumber\\
&-&2e\frac{v_F}{c}g_{s\tau{{\bf{k}}}}^{\left[0\right]} A^{-\tau}
-\frac{4}{c}{\bf A}\cdot{\bf j}_{s\tau{{\bf{k}}}} P_{s\tau{{\bf{k}}}}^{\left[0\right]},
\label{SBEBog}
\\
i\hbar \partial_t \Gamma_{s\tau{{\bf{k}}}}^{\left[1\right]}
&=&0.\end{aligned}$$ Hence, the solutions of Eq. (\[SBEBog\]) can be expanded in terms of the eigenfunctions of the Bogoliubov-Wannier equation $$2{\cal E}_{s\tau{{\bf{k}}}}\psi_{\mu}^{s\tau}({{\bf{k}}})-\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}
\psi_{\mu}^{s\tau}({{\bf{k}}}')=E^{s\tau}_{\mu}\psi_{\mu}^{s\tau}({{\bf{k}}}'),
\label{generalizedwannier}$$ where the dispersion of the noninteracting bands has been replaced by the Bogoliubov quasi-particle dispersion. The groundstate polarization obeys $2{\cal E}_{s\tau{{\bf{k}}}}P_{s\tau{{\bf{k}}}}^{\left[0\right]}-\sum_{{{\bf{k}}}'}V_{{{\bf{k}}},{{\bf{k}}}'}
P_{s\tau{{\bf{k}}}'}^{\left[0\right]}=0$ such that it can be expanded into the eigenfunctions of the Bogoliubov-Wannier equation with the eigenvalue $E^{s\tau}_{\mu}=0$. Hence, we make the ansatz $$\begin{aligned}
P_{s\tau{{\bf{k}}}}^{\left[0\right]}&=&\sum_{\mu,E_\mu=0}
P_{s\tau\mu}^{\left[0\right]} \psi^{s\tau}_\mu({{\bf{k}}}),\\
\Pi_{s\tau{{\bf{k}}}}^{\left[1\right]}&=&\sum_{\mu}\Pi_{s\tau\mu}^{\left[1\right]}\psi^{s\tau}_\mu({{\bf{k}}}),\end{aligned}$$ Inserting the ansatz into Eq. (\[SBEBog\]) yields $$\Pi_{s\tau\mu}^{\left[1\right]}(\omega)=
-2e\frac{v_F}{c}A^{-\tau}\frac{(\gamma_\mu^{s\tau})^*}{\hbar\omega-E_\mu}
-\frac{4}{c}\frac{{\bf A}\cdot{\bf j}^{s\tau}_\mu}{\hbar\omega-E_\mu}.
\label{Pilambda}$$ Here, $\gamma_\mu^{s\tau}=\sum_{{{\bf{k}}}}g_{s\tau{{\bf{k}}}}^{\left[0\right]}\psi_\mu({{\bf{k}}})$ is the coupling strength of the macroscopic interband polarization to the optical field. For an $s$-type groundstate ($1<\alpha< 3$), the populations also have spherical symmetry and the coupling strength vanishes for all non $s$-type excition states, i.e. for all real exciton resonances. Generally, if $\alpha<(2n+1)/2$, the groundstate populations are composed of exciton states with $|j|\le(2n+1)/2$, while real exciton resonances require $|j|\ge(2n+1)/2$. Thus, in the strong Coulomb coupling regime, the first term on the RHS of Eq. (\[Pilambda\]) vanishes.
Reversing the transformation \[trafo\], we can compute the optical susceptibility for the excitonic insulator state $$\begin{aligned}
\chi^\sigma(\omega)_{EI}&=&-\frac{1}{\omega^2}\sum_{s,\mu}
\frac{
|{\bf j}^{s-\sigma}_\mu|^2
}
{\hbar(\omega+i\gamma)-E_{\mu}^{s,-\sigma}}
\nonumber\\
&+&\frac{1}{\omega^2}\sum_{s,\mu}
-\frac{ |{\bf j}^{s\sigma}_\mu|^2}
{\hbar(\omega+i\gamma)+E_{\mu}^{s,\sigma}}
\label{ElliotEI}.\end{aligned}$$ Here, all contributions have intraband character.
As can be recognized from Eq. (\[ElliotEI\]), the poles of the linear susceptibility for the excitonic insulator state occur at the spectrum of the exciton Hamiltonian with Bogoliubov dispersion. The dominant effect of the exciton condensation on the single quasi-particle dispersion is a renormalization of the gap. Hence, the spectrum can be obtained in very good approximation from Eq. \[wannierrel\], provided one uses the appropriate renormalized gap. The major difference as compared to the optical susceptibility, Eq.(\[Elliot\]), for the uncorrelated groundstate is the oscillator strength, which is now given by the intraband current matrix element between the excited exciton state and the excitonic groundstate $${\bf j}^{s\tau}_\mu=\sum_{{{\bf{k}}}}\left.\psi^{s\tau}_{\mu}\right.^*({{\bf{k}}}){\bf j}_{s\tau{{\bf{k}}}} P_{s\tau{{\bf{k}}}}^{\left[0\right]}\equiv \langle \mu s\tau|
{\bf j}|0\rangle.$$ This matrix element connects that part of the groundstate with largest $|j|\le\alpha/2$ to the excited states with $|j'|=|j|+1>\alpha/2$ such that dipole allowed transitions increase (decrease) the total angular momentum by one. Explicitly, in the regime with $1<\alpha< 3$, optical transitions connect the $s$-type excitonic groundstate $j=\tau /2, l=0$ with the excited excitonic states $j'=3\tau/2,l=\tau$, corresponding to $p$-like states. Thus, combining the results of the weakly and strongly interacting regimes, the modified optical selection rules can be summarized by the condition $$(2(|l|-1)< \alpha < (2|l|+1)$$ for the optically active states.
Analysis of Experimental Observations
=====================================
To test our general theory, we compare its predictions with the observations of several recent experiments on WS$_2$ [@Chernikov2014; @Zhu2014; @Ye2014] and WSe$_2$[@Wang2014]. As commonly adopted, we refer to A and B excitons for the lower and higher direct exciton transitions with a spin and valley index combination $s\tau=1,-1$ respectively.
While the effective fine structure constant can be extracted from the tight binding parameters and the dielectric screening, to compute the energetic positions of the bright states requires the additional knowledge of the Coulombic bandgap renormalization. Whereas the bandgap renormalization diverges for an ideal, strictly 2D Coulomb potential, it is finite for any real system where the Coulomb matrix elements must be evaluated from the gap equations with the atomic orbitals used to represent the pseudo spinor space in Eq. (\[Hamiltonian\]). This leads to a quasi-2D Coulomb interaction $$V({{\bf{q}}})=\frac{2\pi e^2}{\cal{A}\kappa}\frac{F(qd)}{q},$$ where $\cal{A}$ is the normalization area and $F(qd)$ is a monotonically decreasing form factor with $F(0)=1$ and $\lim_{x\rightarrow\infty}xF(x)=0$. The form factor leads to a regularization of the $1/r$ singularity at small distances and the parameter $d$ can be interpreted as the effective thickness of the monolayer. Since its explicit expression depends on the specific atomic orbitals contributing to the valence and conduction band, we do not calculate the bandgap but rather use it as a fit parameter.
Linear Reflection experiments
-----------------------------
![(color online) Spectral position of the energetically lowest bright excitonic transitions with $j=\pm 3/2$ in WS$_2$. The blue squares are the computed results for p-type excitons, the red dots show the experimental data taken from [@Chernikov2014], and the green diamonds indicate the theoretical predictions if one assumes $s$- type bright states, respectively. The dashed and dotted lines are guides to the eye. The x-axis label denotes the number $M$ of the experimentally observed bright exciton resonances and is related to the priniple quantum number via $N=M-1$. The inset to the figure shows the results for the energetically higher states with a finer energy resolution.[]{data-label="Comparisontoexperiment"}](Fig1.pdf){width="7.8cm"}
We start with the analysis of the bright excitonic states in WS$_2$ on SiO$_2$ corresponding to the lowest A exciton with spin and valley combination $s\tau=1$[@Chernikov2014]. With the material parameters for WS$_2$ $t=1.37$eV, $2\lambda=0.43$ eV, $a=3.197$ given in Ref. [@Xiao2012], an effective dielectric constant $\kappa=(1+3.9)/2$ for the SiO$_2$ substrate, and using $E_{gap}=2.36$ eV, we find a binding energy of $0.25$ eV for the lowest bright exciton transition with $(n,j)=(0,\pm3/2)$. Using $E_{gap}=2.36$ eV, we present in Fig. \[Comparisontoexperiment\] the computed result for the spectral position of the lowest five bright exciton transitions. The red diamonds show the experimental data points taken from Ref. . For comparison, we also show the predicted values assuming nonrelativistic bright $s$-states. (Note that the $j=\pm1/2$ series gives complex eigenvalues of the relativistic wave equation and thus cannot be used for comparison.) Our computed results show remarkably good agreement with the experimental observations given the fact that only the effective bandgap was used as a single fit parameter. In contrast, the results based on the assumption that the series has $s$-type character are completely off the scale, in particular for the energetically lowest states.
Assuming that the observed spin splitting corresponds to the difference in the renormalized bandgaps, we obtain $E^B_{gap}=2.79eV$ which can be used to compute the spectral positions of the B-excitons. The ratio of the binding energy is fixed by the ratio of the renormalized gaps, i.e. $E^B_0/E^A_0=E^B_{gap}/E^A_{gap}$, giving a binding energy of $0.30$ eV for the lowest bright B exciton. Thus the $2p$ resonance is predicted to occur at $E^B_{2p}=2.49$ eV, which is also in very good agreement with the experimental findings[@Chernikov2014]. In table \[Table1\], a list of the lowest theoretically predicted resonance positions of both A and B exciton is given.
$E^A_2$ $E^A_{3}$ $E^A_{4}$ $ E^A_{G}$ $ E^B_{2}$ $ E^B_{3}$ $E^B_4$ $ E^B_{G}$ method Ref.
-- --------- ----------- ----------- ------------ ------------ ------------- --------- ------------ ------------------- ------------------
2.11 2.27 2.31 2.36 2.49 2.68 2.74 2.79 theory
2.09 2.25 2.3 2.5 linear reflection [@Chernikov2014]
2.02 2.4 linear absorption [[@Zhu2014]]{}
2.4 2.6 2.73 TP-PLE [@Zhu2014]
2.04 2.45 linear reflection [@Ye2014]
x 2.25 2.29 2.35 2.46 TP-PLE [@Ye2014]
1.75 1.91 1.96 2.01 2.15 2.35 2.41 2.47 theory
1.75 PLE [@Wang2014]
x 1.9 x TP-PLE [@Wang2014]
1.75 1.91 1.98 2.02 2.17 2.3 SHG [@Wang2014]
: Excitonic resonance positions extracted from experimental data together with the theoretical predictions where the gap of the A-exciton has been used as single fit parameter to match the experimental data. For WS$_2$, the band gap has been fitted to match the data of Ref. (see Fig. 1 and the discussion in the text). For WSe$_2$, the lowest bright resonance position has been fitted to the data of Ref. . The data of Ref. correspond to room temperature and show a red shift of approximately $0.1$ eV as compared to the linear reflection spectra at 10K. \[Table1\]. The labeling corresponds to the main quantum number $ N$ of the $ Nth$ excitonic level, which is usually used in semiconductor optics.
Two Photon Experiments
----------------------
To provide further evidence for the theoretically predicted selection rules, we also analyze recent experiments probing dark states by two-photon spectroscopy which can only access excitonic states with a parity opposite to those excited by a single photon process[@Zhu2014; @Ye2014; @Wang2014]. In Ref. , linear reflection and TP-PLE have been performed on WS$_2$ flakes having nominally the same material parameters as those in Ref.. The resulting experimental data are shown in Fig. \[spectraWS2experimental\].
![(color online) Comparsion of one and two photon spectra of WS$_2$ measured by different groups. Part a) shows the deravative of the reflection contrast measured by Chernikov et al.[@Chernikov2014], part b) the the linear reflection contrast (black line) together with the TP-PLE spectrum (red dots) measured by Ye at al. [@Ye2014], and part c) the linear absorption (black line) and TP-PLE (red dots) measured by Zhu et al. [@Zhu2014]. To compensate for temperature dependent shifts of the band gaps, we shifted the data such that the lowest resonance of the A excitons are alligned. On the upper axis, we assigned the quantum numbers assuming bright $p$ states. The onsets of the band gaps are denoted by $A_G$ and $B_G$. The dotted line are guides to the eye and show the energetic position of some selected states.[]{data-label="spectraWS2experimental"}](Fig2.pdf){width="7.6cm"}
In Ref. , the linear reflection has been measured for different temperatures, showing two clearly recognizable peaks for $T=10$K at $2.1$ eV and $2.5$ eV, respectively. These peaks agree well with the dominant peaks in the reflection spectrum of Ref. and can be assigned to the A and B exciton respectively. At room temperature, both peaks experience a red shift of approximately $0.1$ eV, most likely resulting from a temperature dependent bandgap renormalization. The TP-PLE measurements have been performed at room temperature and cover the spectral range between $2.4$ and $3.0$ eV. Within this spectral range two peaks in the TP-PLE spectrum are observed at approximately $2.4$ and $2.6$ eV, and a significant gap at approximately $2.73$ eV. In Ref. , the A and B excitonic resonances occur at $ 2.04$ and $2.45$eV in the linear reflection spectrum respectively. Thus, the energetic separation of the two dominant peaks is in good agreement with Refs. and . The small deviations in the absolute values are most likely due to different experimental conditions. The TP-PLE spectrum covers the range between $1.9$ and $2.6$ eV and thus complements the measurements of Ref. . Within this energy range, the TP-PLE spectrum exhibits pronounced maxima between $2.25-2.29$ eV and at $2.45$ eV. Remarkably, both experiments observe a large TP-PLE signal resonant with the lowest bright transition of the B exciton, whereas the TP-PLE in Ref. shows no significant signal at the lowest bright A exciton.
In Ref. , TP-PLE measurements have been performed in conjunction with SHG experiments on WSe$_2$. This material system has very similar properties as WS$_2$, in particular it has the same symmetries and similar spin-splitting of the valence bands. Using PLE, the resonance position of the A exciton is determined as $1.75$ eV, whereas in TP-PLE, the dominant peak occurs at approximately 1.9 eV. In agreement with the observation in Ref. , the TP-PLE spectrum shows no significant signal at the transition energy of the lowest bright A exciton. In contradiction to the results of Ref. , the TP-PLE spectrum does not show a significant signal at the transition energy of the lowest bright B exciton. In contrast, the SHG spectrum exhibits two dominant peaks at $1.75$ eV, i.e, at the lowest bright resonance, and at $2.16$ eV, and less prominent peaks at $1.9$, $2.02$ and $2.3$ eV.
In Refs. , the experiments have been analyzed under the assumption of bright $s$-excitons. This assumption is seemingly supported by the absence of a TP-PLE signal at the resonance frequency of the lowest bright A exciton transition[@Ye2014], and at the lowest A and B exciton[@Wang2014] respectively. Thus, the resonances observed in TP-PLE spectra are assigned to $p$-like states. Since the lowest $p$-state should be energetically well above the lowest $s$- state, all resonances observed in the TP-PLE spectra of WS$_2$ are therefore assigned to excited states of the A exciton, including those resonant with the lowest bright B exciton and above. This assignement directly contradicts the Rydberg series observed by Chernikov et al.[@Chernikov2014], where all excited states and the onset of the band gap of the A-exciton are found [*below*]{} the B- exciton.
Remarkably, all the above mentioned experimental observations can be explained quite naturally by the selection rules predicted by our theory. In the strongly interacting regime, the lowest bright exciton state is $p$-type and nondegerate, as $s$-type wave functions do not exist. The higher states are at least two-fold degenerate, and a two-photon absorption excites a $d$-type excited state. Thus, if the groundstate is excitonic, the lowest excited state should be dark in a two-photon experiment, exactly as is the case for the lowest $s$-exciton in a weakly interacting system. However, if the coupling constant is slightly below the critical value, the $j=1/2$ states become bright and the state with $(n,j)=(0,3/2)$ becomes (nearly) degenerate with the $(n,j)=(1,1/2)$ state, which is the first excited $s$-type state. Simultaneously, the $p$-state can only be reached via a two-photon transition, while the corresponding degenerate $s$-state would be bright. Thus, the observation of a TP-PLE signal at the resonance position of the lowest bright B-exciton in the WS$_2$ sample seems to indicate that the WS$_2$ sample is at the edge of an excitonic insulator, with the lowest B$1s$ exciton transition somewhere in the (far) infrared.
In Table \[Table1\], we summarize the reported observations and compare our theoretically predicted resonance positions of WS$_2$ and WSe$_2$ with the experimentally available data. The theoretical values have been obtained using the material parameters of Ref. [@Xiao2012], $\kappa=(1+3.9)/2$ with the band gap as a single fit parameter. Note, that we assigned quantum numbers to the respective resonance positions that result from our theoretical assignment and do not necessarily correspond to the labeling in the respective original publications. Resonances that could not be resolved by the specific experiment remain as empty spaces in Table \[Table1\] , while those resonances that could in principle occur in the experiment but are not observed are indicated by an ’x’. As can be recognized from the table, not only the agreement between theory and experiment is remarkable, but also the data obtained by the different experimental techniques on different samples is brought into almost perfect mutual agreement as soon as one adopts our interpretation in terms of optically active $p$-states.
Discussion
==========
The results discussed in the previous sections are based on the model Hamiltonian (\[Hamiltonian\]) treating the electrons and holes in the vicinity of the $K$-points as massive Dirac Fermions. The strength of this model Hamiltonian is its simplicity, allowing for an analytical solution of the exciton problem and thus providing insight into the optical properties of Coulomb-interacting chiral quasi-particles that are difficult to obtain numerically. The obvious disadvantage is that it contains several simplifications that might restrict the application of the model to real material systems. In the following, we give a brief discussion of the approximations underlying our analysis, and estimate its validity and restrictions.
Inherent to the massive Dirac model is a single-particle dispersion that has a perfect electron-hole symmetry. The effect of an eventual electron-hole asymmetry is twofold: It modifies the single-particle dispersion and therewith the homogeneous part of the Wannier equation and it alters the dipole matrix element, i.e. the light-matter interaction. The latter effect may weaken the optical selection rules, i.e. unlike for the case of a perfect electron-hole symmetry, one- and two-photon processes can address the same states, while the first effect yields a modified exciton spectrum.
For the exciton equation, the relevant quantity is the energy difference between the conduction and valence band. In a conventional semiconductor with parabolic bands, this leads to an exciton dispersion and binding energy that depend on the reduced mass of the electron-hole pair only. Adding a term to the single-particle Hamiltonian of the form $$H_{as}^{s\tau}=\sum_{{{\bf{k}}}} {\hat{\Psi}}^\dagger_{a\tau{{\bf{k}}}} k^2\left(
\begin{array}{cc}
\alpha_{s\tau}&0\\0&\beta_{s\tau}
\end{array}
\right)
{\hat{\Psi}}_{s\tau{{\bf{k}}}},$$ that has been proposed in Refs. to account for different electron and hole masses within the chiral two-band model, it is straightforward to show that, up to corrections of the order ${\cal O}(k^4)$, the exciton dispersion corresponds to that of a relativistic, symmetric electron-hole pair with a Fermi velocity $v_F^*=v_F\sqrt{\frac{m_0}{2m_r}}$ and effective mass $m^*=\Delta/2 v_F^{*2}$. Here, $m_0=\Delta/2 v_F^2$ and $m_r$ is the reduced mass of the electron-hole pair. It is worth to note that the so defined effective mass is not directly related to the electron and hole masses but describes the [*pair*]{} properties only. Using the parameters for WS$_2$ given in [@Kormanyos2015], this gives $v_F^*=1.066 v_F$ and a nominal coupling constant $\alpha^*=3.08$ which is still well above the critical value. Thus, although a possible electron-hole asymmetry may lead to a violation of the optical selection rules, the massive Fermion model is well suited to describe the exciton spectrum even for asymmetric electrons and holes, provided one uses the effective Fermi-velocity.
To obtain the spectral position of the excitons, we utilize the solutions of the resulting relativistic Wannier equation where we include only the substrate background screening but neglect all other screening effects. In general, screening effects arise both from filled valence bands that are not considered explicitely in the model Hamiltonian, and screening of the conduction and valence band under consideration. The Coulomb matrix elements used to set up the interacting model Hamiltonian should contain only the first type of contributions which can be considered to lead to a constant background dielectric constant. For a film of vanishing thickness $d$, this contribution can be neglected[@Wehling2011]. The second type of contributions should then be computed selfconsistently, using the Coulomb matrix elements screened by the background dielectric constant.
For the massive Dirac-Hamiltonian (\[Hamiltonian\]), the RPA dielectric function of the TMD monolayer is given by [@Rodin2013], $$\epsilon(q)=\kappa\left(
1+\frac{4}{\pi}\frac{1}{qa_B}P\left(\frac{\alpha}{2}qa_B\right)\right),$$ where $\kappa$ is the effective dielectric screening of the substrate and $a_B=2\hbar v_F/\alpha\Delta $ is the $2D$ exciton Bohr radius. An analytical expression for $P(Q)$ is given in Ref. . In the nonrelativistic limit $\frac{\alpha}{2}qa_B\ll 1$, $P(Q)\approx \frac{\pi Q^2}{3}$, yielding $\epsilon(q)=1+\frac{1}{3}\alpha^2 a_B q$. In this regime, the intrinsic screening corresponds to the anti-screening proposed by[@Keldysh1978; @Cudazzo2011] with an anti-screening length $r_0= \frac{1}{3}\alpha^2 a_B$. Using the nominal values of $\alpha$ in suspended TMDs, the anti screening length is typically $\propto 1-1.5 \, a\approx 3-5$ Å, which corresponds roughly to the physical thickness of the monolayer. For supported samples, the anti-screening length is reduced correspondingly.
The anti-screening is typical for a truly 2D-dielectric material and and weakens the strength of the Coulomb interaction on a length scale small compared to the anti-screening length. Thus, for the weakly bound excitons with a spatial extension large compared to the 2D-exciton Bohr radius, we expect the anti-screening effect to be of minor importance. This expectation is indirectly confirmed by the excellent agreement of the experimentally observed resonance positions and our theoretical predictions.
At this point, we mention that the anti-screening model has also been employed to understand the observed non-hydrogenic series in TMDs based on an analysis in terms of optically bright $s$-states, using the anti-screening length as fit parameter[@Chernikov2014; @Berkelbach2015]. For WS$_2$ on SiO$_2$, reasonable agreement with the experimental observations could be obtained using $r_0=75$ Å, which is much larger than the physical thickness.
In the ultrarelativistic regime, $\frac{\alpha}{2}qa_B\gg 1$, $P(Q)\approx \pi^2Q/4$ and the dielectric function $\epsilon(q)=1+2\alpha/\pi$ reduces to a constant, as in graphene. For large $q$-values, intrinsic screening therefore may play a significant role and eventually prevent the phase transition into the excitonic state.
Nevertheless, the weakly bound states are not affected by this screening, nor by other deviations of the full electron spectrum beyond the quadatic approximation. Thus, the excited states still have $|j|>\alpha/2$. Hence, assuming a noninteracting groundstate and a nominal value of $\alpha>1$, the excited exciton states should be dark and the only bright resonance in a linear optical experiment should correspond to the $1s$ exciton. Since the experiments do not observe this feature but rather show a whole series of excitonic resonances, this serves as a clear indication that the groundstate is indeed excitonic and the bright exciton resonances correspond to $p$-type wave functions with $|j|=3/2$.
Conclusion
==========
In conclusion, we presented a general theory which assigns a $p$-like symmetry to the wave functions of the optically active excitonic transitions in recently investigated TMDs. This theory is supported by recently published experimental data, including the observation of a series of resonances in the linear reflectrum spectrum of WS$_2$[@Chernikov2014], two photon absorption in WS$_2$[@Zhu2014; @Ye2014] and WSe$_2$[@Wang2014]. This observation of optically active $p$-transitions is not only of crucial importance for the correct interpretation of experimental data, but impressively demonstrates that in the TMDs investigated here, the Coulomb interaction is strong enough to induce an excitonic ground state. The dipole allowed excited states of such an excitonic insulator correspond to intra-excitonic transitions and are governed by the corresponding optical selection rules. Even for systems in the weakly interacting regime without an excitonic ground state, the lowest excitonic transition should be in the THz or infrared regime and the observed optical resonances correspond to excited exciton levels with main quantum number $N \ge 2$.
Acknowledgements: This work is supported by the Sonderforschungsbereich 1083 funded by the Deutsche Forschungsgemeinschaft. We thank the authors of Ref. for sending us a preprint of their manuscript.
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abstract: 'A total negative field dependence of hole mobility down to low temperature was observed in N,N’-diphenyl-N,N’-bis(3-methylphenyl)-(1,1’-biphenyl)-4,4’diamine (TPD) doped in Polystyrene. The observed field dependence of mobility is explained on the basis of low values of energetic and positional disorder present in the sample. The low value of disorder is attributed to different morphology of the sample due to aggregation/crystallization of TPD. Monte Carlo simulations were also performed to understand the influence of aggregates on charge transport in disordered medium with correlated site energies. The simulation supports our experimental observations and justification on the basis of low values of disorder parameters.'
address: 'Laser Physics Applications Division, Raja Ramanna Centre for Advanced Technology, Indore, India 452013.'
author:
- 'S. Raj Mohan'
- 'M. P. Joshi'
- 'Manoranjan. P. Singh'
title:
- Negative electric field dependence of mobility in TPD doped Polystyrene
- Figure Captions
---
,
,
Molecularly doped polymers ,Disordered system; Morphology ,Electronic transport ,Aggregation ,Monte Carlo simulation
72.80.Le ,72.80.Lg ,71.55.Jv ,73.61.Ph ,68.55.-a
Introduction
============
Films of organic semiconducting materials are widely used in developing various optoelectronic devices like organic light emitting diodes (OLED), organic field effect transistors (OFET), organic solar cells etc\[1,2\]. Spin cast films of molecularly doped polymers (MDP) are used most widely as active layer in these devices because of simple fabrication technique. These MDP films are mostly amorphous/disordered and have low carrier mobility. To improve mobility and transport properties various methods, like annealing, irradiation etc. are employed to reduce the disorder in the active layer (i.e. change film morphology)\[3-6\]. These methods improve structural order in the organic thin film by creating more ordered regions in the otherwise amorphous film. Often, depending upon the processing conditions while film casting, the ordered regions are also formed unintentionally due molecular aggregation / crystallization of dopants in MDP or aggregation of polymer chains in case of conjugated polymers \[7,8\]. Therefore, in most of the cases where the active layers are MDP or pure polymer, films are not purely disordered or amorphous rather they are partially ordered. The charge transport in the partially ordered films therefore demands deeper understanding of transport mechanism. The assumptions made in hopping based charge transport models, for example the Gaussian Disorder Model(GDM) \[9\], developed for a completely isotropic and disordered medium probably not sufficient \[10\]. Moreover these models of charge transport do not consider morphology of the active layers in finer details. The presence of ordered regions can in-fact reduce the overall energetic disorder in the material \[5,7\] and that enhances the mobility but substantially influence the charge transport mechanism \[7,11,12\]. Field and temperature dependence of mobility in such partially ordered samples show wide variation with morphology of the sample and is subject of intense research. In general the field dependence of the mobility in polymer films follows Poole-Frenkel type behavior. However, in certain cases, at low temperature, either very weak field dependence or even negative field dependence of mobility has also been reported \[13-16\]. Reports also suggest dependence of mobility on temperature as $1/T$ as well as $1/T^2$ dependence. In some cases it is hard to distinguish between the two power law behavior of the mobility \[13,14,17\].
In Poole-Frenkel type behavior, the mobility increases with increase of electric field in a $\log\mu Vs. E^{1/2}$ fashion. According to GDM the increase of mobility in $\log\mu Vs. E^{1/2}$ fashion in disordered molecular solids is due to the tilting of density of states by the applied potential that lead to the decrease of energetic barrier as seen by the charge carriers in its transit \[9\]. At higher electric field strength the energetic barrier seen by the carrier is negligibly small and this results in the saturation of drift velocity of the carrier. Once the drift velocity get saturated the mobility decrease with further increase of electric field, i.e. negative field dependence of mobility\[18\]. At intermediate field strength when the temperature is low, the field dependence of mobility remains positive. When temperature increases carrier gains more thermal energy to over come the energetic barrier. Thus the mobility increases and the field dependence of mobility weakens. At higher temperature energetic barrier seen by the carrier is negligibly small which results in negative field dependence of mobility even at intermediate field strengths. Lower the energetic disorder inside the sample lower is the temperature at which negative field dependence can be observed \[7a,11,13b\]. In principle if disorder is very low one can observe negative field dependence of mobility at lower temperature. The above reasoning is also justified on the basis of GDM. GDM predicts the variation of the mobility as given by equation\[9\], $$\mu=\mu_0exp\left(-\left[\frac{2\sigma}{3kT}\right]^2\right)exp\left[C\left\{\left(\frac{\sigma}{kT}\\\right)^2-\Sigma^2\right\}E^{1/2}\right]$$ where $\mu_{0}$ is a prefactor mobility, $\sigma$ is the measure of energetic disorder (width of the Gaussian distribution of site energies), $\Sigma$ is the measure of positional disorder, the measure of geometrical disorder, $k$ is the Boltzmann constant, $T$ is the temperature in Kelvin and *C* is an empirical constant. From Equation (1), the term $ \left\{(\frac{\sigma}{kT})^2-\Sigma^2\right\}$ decides the slope of field dependence of mobility at the intermediate field regime, where the mobility shows $\log\mu Vs. E^{1/2}$ dependence. When temperature increases the term becomes negative. Also, if the value of $\sigma$ is low then the above term become negative at lower temperatures. This suggests that if the overall energetic disorder of the film is very low or if the charge transport occurs through regions of very low energetic disorder then the slope of field dependence of mobility can remain negative down to lower temperatures. Similarly if $(\frac{\sigma}{kT})^2<\Sigma^2$, the mobility shows a negative field dependence. This generally happens only when the positional disorder in the sample is remarkably very high. It has also been shown using Monte Carlo simulations that high positional disorder can lead to negative field dependence of mobility at lower electric field strength \[9\]. In MDPs the negative field dependence has generally been observed at very high temperature and also when the concentration of the dopant is very low \[3\].
In this paper we show negative field dependence of mobility down to low temperatures in MDP. We have investigated the field and temperature dependence of mobility in films of N,N’-diphenyl-N,N’-bis(3-methylphenyl)-(1,1’-biphenyl)-4,4’diamine (TPD) dye. TPD dye is a well known blue emitting laser dye and also a hole transporting dye. Films of TPD have been used widely in fabricating various organic devices \[2,3\]. We used TPD doped in Polystyrene (PS) at 40:60 proportions by weight (TPD:PS) and measured hole mobility using time-of-flight(TOF) transient photoconductivity technique. A total negative field dependence of mobility was observed down to low temperature $\sim$150K. This was attributed to the low value of disorder in the sample due to the presence of aggregation/crystallization of TPD molecules. The study highlights the drastic change in morphology of the sample and the resulting reduction in the overall disorder of a molecularly doped polymer upon aggregation/crystallization of dopant. Monte Carlo simulation was performed to understand the influence of aggregates on charge transport with correlated site energies. Simulation supports our experimental observation as well as the justification provided on the basis of low disorder present in the sample.
Experimental and Simulation Details
===================================
Experimental details
--------------------
Solutions of TPD doped Polystyrene at 40:60 proportions by weight were made by dissolving the required amount of TPD and PS in Chloroform. Films were spin-cast on to a neatly cleaned fluorine doped tin oxide (FTO) coated glass substrate. Samples were kept in vacuum for 24 hours at ambient temperature to remove the residual solvent. A thin layer of amorphous Selenium (a-Se) and an Aluminum top electrode was deposited on to it by thermal evaporation. All coatings were done at base pressure of 10$^{-5}$mbar. Thickness and capacitance of the samples were $\sim$4$\mu$m and $\sim$10pF respectively. Field and temperature dependence of mobility in these samples was determined using conventional small signal Time of Flight (TOF) transient photoconductivity technique \[3\]. Samples were mounted on a homemade cryostat to perform temperature dependent studies. A variable DC potential was applied across the device such that no injection occurs from the electrodes to the sample. A 15ns laser pulse from second harmonic of Nd: YAG laser (532nm) was used for generating a thin sheet of charge in a-Se at a-Se/ TPD:PS interface. Laser intensity is adjusted so that total charge generated is less than 0.05*CV*, where *C* is the capacitance of the device and *V* is the voltage applied across the device. The time resolved photocurrent is acquired using an oscilloscope as a voltage across a load resistance. The transit time, $\tau$, is obtained from photocurrent signal and the mobility is calculated using $\mu= L^2/V\tau$ , where $L$ is the thickness of the sample. Morphology of the samples was characterized using Scanning electron microscope (SEM) images.
Details of Monte Carlo Simulation
---------------------------------
Monte Carlo simulations wereperformed to support our experimental observations and explore the influence of aggregates on field and temperature dependence of mobility. The Monte Carlo simulation is based on the commonly used algorithm reported by Schönherr et al \[19\]. A lattice of 70x70x70, along x, y and z direction, with lattice constant a = 6 was used for computation. Z direction is considered as the direction of the applied field. The size of the lattice is judged by taking into account of the available computational resources. The site energies of lattice were initially taken randomly from a Gaussian distribution of mean $\sim$5.1eV and standard deviation $\sigma$ = 75meV, which gives the energetic disorder parameter $\hat{\sigma}$=$\sigma/kT$ . The value for $\sigma$ was chosen close to the experimental value observed in TPD based MDPs \[3\]. The site energies were made correlated by considering the energy of a site as an average of energies of neighboring sites which is defined as follows \[20\],
$$\varepsilon_{i}=N^{-1/2}\sum_{j} K(r_{ij})\varepsilon_{j}$$
Where the variable $\varepsilon_{j}$ denotes the uncorrelated energies on the neighboring sites, *N* is the normalization factor that results in required standard deviation and *K* is kernel that provides a degree of correlation among the sites \[20\]. In our simulation, kernel *K* is considered as unity within a sphere of radius *a* (*a* is the intersite distance) and zero outside this sphere. Simulation was performed on this energetically disordered lattice with the assumption that the hopping among the lattice sites was controlled by Miller-Abrahams equation \[21\] in which the jump rate $ \nu_{ij}$ of the charge carrier from the site *i* to site *j* is given by
$$\nu _{ij} = \nu _0 \exp \left( { - 2\gamma a\frac{{\Delta R_{ij} }}{a}} \right)\left( {\begin{array}{*{20}c}
{\exp \left( { - \frac{{\varepsilon _i - \varepsilon _j }}{{kT}}} \right)\exp \left( { \pm \frac{{eEa}}{{kT}}} \right)} & { \to \varepsilon _j > \varepsilon _i } \\
1 & { \to \varepsilon _j < \varepsilon _i } \\
\end{array}} \right)$$
where $E$ is the applied electric field, *a* is the intersite distance, $k$ is the Boltzmann constant, $T$ is the temperature in Kelvin, $\Delta R_{ij}=R_{i}-R_{j}$ is the distance between sites *i* and *j* and $2\gamma a$ is the wave function overlap parameter which controls the electronic exchange interaction between sites. Throughout the simulation we assume $2\gamma a=10$\[9,19\].
Film morphology was varied by incorporating cuboids of so called ordered regions (representing molecular aggregates/microcrystallites in MDPs) of varying size that are placed randomly inside the otherwise disordered host lattice. Sizes of ordered regions were limited to a maximum size of 25x25x40 sites along x, y and z directions. The energetic disorder inside the ordered region was kept low compared to the lattice. This is justified because the aggregates are more ordered regions and hence to simulate the charge transport the cuboids must be of low energetic disorder compared to host lattice. The site energies inside the ordered regions were also taken randomly from another Gaussian distribution of standard deviation $\sim$15meV (we chose 5 times less compared to host lattice). Earlier reports have even suggested a ten fold reduction of energetic disorder inside the aggregates \[22\]. Further details of simulation with varying film morphology are provided in Ref \[11\]. The site energy of the ordered region was also correlated as explained above. The mean energies of ordered regions were chosen such that their difference from the mean energy of host lattice is in the order of *kT*. This is justified by the fact that the aggregation of dopants can also lead to change in the energy gap (shift in HOMO, LUMO levels) and hence the mean energy of Gaussian distribution. Simulations were performed by varying the concentration of such ordered regions (varying the percentage of volume of lattice occupied by ordered region), temperature and electric field so as to simulate the field and temperature dependence of mobility.
Results and Discussions
=======================
Experimental Results
--------------------
Fig.1 shows the typical time of flight transient signal obtained for TPD:PS (40:60 wt%) at 290K. Transient showed some plateau suggesting non-dispersive transport. We observe dispersive transport at low electric field strength and low temperature. The transit time was always determined from the double logarithmic plot as shown in the inset of Fig. 1. Fig. 2 shows the field dependence of mobility parametric with temperature. Negative field dependence of mobility was observed down to low temperature and through out the range of electric field studied. The mobility becomes almost field independent at 150K. There is no remarkable increase in mobility above room temperature compared to lower temperatures. Data at low electric field strength, at low temperature, were not recorded due to very low magnitude of photocurrent transient signal. At high temperatures, in the low field regime, a slight increase in mobility was observed with decrease of electric field. Temperature dependence of zero field mobility and the slope ($\beta$) of intermediate field region of $\log\mu(E=0) Vs. E^{1/2}$ follow $T^{-2}$ as predicted by GDM (Figure 2(b) and 2(c) respectively) \[9\]. Based on GDM formalism we estimated the disorder parameters. The obtained values of $\sigma,\Sigma$ and $\mu_0$ are 0.0287eV, 2.277 and 2.8471x10$^{-2}$ cm$^2$/Vs respectively. The value of energetic disorder and positional disorder in our sample is low compared to the reported values for similar and other molecularly doped polymer systems \[3,9\]. Hence the observed negative field dependence of mobility down to low temperature ($\sim$150K) in TPD:PS can be attributed to the presence of very low value of energetic disorder in the sample studied. As explained above (see introduction), low of value of energetic disorder can lead to negative field dependence of mobility down to low temperature. When the disorder is very low then the barrier offered to carrier is also very low. Thus drift velocity of the carrier saturates at lower electric field strength which lead to the decrease of mobility with increases of field strength \[9,16\].
The morphology of TPD:PS in our study was investigated using SEM images. Fig. 3 shows the SEM images of TPD:PS in our study. SEM images showed that TPD has undergone aggregation. Aggregation of TPD resulted in the formation of crystalline regions of few micron sizes and also chains of such microcrystals that are spread all over the sample. Earlier reports of crystallization/aggregation of TPD in TPD doped Polystyrene system, even at low concentration and without annealing\[23\], also supports our observation. Earlier reports of mobility measurement, who apparently report only positive field dependence, in TPD doped polymers assert that their study was performed in totally amorphous sample with no signs of aggregation of dopants at least during the time of experiment \[2,3,24\]. Presence of more ordered regions can drastically change the entire morphology of the sample and can lead to reduction in the effective energetic and positional disorder. This is consistent with our observation of low energetic and positional disorder in TPD:PS. Due to aggregation the charge transport therefore occurs through a combination of ordered and less ordered regions. In such cases the charge transport is highly influenced by the packing and orientation of ordered regions. If the charge transport occurs mostly through these ordered regions, regions of very low disorder, then a negative field dependence of mobility down to low temperature can be expected. So the presence of these microcrystallites can drastically change the morphology with lower energetic and positional disorder and the behavior of charge transport in these samples.
Simulation Results
------------------
In order to justify and support the above explanation a Monte Carlo simulation of charge transport in disordered lattice was also performed. Aim of the simulation was to understand the influence of aggregates on charge transport, in particular the negative field dependence. Simulated field and temperature dependence of mobility for a pure host lattice having DOS with standard deviation $\sim$75meV (without considering any ordered regions) has been discussed in detail in our earlier report \[11\]. Simulation results were as predicted by GDM. To study the influence of embedded micro crystals/aggregation of dopants on charge transport, the simulation was performed after incorporating ordered regions inside the host lattice (as explained in simulation procedure above). Fig. 4 shows the field dependence of mobility, at $\sim$248K, parametric with the concentration of ordered regions having DOS with standard deviation $\sim$15meV and mean energy lower by $\sim$ kT compared to mean energy of host lattice. Magnitude of mobility at all regimes of electric field increases with increase of concentration of ordered regions concomitant with decrease of slope at the intermediate field regime. The saturation of mobility and decrease of mobility with further increase of electric field, at high field regime, was observed when the concentration of ordered region was higher than 60%. With increase in concentration of ordered region the saturation of mobility occurs at lower electric field strength. For the cases when the concentration of ordered region is less that 60% the mobility was not completely saturated even for the maximum strength of electric field used for simulation. When the concentration of the ordered regions inside the host lattice is very high ($\sim$98%) the field dependence of mobility become totally negative even at this low temperature ($\sim$248K). The mobility decreases with increase of electric field right from low electric field strengths used for simulation.
The observed features in the field dependence of the mobility, after incorporating the ordered regions inside the host lattice, can be explained on the basis of decrease of effective energetic disorder in the host lattice when ordered regions are embedded in it. If the energetic disorder is small then the energetic barrier seen by the carriers due to disorder will be small. This results in higher mobility but weaker field dependence. This explains the increase in magnitude of mobility and decrease in the slope of $semi\log\mu Vs. E^{1/2}$ curve at intermediate field regime with increase in concentration of ordered regions in the host lattice. The low value of energetic disorder also can lead to saturation of drift velocity/mobility at lower electric field strength. When the concentration of ordered regions inside the host lattice is very high, the overall energetic disorder will be very low. At such high concentration the carrier will travel mostly through ordered region only. Since the energetic disorder inside the ordered region is small (only $\sim$15meV in our study) the mobility saturates at low electric field strength and even at low temperatures. In such a case one can observe total negative field dependence of mobility down to low temperatures. Moreover for a very low value of energetic disorder the term $ \left\{(\frac{\sigma}{kT})^2-\Sigma^2\right\}$ remains negative down to low temperature. It is also possible to have total negative field dependence down to low temperature when the charge transport occurs totally through ordered regions, i.e. when the whole host lattice is completely occupied with ordered region (Fig. 4, 100%). Fig. 5 shows the field dependence of mobility parametric with temperature when host lattice is completely occupied with ordered region. A total negative field dependence of mobility down to low temperature is observed but the mobility in this case decreases with increase of temperature. Higher magnitude of mobility was observed for lower temperature ($\sim$150K) and low magnitude of mobility at higher temperature ($\sim$390K). This kind of behavior is possible when the overall energetic disorder is very low. When energetic disorder is very low and the thermal energy is comparable to energetic disorder then thermal energy dominates which forces the carriers to move in longer paths. Effectively the transit time increases and mobility decreases with increase of temperature \[16\].
From the simulation it is inferred that charge transport in TPD:PS occurs mostly through aggregates but not completely through aggregates. Hence the effective disorder seen by the carrier is low which results in total negative field dependence of mobility. Experimentally we observed that mobility decreases with decreases of temperature which rules out the possibility that charge transport is occurring completely through aggregates. The influence of small disordered regions present in entire trajectory of charge transport (major part of the trajectory is occupied by ordered region) for each carrier become prominent only at low temperature. When the thermal energy of carrier is low, the carrier will see the effect of disorder and that results in the positive field dependence of mobility at low temperature. Fig. 6(a) shows the field dependence of mobility parametric with temperature when 98% of the host lattice is occupied with ordered regions having DOS with standard deviation $\sim$15meV and mean energy lower by $\sim$ kT compared to mean energy of host lattice. Mobility decreases with decrease of temperature and positive field dependence of mobility was observed at lower temperature (200K). Negative field dependence of mobility was observed down to 248K. At temperature around 300K, there is no variation of mobility with increase of temperature for all electric field strength used for simulation. This observation is similar to what observed experimentally in TPD:PS at higher temperatures. The temperature dependence of zero field mobility and slope of intermediate field region of log versus $E^{1/2}$ plot followed $1/T^2$ dependence as predicted by GDM (Data not shown).
Conclusion
==========
Time of flight mobility measurement was carried out in TPD:PS. A total negative field dependence of mobility down to low temperature was observed. This was explained on the basis of low value of energetic disorder in TPD:PS. The low value of energetic disorder was attributed to aggregation of TPD that results in the formation of crystalline regions of few micron size and chaining of such crystals. The presence of aggregates drastically changes the morphology of the sample and reduces the overall disorder in the sample. Monte Carlo simulation study clearly showed the influence of aggregates on charge transport which supports our experimental observation. The simulation suggests if charge transport occurs mostly through aggregates (crystalline regions of very low energetic disorder) the mobility saturates at low electric field strengths and it can lead to a total negative field dependence of mobility down to low temperature. From the simulation studies it is inferred that the charge transport in TPD:PS in the present study are not occurring totally through aggregates but in combination of aggregates and disordered medium. Thus our experimental and simulation studies showed the influence of morphology of the sample on field and temperature dependence of mobility. The study also highlights the need to consider film morphology in various models used for analyzing charge transport in polymer films.
References
==========
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**Figure 1.** Typical time of flight transient signal for TPD:PS at 4x10$^5$V/cm, 290K. Inset show the double logarithmic plot used for determining the transit time.**Figure 2.** Experimental data for TPD:PS(40:60) (a) Electric field dependence of mobility $semi\log\mu Vs. E^{1/2}$ parametric with temperature. (b) $semi\log\mu(E=0)$ Vs. $T^{-2}$ (c)$ \beta$ Vs. $(\sigma/kT)^2$ plot, where $\beta$ is the slope of high field region of $semi\log\mu Vs. E^{1/2}$ plot.**Figure 3.** SEM image of TPD:PS showing crystallization/aggregation of TPD molecules.**Figure 4.** Simulated field dependence of mobility for host lattice ($\sigma$=75meV), at 248K, embedded with ordered regions of various concentrations having DOS of standard deviation$\sim$15meV.**Figure 5.** Field dependence of mobility of host lattice with $\sim$15meV parametric with temperature. Case is similar when host lattice is completely occupied with ordered regions.**Figure 6.** Field dependence of mobility of host lattice, parametric with temperature, with embedded ordered regions having DOS with standard deviation $\sim$15meV. Concentration of embedded ordered region inside the host lattice is $\sim$98%. Straight line show linear fit to intermediate field region.
{width="15cm"}
\[Fig1\]
{width="15cm"} \[Fig2a\]
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\[Fig2b\]
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\[Fig2c\]
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|
---
abstract: 'We study the regularization of an oriented 1-foliation ${\mathcal{F}}$ on $M \setminus \Sigma$ where $M$ is a smooth manifold and $\Sigma \subset M$ is a closed subset, which can be interpreted as the discontinuity locus of ${\mathcal{F}}$. In the spirit of Filippov’s work, we define a sliding and sewing dynamics on the discontinuity locus $\Sigma$ as some sort of limit of the dynamics of a nearby smooth 1-foliation and obtain conditions to identify whether a point belongs to the sliding or sewing regions.'
address:
- '$^1$ Laboratoire de Mathématiques, Informatique et Applications–UHA, 4 Rue des Frères Lumière - 68093 Mulhouse, France'
- '$^2$ Université de Strasbourg, France'
- '$^3$ Departamento de Matemática – IBILCE–UNESP, Rua C. Colombo, 2265, CEP 15054–000 S. J. Rio Preto, São Paulo, Brazil'
author:
- 'Daniel Panazzolo $^{1,2}$ and Paulo R. da Silva $^3$'
bibliography:
- 'mybibliography.bib'
title: 'Regularization of Discontinuous Foliations: Blowing up and Sliding Conditions via Fenichel Theory'
---
Introduction {#s0}
============
A 1-dimensional (singular) oriented foliation ${\mathcal{F}}$ on a smooth manifold $M$ is defined by exhibiting an open covering of $M$ and a collection of smooth vector fields whose domains are the open sets of this covering, and which agree on the intersections of these open sets up to multiplication by a strictly positive function. A *discontinuous 1-foliation* on $M$ is given by a closed subset $\Sigma \subset M$ with empty interior and a 1-dimensional oriented foliation on $M \setminus \Sigma$.\
To fix the ideas we start with the usual setting which was initially studied by Filippov [@AF]. The foliations considered are determined by flows of vector fields expressed by $$\label{fili}
X = \Big(\frac{1+{\operatorname{sgn}}(f)}{2}\Big) X_+ + \Big(\frac{1-{\operatorname{sgn}}(f)}{2}\Big) X_-$$ for some smooth vector fields $X_+, X_-$ defined on $M$ and a function $f \in C^\infty(M)$ having $0$ as a regular value. The discontinuity locus is the smooth codimension one submanifold $\Sigma = f^{-1}(0)$.\
In this setting, we say that a point $p\in\Sigma$ is *$\Sigma$-regular* if $\mathcal{L}_{X_-}(f)\mathcal{L}_{X_+}(f)\neq 0$ and it is *$\Sigma$-singular* if $\mathcal{L}_{X_-}(f)\mathcal{L}_{X_+}(f)= 0$. Moreover the regular points are classified as *sewing* if $\mathcal{L}_{X_-}(f)\mathcal{L}_{X_+}(f)>0$ or *sliding* if $\mathcal{L}_{X_-}(f)\mathcal{L}_{X_+}(f)<0$.\
According Filippov’s convention, the flow of $X$ is easily determined in the neighborhood of sewing points. Roughly speaking, it behaves like a constant vector field, as in *Flow Box Theorem*. However when a trajectory finds a sliding point, the orbit remains in $\Sigma$ up to a $\Sigma$-singular point. In the sliding region of $\Sigma$ the trajectory follows the flow determined by a convex combination of $X^+$ and $X^-$, called *sliding vector field*.\
These concepts do not have a natural generalization when the discontinuity occurs in singular sets, that is when $\Sigma = f^{-1}(0)$ is the inverse image of a critical value. This is one of subjects which will be discussed in this article.\
Our main tool in the study of discontinuous foliation is the *regularization*. Basically, a regularization is a family of smooth vector fields $ X_{{\varepsilon}} $ depending on a parameter ${\varepsilon}> 0$ and such that $ X_{{\varepsilon}} $ converges uniformly to $ X $ in each compact subset of $M \setminus \Sigma $ as ${\varepsilon}$ goes to zero. One of the most well-known regularization process was introduced by Sotomayor and Teixeira [@LT; @ST]. It is based on the use of *monotonic transition functions* $\varphi:{\ensuremath{\mathbb{R}}}\rightarrow{\ensuremath{\mathbb{R}}}$ ([^1]). The *ST-regularization* of the vector field $X$ given in (\[fili\]) is the one parameter family $$\label{STreg} X_{{\varepsilon}}= \frac{1}{2} \left( 1 + \varphi \left( \dfrac{f}{{\varepsilon}}\right) \right)X_+
+ \frac{1}{2}\left ( 1 - \varphi \left( \dfrac{f}{{\varepsilon}}\right) \right) X_-.$$
The regularized vector field $X_{{\varepsilon}}$ is smooth for ${\varepsilon}> 0$ and satisfies that $X_{{\varepsilon}}= X_+$ on $\{f>{\varepsilon}\}$ and $X_{{\varepsilon}}= X_-$ on $\{f<-{\varepsilon}\}$. With this regularization process Sotomayor and Teixeira developed a systematic study of the singularities of these systems and also developed the Peixoto’s program about structural stability. In particular, Teixeira analyzed the singularity of the kind fold-fold, which was later known as $T$-singularity. We refer also [@Kr1; @Kr2] for related problems.\
In [@BPT], the use of singular perturbation and blow-up techniques were introduced in the study of the ST-regularization. Let us briefly describe this procedure, assuming for simplicity that $M = {\ensuremath{\mathbb{R}}}^2$ and that $\Sigma = \{y=0\}$. If we write $X_+=a_+\frac{\partial}{\partial x} + b_+ \frac{\partial}{\partial y}$ and $X_-=a_-\frac{\partial}{\partial x} + b_- \frac{\partial}{\partial y}$ then $$X_{{\varepsilon}}=\frac{1}{2}\left(a_++a_- +\varphi\left( \dfrac{y}{{\varepsilon}}\right)\big( a_+-a_-\big)\right)\dfrac{\partial}{\partial x}+
\frac{1}{2} \left(b_++b_- +\varphi\left(\dfrac{y}{{\varepsilon}} \right)\big( b_+-b_-\big) \right)\dfrac{\partial}{\partial y}.$$ Considering the directional blow-up $y=\bar{{\varepsilon}}\bar{y}$, ${\varepsilon}=\bar{{\varepsilon}}$ we get the vector field $$\bar{X_{{\varepsilon}}}=\frac{1}{2}\left(a_++a_- +\varphi\left( \bar{y}\right)\big( a_+-a_-\big)\right)\dfrac{\partial}{\partial x}+
\frac{1}{2\bar{{\varepsilon}}} \left(b_++b_- +\varphi\left(\bar{y} \right)\big( b_+-b_-\big) \right)\dfrac{\partial}{\partial \bar{y}}$$ which corresponds to the singular perturbation problem ([^2]).
$$\label{sp-problem}
\left\{ \begin{array}{rcl}
\dot x &=& \frac{1}{2}\left(a_++a_- +\varphi\left( \bar{y}\right)\big( a_+-a_-\big)\right) \vspace{0.5cm}\\
\bar{{\varepsilon}}\,\dot {\bar y} &=& \frac{1}{2} \left(b_++b_- +\varphi\left(\bar{y} \right)\big( b_+-b_-\big) \right)
\end{array}
\right.$$
For $\bar{{\varepsilon}}=0$, the slow manifold of is the set implicitly defined by $$(b_++b_-)+\varphi\left(\bar{y}\right)(b_++b_-)=0$$ with slow flow determined by $$\frac{1}{2}\left(a_++a_- +\varphi\left( \bar{y}\right)\big( a_+-a_-\big) \right)\dfrac{\partial}{\partial x}.$$ Silva et all [@LST2] proved that the set of sliding points, according Filippov convention, is the projection of the slow manifold of on $\Sigma$. Moreover they proved that the slow flow of and the sliding vector field idealized by Filippov have the same equation.\
For better visualization, we use the polar blow up $ y= r \cos \theta, {\varepsilon}= r \sin \theta $ with $ \theta \in (0, \pi) $. In this case the discontinuity $ \Sigma $ is replaced by a semi-cylinder on which we draw the slow manifold and the fast and slow trajectories. See figure \[buNew\].\
The singular perturbation problem which is obtained evidently depends on the choice of the regularization. The sliding vector field idealized by Filippov appears when we consider the ST-regularization, see for instance [@LST; @LST2; @LST3; @LST4]. However Novaes and his collaborators [@NJ] have considered a slightly more general regularization, called non-linear regularization, which produces singular perturbation problem with slow manifold having fold points and thus not defining only one possible sliding flow. It seems evident that other regularizations may produce new sliding regions.\
The techniques of singular perturbation have also been applied to deal with discontinuities on surfaces with singularities. Teixeira and his collaborators realized that in the case where $\Sigma$ has a transverse self-intersection a process of double regularization can be used, and that it generates systems with multiple time scales.\
In this work we intend to unify the different approaches of the previous works. Let us briefly summarize the results proved in this paper.\
Given a smooth manifold $ M $, initially we introduce the concepts of *1*-dimensional oriented foliation on $ M $ and *discontinuous 1-foliation* on $ M $ with discontinuity locus $ \Sigma $. The first question we address is to get conditions so that the foliation can be [*smoothed* ]{}by a sequence of blowing-ups.\
Our first results are the following:\
- If ${\mathcal{F}}$ is a piecewise smooth foliation and the discontinuity locus $\Sigma$ is a smooth submanifold of codimension one then the foliation ${\mathcal{F}}$ is blow-up smoothable. See *Theorem \[theorem-smoothlocus\]*.
- If the discontinuity locus $\Sigma$ is a globally defined analytic subset then there is a piecewise smooth 1-foliation which is related to the initial foliation by a sequence of blow-ups and whose discontinuity locus is smooth. Moreover if we further suppose that the discontinuity locus has codimension one then the foliation is blow-up smoothable. See *Theorem \[theorem-singularlocus\]* and *Corolary \[corr\]*.
The smoothing procedure defined in Section \[s1\] does not allow to define the so-called [*sliding dynamics*]{} along the discontinuity locus. So, we define such sliding dynamics as the limit of the dynamics of a nearby smooth 1-foliations. This leads us to introduce a general notion of regularization for piecewise-smooth 1-foliations.\
Roughly speaking, the regularization is given by a new foliation depending on a parameter ${{\varepsilon}}$, which is smooth for ${{\varepsilon}}> 0$ and which coincides with the original discontinuous 1-foliation when ${{\varepsilon}}$ equals zero. We generalize the notion of ST-regularization to this global context and consider a larger family of regularizations (called of [*transition type*]{}) by dropping the condition of monotonicity of the transition function.
Basically, the sliding region associated to a given regularization is defined as the accumulation set of invariant manifolds of the regularized system. We prove the following results.
- The regularization of transition type is blow-up smoothable. See *Theorem \[theorem-smoothableST\].*
- We obtain conditions on the transition function to identify whether a point lies in the sliding region. See *Theorems \[theorem-slide\]* and *\[theorem-sewingreg\]*.\
The paper is organized as follows. In Section \[s1\] we give the preliminary definitions and prove Theorems \[theorem-smoothlocus\], \[theorem-singularlocus\] and Corolary \[corr\]. In Section \[s2\] we study the regularization and in Section \[s3\] we combine the blowing-up technique and the Fenichel’s theory to give sufficient conditions for identifying the sliding region.\
Figure \[figCORNER\] is a pictorial representation of the blow-up smoothing process. It shows a piecewise smooth discontinuous foliation with analytic discontinuous set having a smooth part and a singular one. Applying a regularization of the kind transition we get a new foliation $\mathcal{F}^r$. In the figure we draw the level $\mathcal{F}^0$ of this foliation. The leaves displayed in $\mathcal{F}^0$ are the trajectories of the singular perturbation problem . The simple arrows correspond to the slow flow and the double arrows correspond to the fast flow, which is obtained after a time reparametrization.
Discontinuous 1-foliations {#s1}
==========================
In this section we present the preliminary definitions related to discontinuous 1-foliations. Also we enunciate and prove results related to blow-up smoothing of discontinuous 1-foliations.
Smoothable discontinous foliations
----------------------------------
We work in the category of manifolds with corners. We briefly recall that a manifold with corners of dimension $n$ is a paracompact Hausdorff space with a smooth structure which is locally modeled by open subsets of $
({\ensuremath{\mathbb{R}}}^+)^k \times {\ensuremath{\mathbb{R}}}^{n-k}
$. We refer the reader to [@Joy; @M] for a careful exposition.
Let $M$ be a smooth manifold (with corners). A smooth vector field defined on $M$ will be called [*non-flat* ]{}if its Taylor expansion is non-vanishing at each point of $M$. From now on, all vector fields we will consider are non-flat.
We say that a pair $(V,Y)$ formed by an open set $V \subset M$ and a smooth vector field $Y$ defined in $V$ is a *local vector field* in $M$.
A *1-dimensional oriented foliation* on $M$ is a collection $${\mathcal{F}}= \{(U_i,X_i)\}_{i \in I}$$ of local vector fields such that:
$\{U_i\}$ is an open covering of $M$.
For each pair $i,j \in I$, $$\label{transition-function}
X_i = \varphi_{ij} X_j$$ for some strictly positive smooth function $\varphi_{ij}$ defined on $U_i\cap U_j$.
The importance to consider [*oriented foliations* ]{}instead of globally defined vector fields in the manifold $M$ will become clear later. Roughly speaking, even if our initial object is a foliation globally defined by a smooth vector field, this property will not necessarily hold after the blowing-up operation.
We say that a local vector field $(V,Y)$ is a *local generator* of the foliation ${\mathcal{F}}$ if the augmented collection $$\{(U_i,X_i)\}_{i \in I} \; \cup \; \{ (V,Y)\}$$ also satisfies conditions 1. and 2. of the above definition. From now on, we will suppose that the collection ${\mathcal{F}}$ is [*saturared*]{}, meaning that it contains all such local generators.
Let $\psi : N \rightarrow M$ be a smooth diffeomorphism between two manifolds $N$ and $M$. We will say that two 1-dimensional oriented foliations ${\mathcal{F}}$ and ${\mathcal{G}}$ defined respectively in $M$ and $N$ are *Êrelated by $\psi$* if for each local vector field $(V,Y)$ which is a generator of ${\mathcal{G}}$, the push-forward of this local vector field under $\psi$, namely $$(U,X) := \big( \psi(V), \psi_* Y \big),$$ is a generator of ${\mathcal{F}}$.
A *possibly discontinuous 1-foliation on a manifold $M$* is given by a closed subset $\Sigma \subset M$ with empty interior and a 1-dimensional oriented foliation ${\mathcal{F}}$ defined in $M \setminus \Sigma$.
The set $\Sigma$ is called the [*discontinuity locus* ]{}of ${\mathcal{F}}$. We can write the decomposition $$\Sigma = \Sigma^{{\ensuremath{\operatorname{smooth}}}} \cup \Sigma^{{\ensuremath{\operatorname{sing}}}}$$ where $\Sigma^{{\ensuremath{\operatorname{smooth}}}}$ denotes the subset of points where $\Sigma$ locally coincides with an embedded submanifold of $M$. We shall say that ${\mathcal{F}}$ has a [*smooth discontinuity locus* ]{}if $\Sigma = \Sigma^{{\ensuremath{\operatorname{smoth}}}}$.
\[example-1\] The vector field in ${\ensuremath{\mathbb{R}}}^2$ given by $$\label{sign-vf}
X = \frac{\partial}{\partial x} - {\operatorname{sgn}}(y) \frac{\partial}{\partial y}$$ defines a discontinuous 1-foliation which has discontinuity locus $\Sigma = \{y = 0\}$.
\[example-2\] Consider the discontinuous 1-foliation defined in ${\ensuremath{\mathbb{R}}}^2$ by the vector field $$\label{radial-vf}
X = \frac{x^2-y^2}{x^2+y^2} \left( -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}\right) +
\left( x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}\right)$$ with discontinuity locus $\Sigma = \{0\}$. Notice that $X$ is not smooth at the origin.
\[example-3\] Consider the discontinuous 1-foliation defined in ${\ensuremath{\mathbb{R}}}^3$ by the vector field $$\label{sign2-vf}
X = - {\operatorname{sgn}}(x) \frac{\partial}{\partial x} - {\operatorname{sgn}}(y) \frac{\partial}{\partial y} + \frac{\partial}{\partial z}$$ with has the (non-smooth) discontinuity locus $\Sigma = \{xy=0\}$.
More generally, let $f$ be an arbitrary smooth function on a manifold $M$ and let $X_+$, $X_-$ be two smooth vector fields defined on $M$. Then $$\label{equation-generaldiscont}
X = \Big(\frac{1+{\operatorname{sgn}}(f)}{2}\Big) X_+ + \Big(\frac{1-{\operatorname{sgn}}(f)}{2}\Big) X_-$$ is a discontinuous 1-foliation with discontinuity locus $\Sigma = f^{-1}(0)$.
\[example-generaldiscont\] The figure \[example4-fig\] illustrates a discontinuous 1-foliation in $M = \mathbb{R}^2$, where $f(x,y) = y^2 - x^2(x + 1)$, $X_+ = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$ and $X_- = Ê-X_+$.
As the above examples show, there is no reason to expect that ${\mathcal{F}}$ can be extended smoothly (or even continuously) to $\Sigma$. In order to circumvent this difficulty, one possibility is to modify the ambient space $M$ by a [*blowing-up* ]{}(or a sequence of blowing-ups) and to expect that the modified foliation extends smoothly to the whole ambient space. We refer the reader to [@M], chapter 5 for a detailed definition of the blowing-up operation in the category of manifold with corners.
We will say that discontinuous 1-foliation ${\mathcal{F}}$ defined in $M$ and with discontinuity locus $\Sigma$ is [*blow-up smoothable* ]{}if there exists a locally finite sequence of blowing-ups (with smooth centers) $$M = M_0 \longleftarrow \cdots \longleftarrow \cdots \longleftarrow \widetilde{M}$$ and a smooth 1-foliation $\widetilde{{\mathcal{F}}}$ defined in $\widetilde{M}$ such that:
The map $\Phi : \widetilde{M} \rightarrow M$ is a diffeomorphism outside $\Phi^{-1}(\Sigma)$, and
$\widetilde{{\mathcal{F}}}$ and ${\mathcal{F}}$ are related by $\Phi$, seen as a map from $\widetilde{M} \setminus \Phi^{-1}(\Sigma)$ to $M \setminus \Sigma$.
Let us show that the examples studied above are blow-up smoothable.
Consider the discontinuous 1-foliation defined in ${\ensuremath{\mathbb{R}}}^2$ by the vector field in (\[sign-vf\]), which has discontinuity locus $\Sigma = \{y = 0\}$. If we denote by ${\mathbbm{S}}^0 = \{\pm 1\}$ the $0^{th}$-sphere, the blowing-up of $\Sigma$ is defined by the map $$\begin{array}{cccl}
\Phi\; : & {\ensuremath{\mathbb{R}}}\times ({\mathbbm{S}}^0\times{\ensuremath{\mathbb{R}}}_{\ge 0}) & \longrightarrow & {\ensuremath{\mathbb{R}}}^2\\
& (u,\;\;(\pm1, r)) & \longmapsto & x = u,\, y = \pm r
\end{array}.$$ The resulting 1-foliation in ${\ensuremath{\mathbb{R}}}\times ({\mathbbm{S}}^0\times{\ensuremath{\mathbb{R}}}_{\ge 0})$, given by $\frac{\partial}{\partial u} \mp \frac{\partial}{\partial r}$ is clearly smooth.
Consider the discontinuous foliation defined by (\[radial-vf\]). The blowing up of the origin is defined by the polar coordinates map $$\begin{array}{cccl}
\Phi\; : \;& {\mathbbm{S}}^1\times{\ensuremath{\mathbb{R}}}_{\ge 0} & \longrightarrow & {\ensuremath{\mathbb{R}}}^2\\
& (\theta,r) & \longmapsto & x = r \cos(\theta),\, y = r \sin(\theta)
\end{array}$$ and an easy computation shows that this foliations is mapped to $$\cos(2\theta) \frac{\partial}{\partial \theta} + r \frac{\partial}{\partial r}$$ which is clearly a $C^\infty$ vector field on ${\mathbbm{S}}^1\times{\ensuremath{\mathbb{R}}}_{\ge 0}$.
Consider the discontinuous foliation defined by (\[sign2-vf\]). This foliation is blow-up smoothable by a sequence of three blowing-ups: $${\ensuremath{\mathbb{R}}}^3 = M_0 \stackrel{\Phi_1}{\longleftarrow} M_1\stackrel{\Phi_1}{\longleftarrow} M_2\stackrel{\Phi_1}{\longleftarrow} M_3$$ with respective centers given by the $z$-axis, the strict transform of the hyperplane $\{x = 0\}$ and the strict transform of hyperplane $\{ y = 0\}$.
Notice however that there are discontinuous 1-foliations which are not blow-up smoothable.
Consider the discontinuous 1-foliation in $\mathbb{R}^2$ defined as in (\[equation-generaldiscont\]), where we take $$f(x,y) = y^2 - e^{-\frac{1}{x^2}} \sin^2\big(1/x)$$ and $X_+ = \partial/\partial y$, $X_- = \partial/\partial x$. The set $\mathbb{R}^2 \setminus f^{-1}(0)$ has an infinite number of open connected components in any neighborhood of the origin, and this property cannot be destroyed by a locally finite sequence of blowing-ups.
Piecewise smooth 1-foliations {#subsect-piecewise-smooth}
-----------------------------
A natural problem which arises is to establish conditions which guarantee that a discontinuous 1-foliation is blow-up smoothable. For this, we will introduce a particular class of discontinuous 1-foliations where, roughly speaking, we require that firstly each local vector field which defines such foliations extends smoothly to the discontinuity locus, and secondly that the discontinuity locus is an analytic subset of $M$.
More formally, let ${\mathcal{F}}$ be a discontinuous 1-foliation defined on a manifold $M$ and with discontinuity locus $\Sigma$. A [*local multi-generator* ]{}of ${\mathcal{F}}$ is a pair $(U,\{X_1,\ldots,X_k\})$ satisfying the following conditions:
$U$ is an open set of $M$ and we can write $U \setminus \Sigma$ as a finite disjoint union $$\label{componentsofSigma}
U \setminus \Sigma = U_1 \sqcup \cdots \sqcup U_k$$ of open sets $U_1,..,U_k$.
For each $i = 1,..,k$, $X_i$ is a smooth vector field [*defined in $U$*]{} and such that $$\label{componentsofSigma-generators}
(U_i, X_i|_{U_i}) \text{Ê is a local generator of ${\mathcal{F}}$.}$$
We will say that ${\mathcal{F}}$ is [*piecewise smooth* ]{}if there exists a collection $\mathcal{C}$ of local multi-generators as above whose domain forms an open covering of $\Sigma$ and the following compatibility condition holds: For each two local multi-generators $$(U,\{X_1,\ldots,X_k\}),\quad (V,\{Y_1,\ldots,Y_l\})$$ belonging to $\mathcal{C}$, there exists a strictly positive smooth function $\varphi$ defined in $U \cap V$ such that $$\label{local-compatibilityofgen}
X_i = \varphi\, Y_j \quad \text{ on }\quad U_i \cap V_j$$ for each pair of indices $i= 1,\ldots,k$ and $j = 1,\ldots,l$.
In what follows, it will be important to require the transition function $\varphi$ to be [*the same* ]{}on all intersections $U_i \cap V_j$.
\[multi-generator\]
The Example \[example-1\] exhibits a piecewise smooth 1-foliation, since ${\mathcal{F}}$ is defined simply by restricting the constant vector fields $$X_+ = \frac{\partial}{\partial x} - \frac{\partial}{\partial y} \qquad \text{and}\qquad X_- = \frac{\partial}{\partial x} + \frac{\partial}{\partial y}$$ to the subsets $U_+ = \{y > 0\}$ and $U_- = \{y < 0\}$, respectively. Similarly, the 1-foliation in Example \[example-3\]Ê is piecewise smooth. On the other hand, the discontinuous vector field $X$ in Example \[example-2\] is not piecewise smooth. In fact, the vector field $X$ is not smooth at the origin and therefore the foliation can not be smoothly extended to $ \Sigma = \{(0,0) \}.$
A simple consequence of the definition of piecewise smooth 1-foliations is the following:
\[theorem-smoothlocus\] Let ${\mathcal{F}}$ be a piecewise smooth 1-foliation on a manifold $M$ whose discontinuity locus $\Sigma$ is an smooth submanifold of codimension one. Then, ${\mathcal{F}}$ is blow-up smoothable.
We consider the smooth map $\Phi : N \rightarrow M$ defined by the blowing-up with center $\Sigma$, and exceptional divisor $D = \Phi^{-1}(\Sigma)$. The smooth foliation ${\mathcal{G}}$ in $N$ is now defined by describing its local generator at each point $q\in N$. We consider separately the case where $q \in N\setminus D$ and $q \in D$. In the former case, we can choose a generator $(U,X)$ of ${\mathcal{F}}$ defined in a sufficiently small neighborhood an of $p = \Phi(q)$ and decree that $$(V,Y) = \big( \Phi^{-1}(V),\Phi^*Y \big)$$ is a local generator of ${\mathcal{G}}$ near $q$. Notice that this construction defines unambiguously the 1-foliation on $N \setminus D$, since it is independent of the choice of $(U,X)$.
Suppose now that $q \in D$, i.e. that $p = \Phi(q)$ lies in $\Sigma$. Then, we can choose local coordinates $(x_1,..,x_{n-1},y)$ in a neighborhood $U$ of $p$ such that $\Sigma = \{y = 0\}$ and the blowing-up map assumes the form $$\begin{array}{rcl}
({\mathbbm{S}}^0\times {\ensuremath{\mathbb{R}}}^+) \times {\ensuremath{\mathbb{R}}}^{n-1} & \longrightarrow & {\ensuremath{\mathbb{R}}}^n\\
(\pm 1,r), (u_1,...,u_{n-1}) & \longmapsto & y = \pm r,\, x_1 = u_1,\,\ldots,\, x_{n-1} = u_{n-1}.
\end{array}$$ Up to reducing $U$ to some smaller neighborhood of $p$, we can assume that $U \setminus \Sigma$ can be written as the disjoint union of connected subsets $U_+ = \{y > 0\}$, $U_- =\{ y < 0\}$ and that there exists two smooth vector fields $X_+,X_-$ defined on $U$ such that $(U_+,X_+)$ and $(U_-,X_-)$ are local generators of ${\mathcal{F}}$. Now, by the expression of the blowing-up map either $V_+ = \Phi^{-1}(U_+)$ or $V_- = \Phi^{-1}(U_-)$ is an open neighborhood of $q$ in $N$. Therefore, according to the choice of the $\pm$ sign, we decree that the local vector field $(V_\pm,\Phi^* X_\pm)$ is a local generator of ${\mathcal{G}}$ at $q$.
This procedure defines in an unambiguous way a smooth 1-foliation ${\mathcal{G}}$ on $N$, which is moreover related to ${\mathcal{F}}$ by $\Phi$.
A natural question which arises is whether the above result can be generalized to the case where $\Sigma$ is a smooth submanifold of higher codimension.
\[example-flatcone\] Consider the piecewise-smooth 1-foliation on $\mathbb{R}^3$ defined by $$X = z^k\frac{\partial}{\partial x} + \varphi\left(\frac{y z^l}{e^{-1/z^2}}\right) \frac{\partial}{\partial y} \quad \text{ if $z \ne 0$}, \quad X = z^k \frac{\partial}{\partial x} + {\operatorname{sgn}}(y) \frac{\partial}{\partial y},\quad \text{ if $z = 0$}.$$ where $\varphi$ is a monotonic transition function as defined in the Introduction and $k,l \in \mathbb{N}$ are arbitrary positive integers. Notice that $X$ is smooth outside the discontinuity locus $\Sigma = \{y=z=0\}$, which is of codimension 2. A blowing up with center the origin will produce (in the $z$-directional chart) the same expression with the integer $l$ replaced by $l+1$. Similarly, a blowing-up with center $\Sigma$ will produce exactly the same expression with $k$ and $l$ replaced by $k+1$ and $l+1$ respectively.
\[flatcone-fig\]
Therefore, no sequence of blowing-ups will allow a $C^\infty$ extension of this vector field to the exceptional divisor.
Our next goal is to obtain a similar result in the case where $\Sigma$ is a codimension one [*singular subvariety*]{}. For this, we need to impose another regularity condition, which will allow us to use the Theorem of Resolution of Singularities.
We will say that ${\mathcal{F}}$ has an [*analytic discontinuity locus* ]{}if the ambient space $M$ is an analytic manifold and the discontinuity locus $\Sigma$ is a [*globally defined analytic subset* ]{}of $M$. In other words, we assume $\Sigma = \Sigma(f)$ is the vanishing locus of a finite collection of global analytic functions $f = (f_1,\ldots,f_m)$ defined on $M$ ([^3]).
Under the above hypothesis, there exists an unique filtration by semianalytic sets (see [@Lo]) $$\label{stratification}
\Sigma^0 \subset \Sigma^1 \subset \cdots \subset \Sigma^d = \Sigma$$ where, for each $k = 1,\ldots,d$, the set $\Sigma^k \setminus \Sigma^{k-1}$ is a smooth manifold of dimension $k$. Using this decomposition, we say that $d$ is the [*dimension* ]{}of $\Sigma$ and that $\Sigma^{{\ensuremath{\operatorname{reg}}}} = \Sigma^d \setminus \Sigma^{d-1}$ is the [*regular part* ]{}of $\Sigma$. The complementary set $\Sigma^{{\ensuremath{\operatorname{exc}}}} = \Sigma \setminus \Sigma^{{\ensuremath{\operatorname{reg}}}}$ is called the [*exceptional locus*]{}.
We observe that, in general, the inclusion $\Sigma^{{\ensuremath{\operatorname{reg}}}}\subset\Sigma^{{\ensuremath{\operatorname{smooth}}}}$ is strict. For instance, the Whitney umbrella $\Sigma= \{z^2 - x y^2 = 0\}$ is such that $\Sigma^{{\ensuremath{\operatorname{exc}}}} = \{y=z=0\}$ contains strictly $\Sigma^{{\ensuremath{\operatorname{sing}}}} = \{x \ge 0, y=z=0\}$. Taking the complementaries it follows that $\Sigma^{{\ensuremath{\operatorname{reg}}}}\subsetneq \Sigma^{{\ensuremath{\operatorname{smooth}}}}$.
Under the above assumptions, we can apply the Theorem of Resolution of Singularities for globally defined real analytic sets (see e.g. [@BM]). As a result, we conclude that there exists a proper analytic map $\Phi: N \rightarrow M$, defined by a locally finite sequence of blowing-ups, such that:
$\Phi$ is a diffeomorphism outside $\Phi^{-1}(\Sigma^{{\ensuremath{\operatorname{exc}}}})$.
$D = \Phi^{-1}(\Sigma^{{\ensuremath{\operatorname{exc}}}})$ is a locally finite union of boundary components $$\bigcup_{i \in I}D_i \subset \partial N$$ of codimension one.
The closure of $\Phi^{-1}(\Sigma^{{\ensuremath{\operatorname{reg}}}})$ is a smooth submanifold $\Omega \subset N$.
The next result states that, under the above conditions, the foliation ${\mathcal{F}}$ pulls-back to a discontinuous foliation in $N$ which has a smooth discontinuity locus.
\[theorem-singularlocus\] Let ${\mathcal{F}}$ is a piecewise smooth 1-foliation with analytic discontinuity locus. Then, using the above notation, there is a piecewise smooth 1-foliation ${\mathcal{G}}$ defined on $N$, which is related to ${\mathcal{F}}$ by $\Phi$, and whose discontinuity locus is $\Omega$ .
We describe separately the local definition of ${\mathcal{G}}$ in points lying in $N\setminus (D\, \cup\, \Omega)$, in points lying in $D \setminus \Omega$, and then in points lying in $\Omega$. If $q \in N\setminus (D \,\cup\, \Omega)$ then choose a generator $(U,X)$ of ${\mathcal{F}}$ defined in a sufficiently small neighborhood of $p = \Phi(q)$ and decree that $$(V,Y) = \big( \Phi^{-1}(V),\Phi^*Y \big)$$ is a local generator of ${\mathcal{G}}$ near $q$. Notice that this construction defines unambiguously the 1-foliation on $N \setminus (D \cup \Omega)$, since it is independent of the choice of $(U,X)$.
Let us show now how to extend ${\mathcal{G}}$ smoothly to $D \setminus \Omega$. By an induction argument, it suffices to consider the case where $\Phi: N \to M$ is defined by a single blowing-up with center on an analytic submanifold $C \subset M$, and such that $D = \Phi^{-1}(C)$.
Given a point $q \in D \setminus \Omega$, let $p = \Phi(q)$ be its image in $C$. According to the definition of piecewise smooth 1-foliation, choose a local multi-generator $(U,\{X_1,\ldots,X_k\})$ of ${\mathcal{F}}$ at $p$. Then, considering the disjoint decomposition (\[componentsofSigma\]), there exists precisely one index $i \in \{1,\ldots,k\}$, say $i = 1$, such that $W = \Phi^{-1}(U_1)$ is an open neighborhood of $q$.
Up to reducing these neighborhoods, we choose local trivializing coordinates $(y_1,\ldots,y_n)$ at $W$ and $(x_1,..,x_n)$ at $U_1$, respectively, such that $C = \{x_1 = \cdots = x_d = 0\}$ and $D = \{y_1 = 0\}$. Further, we can assume that, in these coordinates, the blowing-up map assumes the form $$x_1 = y_1, x_2 = y_1 y_2,\ldots, x_d = y_1 y_d, \quad\text{Êand }\quad \text x_{d+1} = y_{d+1}, \ldots, x_n = y_n.$$ From the assumption that $X_1$ is non-flat, it follows that the set $$E = \{m \in {\ensuremath{\mathbb{Z}}}\; : \; y_1^m\ \Phi^* X_1 \text{Ê extends smoothly to $\{y_1 = 0\}$} \}$$ has the form $m_{\mathrm{min}}+ {\ensuremath{\mathbb{N}}}$, for some minimal element $m_{\mathrm{min}}\in {\ensuremath{\mathbb{Z}}}$. Indeed, if we expand the vector field $X_1$ as $a_1\frac{\partial}{\partial x_1} +
\ldots + a_n \frac{\partial}{\partial x_n}$, with $a_i \in C^\infty(U_1)$, then its pull-back under $\Phi$ has the form $\Phi^* X_1 = b_1\frac{\partial}{\partial y_1} +
\ldots + b_n \frac{\partial}{\partial y_n}$ where $$\begin{array}{l}
b_1 = \Phi^*(a_1), b_2 = \Phi^*\big((a_2 x_1- a_1 x_2)/x_1^2\big), \ldots, b_d = \Phi^*\big((a_d x_1- a_1 x_d)/x_1^2\big), \\
b_{d+1}Ê= \Phi^*(a_{d+1}),
\ldots, b_nÊ= \Phi^*(a_n)
\end{array}$$ In particular, $-m_{\mathrm{min}}$ is algebraically defined as the minimum of all valuations $$\mathrm{val}_{y_1}(\widehat{b}_1),\ldots, \mathrm{val}_{y_1}(\widehat{b}_n)$$ where $\widehat{b}$ denotes the formal series expansion of $b \in C^\infty(W)$ in powers of the $y_1$-variable, seen as element of the field $C^\infty(W \cap D)((y_1))$ of formal Laurent series in $y_1$ with coefficients smooth functions in $y_2,\ldots,y_n$ ([^4]). Using this we define $Y = y_1^{m_{\mathrm{min}}} \Phi^* X_1$ and decree that $(W,Y)$ is a local generator of ${\mathcal{G}}$ at $q$.
Let us show that this local definition is independent of the choice of the local multi-generator $(U,\{X_1,\ldots,X_k\})$. For this, suppose that we choose another local multi-generator $(U',\{X_1',\ldots,X_l'\})$ of ${\mathcal{F}}$ at $p$. Then, up to a reordering of indices and a restriction to some possibly smaller neighborhood of $p$, we can assume that $U_1 = U_1'$ and that $$X_1' = \varphi X_1$$ for some smooth function $\varphi$ which is strictly positive in $U$ (we use here the condition (\[local-compatibilityofgen\]) in the definition of a piecewise smooth 1-foliation). Taking the pull-back of this relations through $\Phi$, one obtains $$\Phi^* X_1'=(\Phi^* \varphi) \, \Phi^* X_1,\quad\text{Êwhere }\quad\Phi^* \varphi \stackrel{\rm def}{=} \varphi\circ\Phi$$ which shows that the set $E$ defined above coincides with the set $E' = \{m \in {\ensuremath{\mathbb{Z}}}: y_1^m \Phi^* X_1' \text{Ê extends smoothly $\{y_1 = 0\}$} \}$. Consequently, $$Y' \stackrel{\rm def}{=} y_1^{m_{\mathrm{min}}} \Phi^* X_1 = y_1^{m_{\mathrm{min}}} (\Phi^* \varphi) \, \Phi^* X_1^\prime = (\Phi^* \varphi)\, Y$$ which shows that the foliation ${\mathcal{G}}$ is well defined at $q$.[^5]
It remains to construct a local generator of ${\mathcal{G}}$ in each point $q \in \Omega$. The reasoning is very similar to the previous cases, and is left to the reader.
Let us show that the previous two Theorems can be combined to give a general smoothing procedure which extends Theorem \[theorem-smoothlocus\] to the case where $\Sigma$ is singular.
\[corr\] Under the assumptions of the Theorem \[theorem-singularlocus\], suppose further that the discontinuity locus of ${\mathcal{F}}$ has codimension one. Then, ${\mathcal{F}}$ is blow-up smoothable.
Regularization and sliding dynamics for piecewise smooth foliations {#s2}
===================================================================
One disadvantage of the smoothing procedure defined in the previous subsection is that it does not allow to define the so-called [*sliding dynamics* ]{}along the discontinuity set.
Our present goal is to define such sliding dynamics as some sort of limit of the dynamics of a nearby smooth 1-foliations. This leads us to introduce the notion of regularization. Later on, we shall see that the blow-up smoothing and the regularization can be combined in a fruitful way.
Let ${\mathcal{F}}$ be a discontinuous 1-foliation on a manifold $M$, with discontinuity locus $\Sigma$. A [*regularization of ${\mathcal{F}}$ (with $p$-parameters)* ]{}is a discontinuous 1-foliation ${\mathcal{F}}^{\mathbf{ r}}$ defined in the product manifold $$M \times (({\ensuremath{\mathbb{R}}}^+)^p,0)$$ ([^6]) which satisfies the three following conditions:
${\mathcal{F}}^{\mathbf{ r}}$ is tangent to the fibers of the canonical projection $$\pi: M \times (({\ensuremath{\mathbb{R}}}^+)^p,0) \rightarrow (({\ensuremath{\mathbb{R}}}^+)^p,0)$$
The restriction ${\mathcal{F}}^{\mathbf{ r}}_0$ of ${\mathcal{F}}^{\mathbf{ r}}$ to the fiber $\pi^{-1}(0)$ coincides with ${\mathcal{F}}$,
The discontinuity locus $\Sigma^{\mathbf{ r}}$ of ${\mathcal{F}}^{\mathbf{ r}}$ is a subset of $\Sigma \times \big\{\prod_i {{\varepsilon}}_i = 0\big\}$, where $({{\varepsilon}}_1,..,{{\varepsilon}}_p)$ are the coordinates in $(({\ensuremath{\mathbb{R}}}^+)^p,0)$.
The last condition implies that, for each ${{\varepsilon}}\in (({\ensuremath{\mathbb{R}}}^+)^p,0)$ such that $\prod_i {{\varepsilon}}_i \ne 0$, the restriction ${\mathcal{F}}^{\mathbf{ r}}_{{\varepsilon}}$ of ${\mathcal{F}}^{\mathbf{ r}}$ to the fiber $\pi^{-1}({{\varepsilon}})$ is a smooth 1-foliation. Furthermore, by the smoothness assumption, $$\lim_{{{\varepsilon}}\rightarrow 0}{\mathcal{F}}^{\mathbf{ r}}_{{\varepsilon}}= {\mathcal{F}}^{\mathbf{ r}}_0$$ uniformly (in the $C^\infty$ topology) on each compact subset of $M \setminus \Sigma$ ([^7]).
In [@H] section 1.4, H" ormander constructs a regularization by convolution. For simplicity, let us assume that ${\mathcal{F}}$ is defined in ${\ensuremath{\mathbbm{R}}}^n$ by a smooth vector field $X$ which extends as a locally bounded measurable function to the discontinuity set $\Sigma$. Given a function $0 \le \chi \in C_0^\infty({\ensuremath{\mathbbm{R}}}^n)$ such that $\int \chi(y) dy = 1$, we define $$X_{{\varepsilon}}(x) = \int_{{\ensuremath{\mathbb{R}}}^n} X(x-{{\varepsilon}}y) \chi(y) dy$$ Then, it is easy to see that $X_{{\varepsilon}}$ is a smooth vector field for each ${{\varepsilon}}> 0$ and that the one-parameter family of 1-foliations ${\mathcal{F}}_{{\varepsilon}}$ defined by these vector fields is a regularization of ${\mathcal{F}}$.
One disadvantage of this regularization by convolution is that some important features of the dynamics of ${\mathcal{F}}$ which appears outside the discontinuity locus can be destroyed by small perturbations, and thus not be seen in ${\mathcal{F}}_{{\varepsilon}}$. For instance, the saddle connection illustrated in figure \[break-saddle-fig\] would be broken by a generic choice of convolution kernel $\chi$ (although it lies outside the discontinuity locus).
In the next subsection, we will describe two regularization methods which keep ${\mathcal{F}}$ unchanged outside $O({{\varepsilon}})$-neighrborhoods of the discontinuity set. As such, we expect to see the full dynamics of ${\mathcal{F}}$ outside $\Sigma$ to be reflected at ${\mathcal{F}}_{{\varepsilon}}$, for each sufficiently small ${{\varepsilon}}$.
Sotomayor-Teixeira regularization and its generalizations {#subsection-ST}
---------------------------------------------------------
Suppose that the discontinuity locus of ${\mathcal{F}}$ is a smooth submanifold $\Sigma \subset M$ of codimension one and that we fix the following data:
A tubular neighborhood map $f: N\Sigma \rightarrow M$, which maps the normal bundle $N\Sigma$ diffeomorphically to an open neighborhood $W=f(N\Sigma)$ of $\Sigma$.
A smoothly varying metric $|\cdot|$ on the fibers of the bundle $N\Sigma \rightarrow \Sigma$ (such that $|p|=0$ iff $p\in \Sigma$).
A [*monotone transition function* ]{}$\phi:{\ensuremath{\mathbbm{R}}}^ \rightarrow [-1,1]$.
Using the map $f$, we pull-back ${\mathcal{F}}$ to a discontinuous 1-foliation ${\mathcal{G}}$ on the normal bundle $N\Sigma$, with discontinuity locus given by the zero section $\Sigma \subset N\Sigma$.
Without loss of generality, we can assume that $N\Sigma$ is covered by local trivialization charts where the bundle map assumes the form $$\begin{array}{rcl}
V \times {\ensuremath{\mathbbm{R}}}&\longrightarrow& V \\
(x,y) & \longmapsto & x
\end{array}$$ for some open set $V\subset \Sigma$, and that ${\mathcal{F}}$ has a local multi-generator of the form $(V\times{\ensuremath{\mathbbm{R}}},\{X_+,X_-\})$, where $X_+$ (resp. $X_-$) is a smooth vector field in $V\times {\ensuremath{\mathbbm{R}}}$ which generate ${\mathcal{G}}$ on $U_+ = \{y >0\}$ (resp. $U_- = \{y < 0\}$). Furthermore, we can assume that the norm on the fibers of $N\Sigma$ is simply the absolute value $|y|$ on ${\ensuremath{\mathbb{R}}}$.
For each ${{\varepsilon}}> 0$, we now define a smooth vector field $X_{{\varepsilon}}$ in $V\times{\ensuremath{\mathbbm{R}}}$ as follows $$X_{{\varepsilon}}\stackrel{\mathrm{def}}{=} \frac{1}{2}\left(1 + \phi\Big(\frac{y}{{{\varepsilon}}}\Big)\, \right) X_+ + \frac{1}{2}\left(1 - \phi\Big(\frac{y}{{{\varepsilon}}}\Big)\, \right) X_-$$ Notice that, by construction $$X_{{\varepsilon}}(x,y) =
\begin{cases}
X_+ (x,y)& \text{Êif }y \ge {{\varepsilon}},\\
X_- (x,y)& \text{Êif }y \le -{{\varepsilon}},\\
\end{cases}$$ Moreover, if we choose another multi-generator of ${\mathcal{G}}$, say $(V\times{\ensuremath{\mathbbm{R}}},\{Y_+,Y_-\})$ then it follows from the condition (\[local-compatibilityofgen\]) in the definition of piecewise smooth 1-foliation that $Y_+ = \varphi X_+$ and $Y_- = \varphi X_-$, for some strictly positive smooth function $\varphi$. Therefore, if we define a family $Y_{{\varepsilon}}$ exactly as above but replacing $X_\pm$ by $Y_\pm$, we conclude that $$Y_{{\varepsilon}}= \varphi X_{{\varepsilon}}, \quad \forall {{\varepsilon}}> 0$$ In other words, the $X_{{\varepsilon}}$ and $Y_{{\varepsilon}}$ define precisely the same smooth 1-foliation in the domain $V \times {\ensuremath{\mathbbm{R}}}$.
By considering an open covering of $N\Sigma$ by these local trivializations, one defines, for each ${{\varepsilon}}> 0$, a smooth foliation ${\mathcal{G}}_{{\varepsilon}}$. By construction, such foliation coincides which the original foliation ${\mathcal{G}}$ outside the region $\{p \in N\Sigma : |p| < {{\varepsilon}}\}$.
The [*Sotomayor-Teixeira regularization* ]{}of ${\mathcal{F}}$ is the discontinuous 1-foliation ${\mathcal{F}}^{\mathbf{ r}}$ defined in the product space $M \times ({\ensuremath{\mathbb{R}}}^+,0)$ as follows: For ${{\varepsilon}}= 0$, we let ${\mathcal{F}}^{\mathbf{ r}}_0 = {\mathcal{F}}$. For ${{\varepsilon}}> 0$, we define the foliation ${\mathcal{F}}^{\mathbf{ r}}_{{\varepsilon}}$ in $M$ by $${\mathcal{F}}^{\mathbf{ r}}_{{\varepsilon}}=
\begin{cases}
{\mathcal{F}}&\text{on }M \setminus W\\
f_* {\mathcal{G}}_{{\varepsilon}}&\text{on }W
\end{cases}$$ It follows from the remark made at the previous paragraph that this defines a globally smooth 1-foliation in $M$. It is easy to verify that the conditions 1.-3. of the definition of a regularization are satisfied.
More generally, under the same assumptions of the previous example, we can define regularization of ${\mathcal{F}}$ by dropping the assumption of monotonicity and $x$-independence of the transition function. Namely, by replacing the choice of function $\phi$ in item 3. by the choice of a smooth function $$\label{psi-function}
\psi : \Sigma \times {\ensuremath{\mathbbm{R}}}_+ \rightarrow [-1,1]$$ such that $\psi(x,t) = -1$ if $t \le -1$ and $\psi(x,t) = 1$ if $t \ge 1$. Correspondingly, we replace the expression of $X_{{\varepsilon}}$ given above by $$\label{def-transitionreg}
X_{{\varepsilon}}\stackrel{\mathrm{def}}{=} \frac{1}{2}\left(1 + \psi\Big(x,\frac{y}{{{\varepsilon}}}\Big)\, \right) X_+ + \frac{1}{2}\left(1 - \psi\Big(x,\frac{y}{{{\varepsilon}}}\Big)\, \right) X_-$$ All the remaining steps in the construction remain the same. The resulting regularization will be called a *regularization of transition type*.
Double regularization of the cross {#subsect-cross}
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Let us show a situation where it is natural to consider a multi-parameter regularization. Consider a discontinuous 1-foliation ${\mathcal{F}}$ in ${\ensuremath{\mathbb{R}}}^3$ with discontinuity locus $\Sigma = \{ xy = 0\}$ (like in Example \[example-3\]). In other words, ${\mathcal{F}}$ is defined by four smooth vector fields $X_{\pm,\pm}$, where the first and the second $\pm$ sign correspond respectively to the sign of the $x$ and $y$ coordinates. In other words, each $X_{\pm,\pm}$ is a generator of ${\mathcal{F}}$ in one of the four quadrants $U_{\pm,\pm} = \{(x,y,z) : {\operatorname{sgn}}(x) = \pm, {\operatorname{sgn}}(y) = \pm\}$.
Choosing monotone transitions functions $\phi,\psi$ as above, we consider the two-parameter family of smooth vector fields $$Y_{{{\varepsilon}},\eta} \stackrel{\mathrm{def}}{=}
\sum_{\alpha,\beta \in \{+,-\}} \left( 1 + \alpha \phi\Big( \frac{x}{{{\varepsilon}}} \Big) \right) \left( 1 + \beta \psi\Big( \frac{y}{\eta} \big) \right) X_{\alpha,\beta}$$ defined for ${{\varepsilon}},\eta > 0$. Similarly, we define the two one-parameter families of discontinuous vector fields $$\begin{array} {ccc}
Z_{\pm,\eta} &\stackrel{\mathrm{def}}{=} &
\displaystyle \sum_{\beta \in \{+,-\}} \left( 1 + \beta \psi\Big( \frac{y}{\eta} \big) \right) X_{\pm,\beta}\\
W_{{{\varepsilon}}, \pm} &\stackrel{\mathrm{def}}{=} &
\displaystyle \sum_{\alpha \in \{+,-\}} \left( 1 + \alpha \phi\Big( \frac{x}{{{\varepsilon}}} \Big) \right) X_{\alpha,\pm}
\end{array}$$ defined respectively for $\eta >0$ and ${{\varepsilon}}> 0$. Notice that the discontinuity locus of $Z_{\pm,\eta}$ and $W_{\pm,\eta}$ is given respectively by $\{x = 0\}$ and $\{y = 0\}$.
The [*double-regularization* ]{}of ${\mathcal{F}}$ is the discontinuous 1-foliation ${\mathcal{F}}^{\mathbf{ r}}$ defined in the product space ${\ensuremath{\mathbb{R}}}^3 \times (({\ensuremath{\mathbbm{R}}}^+)^2,0)$ as follows. For each parameter value ${{\varepsilon}},\eta \ge 0$, the foliation restricted to fiber $\pi^{-1}({{\varepsilon}},\eta)$ is generated by a discontinuous vector field $K_{{{\varepsilon}},\eta}$ chosen as follows $$K_{{{\varepsilon}},\eta} =
\begin{cases}
X_{\pm,\pm} & \text{Êif ${{\varepsilon}}= 0$ or $\eta = 0$,}\\
Z_{\pm,\eta} & \text{Êif ${{\varepsilon}}= 0$ and $\eta > 0$},\\
W_{{{\varepsilon}},\pm} & \text{Êif ${{\varepsilon}}> 0$ and $\eta = 0$},\\
Y_{{{\varepsilon}},\eta} & \text{Êif ${{\varepsilon}}> 0$ and $\eta > 0$},
\end{cases}$$
More generally, assuming that a discontinuous foliation ${\mathcal{F}}$ in ${\ensuremath{\mathbb{R}}}^n$ has a discontinuity locus given by the union of $p$ coordinate hyperplanes, say $$\Sigma = \Big\{\prod_{i=1}^p x_i = 0\Big\}$$ we can define $p$-parameter regularization of ${\mathcal{F}}$ by an easy generalization of the above construction.
Sliding regions
---------------
Let ${\mathcal{F}}$ be a discontinuous $1$-foliation defined on a manifold $M$ and with discontinuity locus $\Sigma$. Given a $p$-parameter regularization ${\mathcal{F}}^{\mathbf{ r}}$ of ${\mathcal{F}}$, our present goal is to define a subset ${\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}})$ of $\Sigma$ where it will be reasonable to study a [*limit dynamics with respect to such given regularization*]{}.
Our definition is local. We will say that point $p \in \Sigma$ is a [*point of sliding for ${\mathcal{F}}^{\mathbf{ r}}$* ]{} if there exists an open neighborhood $U \subset M$ of $p$ and a family of smooth manifolds $$S_{{\varepsilon}}\subset U$$ defined for all ${{\varepsilon}}\in (({\ensuremath{\mathbb{R}}}^\star)^p,0)$ such that:
For each ${{\varepsilon}}$, $S_{{\varepsilon}}$ is invariant by the restriction of ${\mathcal{F}}^{\mathbf{ r}}_{{\varepsilon}}$ to $U$.
For each compact subset $K \subset U$, the sequence $S_{{\varepsilon}}\cap K$ converges to $\Sigma \cap K$ as ${{\varepsilon}}$ goes to zero in some given Hausdorff metric $d_H$ on compact sets of $M$. ([^8])
The set of sliding points for ${\mathcal{F}}^{\mathbf{ r}}$ is a relatively open subset of $\Sigma$, which we denote by ${\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}})$.
Consider the Sotomayor-Teixeira regularization of the discontinuous foliation described in Example \[example-1\]. Then, an easy computation with the expression of $X_{{\varepsilon}}$ defined in the previous subsection (and taking $|y|$ to be the usual absolute value) gives $$X_{{\varepsilon}}= \frac{\partial}{\partial x} - \phi\left( \frac{y}{{{\varepsilon}}}\right) \frac{\partial}{\partial y}.$$ Let $t_0 \in (-1,1)$ be the zero of $\phi$ (which is unique by the monotonicity hypothesis on $\phi$). Then, the family of one-dimensional manifolds $$S_{{{\varepsilon}}} = \{y = t_0 {{\varepsilon}}\}$$ satisfies the above conditions 1. and 2. locally at each point of $\Sigma$. Consequently, ${\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}}) = \Sigma$.
Consider the discontinuous foliation in ${\ensuremath{\mathbb{R}}}^2$ defined by the vector field $$X = \frac{\partial}{\partial x} + \big( (x+1) + sgn(y)(x-1) \big) \frac{\partial}{\partial y}.$$ Then, the Sotomayor-Teixeira regularization is defined by the vector field $$X_{{\varepsilon}}= \frac{\partial}{\partial x} - \big( (x+1) + \phi\left( \frac{y}{{{\varepsilon}}}\right) (x-1) \big)\frac{\partial}{\partial y}.$$
Therefore for each $x \in {\ensuremath{\mathbbm{R}}}$, the coefficient of the $\partial/\partial y$ component of $X_{{\varepsilon}}$ vanishes if and only if $$y = {{\varepsilon}}t_x$$ where $t_x$ is a solution of the equation $\phi(t_x) = (x+1)/(1-x)$. By the assumptions on $\phi$, this equation has a solution (which is necessarily unique) if and only if $x \le 0$. As we shall prove in the next section, it follows that ${\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}}) = \Sigma \cap \{x < 0\}$.
Let us consider the same discontinuous vector field of the previous Example but now use a different regularization. Namely, we let ${\mathcal{F}}^{\mathbf{ r}}$ be a regularization of transition type, with transition function $\psi$ having a graph as illustrated in the figure below.
As a consequence of the results of the next section, we have ${\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}}) = \Sigma \cap \{x < \frac{1}{3} \}$ (because $\frac{1/3 + 1}{1-1/3 } = 2$).
(Stratified Sliding for analytic discontinuity locus) Assume that the discontinuity locus $\Sigma$ is an analytic subset, of dimension $d$. Then, we can define a more refined notion of sliding by considering different strata of $\Sigma$.
More precisely, using the decomposition $
\Sigma^0 \subset \Sigma^1 \subset \cdots \subset \Sigma^d = \Sigma
$ defined in (\[stratification\]), we say that point $p \in \Sigma^k \setminus \Sigma^{k-1}$ is a [*stratified point of sliding* ]{}for ${\mathcal{F}}^{\mathbf{ r}}$ if the conditions 1. and 2. of the above definition holds, when we replace the convergence condition in 2. by $$d_H(S_{{\varepsilon}}\cap K, \Sigma^k \cap K) \rightarrow 0$$ as ${{\varepsilon}}$ goes to zero. The set of all points $\Sigma^k \setminus \Sigma^{k-1}$ satisfying the above condition is called [*sliding region of dimension $k$*]{}, and denoted by ${\mathrm{Slide}}^k({\mathcal{F}}^{\mathbf{ r}})$.
Let us apply the double regularization described in subsection \[subsect-cross\] to the discontinuous 1-foliation defined in Example \[example-3\]. An easy computation shows that the regularized vector field is given by $$X_{{{\varepsilon}},\eta} = - \phi\left(\frac{x}{{{\varepsilon}}}\right) \frac{\partial}{\partial x}Ê- \psi\left(\frac{y}{\eta}\right) \frac{\partial}{\partial y} + \frac{\partial}{\partial z}$$ where $\phi,\psi$ are monotone transition functions.
Consider the one dimensional stratum $\Sigma^1 \subset \Sigma$ given by $\Sigma_1 = \mbox{axis}\, z$. We claim that each point of $\Sigma^1$ is a stratified point of sliding. Indeed, if we denote by $t_0,u_0 \in (-1,1)$ the unique roots of $\phi(t) = 0$ and $\psi(u) = 0$ respectively, then the one-dimensional curve $$S_{{{\varepsilon}},\eta} = \{x = {{\varepsilon}}t_0,\ y = \eta u_0 \}$$ is invariant by $X_{{{\varepsilon}},\eta}$ and clearly converges to $\Sigma^1$ as ${{\varepsilon}},\eta$ converges to zero.
Regularizations of transition type: blowing-up and conditions for sliding {#s3}
=========================================================================
In this section, we consider piecewise smooth 1-foliations whose discontinuity set is a smooth submanifold of codimension one. Our main goal is to describe conditions which guarantee that a point belongs to the sliding region of given a regularization of transition type.
To fix the notation, we choose a piecewise smooth 1-foliation ${\mathcal{F}}$ defined in a manifold $M$, and whose discontinuity locus is a smooth submanifold $\Sigma \subset M$ of codimension one. According to the definition in subsection \[subsect-piecewise-smooth\], at each point $p \in \Sigma$ we can choose local coordinates $(x_1,\ldots,x_{n-1},y)$ and two smooth vector fields $X_+$ and $X_-$ such that $\Sigma = \{ y = 0\}$ and $X_+$ and $X_-$ are generators of ${\mathcal{F}}$ on the sets $\{y > 0\}$ and $\{y < 0\}$, respectively.
First of all, we prove a result which will allow us to use the the theory of smooth dynamical systems to study such regularization.
\[theorem-smoothableST\] Let ${\mathcal{F}}^{\mathbf{ r}}$ be a regularization of transition type of ${\mathcal{F}}$. Then, ${\mathcal{F}}^{\mathbf{ r}}$ is blow-up smoothable.
We will show that a single blowing-up suffices to obtain a smooth foliation. More precisely, consider the blowing up $$\Phi: N \rightarrow M \times ({\ensuremath{\mathbb{R}}}^+,0)$$ with center on $\Sigma$. We claim that there exists a smooth foliation in ${\mathcal{G}}$ in $N$ which is related to ${\mathcal{F}}^{\mathbf{ r}}$ by $\Phi$.
To prove this, we use the trivialization of $\Sigma$ given by the local charts $(x,y) \in V \times {\ensuremath{\mathbb{R}}}$ described in subsection \[subsection-ST\] and the expression for $X_{{\varepsilon}}$ defined at the end of that subsection. Using these coordinates, the blowing-up map can be written (up to restriction to an appropriate subdomain) as $$\begin{array}{ccc}
V \times {\mathbbm{S}}^1 \times {\ensuremath{\mathbbm{R}}}^+ & \longrightarrow & V \times {\ensuremath{\mathbb{R}}}\times {\ensuremath{\mathbb{R}}}\\
x,(\theta,r) & \longmapsto & x,y=r \sin(\theta), {{\varepsilon}}= r \cos(\theta)
\end{array}.$$ In order to make the computations easier, it is better to cover the domain in ${\mathbbm{S}}^1 \times {\ensuremath{\mathbbm{R}}}^+$ by three directional charts, with domains $E = \{\theta \not\equiv 0 (\mathrm{mod}\, \pi {\ensuremath{\mathbb{Z}}})\}$ and $F_\pm = \{\theta \not\equiv \pm \pi/2 (\mathrm{mod}\, 2\pi {\ensuremath{\mathbb{Z}}})\}$. In these charts, the blowing-up map assumes respectively the form $$\begin{array}{c}
\begin{array}{ccc}
V \times {\ensuremath{\mathbb{R}}}\times {\ensuremath{\mathbb{R}}}^+ & \longrightarrow & V \times {\ensuremath{\mathbb{R}}}\times {\ensuremath{\mathbb{R}}}\\
x,(\bar{y},\bar{{{\varepsilon}}}) & \longmapsto & x,y= \bar{{{\varepsilon}}}\bar{y}, {{\varepsilon}}= \bar{{{\varepsilon}}}
\end{array} ,\\ \\
\begin{array}{ccc}
V \times {\ensuremath{\mathbb{R}}}^+ \times {\ensuremath{\mathbb{R}}}^+ & \longrightarrow & V \times {\ensuremath{\mathbb{R}}}\times {\ensuremath{\mathbb{R}}}\\
x,(\tilde{y},\tilde{{{\varepsilon}}}) & \longmapsto & x,y=\pm \tilde{y}, {{\varepsilon}}= \tilde{y} \tilde{{{\varepsilon}}}
\end{array}.
\end{array}$$ Let us compute the pull-back of $X_{{\varepsilon}}$ in each one of these charts.
In the $E$-chart, we have the following transformation rules for the basis vectors of $T(M\times {\ensuremath{\mathbb{R}}})$: $$\frac{\partial}{\partial x_i} = \frac{\partial}{\partial x_i},Ê\quad
\frac{\partial}{\partial y} = \frac{1}{\bar{{{\varepsilon}}}} \frac{\partial}{ \partial \bar{y}} ,\quad
\frac{\partial}{\partial {{\varepsilon}}} = \frac{\partial}{ \partial \bar{{{\varepsilon}}}} - \frac{\bar{y}}{\bar{{{\varepsilon}}}} \frac{\partial}{\partial \bar{y}} .$$ Thus, for instance the vector field $f(x,y,{{\varepsilon}})\frac{\partial}{\partial y}$ is mapped to $\frac{f(x,\bar{y}\bar{{{\varepsilon}}},\bar{{{\varepsilon}}})}{\bar{{{\varepsilon}}}} \frac{\partial}{\partial \bar{y}}$ and son on. Therefore, using the expression in (\[def-transitionreg\]), we obtain $$\Phi^* X_{{\varepsilon}}= \frac{1}{2}\left(1 + \psi\big(x,\bar{y}\big)\; \right) \Phi^* X_+ + \frac{1}{2}\left(1 - \psi\big(x,\bar{y}\big)\; \right) \Phi^* X_-.$$ Therefore, from the above transformation rules, it is clear that the vector field $$\label{vector-fieldY}
Y \stackrel{def}{=} \bar{{{\varepsilon}}}\, \Phi^* X_{{\varepsilon}}$$ has a smooth extension to the exceptional divisor $\{\bar{{{\varepsilon}}} = 0\}$. We take $Y$ to be a local generator of ${\mathcal{G}}$ on this domain.
Similarly, in the $F_\pm$-chart, we have the following transformation rules $$\frac{\partial}{\partial x_i} = \frac{\partial}{\partial x_i},Ê\quad
\frac{\partial}{\partial {{\varepsilon}}} = \pm \frac{1}{\tilde{y}} \frac{\partial}{ \partial \tilde{{{\varepsilon}}}} ,\quad
\frac{\partial}{\partial y} = \pm \left( \frac{\partial}{ \partial \tilde{y}} - \frac{\tilde{{{\varepsilon}}}}{\tilde{y}} \frac{\partial}{\partial \tilde{{{\varepsilon}}}} \right).$$ And the pull-back of $X_{{\varepsilon}}$ in this chart has the form $$\Phi^* X_{{\varepsilon}}= \frac{1}{2}\left(1 + \psi\Big(x,\pm \frac{1}{\tilde{{{\varepsilon}}}}\Big)\; \right) \Phi^* X_+ + \frac{1}{2}\left(1 - \psi\Big(x,\pm \frac{1}{\tilde{{{\varepsilon}}}}\Big)\; \right) \Phi^* X_-.$$ Notice that, for all $0 < \tilde{{{\varepsilon}}} \le 1$, one has $\psi\Big(x,\pm \frac{1}{\tilde{{{\varepsilon}}}}\Big) \equiv \pm 1$ identically, and we can extend this function smoothly to $\tilde{{{\varepsilon}}} = 0$ as being equal to $\pm 1$, according to the domain. Therefore, similarly to the previous case, the vector field $$Z \stackrel{def}{=} \tilde{y}\, \Phi^* X_{{\varepsilon}}$$ has a smooth extension to the exceptional divisor $\{\tilde{y} = 0\}$, and we choose it as a generator of ${\mathcal{G}}$ on the corresponding domain. This concludes the proof.
Now, we will study the sliding regions. The criterion that we are going to describe needs one additional definition: Using the notation introduced above, the [*height function of ${\mathcal{F}}^{\mathbf{ r}}$* ]{}is the smooth function $h^{\mathbf{ r}}$ with domain $(x,t) \in \Sigma\times {\ensuremath{\mathbb{R}}}$ defined by $$h^{\mathbf{ r}}=
\psi\, \mathcal{L}_{(X_+ - X_-)}(y) + {\mathcal{L}_{(X_+ + X_-)}(y)}$$ where $\psi(x,t)$ is the transition function and $\mathcal{L}_X(f)$ denotes the Lie derivative of a function $f$ with respect to a vector field $X$. We remark that that the Lie derivative of $X_+ - X_-$ and $X_+ + X_-$ needs to be evaluated only at points of $\Sigma$.
More explicitly, if we write $X_+$ and $X_-$ in terms of the local trivializing coordinates $(x,y)$ described above as $$\label{expressions-Xpm}
X_\pm = a_\pm \frac{\partial}{\partial y}Ê+ \sum_{i=1}^{n-1} b_{i,\pm} \frac{\partial}{\partial x_i}$$ (for some smooth functions $a_\pm$ and $b_{i,\pm}$) then the height function is given by $$h^{\mathbf{ r}}(x,t) = \psi(x,t)\, \Big(a_+(x,0) - a_-(x,0)\Big) + \Big(a_+(x,0) + a_-(x,0)\Big).$$
Notice that the function $h^{\mathbf{ r}}$ is independent of the choice of local coordinates $(x,y)$ and local generators $(X_+,X_-)$ up to multiplication by a strictly positive function. More precisely, if we replace the local generators $(X_+,X_-)$ by local generators $(Y_+,Y_-)$ such that $Y_\pm = \varphi X_\pm$ then $h^{\mathbf{ r}}$ is transformed to $\varphi h^{\mathbf{ r}}$.
Based on the height function, we define the following subsets in $\Sigma \times {\ensuremath{\mathbb{R}}}$: $$\begin{array}{rcl}
\displaystyle {\mathbf{Z}}^{\mathbf{ r}}&=& \big\{ h^{\mathbf{ r}}(x,t) = 0 \big\},\quad
{\mathbf{W}}^{\mathbf{ r}}= \Big\{ \displaystyle \frac{\partial h^{\mathbf{ r}}}{\partial t}(x,t) \ne 0\Big\},\;\text{Êand }\\
{\mathbf{NH}}^{\mathbf{ r}}&=& {\mathbf{Z}}^{\mathbf{ r}}\cap {\mathbf{W}}^{\mathbf{ r}}.
\end{array}$$ The main result of this section can now be stated as follows:
\[theorem-slide\] Let ${\mathcal{F}}^{\mathbf{ r}}$ be a regularization of transition type of ${\mathcal{F}}$, defined by a transition function $\psi$ as above. Then, $$\pi({\mathbf{NH}}^{\mathbf{ r}}) \subset {\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}}) \subset \pi({\mathbf{Z}}^{\mathbf{ r}}).$$ where $\pi:\Sigma \times {\ensuremath{\mathbb{R}}}\rightarrow \Sigma$ is the canonical projection.
Let us compute in more details the expression of the blowing-up of $X_{{\varepsilon}}$ in the $E$ chart described in the proof of Theorem \[theorem-smoothableST\]. If we write the transformed vector field $Y$ in the form $$\label{vectorfield-Y2}
Y = \alpha\ \frac{\partial}{\partial \bar{y}}Ê+ \bar{{{\varepsilon}}}\; \sum_{i=1}^{n-1} \beta_{i}\ \frac{\partial}{\partial x_i}$$ then, using the expansions of $X_+$ and $X_-$ given in (\[expressions-Xpm\]), we conclude that, for $i = 1,...,n-1$, $$\beta_i = \displaystyle \frac{1}{2} \Big( \psi\, (b_{i,+} - b_{i,-}) + (b_{i,+} + b_{i,-}) \Big),$$ and $$\alpha = \displaystyle \frac{1}{2} \Big( \psi\, (a_+ - a_-) + (a_+ + a_-) \Big)$$ where $\psi$ is the transition function evaluated at $(x,t) = (x,\bar{y})$ and all functions $a_\pm$ and $b_{i,\pm}$ are computed by replacing the variable $y$ by $\bar{{{\varepsilon}}} \bar{y} $. Notice that the restriction $Y|_D$ of $Y$ to the exceptional divisor $D = \{\bar{{{\varepsilon}}} = 0\}$ is simply given by $$\label{YonD}
Y|_D = \frac{1}{2} h^{\mathbf{ r}}(x,\bar{y})\ \frac{\partial}{\partial \bar{y}}Ê$$ where $h^{\mathbf{ r}}$ is the height function defined above.
Suppose now that the coordinates $(x,y,{{\varepsilon}})$ are centered in a point $p \in \Sigma \times \{0\}$ lying in $\pi({\mathbf{NH}}^{\mathbf{ r}})$. Then, it follows from the above definition of the sets ${\mathbf{Z}}^{\mathbf{ r}}$ and ${\mathbf{W}}^{\mathbf{ r}}$ that there exists a $\bar{y}_0 \in {\ensuremath{\mathbbm{R}}}$ lying in the open interval $(-1,1)$ such that $$h^{\mathbf{ r}}(0,\bar{y}_0) = 0, \quad\text{Êand }\quad \frac{\partial h^{\mathbf{ r}}}{\partial y}(0,\bar{y}_0) \ne 0.$$ Looking at the expression of $Y|_D$ given above, it follows from the implicit function theorem that the point $q = (0,\bar{y}_0,0) \in \Phi^{-1}(p)$ lies in a locally defined smooth codimension one submanifold $H_0$ contained in the divisor $D = \{\bar{{{\varepsilon}}} = 0\}$ such that
1. Each point of $H_0$ is an equilibrium point of $Y|_D$.
2. $H_0$ is a normally hyperbolic invariant submanifold of $Y|_D$.
From Fenichel theory [@F] it follows that there exists a local smooth manifold $W \subset N$ of codimension one defined near $p$ which is invariant by the flow of $Y$ and such that $W\cap D = H_0$.
Let $S = \Phi(W)$. Then, it follows that, for each sufficiently small ${{\varepsilon}}> 0$, the set $S_{{\varepsilon}}= S \cap \pi^{-1}({{\varepsilon}})$ is an invariant submanifold of ${\mathcal{F}}^{\mathbf{ r}}_{{\varepsilon}}$ and $S_{{\varepsilon}}$ accumulates on $\Phi(H_0)$ as ${{\varepsilon}}$ goes to zero. Therefore, $p \in {\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}})$.
We have just proved that $\pi({\mathbf{NH}}^{\mathbf{ r}}) \subset {\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}})$. We postpone the proof of the inclusion ${\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}}) \subset \pi({\mathbf{Z}}^{\mathbf{ r}})$ to the end of this section.
Let us now describe the behavior of a regularization in the complement of the sliding set. For this, we introduce the so-called sewing region.
Keeping the above notation, we will say that a point $p \in \Sigma$ is a *point of sewing* for the regularization ${\mathcal{F}}^{\mathbf{ r}}$ if there exists an open neighborhood $U \subset M$ of $p$ and local coordinates $(x,y)$ defined in $U$ such that
$\Sigma = \{y = 0\}$ and,
For each sufficiently small ${{\varepsilon}}> 0$, the *vertical vector field* $\frac{\partial}{\partial y}$ is a generator of ${\mathcal{F}}_{{\varepsilon}}^{\mathbf{ r}}$ in $U$.
We will denote the set of all sewing points by ${\mathrm{Sew}}({\mathcal{F}}^{\mathbf{ r}})$.
\[intersection-slidesew\] Notice that the intersection of the regions ${\mathrm{Sew}}({\mathcal{F}}^{\mathbf{ r}})$ and ${\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}})$ is empty. Indeed, if $p$ lies in ${\mathrm{Sew}}({\mathcal{F}}^{\mathbf{ r}})$ then in the coordinates $(x,y)$ described above, each smooth manifold $S_{{\varepsilon}}$ which is invariant by ${\mathcal{F}}_{{\varepsilon}}^{\mathbf{ r}}$ should have necessarily the form $$S_{{\varepsilon}}= \{f_{{\varepsilon}}(x) = 0\}$$ for some smooth function $f_{{\varepsilon}}$ which is independent of the $y$ variable. In particular, $S_{{\varepsilon}}$ cannot tend to the discontinuity locus $\Sigma = \{y = 0\}$ as ${{\varepsilon}}$ goes to zero. Therefore $p \not\in {\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}})$.
\[theorem-sewingreg\] Let ${\mathcal{F}}^{\mathbf{ r}}$ be a regularization of transition type of ${\mathcal{F}}$, defined by a transition function $\psi$. Then, $$\pi\big({\mathbf{Z}}^{\mathbf{ r}}\big)^\complement \subset {\mathrm{Sew}}({\mathcal{F}}^{\mathbf{ r}})$$ where $\pi({\mathbf{Z}}^{\mathbf{ r}})^\complement$ denotes the complement of $\pi({\mathbf{Z}}^{\mathbf{ r}})$ in $\Sigma$.
Suppose that the coordinates $(x,y,{{\varepsilon}})$ described in the proof of Theorem \[theorem-smoothableST\] are centered in a point $p \in \Sigma\times \{0\}$ which lies in $\pi\big({\mathbf{Z}}^{\mathbf{ r}}\big)^\complement$. We claim that there exists a constant $\mu > 0$ and an open neighborhood $U \subset M \times ({\ensuremath{\mathbbm{R}}}^+,0)$ of $p$ such that the function $$g = \mathcal{L}_{X_{{\varepsilon}}}(y)$$ (which is defined only for ${{\varepsilon}}> 0$) satisfies $|g| > \mu$ on $U \cap \{{{\varepsilon}}> 0\}$. Once we prove this claim, the result is an immediate consequence of the flow-box theorem.
To prove this, we use the following fact: If $f$ and $X$ are respectively a smooth function and vector field defined in an open set $V \subset {\ensuremath{\mathbbm{R}}}^n$ and $\Psi: W \mapsto V$ is a diffeomorphism from another open set $W$ into $V$, then $$\big(\mathcal{L}_X(f)\big) \circ \Psi = \mathcal{L}_{\Psi^* X} \big( f\circ \Psi \big).$$ In other words, the Lie derivative operation commutes with the pull-back operation.
We apply this to the vector field $X_{{\varepsilon}}$ and to blowing-up map $\Phi$ defined in the proof of the Theorem \[theorem-smoothableST\]. Recall that $\Phi$ is a diffeomorphism outside the exceptional divisor $D$, and therefore by the above identity, $$g\circ \Phi = \mathcal{L}_{\Phi^* X_{{\varepsilon}}}\big( y \circ \Phi \big).$$ We are going to compute this expression explicitly using the directional charts. Recall that, in $E$ chart, we have $$\Phi^* X_{{\varepsilon}}= \frac{1}{\bar{{{\varepsilon}}}} Y\quad \text{Êand }\quad y \circ \Phi = \bar{{{\varepsilon}}}\bar{y}$$ where $Y$ is the vector field in (\[vector-fieldY\]). Using the basic properties of the Lie derivative, we get $$\begin{array}{rcll}
g\circ \Phi &=& \displaystyle \mathcal{L}_{\frac{1}{\bar{{{\varepsilon}}}}Y}\big( \bar{{{\varepsilon}}}\bar{y} \big) \\ \vspace{0.2cm}
& = & \displaystyle \frac{1}{\bar{{{\varepsilon}}}}\, \mathcal{L}_{Y}\big( \bar{{{\varepsilon}}}\bar{y}\big) & \text{(because $\mathcal{L}_{fX}(g) = f\mathcal{L}_{X}(g)$)}\\ \vspace{0.2cm}
& = & \displaystyle\frac{1}{\bar{{{\varepsilon}}}}\, \left( \bar{{{\varepsilon}}}\mathcal{L}_{Y}\big( \bar{y}\big) + \bar{y}\mathcal{L}_{Y}\big( \bar{{{\varepsilon}}}\big) \right) & \text{(by Leibniz's rule)}\\
& = & \displaystyle\mathcal{L}_{Y}\big( \bar{y}\big)& \text{(because $\mathcal{L}_{Y}(\bar{{{\varepsilon}}}) = 0$).}\\
\end{array}$$ Now, using the expression of $Y$ given in (\[vectorfield-Y2\]), we obtain $$\mathcal{L}_{Y}\big( \bar{y}\big) = \frac{1}{2}\left(\psi(x,\bar{y})\, \big(a_+ - a_-\big) + a_+ + a_-\right).$$ where $a_+$ and $a_-$ are computed by replacing $x,y$ by $x,\bar{{{\varepsilon}}}\bar{y}$, respectively. By restricting this expression to the divisor $D = \{\bar{{{\varepsilon}}} = 0\}$ and using the expression (\[YonD\]), we easily to see that $|g\circ \Phi| > \mu$ for some $\mu > 0$, uniformly in sufficiently small neighborhood $U_1$ of $\Phi^{-1}(p)$ in the domain of the $E$-chart.
Let us now compute $g\circ \Phi$ in the $F_\pm$ chart. Analogous computations gives $$g\circ \Phi = \frac{1}{\tilde{y}} \mathcal{L}_{Z}\big( \tilde{y} \big) .$$ On the other hand, a simple application of the transformation rules described in the proof of Theorem \[theorem-smoothableST\] shows that $$\mathcal{L}_{Z}(\tilde{y}) = \pm \frac{\tilde{y}}{2}\Big( \psi(x,\frac{1}{\tilde{{{\varepsilon}}}})\, \big(a_+ - a_-\big) + a_+ + a_-\Big)$$ where $a_+$ and $a_-$ are now computed by replacing $x,y$ by $x,\pm\tilde{y}$, respectively. Again, by restricting this expression to the divisor $D = \{\tilde{y} = 0\}$ it is easy to see that $|g\circ \Phi| > \mu$ uniformly in a sufficiently small neighborhood $U_2$ of $\Phi^{-1}(p)$ in the domain of the $F_\pm$-chart.
Since the domains of the $E$ and $F_\pm$ charts covers an entire neighborhood of $\Phi^{-1}(p)$ in the blowed-up space, it follows that $U_1 \, \cup\, U_2$ is a neighborhood of $\Phi^{-1}(p)$ in which $|g \circ \Phi|Ê> \mu$ . Hence the inequality $|g|>\mu$ holds in the neighborhood $U = \Phi(U_1 \cup U_2)$ of $p$. This concludes the proof of Theorem \[theorem-sewingreg\].
We are now ready to conclude the proof of Theorem \[theorem-slide\].
[(end of proof of Theorem \[theorem-slide\])]{} It remains to prove that ${\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}}) \subset \pi({\mathbf{Z}}^{\mathbf{ r}})$. From the Remark \[intersection-slidesew\], we know that ${\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}}) \cap {\mathrm{Sew}}({\mathcal{F}}^{\mathbf{ r}}) = \emptyset$. Combining with the result of the above Theorem, we conclude that ${\mathrm{Slide}}({\mathcal{F}}^{\mathbf{ r}}) \cap \pi({\mathbf{Z}}^{\mathbf{ r}})^\complement = \emptyset$. This concludes the proof.
Recall that in the case of the Sotomayor-Teixeira regularization we require the transition function $\psi$ to be strictly monotone in the interval $(-1,1)$. In this case, the set $\pi({\mathbf{Z}}^{\mathbf{ r}})$ can be alternatively described by the condition $$a_+ \cdot a_- \le 0$$ In other words, we recover the usual sliding condition of Filippov. Correspondingly, in this case $\pi({\mathbf{Z}}^{\mathbf{ r}})^\complement$ is defined by $$a_+ \cdot a_- > 0$$ which corresponds to the Fillipov’s sewing condition.
Notice that in the limit dynamics defined in the sliding region can be highly dependent on the choice of the transition function used in the regularization. To illustrate this, consider the following simple example. Let ${\mathcal{F}}$ be the discontinuous 1-foliation in ${\ensuremath{\mathbb{R}}}^2$ defined by the vector field $$X = \big( x^2 + y \big) \frac{\partial}{\partial x} - {\operatorname{sgn}}(y) \frac{\partial}{\partial y}$$ with discontinuity locus $\Sigma = \{y = 0\}$. Given a monotone transition function $\phi$, the Sotomayor-Teixeira regularization ${\mathcal{F}}^{\mathbf{ r}}$ is defined by the vector field $$X_{{\varepsilon}}= \big( x^2 + y \big) \frac{\partial}{\partial x} - \phi\left( \frac{y}{{{\varepsilon}}} \right) \frac{\partial}{\partial y}$$ and it is easy to see that the sliding region coincides with $\Sigma$. Explicitly, if $t_0 \in (-1,1)$ denotes the unique zero of the transition function $\phi$ then the curve $S_{{\varepsilon}}= \{y = t_0 {{\varepsilon}}\}$ is invariant by the flow of $X_{{\varepsilon}}$, for each ${{\varepsilon}}> 0$. Notice that the flow of $X_{{\varepsilon}}$ [*restricted to $S_{{\varepsilon}}$*]{} is defined by the one-dimensional vector field $$\big( x^2 + t_0 {{\varepsilon}}\big) \frac{\partial}{\partial x}.$$ In particular, we have three completely distinct topological behaviors depending on the sign of $t_0$.
Acknowledgments
===============
The first author is partially supported by FAPESP. The second author is partially supported by CAPES, CNPq, FAPESP, FP7-PEOPLE-2012-IRSES 318999, and PHB 2009-0025-PC.
[99]{} (1997). Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, *Invent. Math.*, **128**, 207–302.
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[^1]: by definition, this is a $C^\infty$ function such that $\varphi(t) = -1$ for $t \le -1$, $\varphi(t) = 1$ for $t \ge 1$ and $\varphi'(t) >0$ for $-1 < t < 1$.
[^2]: System is called slow system and it is equivalent, up to a time reparametrization, to the fast system $$x' = \frac{{\varepsilon}}{2}\left(a_++a_- +\varphi\left( \bar{y}\right)\big( a_+-a_-\big)\right) \quad
\bar{ y} =\frac{1}{2} \left(b_++b_- +\varphi\left(\bar{y} \right)\big( b_+-b_-\big) \right).$$
[^3]: According to [@C], Proposition 15, and Grauert’s embedding theorem [@Gr], this is equivalent to say $\Sigma$ is the vanishing locus of a coherent sheaf of ideals $\mathcal{I}$ defined on $M$.
[^4]: Notice that this minimum is well-defined by the non-flatness assumption. Furthermore, this shows that $m_{\mathrm{min}}$ is uniform, i.e. independent of the choice of the point $q \in W\cap D$.
[^5]: Notice that this construction is a slight generalization of the blowing-up of local foliated vector fields defined in [@DR].
[^6]: We use the notation $(({\ensuremath{\mathbb{R}}}^+)^p,0)$ to indicate in abridged form some open neighborhood of the origin in $({\ensuremath{\mathbb{R}}}^+)^p$.
[^7]: More precisely, given a point $q \in M\setminus \Sigma$, there exists an open neighborhood $U$ of $q$ and a $p$-parameter family of smooth vector field $X_{{\varepsilon}}$ defined on $U \times (({\ensuremath{\mathbb{R}}}^+)^p,0)$ (and depending smoothly on ${{\varepsilon}}$) such that $X_{{\varepsilon}}$ is a local generator of ${\mathcal{F}}^{\mathbf{ r}}_{{\varepsilon}}|_U$ for each ${{\varepsilon}}$.
[^8]: Obviously, this metric depends on the choice of a Riemannian metric on $M$, but the convergence condition is independent of the choice of this metric.
|
---
author:
- 'D. H. Nickeler'
- 'T. Wiegelmann'
- 'M. Karlický'
- 'M. Kraus'
title: MHD flows at astropauses and in astrotails
---
[**ABSTRACT**]{}\
The geometrical shapes and the physical properties of stellar wind – interstellar medium interaction regions form an important stage for studying stellar winds and their embedded magnetic fields as well as cosmic ray modulation. Our goal is to provide a proper representation and classification of counter-flow configurations and counter-flow interfaces in the frame of fluid theory. In addition we calculate flows and large-scale electromagnetic fields based on which the large-scale dynamics and its role as possible background for particle acceleration, e.g. in the form of anomalous cosmic rays, can be studied. We find that for the definition of the boundaries, which are determining the astropause shape, the number and location of magnetic null points and stagnation points is essential. Multiple separatrices can exist, forming a highly complex environment for the interstellar and stellar plasma. Furthermore, the formation of extended tail structures occur naturally, and their stretched field and streamlines provide surroundings and mechanisms for the acceleration of particles by field-aligned electric fields.
When stars move through the interstellar medium (ISM), the material released via their winds collides and interacts with the ISM. This interaction produces several detectable structures, such as stellar wind bow shocks when stars move with supersonic speeds relativ to the ISM. Furthermore, a termination shock can form, where the supersonic stellar wind slows down to subsonic speed, and in between this termination shock and the outer bow shock a contact surface forms, separating the subsonic ISM flow from the subsonic stellar wind flow. This contact surface is called the astropause and has at least one stagnation point at which both flows, the stellar wind and the ISM material stop and diverge.
In downwind direction, a tail like structure can form, which is the astrotail. The tail is not only proposed by the result of simulations, but also by the fact that stretched field or streamlines minimize the corresponding tension forces, so that the configuration is able to approach an equilibrium state. Such bow shocks and astrotails have been directly observed, e.g., around asymptotic giant branch stars [e.g., @2008ApJ...687L..33U; @2010ApJ...711L..53S], and have been proposed to exist also around the Sun [e.g., @2014ApJ...782...25S].
Typically, a magnetic field is embedded in the ISM. For stars with a strong magnetic field, the wind is magnetized as well. Hence, different null points can appear, one of the flow and one of the magnetic field, which are not necessarily at the same location, unless the magnetic field is frozen-in [see, @2008ASTRA...4....7N]. The formation of a stagnation point of the flow or of a magnetic neutral point is crucial for the understanding how an interface in the form of an astropause forms between the very local ISM and a stellar wind. The best object to study counterflow configurations observationally is the heliosphere, where spacecrafts such as [*Voyager 1 & 2*]{} and [*IBEX*]{} perform in situ measurements of the plasma parameters . However, an interpretation of these observations is not always straight forward. While strong indications for the crossing of the termination shock of [*Voyager 1*]{} exist, it is yet unclear and contradictory whether the heliopause region was already left [@2013Sci...341..147B; @2014ApJ...784..146B; @2013ApJ...776...79F; @2013Sci...341.1489G].
The problematics of computing counterflow configurations have been attacked from different viewpoints, such as hydrodynamics , kinematical magnetohydrodynamics (MHD) , and self-consistent MHD , focusing on different aspects and regions within the astrosphere. We are aware that fluid approaches like MHD are strictly valid only for collisional plasmas, but are frequently applied to collisionless configurations like magnetospheres, coronae or astrospheres. The alternative approach, kinetic theory, is not applicable because of the difference of kinetic and macroscopic scales.
It is evident that numerical simulations allow to study more involved physical models (like full MHD), which cannot be solved analytically. On the other hand, the above cited analytical studies, including the present, have the advantage that they provide exact solutions, and they allow us to analyse physical and mathematical effects in great detail. In particular, analytical investigations are indispensable for studying how the distribution of the magnetic null points/stagnation points determines both the topology of the streamlines/magnetic field lines and the geometrical shape of the heliopause.
In this paper, we provide exact mathematical definitions of astrospheres and astropauses. We discuss possible shapes with respect to the spatial arrangement of the stagnation and/or null points and emphasize the role of the directions of flow and field of both the ISM and the star. Furthermore, we discuss the role of magnetic shear flows as possible trigger for particle acceleration (e.g., ACRs) in the heliotail.
Geometrical shapes and topological properties of astropauses
============================================================
The geometrical shapes of astropauses depend on the strengths and directions of the two involved flows (stellar wind and ISM flow) and their electromagnetic fields. The fact that most of the ISM flow cannot penetrate the stellar wind region but is forced to flow around defines the boundary called astropause. In the mathematical sense, the streamlines in the local ISM and in the stellar wind region are topologically disjoint. Still, different possibilities for the definition of the physical astropause exist. One way to define the location of this boundary might be to use the stagnation point of the flow as criterion. This stagnation point is the intersection of the stagnation line and the astropause (separatrix) surface. The stagnation line is this streamline along which all fluid elements heading towards the astropause and coming from opposite directions, i.e., on the one hand from the ISM and on the other hand from the star, are decelerated down to zero velocity. However, such a definition is only useful in a single-fluid (i.e. classical HD or MHD) theory, while in a more general multi-fluid model different pauses (i.e. one for each component) exist. Here a definition via a stagnation point of the flow is not unique. A precise way is thus to use the null point of the magnetic field, and to transfer the concept of the stagnation point of the flow to the magnetic field configuration. Such a definition via the null point of the magnetic field makes only sense as long as the star itself has a magnetic field, strong enough to be dynamically important. Here, we focus on stars with magnetic fields, for which the astropause can be uniquely defined as the outermost magnetic separatrix. The most well-known magetic star is our Sun with its boundary, the heliopause.
Definition of a separatrix
--------------------------
The field lines of a vector field are typically represented by the trajectories of a corresponding system of ordinary differential equations, the so-called phase portrait. Physically interesting phase portraits are those in which null points exist. Then, a topological classification of the local vector field can be performed by analysing the eigenvalues of the corresponding Jacobian matrix of the vector field at the null point [see, e.g., @1990ApJ...350..672L; @Arnold; @1996PhPl....3..759P]. The configuration of the vector field in the vicinity of the magnetic null point depends on the dimension of the considered problem. For instance, in 3D, the magnetic null point is the intersection of the 1D stable or unstable submanifold termed the spine, with a 2D unstable or stable manifold termed the fan. This 2D manifold is termed the separatrix (or pause). In cartesian 2D, the magnetic null point is the intersection of two separatrix field lines of which one is called stable and the other one is called unstable. This null point in 2D is of X-point type. Trajectories, i.e., streamlines or field lines, of the vector field are called stable, when the streamline or field line points towards the null point, while they are called unstable when they point away from the null point. Such configurations can only be obtained if the eigenvalues of the Jacobian matrix of the vector field at the null point are of so-called hyperbolic or saddle point type. More specifically, in 2D the eigenvalues are real and have opposite signs, while in 3D the situation is more complex. Here, the real parts of the eigenvalues should not vanish and always two of their real parts have the identical sign. Basically, this separatrix concept is valid also concerning the flow field in HD or MHD.
The definition of a separatrix was so far only restricted to a scenario with a single, isolated null point. In real astrophysical scenarios, multiple null points may exist, depending on the complexity of the stellar wind and its magnetic field. Therefore, multiple and highly complex, maybe even nested separatrices may occur. For instance, @2013ApJ...774L...8S find that the heliopause can be considered as a region consisting of bundles of separatrices and magnetic islands, resulting from magnetic reconnection processes, which form a porous, multi-layered structure. Examples of multiple and nested separatrices originating from X-type null points are shown and discussed in the following.
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Determination of the global shape of astrospheres: distribution of null points for pure potential fields
--------------------------------------------------------------------------------------------------------
To compute non-linear MHD flows with separatrices as we will do in Sect.\[flows\], geometrical patterns are needed for the structure of the corresponding flows and fields. To obtain such patterns, we start from the simplest possible fields, the potential fields, and investigate how separatrices form and how they are shaped by different spatial distributions of null points. The advantage of using potential fields is given by the fact that these fields obey a superposition principle, i.e., they are linear, and every null point is automatically an X-point of the magnetic field. In addition, potential fields have no free magnetic energy (i.e., they are stable) and can easily be mapped to non-linear fields by either generalized contact transformations or algebraic transformations. The concept of contact transformations was presented, e.g., by @1992PhFlB...4.1689G and later on used and refined by to describe the MHD fields of different space plasma environments, while the algebraic transformations were introduced by @2000JMP....41.2043B [@2000PhRvE..62.8616B; @2001PhLA..291..256B; @2002PhRvE..66e6410B].
As was shown by in the case of pure HD and by in the case of MHD, the number and spatial distribution of the hyperbolic null points determine the topological scaffold of an astrosphere. This, in combination with the assumption of a homogeneous background field as asymptotic boundary condition, allows us to describe the general shape of its field and streamlines. In the current paper, we aim at giving a qualitative overview of the different scenarios. For the detailed theoretical description and treatment we refer to .
If several hyperbolic null points exist, which of the separatrices defines then the real pause? What can be said about multiple null points is that they all have to be non-degenerate, because double or higher order null points are topologically unstable [e.g., @1996PhPl....3..781H].
Having discussed the general topological aspects, we now focus on the geometrical ones, considering a simplified 2D scenario, in analogy to typical flows in aero and fluid dynamics. While in the vicinity of the star the fields are full 2D to account for a variety of multipolar field structures, asymptotically, i.e., far away from the star in downwind direction, the field converges to a tail-like (1D) structure. Let us start with the case of two null points in the frame of a 2D cartesian potential field. We assume that the $z$-direction is the invariant direction, which means that for all parameters $\partial/\partial z = 0$. Further, $x$ points to the downwind (i.e. tail) direction, and $y$ in perpendicular direction. The global magnetic field, $\vec B = \vec\nabla A(x,y)
\times \vec e_{z}$, with the unit vector $\vec e_{z}$ in $z$-direction, can be described via complex analysis. As $\vec B$ should be a potential field, it follows that $\Delta A = 0$. To solve this Laplace equation, we define a stream or, here, the magnetic flux function $A$ by $A= \Im({\cal A})$, where ${\cal A}$ is the complex stream or magnetic flux function. This complex flux function is obtained from a Laurent series of the form $${\cal A} = B_{S\infty} u + C_{0} \ln u + \frac{C_{1}}{u} + \textrm{terms of higher order}\, .$$ In the following, we will neglect the higher order terms, so that the complex magnetic flux function consists purely of a monopole[^1] and dipole. Both multipoles are located in the origin. Here, $u= x + i y$ is the complex coordinate, $B_{S\infty}$ is the asymptotical boundary condition $\lim B = B_{S\infty}$ for $|u|\rightarrow \infty$, i.e., the background field, and $C_{0}$ and $C_{1}$ are the monopole and dipole moments, respectively. For the case of two null points, these moments have the following form $$C_{0} = -B_{S\infty} (u_{1} + u_{2}) \qquad \textrm{and} \qquad C_{1} = -B_{S\infty} u_{1} u_{2}$$ where $u_{1}$ and $u_{2}$ are the complex coordinates of the two null points.
![As Fig.\[series\], but comparing separatrix shapes with only one null point. Top: the symmetric Parker case ($u_{1} = x = -1$), and bottom: the asymmetric case where $u_{1}$ is rotated off the $x$-axis by $-\pi/4$, i.e., $u_{1} = -1 (\cos(-\pi/4)+ i \sin(-\pi/4))$. The apparent non-connectivity of some field lines at $y=0$ and $x < 0$ is not real but caused by the flip between Riemann surfaces.[]{data-label="axisym"}](helioparkergross.pdf "fig:"){width="8.3cm"} ![As Fig.\[series\], but comparing separatrix shapes with only one null point. Top: the symmetric Parker case ($u_{1} = x = -1$), and bottom: the asymmetric case where $u_{1}$ is rotated off the $x$-axis by $-\pi/4$, i.e., $u_{1} = -1 (\cos(-\pi/4)+ i \sin(-\pi/4))$. The apparent non-connectivity of some field lines at $y=0$ and $x < 0$ is not real but caused by the flip between Riemann surfaces.[]{data-label="axisym"}](heliotwistparker.pdf "fig:"){width="8.3cm"}
![As Fig.\[series\], but for an asymmetric case where the front null point is rotated off the $x$-axis by $-\pi/4$, i.e., $u_{1} = -1 (\cos(-\pi/4)+
i \sin(-\pi/4))$, and the other one is on the $x$-axis at $u_{2} = x=0.4$. The apparent non-connectivity of some field lines at $y=0$ and $x < 0$ is not real but caused by the flip between Riemann surfaces.[]{data-label="twist"}](heliotwist5.pdf){width="8.3cm"}
The simplest scenario is the one with two symmetric null points ($u_{1}= -u_{2}$). In this case, only the dipole moment exists, and we can interprete this with a star with a dipole magnetic field embedded in a homogeneous magnetic background field. Such a scenario is an analogy to the classical hydrodynamical example of a cylindrical obstacle in a homogeneous flow. If we assume the star is located at the origin of a cartesian coordinate system and the flow is parallel to the $x$-axis and streams in positive $x$-direction, the separatrices form a circle in the $(x,y)$-plane, where the stagnation lines lie on the $x$-axis and intersect the circular separatrix from both sides at the two null points and pass through the pole. This is shown in the upper left panel of Fig.\[series\]. The radius $R$ of the circular separatrix, and hence the location of the two symmetric null points, depends on the strength of the background magnetic field ($B_{S\infty}$) and the dipole field via $R^{2} = B_0 R_0^2 / B_{S\infty}$. The term $B_0 R_0^2$ is hereby the dipole moment. Such a symmetric scenario is not very realistic, because it would imply a completely closed separatrix. Hence, no plasma can escape via the stellar wind.
To enable at least a half-open astrosphere, the second null point (the one in the downwind region) must be located closer to the pole. The shape of the resulting separatrices for different pole distances are depicted in the series of plots in Fig.\[series\]. When moving the second null point towards the pole, one can notice several effects. First, the separatrix resulting from the first null point opens in the downwind region. Second, an inner, closed separatrix is formed that passes through the second null point and through the pole. This separatrix encloses now the dipolar field region, which became smaller, and the dipole field weaker, if the same background field is present. This inner separatrix meets the stagnation line from the first null point at the pole. A measurable bundle of field lines can leave the inner dipole region upstream, i.e., between the inner separatrix and the stagnation line. These field lines origin from the monopole part of the field. They are deflected at the outer separatrix so that they bend around the inner separatrix and extend as open field lines into the astrotail, forming the inner astrosheath field lines. The closer the second null point is located to the pole, the more opens the tail region.
As soon as the second null point reaches the pole, it vanishes. Hence, only one real null point remains and the field has only a monopole moment. This scene is shown in the middle right panel of Fig.\[series\] and is similar to the Parker scenario [@1961ApJ...134...20P] for subsonic flows. If the second null point is located on the same side as the first one, i.e., in upwind direction (lower right panel of Fig.\[series\]), the second separatrix is also located in upwind direction and both null points are physically connected by the stagnation line. In addition, the monopole moment increases so that the astrotail becomes even wider.
Restricting for the moment to a single null point, it is, of course, not necessary that this null point is located on the $x$-axis. For instance, with respect to the heliopause, measurements from [*Voyager 1*]{} indicate an asymmetry [@2013Sci...341..147B]. A natural way to displace the null point is provided by the solar (or stellar, in general) rotation, which results in a winding-up of the field lines and hence to a spiral structure of the magnetic field. While the monopole moment in the symmetric examples is a pure real number, it now becomes complex. Hence, an azimuthal component of the outflow or field occurs. The result is a displacement of the null point off the $x$-axis. This is demonstrated in Fig.\[axisym\], where we plot the asymmetric configuration[^2] in comparison to the symmetric one. An even more complex situation is achieved when a second null point exists in the asymmetric scene. Such an example is shown in Fig.\[twist\].
Having multiple, nested separatrices, the real astropause can be uniquely defined by the outermost magnetic separatrix between the magnetized interstellar medium and the magnetized stellar wind.
Self-consistent non-linear MHD flows {#flows}
====================================
The pattern of the potential fields, as we calculated in the previous section, serve now as static MHD equilibria (MHS). These are then mapped with the non-canonical transformation method to self-consistent steady-state MHD flows. Thereby we make use of estimated or observed physical quantities, such as density, magnetic field strength, etc., within and outside the heliosphere. These quantities serve as asymptotical boundary conditions, based on which some of the coefficients of the mapping can be fixed.
Observations from [*Voyager 1*]{} suggest that the plasma flow in the vicinity of the heliopause and in the heliotail region is approximately parallel to the magnetic field [@2013Sci...341..147B; @2013ApJ...776...79F]. In addition, the plasma within a stagnation region is incompressible. This can be understood in terms of the steady-state mass continuity equation $$\begin{aligned}
\vec\nabla\cdot\left(\rho\,\vec v\right)=0
\quad\Leftrightarrow
\quad \vec v\cdot\vec\nabla\rho + \rho \vec\nabla\cdot\vec v=0 \, . \end{aligned}$$ When approaching the stagnation point, i.e. $\vec v\rightarrow \vec 0$, the term $\vec v\cdot\vec\nabla\rho$ in the second equation vanishes, implying that $\rho\vec\nabla\cdot\vec v$, and in particular, as the density reaches a maximum, $\vec\nabla\cdot\vec v$ has to vanish as well.[^3] The plasma flow on streamlines, which originate in such stagnation point regions, transports the property of incompressibility further into the tail region. Hence, it is reasonable to investigate the heliotail and heliopause region using field-aligned, incompressible flows, and the basic ideal MHD equations are given by $$\begin{aligned}
\vec\nabla\cdot\left( \rho\vec{\rm v} \right) &=& 0\, ,\label{mce}\\
\rho\left( \vec{\rm v}\cdot\vec\nabla\right)\vec{\rm v} &=& \vec
j\times\vec B - \vec\nabla P\, \label{ee},\\
\vec\nabla\times\left(\vec{\rm v}\times\vec B\right) &=&\vec 0\, ,\label{ie}\\
\vec\nabla\times\vec B &=& \mu_{0}\vec j\, ,\label{al}\\
\vec\nabla\cdot\vec B &=& 0\, ,\label{sc}\\
\vec\nabla\cdot\vec{\rm v} &=& 0\, ,\label{ice}\\
\vec{\rm v} &=& \pm|M_{A}| \vec{\rm v_{A}} \\
\vec{\rm v_{A}}& := &\frac{\vec B}{\sqrt{\mu_{0}\rho}}\, , \label{letzte}
\end{aligned}$$ where $\rho$ is the mass density, $\vec{\rm v}$ is the plasma velocity, $\vec B$ is the magnetic flux density, $\vec j$ is the current density, $P$ is the plasma pressure, $M_{A}$ is the Alfvén Mach number, $\vec{\rm v_{A}}$ is the Alfvén velocity, and $\mu_{0}$ is the magnetic permeability of the vacuum.
Given solutions for $p_{S}$ and $B_{S}$ of the MHS equations $$\vec\nabla p_{S} = \vec j_{S}\times\vec B_{S}\, ,$$ and additional solutions for $M_{A}$ and $\rho$ of the systems $$\begin{aligned}
\vec B_{S}\cdot\vec\nabla\rho & = & 0\, , \\
\vec B_{S}\cdot\vec\nabla M_{A} & = & 0\, ,\end{aligned}$$ are the parameters needed to perform the transformation, i.e., to compute the general solution [see, e.g., @2012AnGeo..30..545N] of the system Eq.(\[mce\])-(\[letzte\]) $$\begin{aligned}
\vec B &= &\frac{\vec B_{S}}{\sqrt{1-M_{A}^2}}\, , \\
p & = & p_{S} - \frac{1}{2\mu_{0}}\frac{M_{A}^2\left|\vec B_{S}\right|^2}{1-M_{A}^2}\, , \label{magpressuretrafo}\\
\sqrt{\rho} \vec{\rm v} &= & \frac{1}{\sqrt{\mu_{0}}}\,
\frac{M_{A}\vec B_{S}}{\sqrt{1-M_{A}^2}}\, ,\\
\vec j & = & \frac{M_{A}}{\mu_{0}}\frac{\vec\nabla M_{A}\times\vec B_{S}}{\left(1-M_{A}^2
\right)^{\frac{3}{2}}} +\frac{\vec j_{S}}{\left(1-M_{A}^2\right)^{\frac{1}{2}}}\, . \label{strom}
\label{streaming2mhs1}\end{aligned}$$ Properties of these solutions are that the plasma density $\rho$, the Alfvén Mach number $M_{A}$, and the Bernoulli-pressure $\varPi = P+\frac{1}{2}\rho \vec {\rm v}^2$ are constant on field lines. However, these parameters can vary perpendicular to the field lines, which means that for example strong shear flows, implying a strong gradient of the Alfvén Mach number, produce strong current densities (curent sheets). This is obvious from Eq.(\[strom\]), even if we start from a potential field, which implies $\vec j_{S} = \vec 0$. The occurrence of current sheets, especially around multiple separatrices, is known, e.g., in solar flare physics, as current fragmentation [e.g., @2008CEAB...32...35K; @2008SoPh..247..335K; @2010AdSpR..45...10B], or, in steady-state as fragmented currents . It should be emphasized that a similar transformation can also be performed using super-Alfvénic flows.
In the following, we concentrate on the tail region, taking the symmetric Parker-like tail (top panel of Fig.\[axisym\]). The mapping ansatz we use was described in detail in . The Alfvén Mach number is defined via $M_{A}^{2} = 1-1/(\alpha'(A))^{2}$, where the prime denotes the derivation with respect to $A$, and $\alpha$ is the mapped $z$-component of the vector potential of $A$. This means that if $\vec B_{S} = \vec\nabla A \times \vec e_{z}$ and $\alpha = \alpha(A)$, then $\vec B = \vec\nabla\alpha(A) \times \vec e_{z} = \alpha'(A)
\vec\nabla A \times \vec e_{z}$. Hence, $\alpha'(A)$ is the amplification factor for the magnetic field strength, and it results to $$\begin{aligned}
\alpha'(A) & = & \frac{1}{2}\left( \frac{1}{\sqrt{1-M_{A\infty}^2}} - \frac{1}{\sqrt{1-M_{A,i}^2}} \right)\\
& & \cdot \left(\tanh{\frac{\frac{A}{\sqrt{1 - M_{A\infty}^2}\, B_{\infty}}
- y_{1}}{d_{1}}} -\tanh{\frac{\frac{A}{\sqrt{1 - M_{A\infty}^2}\, B_{\infty}}+
y_{1}}{d_{1}}}\right)\, .\nonumber
\end{aligned}$$ The ansatz for $\alpha'(A)$ is chosen in order to mimic two current sheets located symmetrically at $\pm y_{1}$ around the heliopause field lines, with oppositely directed currents. To compute the Mach number profile and the current sheets, we use the following set of values [see, e.g. @2012ApJ...760..106F; @2013Sci...341..147B]: For the magnetic field of the interstellar medium $B_{\infty} = 5\,\mu$G, for the inner magnetic field strength in the tail $B_{\rm i} = 4\,\mu$G, the particle number densities of electrons and ions are about equal and $n_{\rm e} \approx n_{\rm i} = 0.1$cm$^{-3}$, the velocity of the ISM plasma relative to the sun is ${\rm v}_{\infty} = 25$kms$^{-1}$, and the Mach numbers of the ISM plasma and of the heliotail result to $M_{\rm A\infty} =
0.72$ and $M_{\rm A,i} = 0.52$. These values might not be absolutely true, but they are used here to provide a rough idea of the global shape of the current sheets.
The Alfvén Mach number profile for the chosen parameters is shown in the top panel of Fig.\[currentsheets\]. It is computed for a thickness of the current sheets in the tail of $d_{1} = 100$AU. This is an unrealistic scenario, and is only shown to highlight the current distribution in the tail (middle panel of Fig.\[currentsheets\]). A more realistic case for the current sheets requires much narrower widths. This is depicted in the lower panel of Fig.\[currentsheets\] where we use a width of only 10AU. Obviously, a reduction in the width by a factor of ten leads to an increase of the current strength by a factor of ten. At the boundary between ISM and solar wind flows, Kelvin-Helmholtz-like or current driven reconnection instabilities can occur due to shear flows and extremely high current densities, respectively. These instability regions are thus ideal locations for plasma heating and particle acceleration.
![Alfvén Mach number profile (top, restricted to $M_{\rm A} > 0.6$ for displaying purposes) and resulting current sheets for a sheet width of 100AU (middle) and 10AU (bottom). The current density of the poloidal magnetic field is here given in units of $2.65\times 10^{-17}$Am$^{-2}$.[]{data-label="currentsheets"}](Mach.pdf "fig:"){width="7.5cm"} ![Alfvén Mach number profile (top, restricted to $M_{\rm A} > 0.6$ for displaying purposes) and resulting current sheets for a sheet width of 100AU (middle) and 10AU (bottom). The current density of the poloidal magnetic field is here given in units of $2.65\times 10^{-17}$Am$^{-2}$.[]{data-label="currentsheets"}](current1.pdf "fig:"){width="7.5cm"} ![Alfvén Mach number profile (top, restricted to $M_{\rm A} > 0.6$ for displaying purposes) and resulting current sheets for a sheet width of 100AU (middle) and 10AU (bottom). The current density of the poloidal magnetic field is here given in units of $2.65\times 10^{-17}$Am$^{-2}$.[]{data-label="currentsheets"}](current2.pdf "fig:"){width="7.5cm"}
Magnetic shear as trigger for particle acceleration
===================================================
The separatrix regions and, in particular, the heliotail regions can serve as important particle acceleration locations. This assumption is supported by observations of both the cosmic ray anisotropy and the broad excess of sub-TeV cosmic rays in the direction of the heliotail. @2010ApJ...722..188L propose that this excess originates from magnetic reconnection in the magnetotail. The heliotail/heliopause region was also considered by @2009ApJ...703....8L as an important region for particle acceleration. In their approach, @2009ApJ...703....8L use a Spitzer-like resistivity and propose first-order Fermi acceleration as the dominant acceleration process of energetic particles along the magnetotail. In contrast, to approach the problematics of particle acceleration we focus on the magnetic shear generated by magnetic shear flows across the heliopause boundary. To reduce the complexity of the problem, we shear here the magnetic field only in $z$-direction. This guarantees that the current, and, therefore, also the electric field, are aligned with the tail direction.
We investigate the generation of parallel electric fields and consider them as acceleration engines which, in solar physics, is typically referred to as Direct Current (DC) field acceleration [see, e.g., @2002SSRv..101....1A]. The link to solar physics scenarios is obvious, as the acceleration process takes place in a diluted plasma environment, i.e., in regions of low density.
As the magnetic field jumps across the heliopause, the magnetic shear in $z$-direction, $B_{z}$, is connected with a narrow current sheet located around the heliopause and extending into the heliotail. As in the tail region the flow is directed along the field lines, the presence of a current density immediately implies that even in such a field-aligned flow scenario an electric field can exist due to the validity of resistive Ohm’s law $\vec E + \vec{\rm v} \times \vec B = \eta \vec j$, as long as the resistivity $\eta
\neq 0$. Furthermore, as was shown by @2014arXiv1407.3227N, the solution of non-ideal Ohm’s law decouples from the rest of the MHD equations as the flow is field-aligned. Therefore, the only additional equation to be solved is $$\vec\nabla \times (\eta \vec j) = \vec 0\, ,$$ as $\vec E = \eta \vec j$ and $\vec\nabla \times \vec E = \vec 0$ (stationary approximation).
A reasonable resistivity should be valid in our steady-state model and account for the case of a collisionless plasma. While the Spitzer resistivity is effective only in collisional plasmas, the turbulent collisonless resistivity (anomalous resistivity due to wave-particle interactions) is usually not stationary.
The usual approach for the resistivity is to use the electric force and to introduce some frictional force $\nu v_{D}$ acting on the charges $q$, with the collision frequency $\nu$ and the drift velocity $v_{D}$ [e.g. @1977RvGSP..15..113P] $$\begin{aligned}
&&\frac{d v_{D}}{dt}=\frac{q}{m} E - \nu v_{D}=0 \quad\land\quad E=\eta j=\eta n q v_{D}\\
\Rightarrow && \eta=\frac{m\nu}{n q^2} \, . \end{aligned}$$ For our collisionless plasma, we consider the interaction between the electromagnetic field and charged particles as a substitute for collisions. This ansatz is motivated by the fact that the interaction time, which is limited by the time the particle needs to cross the current sheet, i.e. the “transit time” of the particle within the system, is much shorter than the collision time. Hence the collision time (collision frequency $\nu$) has to be replaced by the gyro-time (gyro-frequency $q B/m$), delivering $$\eta = \frac{1}{\sigma_{g}} \qquad \textrm{with} \qquad \sigma_{g}=\frac{n q}{|\vec B|}\, .$$ where $\sigma_{g}$ is the gyroconductivity, $n$ and $q$ are the particle number density and charge, respectively. This approach was introduced by and @1985JGR....90.8543L and the resulting resistivity is called inertial or gyro-resistivity. We want to emphasize that the substitution of the collision frequency by the gyro-frequency automatically delivers huge resistivity values in the case of a diluted plasma, which are only important for the generation of an electric field in regions of strong current density and not in regions where the field has the character of a potential field.
We use an approximate value for the magnetic shear of $B_{z}\approx 10^{-11}$Tesla, which is of the order of 10% of the heliotail (or ISM) magnetic field strength and can be regarded as a lower bound. For the typical lengthscale of the shear layer we set $l\approx 10^{3}$km [e.g., @1983MNRAS.205..839F]. Hence, we can estimate the resulting current density via $$|\vec j| \approx \frac{|B_{z}|}{\mu_{0} l} = 5.3\times 10^{-11}\,\textrm{A\,m}^{-2}\, .$$ For density values of $10^{4}$m$^{-3}$ [@1986SSRv...43..329F] and magnetic field strengths of $2\times 10^{-10}$Tesla typical for heliotail conditions, the gyroresisitivity $\eta$ is on the order of $10^{5}$Ohmm. Consequently, the parallel electric field becomes $$E_{\parallel} = \frac{\vec E\cdot \vec B}{|\vec B|} \approx
\frac{\eta}{\mu_{0}}\,|\vec j| \approx 10^{-6}\,\textrm{Volt\,m}^{-1}\, .$$
Within the tail, the field lines are stretched, and the current is concentrated around the heliopause region, providing a sufficiently extended environment for accelerating particles continuously along the magnetic field lines. Outside this narrow region, the current vanishes and hence also the electric field, so that those astrosphere regions can be ideal.
Considering a relatively conservative case, in which the field aligned electric field extends to about 100AU, only, meaning that the tail extends to just twice the distance than the heliopause nose, the voltage seen by the particles is $$\int E_{\parallel} ds \approx E_{\parallel} \cdot s \approx 10^{7}\,\textrm{Volt}\, .$$ This voltage can contribute to cosmic ray acceleration.
Discussion and Conclusions
==========================
We have shown that the distribution of null points and stagnation points defines the global topology and the large-scale structure of an astrosphere. Multiple separatrices can exist implying jumps (tangential discontinuities) of several physical parameters, such as the magnetic field strength, particle density, etc. As the outermost separatrix defines the astropause, its global geometrical shape is hence also determined.
With respect to the heliosphere, the multiple decreases and increases in the magnetic field strength as well as in other physical parameters measured by [*Voyager 1*]{} [@2013Sci...341..147B] indicates several crossings of either one or several individual separatrices. Such a scenario is in good qualitative agreement with the multiple separatrix structures due to more than one null point as proposed here and formerly by . A similar scene considering multiple, nested separatrices and magnetic islands was recently suggested based on detailed numerical simulations by @2013ApJ...774L...8S.
Interestingly, our results for the two null point scenarios also agree with the recently proposed presence of a heliocliff region inside the heliopause [@2013ApJ...776...79F]. In particular, the heliocliff might be interpreted as the separatrix resulting from the second null point (as shown in the middle left panel of Fig.\[series\]), and the streamlines originating from the monopole part, which bend into the heliotail, would represent the open heliosheath as introduced by @2013ApJ...776...79F. In the heliocliff region, the model of @2013ApJ...776...79F turns out to produce a super-Alfvénic field-aligned flow, while in our model the flow close to the heliopause and in the heliotail region is field-aligned but can also be sub-Alfvénic.
Furthermore, the presence of magnetic shear flows can produce vortex current sheets [@2012AnGeo..30..545N] leading to the generation of instabilities and magnetic reconnection close to separatrices. In the current work we restrict our analysis to a maximum of two separatrices and we apply the mapping only to the heliotail with one symmetric separatrix (top panel of Fig.\[axisym\]) with two current sheets. As multiple separatrices can exist in the heliosphere, the presence of multiple curent sheets in the vicinity of these separatrices can lead to fragmented structures . Introducing a non-collisional resistivity, strong electric (DC) fields parallel to the magnetic field can be generated, which can contribute to cosmic ray acceleration as suggested by @2005PhDT........85N.
We thank Andreas Kopp and two more, anonymous referees for helpful comments on the paper draft. This research made use of the NASA Astrophysics Data System (ADS). D.H.N. and M.K. acknowledge financial support from GAČR under grant numbers 13-24782S and P209/12/0103, respectively. The Astronomical Institute Ondřejov is supported by the project RVO:67985815.
[99]{} \[1\][\#1]{}
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[^1]: The monopole is introduced as a mathematical tool, used to generate radial streamlines (i.e. the stellar wind) and radial (i.e. open) field lines. Without it, the streamlines and magnetic field lines would otherwise always be closed.
[^2]: Considering that [*Voyager 1*]{} might have passed the stagnation region at a distance of 123AU [@2013Sci...341..144K], meaning that within our scenario $u_{1}$ would be at roughly 123AU, our configuration shown in Fig.\[axisym\] (bottom) and Fig.\[twist\] might be approximately scaled with 1:87AU.
[^3]: This remains valid, even if $\vec\nabla\rho$ happens to become extremely large across the heliopause boundary layer.
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